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Last night Democrats scored big wins in Virginia.
Ralph Northam won the governorship with an 8.5-point victory over the establishment GOP (GOPe) candidate Ed Gillespie. This was not particularly surprising.
What happened in the Virginia House of Delegates was another matter entirely.
The Democrats began the night just one net loss away from giving Republicans a super-majority. This morning – with results still being tallied – they stand one seat away from parity.
Democrats might end up with a majority in the Virginia House.
I came away from this election with two primary thoughts.
Democrats are energized – despite numerous party divisions and chaos at the leadership level. Anti-Trump is very real – and a force to be reckoned with.
Conservative voters are not willing to mobilize for a middle of the road establishment Republican candidate.
Gillespie received less votes (1.18mm) than the Republican candidates for Lieutenant Governor (Jill Vogel – 1.22 million) and Attorney General (John Adams – 1.21 mil).
Gillespie was essentially from the NeverTrump camp – and a firm member of the GOP establishment.
Vogel and Adams are not. Both have been far more supportive of President Trump and his policies.
Let's call it like it is. I'm in VA, voted for Ed, and couldn't figure out the difference between the two candidates. Ed had no platform, and was probably one of the few human beings in the country who is more boring than Jeb Bush. Last time I will ever vote for a "Hold Your Nose Republican".
I like that term. Think I may use it.
Meanwhile, Democrats are excited over the victories in Virginia – and rightly so.
An enormous storm is coming. Democrats are wildly motivated to register their hatred for Trump, moderates are disgusted, and his base is depressed by failure to deliver.
I agree with Jeff on all points. But the storm I see coming pertains to the GOP establishment.
GOPe has done nothing but sit on their hands.
Conservatives unwilling to vote for GOPe RINOs.
Richard Baris is the Editor-in-Chief at People's Pundit Daily and the Director of the PPD Poll.
On Election Day, the PPD Poll had President-Elect Trump winning by a 0.6% margin. I find him worth listening to.
1. Final thought: there're definitely danger signs when we look at everything in it's totality. But not what you're hearing. GOP isn't getting anything done and they are resisting the evolution from failed Bushism to Trumpism. If they keep resisting, they're going to get creamed.
2. Political coalitions change & Va. shift is a reaction to a realignment. GOP has no future in Bushism, the Gillespie wing. They're resisting Trumpism, which won them more than enough "un-winnable" states to make up for losing Va. Now they're depressing vote by doing nothing.
3. (Last) – As I've been arguing for years, GOP will never out-pander Ds. Their new electoral coalition runs from NC, around Va., down to FL, up throughout the South and into the Midwest. If they screw over voters @realDonaldTrump brought into the fold, they deserve their fate.
I agree with Baris' take completely.
After last night, GOPe pushes for @RepMcSally vs. @kelliwardaz primary in AZ. The lack of GOPe ability to understand what's happening in this country is breathtaking. Breathtaking. Same D.C. consultants. Same pollsters. Same stale campaigns. Same stale message. Same outcome.
The Virginia election was not a referendum on President Trump.
It was a referendum on the GOPe.
I'm not worried in the least about yesterday's results. We learned a year ago that there is no voter enthusiasm for the GOP establishment.
The GOP needs to realize it's campaigning during a culture war. That and Congress doing their job will win 2018.
Conservatives are done with "Hold Your Nose Republicans". | {
"redpajama_set_name": "RedPajamaC4"
} | 9,258 |
\section{Introduction}
Monotonicity testing, especially over hypergrid domains, is one of the most well studied problems in property testing.
We use $[n]$ to denote the set $\{1,2,\ldots, n\}$. The set $[n]^d$ is the $d$-dimensional hypergrid where $\mathbf{x} \in [n]^d$ is a $d$-dimensional vector with $\mathbf{x}_i \in [n]$.
The hypergrid is equipped with the natural partial order $\mathbf{x} \preceq \mathbf{y}$ iff $\mathbf{x}_i \leq \mathbf{y}_i$ for all $i\in [d]$. Note that when $n=2$, the hypergrid $[n]^d$ is isomorphic to the
hypercube $\{0,1\}^d$.
Let $f:[n]^d \to \{0,1\}$ be a Boolean function defined on the hypergrid. The function $f$ is monotone if $f(\mathbf{x}) \leq f(\mathbf{y})$ whenever $\mathbf{x} \preceq \mathbf{y}$.
The Hamming distance between two Boolean functions $f$ and $g$,
denoted as $\Delta(f,g)$, is the fraction of points where they differ.
The {\em distance to monotonicity} of a function $f:[n]^d \to \{0,1\}$ is defined as
$\varepsilon_f := \min_{g~\textrm{monotone}} \Delta(f,g)$.
The Boolean monotonicity testing problem on the hypergrid takes parameter $\varepsilon$ and oracle access to $f:[n]^d \to \{0,1\}$. The objective is to design a randomized algorithm, called the tester,
that accepts a monotone function with probability $\geq 2/3$ and rejects a function $f$ with $\varepsilon_f \geq \varepsilon$ with probability $\geq 2/3$.
A tester is one-sided if it accepts a monotone function with probability $1$. A tester is non-adaptive if all its queries are made in one round before seeing any responses.
There has been a rich history of results on monotonicity testing over hypergrids,
with a significant focus on hypercubes~\cite{GGLRS00,DGLRRS99,ChSe13,ChSe13-j,BeRaYa14,ChenST14,ChDi+15,ChenDST15,KMS15,BeBl16,Chen17,BlackCS18,BlackCS20,BKR20,HY22}.
We discuss the history more in \Cref{sec:related}, but for now, we give the state of the art.
For hypercubes, after a long line of work, the breakthrough result~\cite{KMS15} of Khot, Minzer, and Safra gave
an $\widetilde{O}_\varepsilon(\sqrt{d})$-query non-adaptive, one-sided tester. This result is tight due to a nearly matching $\widetilde{\Omega}(\sqrt{d})$-query lower bound for non-adaptive testers due to Chen, Waingarten, and Xie~\cite{Chen17}.
For general hypergrids, the best upper bound is the $\widetilde{O}_\varepsilon(d^{5/6})$-query
tester of the authors~\cite{BlackCS18,BlackCS20}.
This $\widetilde{\Omega}(\sqrt{d})$ vs $\widetilde{O}(d^{5/6})$ gap for non-adaptive testers is a tantalizing and important open question in property testing.
Even for the domain $[3]^d$, the optimal non-adaptive monotonicity testing bound is unknown.
One of the main questions driving our work is:
\begin{center}
\emph{Are there $\widetilde{O}_\varepsilon(\sqrt{d})$-query monotonicity testers for domains beyond the hypercube?}
\end{center}
\paragraph{Directed isoperimetric theorems.} The initial seminal work on monotonicity testing,
by Goldreich, Goldwasser, Lehman, Ron, and Samorodnitsky~\cite{GGLRS00}
and Dodis, Goldreich, Lehman, Ron, Raskhodnikova and Samorodnitsky~\cite{DGLRRS99}
prove the existence of $\widetilde{O}_\varepsilon(d)$-query testers.
For almost a decade, it was not clear whether $o(d)$-query testers were possible.
In~\cite{ChSe13-j}, the last two authors gave the first such tester via an exciting connection
with \emph{robust directed isoperimetric theorems}. Indeed, all $o(d)$-query testers
are achieved through such theorems.
Think of a Boolean function $f$ as the indicator for a subset of the domain.
The variance of $f$, $\mathrm{var}(f)$, is a measure of the volume of the indicated subset.
An isoperimetric theorem for Boolean functions relates the variance of $f$ to the ``boundary'' of the function which corresponds
to the sensitive edges and/or their endpoints.
The deep insight of these theorems comes from sophisticated ways of measuring boundary size, involving both the vertex and edge boundary.
A \emph{directed} isoperimetric theorem is an analog where we only measure ``up-boundary'' formed by monotonicity violations.
Rather surprisingly, in the directed case, one can replace the variance as a measure of volume by the distance to monotonicity.
In \Cref{tab:dir-iso}, we list some classic isoperimetric results and their directed analogues for the hypercube.
For a point $\mathbf{x}$, $I_f(\mathbf{x})$ is the number of sensitive edges incident to $\mathbf{x}$.
We use $I_f$ to denote $\mathbf{E}_{\mathbf{x}}[I_f(\mathbf{x})]$, the total influence of $f$, which the number of sensitive edges in $f$ divided by the domain size $2^d$.
The quantity $\Gamma_f$ is the vertex boundary size divided by $2^d$. The directed analogues of these, $I^-_f, \Gamma^-_f, I^-_f(\mathbf{x})$,
only consider sensitive edges that violate monotonicity.
\begin{table}[ht!]
\begin{center}
\def1.5{1.5}
\begin{tabular} {| c | c | }
\hline
Undirected Isoperimetry & Directed Isoperimetry \\ \hline
$I_f \geq \Omega(\mathrm{var}(f))$ ~~~~ (\emph{Poincar\'{e} inequality, Folklore})& $I^-_f \geq \Omega(\varepsilon_f)$ ~~~ (\emph{Goldreich et al.\cite{GGLRS00}})
\\ \hline
$I_f\cdot \Gamma_f \geq \Omega(\mathrm{var}(f)^2)$ ~~~(\emph{Margulis~\cite{Mar74}})
& $I^-_f \cdot \Gamma^-_f \geq \Omega(\varepsilon^2_f)$ ~~(\emph{Chakrabarty, Seshadhri~\cite{ChSe13-j}})
\\ \hline
$\mathbf{E}_\mathbf{x}\left[\sqrt{I_f(\mathbf{x})}\right] \geq \Omega(\mathrm{var}(f))$ (\emph{Talagrand ~\cite{Tal93}})
& $\mathbf{E}_\mathbf{x}\left[\sqrt{I^-_f(\mathbf{x})}\right] = \Omega(\frac{\varepsilon_f}{\log d})$ ~~(\emph{Khot, Minzer, Safra~\cite{KMS15}})
\\ \hline
\end{tabular}
\end{center}
\caption{\em Boolean hypercube isoperimetry results and their directed analogues.
Pallavoor, Raskhodnikova, and Waingarten~\cite{PRW22} removed the $\log d$-dependence in the directed Talagrand inequality.}\label{tab:dir-iso}
\end{table}
Observe the remarkable parallel between the standard isoperimetric results and their directed versions.
The Talagrand inequality is the strongest statement, and implies all other bounds. The directed versions
imply the undirected versions, using standard inequalities regarding monotone functions. The~\cite{KMS15}
$\widetilde{O}_\varepsilon(\sqrt{d})$-query tester is based on the directed Talagrand inequality.
The story for hypergrids is much more complicated. From an isoperimetric perspective, a common approach is to consider the \emph{augmented hypergrid},
wherein we add edges between pairs in the same line.
The dimension reduction technique in~\cite{DGLRRS99} used to prove the $\widetilde{O}_\varepsilon(d)$ testers can be thought of as establishing a directed Poincar\'{e} inequality~.
In previous work~\cite{BlackCS18}, the authors proved a directed Margulis inequality, which led to the $\widetilde{O}_\varepsilon(d^{5/6})$ query tester.
Another motivating question for our work is:
\begin{center}
\emph{Can the directed Talagrand inequality be generalized beyond the hypercube?}
\end{center}
\subsection{Main results} \label{sec:results}
We answer both questions mentioned above in the affirmative. To state our results more formally, we begin with some notation.
For any $i\in [d]$, we use $\mathbf{e}_i$ to denote the $d$-dimensional vector which has $1$ on the $i$th coordinate and zero everywhere else.
For a dimension $i$, a pair $(\mathbf{x},\mathbf{y})$ is called \emph{$i$-aligned} if $\mathbf{x}$ and $\mathbf{y}$ only differ on their
$i$-coordinate. An \emph{$i$-line} is a 1D line of $n$ points obtained by fixing all but the $i$th coordinate.
We define a notion of directed influence of Boolean functions on hypergrids, which generalizes the notion for Boolean functions on hypercubes.
In plain English, for a point $\mathbf{x}$ we count the number of {\em dimensions} in which $\mathbf{x}$ takes part in a violation.
We call this the {\em thresholded negative influence} of $\mathbf{x}$. Note that $\mathbf{x}$ could participate in multiple violations along the same dimension.
Throughout this paper, we will be only talking about negative influences of functions on the hypergrid, and thus will often refer to the above
as just thresholded influence, and for brevity's sake we also don't use the superscript ``$-$'' in the notation below to denote the negative aspect.
\begin{definition}[Thresholded Influence]\label{def:phi-f}
Fix $f:[n]^d \to \{0,1\}$ and a dimension $i\in [d]$. Fix a point $\mathbf{x} \in [n]^d$. The thresholded influence of $\mathbf{x}$ along coordinate $i$ is
denoted $\Phi_f(\mathbf{x};i)$, and has value $1$ if there exists an $i$-aligned violation $(\mathbf{x},\mathbf{y})$.
The thresholded influence of $\mathbf{x}$ is $\Phi_f(\mathbf{x}) = \sum_{i=1}^d \Phi_f(\mathbf{x};i)$.
\end{definition}
Note that the thresholded influence coincides with the hypercube directed influence when $n=2$.
Also note that for any $\mathbf{x}$, $\Phi_f(\mathbf{x}) \in \{0,1,\ldots, d\}$ and is independent of $n$.
We prove the following theorem, a directed Talagrand theorem for hypergrids, which generalizes the~\cite{KMS15} result.
\begin{theorem}\label{thm:dir-tal-uncolored}~
Let $f:[n]^d \to \{0,1\}$ be $\varepsilon$-far from monotone.
\[
\mathbf{E}_{\mathbf{x}\in [n]^d}~\left[\sqrt{\Phi_{f}(\mathbf{x})} \right] = \Omega\left(\frac{\varepsilon}{\log n}\right)
\]
\end{theorem}
\paragraph{Robust isoperimetric theorems and monotonicity testing.} For the application to monotonicity testing,
as~\cite{KMS15} showed, a significant strengthening of \Thm{dir-tal-uncolored} is required.
The weakness of \Thm{dir-tal-uncolored}, as stated, is that the same violation/influence
is ``double-counted" at both its endpoints.
The LHS can significantly vary depending on whether we choose to only ``count" influences at zero-valued or one-valued points, and this is true even on the hypercube.
As a simple illustration,
consider the function $f$ that is $1$ at the all zeros point and $0$ everywhere else.
Suppose we only count influences at one-valued points.
Then the only vertex with any $I_f^-(\mathbf{x})$ is the all $0$'s point, and this value is $d$. Therefore, the Talagrand objective is $\frac{\sqrt{d}}{2^d}$.
On the other hand, if we count influences at zero-valued points, then $I_f^-(\mathbf{x}) = 1$ for the $d$ points $\mathbf{e}_1$ to $\mathbf{e}_d$,
and $0$ everywhere else. The Talagrand objective counted from zero-valued points is now much larger: $\frac{d}{2^d}$.
Therefore, depending on how we count, one can potentially reduce the Talagrand objective, $\mathbf{E}_\mathbf{x}[\sqrt{I_f^-(\mathbf{x})}]$.
~\cite{KMS15} define a general way of deciding which endpoint ``pays'' for a violated edge. Consider a {\em coloring}\footnote{\cite{KMS15} considered the colorings to be red/blue, but we find the $0,1$-coloring more natural.} $\chi: E \to \{0,1\}$ of every edge $(\mathbf{x},\mathbf{y}) \in E$ of the hypercube to either $0$ or $1$. Now, given a violated edge $(\mathbf{x}, \mathbf{y})$, we use this coloring to decide whose influence this edge contributes towards. More precisely, given this coloring $\chi$, the {\em colored} directed influence $I^-_{f,\chi}(\mathbf{x})$ of $\mathbf{x}$ is defined as
the number of violated edges $(\mathbf{x}, \mathbf{y})$ incident on $\mathbf{x}$ which have the same color as $f(\mathbf{x})$.
Given a coloring, the {\em colorful} Talagrand objective equals the expected root colored directed influence.
What~\cite{KMS15} prove is that no matter what coloring $\chi$ one chooses,
the Talagrand objective is still large, and in particular $\mathbf{E}_\mathbf{x}\left[\sqrt{I^-_{f,\chi}(\mathbf{x})}\right] = \Omega(\frac{\varepsilon_f}{\log d})$.
We define the robust/colorful generalizations of the thresholded negative influence on hypergrids.
Consider the \emph{fully augmented hypergrid}, where we put the edge $(\mathbf{x},\mathbf{y})$ if $\mathbf{x}$
and $\mathbf{y}$ differ on only one coordinate.
Let $E$ be the set of edges in the fully augmented hypergrid.
\begin{definition}[Colorful Thresholded Influence]\label{def:phi-f-chi}
Fix $f:[n]^d \to \{0,1\}$ and $\chi:E\to \{0,1\}$. Fix a dimension $i\in [d]$ and a point $\mathbf{x} \in [n]^d$. The colorful thresholded negative influence of $\mathbf{x}$ along coordinate $i$ is denoted $\Phi_{f,\chi}(\mathbf{x};i)$, and has value $1$ if there exists an $i$-aligned violation $(\mathbf{x},\mathbf{y})$
such that $\chi(\mathbf{x},\mathbf{y}) = f(\mathbf{x})$, and has value $0$ otherwise.
The colorful thresholded negative influence of $\mathbf{x}$ is $\Phi_{f,\chi}(\mathbf{x}) = \sum_{i=1}^d \Phi_{f,\chi}(\mathbf{x};i)$.
\end{definition}
The main result of our paper is a robust directed Talagrand isoperimetry theorem for Boolean functions on the hypergrid.
It is a strict generalization of the KMS Talagrand theorem for hypercubes.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{theorem}\label{thm:dir-tal}~
Let $f:[n]^d \to \{0,1\}$ be $\varepsilon$-far from monotone, and let $\chi:E\to \{0,1\}$ be an arbitrary coloring of the edges of the augmented hypergrid.
\[
\mathbf{E}_{\mathbf{x}\in [n]^d}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right] = \Omega\left(\frac{\varepsilon}{\log n}\right)
\]
\end{theorem}
\end{mdframed}
As a consequence of this theorem, we can (up to log factors) resolve the question of non-adaptive
monotonicity testing on hypergrids with constant $n$. We note that the best bound for any $n > 2$
was $\widetilde{O}(d^{5/6})$. Even for the simplest non-hypercube case of $n=3$, it was open whether the optimal non-adaptive complexity of monotonicity testing is $\sqrt{d}$.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{theorem}\label{thm:mono-testing}~
Consider Boolean functions over the hypergrid, $f:[n]^d \to \{0,1\}$.
There is a one-sided, non-adaptive tester for monotonicity that makes $O(\varepsilon^{-2} n \sqrt{d} \log^5(nd))$ queries.
\end{theorem}
\end{mdframed}
\paragraph{The importance of being robust.} We briefly explain why the
robust Talagrand version is central
to the monotonicity testing application. All testers that have a $o(d)$-query complexity are versions of a \emph{path tester},
which can be thought of as querying endpoints of a directed random walk in the hypercube. Consider a function $f$
as the indicator for a set ${\bf 1}_f$, where the violating edges
form the ``up-boundary" between ${\bf 1}_f$ and its complement. To analyze the random walk,
we would like to lower bound the probability that a random walk starts in ${\bf 1}_f$, crosses over the boundary, and stays
in $\overline{{\bf 1}_f}$, that is, the set of $0$'s. To analyze this, one needs some structural properties in the graph induced by the boundary edges, which~\cite{KMS15}
express via their notion of a ``good subgraph''. In particular, one needs that there be a large number of edges, but also that
they are regularly spread out among the vertices. It doesn't seem that the ``uncolored'' Talagrand versions (like~\Cref{thm:dir-tal-uncolored})
are strong enough to prove this regularity, but the robust version can ``weed out'' high-degree vertices via a definition of a suitable coloring function $\chi$.
In short, the robust version of the Talagrand-style isoperimetric theorem is much more expressive. Indeed, these style of robust results
have found other applications in distribution testing~\cite{CaChGa+21} as well.
\paragraph{The dependence on $n$.} Given \Thm{mono-testing}, it is natural to ask whether the dependence
on $n$ is necessary. Previous \emph{domain reduction} theorems have
shown that one can reduce $n$ to $\mathrm{poly}(d)$ in a black box manner~\cite{BlackCS20,HY22}. The monotonicity tester
based on the directed Margulis inequality for hypergrids has a logarithmic dependence on $n$~\cite{BlackCS18}.
Combining with domain reduction, we get a $\widetilde{O}(\mathrm{poly}(\varepsilon^{-1}) d^{5/6})$-query tester.
It is an outstanding open problem to remove the dependence on $n$ from \Thm{mono-testing}.
In \Cref{sec:no-n}, we outline an approach to do so using the directed Talagrand inequality of \Thm{dir-tal}.
\iffalse
\subsection{Older text}
If we consider $I^{-}(\mathbf{x})$ as we have defined it, we notice that the same violated edge $(\mathbf{x}, \mathbf{y})$
is counted twice: once in $I^{-}(\mathbf{x})$ and once in $I^{-}(\mathbf{y})$. Indeed, to avoid this ``double counting'', many works only count $I^{-}(\mathbf{x})$ only for $\mathbf{x}$'s such that $f(\mathbf{x}) = 1$.
Two things should be clear. One, the Talagrand objective with respect to this new definition decreases, and two, that choosing only the $f(\mathbf{x}) = 1$ nodes seems arbitrary. Indeed, if we
counted $I^{-}(\mathbf{y})$ only for $f(\mathbf{y}) = 0$ nodes, the Talagrand objective may be different since we are taking a sum of square roots.
It is important to note that the {\em sum} of the influences is completely independent of how we color the edges of the hypercube; irrespective of the coloring, the sum is just the number of violated edges.
On the other hand, the Talagrand objective which is the sum of square-roots of the influences can be very different. To use a silly example, consider the function $f$ which is $1$ at the all zeros point and $0$ everywhere else. If the coloring is $\chi \equiv 1$, that is all edges are colored $1$, then the only vertex with any $I^{-}_\chi(\mathbf{x})$ is the all $0$'s point, and this value is $d$. Therefore, the colorful Talagrand objective with respect to this coloring is $\frac{\sqrt{d}}{2^d}$. On the other hand, if the coloring is $\chi \equiv 0$, then $I^-_\chi(\mathbf{e}_i) = 1$ for all $1\leq i\leq d$
and $0$ everywhere else. Note that the colorful Talagrand objective is now much larger: $\frac{d}{2^d}$. What Khot, Minzer, and Safra~\cite{KMS15} prove, and what is really crucially needed for the monotonicity testing application, is that no matter what the coloring $\chi$ is, the Talagrand objective is always $\widetilde{\Omega}(\varepsilon_f)$. In this way, this is a much more robust version of Talagrand's isoperimetry theorem, and indeed, to the best of our knowledge even the undirected version was not known before their work. Since we need this {\em undirected} colorful Talagrand statement, we explicitly state it below.
Note that unlike normal directed influence, the thresholded influence is {\em not} simply the out-degree or in-degree of the vertex $\mathbf{x}$ in the graph of violating edges of the augmented hypergrid.
In particular, if a vertex participates in multiple violations on the same line, this count is thresholded at one, and not counted multiple times.
As stated in the Introduction, this definition keeps $\Phi_f(\mathbf{x})$ between $0$ and $d$, and doesn't depend on $n$, the granularity of the hypergrid.
that is, the number of neighboring pairs, or edges, in the poset which differ in value. A {\em directed} isoperimetric theorem relates
the distance to monotonicity to the number of {\em violating edges}. If one considers edges being directed from smaller to bigger vertices in the poset, then
this number is the ``out-edge boundary'' of the set of points which evaluate to $1$.
In the past decade, a deep connection between
the distance to monotonicity and {\em directed isoperimetric} theorems have emerged. An isoperimetric theorem for a Boolean function relates the {\em variance} of the function, that is the product of the fraction of $0$s and $1$s, to the ``edge boundary'' of the function, that is, the number of neighboring pairs, or edges, in the poset which differ in value. A {\em directed} isoperimetric theorem relates
the distance to monotonicity to the number of {\em violating edges}. If one considers edges being directed from smaller to bigger vertices in the poset, then
this number is the ``out-edge boundary'' of the set of points which evaluate to $1$.
For example, it is a folklore isoperimetric theorem that for any Boolean function $f:\{0,1\}^d \to \{0,1\}$, the number of sensitive edges divided by $2^d$ is $\Omega(\mathrm{var}(f))$.
In the paper~\cite{GGLRS00}, Goldreich, Goldwasser, Lehman, Ron, and Samorodnitsky proved the directed analog of this theorem;
they proved that if $f$ is $\varepsilon$-far from being monotone, then the number of violating edges divided by $2^d$ is at least $\Omega(\varepsilon)$. This fact was then used to give a simple $O(d/\varepsilon)$-tester.
Interestingly, when the range of the function is not Boolean, there is a $\Omega(d)$~\cite{BBM11} lower bound on the query complexity of any tester. For the Boolean range, however, the first $o(d)$ tester was
obtained by the last two authors~\cite{ChSe13-j}. In fact, this paper explicitly suggested the connection to directed isoperimetric theorems, and one of the main contributions was
proving a directed analog of a more general isoperimetric theorem due to Margulis~\cite{Mar74}. \cite{ChSe13-j} used this directed version to derive a $O(d^{7/8}\varepsilon^{-3/2}\ln(1/\varepsilon))$ query tester.
Subsequently, in a remarkable paper~\cite{KMS15}, Khot, Minzer, and Safra proved a directed analog of Talagrand's isoperimetric theorem, which we mention below, and used this to design a $\widetilde{O}(\sqrt{d}/\varepsilon^2)$ tester. This result is the best possible for non-adaptive testers.
For a point $\mathbf{x} \in \{0,1\}^d$, let's use $I(\mathbf{x})$ to denote the number of sensitive edges incident on $\mathbf{x}$. Talagrand's isoperimetric theorem~\cite{Tal93} states
\begin{equation}
\text{For any $f:\{0,1\}^d \to \{0,1\}$,}~~~ \mathbf{E}_{\mathbf{x} \in \{0,1\}^d}\left[\sqrt{I(\mathbf{x})}\right] = \Omega(\mathrm{var}(f)) \tag{Talagrand}
\end{equation}
\cite{KMS15} proved a directed version where $I(\mathbf{x})$ is replaced by $I^-(\mathbf{x})$, the number of violating edges incident on $\mathbf{x}$, and $\mathrm{var}(f)$ replaced by $\varepsilon_f/\log d$, where $\varepsilon_f$ is the distance of monotonicity. In a subsequent work~\cite{PRW22}, Pallavoor, Raskhodnikova, and Waingarten removed the $\log d$-dependence. \smallskip
\noindent
Our goal in this paper is to investigate directed isoperimetry theorems on the {\em hypergrid} domain $[n]^d$ and use them to obtain Boolean monotonicity testers over the hypergrid.
The hypergrid domain is richer than the hypercube domain. This manifests itself even in the one-dimensional setting; while the situation is trivial in the hypercube case,
the property testing question on a line is non-trivial. Similarly, when one considers isoperimetry theorems, one needs to establish what the notion of ``neighboring pairs'' are, and what
the right generalization of $I(\mathbf{x})$ and $I^-(\mathbf{x})$ should be.
It is not too hard to see that if we only define pairs which differ in only one coordinate and that too only by a value of $1$, which is the usual notion of neighbors in the hypergrid graph, then the number of violating neighbors could be much smaller than the variance/distance to monotonicity. For example, take a function which is $1$ s on the ``left half'', that is, on all $\mathbf{x}$ with $\mathbf{x}_1 \leq n/2$, and $0$s on all $\mathbf{x}$'s with $\mathbf{x}_1 > n/2$. This function is $1/2$-far from being monotone, and has $\mathrm{var}(f) = 1/4$. The number of violating edges (divided by $n^d$, the domain size) however is
$1/n$. One could move to a weaker notion where we call any two pairs differing in one coordinate a neighbor, but even on a single line the influence of $\mathbf{x}$ could become as large as $\Theta(n)$.
We would like a notion of (directed) influence which generalizes the notion in hypercubes, and yet is not something which blows up when $n\to \infty$.
To this end, we define what we call the {\em thresholded} influence. While we give the formal definition in~\Cref{def:phi-f}, the informal definition is easy to explain in English.
Every point $\mathbf{x}$ considers the $d$ different lines passing through it and defines $\Phi_f(\mathbf{x})$ to be the number of lines in which it participates in at least one violation.
Note that in the hypercube, $\Phi_f(\mathbf{x})$ and $I^{-}(\mathbf{x})$ coincide (we have directly defined the {\em directed} version and we eschew putting the negative sign as superscript on $\Phi$ mainly because the notations become quite hairy even without them).
Dodis, Goldreich, Lehman, Ron, Raskhodnikova and Samorodnitsky~\cite{DGLRRS99} were the first to consider the Boolean monotonicity testing problem on hypergrids. In their paper, they proved a ``dimension reduction'' theorem which, when recast in terms of the above definition, proves that $\mathbf{E}_{\mathbf{x} \in [n]^d} [\Phi_f(\mathbf{x})] = \Omega(\varepsilon_f)$. This generalizes the \cite{GGLRS00} result, and also gives
a $O(d/\varepsilon)$-query tester.
In the paper~\cite{BlackCS18}, the authors generalized the directed Margulis\footnote{We have deliberately not stated the Margulis type isoperimetric statement at all since it requires the definition of a vertex boundary, and follows via Cauchy-Schwarz from the Talagrand statement. We direct the interested reader to~\cite{KMS15} for this connection.}-type isoperimetric theorem proved in~\cite{ChSe13-j} (with a loss of $\log n$), and used it to give an $O(d^{5/6}\mathrm{poly}(\log n,1/\varepsilon))$-query tester. In a later work~\cite{BlackCS20}, the dependence on $n$ was removed; see also a recent paper~\cite{HY22} by Harms and Yoshida. The dependence on $d$, however, was still not optimal. \medskip
The main result of this paper is a Talagrand style isometric theorem on $\Phi_f(\mathbf{x})$. In particular, we prove $\mathbf{E}_{\mathbf{x} \in [n]^d} \left[\sqrt{\Phi_f(\mathbf{x})}\right] = \Omega(\varepsilon/\log n)$ which vastly generalizes the result of~\cite{KMS15}.
Using this result, we can then piggyback on the random walk tester of~\cite{KMS15} to obtain an $\widetilde{O}(\sqrt{d} \cdot n^2/\varepsilon^2)$ query monotonicity tester.
As far as the application to testing goes, the polynomial dependence on $n$ is quite unsatisfactory. However, as far as we know, $\widetilde{O}(\sqrt{d}\mathrm{poly}(1/\varepsilon))$-tester was open even for $n=3$; the only analysis route we currently know for analyzing these testers are the isoperimetry theorems, and the~\cite{KMS15} technique doesn't seem to readily generalize even to
$\{0,1,2\}^d$. \DeepC{We have to be careful in what we say here. I think I have not misstated anything.}
In~\Cref{sec:def-plus-res} we describe the main notations and definitions we use in our admittedly notation-heavy paper. We also
formally state our directed isoperimetry theorem and it is actually stronger than what's mentioned above (as is the result of~\cite{KMS15} than what we have mentioned above). In~\Cref{sec:main-ideas} we describe the challenges that one encounters in the hypergrid domain, and in particular the challenges in generalizing~\cite{KMS15}. We describe the main conceptual contribution of ours that does allow us to overcome these challenges, and also sketch the main ideas behind our proof.
\fi
\subsection{Challenges} \label{sec:challenges}
We explain the challenges faced in proving \Thm{dir-tal} and \Thm{mono-testing}. The KMS proof of the directed Talagrand inequality for the hypercube is a
tour-de-force~\cite{KMS15}, and there are many parts of their proof that do not generalize for $n > 2$.
We begin by giving an overview of the KMS proof for the hypercube case.
For the time being, let us focus on the uncolored case.
For convenience, let $T(f) = \mathbf{E}_\mathbf{x}[\sqrt{I^-_f(\mathbf{x})}]$ denote the hypercube directed Talagrand objective for a $f:\hyp{d} \to \hyp{}$.
To lower bound $T(f)$,~\cite{KMS15} transform the function $f$ to a function $g$ using a sequence of what they call {\em split} operators.
The $i$th split operator applied to $f$ replaces the $i$th coordinate/dimension by two new coordinates $(i,+)$ and $(i,-)$.
One way to think of the split operator is that takes the $\left((0,\mathbf{x}_{-i}), (1, \mathbf{x}_{-i})\right)$ edge and converts it into a square.
(Here, $\mathbf{x}_{-i}$ denotes the collection of coordinates in $\mathbf{x}$ skipping $\mathbf{x}_i$.) The ``bottom" and ``top" corners
of the square store the original values of the edge, while the ``diagonal" corners store the min and max values (of the edge).
The definition of this remarkably ingenious operator ensures that the split function
is monotone in $(i,+)$ and anti-monotone in $(i,-)$. The final function $g:\{0,1\}^{2d} \to \{0,1\}$ obtained by splitting
on all coordinates has the property that it is either monotone or anti-monotone on all coordinates. That is, $g$ is unate (or pure, as~\cite{KMS15} call them),
and for such functions the directed Talagrand inequality can be proved via a short reduction to the undirected case.
The utility of the split operator comes from the main technical contribution of~\cite{KMS15} (Section 3.4),
where it is shown that splitting cannot increase the directed Talagrand objective. This is a ``roll-your-sleeve-and-calculate'' argument that follows a case-by-case analysis.
So, we can lower bound $T(f) \geq T(g)$. Since $g$ is unate, one can prove $T(g) = \Omega(\varepsilon_g)$ (the distance of $g$ to monotonicity).
But how does one handle $\varepsilon_g$, or $g$ more generally?
This is done by relating splitting to the classic {\em switch operator} in monotonicity testing, introduced
in \cite{GGLRS00}.
The switch operator for the $i$th coordinate can be thought of as modifying the edges along the $i$-dimension:
for any $i$-edge violation $(\mathbf{x},\mathbf{y})$, this operator switches the values, thereby fixing the violation.
The switching operator has the remarkable property of never increasing monotonicity violations in other dimensions;
hence, switching in all dimensions leads to a monotone function.
\cite{KMS15} observe that the function $g$ basically ``embeds" disjoint variations of $f$, wherein
each variation is obtained by performing a distinct sequence of switches on $f$.
The function $g$ contains all possible such variations of $f$, stored
cleverly so that $g$ is unate. One can then use properties of the switch
operators to relate $\varepsilon_g$ to $\varepsilon_f$. (The truth is more complicated; we will come
back to this point later.)\smallskip
\noindent
{\bf Challenge \#1, splitting on hypergrids?} The biggest challenge in trying to generalize the~\cite{KMS15} argument
is to generalize the split operator. One natural
starting point would be to consider the \emph{sort} operator, defined in~\cite{DGLRRS99}, which generalizes the switch operator:
the sort operator in the $i$th coordinate sorts the function along all $i$-lines.
But it is not at all clear how to split the $i$th coordinate into a set of coordinates that
contains the information about the sort operator thereby leading to a pure/unate function.
In short, sorting is a much more complicated operation than switching, and it is not clear how to succinctly encode this information
using a single operator.
We address this challenge by a reorientation of the KMS proof. Instead of looking at operators on dimensions to understand effects of switching/sorting,
we do this via what we call ``tracker functions'' which are $n^d$ different Boolean functions tracking the changes in $f$. We discuss this more in~\Cref{sec:main-ideas}. \smallskip
\noindent
{\bf Challenge \#2, the case analysis for decreasing Talagrand objective.} As mentioned earlier,
the central calculation of KMS is in showing that splitting does not increase the directed
Talagrand objective. This is related (not quite, but close enough) to showing
that the switch operator does not increase the Talagrand objective.
A statement like this is proven in KMS by case analysis; there are $4$ cases, for the possible values
a Boolean function takes on an edge. One immediately sees that such an approach
cannot scale for general $n$, since the number of possible Boolean
functions on a line is $2^n$. Even with our new idea of tracking functions,
we cannot escape this complexity of arguing how the Talagrand-style objective decreases upon a sorting
operation, and a case-by-case analysis depending on the values of the function is infeasible.
We address this challenge by a connection to the theory of majorization. We show
that the sort operator is (roughly) a majorizing operator on the vector of influences.
The concavity of the square root function implies that sorting along lines cannot increase
the Talagrand objective. More details are given in the next section. \smallskip
\noindent
{\bf Challenge \#3, the colorings.} Even if we circumvented the above
issues, the robust colored Talagrand objective brings a new set of issues.
Roughly speaking, colorings decide which points ``pay" for violations of the Talagrand objective, the switching/sorting operator
move points around by changing values, and the high-level argument to prove $T(f)$ drops is showing that these violations ``pay'' for the moves.
In the hypercube, a switch either changes the values on all the points
of the edge or none of the points, and this binary nature makes the handling of colors in the KMS proof fairly
easy, merely introducing a few extra cases in their argument.
Sorting, on the other hand, can change an arbitrary set of points, and in particular,
even in the case of $n=3$, a point participating in a violation may not change value in a sort.
To address this challenge, as we apply the sort operators to obtain a handle on our function,
we also need to {\em recolor} the edges such that we obtain the drop in the $T$-objective.
Once again, the theory of majorization is the guide. This part of the proof is perhaps the most technical portion of our paper. \smallskip
\noindent
{\bf Other minor challenges: the telescoping argument and tester analysis:} The issues
detailed here are not really conceptual challenges, but they do require some work
to handle the richer hypergrid domain.
Recall that the KMS analysis proves the chain of inequalities, $T(f) \geq T(g) = \Omega(\varepsilon_g)$.
Unfortunately, it can happen that $\varepsilon_g \ll \varepsilon_f$. In this case, KMS observe
that one could redo the entire argument on random restrictions of $f$ to half the coordinates.
If the corresponding $\varepsilon_g$ is still too small, then one restricts on one-fourth of the coordinates,
so on and so forth. One can prove that somewhere along these $\log d$ restrictions, one must have $\varepsilon_g = \Omega(\varepsilon_f)$.
Pallavoor, Raskhodnikova, and Waingarten~\cite{PRW22} improve this analysis to remove a $\log d$ loss from the final bound.
We face the same problems in our analysis, and have to adapt the analysis to our setting.
Finally, the tester analysis of KMS for the hypercube can be ported to the hypergrid path tester,
with some suitable adaptations of their argument. It is convenient to think of the \emph{fully augmented hypergrid},
where all pairs that lie along a line are connected by an edge. We can essentially view the hypergrid tester
as sampling a random hypercube from the fully augmented hypergrid, and then performing
a directed random walk on this hypercube. We can then piggyback on various tools from KMS for the hypercube tester,
to bound the rejection probability of the path tester for hypergrids.
\subsection{Main Ideas}\label{sec:main-ideas}
We sketch some key ideas needed to prove~\Cref{thm:dir-tal} and address the challenges detailed earlier.
We begin with a key conceptual contribution of this paper. Given a function $f:[n]^d \to \hyp{}$, we define a collection of Boolean functions on the hypercube
called {\em tracker functions}.
We will lower bound the directed Talagrand objective on the hypergrid by the undirected Talagrand objective on these tracker functions.
Indeed, the inspiration of these tracker functions arose out of understanding the analysis in~\cite{KMS15}, in particular, the intermediate ``$g$'' function in their Section $4$.
As an homage, we also denote our tracker functions with the same Roman letter, even though it is different from their function.
\subsubsection{Tracker functions $g_\mathbf{x}$ for all $\mathbf{x} \in [n]^d$}
Let us begin with the sort operator discussed earlier.
Without loss of generality, fix the ordering of coordinates in $[d]$ to be $(1,2,\ldots,d)$.
The operator $\mathtt{sort}_i$ for $i\in [d]$ sorts the function on every $i$-line. Given a subset $S\subseteq [d]$ of coordinates,
the function $(S\circ f)$ is obtained by sorting $f$ on the coordinates in $S$ in that order.
Sorting along any dimension
cannot increase the number of violations along any other dimension, and therefore upon sorting on all dimensions, the result is a monotone function~\cite{DGLRRS99}.
Suppose $f$ is $\varepsilon$-far from monotone.
Clearly, the total number of points changed by sorting along all dimensions must be at least $\varepsilon n^d$.
While this is not obvious here, it will be useful to
to {\em track} how the function value changes when we sort along a
certain subset $S$ of coordinates. The intuitive idea is: if the function value changes for most such partial sortings, then perhaps the function is far from being monotone.
To this end, for every point $\mathbf{x} \in [n]^d$, we define a Boolean function $g_\mathbf{x} : 2^{[d]} \to \{0,1\}$ that tracks how the function value $f$ changes
as we apply the sort operator a subset $S$ of the coordinates. It is best to think of the domain of $g_\mathbf{x}$ as subsets $S\subseteq [d]$.
\begin{definition}[Tracker Functions $g_\mathbf{x}$]\label{def:gx}
Fix an $\mathbf{x}\in [n]^d$. The tracker function $g_\mathbf{x} : \{0,1\}^d \to \{0,1\}$ is defined as
\[
\forall S\subseteq [d], ~~~~ g_\mathbf{x}(S) := \left(S\circ f\right) (\mathbf{x})
\]
\end{definition}
\noindent
We provide an illustration of this definition in~\Cref{fig:tracker-illus}.
\begin{figure}[ht!]
\begin{center}
\includegraphics[trim = 200 150 200 10, clip, scale=0.5]{figs/tracker-function-illus}
\end{center}
\caption{\em The blue function $f:[n]^d\to\{0,1\}$ is defined in the middle using bold, gothic characters. We have $d=2$ and $n=2$.
For each of the $4$ points of this square, we have four different $g_\mathbf{x}:\{0,1\}^2 \to \{0,1\}$ and they are described in the
four green squares.
For any $S\subseteq \{1,2\}$, if we focus on the corresponding corners of the four squares, then we get the function $(S\circ f)$.
For instance, if $S = \{2\}$, then if we focus on the top left corners, then starting from $g_{00}$ and moving clockwise we get $(0, 1, 1, 0)$.
These will precisely the function $f$ (read clockwise from $00$) after we sort along dimension $2$.
}\label{fig:tracker-illus}
\end{figure}
\noindent
Note that when $f$ is a monotone function, all the functions $g_\mathbf{x}$ are constants.
Sorting does not change any values, so $g_\mathbf{x}(S)$ is always $f(\mathbf{x})$.
On the other hand, if $f$ is not monotone along dimension $i$, then there are points such that $g_\mathbf{x}(\{i\}) \neq f(\mathbf{x})$.
Indeed, one would expect the typical variance of these $g_\mathbf{x}$ functions
to be related to the distance to monotonicity of $f$ (technically not true, but we come to this point later).
The tracker functions help us lower bound the (colorful) Talagrand objective for thresholded influence, in particular, the LHS in~\Cref{thm:dir-tal}.
Recall that the Talagrand objective is the expected square root of the colorful thresholded influences on the hypergrid function $f$.
We lower bound this quantity by the expected Talagrand objective on the {\em undirected} (colorful, however) influence of the various $g_\mathbf{x}$ functions.
Note that $g_\mathbf{x}$ functions are defined on hypercubes.
So we reduce the robust directed Talagrand inequality on hypergrids to
robust undirected Talagrand inequalities on hypercubes.
This is the main technical contribution of our paper. Let us define the (colored) influences of these $g_\mathbf{x}$ functions.
\begin{definition}[Influence of the Tracking Functions]
Fix a $\mathbf{x} \in [n]^d$ and consider the tracking function $g_\mathbf{x} : \{0,1\}^d \to \{0,1\}$.
Fix a coordinate $j\in [d]$. The influence of $g_\mathbf{x}$ at a subset $S$ along the $j$th coordinate is defined as
\[
I^{= j}_{g_\mathbf{x}}(S) = 1 ~~\textrm{iff}~~g_\mathbf{x}(S) \neq g_\mathbf{x}(S\oplus j)~~~~\textrm{that is}~~~ (S\circ f)(\mathbf{x}) \neq (S\oplus j~~\circ f)(\mathbf{x})
\]
\end{definition}
\noindent
In plain English, the influence of the $j$th coordinate at a subset $S$ is $1$ if the function value (the hypergrid function) changes when we include the dimension $j$
to be sorted.
Once again, note that the same sensitive edge $(S, S\oplus j)$ is contributing towards both $I^{=j}_{g_\mathbf{x}}(S)$ and $I^{=j}_{g_\mathbf{x}}(S\oplus j)$.
We define a robust, colored version of these influences.
\begin{definition}[Colorful Influence of the Tracking Functions]
Fix a $\mathbf{x} \in [n]^d$ and consider the tracking function $g_\mathbf{x} : \{0,1\}^d \to \{0,1\}$.
Fix any arbitrary coloring $\xi_\mathbf{x} : E(2^{[d]}) \to \{0,1\}$ of the Boolean {\em hypercube}.
Fix a coordinate $j\in [d]$. The influence of $g_\mathbf{x}$ at a subset $S$ along the $j$th coordinate is defined as
\[
I^{= j}_{g_\mathbf{x}, \xi_\mathbf{x}}(S) = 1 ~~\textrm{iff}~~g_\mathbf{x}(S) \neq g_\mathbf{x}(S\oplus j)~~~~\textbf{and}~~~ g_\mathbf{x}(S) = \xi_\mathbf{x}(S, S\oplus j)
\]
The colorful total influence at the point $S$ in $g_\mathbf{x}$ is defined as
\begin{equation}\label{eq:def-colorful-total-infl-gx}
I_{g_\mathbf{x}, \xi_\mathbf{x}}(S) := \sum_{j=1}^d I^{=j}_{g_\mathbf{x}, \xi_\mathbf{x}} (S) \notag
\end{equation}
\end{definition}
\noindent
As before, for a sensitive edge $(S, S\oplus j)$ of $g_\mathbf{x}$, we count it towards the influence of the endpoint whose value equals the color $\xi_\mathbf{x}(S, S\oplus j)$.
The main technical contribution of this paper is proving that for any function $f:[n]^d \to \{0,1\}$ and any arbitrary coloring $\chi: E\to \{0,1\}$ of the
hypergrid edges, for every $\mathbf{x}\in [n]^d$ there \underline{exists} a coloring $\xi_\mathbf{x}:E(2^{[d]}) \to \{0,1\}$ of the Boolean {\em hypercube} edges, such that
\begin{equation}\label{eq:hope2}
T_{\Phi_\chi}(f) := \mathbf{E}_{\mathbf{x}\in [n]^d}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right] ~~\gtrapprox~~ \mathbf{E}_{\mathbf{x}\in [n]^d} \mathbf{E}_{S\subseteq [d]}~~[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}] \tag{H1}
\end{equation}
We explain the $\approx$ in the above inequality in the next subsection.
\smallskip
Why is a statement like~\eqref{eq:hope2} useful? Because the RHS terms are Talagrand objectives on colored influences on the usual undirected hypercube.
Therefore, we can apply undirected Talagrand bounds (known from KMS, \Cref{thm:kms-und}) to get an upper bound on the variance.
\begin{restatable}[Corollary of Theorem 1.8 in~\cite{KMS15}]{corollary}{corkms}
\label{cor:kms}
Fix $f:[n]^d \to \{0,1\}$. Fix an $\mathbf{x} \in [n]^d$ and consider the tracking function $g_\mathbf{x} : \{0,1\}^d \to \{0,1\}$.
Consider any {\em arbitrary} coloring $\xi_\mathbf{x} : E(2^{[d]}) \to \{0,1\}$ of the Boolean {\em hypercube}. Then, for every $\mathbf{x} \in [n]^d$, we have
\[
\mathbf{E}_{S\subseteq [d]}~~[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}] = \Omega(\mathrm{var}(g_\mathbf{x}))
\]
\end{restatable}
\noindent
The final piece of the puzzle connects $\mathrm{var}(g_\mathbf{x})$'s with the distance to monotonicity. Ideally, we would have liked to have a statement such as the following true.
\begin{equation}\label{eq:hope1}
\mathbf{E}_{\mathbf{x}\in [n]^d} \left[\mathrm{var}(g_\mathbf{x})\right] \approx \Omega(\varepsilon_f) \tag{H2}
\end{equation}
We now see that \eqref{eq:hope2}, \Cref{cor:kms}, and \eqref{eq:hope1} together implies~\Cref{thm:dir-tal} (indeed without the $\log n$).
\subsubsection{High level description of our approaches}
\paragraph{Addressing the $\approx$ in \eqref{eq:hope2} via semisorting.} As stated, we do not know if \eqref{eq:hope2} is true. However, we establish \eqref{eq:hope2} for
{\em semisorted} functions $f:[n]^d \to \{0,1\}$. A function $f$ is semisorted if on any line $\ell$, the restriction of the function on the first half is sorted and the restriction
on the second half is sorted. This may seem like a simple subclass of functions, but note that all functions on the Boolean hypercube ($n=2$)
are vacuously semisorted. Thus, proving \Cref{thm:dir-tal} on semi-sorted functions is already a generalization of the~\cite{KMS15} result. \Cref{thm:semisorted-reduce-to-g} is the formal
restatement of \eqref{eq:hope2}.
We reduce \Thm{dir-tal} on general functions to the same bound for semisorted functions.
Consider semisorting $f$, which means we sort $f$ on each half of every line.
Suppose the Talagrand objective did not increase \emph{and} the distance to monotonicity
did not decrease. Then \Thm{dir-tal} on the semisorted version of $f$ implies
\Thm{dir-tal} on $f$.
What we can prove is that: given the semisorted function, one can find a {\em recoloring} of the hypergrid edges such that the Talagrand objective doesn't increase. The precise statement is given in~\Cref{lem:semisorting-decreases}. We comment on our techniques to prove such a statement in a later paragraph.
Although semisorting can't increase the Talagrand objective, it can clearly reduce the distance to monotonicity. However, a relatively simple inductive
argument proves \Thm{dir-tal} with a $\log n$ loss.
Any function can be turned into a completely sorted (aka monotone) function by performing ``$\log n$ semisorting steps'' at varying scales.
In each scale, we consider many disjoint small hypergrids, and
convert a semisorted function defined over a small hypergrid to another semisorted function over a hypergrid of double the size (the next scale).
In one of these scales, we will find a semisorted function that has $\Omega(\varepsilon/\log n)$ distance from its sorted version.
One can average \Thm{dir-tal} over all the small hypergrids at this scale to bound the Talagrand objective of the whole function by $\Omega(\varepsilon/\log n)$. This is the step where we incur the $\log n$-factor loss.
This argument is not complicated, and we provide illustrated details in~\Cref{sec:semisorted}.
The real work happens in proving \Cref{thm:dir-tal-semisorted}, that is,~\eqref{eq:hope2} for semisorted functions.
\paragraph{Approach to proving \eqref{eq:hope2} for semisorted functions.}
Recall, we have a fixed adversarial coloring $\chi:E \to \{0,1\}$.
The proof follows a ``hybrid argument'' where we define a potential that is modified over $d+1$ rounds.
At the beginning of round $0$ it takes the value $\mathbf{E}_{\mathbf{x} \in [n]^d}[\sqrt{\Phi_{f,\chi}(\mathbf{x})}]$ which is the LHS of~\eqref{eq:hope2}. At the end of round $d$ it takes the value $\mathbf{E}_{\mathbf{x} \in [n]^d}\mathbf{E}_{S\subseteq [d]}[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}]$ which is the RHS of \eqref{eq:hope2}.
The proof follows by showing that the potential decreases in each round.
Let us describe the potential. Let us first write this without any reference to the colorings (so no $\chi$'s and $\xi_\mathbf{x}$'s), and then subsequently address the colorings.
At stage $i$, fix a subset $S \subseteq [i]$. Define
\begin{equation}\label{eq:hybrid}
R_{i}(S) := \mathbf{E}_{\mathbf{x} \in [n]^d} \left[\sqrt{~\sum_{j=1}^{i} I^{=j}_{g_\mathbf{x}} (S) ~~+~~ \sum_{j=i+1}^d \Phi_{S\circ f}(\mathbf{x}; j) }~\right] \tag{Hybrid}
\end{equation}
We remind the reader that $S\circ f$ is the function $f$ after the dimensions corresponding to $i\in S$ have been sorted.
Thus, $R_i(S)$ is a ``hybrid" Talagrand objective, with two different kinds of influences being summed.
Consider point $\mathbf{x} \in [n]^d$. On the first $i$ coordinates,
we sum the undirected influence (along these coordinates) of $S$ on the function $g_\mathbf{x}$. On the coordinates $i+1$ to $d$,
we sum to directed influence along these coordinates in the function $S \circ f$.
The potential is $\Lambda_i := \mathbf{E}_{S\subseteq [i]} [R_i(S)]$.
To make some sense of this, consider the extreme cases of $i=0$ and $i=d$. When $i=0$,
we only have the second $\Phi_{S \circ f}$ term. Furthermore, $S$ is empty since $S \subseteq [i]$.
So $\Lambda_0$ is precisely the original directed Talagrand objective, the LHS of \eqref{eq:hope2}.
When $i = d$, we only have the $I^{=j}_{g_\mathbf{x}}$ terms. Taking expectation
over $S \subseteq [d]$ to get $\Lambda_d$, we deduce that $\Lambda_d$ is the RHS of \eqref{eq:hope2}.
We will prove $\Lambda_{i-1} \geq \Lambda_i$ for all $1\leq i\leq d$.
To choose a uar set in $[i]$, we can choose a uar subset of $[i-1]$
and then add $i$ with $1/2$ probability.
Hence, $\Lambda_i = (\mathbf{E}_{S \subseteq [i-1]} [R_i(S) + R_i(S+i)])/2$,
while $\Lambda_{i-1} = \mathbf{E}_{S \subseteq [i-1]} [R_{i-1}(S)]$.
So, if we prove that $R_{i-1}(S)$ is at least both $R_i(S)$ and $R_i(S+i)$, then $\Lambda_{i-1} \geq \Lambda_i$.
The bulk of the technical work in this paper is involved in proving these two inequalities, so let us spend a little time explaining what proving this entails.
Let's take the inequality $R_{i-1}(S) \geq R_i(S)$. Refer again to \eqref{eq:hybrid}. When we go from $R_{i-1}(S)$ to $R_i(S)$,
under the square root, the term $\Phi_{S\circ f}(\mathbf{x};i)$ is replaced by $I^{=i}_{g_\mathbf{x}} (S)$.
To remind the reader, the former term is the indicator of whether $\mathbf{x}$ participates in a $i$-violation after the coordinates in $S \subseteq [i-1]$ have been sorted.
The latter term is whether $g_\mathbf{x}(S+i)$ equals $g_\mathbf{x}(S)$, that is, whether the (hypergrid) function value at $\mathbf{x}$ changes between sorting on coordinates in $S$ and $S+i$.
Just by parsing the definitions, one can observe that $\Phi_{S \circ f}(\mathbf{x};i) \geq I^{=i}_{g_\mathbf{x}}(S)$; if a point is modified on sorting in the $i$-coordinate,
then it must be participating in some $i$-violation (note that vice-versa may not be true and thus we have an inequality and not an equality). The quantity under the square-root {\em point-wise} dominates (ie, for every $\mathbf{x}$) when we move from $R_{i-1}(S)$ to $R_i(S)$. Thus, $R_{i-1}(S) \geq R_i(S)$.
The other inequality $R_{i-1}(S) \geq R_i(S+i)$, however, is much trickier to establish.
In $R_i(S+i)$, the second summation under the square-root, the $\Phi$ terms, are actually on a {\em different} function.
The $\Phi_{S\circ f}(\mathbf{x}; j)$ terms in $R_{i-1}(S)$ are the thresholded influences of the function after sorting on coordinates in $S$.
But in $R_i(S+i)$, these terms are $\Phi_{(S+i)\circ f} (\mathbf{x};j)$, the thresholded influences of $\mathbf{x}$ for the function after sorting on $S+i$.
Although, it is true that sorting on more coordinates cannot increase the total number of violations along any dimension,
this fact is {\em not} true point-wise. So, a point-wise argument as in the previous inequality is not possible.
The argument for this inequality proceeds {\em line-by-line}. One fixes an $i$-line $\ell$ and considers the vector of ``hybrid function'' values on this line.
We then consider this vector when moving from $R_{i-1}(S)$ to $R_i(S+i)$, and we need to show that the {\em sum of square roots} can get only smaller.
This is where one of our key insights comes in: the theory of majorization can be used to assert these bounds.
Roughly speaking, a vector $\mathbf{a}$ (weakly) majorizes a vector $\mathbf{b}$ if the sum of the $k$-largest coordinates of $\mathbf{a}$
dominates the sum of the $k$-largest coordinates of $\mathbf{b}$, for every $k$.
A less balanced vector majorizes a more balanced vector.
If the $\ell_1$-norms of these vectors are the same, then the sum of square roots
of the entries of $\mathbf{a}$ is at most the sum of square roots of that of $\mathbf{b}$.
This follows from concavity of the square-root function.
Our overarching mantra throughout this paper is this: whenever we perform an operation and the hybrid-influence-vector induced by a line changes, the new vector majorizes the old vector.
Specifically, these vectors are generated by look at the terms of $R_{i-1}(S)$ and $R_i(S+i)$
restricted to $i$-lines.
To prove this vector-after-operation majorizes vector-before-operation, we need some structural assumptions on the function. Otherwise, it's not hard to construct examples where this just fails.
The structure we need is precisely the {\em semisortedness} of $f$. When a function is semisorted,
the majorization argument goes through. At a high level, when $f$ is semisorted,
the vector of influences (along a line) satisfy various monotonicity properties.
In particular, when we (fully) sort on some coordinate $i$, we can show
the points losing violations had low violations to begin with. That is,
the vector of violations becomes less balanced, and the majorization follows.
The above discussion disregarded the colors. With colors, the situation is noticeably more difficult. Although the function $f$ is assumed to be semisorted, the coloring $\chi:E \to \{0,1\}$ is adversarial. So even though the vector
of influences may have monotonicity properties, the colored influences may not have this structure.
So a point with high influence could have much lower colored influence. Note that the sort operator
is insensitive to the coloring. So the majorization argument discussed above might not hold when
looking at colored influences.
With colors, \eqref{eq:hybrid} is replaced by the actual quantity~\eqref{eq:rhs-quantity} described in~\Cref{sec:mainworkhorse}.
To carry out the majorization argument, we need to construct a family of colorings $\xi_\mathbf{x}$ on the $n^d$
different hypercubes. We also need
$2^d$ many different auxiliary colorings $\chi_S$ of the hypergrid, constructed after every sort operation.
The argument is highly technical. But all colorings are
chosen to follow our mantra: vector after operation should majorize vector before operation.
The same principle is also used to prove~\Cref{lem:semisorting-decreases} which claims that semisorting an interval can only decrease the Talagrand objective, after a recoloring.
The details of the actual $R_i(S)$ hybrid function and the strategy to use them is presented in~\Cref{sec:mainworkhorse}. The most technical part of the paper
is in~\Cref{sec:proofoflemma6}, which proves that
the potential decreases in each round.
\paragraph{Addressing the $\approx$ in \eqref{eq:hope1} via random sorts.} To finally complete the argument, we need \eqref{eq:hope1}
that relates the average variance of the $g_\mathbf{x}$ functions to the distance to monotonicity of $f$.
As discussed earlier, \eqref{eq:hope1} is false, even for the case of hypercubes.
Nevertheless, one can use \eqref{eq:hope2} and \Cref{cor:kms} to prove a lower bound on $T_{\Phi_{\chi}}(f)$ with respect to $\varepsilon_f$.
This is the telescoping argument of KMS, refined in~\cite{PRW22}. We describe the main ideas below.
The first observation (see~\Cref{thm:semisorted-reduce-conv}) is that $ \mathbf{E}_{\mathbf{x}\in [n]^d} \left[\mathrm{var}(g_\mathbf{x})\right]$ is roughly
$\mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f]$
where $S$ is a uniform random subset of coordinates.
The distance to monotonicity $\varepsilon_f$ is approximated by $\Delta\left(f, S\circ \overline{S}\circ f\right)$ which, by the triangle inequality, is at most
$\Delta(f, S\circ f) + \Delta(S\circ f, \overline{S}\circ f)$. Thus, we get a relation between $\varepsilon_f$, the expected $\mathrm{var}(g_\mathbf{x})$, and the distance between $f$ and a ``random sort'' of $f$.
Therefore, if \eqref{eq:hope1} is not true, then a random sort of $f$ must be still far from being monotone, and then one can repeat the whole argument on just this random sort itself.
In one of these $\log d$ ``repetitions'', the \eqref{eq:hope1} must be true since in the end we get a monotone function (which can't be far from being monotone).
And this suffices to establish \Cref{thm:dir-tal}. We re-assert that the main ideas are already present in~\cite{KMS15,PRW22}. However, we require a more general presentation
to make things work for hypergrids. These details can be found in~\Cref{sec:semisorted-tal-dist}.
\subsection{Related Work} \label{sec:related}
Monotonicity testing has seen much activity since its introduction around 25 years ago~\cite{Ras99,EKK+00,GGLRS00,DGLRRS99,LR01,FLNRRS02,HK03,AC04,HK04,ACCL04,E04,SS08,Bha08,BCG+10,FR,BBM11,RRSW11,BGJ+12,ChSe13,ChSe13-j,ChenST14,BeRaYa14,BlRY14,ChenDST15,ChDi+15,KMS15,BeBl16,Chen17,BlackCS18,BlackCS20,BKR20,HY22}.
We have already covered much of the previous work on Boolean monotonicity testing
over the hypercube, but give a short recap. For convenience of presentation, in some results, we subsume $\varepsilon$-dependencies
using the notation $O_\varepsilon$.
The problem was introduced by Goldreich et al.~\cite{GGLRS00} and Raskhodnikova \cite{Ras99}, who described an $O(d/\varepsilon)$-query tester.
Chakrabarty and Seshadhri~\cite{ChSe13-j} achieved the first sublinear in dimension query complexity of $\widetilde{O}_\varepsilon(d^{7/8})$ using directed isoperimetric inequalities.
Chen, Servedio, and Tan~\cite{ChenST14} improved the analysis to $\widetilde{O}_\varepsilon(d^{5/6})$ queries. Fischer et al.~\cite{FLNRRS02} had first shown an $\Omega(\sqrt{d})$-query
lower bound for non-adaptive, one-sided testers, by a short and neat construction.
The non-adaptive, two-sided $\widetilde{\Omega}(\sqrt{d})$ lower bound is much harder to attain, and was done by Chen, Waingarten, and Xie~\cite{Chen17},
improving on the $\Omega(d^{1/2-c})$ bound from~\cite{ChenDST15}, which itself improved on the $\widetilde{\Omega}(d^{1/5})$ bound of \cite{ChenST14}. ~\cite{KMS15} gave an $\widetilde{O}_\varepsilon(\sqrt{d})$-query tester, via the
robust directed Talagrand inequality.
While this resolves the non-adaptive testing complexity (up to $\mathrm{poly}(\varepsilon^{-1}\log d)$ factors) for the hypercube,
the adaptive complexity is still open.
The first polynomial lower bound of $\widetilde{\Omega}(d^{1/4})$ for adaptive testers was given
by Belovs and Blais~\cite{BeBl16} and has since been improved to $\widetilde{\Omega}(d^{1/3})$ by Chen, Waingarten, and Xie~\cite{Chen17}.
Chakrabarty and Seshadhri~\cite{ChSe19} gave an adaptive $\widetilde{O}_\varepsilon(I_f)$-query tester, thereby showing that adaptivity
can help in monotonicity testing. The $d^{1/3}$ vs $\sqrt{d}$ query complexity gap is an outstanding open question
in property testing.
There has been work on approximating the distance to monotonicity in $\mathrm{poly}(d,\varepsilon_f)$-queries.
Fattal and Ron~\cite{FR} gave the first non-trivial result of an $O(d)$-approximation, and Pallavoor, Raskhodnikova, and Waingarten~\cite{PRW22}
gave a non-adaptive $O(\sqrt{d})$-approximation (all running in $\mathrm{poly}(d,\varepsilon_f)$ time). They also
show that non-adaptive $\mathrm{poly}(d)$-time algorithms cannot beat this approximation factor.
The above discussion is only for Boolean valued functions on the hypercube. For arbitrary ranges,
the original results on monotonicity testing gave an $O(d^2/\varepsilon)$-query tester~\cite{GGLRS00,DGLRRS99}.
Chakrabarty and Seshadhri~\cite{ChSe13} proved that $O(d/\varepsilon)$-queries suffices for monotonicity testing,
matching the lower bound of $\Omega(d)$ of Blais, Brody, and Matulef~\cite{BBM11}. The latter bound
holds even when the range size is $\sqrt{d}$.
A recent result of Black, Kalemaj, and Raskhodnikova showed a smooth trade-off between the $\sqrt{d}$ bound
for the Boolean range and the $d$ bound for arbitrary ranges (\cite{BKR20}). Consider functions $f:\hyp{d} \to [r]$.
They gave a tester with query complexity $\widetilde{O}_\varepsilon(r\sqrt{d})$, achieved by extending
the directed Talagrand inequality to arbitrary range functions. Their techniques are quite black-box
and carry over to other posets. We note that their techniques
can also be ported to our setting, so we can get an $\widetilde{O}_\varepsilon(rn\sqrt{d})$-query monotonicity
tester for functions $f:[n]^d \to [r]$.
We now discuss monotonicity testing on the hypergrid. We discuss more about the $\varepsilon$-dependencies,
since there have been interesting relevant discoveries. As mentioned above, \cite{DGLRRS99}
gives a non-adaptive, one-sided $O((d/\varepsilon)\log^2(d/\varepsilon))$-query tester. This was improved to $O((d/\varepsilon)\log(d/\varepsilon))$ by Berman, Raskhodnikova, and Yaroslavtsev~\cite{BeRaYa14}.
This paper also showed an interesting adaptivity gap for 2D functions $f:[n]^2 \to \hyp{}$:
there exists an $O(1/\varepsilon)$-query adaptive tester (in fact, for any constant dimension $d$), and they show
an $\Omega(\log(1/\varepsilon)/\varepsilon)$ lower bound for non-adaptive testers.
Previous work~\cite{BlackCS18} by the authors gave an $\widetilde{O}_\varepsilon(d^{5/6}\log n)$-query tester, by proving
a directed Margulis inequality on augmented hypergrids. Another work~\cite{BlackCS20}
of the authors, and subsequently a work~\cite{HY22} by Harms and Yoshida, designed domain reduction methods for monotonicity testing, showing how $n$ can be reduced to $\mathrm{poly}(\varepsilon^{-1},d)$ by subsampling
the hypergrid.
For hypergrid functions with arbitrary ranges, the optimal complexity is known to be $\Theta(d\log n)$~\cite{ChSe13,ChSe14}.
When the range is $[r]$ and $d=1$, one can get $O(\log r)$-query testers~\cite{PaRaVa18}.
\section*{Majorization and Talagrand Objective}
\section{Preliminaries} \label{sec:prelims}
A central construct in our proof is the \emph{sort} operator.
\begin{definition} \label{def:sortline} Consider a Boolean function on the line $h:[n] \to \hyp{}$.
The sort operator $\sortline{}$ is defined as follows.
$$ \sortline{h}(b) =
\begin{cases}
0 & \textrm{if} \ b < n-\|h\|_1 \\
1 & \textrm{if} \ b \geq n-\|h\|_1
\end{cases}$$
\end{definition}
Thus, the sort operator ``moves" the values on a line to ensure that it is sorted.
Note that $\sortline{h}$ and $h$ have exactly the same number of zero/one valued points.
We can now define the sort operator for any dimension $i$. This operator
takes a hypergrid function and applies the sort operator on every $i$-line.
\begin{definition} \label{def:sortop} Let $i$ be a dimension
and $f:[n]^d \to \hyp{}$. The sort operator for dimension $i$, $\sorti{}{i}$, is defined as follows.
For every $i$-line $\ell$, $\sorti{f}{i}|_\ell = \sortline{f|_\ell}$.
Let $S$ be an ordered list of dimensions, denoted $(i_1, i_2, \ldots, i_k)$.
The function $S \circ f$
is obtained by applying the $\sorti{}{i}$ operator in the order given by $S$. Namely,
$$ S \circ f = \sorti{\sorti{\ldots \sorti{f}{{i_1}}}{{i_{k-1}}}}{{i_k}} $$
\end{definition}
Somewhat abusing notation, we will treat the ordered list of dimensions $S$
as a set, with respect to containing elements. The key property of the sort
operator is that it preserves the sortedness of \emph{other} dimensions.
\begin{claim} \label{clm:sortS} The function $S \circ f$ is monotone
along all dimensions in $S$.
\end{claim}
\begin{proof} We will prove the following statement: if $f$
is monotone along dimension $i$, then $\sorti{f}{j}$ is monotone
along both dimensions $i$ and $j$. A straightforward induction (which we omit)
proves the claim.
By construction, the function $\sorti{f}{j}$ is monotone along dimension $j$.
Consider two arbitrary points $\mathbf{x} \preceq \mathbf{x}'$ that are $i$-aligned (meaning
that they only differ in their $i$-coordinates).
We will prove that $\sorti{f}{j}(\mathbf{x}) \leq \sorti{f}{j}(\mathbf{x}')$, which will prove
that $\sorti{f}{j}$ is monotone along dimension $i$.
For convenience, let the $j$-lines containing $\mathbf{x}$ and $\mathbf{x}'$
be $\ell$ and $\ell'$, respectively. Note that these $j$-lines
only differ in their $i$-coordinates. Let $c$ denote the $j$-coordinate
of $\mathbf{x}$ (and $\mathbf{x}'$). Observe that $\sorti{f}{j}{\mathbf{x}} = \sorti{f}{j}|_\ell(c)$
(analogously for $\mathbf{x}'$).
Note that, $\forall c \in [n]$, $f|_\ell(c) \leq f|_{\ell'}(c)$.
This is because $f$ is monotone along dimension $i$, and $\ell$
has a lower $i$-coordinate than that of $\ell'$.
Hence, $\|f|_{\ell}\|_1 \leq \|f|_{\ell'}\|_1$.
By the definition of the sort operator, $\forall c \in [n], \sortline{f|_{\ell}}(c) \leq \sortline{f|_{\ell'}}(c)$.
Thus, $\sorti{f}{j}|_\ell(c) \leq \sorti{f}{j}|_{\ell'}(c)$, implying
$\sorti{f}{j}(\mathbf{x}) \leq \sorti{f}{j}(\mathbf{x}')$.
\end{proof}
A crucial property of the sort operator is that it can never increase
the distance between functions. This property, which was first established in~\cite{DGLRRS99} (Lemma 4), will be used in
\Cref{sec:semisorted-tal-dist}, where we apply our main isoperimetric theorem
on random restrictions
We provide a proof for completeness.
\begin{claim} \label{clm:sort-hamm} Let $f, f': [n]^d \to \hyp{}$
be two Boolean functions. For any ordered set $S \subseteq [d]$,
$$ \Delta(S \circ f, S \circ f') \leq \Delta(f,f')$$
\end{claim}
\begin{proof} It suffices to prove this bound when $S$
is a singleton. We prove that for any $i \in [d]$, $\Delta(\sorti{f}{i}, \sorti{f'}{i}) \leq \Delta(f,f')$. In the following, we will use the simple fact that for monotone functions $h, h':[n] \to \hyp{}$, $\Delta(h,h') = \Big| \|h\|_1 - \|h'\|_1 \Big|$.
Also, we use the equality $\|\sortline{h}\|_1 = \|h\|_1$.
\begin{eqnarray*}
\Delta(\sorti{f}{i}, \sorti{f'}{i}) & = & \sum_{\ell \ \textrm{$i$-line}} \Delta(\sorti{f}{i}|_\ell,
\sorti{f'}{i}|_\ell) = \sum_{\ell} \Big| \| \sorti{f}{i}|_\ell\|_1 - \| \sorti{f'}{i}|_{\ell}\|_1 \Big| \nonumber \\
& = & \sum_{\ell} \Big| \|f|_\ell\|_1 - \|f'|_{\ell}\|_1 \Big| \\
& = & \sum_{\ell} \Big| \sum_{c \in [n]} f|_\ell(c) - \sum_{c \in [n]} f|_{\ell}(c) \Big| \\
& \leq & \sum_{\ell} \sum_{c \in [n]} \Big|f|_\ell(c) - f'|_\ell(c) \Big| = \Delta(f,f')
\end{eqnarray*}
\end{proof}
The method of obtaining a monotone function via repeated sorting is close to being optimal.
For hypercubes, this result was established by~\cite{FR} (Lemma 4.3) and also present in~\cite{KMS15} (Lemma 3.5).
The proofs goes through word-for-word applied to hypergrids.
\begin{claim}\label{clm:2appx}
For any function $f:[n]^d \to \{0,1\}$,
\[
\varepsilon_f \leq \Delta(f, [d]\circ f) \leq 2\varepsilon_f
\]
\end{claim}
\begin{proof}
The first inequality is obvious since $[d]\circ f$ is monotone as established in~\Cref{clm:sortS}.
Let $h$ be the monotone function closest to $f$, that is, $\varepsilon_f =\Delta(f, h)$.
So,
\[
\Delta(f, [d]\circ f) \underbrace{\leq}_{\text{triangle ineq}} \Delta(f, h) + \Delta([d]\circ f, h)
\underbrace{=}_{\text{since}~h = [d]\circ h} \Delta(f, h) + \underbrace{\Delta([d]\circ f, [d]\circ h)}_{\leq \Delta(f,h) ~ \text{by~\Cref{clm:sort-hamm}}}~ \leq 2\Delta(f,h) = 2\varepsilon_f
\]
\end{proof}
We provide one more simple claim about the sort operator that will be used throughout \Cref{sec:proofoflemma6}. Given $h,h' \colon [n] \to \{0,1\}$, define
\[
\Delta^-(h,h') = |\{c \in [n] \colon h(c) > h'(c)\}| \text{ and } \Delta^+(h,h') = |\{c \in [n] \colon h(c) < h'(c)\}| \text{.}
\]
\begin{claim} \label{clm:sort-violations} Let $h,h' \colon [n] \to \{0,1\}$ be any two functions. Then, $\Delta^-(\sortline{h},\sortline{h'}) \leq \Delta^-(h,h')$. \end{claim}
\begin{proof} Observe that if $\norm{h}_1 \leq \norm{h'}_1$, then $\Delta^-(\sortline{h},\sortline{h'}) = 0$ and so we are done. On the other hand if $\norm{h}_1 \geq \norm{h'}_1$, then we have
\[
\Delta^-(\sortline{h},\sortline{h'}) = \norm{h}_1 - \norm{h'}_1 = \sum_{c\in[n]} h(c) - h'(c) = \Delta^-(h,h') - \Delta^+(h,h') \leq \Delta^-(h,h') \text{.}
\]
\end{proof}
\subsection{Colorful Influences and the Talagrand Objective} \label{sec:tal-obj}
We will need undirected, colorful Talagrand inequalities for proving \Thm{dir-tal}.
For the sake of completeness, we explicitly define the undirected colored influence.
\begin{definition} \label{def:col-inf} Consider a function $g:\hyp{d} \to \hyp{}$
and a $0$-$1$ coloring $\xi$ of the edges of the hypercube $\hyp{d}$.
The influence of $\mathbf{z} \in \hyp{d}$, denoted $I_{g,\xi}(\mathbf{z})$,
is the number of sensitive edges incident to $\mathbf{z}$ whose color has value $f(\mathbf{z})$.
(An edge is sensitive if both endpoints have different values.)
\end{definition}
Talagrand's theorem asserts that $\mathbf{E}_\mathbf{z}[\sqrt{I_g(\mathbf{z})}] = \Omega(\mathrm{var}(g))$~\cite{Tal93}.
The robust/colored version proven by KMS asserts this to be true for arbitrary colored
influences.
\begin{theorem}[Paraphrasing Theorem 1.8 of~\cite{KMS15}] (Colored Talagrand Theorem on the Undirected Hypercube)\label{thm:kms-und}
There exists an absolute constant $C > 0$ such that
for any function $g:\{0,1\}^d \to \{0,1\}$ and any $0$-$1$ coloring
$\xi$ of the edges of the hypercube,
\[
\mathbf{E}_{\mathbf{z}\in \{0,1\}^d} \left[\sqrt{I_\xi(\mathbf{z})} \right] \geq C \cdot \mathrm{var}(g)
\]
\end{theorem}
It will be convenient in our analysis to formally define the Talagrand objective for colored, thresholded influences
on the hypergrid.
\begin{definition}[Colored Thresholded Talagrand Objective]
Given any Boolean function $f:[n]^d \to \{0,1\}$ and $\chi:E \to \{0,1\}$,
we define the Talagrand objective with respect to the colorful thresholded influence as
\[
T_{\Phi_\chi}(f) := \mathbf{E}_\mathbf{x}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right]
\]
where, $\Phi_{f,\chi}$ is defined in~\Cref{def:phi-f-chi}.
\end{definition}
\subsection{Majorization} \label{sec:major}
It is convenient to think of the Talagrand objective as a ``norm'' of a vector. Throughout the paper, we (ab)use the following notation:
\[
\textrm{given a vector $\mathbf{v} \in \mathbb{R}^t_{\geq 0}$},~~~ \norm{\mathbf{v}}_{1/2} := \sum_{i=1}^t \sqrt{\mathbf{v}_i}\text{.}
\]
If we imagine an $n^d$-dimensional vector indexed by the points of the hypergrid, we see that the Talagrand objective is precisely the norm of
the vector whose $\mathbf{x}$'th entry is $\Phi_{f,\chi}(\mathbf{x})$. Most often, however, we would be considering the Talagrand objective line-by-line, with the
natural ordering of the line defining a natural ordering on the vector. To be more precise, fix a dimension $i\in [d]$ and fix an $i$-line $\ell$.
An $i$-line is a set of $n$ points which only differ in the $i$th coordinate. This line $\ell$ defines a vector $\vv{\Phi_\ell(f)} \in \mathbb{R}_{\geq 0}^n$
whose $j$th coordinate, for $1\leq j\leq n$ is precisely $\Phi_{f,\chi}(\mathbf{x})$ where $\mathbf{x} \in \ell$ has $\mathbf{x}_i = j$. Note that
\[
\forall i\in [d],~~~ T_{\Phi_{\chi}}(f) = \frac{1}{n^d} \sum_{i\text{-lines}~\ell} \norm{\vv{\Phi_\ell(f)}}_{1/2}\text{.}
\]
Our proof to establish (the correct version of)~\eqref{eq:hope2} proceeds via a hybrid argument that modifies the function and the coloring in various stages.
In each stage, we prove that the norm decreases. We use the following facts from the theory of majorization.
In the rest of this subsection all vectors, unless explicitly mentioned, live in $\mathbb{R}^t_{\geq 0}$ for some positive integer $t$.
Given a vector $\mathbf{a}$, we use $\sortdown{\mathbf{a}}$ and $\sortup{\mathbf{a}}$ to denote the vectors obtained by sorting $\mathbf{a}$ in decreasing and increasing order, respectively. Given two vectors
$\mathbf{a}$ and $\mathbf{b}$ with the same $\ell_1$ norm,
we say $\mathbf{a} \succeq_{\mathsf{maj}} \mathbf{b}$ if for all $1\leq k \leq t$, $\sum_{i\leq k} \sortdown{\mathbf{a}}_i \geq \sum_{i\leq k} \sortdown{\mathbf{b}}_i$.
Throughout this paper, when we apply majorization the LHS vector would be sorted (either increasing or decreasing) while the RHS vector would be unsorted.
To be absolutely clear which is which, when $\mathbf{a}$ is sorted decreasing, we use the notation $\mathbf{a} \succeq_{\mathsf{maj}} \sortdown{\mathbf{b}}$ and when $\mathbf{a}$ is sorted increasing we use the notation
$\mathbf{a} \succeq_{\mathsf{maj}} \sortup{\mathbf{b}}$. Here is a simple standard fact that connects majorization to the Talagrand objective;
it uses the fact that the sum of square roots is a symmetric concave function, and is thus Schur-concave.
\begin{fact}[Chapter 3,~\cite{MarshallOA11}]
Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \succeq_{\mathsf{maj}} \mathbf{b}$. Then, $\norm{\mathbf{a}}_{1/2} \leq \norm{\mathbf{b}}_{1/2}$.
\end{fact}
\noindent
Next, we state and prove a simple but key lemma repeatedly used throughout the analysis.
\begin{mdframed}[backgroundcolor=blue!10,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{lemma}\label{lem:sum-of-vectors}
Let $\vv{U} = \sum_i \mathbf{w}_i$ be a finite sum of $t$-dimensional non-negative vectors.
Let $\vv{S} := \sum_i \sortdown{\mathbf{w}_i}$. Then, $\vv{S} \succeq_{\mathsf{maj}} \sortdown{\vv{U}}$.
Analogously, if $\vv{S} := \sum_i \sortup{\mathbf{w}_i}$, then $\vv{S} \succeq_{\mathsf{maj}} \sortup{\vv{U}}$.
\end{lemma}
\end{mdframed}
\begin{proof}
We prove the first statement; the second analogous statement has an absolutely analogous proof.
We begin by noting $\vv{S}$ is a sorted decreasing vector since it is a sum of sorted decreasing vectors.
For brevity, let's use $\vv{V} := \sortdown{\vv{U}}$. Next, we note that
$\norm{\vv{S}}_1 = \norm{\vv{V}}_1 = \sum_i \norm{\mathbf{w}_i}_1$.
Now fix a $1\leq \tau \leq t$. We need to show $\sum_{j=1}^\tau \vv{S}_j\geq \sum_{j=1}^\tau \vv{V}_j$.
Consider the $\tau$ largest coordinates of $\vv{U}$, and let them comprise $T\subseteq [t]$ where $|T| = \tau$.
Consider the $\tau$-dimensional vectors $\mathbf{w}_i[T]$ where we restrict our attention to only these coordinates.
Let $\vv{S}'$ be the $\tau$-dimensional vector formed by the sum of the sorted versions $\sortdown{\mathbf{w}_i[T]}$.
Note that $\sum_{j=1}^{\tau} \vv{S}'_j = \sum_{j=1}^\tau \vv{V}_j$.
Also note that for any $1\le j\leq \tau$, the number $\vv{S}'_j$ equals $\sum_i(\text{$j$th max of $\mathbf{w}_i[T]$})$ and $\vv{S}_j$ equals $\sum_{i} (\text{$j$th max of $\mathbf{w}_i$})$.
Thus, $\vv{S}_j \geq \vv{S}'_j$, proving that $\sum_{j=1}^\tau \vv{S}_j\geq \sum_{j=1}^\tau \vv{V}_j$. \end{proof}
\section{Connecting Talagrand Objectives of $f$ and the Tracker Functions}\label{sec:mainworkhorse}
In this section and the next, we establish our main technical result~\Cref{thm:semisorted-reduce-to-g} relating the Talagrand objectives
on the colorful thresholded influence of the hypergrid function $f:[n]^d \to \{0,1\}$ and the Talagrand objectives on the undirected influence of the
tracker functions. We restate the theorem below for convenience.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\semisorted*
\end{mdframed}
\noindent
To prove \Cref{thm:semisorted-reduce-to-g} we need to describe the coloring $\xi_\mathbf{x}$ for each $\mathbf{x}$ in $[n]^d$. We proceed doing so in $d$ stages.
\begin{itemize}
\item For every $i \in \{0,1,\ldots,d\}$ and for every $\mathbf{x} \in [n]^d$, we define a {\em partial} edge coloring $\xi^{(i)}_\mathbf{x}$ of the hypercube which assigns a $\{0,1\}$ value to every hypercube edge of the form $(T, T\oplus j)$ for all $j \leq i$, and for all $T\subseteq [i]$. The process will begin with the null coloring, $\xi^{(0)}_{\mathbf{x}}$, and end with a complete coloring, $\xi_\mathbf{x} := \xi^{(d)}_\mathbf{x}$, for every $\mathbf{x} \in [n]^d$.
\item For every $i \in \{0,1,\ldots,d\}$ and every $S \subseteq [i]$ we will also define a coloring $\chi_S^{(i)}$ of the edges of the augmented hypergrid. We start with $\chi^{(0)}_{\emptyset} := \chi$ where $\chi$ is the original coloring which, recall, is adversarially chosen.
\end{itemize}
For every $i \in \{0,1,\ldots,d\}$ and $S \subseteq [i]$ we will use the above colorings to define the $(i,S)$-\emph{hybrid Talagrand objective}
\begin{equation}\label{eq:rhs-quantity}
R_{i}(S) := \mathbf{E}_{\mathbf{x} \in [n]^d} \sqrt{~\sum_{j=1}^{i} I^{=j}_{g_\mathbf{x}, \xi^{(i)}_\mathbf{x}} (S) ~~+~~ \sum_{j=i+1}^d \Phi_{S\circ f,\chi_S^{(i)}}(\mathbf{x}; j) }\text{.} \tag{Colorful Hybrid}
\end{equation}
Recall that $S\circ f$ is the function obtained after sorting $f$ on the coordinates in $S$. Note that $R_i(S)$ is well-defined given the partial colorings $\xi_\mathbf{x}^{(i)}$ for each $\mathbf{x} \in [n]^d$ as defined above. Also
observe that since $\chi_\emptyset^{(0)} := \chi$, the arbitrary coloring specified in the theorem statement, we have that
$R_0(\emptyset)$ is precisely the LHS in the statement of \Cref{thm:semisorted-reduce-to-g}, that is, $R_0(\emptyset) = \mathbf{E}_{\mathbf{x} \in [n]^d}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right]$. Additionally, since we use $\xi_\mathbf{x} := \xi_\mathbf{x}^{(d)}$, observe that $\mathbf{E}_{S\subseteq [d]}[R_d(S)]$ is precisely the RHS in the statement of \Cref{thm:semisorted-reduce-to-g}.
With the above setup in mind, we show that the following \Cref{lem:lhs2-and-lhs3} suffices to prove \Cref{thm:semisorted-reduce-to-g}.
\begin{mdframed}[backgroundcolor=blue!10,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{lemma}[Potential Drop Lemma]\label{lem:lhs2-and-lhs3}
Fix $i \in \{1,\ldots,d\}$, $\xi^{(i-1)}_\mathbf{x}$ for all $\mathbf{x} \in [n]^d$, and $\chi_S^{(i-1)}$ for every $S \subseteq [i-1]$, which all satisfy the specifications described in the previous paragraph.
There exists a choice of $\xi^{(i)}_\mathbf{x}$ for every $\mathbf{x}\in [n]^d$ and $\chi_S^{(i)}$, $\chi_{S+i}^{(i)}$ for every $S \subseteq [i-1]$ all satisfying the specifications described in the previous paragraph, such that for all $S \subseteq [i-1]$, we have (a) $R_{i-1}(S) \geq R_i(S)$ and (b) $R_{i-1}(S)\geq R_i(S+i)$.
\end{lemma}
\end{mdframed}
\begin{proof}[\bf Proof of \Cref{thm:semisorted-reduce-to-g}:]
Consider the following binary tree with $d+1$ levels. Each level $i \in \{0,1,\ldots,d\}$ has $2^i$ nodes indexed by subsets $S\subseteq [i]$.
Every such node is associated with a coloring $\chi_S^{(i)}$ of the augmented hypergrid edges. The level $i$ is also associated with a partial coloring $\xi^{(i)}_\mathbf{x}$ for every $\mathbf{x}\in [n]^d$.
The $0$'th level contains a single node indexed by $\emptyset$. The associated augmented hypergrid coloring
is $\chi_\emptyset^{(0)} := \chi$. The partial coloring $\xi^{(0)}_\mathbf{x}$ is null for all $\mathbf{x}\in [n]^d$.
We associate the value $R_0(\emptyset) = T_{\Phi_{\chi}}(f)$ with the root.
For $1\leq i\leq d$, we describe the children of each node in level $i-1$. Each node in level $i-1$ is indexed by some $S\subseteq [i-1]$. We associate this node with the value $R_{i-1}(S)$. This node has two children at level $i$: one, the left child, indexed by $S$ and the other, the right child, indexed by $S+i$. The coloring of the hypergrid edges at the left child is defined as $\chi_S^{(i)}$ from the lemma, and that of the hypergrid edges at the right child is defined as $\chi_{S+i}^{(i)}$ from the lemma. The left and right children hold the quantites $R_{i}(S)$ and $R_i(S+i)$, respectively. At level $i$, the partial coloring $\xi_\mathbf{x}^{(i-1)}$ is also extended to $\xi_\mathbf{x}^{(i)}$ for every $\mathbf{x} \in [n]^d$ as stated in the lemma. From the lemma, we have $R_{i-1}(S) \geq R_i(S)$ and $R_{i-1}(S) \geq R_{i}(S+i)$. This immediately implies the following:
\[
\text{For all } i \in \{1,\ldots,d\} \text{, we have } \mathbf{E}_{S \subseteq [i-1]}[R_{i-1}(S)] \geq \mathbf{E}_{S \subseteq [i]}[R_{i}(S)]
\]
and chaining these $d$ inequalities together yields $R_0(\emptyset) \geq \mathbf{E}_{S \subseteq [d]}[R_d(S)]$.
Now consider the leaf nodes of this tree, which hold the values $R_d(S)$ for every $S \subseteq [d]$.
Observe that $R_d(S) = \mathbf{E}_{\mathbf{x} \in [n]^d} \left[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}\right]$ since $\xi_\mathbf{x} := \xi_\mathbf{x}^{(d)}$. Recalling that $R_0(\emptyset) = T_{\Phi_{\chi}}(f)$ yields
\[
T_{\Phi_{\chi}}(f) = R_0(\emptyset) \geq \mathbf{E}_{S\subseteq [d]} [R_d(S)] = \mathbf{E}_{S\subseteq [d]}\mathbf{E}_{\mathbf{x} \in [n]^d} \left[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}\right]
\]
and this establishes the claim after exchanging the expectations. \end{proof}
\section{Proof of Potential Drop~\Cref{lem:lhs2-and-lhs3}}\label{sec:proofoflemma6}
Recall $i \in \{1,\ldots,d\}$ is fixed. For brevity's sake, we will fix a set $S \subseteq [i-1]$ and call $h := (S\circ f)$. Let's refer to $\chi^{(i-1)}_S$ as simply $\chi$ without confusing with the original $\chi$ in the theorem.
The two colorings $\chi^{(i)}_S$ and $\chi^{(i)}_{S+i}$ that we construct will be simply called $\chi'$ and $\chi''$, respectively.
Let's call the partial colorings $\xi^{(i-1)}_\mathbf{x}$ as simply $\xi_\mathbf{x}$.
We will call the coloring $\xi^{(i)}_\mathbf{x}$ which we need to construct simply $\xi'_\mathbf{x}$ in the latter. Recall that $\xi_{\mathbf{x}}$ is defined on all edges $(T,T\oplus j)$ for $T \subseteq [i-1]$ and $j \leq i-1$ and in order to prove the lemma we will need to define $\xi_\mathbf{x}'$ on all edges $(T \oplus j)$ for $T \subseteq [i]$ and $j \leq i$.
Fix an $i$-line $\ell$. We prove the lemma line-by-line. To be precise, let us consider the following vectors. First,
\begin{equation}\label{eq:rhs-vector-def}
\vv{L}_{\ell} := \left(~~ \underbrace{\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)}_{\vv{L^{(1)}}_{\ell}} ~+~ \underbrace{\Phi_{h,\chi}(\mathbf{x};i)}_{_{\vv{L^{(2)}}_{\ell}}} ~+~\underbrace{\sum_{j=i+1}^d \Phi_{h,\chi}(\mathbf{x};j)}_{\vv{L^{(3)}}_{\ell}}~~~~:~\mathbf{x}\in \ell\right)
\end{equation}
Observe that
\begin{equation}\label{eq:rhs}
R_{i-1}(S) = \frac{1}{n^d} \sum_{i\text{-lines } \ell}\norm{\vv{L}_{\ell}}_{1/2} = \frac{1}{n^d} \sum_{i\text{-lines } \ell} \norm{\vv{L^{(1)}}_{\ell} + \vv{L^{(2)}}_{\ell} + \vv{L^{(3)}}_{\ell}}_{1/2}
\end{equation}
where, recall, we are (ab)using the notation $\norm{v}_{1/2} := \sum_i \sqrt{v_i}$.
\noindent
Define
\begin{equation}\label{eq:lhs2-vector-def}
\vv{R}_{\ell} := \left(~~\underbrace{\sum_{j=1}^{i-1} I_{g_\mathbf{x},\color{red} \xi'_\mathbf{x}}^{=j}(S)}_{\vv{R^{(1)}}_{\ell}} ~+~ \underbrace{ I_{g_\mathbf{x},{\color{red} \xi'_\mathbf{x}}}^{=i}(S)}_{\vv{R^{(2)}}_{\ell}} ~+~
\underbrace{\sum_{j=i+1}^d \Phi_{h,\color{red} \chi'}(\mathbf{x};j)}_{\vv{R^{(3)}}_{\ell}}~ ~~~:~\mathbf{x}\in \ell\right)
\end{equation}
where we have denoted, in red, the recolorings that we need to define.
The ``first'' RHS term is
\begin{equation}\label{eq:rhs}
R_{i}(S) := \frac{1}{n^d} \sum_{i\text{-lines } \ell} \norm{\vv{R}_{\ell}}_{1/2} = \frac{1}{n^d} \sum_{i\text{-lines } \ell} \norm{\vv{R^{(1)}}_{\ell} + \vv{R^{(2)}}_{\ell} + \vv{R^{(3)}}_{\ell}}_{1/2}
\end{equation}
Similarly, define
\begin{equation}\label{eq:lhs3-vector-def}
\vv{M}_{\ell} := \left(~~\underbrace{\sum_{j=1}^{i-1} I^{=j}_{g_\mathbf{x},\color{red} \xi'_\mathbf{x}}(S+i)}_{\vv{M^{(1)}}_{\ell}} ~+~ \underbrace{I^{=i}_{g_\mathbf{x},{\color{red} \xi'_\mathbf{x}}}(S+i)}_{\vv{M^{(2)}}_{\ell}} ~+~
\underbrace{\sum_{j=i+1}^d \Phi_{i\circ h,\color{red} \chi''}(\mathbf{x};j)}_{\vv{M^{(3)}}_{\ell}}~ ~~~:~\mathbf{x}\in \ell\right)
\end{equation}
and notice that the ``second'' RHS term is
\begin{equation}\label{eq:rhs}
R_{i}(S+i) := \frac{1}{n^d} \sum_{i\text{-lines } \ell}\norm{\vv{M}_{\ell}}_{1/2} = \frac{1}{n^d} \sum_{i\text{-lines } \ell} \norm{\vv{M^{(1)}}_{\ell} + \vv{M^{(2)}}_{\ell} + \vv{M^{(3)}}_{\ell}}_{1/2}
\end{equation}
Observe now that it suffices to prove that there exists colorings $\chi', \chi''$, and $\xi'_\mathbf{x}$'s such that $\norm{\vv{L}_{\ell}}_{1/2} \geq \norm{\vv{R}_{\ell}}_{1/2}$ and $\norm{\vv{L}_{\ell}}_{1/2} \geq \norm{\vv{M}_{\ell}}_{1/2}$ for all $i$-lines $\ell$. Thus, we now fix an $i$-line $\ell$ and drop the subscript, $\ell$, from all the previously defined vectors for brevity. We define $\mathsf{LHS} := \norm{\vv{L}}_{1/2}$, $\mathsf{RHS}_1 := \norm{\vv{R}}_{1/2}$, $\mathsf{RHS}_2 := \norm{\vv{M}}_{1/2}$, and set out to prove $\mathsf{LHS} \geq \mathsf{RHS}_1$ and $\mathsf{LHS} \geq \mathsf{RHS}_2$.
\paragraph{A Picture of the Line.}
Since $h$ is semisorted, the picture of $h$ restricted to $\ell$ looks like this. The green zone is where the function is $1$.
Without loss of generality we assume $\ell$ has more ones than zeros.
We use $A$ to denote the ones on the left and $C$ to denote the zeros on the right. We use $k := |C|$, and $B\subseteq A$ are the $k$ right most ones
in the left side.
\begin{figure}[h!]
\includegraphics*[trim = 0 320 0 100, clip, scale = 0.5]{figs/semisorted-k-try2}
\end{figure}
Throughout, we will use the notation $\vv{A}_X$ to denote the sub-vector of $\vv{A}$ defined on $\ell$ with coordinates restricted to $x\in X$; we will always use this notation when $X$ is a contiguous interval. Indeed, these $X$'s will be always picked from $\{W, A, C, O, B, A\setminus B\}$ or unions of these, always making sure they form a contiguous interval.
\paragraph{High Level Idea.} Before we venture into proving the inequalities, we would like to remind the reader again
of the proof strategy discussed in~\Cref{sec:main-ideas}. We need to define the colorings $\chi'$, $\chi''$, and
also $\xi_\mathbf{x}^{(i)}$'s such that the objective after recoloring satisfy the inequality we desire to prove. This going to hinge upon showing that the vector
obtained after operation either majorizes or is coordinate-wise dominated by a vector that majorizes the vector before the operation.
In particular, these are the conditions (a)-(d) and (e)-(h) mentioned below in the grey boxes.
To show these properties, we would be crucially using the property that the function $f$ is semi-sorted which leads to certain monotonicity properties that allows us to claim them. In particular, we would be using~\Cref{lem:sum-of-vectors} when establishing almost all the conditions mentioned above.
There is a certain sense of repetition in which these arguments are made, however, we have provided all the details for completeness.
\subsection{Proving $\mathsf{LHS} \geq \mathsf{RHS}_1$}
During the proof of $\mathsf{LHS} \geq \mathsf{RHS}_1$, we will define the coloring $\chi'$ on all edges of the fully augmented hypergrid and $\xi'_\mathbf{x}(S, S\oplus j)$ where $j\leq i$ for all $\mathbf{x} \in [n]^d$.
We will not specify $\xi'_\mathbf{x}(S + i, S + i \oplus j)$ since these won't be needed to prove this inequality; we will describe them when we prove $\mathsf{LHS} \geq \mathsf{RHS}_2$.
Before we describe the recolorings, it is useful to describe the plan of the proof. This will motivate why we recolor as we do.
We will actually consider
\[
\mathsf{LHS} = \norm{\vv{L}_{W}}_{1/2} + \norm{\vv{L}_{A}}_{1/2} + \norm{\vv{L}_C}_{1/2} + \norm{\vv{L}_{O}}_{1/2} \]
and
\[
\mathsf{RHS}_1 = \norm{\vv{R}_{W}}_{1/2} + \norm{\vv{R}_{A}}_{1/2} + \norm{\vv{R}_C}_{1/2} + \norm{\vv{R}_{O}}_{1/2}
\]
and argue domination term-by-term.
More precisely, we find recolorings $\chi', \xi'$ such that
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{center}
\begin{enumerate}
\item[(a)] $\vv{R^{(q)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(q)}_A}}$ and $\vv{R^{(q)}_O} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(q)}_O}}$, for $q\in \{1,3\}$,
\item[(b)] $\exists \vv{L'^{(2)}_A}$ such that $\vv{L'^{(2)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(2)}_A}}$ and $\vv{L'^{(2)}_A} \succeq_{\mathsf{coor}} \vv{R^{(2)}_A}$,
\item[(c)] $\vv{R^{(q)}_W} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(q)}_W}}$ and $\vv{R^{(q)}_C} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(q)}_C}}$, for $q\in \{1,3\}$,
\item[(d)] $\exists \vv{L'^{(2)}_C}$ such that $\vv{L'^{(2)}_C} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(2)}_C}}$ and $\vv{L'^{(2)}_C} \succeq_{\mathsf{coor}} \vv{R^{(2)}_C}$.
\end{enumerate}
\end{center}
\end{mdframed}
Let us see why the above conditions suffice to prove the inequality.
The second part of (b) implies that $\norm{\vv{R_A}}_{1/2} \leq \norm{\vv{R^{(1)}_A} + \vv{R^{(3)}_A} + \vv{L'^{(2)}_A}}_{1/2}$.
Part (a) and the first part of (b), along with~\Cref{lem:sum-of-vectors}, implies $\vv{R^{(1)}_A} + \vv{R^{(3)}_A} + \vv{L'^{(2)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L_A}}$.
And so, $\norm{L_A}_{1/2} \geq \norm{R_A}_{1/2}$. A similar argument using (c) and (d) implies $\norm{L_C}_{1/2} \geq \norm{R_C}_{1/2}$.
One last observation is needed to complete the proof. Note that $R^{(2)}_W$ is the {\bf zero} vector: the points $x\in W$ don't change value even when $\ell$ is sorted.
Also note that $L^{(2)}_W$ is the zero vector; the points $x\in W$ don't participate in a violation in direction $i$.
And therefore, part (c) along with~\Cref{lem:sum-of-vectors} implies $\vv{R_W} \succeq_{\mathsf{maj}} \sortup{\vv{L_W}}$ implying $\norm{L_W}_{1/2} \geq \norm{R_W}_{1/2}$.
Similarly, $R^{(2)}_O \equiv L^{(2)}_O \equiv \mathbf{0}$, and thus part (a) along with~\Cref{lem:sum-of-vectors} implies $\norm{L_O}_{1/2} \geq \norm{R_O}_{1/2}$.
\subsubsection{Proving (a) and (c) for $q=3$} \label{sec:ac3}
\paragraph{Defining the Coloring $\chi'$:}
We will now describe the coloring $\chi'$ on all edges of the form $(\mathbf{x},\mathbf{x} + a\mathbf{e}_j)$ where $j \geq i+1$, $h(\mathbf{x}) = 1$ and $h(\mathbf{x}+a\mathbf{e}_j) = 0$. For all other edges $e$, we simply define $\chi'(e) = \chi(e)$ as these edges do not play a role in proving the inequality.
Given a pair of $i$-lines $\ell$ and $\ell' = \ell + a\mathbf{e}_j$ for $j\geq i+1$ and $a > 0$, we consider the set of violations from $\ell$ to $\ell'$ in $h$:
\begin{align} \label{eq:violations}
V := \{(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in \ell \text{, } h(\mathbf{x}) = 1\text{, and } h(\mathbf{x}+a\mathbf{e}_j) = 0\} \text{.}
\end{align}
Since $h$ is semi-sorted, it's clear that we can write $V = V_L \cup V_R$ as a union of two intervals, in the sense that $\{\mathbf{x} \colon (\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \in V_L\}$ is an interval in the lower half of $\ell$ and $\{\mathbf{x} \colon (\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \in V_R\}$ is an interval in the upper half of $\ell$. Similarly, the upper endpoints form two intervals in $\ell'$. We then obtain $\chi'$ by down-sorting $\chi$ on each of these intervals, moving left-to-right:
\[
(\chi'(e) \colon e \in V_L) = \sortdown{\chi(e) \colon e \in V_L} \text{ and } (\chi'(e) \colon e \in V_R) = \sortdown{\chi(e) \colon e \in V_R}\text{.}
\]
We provide the following illustration for clarity. The white and green intervals represent where $h = 0$ and $h=1$, respectively. The vertical arrows represent violated edges. Blue edges have color $0$ and red edges have color $1$. The left picture depicts the original coloring, $\chi$, and the right picture depicts the recoloring $\chi'$.
\hspace*{-1cm}\includegraphics*[clip, scale = .5, trim = 0 300 0 40]{figs/RHS1-chi-recoloring}
We now return to our fixed $i$-line $\ell$ and set out to prove parts (a) and (c) for $q=3$, given this coloring $\chi'$. Let's recall our illustration of $\ell$ and our definition of the intervals $W,A,C,O$.
\begin{figure}[h!]
\includegraphics*[trim = 20 320 0 100, clip, scale = 0.5]{figs/semisorted-k-try2}
\end{figure}
\paragraph{Proving (a) for $q=3$:}
Fix $j \geq i+1$ and a $i$-line $\ell' := \ell + a\mathbf{e}_j$. Let $A' := \{\mathbf{x}\in A~:~ h(\mathbf{x}+a\mathbf{e}_j) = 0\}$ and $O' := \{\mathbf{x}\in O~:~ h(\mathbf{x}+a\mathbf{e}_j) = 0\}$.
Since $h$ is semi-sorted, it is not hard to see that $A'$ and $O'$ are prefixes of $A$ and $O$, respectively.
\begin{claim}
If $\mathbf{x}_i < \mathbf{x}'_i$ in $A$ such that $\mathbf{x}' \in A'$, then $\mathbf{x}\in A'$. The same is true for $O$ and $O'$.
\end{claim}
\begin{proof}
Since $h$ is semisorted, $h(\mathbf{x}' + a\mathbf{e}_j) = 0$ implies $h(\mathbf{x}+a\mathbf{e}_j) = 0$.
\end{proof}
Moreover, observe that our definition of $\chi'$ gives us
\[
(\chi'(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in A') = \sortdown{\chi(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in A'}
\]
and
\[
(\chi'(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in O') = \sortdown{\chi(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in O'} \text{.}
\]
Let's investigate what this leads to. These are key properties.
\begin{definition}
Fix $j \geq i + 1$ and fix an $i$-line $\ell' := \ell + a\mathbf{e}_j$ for $a > 0$. Define the following two boolean vectors
\[
\mathbf{v}^{R}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} + a\mathbf{e}_j) = 0 ~~\textbf{and}~~~\chi'(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1\right) }~~:~\mathbf{x}\in A \right)
\]
and
\[
\mathbf{v}^{L}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} + a\mathbf{e}_j) = 0 ~~\textbf{and}~~~\chi(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1\right) }~~:~\mathbf{x}\in A\right)
\]
\end{definition}
\noindent
Observe, for $\mathbf{x}\in A$,
\begin{equation}\label{eq:L3R3-piece1}
\Phi_{h,\chi'}(\mathbf{x};j) = \min\left(1, \sum_a \mathbf{v}^{R}_{j,a}(\mathbf{x})\right) ~~\textrm{and}~~~ \Phi_{h,\chi}(\mathbf{x}; j) = \min\left(1, \sum_a \mathbf{v}^{L}_{j,a}(\mathbf{x}) \right)
\end{equation}
\begin{claim}
Fix a $j \geq i + 1$ and $a > 0$. For any two $\mathbf{x}_i < \mathbf{x}'_i$ in $A$, we have $\mathbf{v}^{R}_{j,a}(\mathbf{x}) \geq \mathbf{v}^{R}_{j,a}(\mathbf{x}')$.
That is, the vector $\mathbf{v}^{R}_{j,a}$ is sorted decreasing.
\end{claim}
\begin{proof}
Since $h$ is semisorted $h(\mathbf{x}' + a\mathbf{e}_j) = 0$ implies $h(\mathbf{x} + a\mathbf{e}_j) = 0$.
Furthermore, since both these are violations, by design $\chi'(\mathbf{x}', \mathbf{x}'+a\mathbf{e}_j) = 1$ implies $\chi'(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1$.
\end{proof}
\begin{claim}\label{clm:L2-permutation}
Fix a $j \geq i + 1$ and $a > 0$. The vectors $\mathbf{v}^{R}_{j,a}$ and $\mathbf{v}^{L}_{j,a}$ are permutations of one another.
\end{claim}
\begin{proof}
This is precisely how $\chi'$ is defined: it only permutes the colorings on the violations incident on $A$.
\end{proof}
\noindent
In conclusion, using the observation~\eqref{eq:L3R3-piece1}, we conclude that we can write
\[
\vv{L^{(3)}_A} = \left(\sum_{j=i+1}^d \Phi_{h,\chi}(\mathbf{x};j)~~:~~\mathbf{x} \in A\right)
\]
as a weighted sum of Boolean vectors, and the above two claims imply that the vector
\[
\vv{R^{(3)}_A} = \left(\sum_{j=i+1}^d \Phi_{h,\chi'}(\mathbf{x};j)~~:~~\mathbf{x}\in A\right)
\]
is the same weighted sum of the {\em sorted decreasing} orders of those Boolean vectors.
Therefore, we can conclude using~\Cref{lem:sum-of-vectors},
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L3R3A}
\vv{R^{(3)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(3)}_A}}
\end{equation}
\end{mdframed}
An absolutely analogous argument with $O$'s replacing $A$'s gives us
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L3R3O}
\vv{R^{(3)}_O} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(3)}_O }}
\end{equation}
\end{mdframed}
\paragraph{Proving (c) for $q=3$:}
The picture is similar, but reversed, when we consider the points in $W\cup C$, where $h(\mathbf{x}) = 0$. Recall the definition of $W$ and $C$ as in the illustration. Fix $j \geq i+1$ and a $i$-line $\ell'' := \ell - a\mathbf{e}_j$. Let $W' := \{\mathbf{x}\in W~:~ h(\mathbf{x}-a\mathbf{e}_j) = 1\}$ and $C' := \{\mathbf{x}\in C~:~ h(\mathbf{x}-a\mathbf{e}_j) = 1\}$.
It is not hard to see that $W'$ and $C'$ are suffixes of $W$ and $C$, respectively.
\begin{claim}
If $\mathbf{x}_i < \mathbf{x}'_i$ in $W$ such that $\mathbf{x} \in W'$, then $\mathbf{x}'\in W'$. The same is true for $C$ and $C'$.
\end{claim}
\begin{proof}
Since $h$ is semisorted, $h(\mathbf{x} - a\mathbf{e}_j) = 1$ implies $h(\mathbf{x}'-a\mathbf{e}_j) = 1$.
\end{proof}
Again, observe that our definition of $\chi'$ gives us
\[
(\chi'(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in W') = \sortdown{\chi(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in W'}
\]
and
\[
(\chi'(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in C') = \sortdown{\chi(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in C'}
\]
\begin{definition}
Fix $j \geq i + 1$ and fix an $i$-line $\ell'' := \ell - a\mathbf{e}_j$ for $a > 0$. Define the following two boolean vectors
\[
\mathbf{v}^{R}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} - a\mathbf{e}_j) = 1 ~~\textbf{and}~~~\chi'(\mathbf{x}-a\mathbf{e}_j, \mathbf{x}) = 0\right)}~~:~\mathbf{x}\in C\right)
\]
and
\[
\mathbf{v}^{L}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} - a\mathbf{e}_j) = 1 ~~\textbf{and}~~~\chi(\mathbf{x}-a\mathbf{e}_j, \mathbf{x}) = 0\right) }~~:~\mathbf{x}\in C\right)
\]
\end{definition}
Observe, for $\mathbf{x}\in C$,
\begin{equation}
\Phi_{h,\chi'}(\mathbf{x};j) = \min\left(1, \sum_a \mathbf{v}^{R}_{j,a}(\mathbf{x})\right) ~~\textrm{and}~~~ \Phi_{h,\chi}(\mathbf{x};j) = \min\left(1, \sum_a \mathbf{v}^{L}_{j,a}(\mathbf{x}) \right)
\end{equation}
\begin{claim}
Fix a $j \geq i + 1$ and $a > 0$. For any two $\mathbf{x}_i > \mathbf{x}'_i$ in $C$, we have $\mathbf{v}^{R}_{j,a}(\mathbf{x}) \geq \mathbf{v}^{R}_{j,a}(\mathbf{x}')$.
That is, the vector $\mathbf{v}^{R}_{j,a}$ is sorted increasing when considered left to right.
\end{claim}
\begin{proof}
Since $h$ is semisorted $h(\mathbf{x}' - a\mathbf{e}_j) = 1$ implies $h(\mathbf{x} - a\mathbf{e}_j) = 1$.
Furthermore, since both these are violations, by design $\chi'(\mathbf{x}', \mathbf{x}'+a\mathbf{e}_j) = 0$ implies $\chi'(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 0$.
\end{proof}
\begin{claim}\label{clm:L2-permutation}
Fix a $j \geq i + 1$ and $a > 0$. The vectors $\mathbf{v}^{R}_{j,a}$ and $\mathbf{v}^{L}_{j,a}$ are permutations of one another.
\end{claim}
A similar argument to the one given above now implies
$\vv{L^{(3)}_C}$
is a sum of Boolean vectors, and
$\vv{R^{(3)}_C}$
is the sum of the {\em sorted increasing} orders of those Boolean vectors.
Using~\Cref{lem:sum-of-vectors}, we can conclude
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L3R3C}
\vv{R^{(3)}_C} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(3)}_C}}
\end{equation}
\end{mdframed}
And an absolutely analogous argument gives
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L3R3W}
\vv{R^{(3)}_W} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(3)}_W}}
\end{equation}
\end{mdframed}
This finishes the proofs of $q=3$ for (a) and (c). \medskip
\subsubsection{Proving (a) and (c) for $q=1$}
\paragraph{Defining $\xi'_{\mathbf{x}}(S,S\oplus j)$ for $S \subseteq [i-1]$ and $j \leq i-1$:}
We now define the partial coloring $\xi'_{\mathbf{x}} := \xi^{(i)}_{\mathbf{x}}$ on all edges $(S,S\oplus j)$ where $S \subseteq [i-1]$ and $j\leq i-1$ for all $\mathbf{x} \in [n]^d$. These are exactly the relevant edges for the proof of parts (a) and (c) for $q=1$. Note that the partial coloring $\xi_{\mathbf{x}} := \xi_{\mathbf{x}}^{(i-1)}$ is defined over precisely these edges for each $\mathbf{x} \in [n]^d$. The color of $\xi'_{\mathbf{x}}$ on the edges $(S,S+i)$ for $S \subseteq [i-1]$ will be defined when we prove parts (b) and (d). The color of $\xi'_{\mathbf{x}}$ on the edges $(S+i,S+i \oplus j)$ for $S \subseteq [i-1]$ and $j \leq i-1$ will be defined when we prove $\mathsf{LHS} \geq \mathsf{RHS}_2$.
Fix $j \leq i-1$, $S \subseteq [i-1]$, and a $i$-line $\ell$. We consider the set of $\mathbf{x} \in \ell$ such that $(S,S\oplus j)$ is influential in $g_{\mathbf{x}}$:
\begin{align} \label{eq:V_gx}
V := \left\{\mathbf{x} \in \ell \colon g_{\mathbf{x}}(S) = 1 \text{ and } g_{\mathbf{x}}(S \oplus j) = 0 \right\} \text{.}
\end{align}
Note that since $f$ is semi-sorted, we have that $(S \circ f)$ and $(S \oplus j \circ f)$ are both semi-sorted.
Thus, we can write $V = V_L \cup V_R$ where $V_L$ and $V_R$ are intervals contained in the left and right half of $\ell$, respectively. We again obtain $\xi'_{\mathbf{x}}$ by down-sorting the original coloring on these intervals:
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in V_L) = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in V_L}
\]
and similarly
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in V_R) = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in V_R} \text{.}
\]
For all $\mathbf{x} \in \ell \setminus V$, we define $\xi'_{\mathbf{x}}(S,S\oplus j) := \xi_{\mathbf{x}}(S,S\oplus j)$. This completely describes $\xi'_{\mathbf{x}}(S,S\oplus j)$ for every $\mathbf{x} \in [n]^d$.
We provide the following illustration for clarity. Note that the picture is quite similar to the one provided in \Cref{sec:ac3}, when we defined $\chi'$. The key difference is that the bottom and top segments represent the same line $\ell$, but with different functions $S \circ f$ and $(S \oplus j) \circ f$, respectively. The vertical lines are no longer arrows to emphasize that they represent \emph{undirected edges in the hypercube} as opposed to directed edges in the augmented hypergrid.
\hspace*{-1cm}\includegraphics*[clip, scale = .5, trim = 0 300 0 20]{figs/RHS1-xi-recoloring}
We now return to our fixed $i$-line $\ell$ and set out to prove parts (a) and (c) for $q=1$, given the colorings $\xi'_{\mathbf{x}}$. Let's recall our illustration of $\ell$ and our definition of the intervals $W,A,C,O$. Recall that $g_{\mathbf{x}} = h(\mathbf{x})$ and so the definition of these intervals is the same.
\begin{figure}[h!]
\includegraphics*[trim = 0 320 0 100, clip, scale = 0.5]{figs/semisorted-k-try2}
\end{figure}
\paragraph{Proof of Part (a) for $q=1$:}
Fix $j \leq i-1$ and let $A' = \{\mathbf{x} \in A \colon g_{\mathbf{x}}(S\oplus j) = 0\}$ and $O' = \{\mathbf{x} \in O \colon g_{\mathbf{x}}(S\oplus j) = 0\}$, which are prefixes of $A$ and $O$, respectively. From our definition of $\xi'_{\mathbf{x}}(S,S\oplus j)$ above, we have
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in A') = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in A'}
\]
and similarly
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in O') = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in O'} \text{.}
\]
\begin{claim} \label{clm:A_sorted}
$\left(I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in A\right)$ is a sorted decreasing vector, and is a permutation of $\left(I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in A\right)$.
\end{claim}
\begin{proof}
Take $\mathbf{x}_i < \mathbf{x}'_i$ in $A$. Note that $g_\mathbf{x}(S) = 1$ for both $\mathbf{x}, \mathbf{x}'$. Thus,
\[
I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S) = {\bf 1}\left(g_\mathbf{x}(S\oplus j) = 0 ~\textbf{and}~\xi'_\mathbf{x}(S,S\oplus j) = 1 \right)
\]
and
\[
I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S) = {\bf 1}\left(g_\mathbf{x}(S\oplus j) = 0 ~\textbf{and}~\xi_\mathbf{x}(S,S\oplus j) = 1 \right)
\]
The two vectors are Boolean vectors with number of ones equal to the number of ones in $(\xi_\mathbf{x}(S, S\oplus j) ~:~\mathbf{x}\in A')$ which equals
the number of ones in $(\xi'_\mathbf{x}(S, S\oplus j)~:~\mathbf{x}\in A')$. Thus, they are permutations.
By design of $\xi'_\mathbf{x}$'s, this vector is sorted decreasing on $A'$, and all zeros in $A\setminus A'$ (which come to the right of $A'$).
\end{proof}
Observing that
\[
\vv{L^{(1)}_A} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~:~\mathbf{x}\in A\right) ~~\textrm{and}~~\vv{R^{(1)}_A} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S)~:~\mathbf{x}\in A\right)
\]
using~\Cref{lem:sum-of-vectors} and the claim above, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L1R1A}
\vv{R^{(1)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(1)}_A}}
\end{equation}
\end{mdframed}
Absolutely analogously, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L1R1O}
\vv{R^{(1)}_O} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(1)}_O}}
\end{equation}
\end{mdframed}
\paragraph{Proof of Part (c) for $q=1$:}
The picture is similar, but reversed when we consider the points in $W \cup C$, where $g_{\mathbf{x}}(S) = 0$. Fix $j \leq i-1$ and define $W' := \{\mathbf{x} \in W \colon g_{\mathbf{x}}(S \oplus j) = 1\}$ and $C' := \{\mathbf{x} \in C \colon g_{\mathbf{x}}(S \oplus j) = 1\}$ which are suffixes of $W$ and $C$, respectively. From our definition of $\xi'_{\mathbf{x}}(S,S\oplus j)$ above, made from the perspective of the set $S \oplus j$,
we have
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in W') = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in W'}
\]
and similarly
\[
(\xi'_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in C') = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in C'} \text{.}
\]
Analogous to \Cref{clm:A_sorted}, we have the following claim.
\begin{claim}
$\left(I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in W\right)$ is a sorted increasing vector, and is a permutation of $\left(I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in W\right)$.
\end{claim}
Arguing similarly to the proof of \Cref{eq:L1R1A} we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L1R1W}
\vv{R^{(1)}_W} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(1)}_W}}
\end{equation}
\end{mdframed}
and absolutely analogously, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:L1R1C}
\vv{R^{(1)}_C} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(1)}_C}}
\end{equation}
\end{mdframed}
\eqref{eq:L3R3A}, \eqref{eq:L3R3O}, \eqref{eq:L3R3C}, \eqref{eq:L3R3W}, and \eqref{eq:L1R1A}, \eqref{eq:L1R1O}, \eqref{eq:L1R1W}, \eqref{eq:L1R1C}
establish (a) and (c). \medski
\subsubsection{Proving (b) and (d):}
Finally, we need to establish (b) and (d). Let us recall these and also draw the picture of $\ell$ that we have been using.
\begin{itemize}
\item[(b)] $\exists \vv{L'^{(2)}_A}$ such that $\vv{L'^{(2)}_A} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(2)}_A}}$ and $\vv{L'^{(2)}_A} \succeq_{\mathsf{coor}} \vv{R^{(2)}_A}$.
\item[(d)] $\exists \vv{L'^{(2)}_C}$ such that $\vv{L'^{(2)}_C} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(2)}_C}}$ and $\vv{L'^{(2)}_C} \succeq_{\mathsf{coor}} \vv{R^{(2)}_C}$.
\end{itemize}
\begin{figure}[h!]
\includegraphics*[trim = 0 320 0 100, clip, scale = 0.5]{figs/semisorted-k-try2}
\end{figure}
\noindent
We remind the reader that
$\vv{L^{(2)}}(\mathbf{x}) = \Phi_{h,\chi}(\mathbf{x};i)$ for all $\mathbf{x}\in \ell$.
We begin with an observation which strongly uses the ``thresholded'' nature of the definition of $\Phi$.
\begin{claim}
No matter how $\chi$ is defined, either $\vv{L^{(2)}_A}$ is the all $1$s vector, or $\vv{L^{(2)}_C}$ is the all $1$s vector.
\end{claim}
\begin{proof}
Suppose for the sake of contradiction, there exists $\mathbf{x} \in A$ and $\mathbf{y} \in C$ such that $\Phi_{h,\chi}(\mathbf{x};i) = \Phi_{h,\chi}(\mathbf{y};i) = 0$.
But the edge $(\mathbf{x}, \mathbf{y})$ is a violation, and if $\chi(\mathbf{x}, \mathbf{y}) = 1$ then $\Phi_{h,\chi}(\mathbf{x};i) = 1$, otherwise $\Phi_{h,\chi}(\mathbf{y};i) = 1$. Contradiction.
\end{proof}
Next we remind the reader that $\vv{R^{(2)}}(\mathbf{x}) = I^{=i}_{g_\mathbf{x}, \xi'_\mathbf{x}}(S)$.
We now define the $\xi'_\mathbf{x}(S, S+i)$ colorings for $\mathbf{x} \in A\cup C$ using the above claim in the following simple manner.
\begin{equation}\label{eq:xi1}
\textrm{If}~~\vv{L^{(2)}_A} \equiv {\bf 1}, ~~\textrm{then}~~ \xi'_\mathbf{x}(S, S+i) = 1~~\forall \mathbf{x} \in A \cup C
\end{equation}
otherwise,
\begin{equation}\label{eq:xi2}
\textrm{we have}~~\vv{L^{(2)}_C} \equiv {\bf 1}, ~~\textrm{and so we define}~~ \xi'_\mathbf{x}(S, S+i) = 0~~\forall \mathbf{x} \in A \cup C
\end{equation}
In the former case, we have $\vv{R^{(2)}_A} = (\underbrace{111\cdots1}_{k~\text{many}}0000)$ and $\vv{L^{(2)}_A} \equiv {\bf 1}$ and so we pick $\vv{L'^{(2)}_A} = \vv{L^{(2)}_A}$.
Also note that we have $\vv{R^{(2)}_C}$ as the all zeros vector, and so we pick $\vv{L'^{(2)}_C} = \sortup{\vv{L^{(2)}_C}}$. These satisfy (b) and (d).
In the latter case the argument is analogous. Thus, in either case we have established (b) and (d), and this completes the proof of $\mathsf{LHS} \geq \mathsf{RHS}_1$. \medskip
We remind the reader that we have now defined $\xi'_\mathbf{x}(S, S\oplus j)$ for all subsets $S\subseteq [i-1]$ and $1\leq j\leq i$. In the next subsection, when we prove $\mathsf{LHS} \geq \mathsf{RHS}_2$, we will need to define $\xi'_\mathbf{x}(S+i, S+i\oplus j)$ for all $j \leq i-1$. Note that for $j=i$, we have $(S+i, S+i\oplus j) = (S+i,S)$ and the coloring $\xi_{\mathbf{x}}'$ has already been defined for these edges in \eqref{eq:xi1} or \eqref{eq:xi2}.
\subsection{Proving $\mathsf{LHS} \geq \mathsf{RHS}_2$} This inequality is a bit trickier to establish because the function $h$ itself now changes to $i\circ h$ in $\mathsf{RHS}_2$.
For instance, focusing on the illustration we have been using, upon sorting the picture looks like this.
\begin{figure}[ht!]
\includegraphics*[trim = 0 210 0 100, clip, scale = 0.5]{figs/semisorted-to-sorted-try2}
\end{figure}
We have now partitioned the interval $A$ into $I \cup B$ where $B$ is the $k$-ones closest to the semi-sorting boundary.
After sorting, we think of the ones in $B$ moving into $C$, and the ones in $I$ shifting and moving to $Q \subseteq A$.
The first $k$ entries of $A$, which we call $Z$, takes the value $0$ after sorting this line.
To argue $\mathsf{LHS} \geq \mathsf{RHS}_2$, we break the vector $\vv{L}$ as
\[
\norm{\vv{L}}_{1/2} = \norm{\vv{L}_W}_{1/2} + \norm{\vv{L}_{I \cup B\cup O}}_{1/2} + \norm{\vv{L}_C}_{1/2}
\]
and the vector $\vv{M}$ as
\[
\norm{\vv{M}}_{1/2} = \norm{\vv{M_W}}_{1/2} + \norm{\vv{M_{Q\cup C\cup O}}}_{1/2} + \norm{\vv{M_Z}}_{1/2}
\]
and argue vector-by-vector. The plan of the proof is similar to the previous case. We want to find recolorings $\chi''$ and $\xi'_\mathbf{x}$ such that
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{center}
\begin{enumerate}
\item[(e)] $\exists \vv{M'^{(q)}_{QCO}}$ such that $\vv{M'^{(q)}_{QCO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(q)}_{IBO}}}$ and $\vv{M'^{(q)}_{QCO}} \succeq_{\mathsf{coor}} \vv{M^{(q)}_{QCO}}$, for $q\in \{1,3\}$.
\item[(f)] $\exists \vv{L'^{(2)}_{IBO}}$ such that $\vv{L'^{(2)}_{IBO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(2)}_{IBO}}}$ and $\vv{L'^{(2)}_{IBO}} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{QCO}}$.
\item[(g)] $\exists \vv{M'^{(q)}_{WZ}}$ such that $\vv{M'^{(q)}_{WZ}} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(q)}_{WC}}}$ and $\vv{M'^{(q)}_{WZ}} \succeq_{\mathsf{coor}} \vv{M^{(q)}_{WZ}}$, for $q\in \{1,3\}$.
\item[(h)] $\exists \vv{L'^{(2)}_{WC}}$ such that $\vv{L'^{(2)}_{WC}} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(2)}_{WC}}}$ and $\vv{L'^{(2)}_{WC}} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{WZ}}$.
\end{enumerate}
\end{center}
\end{mdframed}
Let us see why the above conditions suffice to prove the inequality.
The second part of (f) implies that $\norm{\vv{M_{QCO}}}_{1/2} \leq \norm{\vv{M^{(1)}_{QCO}} + \vv{M^{(3)}_{QCO}} + \vv{L'^{(2)}_{IBO}}}_{1/2}$.
Part (e) and the first part of (f), along with~\Cref{lem:sum-of-vectors}, implies $\vv{M^{(1)}_{QCO}} + \vv{M^{(3)}_{QCO}} + \vv{L'^{(2)}_{IBO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L_{IBO}}}$.
And so, $\norm{L_{IBO}}_{1/2} \geq \norm{M_{IBO}}_{1/2}$. Now, by the second part of (g) and the second part of (h) we have $\norm{\vv{M_{WC}}}_{1/2} \leq \norm{\vv{M'^{(1)}_{WC}} + \vv{M'^{(3)}_{WC}} + \vv{L'^{(2)}_{WC}}}_{1/2}$ and by the first part of (g) and (h) we have $\norm{\vv{M'^{(1)}_{WC}} + \vv{M'^{(3)}_{WC}} + \vv{L'^{(2)}_{WC}}}_{1/2} \succeq_{\mathsf{maj}} \sortup{\vv{L_{WC}}}$. Thus, $\norm{L_{WZ}}_{1/2} \geq \norm{M_{WZ}}_{1/2}$.
\subsubsection{Proving (e) and (g) for $q=3$} \label{sec:eg3}
\paragraph{Defining the Coloring $\chi''$:} We now describe the coloring $\chi''$ on all edges of the form $(\mathbf{x},\mathbf{x} + a\mathbf{e}_j)$ where $j \geq i+1$, $(i\circ h)(\mathbf{x}) = 1$ and $(i \circ h)(\mathbf{x} + a\mathbf{e}_j) = 0$. For all other edges $e$, we simply define $\chi''(e) = \chi(e)$.
Given a pair of $i$-lines $\ell$ and $\ell' = \ell + a\mathbf{e}_j$ for $j \geq i+1$ and $a > 0$ we consider the set of violations from $\ell$ to $\ell'$ in $h$ and in $i \circ h$. As before, the violations in $h$ form two a union of two intervals $V = V_L \cup V_R$. Recall the definition of $V$ in \Eqn{violations}. Since $(i \circ h)$ is sorted in dimension $i$, the violations from $\ell$ to $\ell'$ in $(i \circ h)$ form a single interval which we will call $U$:
\[
U := \left\{(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in \ell \text{, } (i\circ h)(\mathbf{x}) = 1)\text{, and } (i\circ h)(\mathbf{x} + a\mathbf{e}_j) = 0\right\} \text{.}
\]
Since the sort operator can only reduce the number of violations in a dimension, we have $|U| \leq |V|$ (\Cref{clm:sort-violations} applied to $h|_{\ell}$ and $h|_{\ell'}$). We define $J$ to be the interval of $|V|-|U|$ points directly to the right of $U$ so that $U \cup J$ is an interval of size $|V|$. We then define
\[
(\chi''(e) \colon \mathbf{x} \in U \cup J) = \sortdown{\chi(e) \colon e \in V} \text{.}
\]
We now have a complete description of $\chi''$. We provide the following illustration for clarity. The white and green intervals represent where $h = 0$ and $h=1$, respectively. The vertical arrows represent violated edges. Blue edges have color $0$ and red edges have color $1$. The left picture depicts the original coloring, $\chi$, and the original function, $h$. The right picture depicts the recoloring, $\chi''$, and the function after sorting, $i \circ h$.
\includegraphics*[trim = 60 260 0 20, clip, scale = 0.5]{figs/RHS2-chi-recoloring}
We now return to our fixed $i$-line $\ell$ and set out to prove (e) and (g) for $q=3$, given this coloring $\chi''$. Let's recall our illustration of $h$ and $(i \circ h)$ restricted to $\ell$ and our definition of the intervals $W,I,B,C,O,Z,Q$.
\hspace{1cm}\includegraphics*[trim = 0 210 0 80, clip, scale = 0.4]{figs/semisorted-to-sorted-try2}
\paragraph{Proving (e) for $q=3$:} Recall the definition of $A = I \cup B$, $O$, and $Q \cup C \cup O$ as in the illustration. Fix $j \geq i+1$ and a $i$-line $\ell' = \ell + a\mathbf{e}_j$. Let $A' := \{\mathbf{x} \in A \colon h(\mathbf{x} + a\mathbf{e}_j) = 0\}$, $O' := \{\mathbf{x} \in O \colon h(\mathbf{x} + a\mathbf{e}_j) = 0\}$, and $U := \{\mathbf{x} \in Q \cup C \cup O \colon (i\circ h)(\mathbf{x} + a\mathbf{e}_j) = 0\}$. Again, applying \Cref{clm:sort-violations} to $h|_{\ell}$ and $h|_{\ell'}$, we have $|U| \leq |A'| + |O'|$. Let $J$ denote the interval of size $|A'| + |O'| - |U|$ directly to the right of $U$ so that $U \cup J$ is an interval of size $|A'| + |O'|$. Observe that by our definition of $\chi''$ above, we have
\[
(\chi''(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in U \cup J) = \sortdown{\chi(\mathbf{x},\mathbf{x}+a\mathbf{e}_j) \colon \mathbf{x} \in A' \cup O'} \text{.}
\]
Let's see what this leads to.
\begin{definition}
Fix $j \geq i + 1$ and fix an $i$-line $\ell' := \ell + a\mathbf{e}_j$ for $a > 0$. Define the following two boolean vectors:
\[
\mathbf{v}^{M}_{j,a} := \left({\bf 1}{\left((i\circ h)(\mathbf{x} + a\mathbf{e}_j) = 0 ~~\textbf{and}~~~\chi''(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1\right) }~~:~\mathbf{x}\in Q\cup C\cup O \right)
\]
and
\[
\mathbf{v}^{L}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} + a\mathbf{e}_j) = 0 ~~\textbf{and}~~~\chi(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1\right) }~~:~\mathbf{x}\in I\cup B\cup O\right)\text{.}
\]
\end{definition}
\noindent
Observe, for $\mathbf{x}\in Q\cup C\cup O$,
\begin{equation}\label{eq:M3R3-piece1.1}
\Phi_{i\circ h,\chi''}(\mathbf{x};j) = \min\left(1, \sum_a \mathbf{v}^{M}_{j,a}(\mathbf{x})\right)
\end{equation}
and for $\mathbf{x} \in I\cup B\cup O$,
\begin{equation}\label{eq:M3R3-piece1.2}
\Phi_{h,\chi}(\mathbf{x}; j) = \min\left(1, \sum_a \mathbf{v}^{L}_{j,a}(\mathbf{x}) \right) \text{.}
\end{equation}
\begin{claim}
Fix $j \geq i + 1$ and $a > 0$. For any two $\mathbf{x}_i < \mathbf{x}'_i$ in $Q\cup C\cup O$, we have $\mathbf{v}^{M}_{j,a}(\mathbf{x}) \geq \mathbf{v}^{M}_{j,a}(\mathbf{x}')$.
That is, the vector $\mathbf{v}^{M}_{j,a}$ is sorted decreasing.
\end{claim}
\begin{proof}
Since $(i\circ h)$ is semisorted $(i\circ h)(\mathbf{x}' + a\mathbf{e}_j) = 0$ implies $(i\circ h)(\mathbf{x} + a\mathbf{e}_j) = 0$.
Furthermore, since both these are violations, by design $\chi''(\mathbf{x}', \mathbf{x}'+a\mathbf{e}_j) = 1$ implies $\chi''(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1$.
\end{proof}
\begin{claim}\label{clm:L3-permutation}
Fix $j \geq i + 1$ and $a > 0$. The vector $\mathbf{v}^{M}_{j,a}$ has at most as many $1$s as $\mathbf{v}^{L}_{j,a}$ and thus $ \sortdown{\mathbf{v}^{L}_{j,a}}\succeq_{\mathsf{coor}} \mathbf{v}^{M}_{j,a}$.
\end{claim}
\begin{proof}
This is precisely how $\chi''$ is defined: it only permutes the colorings on the violations incident on $I\cup B\cup O$, and this number can only decrease upon sorting (\Cref{clm:sort-violations} applied to $\chi$ restricted to the edges going from $\ell$ to $\ell'$).
\end{proof}
In conclusion, we can write
\[
\vv{L^{(3)}_{IBO}} = \left(\sum_{j=i+1}^d \Phi_{h,\chi}(\mathbf{x};j)~~:~~\mathbf{x} \in I\cup B\cup O\right)
\]
as a sum of Boolean vectors, and the above two claims imply that the vector
\[
\vv{M^{(3)}_{QCO}} = \left(\sum_{j=i+1}^d \Phi_{(i\circ h),\chi''}(\mathbf{x};j)~~:~~\mathbf{x}\in Q\cup C\cup O\right)
\]
is {\em coordinate wise} dominated by the
sum of the {\em sorted decreasing} orders of those Boolean vectors. Defining $\vv{M'^{(3)}_{QCO}}$ to be the sum of the sorted decreasing orders, using~\Cref{lem:sum-of-vectors}, we establish part (e) for $q=3$. Namely, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:M3R3-QCO}
\exists \vv{M'^{(3)}_{QCO}}:~~\vv{M'^{(3)}_{QCO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(3)}_{IBO}}}~~\text{and}~~ \vv{M'^{(3)}_{QCO}} \succeq_{\mathsf{coor}} \vv{M^{(3)}_{QCO}}
\end{equation}
\end{mdframed}
\paragraph{Proving (g) for $q=3$:}
A similar argument but working with the zeros establishes part (g) for $q=3$. The picture is similar, but reversed, when we consider the points in $W\cup C$, where $h(\mathbf{x}) = 0$. Fix a dimension $j \geq i+1$ and some $\ell'' = \ell - ae_j$. Let $W' = \{\mathbf{x} \in W \colon h(\mathbf{x}-a\mathbf{e}_j) = 1\}$, $C' = \{\mathbf{x} \in C \colon h(\mathbf{x}-a\mathbf{e}_j) = 1\}$, and $U = \{\mathbf{x} \in W \cup Z \colon (i \circ h)(\mathbf{x} - a\mathbf{e}_j) = 1\}$. Note that $|U| \leq |W'| + |C'|$ (\Cref{clm:sort-violations} applied to $h|_{\ell''}$ and $h|_{\ell}$). Let $J$ denote the interval of $|W'| + |C'|$ directly to the right of $|U|$ so that $U \cup J$ is an interval of size $|W'| + |C'|$. Observe that by our definition of $\chi''$ above, we have
\[
(\chi''(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in U \cup J) = \sortdown{\chi(\mathbf{x}-a\mathbf{e}_j,\mathbf{x}) \colon \mathbf{x} \in W' \cup C'} \text{.}
\]
Let's see what this leads to.
\begin{definition}
Fix $j \geq i + 1$ and fix an $i$-line $\ell'' := \ell - a\mathbf{e}_j$ for $a > 0$. Define the following two boolean vectors:
\[
\mathbf{v}^{M}_{j,a} := \left({\bf 1}{\left((i\circ h)(\mathbf{x} - a\mathbf{e}_j) = 1 ~~\textbf{and}~~~\chi''(\mathbf{x}-a\mathbf{e}_j, \mathbf{x}) = 0\right) }~~:~\mathbf{x}\in W\cup Z \right)
\]
and
\[
\mathbf{v}^{L}_{j,a} := \left({\bf 1}{\left(h(\mathbf{x} - a\mathbf{e}_j) = 1 ~~\textbf{and}~~~\chi(\mathbf{x}-a\mathbf{e}_j, \mathbf{x}) = 0\right) }~~:~\mathbf{x}\in W \cup C\right)\text{.}
\]
\end{definition}
\noindent
Observe, for $\mathbf{x}\in W \cup Z$,
\begin{equation}\label{eq:M3R3-piece1.1}
\Phi_{i\circ h,\chi''}(\mathbf{x};j) = \min\left(1, \sum_a \mathbf{v}^{M}_{j,a}(\mathbf{x})\right)
\end{equation}
and for $\mathbf{x} \in W \cup C$,
\begin{equation}\label{eq:M3R3-piece1.2}
\Phi_{h,\chi}(\mathbf{x}; j) = \min\left(1, \sum_a \mathbf{v}^{L}_{j,a}(\mathbf{x}) \right) \text{.}
\end{equation}
\begin{claim}
Fix $j \geq i + 1$ and $a > 0$. For any two $\mathbf{x}_i < \mathbf{x}'_i$ in $W \cup Z$, we have $\mathbf{v}^{M}_{j,a}(\mathbf{x}) \leq \mathbf{v}^{M}_{j,a}(\mathbf{x}')$.
That is, the vector $\mathbf{v}^{M}_{j,a}$ is sorted increasing.
\end{claim}
\begin{proof}
Since $(i\circ h)$ is sorted in dimension $i$, we have $(i\circ h)(\mathbf{x} - a\mathbf{e}_j) = 1$ implies $(i\circ h)(\mathbf{x}'-a\mathbf{e}_j) = 1$.
Furthermore, since both these are violations, by design $\chi''(\mathbf{x}-a\mathbf{e}_j, \mathbf{x}) = 0$ implies $\chi''(\mathbf{x}'-a\mathbf{e}_j, \mathbf{x}') = 0$.
\end{proof}
\begin{claim}\label{clm:L3-permutation}
Fix $j \geq i + 1$ and $a > 0$. The vector $\mathbf{v}^{M}_{j,a}$ has at most as many $1$s as $\mathbf{v}^{L}_{j,a}$ and thus $ \sortup{\mathbf{v}^{L}_{j,a}}\succeq_{\mathsf{coor}} \mathbf{v}^{M}_{j,a}$.
\end{claim}
\begin{proof}
This is precisely how $\chi''$ is defined: it only permutes the colorings on the violations incident on $W \cup C$, and this number can only decrease upon sorting (\Cref{clm:sort-violations} applied to $\chi$ restricted to the edges going from $\ell''$ to $\ell$).
\end{proof}
In conclusion,
we can write
\[
\vv{L^{(3)}_{WC}} = \left(\sum_{j=i+1}^d \Phi_{h,\chi}(\mathbf{x};j)~~:~~\mathbf{x} \in W\cup C\right)
\]
as a sum of Boolean vectors, and the above two claims imply that the vector
\[
\vv{M^{(3)}_{WZ}} = \left(\sum_{j=i+1}^d \Phi_{(i\circ h),\chi''}(\mathbf{x};j)~~:~~\mathbf{x}\in W \cup Z\right)
\]
is {\em coordinate wise} dominated by the
sum of the {\em sorted increasing} orders of those Boolean vectors. Defining $\vv{M'^{(3)}_{WZ}}$ to be the sum of the sorted decreasing orders, using~\Cref{lem:sum-of-vectors}, we establish part (g) for $q=3$. Namely, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:M3R3-QCO}
\exists \vv{M'^{(3)}_{MZ}}:~~\vv{M'^{(3)}_{MZ}} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(3)}_{WC}}}~~\text{and}~~ \vv{M'^{(3)}_{WZ}} \succeq_{\mathsf{coor}} \vv{M^{(3)}_{WZ}}
\end{equation}
\end{mdframed}
\subsubsection{Proving (e) and (g) for $q=1$}
\paragraph{Defining $\xi'_{\mathbf{x}}(S+i,S+i\oplus j)$ for $S\subseteq [i-1]$ and $j \leq i-1$:} We now define the partial coloring $\xi'_{\mathbf{x}} := \xi^{(i)}_{\mathbf{x}}$ on all edges $(S+i,S+i\oplus j)$ where $S \subseteq [i-1]$ and $j\leq i-1$ for all $\mathbf{x} \in [n]^d$. These are exactly the relevant edges for the proof of parts (e) and (g) for $q=1$. Note that the partial coloring $\xi_{\mathbf{x}} := \xi_{\mathbf{x}}^{(i-1)}$ is undefined over these edges.
Fix $S\subseteq [i-1]$, $j \leq i-1$, and a $i$-line $\ell$. We consider the set of $\mathbf{x} \in \ell$ such that $(S,S\oplus j)$ is influential in $g_{\mathbf{x}}$ and the set of edges where $(S+i,S+i\oplus j)$ is influential in $g_{\mathbf{x}}$. As before, the former is a union of two intervals $V = V_L \cup V_R$. Recall the definition of $V$ in \Eqn{V_gx}. Since $(S+i) \circ f$ and $(S+i \oplus j) \circ f$ are both sorted in dimension $i$, the set of $\mathbf{x} \in \ell$ such that $(S+i,S+i\oplus j)$ is influential forms a single interval which we will call $U$:
\[
U := \left\{\mathbf{x} \in \ell \colon g_\mathbf{x}(S + i) = 1 \text{ and } g_{\mathbf{x}}(S+i \oplus j) = 0\right\}\text{.}
\]
Again, we have $|U| \leq |V|$ (\Cref{clm:sort-violations} applied to $(S \circ f)|_{\ell}$ and $((S \oplus j) \circ f)|_{\ell}$) and we let $J$ denote the $|V| - |U|$ points directly right of $U$, so that $U \cup J$ is an interval of length $|V|$. We then define
\[
(\xi'_{\mathbf{x}}(S+i,S+i\oplus j) \colon \mathbf{x} \in U \cup J) = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in V} \text{.}
\]
For all $x \in \ell \setminus (U \cup J)$ we define $\xi'_{\mathbf{x}}(S+i,S+i\oplus j) = 1$. Note that this is an arbitrary choice since such edges are not influential and so they do not come in to play in the rest of the proof.
We now have a complete description of $\xi'_{\mathbf{x}}$ on $(S+i,S+i\oplus j)$ for all $\mathbf{x} \in [n]^d$. We provide the following illustration for clarity, which is quite similar to the illustration provided in \Sec{eg3} when we defined $\chi''$. The left picture depicts the original colorings, $\xi_{\mathbf{x}}$, and the relevant functions before applying the sort operator in dimension $i$. The right picture depicts the recoloring, $\xi'_{\mathbf{x}}$, and the relevant functions after applying the sort operator in dimension $i$.
\includegraphics*[trim = 50 280 0 20, clip, scale = 0.5]{figs/RHS2-xi-recoloring}
We now return to our fixed $i$-line $\ell$ and set out to prove (e) and (g) for $q=1$, given the colorings $\xi'_{\mathbf{x}}$. Recall $g_{\mathbf{x}}(S) = h(\mathbf{x})$ and $g_{\mathbf{x}}(S+i) = (i \circ h)(\mathbf{x})$ and so we can reference the same illustration and our definition of the intervals $W,I,B,C,O,Z,Q$.
\hspace{1cm}\includegraphics*[trim = 0 210 0 80, clip, scale = 0.4]{figs/semisorted-to-sorted-try2}
\paragraph{Proving (e) for $q=1$:}
Fix $j \leq i-1$ and let $A' := \{\mathbf{x} \in A \colon g_{\mathbf{x}}(S \oplus j) = 0\}$, $O' := \{\mathbf{x} \in O \colon g_{\mathbf{x}}(S \oplus j) = 0$, and $U := \{\mathbf{x} \in Q \cup C \cup O \colon g_{\mathbf{x}}(S+i\oplus j) = 0\}$. As before, $|U| \leq |A'| + |O'|$ (applying \Cref{clm:sort-violations} to $(S \circ f)|_{\ell}$ and $((S \oplus j) \circ f)|_{\ell}$) and we define $J$ to be the $|A'| - |O'|$ points directly to the right of $U$ so that $U \cup J$ is a prefix of $Q\cup C \cup O$ of size $|A'| + |O'|$. From our definition of $\xi'_{\mathbf{x}}$ from above we have
\[
(\xi'_{\mathbf{x}}(S+i,S+i\oplus j) \colon \mathbf{x} \in U \cup J) = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in A' \cup O'} \text{.}
\]
We now get the following claim.
\begin{claim}
$\left(I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S+i)~~:~~ \mathbf{x}\in Q\cup C\cup O\right)$ is a sorted decreasing vector, and has at most as many ones as the vector $\left(I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in I\cup B\cup O\right)$.
\end{claim}
\begin{proof}
Take $\mathbf{x}_i < \mathbf{x}'_i$ in $Q\cup C\cup O$. Note that $g_\mathbf{x}(S+i) = g_{\mathbf{x}'}(S+i) = 1$ by definition $Q\cup C\cup O$.
Thus,
\[
I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S + i) = {\bf 1}\left(g_\mathbf{x}(S + i\oplus j) = 0 ~\textbf{and}~\xi'_\mathbf{x}(S+i,S+i\oplus j) = 1 \right)
\]
and
\[
I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S) = {\bf 1}\left(g_\mathbf{x}(S\oplus j) = 0 ~\textbf{and}~\xi_\mathbf{x}(S,S\oplus j) = 1 \right)
\]
By design of the $\xi'_\mathbf{x}$'s, the first vector is sorted decreasing on $Q\cup C\cup O$ (it takes value $0$ after $U$).
Also by design, the number of ones in the latter vector can only be larger since we obtain $\xi'$ by taking a permutation and possibly discarding some ones (the ones corresponding to $J$).
\end{proof}
Observing that
\[
\vv{L^{(1)}_{IBO}} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~:~\mathbf{x}\in I\cup B\cup O\right) ~~\textrm{and}~~\vv{M^{(1)}_{QCO}} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S+i)~:~\mathbf{x}\in Q\cup C\cup O\right)
\]
we see that the latter vector is coordinate-wise dominated by a vector which is a sum of \emph{sorted decreasing} versions of Boolean vectors which add up to the former one. Defining $\vv{M'^{(1)}_{QCO}}$ to be the sum of the sorted decreasing orders, using~\Cref{lem:sum-of-vectors}, we establish part (e) for $q=3$. Namely, we get
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:M1R1-QCO}
\exists \vv{M'^{(1)}_{QCO}}:~~\vv{M'^{(1)}_{QCO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(1)}_{IBO}}}~~\text{and}~~ \vv{M'^{(1)}_{QCO}} \succeq_{\mathsf{coor}} \vv{M^{(1)}_{QCO}} \text{.}
\end{equation}
\end{mdframed}
\paragraph{Proof of Part (g) for $q=1$:}
A similar argument but working with the zeros establishes part (g) for $q=1$. Recall the definition of the sets $W$, $C$, and $Z$. Let $W' = \{\mathbf{x} \in W \colon g_{\mathbf{x}}(S\oplus j) = 1\}$, $C' = \{\mathbf{x} \in C \colon g_{\mathbf{x}}(S \oplus j) = 1\}$, and $U = \{x \in W \cup Z \colon g_{\mathbf{x}}(S+i\oplus j) = 1\}$. As before $|U| \leq |W'| + |C'|$ (applying \Cref{clm:sort-violations} to $((S \oplus j) \circ f)|_{\ell}$ and $(S \circ f)|_{\ell}$) and we define $J$ to be the set of $|W'|+|C'| - |U|$ points directly to the right of $U$ so that $U \cup J$ is an interval of size $|W'| + |C'|$. Note that $U$ is a suffix of $W \cup Z$ and $J$ is a prefix of $Q \cup C \cup O$.
From our definition of $\xi'_{\mathbf{x}}$ above, made with the set $S \oplus j$, we have
\[
(\xi'_{\mathbf{x}}(S+i,S+i\oplus j) \colon \mathbf{x} \in U \cup J) = \sortdown{\xi_{\mathbf{x}}(S,S\oplus j) \colon \mathbf{x} \in W' \cup C'} \text{.}
\]
\begin{claim}
$\left(I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S+i)~~:~~ \mathbf{x}\in W \cup Z\right)$ is a sorted increasing vector, and has at most as many ones as the vector $\left(I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~~:~~ \mathbf{x}\in W \cup C\right)$.
\end{claim}
\begin{proof}
Take $\mathbf{x}_i < \mathbf{x}'_i$ in $W \cup Z$. Note that $g_\mathbf{x}(S+i) = g_{\mathbf{x}'}(S+i) = 0$ by definition $W \cup Z$.
Thus,
\[
I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S + i) = {\bf 1}\left(g_\mathbf{x}(S + i\oplus j) = 1 ~\textbf{and}~\xi'_\mathbf{x}(S+i,S+i\oplus j) = 0 \right)
\]
and
\[
I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S) = {\bf 1}\left(g_\mathbf{x}(S\oplus j) = 1 ~\textbf{and}~\xi_\mathbf{x}(S,S\oplus j) = 0 \right)
\]
By design of the $\xi'_\mathbf{x}$'s, the first vector is sorted increasing on $W \cup Z$.
Also by design, the number of ones in the latter vector can only be larger since we obtain $\xi'$ by taking a permutation and possibly discarding some ones (the ones corresponding to $J$)
\end{proof}
Observing that
\[
\vv{L^{(1)}_{WC}} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi_\mathbf{x}}^{=j}(S)~:~\mathbf{x}\in W \cup C\right) ~~\textrm{and}~~\vv{M^{(1)}_{WZ}} = \left(\sum_{j=1}^{i-1} I_{g_\mathbf{x},\xi'_\mathbf{x}}^{=j}(S+i)~:~\mathbf{x}\in W \cup Z\right)
\]
we see that the latter vector is coordinate-wise dominated by a vector which is a sum of \emph{sorted increasing} versions of Boolean vectors which add up to the former one. Defining $\vv{M'^{(1)}_{WZ}}$ to be the sum of the sorted increasing orders, using~\Cref{lem:sum-of-vectors}, we establish part (e) for $q=3$. Namely,
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{equation}\label{eq:M1R1-QCO}
\exists \vv{M'^{(1)}_{WZ}}:~~\vv{M'^{(1)}_{WZ}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(1)}_{WC}}}~~\text{and}~~ \vv{M'^{(1)}_{WZ}} \succeq_{\mathsf{coor}} \vv{M^{(1)}_{WZ}} \text{.}
\end{equation}
\end{mdframed}
\subsubsection{Proving (f) and (h):}
Let us now prove part (f) and (h). Note, at this point, $\xi'_\mathbf{x}$ is fully defined on all pairs $(S, S\oplus j)$ for $S\subseteq [i]$ and $j \leq i$. We don't have the freedom to redefine.
However, we see that the definition we made in~\eqref{eq:xi1} and~\eqref{eq:xi2} suffices. Let us recall what we want to establish.
\begin{enumerate}
\item[(f)] $\exists \vv{L'^{(2)}_{IBO}}$ such that $\vv{L'^{(2)}_{IBO}} \succeq_{\mathsf{maj}} \sortdown{\vv{L^{(2)}_{IBO}}}$ and $\vv{L'^{(2)}_{IBO}} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{QCO}}$.
\item[(h)] $\exists \vv{L'^{(2)}_{WC}}$ such that $\vv{L'^{(2)}_{WC}} \succeq_{\mathsf{maj}} \sortup{\vv{L^{(2)}_{WC}}}$ and $\vv{L'^{(2)}_{WC}} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{WZ}}$.
\end{enumerate}
We remind the reader that $\vv{L^{(2)}}(\mathbf{x}) = \Phi_{h,\chi}(\mathbf{x};i)$ for all $\mathbf{x}\in \ell$ and the coloring was defined as follows:
\begin{equation
\textrm{If}~~\vv{L^{(2)}_{IB}} \equiv {\bf 1}, ~~\textrm{then}~~ \xi'_\mathbf{x}(S, S+i) = 1~~\forall \mathbf{x} \in I\cup B\cup C \notag
\end{equation}
otherwise,
\begin{equation
\textrm{we have}~~\vv{L^{(2)}_C} \equiv {\bf 1}, ~~\textrm{and so}~~ \xi'_\mathbf{x}(S, S+i) = 0~~\forall \mathbf{x} \in I\cup B \cup C \notag
\end{equation}
\noindent
We remind the reader that $\vv{M^{(2)}}(\mathbf{x}) = I^{=i}_{g_\mathbf{x}, \xi'_\mathbf{x}}(S+i)$ and therefore this is $1$
iff $g_\mathbf{x}(S + i) \neq g_\mathbf{x}(S)$ and $\xi'_\mathbf{x}(S, S+i) = g_\mathbf{x}(S+i)$. The former implies $\mathbf{x} \in Z\cup C$.
Suppose we are in the first case. Then, $\vv{M^{(2)}}(\mathbf{x}) = 1$ if and only if $\mathbf{x} \in C$.
Since $\vv{L^{(2)}_{IB}} \equiv {\bf 1} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{QC}}$, we can set $\vv{L'^{(2)}_{IBO}}$ to be the vector that is $1$s in $I\cup B$ and $0$'s in $O$. This establishes (f).
To establish (h), we observe that $\vv{M^{(2)}_{WZ}}$ is the zero vector, and thus we can choose $\vv{L'^{(2)}_{WC}}$ to be $\sortup{\vv{L^{(2)}_{WC}}}$.
Suppose we are in the second case. Then, $\vv{M^{(2)}}(\mathbf{x}) = 1$ if and only if $\mathbf{x} \in Z$.
Since $\vv{L^{(2)}_{C}} \equiv {\bf 1} \succeq_{\mathsf{coor}} \vv{M^{(2)}_{Z}}$, we can set $\vv{L'^{(2)}_{WC}}$ to be the vector that is $1$s in $C$ and $0$'s in $W$. This establishes (h).
To establish (f), we observe that $\vv{M^{(2)}_{QCO}}$ is the zero vector, and thus we can choose $\vv{L'^{(2)}_{IBO}}$ to be $\sortdown{\vv{L^{(2)}_{IBO}}}$.
In either case, we have established (f) and (h), and thus completed the proof.
\subsection{Semisorting only decreases the Talagrand objective: Proof of~\Cref{lem:semisorting-decreases}}\label{sec:semisorting-can-only-reduce}
Let us first describe the coloring $\chi'$.
\begin{itemize}
\item First let us describe the recoloring of pairs of points $(\mathbf{x}, \mathbf{x}')$ which differ only in some coordinate $j\neq i$
and $\mathbf{x}_i = \mathbf{x}'_i$ lies in the interval $[a,b]$. We go over all these edges by considering pairs of $i$-lines which differ on a single coordinate $j\neq i$.
More precisely, if $\ell = \mathbf{x} \pm t\mathbf{e}_i$ then $\ell' = \mathbf{x}' \pm t\mathbf{e}_i$ for some $\mathbf{x}' = \mathbf{x} + a\mathbf{e}_j$ with $a > 0$. We now consider re-coloring the pairs
$(\mathbf{x}, \mathbf{x}' = \mathbf{x} + a\mathbf{e}_j)$ as follows.
Let $V$ denote the points $\mathbf{x} \in \ell$ such that (a) $\mathbf{x}_i \in I$, (b) $f(\mathbf{x}) = 1$, but (c) $f(\mathbf{x}+ a\mathbf{e}_j) = 0$. That is $(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j)$ is a violation.
Consider all edges $E_V := \{(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j)~:~\mathbf{x}\in V\}$ and let $\vv{\chi}$ be the $|E_V|$ dimensional $0,1$-vector which are the $\chi$ values of edges in $E_V$ going left to right.
Now consider the function $h$ where $I$ has been sorted on both $\ell$ and $\ell'$. Let $U$ denote the points $\mathbf{x} \in \ell$ such that (a) $\mathbf{x}_i \in I$, (b) $h(\mathbf{x}) = 1$, but (c) $h(\mathbf{x}+ a\mathbf{e}_j) = 0$. That is $(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j)$ is a violation in $h$. Firstly note that $|U| \leq |V|$ and furthermore, these $|U|$ points form a contiguous interval of $I$.
We now describe the recoloring $\chi'$ of the edges in $E_U := \{(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j)~:~\mathbf{x}\in V\}$; all the other recolorings are immaterial since they don't contribute to $T_{\Phi_{\chi'}}(h)$ since the edges are not violating. We take the $|V|$-dimensional vector $\vv{\chi}$, sort in {\em decreasing} order, and then take the first $|U|$ coordinates and use them to define $\chi'(e)$ for $e\in E_U$, left to right. See~\Cref{fig:semisorting-reduces} for an illustration.
\begin{figure}[ht!]
\includegraphics[trim = 0 300 0 50, clip, scale=0.5]{figs/semisorting-reduces}
\caption{\em We are considering only the interval $I$. The line below is $\ell$ and the line above is $\ell'$. The green shaded zones correspond to where the function evaluates to $1$s.
The situation to the right is after sorting. Only the violating edges are marked. On the left,
the red solid edges are colored $\chi(e) = 1$ while the blue dashed
are colored $\chi(e) = 0$. On the right, the color-coding is the same but for $\chi'$. All other unmarked edges inherit the same colors as $\chi$. }\label{fig:semisorting-reduces}
\end{figure}
\item Now we describe recoloring of pairs of points $(\mathbf{x}, \mathbf{y})$ which only differ in coordinate $i$. First, if both $\mathbf{x}_i$ and $\mathbf{y}_i$ lie in $I$, or if they both lie outside $I$, then we leave their colors unchanged. Furthermore, if $(\mathbf{x}, \mathbf{y})$ is {\em not} a violating pair in $f$, then we leave its color unchanged.
Now consider a $\mathbf{y}$ to the right of $I$, that is, $\mathbf{y}_i > b$ and $f(\mathbf{y}) = 0$. Consider the $\mathbf{x}$'s with $\mathbf{x}_i$ in $I$ with $f(\mathbf{x}) = 1$, each of which forms a violation with $\mathbf{y}$. Suppose there are $k$ many of them, of which $k_0$ of them are colored $0$
and $k_1$ of them are colored $1$. We now consider the picture in $h$, and once again there are exactly $k$ (possibly different) points in the interval which are violating with $\mathbf{y}$ in $h$.
Going from left to right, we color the first $k_1$ of them $1$ and the next $k_0$ of them $0$, in $\chi'$.
We now do a similar thing for a $\mathbf{z}$ to the left of $I$, that is, $\mathbf{z}_i < a$ and $f(\mathbf{z}) = 1$. We now consider the $\mathbf{x}$'s with $\mathbf{x}_i \in I$ with $f(\mathbf{x}) = 0$, each of which forms a violation with $\mathbf{z}$.
As before, suppose there are $k$ many of them $k_1$ of them colored $1$ and $k_0$ of them colored $0$. In $g$ also there are $k$ locations with which $\mathbf{z}$ is a violation.
We, once again, going from left to right, color the first $k_1$ of them $1$ and the next $k_0$ of them $0$, in $\chi'$.
See~\Cref{fig:semisorting-reduces-2} for an illustration.
\begin{figure}[ht!]
\includegraphics[trim = 0 300 0 0, clip, scale=0.5]{figs/semisorting-reduces-2}
\caption{\em The two vertical black lines demarcate $I$.
The green shaded zones correspond to where the function evaluates to $1$s. The situation to the right is after sorting.
$\mathbf{y}$ is a point with $f(\mathbf{y}) = 0$ to the right of $I$; $\mathbf{z}$ is a point with $f(\mathbf{z}) = 1$ to the left of $I$.
Only the violating edges incident to $\mathbf{y}$ and $\mathbf{z}$ are marked. On the left,
the red solid edges are colored $\chi(e) = 1$ while the blue dashed
are colored $\chi(e) = 0$. On the right, the color-coding is the same but for the recoloring $\chi'$. All other unmarked edges incident of $\mathbf{y}$ or $\mathbf{z}$ inherit the same colors as $\chi$.
Edges with both endpoints in $I$ or both endpoints outside $I$ also inherit the same color.}\label{fig:semisorting-reduces-2}
\end{figure}
\end{itemize}
Now we prove the lemma ``line-by-line''. In particular, we want to prove for any $i$-line $\ell$, we have
\[
\sum_{\mathbf{x}\in \ell} \sqrt{\Phi_{f,\chi}(\mathbf{x})} \geq \sum_{\mathbf{x}\in \ell} \sqrt{\Phi_{h,\chi'}(\mathbf{x})}
\]
Note that it suffices to prove the above for $\mathbf{x}$ whose $\mathbf{x}_i \in I$.
To prove the above inequality, it is best to consider the two vectors $\vv{\Phi_\chi(f)}$ and $\vv{\Phi_{\chi'}(h)}$ which are $|I|$-dimensional whose $\mathbf{x}$th coordinate is precisely $\Phi_{f,\chi}(\mathbf{x})$
and $\Phi_{h,\chi'}(\mathbf{x})$ respectively. We want to prove
\begin{equation}\label{eq:line}
\norm{\vv{\Phi_\chi(f)}}_{1/2} \geq \norm{\vv{\Phi_{\chi'}(h)}}_{1/2}
\end{equation}
First we divide the $|I|$ coordinates of $\vv{\Phi_\chi(f)}$ into $O \cup Z$ corresponding to when $f(\mathbf{x}) = 1$ and $f(\mathbf{x}) = 0$. Let's call these two vectors $\vv{\Phi^{(1)}_\chi(f)}$ and $\vv{\Phi^{(0)}_\chi(f)}$. The former vector is $|O|$ dimensional, the latter is $|Z|$ dimensional, and $\vv{\Phi_\chi(f)}$ is obtained by some splicing of these two vectors.
We will do the same for the coordinates of $\vv{\Phi_{\chi'}(h)}$ to obtain $\vv{\Phi^{(1)}_{\chi'}(h)}$ and $\vv{\Phi^{(0)}_{\chi'}(h)}$. Note that since sorting doesn't change the number of $0$s or $1$,
both these vectors are $|O|$ and $|Z|$ dimensional, respectively.
We now set to prove
\begin{equation}\label{eq:2cases}
\norm{\vv{\Phi^{(1)}_\chi(f)}}_{1/2} \geq \norm{\vv{\Phi^{(1)}_{\chi'}(h)}}_{1/2} ~~~\textrm{and}~~~ \norm{\vv{\Phi^{(0)}_\chi(f)}}_{1/2} \geq \norm{\vv{\Phi^{(0)}_{\chi'}(h)}}_{1/2}
\end{equation}
and this will prove \eqref{eq:line}. We prove the first inequality; the proof of the second is analogous.
For brevity's sake, for the rest of the section we drop the superscript $(1)$ from $\vv{\Phi^{(1)}}$. \medskip
\noindent
The plan is to write $\vv{\Phi_\chi(f)}$ as a sum of (Boolean) vectors, and then show that $\vv{\Phi_{\chi'}(h)}$ is dominated by the sum of sorts of those Boolean vectors. Then we invoke~\Cref{lem:sum-of-vectors}.
We write $\vv{\Phi_\chi(f)}$ as a sum of Boolean vectors as follows.
Fix any other $i$-line $\ell' := \ell + a\mathbf{e}_j$ for some $j\neq i$ and $a > 0$. Define the following
$(0,1)$-vector also indexed by elements of $O$.
\[
\mathbf{u}_{\ell'}(\mathbf{x}) = 1 ~~\text{if $f(\mathbf{x} + a\mathbf{e}_j) = 0$ and $\chi(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1$}
\]
That is, $\mathbf{u}_{\ell'}(\mathbf{x}) = 1$ if the projection of $\mathbf{x}$ onto $\ell'$, $(\mathbf{x}, \mathbf{x}' := \mathbf{x} + a\mathbf{e}_j)$, is a violating edge in $f$ with $\chi$-color $1$.
Define the following vector $\boldsymbol{A}$ as follows.
\begin{definition} \label{eq:def1}
For any $\mathbf{x}\in O$,
\[
\boldsymbol{A}(\mathbf{x}) = \sum_{j\neq i} \underbrace{\min\left(1, \sum_{\ell' = \ell + a\mathbf{e}_j} \mathbf{u}_{\ell'}(\mathbf{x}) \right)}_{\text{let's call this}~\mathbf{w}_j(\mathbf{x}) \in \{0,1\}} ~~=:\sum_{j\neq i} \mathbf{w}_j(\mathbf{x})
\]
\end{definition}
Finally, for $\mathbf{x} \in O$, define
\[
\vv{A^{\parallel}}(\mathbf{x}) = 1~~\textrm{if there is some $\mathbf{y}$ to its right, potentially outside the interval $I$ with $f(\mathbf{y}) = 0$ and $\chi(\mathbf{x},\mathbf{y}) = 1$}
\]
Using the vectors, we can write
\begin{observation}\label{obs:phiA}
For any $\mathbf{x} \in O$,
\[
\vv{\Phi_\chi(f)}(\mathbf{x}) = \boldsymbol{A}(\mathbf{x}) + \vv{A^{\parallel}}(\mathbf{x})
\]
\end{observation}
Now let's consider the situation after $I$ is sorted. The ones of $O$ now ``shift around''; indeed, they are the $|O|$ many right most points.
Let's call these locations $O'$ and note $|O'| = |O|$.
Now define the $|O'|=|O|$ dimensional vector $\mathbf{v}_{\ell'}$ where for $\mathbf{x} \in O'$
\[
\mathbf{v}_{\ell'}(\mathbf{x}) = 1 ~~\text{if $h(\mathbf{x} + a\mathbf{e}_j) = 0$ and $\chi'(\mathbf{x}, \mathbf{x}+a\mathbf{e}_j) = 1$}
\]
Now we will use the property of the recoloring we performed. We claim two things:
\begin{claim}\label{clm:crucial-1}
The number of $1$s in $\mathbf{v}_{\ell'}$ is at most the number of $1$s in $\mathbf{u}_{\ell'}$, and $\mathbf{v}_{\ell'}$ is sorted decreasing.
\end{claim}
\begin{proof}
The number of $1$s in $\mathbf{u}_{\ell'}$ is precisely the number of violating edges of the form $(\mathbf{x}, \mathbf{x}')$ in $f$, where $\mathbf{x}_i \in I$ and $\mathbf{x}' = \mathbf{x} + a\mathbf{e}_j$ and $\chi(\mathbf{x},\mathbf{x}') = 1$.
Similarly, the number of $1$s in $\mathbf{u}_{\ell'}$ are precisely the number of violating edges of the form $(\mathbf{x}, \mathbf{x}')$ in $h$, where $\mathbf{x}_i \in I$ and $\mathbf{x}' = \mathbf{x} + a\mathbf{e}_j$ and $\chi'(\mathbf{x},\mathbf{x}') = 1$.
When we recolored to get $\chi'$ we made sure by property (a) that the latter number is smaller.
Take $\mathbf{x}$ and $\mathbf{y}$ in $O$, with $\mathbf{x}_i < \mathbf{y}_i$, but suppose, for the sake of contradiction, $\mathbf{v}_{\ell'}(\mathbf{x}) = 0$ and $\mathbf{v}_{\ell'}(\mathbf{y}) = 1$.
The latter implies $h(\mathbf{y}' := \mathbf{y} + a\mathbf{e}_j) = 0$ and $\chi'(\mathbf{y},\mathbf{y}') = 1$. Since $h$ is sorted on $\ell'$, $h(\mathbf{x}' := \mathbf{x} + a\mathbf{e}_j) = 0$ as well.
Since $\mathbf{x} \in O$, $h(\mathbf{x}) = 1$ which means $(\mathbf{x}, \mathbf{x}')$ is a violating edge in $h$. $\mathbf{v}_{\ell'}(\mathbf{x}) = 0$ implies $\chi'(\mathbf{x},\mathbf{x}') = 0$.
But this violates property (b) of $\chi'$.
\end{proof}
What we need is the following corollary.
\begin{equation}\label{eq:coordom-sort}
\textrm{For any $\ell' = \ell + a\mathbf{e}_j$}, ~~~ \mathbf{v}_{\ell'} ~\leq_{\mathrm{coor}} \sortdown{\mathbf{u}_{\ell'}}
\end{equation}
where recall that $\sortdown{z}$ is the sorted-decreasing version of $z$.
Just as we defined $\boldsymbol{A}$, define the $|O|$-dimensional vector $\vv{B^{\bot}}$ as follows.
\begin{definition}\label{eq:def2}
For any $\mathbf{x}\in O'$,
\[
\vv{B^{\bot}}(\mathbf{x}) = \sum_{j\neq i} \underbrace{\min\left(1, \sum_{\ell' = \ell + a\mathbf{e}_j} \mathbf{v}_{\ell'}(\mathbf{x}) \right)}_{\text{let's call this}~\mathbf{z}_j(\mathbf{x}) \in \{0,1\}} ~~=:\sum_{j\neq i} \mathbf{z}_j(\mathbf{x})
\]
\end{definition}
Note that for every $j \neq i$, $\mathbf{w}_j$ and $\mathbf{z}_j$ are $|O|=|O'|$ dimensional Boolean vectors which we index by $\mathbf{x} \in O$ and $\mathbf{x} \in O'$, respectively.
\begin{claim}\label{clm:z-vs-w}
For all $j$, $\mathbf{z}_j \leq_{\mathrm{coor}} \sortdown{\mathbf{w}_j}$.
\end{claim}
\begin{proof}
Follows from \eqref{eq:coordom-sort}, and the defintions of $\mathbf{z}_j$ and $\mathbf{w}_j$ as described in~\Cref{eq:def1} and~\Cref{eq:def2}.
\end{proof}
Finally, for $\mathbf{x} \in O'$, define the $|O'| = |O|$ dimensional vector $\vv{B^{\parallel}}$ as
\[
\vv{B^{\parallel}}(\mathbf{x}) = 1~~\textrm{if there is some $\mathbf{y}$ to its right, outside the interval $I$ with $h(\mathbf{y}) = f(\mathbf{y}) = 0$ and $\chi'(\mathbf{x},\mathbf{y}) = 1$\text{.}}
\]
Just as in~\Cref{obs:phiA}, note that
\begin{observation}\label{obs:Phib}
For any $\mathbf{x} \in O'$,
\[
\vv{\Phi_{\chi'}(h)}(\mathbf{x}) = \vv{B^{\bot}}(\mathbf{x}) + \vv{B^{\parallel}}(\mathbf{x})
\]
\end{observation}
We now connect $\vv{A^{\parallel}}$ and $\vv{B^{\parallel}}$ as follows.
\begin{claim}\label{eq:googoo}
$\vv{B^{\parallel}} \leq_{\mathrm{coor}} \sortdown{\vv{A^{\parallel}}}$
\end{claim}
\begin{proof}
Similar to~\Cref{clm:crucial-1}, this follows from the following claim.
\begin{claim}\label{clm:crucial-2}
The number of $1$s in $\vv{B^{\parallel}}$ is at most that in $\vv{A^{\parallel}}$, and $\vv{B^{\parallel}}$ is sorted decreasing.
\end{claim}
\begin{proof}
This also follows from the way we recolor $\chi'$ the pairs of the form $(\mathbf{x}, \mathbf{y})$ with $\mathbf{y}$ lying to the right of $I$ and $f(\mathbf{y}) = 0$.
First let's show $\vv{B^{\parallel}}$ is sorted decreasing. Take two points $\mathbf{x}$ and $\mathbf{z}$ with $a < \mathbf{x}_i < \mathbf{z}_i < b$ both evaluating to $1$ in $g$.
Say, $\vv{B^{\parallel}}(\mathbf{z}) = 1$ implying there is some $\mathbf{y}$ with $g(\mathbf{y}) = f(\mathbf{y}) = 0$ to the right of $I$ s.t. $\chi'(\mathbf{z},\mathbf{y}) = 1$.
However, the way we recolor the edges incident on $\mathbf{y}$, this implies $\chi'(\mathbf{x}, \mathbf{y}) = 1$ as well. But that would imply $\vv{B^{\parallel}}(\mathbf{x}) = 1$.
The first part of the claim also follows from the way we recolor. Suppose the number of ones in $\vv{A^{\parallel}}$ is $t$.
That is, only $t$ of the points in $O$ have $1$-colored edges going to the right of the interval. Consider the subset $W$ of these outer endpoints.
The function value, both $f$ and $g$, are $0$ here. Note that none of these points in $W$ have more than $t$ edges incident on them which are colored $1$ in $\chi$.
Now note that in $\chi'$, this number of $1$-edges are conserved, and so for every $\mathbf{w} \in W$, the number of $1$-colored violating edges is still $\leq t$.
Now suppose for contradiction $\vv{B^{\parallel}}$ has $(t+1)$ ones. Take the right most point $\mathbf{x}$ and consider the violating edge $(\mathbf{x}, \mathbf{y})$ which is colored $1$ in $\chi'$.
By construction, this $\mathbf{y}$ must have $1$-colored edges to all the $(t+1)$ points (since we color them $1$ left-to-right). This contradicts the number of $1$-edges incident on $\mathbf{y}$.
\end{proof}\end{proof}
To summarize, we have from~\Cref{obs:phiA} and~\Cref{eq:def1},
\[
\vv{\Phi_\chi(f)} = \sum_{j\neq i} \mathbf{w}_j + \vv{A^{\parallel}}
\]
that is, we have written the LHS as a sum of Boolean vectors.
And, we have from~\Cref{obs:Phib} and~\Cref{eq:def2}, followed by~\Cref{clm:z-vs-w} and~\eqref{eq:googoo} that
\[
\vv{\Phi_{\chi'}(h)} = \sum_{j\neq i} \mathbf{z}_j + \vv{B^{\parallel}} ~~~\leq_{\mathrm{coor}}~~~ \underbrace{\sum_{j\neq i} \sortdown{\mathbf{w}_j} + \sortdown{\vv{A^{\parallel}}}}_{\text{call this $\vv{s\Phi}$}}
\]
Trivially, we have $\norm{\vv{\Phi_{\chi'}(h)}}_{1/2} \leq \norm{\vv{s\Phi}}_{1/2}$, and from~\Cref{lem:sum-of-vectors}, we get $\norm{\vv{s\Phi}}_{1/2} \leq \norm{ \vv{\Phi_\chi(f)}}_{1/2}$, completing the proof of the first part of~\eqref{eq:2cases}.
\section{Connecting with the Distance to Monotonicity: Proof of~\Cref{thm:dir-tal-semisorted}} \label{sec:semisorted-tal-dist}
In this section, we set the intuition behind~\eqref{eq:hope1} straight. We show how the isoperimetric theorem~\Cref{thm:semisorted-reduce-to-g}
on semisorted functions can be used to prove~\Cref{thm:dir-tal-semisorted}.
We begin by recalling the corollary of the undirected, colored Talagrand
objective on the hypercube.
\corkms*
\noindent
As mentioned earlier, one can't show \eqref{eq:hope1}, that is,
$\mathbf{E}_\mathbf{x} [\mathrm{var}(g_{\mathbf{x}})] = \Omega(\varepsilon_f)$. Indeed, there are examples
of functions even over the hypercube where the above bound does \emph{not} hold.
KMS deal with this problem by applying \Thm{semisorted-reduce-to-g}
to random restrictions of $f$. One can show that there
is some restriction where the corresponding $\mathbf{E}_\mathbf{x} [\mathrm{var}(g_{\mathbf{x}})]$ is large. They referred
to these calculations as the ``telescoping argument". This argument was quantitatively improved
by Pallavoor-Raskhodnikova-Waingarten~\cite{PRW22}.
In this section, we port that argument to the hypergrid setting. Our proof
is different in its presentation, though the key ideas are the same as KMS.
Our first step is to convert \Thm{semisorted-reduce-to-g} to a more convenient form,
using the undirected~\Thm{kms-und}.
\begin{theorem}\label{thm:semisorted-reduce-conv}
There exists a constant $C' > 0$ such that for any
semisorted function $f:[n]^d \to \{0,1\}$ and any arbitrary coloring $\chi:E\to \{0,1\}$ of the augmented hypergrid, we have
\[
T_{\Phi_\chi}(f) \geq C' \cdot \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]\text{.}
\]
\end{theorem}
\begin{proof} By \Thm{semisorted-reduce-to-g}, there exists some colorings $\xi_\mathbf{x}$ such that $T_{\Phi_\chi}(f) \geq \mathbf{E}_\mathbf{x}\mathbf{E}_S[\sqrt{\infl{{g_\mathbf{x},\xi_\mathbf{x}}}{S}}]$.
By the undirected Talagrand bound \Thm{kms-und},
$\mathbf{E}_S[\sqrt{\infl{{g_\mathbf{x},\xi_\mathbf{x}}}{S}}] \geq C\cdot \mathrm{var}(g_\mathbf{x})$.
\begin{eqnarray}
\mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)] & = & \mathbf{E}_S \mathbf{E}_\mathbf{x}[\mathbbm{1}((S \circ f)(\mathbf{x}) \neq (\overline{S} \circ f)(\mathbf{x}))] \nonumber \\
& = & \mathbf{E}_S \mathbf{E}_\mathbf{x}[\mathbbm{1}(g_\mathbf{x}(S) \neq g_\mathbf{x}(\overline{S}))] \nonumber \\
& = & \mathbf{E}_\mathbf{x} \mathbf{E}_S[\mathbbm{1}(g_\mathbf{x}(S) \neq g_\mathbf{x}(\overline{S}))] \leq 4\mathbf{E}_\mathbf{x}[\mathrm{var}(g_\mathbf{x})] \label{eq:delta-var}
\end{eqnarray}
(The final inequality uses \Clm{var-prob}, stated below.)
Hence, $\mathbf{E}_\mathbf{x}\mathbf{E}_S[\sqrt{\infl{{g_\mathbf{x},\xi}}{S}}] \geq (C/4)\mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]$.
\begin{claim} \label{clm:var-prob} For any Boolean function $h:\hyp{d} \to \hyp{}$, $\Pr_S[h(S) \neq h(\overline{S})] \leq 4\mathrm{var}(h)$.
\end{claim}
\begin{proof} Recall that $\mathrm{var}(h) = 4\Pr_S[h(S) = 0] \Pr_S[h(S) = 1]$.
Hence, $\mathrm{var}(h) = 4\max_{b \in \{0,1\}} \Pr_S[h(S) = b] \min_{b \in \{0,1\}} \Pr_S[h(S) = b]$.
Since one of the values is taken with probability at least $1/2$, $\mathrm{var}(h) \geq 2 \min_{b \in \{0,1\}} \Pr_S[h(S) = b]$.
Let $\boldsymbol{S} = \{S \ | \ h(S) \neq h(\overline{S})\}$. Observe that half the sets in $\boldsymbol{S}$ have an $h$-value of $1$,
and the other half have value zero. Hence, $\Pr_S[h(S) \neq h(\overline{S})] \leq 2 \min_{b \in \{0,1\}} \Pr_S[h(S) = b]$.
Combining with the bound from the previous paragraph, $\Pr_S[h(S) \neq h(\overline{S})] \leq 4\mathrm{var}(h)$.
\end{proof}
\end{proof}
We now give some definitions and claim regarding the Talagrand objective of random restrictions
of functions.
\begin{definition} \label{def:restrict} Let $S \subseteq [d]$ be a subset of coordinates.
The \emph{distribution of restrictions on $S$}, denoted ${\cal R}_S$, is supported over functions and generated
as follows. We pick a uar setting of the coordinates in $\overline{S}$, and output the function under this restriction.
(Hence, $h \sim {\cal R}_S$ has domain $[n]^S$.)
\end{definition}
The isoperimetric theorem of \Thm{semisorted-reduce-to-g} holds for any
ordering of the coordinates. In this section, we will need to randomize the ordering
of the sort operators.
We will represent an ordering
as a permutation $\pi$ over $[d]$. Abusing notation, for any subset $S \subseteq [d]$,
$\pi(S)$ is the induced ordered list of $S$.
\begin{definition} \label{def:delta} For any function $h: [n]^k \to \hyp{}$, define $\delta(h)$
to be $\mathbf{E}_\pi[\Delta(h, \pi([k]) \circ h)]$.
\end{definition}
By \Clm{sortS}, sorting on all coordinates leads to a monotone function.
Thus, $\delta(h)$ is at least the distance of $h$ to monotonicity.
We will perform our analyses in terms of $\delta(f)$, since it is more
amenable to a proof by induction over domain size.
The following claim is central to the final induction, and relates $\delta(f)$
to $\mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]$. This is the (only) claim
where we need to permute the coordinates. All other claims and theorems
hold for an arbitrary ordering of the coordinates (when defining $S \circ f$).
\begin{claim} \label{clm:triangle} $\delta(f) \leq \mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)}[\delta(h)] + \mathbf{E}_\pi \mathbf{E}_S[\Delta(\pi(S) \circ f, \pi(\overline{S}) \circ f)]$
\end{claim}
\begin{proof} Let us consider an arbitrary ordering of dimensions.
By triangle inequality,
\begin{eqnarray*}
\Delta(f, S \circ \overline{S} \circ f) \leq \Delta(f, S \circ f) + \Delta(S \circ f, S \circ \overline{S} \circ f)
\end{eqnarray*}
Observe that $S \circ S \circ f = S \circ f$, since sorting repeatedly on a dimension does not modify a function.
Hence, $\Delta(S \circ f, S \circ \overline{S} \circ f) = \Delta(S \circ S \circ f, S \circ \overline{S} \circ f)
\leq \Delta(S \circ f, \overline{S} \circ f)$. The latter inequality holds because sorting only reduces the Hamming distance
between functions (\Clm{sort-hamm}). Plugging this bound in and taking expectations over ordered subset $S$ of dimensions:
\begin{equation} \label{eq:restrict}
\mathbf{E}_S[\Delta(f, S \circ \overline{S} \circ f)] \leq \mathbf{E}_S[\Delta(f, S \circ f)] + \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]
\end{equation}
Observe that $S \circ f$ only changes the function in the dimensions in $S$, and can be thought
to act on the restrictions of $f$ (to $S$). Hence $\mathbf{E}_S[\Delta(f, S \circ f)] = \mathbf{E}_{h \sim {\cal R}(S)}[\Delta(h, S \circ h)]$.
Roughly speaking, the quantity $\Delta(f, S \circ \overline{S} \circ f)$ is $\varepsilon(f)$
and $\mathbf{E}_{h \sim {\cal R}(S)}[\Delta(h, S \circ h)]$ is $\mathbf{E}_{h \sim {\cal R}{S}} \varepsilon(h)$. So we would
hope that \Eqn{restrict} implies $\varepsilon(f) \leq \varepsilon(h) + \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]$.
Unfortunately, the quantities are only constant factor approximations of $\varepsilon(f), \varepsilon(h)$.
So by converting \Eqn{restrict} in terms of $\varepsilon(f)$, we would potentially lose a constant factor
in \Eqn{restrict}.
To avoid this problem, we deal with $\delta(f)$ instead. By randomly permuting $S$ and taking expectations,
the quantities in \Eqn{restrict} can be replaced by $\delta(\cdot)$ terms.
Taking expectations over a uar $\pi$, \Eqn{restrict} implies
\begin{equation}
\mathbf{E}_\pi\mathbf{E}_S[\Delta(f, \pi(S) \circ \pi(\overline{S}) \circ f)] \leq \mathbf{E}_\pi\mathbf{E}_S[\Delta(f, \pi(S) \circ f)] + \mathbf{E}_\pi\mathbf{E}_S[\Delta(\pi(S) \circ f, \pi(\overline{S}) \circ f)]
\end{equation}
Note that the switching order in the LHS, $\pi(S) \circ \pi(\overline{S})$, is uniformly random.
Moreover,
$$ \mathbf{E}_\pi \mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)}[\Delta(h, \pi(S) \circ h)] = \mathbf{E}_S \mathbf{E}_h \mathbf{E}_\pi [\Delta(h, \pi(S) \circ h)] = \mathbf{E}_S \mathbf{E}_h[\delta(h)] $$
Combining all our bounds, we get that $\delta(f) \leq \mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)}[\delta(h)] + \mathbf{E}_\pi \mathbf{E}_S[\Delta(\pi(S) \circ f, \pi(\overline{S}) \circ f)]$.
\end{proof}
We prove a useful claim about the Talagrand objective of restrictions, made in~\cite{PRW22}.
\begin{claim} \label{clm:tal-restrict} Let $p \in (0,1)$, and ${\cal H}(p)$ be the distribution
of subsets of $[d]$ generated by selecting each element with iid probability $p$. Then,
$\dtal{f} \geq (1/\sqrt{p}) \cdot \mathbf{E}_{S \sim {\cal H}(p)} \mathbf{E}_{h \sim {\cal R}_S} [\dtal{h}]$.
\end{claim}
\begin{proof} Fix a set $S$. For any subset $S$ of coordinates, let the define the influence in $S$ as $\Phiinfl{f,\chi}{\mathbf{x};S} := \sum_{i \in S} \Phiinfl{f,\chi}{\mathbf{x};i}$.
We are just summing the influences over the coordinates of $S$.
Consider the quantity $\mathbf{E}_{h \sim {\cal R}_S} [\dtal{h}] = \mathbf{E}_{h \sim {{\cal R}_S}} \mathbf{E}_{\mathbf{z}} [\sqrt{\Phiinfl{h,\chi}{\mathbf{z}}}]$.
Note that $\mathbf{z}$ denotes a uar setting of the coordinates in $S$. The colorings of $h$ are inherited from the coloring of $f$.
Each function $h$ is indexed
by a (uar) setting of $\overline{S}$. Hence,
\begin{equation}
\mathbf{E}_{h \sim {{\cal R}_S}} \mathbf{E}_{\mathbf{z}} [\sqrt{\Phiinfl{h,\chi}{\mathbf{z}}}] = \mathbf{E}_{\mathbf{x}} [\sqrt{\Phiinfl{f,\chi}{\mathbf{x}; S}}]
\end{equation}
The point $\mathbf{x}$ is uar in the entire domain $[n]^d$.
Note that $\mathbf{E}_{S \sim {\cal H}(p)} [\Phiinfl{f,\chi}{\mathbf{x}; S}]$
is precisely $p \cdot \Phiinfl{f,\chi}{\mathbf{x}; S}$, since each coordinate is independently picked in $S$ with probability $p$.
\begin{eqnarray*}
\mathbf{E}_{S \sim {\cal H}(p)} \mathbf{E}_{h \sim {\cal R}_S} [\dtal{h}] & = & \mathbf{E}_{S} \mathbf{E}_\mathbf{x} [\sqrt{\Phiinfl{f,\chi}{\mathbf{x}; S}}] \\
& = & \mathbf{E}_\mathbf{x} \mathbf{E}_{S}[\sqrt{\Phiinfl{f,\chi}{\mathbf{x}; S}}] \\
& \leq & \mathbf{E}_\mathbf{x} \Big[\sqrt{\mathbf{E}_{S}[\Phiinfl{f,\chi}{\mathbf{x}; S}]}\Big]
= \mathbf{E}_\mathbf{x} \Big[\sqrt{p \cdot \Phiinfl{f,\chi}{\mathbf{x}; S}}\Big] = \sqrt{p} \cdot \dtal{f}
\end{eqnarray*}
The inequality above is a consequence of the concavity of the square root function and Jensen's inequality.
\end{proof}
Now we have all the ingredients to prove~\Cref{thm:dir-tal-semisorted} whice we restate below for convenience.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\dirtalsemisorted*
\end{mdframed}
\begin{proof} The proof is by induction over the dimension $d$ of the domain.
Formally, we will prove a lower bound of $(C'/10)\varepsilon$, where $C'$ is the constat of \Thm{semisorted-reduce-conv}.
Let us first prove the base case, when $d \leq 10$.
Note that $\Phi_{f,\chi}(\mathbf{x}) = \sum_{i=1}^d \Phi_{f,\chi}(\mathbf{x};i)$,
where each term in the summation is 0-1 valued. Hence,
by the $l_1$-$l_2$-inequality, $\sqrt{\Phi_{f,\chi}(\mathbf{x})} \geq \sum_{i=1}^d \Phi_{f,\chi}(\mathbf{x};i)/d = \Phi_{f,\chi}(\mathbf{x})/d$.
Thus, $T_{\Phi_\chi}(f) \geq \mathbf{E}_\mathbf{x}[\Phi_{f,\chi}(\mathbf{x})]/d$. Furthermore,
$\mathbf{E}_\mathbf{x}[\Phi_{f,\chi}(\mathbf{x})] = \sum_{i=1}^d \mathbf{E}_{\mathbf{x}}[\Phi_{f,\chi}(\mathbf{x};i)]$. We can
break the expectation over $\mathbf{x}$ into lines as follows.
$$\mathbf{E}_\mathbf{x}[\Phi_{f,\chi}(\mathbf{x})] = \sum_{i=1}^d \mathbf{E}_{\ell \ \textrm{uar $i$-line}} \mathbf{E}_c [\Phi_{{f|_\ell},\chi}(c)]$$
(The coordinate $c$ is uar in $[n]$.) Now, for a Boolean function $f|_\ell$ on a line, if the distance to monotonicity
is $\varepsilon$, then there are at least $\varepsilon n$ violating pairs~\cite{EKK+00}, and thus for any coloring $\chi$, we have
$\mathbf{E}_c [\Phi_{{f|_\ell},\chi}(c)] \geq \varepsilon(f|_\ell)$,
and $\sum_{i=1}^d \mathbf{E}_{\ell \ \textrm{uar $i$-line}} \varepsilon(f|_\ell) = \Omega(\varepsilon(f))$.
Hence, $T_{\Phi_\chi}(f) = \Omega(\varepsilon/d)$. For $d \leq 10$, the lemma holds, and so henceforth we assume $d\geq 10$.
Now for the induction step. We now break into cases.
\underline{Case 1, $\mathbf{E}_\pi \mathbf{E}_S[\Delta(\pi(S) \circ f, \pi(\overline{S}) \circ f)] \geq \delta(f)/10$:} By \Thm{semisorted-reduce-conv},
$\dtal{f} \geq c\cdot \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]$ (for any ordering of coordinates). So
$\dtal{f} \geq c\cdot \mathbf{E}_\pi \mathbf{E}_S[\Delta(\pi(S) \circ f, \pi(\overline{S}) \circ f)] \geq (c/10) \cdot \delta(f)$.
\medskip
\underline{Case 2, $\mathbf{E}_\pi \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)] < \delta(f)/10$:} By \Clm{triangle},
$\mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)}[\delta(h)] \geq \delta(f) - \mathbf{E}_\pi \mathbf{E}_S[\Delta(S \circ f, \overline{S} \circ f)]$.
In this case, we can lower bound $\mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)}[\delta(h)] \geq (9/10)\delta(f)$.
Note that $S$ is drawn from the distribution ${\cal H}(1/2)$.
When $S \neq [d]$, we can apply induction to $\dtal{h}$ for $h \sim {\cal R}(S)$. Hence,
\begin{eqnarray}
\mathbf{E}_{S \sim {\cal H}(1/2)} \mathbf{E}_{h \sim {\cal R}(S)} [\dtal{h}] & \geq & 2^{-d} \sum_{S \neq [d]} \mathbf{E}_{h \sim {\cal R}(S)}[\dtal{h}] \geq 2^{-d} \cdot (c/10) \cdot \sum_{S \neq [d]} \mathbf{E}_{h \sim {\cal R}(S)} [\delta(h)] \notag \\
& = & 2^{-d} \cdot (c/10) \cdot \Big(\sum_{S \subseteq [d]} \mathbf{E}_{h \sim {\cal R}(S)} [\delta(h)] - \mathbf{E}_{h \sim {\cal R}([d])} [\delta(h)]\Big)\notag \\
& = & (c/10) \Big(\mathbf{E}_S \mathbf{E}_{h \sim {\cal R}(S)} [\delta(h)] - 2^{-d} \delta(f)\Big) \ \ \ \ \textrm{($h \sim {\cal R}([d])$ is $f$)} \notag \\
&\geq& (c/10)\cdot(9/10)\cdot \delta(f) - 2^{-d}\cdot(c/10)\cdot \delta(f) \ \ \ \textrm{(by case condition)}\notag \\
& = & (9/10 - 2^{-d}) \cdot (c/10) \cdot \delta(f) \geq (4/5) \cdot (c/10) \cdot \delta(f) \label{eq:tal-restrict}
\end{eqnarray}
By \Clm{tal-restrict}, $\dtal{f} \geq \sqrt{2}\cdot \mathbf{E}_{S \sim {\cal H}(1/2)} \mathbf{E}_{h \sim {\cal R}(S)} [\dtal{h}]$.
Combining with the inequality of \Eqn{tal-restrict}, $\dtal{f} \geq (\sqrt{2}\cdot 4/5) \cdot (c/10) \cdot \delta(f) \geq (c/10) \cdot \delta(f)$.
\end{proof}
\section{The Tester and it's Analysis: Proof of~\Cref{thm:mono-testing}} \label{sec:tester}
With the isoperimetric theorem of \Thm{dir-tal} in place, we can now design and analyze
the monotonicity tester for Boolean hypergrid functions. This section closely
follows the analogous analysis in~\cite{KMS15}, and will lift certain notions from that paper.
We do have to make slight adaptations to various
arguments therein to account for the hypergrid domain.
We first describe the path tester for the hypergrid.
\begin{figure}[h]
\begin{framed}
\noindent \textbf{Input:} A Boolean function $f: [n]^d \to \{0,1\}$.
\smallskip
\begin{asparaenum}
\item Choose $k\in_R \{0,1,2,\ldots,\ceil{\log d}\}$ uniformly at random. Set $\tau := 2^k$.
\item Choose $\mathbf{x}\in [n]^d$ uniformly at random. Denote $\mathbf{x} = (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_d)$.
\item Pick a uniform random subset $R\subseteq [d]$ of $\tau$ coordinates.
\item For each $r \in R$, pick uar value $c_r \in [n] \setminus \{\mathbf{x}_r\}$.
\item Generate $\mathbf{z}$ as follows. For every $r \in [d]$, if $r \in R$ \emph{and} $c_r > \mathbf{x}_r$, set $\mathbf{z}_r = c_r$.
Else, set $\mathbf{z}_r = \mathbf{x}_r$.
\item If $f(\mathbf{z}) > f(\mathbf{x})$, REJECT.
\end{asparaenum}
\end{framed}
\caption{\small{\textbf{Path Tester for Hypergrid Functions}}}
\label{fig:alg}
\end{figure}
Clearly, the tester doesn't reject any monotone function.
Our main theorem regarding the tester follows.
A standard boosting argument gives us the $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$-query Boolean monotonicity tester on $[n]^d$, proving~\Cref{thm:mono-testing}.
\begin{theorem} \label{thm:tester} If $f$ is $\varepsilon$-far from monotone,
then the path tester for hypergrid functions rejects with probability $\Omega(\frac{\varepsilon^2}{n\sqrt{d}\log^5(nd)})$.
\end{theorem}
\subsection{Directed random walks, influences, and persistence} \label{sec:dirwalk}
One can think of the above tester as obtaining $\mathbf{z}$ by a lazy directed random walk of $\tau$ steps from a uniform random $\mathbf{x}$.
Note that in some steps we may not move at all; these correspond to coordinates $r\in R$ such that $c_r \leq \mathbf{x}_r$.
It will be convenient to define an alternate, but equivalent process that generates the pair $(\mathbf{x},\mathbf{z})$.
\begin{figure}[h]
\begin{framed}
\noindent \textbf{Input:} A length parameter $\tau$.
\smallskip
\begin{asparaenum}
\item In each dimension $i$, sample a uniform random pair $a_i < b_i$ from $[n]$.
\item Let $H$ be the hypercube formed by $\prod_{i=1}^d \{a_i, b_i\}$.
\item Pick a uar point $\mathbf{x}$ from $H$.
\item Pick a uar subset $R$ of $\tau$ coordinates, permuted randomly.
\item Generate $\mathbf{z}$ as follows. For every $r \in [d]$, if $r \in R$ and $\mathbf{x}_r = a_r$, set $\mathbf{z}_r = b_r$. Else, set $\mathbf{z}_r = \mathbf{x}_r$.
\end{asparaenum}
\end{framed}
\caption{\small{\textbf{Directed random walks, by sampling hypercubes}}}
\label{fig:dirwalk}
\end{figure}
This process generates walks by first sampling a random hypercube $H$, and then
doing a lazy directed walk on $H$. We first observe that conditioned on the walk length $\tau$, the distribution
of $(\mathbf{x}, \mathbf{z})$ pairs
generated by the path tester and the above process are identical.
\begin{observation} \label{obs:walk-distributions} Fix $\tau \in [d]$. Let $\mathcal{D}_1,\mathcal{D}_2$ denote the distributions over $[n]^d \times [n]^d$ described in \Fig{alg} and \Fig{dirwalk}, respectively, conditioned on the walk length $\tau$. Then, for any pair $(\mathbf{x},\mathbf{z})$ where $\mathbf{x} \preceq \mathbf{z}$, we have $\Pr[(\mathbf{x},\mathbf{z}) \sim \mathcal{D}_1] = \Pr[(\mathbf{x},\mathbf{z}) \sim \mathcal{D}_2]$. \end{observation}
\begin{proof}
Let $S = \{i \in [d] \colon \mathbf{z}_i > \mathbf{x}_i\}$. The probability from $\mathcal{D}_1$ is given by
\begin{alignat}{4}
\Pr[(\mathbf{x}, \mathbf{z}) \sim \mathcal{D}_1] &&~=~& \frac{1}{n^d} \cdot \sum_{R\supseteq S~:~|R|=\tau} {d \choose \tau}^{-1} \prod_{i\in S} \Pr[c_i = \mathbf{z}_i] \prod_{i\in R\setminus S} \Pr[c_i \leq \mathbf{x}_i] \notag \\
&&~=~& \frac{1}{n^d} \cdot {d \choose \tau}^{-1} \cdot \sum_{R\supseteq S~:~|R|=\tau} \prod_{i\in S} \frac{1}{n-1} \prod_{i\in R\setminus S} \frac{\mathbf{x}_i-1}{n-1} \text{.}\notag
\end{alignat}
We now compute the probability for $\mathcal{D}_2$. For $i \in [d]$, let ${\cal E}_i$ be the event that $a_i = \mathbf{x}_i$ or $b_i = \mathbf{x}_i$. Note
\[
\Pr[\neg {\cal E}_i] = \frac{\binom{n-1}{2}}{\binom{n}{2}} = \frac{n-2}{n } ~~\Rightarrow \Pr[{\cal E}_i] = \frac{2}{n} \text{.}
\]
Let ${\cal E}_{\mathbf{x}}$ denote the event that the first point sampled according to $\mathcal{D}_2$ is $\mathbf{x}$. We have
\[
\Pr[{\cal E}_{\mathbf{x}}] = \prod_{i=1}^d \Pr[{\cal E}_i]\cdot \frac{1}{2^d} = \left(\frac{2}{n}\right)^d \frac{1}{2^d} = \frac{1}{n^d} \text{.}
\]
Let ${\cal E}_\mathbf{z}$ denote the event that the second point is $\mathbf{z}$. We have
\[
\Pr\left[{\cal E}_{\mathbf{z}} | {\cal E}_{\mathbf{x}}\right] = \sum_{R\supseteq S~:~|R|=\tau} {d \choose \tau}^{-1} \prod_{i\in S} \Pr[a_i = \mathbf{x}_i ~\textrm{and}~b_i = \mathbf{z}_i~|~{\cal E}_\mathbf{x}] \cdot \prod_{i\in R\setminus S} \Pr[b_i = \mathbf{x}_i~|~{\cal E}_\mathbf{x}]
\]
Fix an $i\in S$. We hav
\[
\Pr[a_i = \mathbf{x}_i ~\textrm{and}~b_i = \mathbf{z}_i~|~{\cal E}_\mathbf{x}] = \Pr[a_i = \mathbf{x}_i ~\textrm{and}~b_i = \mathbf{z}_i~|~{\cal E}_i] = \frac{\Pr[a_i = \mathbf{x}_i ~\textrm{and}~b_i = \mathbf{z}_i]}{\Pr[{\cal E}_i]} = \frac{1/{n \choose 2}}{2/n} = \frac{1}{n-1} \text{.}
\]
Now fix an $i\in R\setminus S$. We have
\[
\Pr[b_i = \mathbf{x}_i~|~{\cal E}_{\mathbf{x}}] = \Pr[b_i = \mathbf{x}_i~|~{\cal E}_i] = \frac{\Pr[b_i = \mathbf{x}_i]}{\Pr[{\cal E}_i]} = \frac{\left(\frac{\mathbf{x}_i-1}{{n \choose 2}}\right)}{2/n} = \frac{\mathbf{x}_i-1}{n-1} \text{.}
\]
Therefore, $\Pr[(\mathbf{x},\mathbf{z}) \sim \mathcal{D}_2] = \Pr[{\cal E}_\mathbf{x}] \cdot \Pr[{\cal E}_{\mathbf{z}} | {\cal E}_{\mathbf{x}}] = \Pr[(\mathbf{x},\mathbf{z}) \sim \mathcal{D}_1]$. \end{proof}
\noindent
We now have that the random walk distributions described in \Fig{alg} and \Fig{dirwalk} are equivalent. The former is more convenient to analyze for the final rejection probability, but the latter
perspective allows us to prove various influence bounds by piggybacking on the \cite{KMS15} hypercube analysis.
\begin{definition} \label{def:totalinf} We define the total influence and total negative
influence of $f$ as follows.
$$ I_f = n^{-d} \sum_{\mathbf{x} \in [n]^d} \sum_{i=1}^d \sum_{c = 1}^n {\bf 1}(f(\mathbf{x}) \neq f(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_{i-1}, c, \mathbf{x}_{i+1}, \ldots, \mathbf{x}_d))$$
$$ I^-_f = n^{-d} \sum_{\mathbf{x} \in [n]^d: f(\mathbf{x}) = 1} \sum_{i=1}^d \sum_{c > \mathbf{x}_i} {\bf 1}(f(\mathbf{x}) \neq f(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_{i-1}, c, \mathbf{x}_{i+1}, \ldots, \mathbf{x}_d))$$
Note these are different from the thresholded influences and for most functions will grow as $n$ grows.
\end{definition}
We can analogously define these influences on the hypercubes $H$ sampled by the process described in \Fig{dirwalk}.
Abusing notation, we will denote these influences as $I_H := I_{f|_{H}}$ and $I_H^- := I^-_{f|_H}$. A simple, yet important claim
relates the expected influence on $H$ to the total influence on the hypergrid. All expectations over $H$ are taken with respect to the distribution described in steps 1-2 of \Fig{dirwalk}.
\begin{claim} \label{clm:inf-hyp} $\mathbf{E}_H[I_H] = I_f/(n-1)$ and $\mathbf{E}_H[I^-_H] = I^-_f/(n-1)$.
\end{claim}
\begin{proof} Consider any $i$-edge $(\mathbf{x},\mathbf{y})$ of the fully augmented hypergrid.
The probability that this edge is present in $H$ can be computed as follows.
Firstly, we need $a_i = \mathbf{x}_i$ and $b_i = \mathbf{y}_i$. This happens with probability ${n \choose 2}^{-1}$. Then, for all $j \neq i$,
one of $\{a_j,b_j\}$ needs to be $\mathbf{x}_j$ (note that $\mathbf{x}_j = \mathbf{y}_j$ since $(\mathbf{x},\mathbf{y})$ is an $i$-edge). This happens with probability $(1-({n-1 \choose 2}/{n \choose 2}))^{d-1} = (2/n)^{d-1}$.
The total number of influential edges in $f$ is $n^d I_f$. Thus, by linearity
of expectation, the expected number of influential edges in $H$
is ${n \choose 2}^{-1} \times (2/n)^{d-1} \times n^d I_f = 2^d I_f/(n-1)$.
An analogous proof holds for the negative influence. \end{proof}
By using a lemma
of \cite{KMS15}, we can prove that if the total influence of $f$ is too large, then the negative influence
is also large.
\begin{claim} \label{clm:tot-neg-inf} If $I_f > 9(n-1)\sqrt{d}$, then $I^-_f > (n-1)\sqrt{d}$.
\end{claim}
\begin{proof} Theorem 9.1 of \cite{KMS15} asserts that, if $I_H > 6\sqrt{d}$, then $I^-_H > I_H/3$.
(This holds for any Boolean hypercube function.) If $I_f > 9(n-1)\sqrt{d}$, then
by \Clm{inf-hyp}, $\mathbf{E}_H[I_H] > 9\sqrt{d}$. Hence,
\begin{eqnarray}
9\sqrt{d} < \mathbf{E}_H[I_H] & = & \Pr[I_H \leq 6\sqrt{d}] \ \mathbf{E}_H[I_H | I_H \leq 6\sqrt{d}] \ + \ \Pr[I_H > 6\sqrt{d}] \ \mathbf{E}_H[I_H | I_H > 6\sqrt{d}] \nonumber \\
& < & 6\sqrt{d} + \Pr[I_H > 6\sqrt{d}] \mathbf{E}_H[3 I^-_H | I_H > 6\sqrt{d}] \leq 6\sqrt{d} + 3\mathbf{E}_H[I^-_H]
\end{eqnarray}
Hence, $\mathbf{E}_H[I^-_H] > \sqrt{d}$. By \Clm{inf-hyp}, $I^-_f > (n-1)\sqrt{d}$.
\end{proof}
One of the crucial definitions is that of \emph{persistence}.
\begin{definition} \label{def:persist} A point $\mathbf{x} \in [n]^d$ is called $\tau$-persistent
if $\Pr_{\mathbf{z}}[f(\mathbf{x}) = f(\mathbf{z})] \geq 1/2$ where $\mathbf{z}$ is chosen by a $\tau$-length directed random walk from $\mathbf{x}$. \end{definition}
\begin{lemma} \label{lem:persist} If $I_f \leq 9(n-1)\sqrt{d}$, then the fraction of vertices that are not $\tau$-persistent is at most $C_{per} \tau/\sqrt{d}$ (where $C_{per}$ is an absolute constant). \end{lemma}
\begin{proof} We will analyze the random walk using the distribution described in \Fig{dirwalk} and leverage the analysis that \cite{KMS15} use to prove their Lemma 9.3. Let $\alpha$ denote the fraction of $\tau$-non-persistent vertices in the fully augmented hypergrid with respect to our function $f$. Again, let $\mathcal{D}_1$, $\mathcal{D}_2$ denote the distributions over pairs $(\mathbf{x},\mathbf{z})$ described in \Fig{alg}, \Fig{dirwalk}, respectively, conditioned on the walk length $\tau$. For a fixed $H$ sampled in steps 1-2 of \Fig{dirwalk}, let $\mathcal{D}_{2,H}$ denote the distribution over $(\mathbf{x},\mathbf{z})$ described in steps 3-5. Using the definition of persistence and \Obs{walk-distributions}, we have
\begin{align} \label{eq:persistence1}
\frac{\alpha}{2} \leq \Pr_{(\mathbf{x},\mathbf{z})\sim \mathcal{D}_1}(f(\mathbf{x}) \neq f(\mathbf{z})) = \mathbf{E}_H\left[\Pr_{(\mathbf{x},\mathbf{z})\sim \mathcal{D}_{2,H}}\left(f(\mathbf{x}) \neq f(\mathbf{z})\right)\right] \text{.}
\end{align}
Let $\widehat{\mathcal{D}}_{2,H}$ denote the same distribution as $\mathcal{D}_{2,H}$ except with $R$ being a uar subset of the $0$-coordinates of $\mathbf{x}$ (recall step 4 of \Fig{dirwalk}). I.e. $\widehat{\mathcal{D}}_{2,H}$ is the \emph{non-lazy} walk distribution on $H$. Let $\mathbf{x} = \mathbf{x}^0, \mathbf{x}^1, \ldots, \mathbf{x}^{\tau} = \mathbf{z}$ be the $\tau$ steps taken on the walk sampled by $\mathcal{D}_{2,H}$ and let $\mathbf{x} = \widehat{\mathbf{x}}^0, \widehat{\mathbf{x}}^1, \ldots, \widehat{\mathbf{x}}^{\tau} = \mathbf{z}$ be the $\tau$ steps taken on the walk sampled by $\widehat{\mathcal{D}}_{2,H}$. For a fixed $H$ we have
\begin{align} \label{eq:persistence2}
\Pr_{(\mathbf{x},\mathbf{z})\sim \mathcal{D}_{2,H}}\left(f(\mathbf{x}) \neq f(\mathbf{z})\right) \leq \sum_{\ell=0}^{\tau - 1} \Pr\left(f(\mathbf{x}^{\ell}) \neq f(\mathbf{x}^{\ell+1})\right) \leq \sum_{\ell=0}^{\tau - 1} \Pr\left(f(\widehat{\mathbf{x}}^{\ell}) \neq f(\widehat{\mathbf{x}}^{\ell+1})\right) \text{.}
\end{align}
The first inequality is by a union bound and the second inequality holds because the first walk is lazy and the second is not. More precisely, we can couple the $\tau' \leq \tau$ steps
of the lazy-random walk where the point actually moves to the first $\tau'$ steps of the second non-lazy walk, and the remaining $\tau-\tau'$ terms of the non-lazy walk can only increase the RHS.
By Lemma 9.4 of \cite{KMS15}, the edge $(\widehat{\mathbf{x}}^{\ell},\widehat{\mathbf{x}}^{\ell+1})$ is distributed approximately as a uniform random edge in $H$. In particular, this implies $\Pr\left(f(\widehat{\mathbf{x}}^{\ell}) \neq f(\widehat{\mathbf{x}}^{\ell+1})\right) \leq C \cdot 2I_H/d$ for an absolute constant $C$. (Note $2I_H/d$ is the probability of a uniform random edge in $H$ being influential.) Putting \Eqn{persistence1} and \Eqn{persistence2} together yields $\alpha \leq \frac{4 C \tau }{d}\mathbf{E}_H[I_H]$. By \Clm{inf-hyp} we have $\mathbf{E}_H[I_H] \leq 9\sqrt{d}$ and so setting $C_{per} := 36C$ completes the proof. \end{proof}
\subsection{The good subgraph and capturing violations} \label{sec:good-sub}
We now use the isoperimetric theorem of \Thm{dir-tal} to construct a \emph{good subgraph},
in the parlance of~\cite{KMS15}.
\begin{theorem} \label{thm:good-sub} There exists a bipartite subgraph $G = (X,Y,E)$
of the fully augmented hypergrid with the following properties.
\begin{asparaitem}
\item $|X| = \sigma n^d$ or $|Y| = \sigma n^d$.
\item Every vertex has degree at most $k$.
\item For all $\mathbf{x} \in X$, $f(\mathbf{x}) = 1$. For all $\mathbf{y} \in Y$, $f(\mathbf{y}) = 0$.
\item $|E| \geq \sigma k n^d/2$.
\item $\sigma\sqrt{k} = \Theta(\varepsilon/\log^2(nd))$.
\end{asparaitem}
\end{theorem}
\begin{proof} Consider the bipartite subgraph consisting of all violations
of the fully augmented hypergrid. Consider any bi-coloring $\chi$ of the edges
of this subgraph. Let $\deg_{\chi}(\mathbf{x})$ denote the number of violating edges $e$ incident to $\mathbf{x}$ for which $\chi(e) = f(\mathbf{x})$. Note that $\deg_{\chi}(\mathbf{x}) \geq \Phi_{f,\chi}(\mathbf{x})$.
Hence, \Thm{dir-tal} asserts that $\sum_\mathbf{x} \sqrt{\deg_{\chi}(\mathbf{x})} \geq C' n^d \varepsilon/\log n$,
for some absolute constant $C'$.
According to Def. 6.4 of \cite{KMS15}, the bipartite graph is $C' n^d \varepsilon/\log n$-robust.
By Lemma 6.5 of \cite{KMS15}, any robust bipartite graph has a ``good subgraph" satisfying
the following bound. (Below, $C''$ is a constant.)
$$ \sigma n^d \times \sqrt{k} \geq \frac{C' \varepsilon n^d}{8\log(nd)\log n} \ \ \ \Longrightarrow \ \ \ \sigma\sqrt{k} \geq \frac{C''\varepsilon}{\log^2(nd)} $$
One can remove vertices from this good subgraph to ensure that $\sigma \sqrt{k} = \Theta(\varepsilon/\log^2(nd))$.
\end{proof}
For the rest of the analysis we will assume $|X| = \sigma n^d$, without loss of generality. The edges of the good subgraph of \Thm{good-sub} are central to the tester analysis. We will need
to choose $\tau$ carefully to ensure that the analysis carries through. Towards that choice,
we will set a convenient bound on $\sigma$. We will use $C_{lar}$ to denote a sufficiently
large constant that is at least $100C_{per}$ and the constants of \Thm{dir-tal}. (The constant $C_{per}$ is from \Lem{persist}.)
\begin{claim} \label{clm:small-sigma} If $\sigma < C_{lar} /\sqrt{d}$, then $I^-_f = \Omega(\varepsilon^2\sqrt{d}/\log^4(nd))$.
\end{claim}
\begin{proof} By the good subgraph properties in \Thm{good-sub}, $\sqrt{k} = \Omega(\varepsilon\sqrt{d}/\log^2(nd))$.
Hence, the number of edges of the good subgraph is at least $\sigma k n^d/2 = (\sigma \sqrt{k}) \times \sqrt{k} \times n^d/2
= \Omega(\varepsilon^2 \sqrt{d} n^d/\log^4(nd))$. We divide this bound by $n^d$ to bound $I^-_f$.
\end{proof}
Essentially, for the analysis, we can ignore the case when $\sigma$ is too small. With this bound in place,
we can now set the right choice of $\tau$ based on the good subgraph parameters.
\begin{definition} \label{def:tau-sigma} For any $\sigma \geq C_{lar}/\sqrt{d}$, define $\tau_\sigma$ to be the power of $2$
in the range $[\sigma \sqrt{d}/C_{lar}, \sigma 2\sqrt{d}/C_{lar}]$. (Since $\sigma \sqrt{d}/C_{lar} \geq 1$,
the choice of $\tau_\sigma$ exists.)
\end{definition}
We will now define a particular ``edge capturing event" that ensures that the tester
finds a violation to monotonicity. The crucial property is that these events
are uniquely associated with edges of the good subgraph, and are all disjoint.
So, we can lower bound the probability of this event and multiply by the
number of edges of the good subgraph.
\begin{definition} \label{def:capture} Let $\mathbf{x} \in X$. We call an edge $(\mathbf{x},\mathbf{y})$ of the good
subgraph \emph{viable} if $\mathbf{y}$ is $\tau_\sigma$-persistent. The set of
\emph{viable coordinates} of $\mathbf{x}$ are the dimensions containing the viable
edges incident to $\mathbf{x}$.
For a viable $i$-edge $e = (\mathbf{x},\mathbf{y})$, the \emph{capturing event} ${\cal C}_e$ is defined as follows.
Consider the sampling process of the tester, and condition on $\tau := \tau_\sigma$. We define ${\cal C}_e = {\cal E}_1 \wedge {\cal E}_2 \wedge {\cal E}_3 \wedge {\cal E}_4 \wedge {\cal E}_5$ where:
\begin{asparaitem}
\item ${\cal E}_1$: The point $\mathbf{x}$ is chosen (as the first point).
\item ${\cal E}_2$: The coordinate set $R$ contains $i$.
\item ${\cal E}_3$: The coordinate $c_i$ is $\mathbf{y}_i$.
\item ${\cal E}_4$: $R\setminus i$ does not contain any viable coordinates of $\mathbf{x}$.
\item ${\cal E}_5$: $f(\mathbf{z}) = 0$.
\end{asparaitem}
\end{definition}
The main calculation is to lower bound the probability of the event ${\cal C}_e$,
for any viable edge.
\begin{lemma} \label{lem:viable} For any viable edge $e = (\mathbf{x},\mathbf{y})$, $\Pr[{\cal C}_e | \tau = \tau_\sigma] = \Omega(n^{-d} \times n^{-1} \times \tau_\sigma/d)$.
\end{lemma}
\begin{proof} The probability of choosing $\mathbf{x}$ is $n^{-d}$. The probability that $i$ lies in $R$
is ${{d-1}\choose {{\tau_\sigma}-1}}/{d\choose {\tau_\sigma}} = \tau_\sigma/d$. The probability
that $c_i$ equals $\mathbf{y}_i$ is $\frac{1}{n-1}$. That is, $\Pr[{\cal E}_1 \wedge {\cal E}_2 \wedge {\cal E}_3] = \Omega(n^{-d} \times n^{-1} \times \tau_\sigma/d)$. Thus, it remains to show that $\Pr[{\cal E}_4 \wedge {\cal E}_5 ~|~ {\cal E}_1 \wedge {\cal E}_2 \wedge {\cal E}_3] = \Omega(1)$.
Let $T$ denote the set of viable coordinates of $\mathbf{x}$. Since the maximum degree in the good subgraph $G$ is $k$, we have $|T| \leq k$. Thus,
\begin{align}
\Pr[(R \setminus i) \cap T = \emptyset] \geq \frac{{d - k \choose \tau_{\sigma} - 1}}{{d \choose \tau_{\sigma} - 1}} = \prod_{\ell=0}^{k-1} \left(1-\frac{\tau_{\sigma}-1}{d-\ell}\right) \geq \left(1-\frac{\tau_{\sigma}}{d-k}\right)^k \geq \exp\left(-\frac{k \cdot \tau_{\sigma}}{d-k}\right)
\end{align}
Recall from \Thm{good-sub} that $k = \Theta(\frac{\varepsilon^2}{\log^4(nd)} \cdot \frac{1}{\sigma^2})$ and from \Clm{small-sigma} we may assume that $\sigma \geq C_{lar}/\sqrt{d}$. Thus, we have $k = O(d \cdot \frac{\varepsilon^2}{\log^4(nd)})$ and in particular, $k = o(d)$ and so
\begin{align}
\Pr[(R \setminus i) \cap T = \emptyset] \geq \exp\left(-O\left(\frac{k \cdot \tau_\sigma}{d}\right)\right) = \exp\left(-O\left(\frac{k}{d} \cdot \sigma\sqrt{d}\right)\right) = \exp(-O(\sigma\sqrt{k}))
\end{align}
where the second step used our definition of $\tau_\sigma$ and the last step simply used the fact that $k \leq d$. By \Thm{good-sub} we have $\sigma\sqrt{k} = \Theta(\varepsilon/\log^2(nd))$ and so we have $\Pr[(R \setminus i) \cap T = \emptyset] \geq 9/10$ as long as $nd$ is at least some constant.
Finally, since $\mathbf{y}$ is $\tau_\sigma$-persistent, the probability that this random walk from $\mathbf{y}$ ends at $\mathbf{z}$ where $f(\mathbf{z}) = 1$ is at most $1/2$. Thus, by the union bound
\[
\Pr[{\cal E}_4 \wedge {\cal E}_5 ~|~ {\cal E}_1 \wedge {\cal E}_2 \wedge {\cal E}_3] \geq 1-(1/10 + 1/2) = 2/5
\]
and this completes the proof. \end{proof}
\subsection{Wrapping it all up} \label{sec:tester-wrap}
We combine all the bounds and calculations to prove that the path tester
has an $\widetilde{\Omega}(\varepsilon^2/n\sqrt{d})$ probability of success.
\begin{proof} (of \Thm{tester}) We first take care of some edge cases.
{\em Case 1, $I_f > 9(n-1)\sqrt{d}$:} By \Clm{tot-neg-inf}, $I^-_f > (n-1)\sqrt{d}$.
Thus, the total number of violated edges of the augmented hypergrid is at least $(n-1)\sqrt{d} n^d$. The total number of edges is $\frac{1}{2}(n-1)dn^d$. Hence,
a uniform random edge is a violation with probability at least $1/(2\sqrt{d})$.
The path tester selects $\tau = 1$ with probability at least $1/\log d$,
so the rejection probability is at least $1/(2\sqrt{d}\log d)$.
{\em Case 2, $\sigma < C_{lar}/\sqrt{d}$:} By \Clm{small-sigma},
$I^-_f = \Omega(\varepsilon^2\sqrt{d}/\log^4(nd))$. Thus the probability
that a uniform random edge is a violation is $\Omega(\varepsilon^2/(n\sqrt{d}\log^4(nd)))$.
Similar to the above case, the rejection probability is $\Omega(\varepsilon^2/(n\sqrt{d}\log^5(nd)))$.
{\em Case 3, $I_f \leq 9(n-1)\sqrt{d}$ and $\sigma \geq C_{lar}/\sqrt{d}$:}
This is the interesting case, where all the previous claims and lemmas are used.
Since $\sigma \geq C_{lar}/\sqrt{d}$, we can define $\tau_\sigma$ using \Def{tau-sigma}.
By \Lem{persist}, since $I_f \leq 9(n-1)\sqrt{d}$,
the fraction of vertices that are not $\tau_\sigma$-persistent
is at most
\[
C_{per} \tau_\sigma/\sqrt{d} \leq (2C_{per}/C_{lar})\cdot \sigma < \sigma/4\text{.}
\]
Let us now count the number of viable edges in the good subgraph promised by \Thm{good-sub}. There are at least $\sigma k n^d/2$ edges in the good subgraph. There are at most $(\sigma/4) \cdot n^d$ non-persistent vertices in the good subgraph, each of which has degree at most $k$. Thus, removing all non-persistent vertices leaves us with
\[
\sigma k n^d/2 - (\sigma/4) \times n^d \times k = \sigma k n^d/4
\]
edges, all of which are viable. Let us now lower bound the probability of the tester
rejecting using \Lem{viable}. Recall that all the events ${\cal C}_e$ are disjoint and $\tau_\sigma = \Theta(\sigma\sqrt{d})$.
\begin{eqnarray*}
\Pr\left[\bigcup_{e \ \textrm{viable}} {\cal C}_e | \tau = \tau_\sigma\right] & = & \sum_{e \ \textrm{viable}} \Pr[{\cal C}_e | \tau = \tau_\sigma]\\
& \geq & (\sigma k/4) n^d \times n^{-d} \times n^{-1} \times (\tau_\sigma/d) \\
& = & \Omega(\sigma^2 k/(n\sqrt{d})) = \Omega(\varepsilon^2/(n\sqrt{d}\log^4(nd))) \text{.}
\end{eqnarray*}
The probability of setting $\tau$ to be $\tau_\sigma$
is $1/\log d$, so we multiply the bound above by $1/\log d$ to complete the proof. \end{proof}
\section{Towards a $\widetilde{O}(\sqrt{d})$ tester} \label{sec:no-n}
In this section we describe a different notion of influence of Boolean functions on hypergrids. We conjecture a Talagrand style isoperimetric theorem for the colored version of this influence is true. If so, then we can design a tester whose query complexity has no polynomial dependence on $n$. More precisely, the dependence on $n$ is only polylogarithmic, and since by results of~\cite{BlackCS20,HY22} one can assume $n = \mathrm{poly}(d)$, the final tester's query complexity is $\widetilde{O}(\sqrt{d})$.
\begin{definition}[Weighted Influence]\label{def:psi-f}
Fix $f:[n]^d \to \{0,1\}$ and a dimension $i\in [d]$. Fix a point $\mathbf{x} \in [n]^d$. The {\em weighted} influence of $\mathbf{x}$ along coordinate $i$ is defined as
\[
\Psi_f(\mathbf{x};i) := \sum_{\mathbf{y} = \mathbf{x}\pm a\mathbf{e}_i~:~ \textrm{such that} ~(\mathbf{x}, \mathbf{y})~ \textrm{is violating}} ~~\frac{1}{a}
\]
The weighted influence of $\mathbf{x}$ is $\Psi_f(\mathbf{x}) = \sum_{i=1}^d \Psi_f(\mathbf{x};i)$.
\end{definition}
Consider giving a {\em weight} to every edge $(\mathbf{x}, \mathbf{x} \pm a\mathbf{e}_i)$ equal to $1/a$, the reciprocal of the length of the edge. The weighted influence of a point $\mathbf{x}$ with $f(\mathbf{x}) = 1$ is
the sum of the weights of {\em out}-edges which are violating. This is another generalization of the notion of influence in hypercubes. Also note that for any $\mathbf{x}$, the thresholded influence
of any point can't be much smaller than the weighted influence; indeed, $\Phi_f(\mathbf{x}) \geq \frac{1}{H_n} \Psi_f(\mathbf{x})$ where $H_n$ is the $n$th Harmonic number and is $\Theta(\log n)$.
Therefore, lower bounding $\Psi$'s also lower bounds $\Phi$'s. On the other hand, $\Phi_f(\mathbf{x};i)$ could be as large as $(n-1)\cdot \Psi_f(\mathbf{x};i)$ for a particular $\mathbf{x}$.
The Talagrand objective with respect to this notion is defined as the expected {\em square root} of the weighted influence.
\begin{definition}[Talagrand Objective wrt Weighted Influence]
Given any Boolean function $f:[n]^d \to \{0,1\}$, we define the Talagrand objective with respect to the weighted influence as
\[
T_{\Psi}(f) := \mathbf{E}_{\mathbf{x}\in [n]^d} \left[ \sqrt{\Psi_f(\mathbf{x})}\right]
\]
where $\Psi_f$ is as defined in~\Cref{def:psi-f}.
\end{definition}
We also have a notion of colorful weighted influence and the corresponding Talagrand objective.
\begin{definition}[Colorful Weighted Influence]\label{def:psi-f-chi}
Fix $f:[n]^d \to \{0,1\}$ and $\chi:E\to \{0,1\}$. Fix a dimension $i\in [d]$. Fix a point $\mathbf{x} \in [n]^d$. The colorful weighted influence of $\mathbf{x}$ along coordinate $i$ is defined as
\[
\Psi_{f,\chi}(\mathbf{x};i) := \sum_{\mathbf{y} = \mathbf{x}\pm a\mathbf{e}_i~:~ \textrm{such that} ~(\mathbf{x}, \mathbf{y})~ \textrm{is violating and}~ \chi(\mathbf{x}, \mathbf{y}) = f(\mathbf{x})} ~~\frac{1}{a}
\]
The colorful weighted influence of $\mathbf{x}$ is $\Psi_{f,\chi}(\mathbf{x}) = \sum_{i=1}^d \Psi_{f,\chi}(\mathbf{x};i)$.
\end{definition}
\begin{definition}[Colorful Thresholded Talagrand Objective]
Given any Boolean function $f:[n]^d \to \{0,1\}$ and $\chi:E \to \{0,1\}$,
we define the Talagrand objective with respect to the colorful weighted influence as
\[
T_{\Psi_\chi}(f) := \mathbf{E}_\mathbf{x}~\left[\sqrt{\Psi_{f,\chi}(\mathbf{x})} \right]
\]
where $\Psi_{f,\chi}$ is as defined in~\Cref{def:psi-f-chi}.
\end{definition}
We are now ready to state our main conjecture. Note that due to the fact that $\Psi_{f,\chi}(\mathbf{x}) = O(\log n)\Phi_{f,\chi}(\mathbf{x})$, the conjecture below generalizes~\Cref{thm:dir-tal} up to $\mathrm{poly}\log n$ factors.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{conjecture}\label{conj:dir-tal}~
Let $f:[n]^d \to \{0,1\}$ which is $\varepsilon$-far from monotone, and let $\chi:E\to \{0,1\}$ be an arbitrary coloring of the edges of the augmented hypergrid. Then there exists a constant $c \geq 0$ such that
\[
T_{\Psi_\chi}(f) := \mathbf{E}_{\mathbf{x}\in [n]^d}~\left[\sqrt{\Psi_{f,\chi}(\mathbf{x})} \right] = \Omega\left(\frac{\varepsilon}{\log^c n}\right)
\]
where $\Psi_{f,\chi}(\mathbf{x})$ is as defined in~\Cref{def:psi-f-chi}.
\end{conjecture}
\end{mdframed}
If the above conjecture is true, then we can design a $\widetilde{O}(\sqrt{d})$ tester. Before we state this tester, and indeed to motivate it, we first note that the analysis of the tester described in~\Cref{fig:alg} is tight on its dependence on $n$.
An example is the so-called {\em centrist} function defined in~\cite{BlackCS20} (Section 8) defined as follows for the case $n=d$, that is, $f:[d]^d \to \{0,1\}$.
\[
f(\mathbf{x}) = \begin{cases}
0 & \textrm{if}~~\exists i\in [d]~~~ \mathbf{x}_i = 2 \\
1 & \textrm{otherwise}
\end{cases}
\]
\cite{BlackCS20} (Claim 8.2) showed that $\varepsilon_f = \Omega(1)$ for the above function. To see this, observe that the probability of a random point having $\mathbf{x}_i \neq 2$ for all $i \in [d]$ is
$\left(1-\frac{1}{d}\right)^d = \Theta(1)$. Now let's consider the rejection probability of the algorithm in~\Cref{fig:alg}. Note that a violating pair $(\mathbf{x}, \mathbf{y})$ must satisfy $\mathbf{x}_i = 1$
and $\mathbf{y}_i = 2$ for some $i$, and that $i$ needs to be picked in the random set $R$. Even conditioning on picking that $i\in R$, the probability $c_i$ would be set to $2$ is $\frac{1}{n-1} \approx \frac{1}{d}$ in this case. Therefore, the algorithm needs to be modified.
To fix this, think about the tester as performing a (lazy) directed random walk. And now when one picks a dimension $i$ to move in, one doesn't have a {\em uniform} distribution over the length it moves in this dimension, as currently is the case in~\Cref{fig:alg}, rather one chooses a length using a discrete Pareto distribution. That is, smaller lengths are given more weight than longer lengths. More precisely, we pick a length $a$ to move with probability proportional to $\frac{1}{a}$; note that the constant of proportionality that we need to scale down by is at most $H_n = \Theta(\log n)$. This is the reason we defined the weighted influence as we did in~\Cref{def:psi-f}, and indeed, if~\Cref{conj:dir-tal} is true, then we can prove that this tester is an $\widetilde{O}(\sqrt{d})$ tester; the proof technique is similar to that described in~\Cref{sec:tester} which itself is a modification of the analysis in~\cite{KMS15}. \smallskip
We now show that the {\em uncolored} version of~\Cref{conj:dir-tal} is in fact true, and follows easily using the colorful version~\Cref{thm:dir-tal} for thresholded influence.
\begin{theorem}
Let $f:[n]^d \to \{0,1\}$ which is $\varepsilon$-far from monotone. Then,
\[
T_{\Psi}(f) := \mathbf{E}_{\mathbf{x}\in [n]^d}~\left[\sqrt{\Psi_{f}(\mathbf{x})} \right] = \Omega\left(\frac{\varepsilon}{\log n}\right)
\]
where $\Psi_{f}(\mathbf{x})$ is as defined in~\Cref{def:psi-f}.
\end{theorem}
\begin{proof} The key is that we can always define a coloring $\chi$ for which $T_{\Psi}(f) = \Omega(T_{\Phi_{\chi}}(f))$ simply as follows. For every edge $(\mathbf{x},\mathbf{x}+a\mathbf{e}_i)$ of the fully augmented hypergrid, define $\chi(\mathbf{x},\mathbf{x}+a\mathbf{e}_i) := 1$ if the interval $[\mathbf{x},\mathbf{x}+a\mathbf{e}_i]$ is at least half $0$'s and $\chi(\mathbf{x},\mathbf{x}+a\mathbf{e}_i) := 0$ otherwise. This coloring achieves the desired property because of the following simple fact.
\begin{fact} Let $S \subseteq \{1,\ldots,n\}$ be of size $|S| \geq \lceil n/2 \rceil$. Then $\sum_{a \in S} \frac{1}{a} = \Omega(1)$. \end{fact}
\begin{proof} For simplicity let $n$ be even. The sum is minimized when $S = [n] \setminus [n/2]$ and in this case the sum is equal to $H_n - H_{n/2} = \Omega(1)$. \end{proof}
Now consider $\mathbf{x}$ and $i \in [d]$ such that $f(\mathbf{x}) = 1$ and $\Phi_{f,\chi}(\mathbf{x};i) = 1$. This implies that there exists an $i$-edge $(\mathbf{x},\mathbf{y})$ for which $f(\mathbf{y}) = 0$ and $\chi(\mathbf{x},\mathbf{y}) = 1$. By definition of the coloring, the interval $[\mathbf{x},\mathbf{y}]$ is at least half $0$'s and so $\Psi_f(\mathbf{x};i) = \Omega(1)$. Similarly, consider $\mathbf{y}$ and $i \in [d]$ such that $f(\mathbf{y}) = 0$ and $\Phi_{f,\chi}(\mathbf{y};i) = 1$. This implies that there exists an $i$-edge $(\mathbf{x},\mathbf{y})$ for which $f(\mathbf{x}) = 1$ and $\chi(\mathbf{x},\mathbf{y}) = 0$. By definition of the coloring, the interval $[\mathbf{x},\mathbf{y}]$ is at least half $1$'s and so $\Psi_f(\mathbf{y};i) = \Omega(1)$. Thus, $T_{\Psi}(f) = \Omega(T_{\Phi_{\chi}}(f))$ and so invoking \Cref{thm:dir-tal} completes the proof. \end{proof}
It is worthwhile to point out the challenge in generalizing the above theorem to the colored version. Note the above was a ``point-by-point'' argument in that we found a coloring $\chi$ of the fully augmented hypergrid edges such that for every $\mathbf{x}$ and every $i\in [d]$, we could prove $\Psi_f(\mathbf{x};i) = \Omega(\Phi_{f,\chi}(\mathbf{x};i))$. One would wonder if such a point-by-point analysis is possible even when we have an arbitrary coloring $\chi'$, and the LHS in the previous statement is replaced with the colored version. Unfortunately, this is not possible. One can find examples of $f$ and $\chi'$ such that no matter how you define $\chi$, there will be some point $\mathbf{x}$ and some dimension $i$ such that $\Phi_{f, \chi}(\mathbf{x};i) = 1$, but $\Psi_{f,\chi'}(\mathbf{x};i) \approx 1/n$. These examples do not disprove the conjecture since, in these examples, for a constant fraction of $(\mathbf{x}, i)$ pairs, we do have $\Psi_{f,\chi'}(\mathbf{x};i) = \Omega(\Phi_{f,\chi}(\mathbf{x}))$, but it does point to the need of a new argument. One could also wonder if the proof technique used to prove~\Cref{thm:dir-tal} can bear upon the proof of the conjecture. There are many roadblocks here, one of them primarily being that semisorting can {\em increase} the Talagrand objective with respect to the $\Psi$-influences mainly because it can bring violations closer which bumps up the $\Psi$-influence. Nevertheless, the authors believe that~\Cref{conj:dir-tal} is true, and we leave the resolution of this as a promising direction towards getting rid of the polynomial dependence on $n$ thereby resolving the Boolean monotonicity testing question on hypergrids.
\section{Semisorting and Reduction to Semisorted Functions}\label{sec:semisorted}
As we mentioned earlier when we stated~\eqref{eq:hope2}, we do not know if this is a true statement for an arbitrary function.
It is true for what we call semisorted functions, and proving this would be the bulk of the work. In this section, we define what semisorted
functions are, we prove that the Talagrand objective can only decrease when one moves to a semisorted function, and therefore how one can reduce to proving~\Cref{thm:dir-tal}
only for semisorted functions. \smallskip
Fix a function $f:[n]^d \to \{0,1\}$. Fix a coordinate $i$ and fix an interval $I = [a,b]$.
Semisorting $f$ on this interval in dimension $i$ leads to a function $h:[n]^d \to \{0,1\}$ as follows. We take every $i$-line $\ell$
and consider the function restricted on the interval $I$ on this line, and we sort it. The following lemma shows that semisorting on any $(i,I)$ pair
can only reduce the Talagrand objective. We defer its proof to~\Cref{sec:semisorting-can-only-reduce}.
\begin{mdframed}[backgroundcolor=blue!10,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{lemma}[Semisorting only decreases $T_\Phi$.]\label{lem:semisorting-decreases}
Let $f$ be any hypergrid function and let $\chi$ be any bicoloring of the augmented hypergrid edges. Let $i\in [d]$ be any dimension and $I$ be any interval $[a,b]$.
There exists a (re)-coloring $\chi'$ of the edges of the augmented hypergrid such that
\[
T_{\Phi_\chi}(f) \geq T_{\Phi_{\chi'}}(h)
\]
where $h$ is the function obtained upon semisorting $f$ in dimension $i$ on the interval $I$.
\end{lemma}
\end{mdframed}
\noindent
A function $f:[n]^d \to \{0,1\}$ is called {\em semisorted} if for any $i\in [d]$ and any $i$-line $\ell$, the function restricted to the first $n/2$ points is sorted increasing
and the function restricted to the second half is also sorted increasing. It is instructive to note that when $n=2$, that is when the domain is the hypercube, every function is semisorted.
This shows that semisorted functions form a non-trivial family. However, the semisortedness is a property that allows us to prove that
\eqref{eq:hope2} holds. In particular, we prove this theorem.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{restatable}[Connecting Talagrand Objectives of $f$ and Tracker Functions]{theorem}{semisorted}
\label{thm:semisorted-reduce-to-g}
Let $f \colon [n]^d \to \{0,1\}$ be a {\bf \em semisorted} function and let $\chi \colon E\to \{0,1\}$ be an arbitrary coloring of the edges of the fully augmented hypergrid.
Then for every $\mathbf{x} \in [n]^d$, one can find a coloring $\xi_\mathbf{x}$ of the edges of the Boolean hypercube such that
\[
T_{\Phi_\chi}(f) := \mathbf{E}_{\mathbf{x} \in [n]^d}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right] ~~\geq~~ \mathbf{E}_{\mathbf{x} \in [n]^d} ~\mathbf{E}_{S\subseteq [d]}~~[\sqrt{I_{g_\mathbf{x}, \xi_\mathbf{x}}(S)}] \text{.}
\]
\end{restatable}
\end{mdframed}
\noindent
We can use the above theorem to get set the intuition behind~\eqref{eq:hope1} correct, and prove~\Cref{thm:dir-tal} for semisorted functions. We state this below, but we defer the proof
of this to~\Cref{sec:semisorted-tal-dist}. At this point we remind the reader again that this is not at all trivial, but the proof ideas are generalizations of those present in~\cite{KMS15,PRW22}
for the hypercube case.
\begin{mdframed}[backgroundcolor=gray!20,topline=false,bottomline=false,leftline=false,rightline=false]
\begin{restatable}[\Cref{thm:dir-tal} for semisorted functions.]{theorem}{dirtalsemisorted}\label{thm:dir-tal-semisorted}
Let $f:[n]^d \to \{0,1\}$ be a {\bf \em semisorted} function that is $\varepsilon$-far from monotone. Let $\chi:E\to \{0,1\}$ be an arbitrary coloring of the edges of the augmented hypergrid.
Then there is a constant $C''$ such that
\[
T_{\Phi_\chi}(f) := \mathbf{E}_\mathbf{x}~\left[\sqrt{\Phi_{f,\chi}(\mathbf{x})} \right] \geq C''\varepsilon
\]
\end{restatable}
\end{mdframed}
\Cref{lem:semisorting-decreases} shows that the Talagrand objective can't rise on semisorting. The distance to monotonicty, however, can fall. In the remainder of the section we show
how we can reduce to the semisorted case with a loss of $\log n$, and in particular, we use~\Cref{thm:dir-tal-semisorted} to prove~\Cref{thm:dir-tal}.
\paragraph{Sequence of Semisorted Functions and Reduction to the Semisorted Case.}
We now describe a semi-sorting process which gives a way of getting from $f$ to a monotone function. Without much loss of generality, let us assume $n=2^k$ which we can assume by padding.
Iteratively coarsen the domain $[n]^d = [2^k]^d$ as follows.
First ``chop'' this hypergrid into $2^d$ many $[n/2]^d = [2^{k - 1}]^d$ hypergrids by slicing through the ``middle'' in each of the $d$-coordinates.
More precisely, these $2^d$ hypergrids can be indexed via $\mathbf{v} \in \{0,1\}^d$, where given such a vector, the corresponding hypergrid is
\[
H_\mathbf{v} = \prod_{i=1}^d \{\mathbf{v}_i \cdot \frac{n}{2} + 1, \mathbf{v}_i \cdot \frac{n}{2} + 2, \cdots, \mathbf{v}_i \cdot \frac{n}{2} + \frac{n}{2} \}
\]
Each hypergrid $H_\mathbf{v}$ is an $[n/2]^d = [2^{k-1}]^d$ hypergrid. Let us denote the collection of all these hypergrids as the set ${\cal H}_1$.
So, ${\cal H}_1$ has $2^d$ many hypergrids and each hypergrid has dimension $[n/2]^d = [2^{k-1}]^d$.
Repeat the above operation on each hypergrid in ${\cal H}_1$. More precisely, each hypergrid $H_\mathbf{v}$ in ${\cal H}_1$ will lead to $2^d$ hypergrids each with dimension $[n/4]^d = [2^{k-2}]^d$.
The total number of such hypergrids, which we collect in the collection ${\cal H}_2$, is $2^d \times 2^d = (2^2)^d$. More generally, we have a family ${\cal H}_i$ consisting of $\left(2^i\right)^d$ many
hypergrids of dimension $[n/2^i]^d = [2^{k - i}]^d$. The collection ${\cal H}_{k-1}$ consists of $(2^{k-1})^d$ many $d$-dimensional hypercubes.
\begin{figure}[ht!]
\begin{center}
\includegraphics[trim = 0 150 0 60, clip, scale=0.4]{figs/semisort.pdf}
\caption{\em In the figure, we see an example with $d = 2$ and $n = 8 =2^{3}$. There are $2^2$ many $4\times 4$ green (hyper)-grids,
and $4^2$ many $2\times 2$ red squares.}
\end{center}
\end{figure}
\noindent
Note that in any family ${\cal H}_i$ for $1\leq i\leq k-1$, each $H\in {\cal H}_i$ is a sub-hypergrid of $[n]^d$. We let $f_H$ denote the restriction of $f$ only to this subset $H$ of the domain.
Also, let ${\cal H}_0$ denote the singleton set containing only one hypergrid, $[n]^d$.
Define the function $f_1 : [n]^d \to \{0,1\}$ as follows: consider every hypergrid\footnote{these will be hypercubes} $H$ in ${\cal H}_{k-1}$ and apply the sort operator on $f_H$ for all these hypergrids. Note that $f_1$ is a monotone function when restricted to $H\in {\cal H}_{k-1}$. Recursively define $f_i$ as follows: consider every hypergrid $H\in {\cal H}_{k-i}$ and apply the sort operator on $(f_{i-1})_H$ for all these hypergrids.
\Cref{fig:semisortwithvals}~is an illustration for $d = 2$ and $k = 3$, i.e. $n=8$.
\begin{figure}[ht!]
\begin{center}
\includegraphics[trim = 0 45 0 30, clip, scale=0.5]{figs/semisortwithvals.pdf}
\caption{\em The function $f=f_0$ is described to the left, and then one obtains $f_1, f_2$ and $f_3$. The function $h$ which is obtained doing sort on the whole of $f$
is described below. Note $h\neq f_3$.}\label{fig:semisortwithvals}
\end{center}
\end{figure}
\begin{claim}\label{clm:triangle-ineq}
There must exist an $0\leq j \leq k-1$ such that $\Delta(f_{j}, f_{j+1}) \geq \varepsilon_f/k$.
\end{claim}
\begin{proof}
This follows from triangle inequality and the fact that $\Delta(f_0, f_k) \geq \varepsilon_f$.
\end{proof}
\begin{proof}[\bf Proof of~\Cref{thm:dir-tal}]
We now show how~\Cref{thm:dir-tal} follows from~\Cref{lem:semisorting-decreases} and~\Cref{thm:dir-tal-semisorted} via an averaging argument.
We fix the $j$ as in~\Cref{clm:triangle-ineq}.
By~\Cref{lem:semisorting-decreases} we get that for any function $f$ and any coloring $\chi$, there exists a recoloring $\chi'$ such that
$T_{\Phi_{\chi}}(f) \geq T_{\Phi_{\chi'}}(f_{j})$. Now consider the hypergrids in $H\in {\cal H}_{k-j-1}$. Let $f_j|_H$ be the function restricted to this sub-domain $H$.
Note that the function $f_j|_H$ is indeed semisorted by construction. Therefore, by~\Cref{thm:dir-tal-semisorted} (on the coloring $\chi'$) we know that for all $H\in {\cal H}_{k-j-1}$,
\[
T_{\Phi_{\chi'}}({f_j|_H}) \geq C''\cdot \varepsilon_{f_{j}|_H}
\]
By~\Cref{clm:2appx}, we know that $2\varepsilon_{f_{j}|_H} \geq \Delta(f_{j}|_H, f_{j+1}|_H)$. Taking expectation over $H\in {\cal H}_{k-j-1}$,
we see that the LHS is at most (at most since we only consider violations staying in $H$) $T_{\Phi_{\chi'}}(f_j)$, while the RHS is precisely $\Delta(f_j,f_{j+1})/2 \geq \varepsilon_f/2k$.
Putting everything together, we get
$
T_{\Phi_\chi}(f) \geq \frac{C''\varepsilon_f}{2\log n}
$
proving~\Cref{thm:dir-tal}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,247 |
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This Delta Air Lines flight traveled from San Francisco International Airport to Salt Lake City International Airport, on November 18, 2019. It departed from gate C10. Passengers were flying aboard an aircraft with tail number N110DU. See all the flights for DL1390.
VS4531 (SFO-SLC)
This Virgin Atlantic Airways flight between San Francisco International Airport and Salt Lake City International Airport, on November 18, 2019, was operated by another airline. It departed from gate C10. See all the flights for VS4531.
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This KLM (Royal Dutch Airlines) flight between San Francisco International Airport and Salt Lake City International Airport, on November 18, 2019, was operated by another airline. It departed from gate C10. See all the flights for KL7738.
UA1061 (SFO-BWI)
This United Airlines flight traveled from San Francisco International Airport to Baltimore-Washington International Thurgood Marshall Airport, on November 18, 2019. It departed from gate E8. Passengers were flying aboard a Boeing 737-800 with tail number N37298. See all the flights for UA1061. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,686 |
Product information "Park Tool SZR-1 Scissors"
Shop quality scissors with sturdy, stainless steel blades and a dual density grip. Perfect for a wide range of mechanic's duties including cutting zip-ties, boxes, bar tape and handle grips.
Related links to "Park Tool SZR-1 Scissors"
Customer evaluation for "Park Tool SZR-1 Scissors" | {
"redpajama_set_name": "RedPajamaC4"
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Chris Cuomo Destroyed by St Louis Homeowner Who Armed Himself Against BLM Crowd: "That's an entirely false concept"
By Leisa Audette | Jul 1, 2020
Chris Cuomo tries to mock and ridicule the homeowner who was defending his home and family by brandishing a gun.
Mark McCloskey and his wife were on video trying to protect their home from a mob of protesters who had come on their property. The video went viral, and the left is targeting the couple who happen to be Democrats.
McCloskey joined Chris Cuomo to try and explain what happened and ended up going head to head with a contentious Cuomo.
The St. Louis homeowner did a fantastic job in his effort to get the truth out by even calling out a local radical group by name. McCloskey also knew exactly how far the mayor's house is from his so he could point out that this crowd was up to no good.
Watch below as Chris Cuomo gets hammered by a very good lawyer:
Mark McCloskey—the armed husband in the viral video defending his St. Louis home—talks to CNN's Chris Cuomo: pic.twitter.com/1JWsMpZc5U
Trending: Have You Been Thinking About Quitting The Fight To Save America After The Disappointing 2022 Election Results? I Almost Did, and So Did Many Others...Here's Why We Decided To Stay And Fight
— Alex Salvi (@alexsalvinews) July 1, 2020
The McCloskey family has been a target of the left since this incident and is even being targeted by the local prosecutor who is trying to file criminal charges.
OUR PREVIOUS REPORT ON THE INCIDENT:
Another statue protest took place in St. Louis over the weekend, where a mob of protesters descended on the Mayor of St. Louis' home.
A group of BLM protesters also came onto the front lawn of a beautiful historic home where the frightened homeowners had come out to protect their property. The protesters came through a private entrance of the property that is also part of a gated community.
With all of the vandalism and looting taking place, who can blame this couple for protecting their home?
The owners purchased the rundown mansion in 1988, and it took 30 years to bring it back to its former glory. Wouldn't anyone be protective of this very special home?
The homeowners were rightfully exercising their Second Amendment Right, but the left made them into the bad guys:
"A couple has come out of their house and is pointing guns at protesters in their neighborhood."
A couple has come out of their house and is pointing guns at protesters in their neighborhood #StLouis #lydakrewson pic.twitter.com/ZJ8a553PAU
— Daniel Shular (@xshularx) June 29, 2020
Now the leftist anti-gun crowd is coming after this couple who were just trying to defend themselves. Soon after the call for their personal information was put out, their information was published for all to see. This should be frightening to anyone because the left will come after you:
Someone identify these people. I don't know the area. Get an address. I'll help.
Sue them, ruin their businesses, help their employees sabotage their capital, call their kids, call their family members, call their country club— use capitalism for the weapon it's designed to be.
The left tries to make the couple out to be the aggressors against "peaceful protesters":
https://twitter.com/RyanzoPerez/status/1277404541693169672?s=20
Save up on MyPillow products. Use promo code FedUp at checkout and save 50% on individual MyPillow Towels.
Large Mob of Single, Male, Adult Illegal Aliens REFUSE To Leave Manhattan Hotel Where They Are Getting Free Rooms, Food, Health Care and More..
Mom SHOCKS School Board After Reading Disgusting, Pornographic Book Available in School Library for Students [VIDEO]
Tyre Nichols' Mom Condemns the Black Officers Who Beat Her Son to Death... "I hate it was 5 Black men that did this" [VIDEO]
"Cheers to those who lick us where we pee"...Drag Queen Gives Vulgar Toast at "Family-Friendly" Drag Show In Front of Small Children...Tells Them: "Close your ears!" [VIDEO]
Tennis Legend John McEnroe Is Cut Off By ESPN's Chris Fowler During Heated Exchange Over Novak Djokovic's Refusal to Be Vaccinated Moments After His Australian Open Win [VIDEO] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,593 |
Q: What's the significance of TLD-first domain-like identifiers? "TLD-first domain-like identifiers" is a mouthful but that's all I can come up with.
I've seen these used in various places over the years and wondered what the history/reason behind this convention is, since you might be forgiven in thinking that there is one true way to mention a domain.
I don't use Java but I recall from poking around that namespaces are often done like this:
uk.co.tophats.stitchkit
A specification file for a "Launch Agent" on Mac OS X:
ws.agile.1PasswordAgent.plist
A preferences file on Mac OS X:
com.apple.iTunesHelper.plist
Why is the TLD first? Is it just hierarchical pedantry like UK vs. US date formats?
A: For the case of Java, the packages correspond directly to directory hierarchy, and it makes far more sense for the directory hierarchy to be rooted at the most general, rather than the most specific, domain identifier. Also, when reading off directory hierarchy, it is most common to read it from the top down. So I'd say the convention of flipping the order of domain components makes sense there.
A: In Java and other programming languages, the package identifier dictates how the directory hierarchy of the project is.
So if you have two packages com.stackoverflow.server and com.stackoverflow.client, you'll end up with this directory layout:
com/
stackoverflow/
client/
server/
which is good and logical, while the other way around would give you
client/
stackoverflow/
com/
server/
stackoverflow/
com/
which is impratical.
A: Its a way of using a globally unique name as the prefix to all your namespaces and thus keep all your namespaces private to the globally unique name.
A: In sience, notation is normally started with the most significant bit of information, to make ordering and grouping of the information easier. The TLD is the root of the domain structure and thus the most significant bit of information. So it makes sense to structure the packages this way. Sure, it doesn't matter if it's com.example.mypackage or example.com.mypackage. But SUN decided to prefer the more sientific way.
Regarding the date format, be aware that there is an ISO describing an "interchange format" as: YYYY-MM-DD
It's the same scheme, because the year is the most significant part, followed by the Month and Date.
So, when looking from this point, the one who is using the "wrong" notation is the DNS system itself. But I think they tried to optimize the whole thing for parsing of an url (e.g. The "www" first, to indicate a WebServer)
A: It's a bit subjective. You can be used to conventions like "stackoverflow.com", and wonder what the heck all this "com.apple" stuff is about. Or you can be a programmer with years of experience and stuff like "System.out" could be the most natural thing for you.
com.apple is like saying "look into the com domain, inside this, look for apple".
apple.com is like saying "look for apple, which can be found in the com domain".
So it all depends on the environment/situation you're using. Just my two cents!
A: No, it's not pedantry, it's to create a namespace.
There might be other 3 party developers/programs/etc. that also could create a iTunesHelper.plist
or a SQLRunner.java class. You prefix these with your own namespace, e.g. your domain name
to create a reasonably unique name so com.oracle.SQLRunner.java is different and doesn't clash with org.postgresql.SQLRunner.java
| {
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Patrick Hoo, a physics enthusiast.
TeX, Vim, Emacs, JabRef, Wiki, etc.
syn match cMathOperator display "[-+\*/%=]"
syn match cPointerOperator display "->\|\."
syn match cLogicalOperator display "[!<>]=\="
syn match cLogicalOperator display "=="
syn match cBinaryOperator display "\(&\||\|\^\|<<\|>>\)=\="
syn match cBinaryOperator display "\~"
syn match cBinaryOperatorError display "\~="
syn match cLogicalOperator display "&&\|||"
syn match cLogicalOperatorError display "\(&&\|||\)=" | {
"redpajama_set_name": "RedPajamaC4"
} | 1,087 |
\section{Introduction}
The first strong hints of the deconfined quarks at high temperature
appeared more than ten years ago~\cite{polsus}, the numerical
confirmation~\cite{num} followed soon. Subsequently a large
number of details has been clarified but the
driving force of the deconfinement transition, the
confinement-deconfinement mechanism, remained elusive.
A subjective and a rather sketchy list of remarks is
presented here to indicate a unique and challenging
aspect of this phase transition. Only one detail will
be discussed in a slightly more detailed manner,
the formal similarity between the density matrix of
the Aharonov-Bohm (A-B) system and of QCD.
One can distinguish two confinement mechanisms~\cite{como},
a hard and a soft one. The hard mechanism is responsible
for the linear potential between static test quarks in the
absence of dynamical quarks. The soft, low energy mechanism is
which screens a test quark and was thought by V. Gribov
to be similar to the supercritical vacuum of QED. More
precisely the infrared instability of the perturbative
QCD, the source of the hard confinement mechanism, leads
to strong gluon interactions at large distances. Sufficiently
far from a test quark the coupling constants reaches a
large enough value to ignite the spontaneous creation
of the quark-anti quark pairs which in turn shield the
test quark charge. Most of the remarks mentioned here
refers to the hard confinement mechanism which is
more elementary and should be clarified before
embarking the study of the soft mechanism of full QCD.
\section{Unusual or unique features}
\underbar{1. Different degrees of freedom:} We find different
degrees of freedom at the two sides of the phase transition.
This happens in a number of other phase transitions,
the Mott or the localisation-delocalisation transitions
may serve as examples. Observe that the elementary degrees
of freedom are recovered among the highly excited states
in these cases. This does not
happen in the hadronic phase. The relevance of this
obvious remark becomes clear by considering the
thermal average of an observable $A$,
$\langle A\rangle=Z^{-1}\sum_n\langle n|A|n\rangle e^{-{E_n/T}}$.
In order to reproduce the thermal averages we use the
color singlet asymptotic states $|n\rangle$ of the hamiltonian of the
strong interactions. How can
we recover the contributions of an isolated, deconfined
quark to the given observable? The only way out of this problem
is the modification of the Hilbert space at the phase transition.
\underbar{2. Weakly or strongly coupled phase?}
The asymptotically free running coupling constants becomes small
around the typical energy scale $p=T$ at high energies,
$T>\Lambda_{QCD}\approx T_c$. Does that mean that
the deconfined phase is weakly coupled at high enough temperature?
The answer is known to be negative since long time~\cite{linde}.
The small parameter of the perturbation expansion at high
temperature stems from a non-perturbative quantity, the
magnetic screening mass. This can be understood by recalling that
the thermal bath breaks the Lorentz invariance.
Though the typical energy scale is pushed up at high
temperature $E\approx T$, the (off-shell) momentum scale in the
loop integrals is effected differently by the temperature and
the infrared stabilization of the long wavelength
modes remains a difficult question.
This is because the partition function of the
high temperature $3+1$ dimensional
QCD can be approximated by a $3$ dimensional (classical)
Yang-Mills-Higgs system and the infrared sensitivity
of the partition function increases by lowering the
dimension. Thus the fate of the perturbation expansion which is
based on massless gluons depends on the screening mechanism.
The usual strategy of dealing with the IR divergences,
the separation of the scales $T$, $gT$
and $g^2T$, can not solve this problem because $g$
does not reach small enough values, $g(m_{Planck})\approx1/2$.
\underbar{3. Order parameter:}
The order parameter related to the hard confinement mechanism
is the trace of the heavy quark propagator continued over complex time,
\begin{equation}
\omega(\vec x,t)=\la0|\psi_\alpha\left(\vec x,t+{i\over T}\right)
\bar\psi_\alpha(\vec x,t)|0\rangle.
\end{equation}
In the high temperature phase
where the time extent of the Euclidean space-time is shorter
than the correlation length, $1/T<\xi\approx\Lambda^{-1}_{QCD}$,
the gluon field variables are correlated along the world line
of the heavy quark and the order parameter develops a non-vanishing
expectation value. A distinguishing feature of the deconfining
transition is that its order parameter is not a canonical variable.
It controls the symmetry with respect the
global center
\footnote{The center $C(G)$ of the group $G$ is a subgroup of $G$. It consists of the
elements which commute with $G$, $[C,G]=0$, e.g.
$C(SU(N))=Z_N$.} gauge transformations performed at the initial or
the final state of a transition amplitude. It is important to keep in mind
that the center of the global gauge transformations is
the fundamental group of the gluonic configuration space~\cite{mech},
$Z_3=\pi_1(SU(3)/Z_3)$
\footnote{Consider the gauge transformation
$\vec A(\vec x)\to g(\vec x)(\vec\partial+\vec A(\vec x))g^\dagger(\vec x)$
acting on the anti-hermitean gauge field in the temporal gauge.
The global gauge transformations which commute with other gauge
transformations leave $\vec A(\vec x)$ invariant.}.
The only other known dynamical breakdown of the
fundamental group symmetry is the liquid-droplet quantum phase
transition.
\underbar{4. Finite volume effects:}
The ratio of the
gluonic partition functions with and without a static
quark is given by the expectation value of the order
parameter, $e^{-(F_q-F)/T}=\langle\omega\rangle$. Since the
spontaneous symmetry breaking does not occur in a finite system,
$\langle\omega\rangle=0$ and the static quarks always appear
confined, $F_q=\infty$, in finite volume. Where does the
singular free energy density, $F_q/V=\infty$, come from?
This problem is solved by taking into account the destructive
interference between the homotopy classes in the gluonic configuration
space.
\underbar{5. Symmetry breaking by the kinetic energy:}
The spontaneous symmetry breaking mechanism is operating
at low energy where the order parameter is driven to a non-symmetrical
value due to the degenerate minima of the potential energy.
The kinetic energy might drive a spontaneous, or more precisely
dynamical symmetry breaking at high energies.
The dynamical breakdown of the center symmetry results
from such a mechanism~\cite{mech}. This can be understood
by inspecting a quantum top, the baby version of the $SU(2)$
Yang-Mills model. The configuration space
which consists of the $3\times3$ orthogonal matrices,
$\{R\}=SO(3)=SU(2)/Z_2$,
is doubly connected and the wave functions are single
and double valued in the integer (gluons) and the
half-integer (quarks) spin subspaces, respectively.
Consider now the transition amplitude
${\cal A}(R',R)=\langle R'|e^{-itH/\hbar}|R\rangle$
as the function of the
final state $R'$. Since an orientation of the top is
undistinguishable from its $2\pi$ rotated copy
the integer spin amplitude is doubly degenerate
on the covering space $SU(2)$, ${\cal A}(r',r)={\cal A}(r'',r)$
where the final points $r',r''\in SU(2)$ differ in a rotation
by $2\pi$, $r'=-r''$ (center symmetry). Suppose that $r'$ is
closer to the initial point $r$ than $r''$. Then the kinetic energy tends
to suppress the propagation to $r''$ if the time available for the
propagation is short (high temperature or energy). The result for an infinite
top whose coordinate $r$ influences infinitely many degrees
of freedom (global gauge transformations) is that the
propagation to $r''$ is totally suppressed (center symmetry breakdown).
The confinement can be understood as the destructive interference
in the quark propagator between the different homotopy classes.
In fact, the center symmetry of the pure gluon system yields
identical amplitudes in different homotopy classes. But a particle
in the fundamental representation of the gauge group $SU(N)$
propagating along the system picks up the phases $e^{2i\pi n/N}$,
$n=1,\cdots,N$
which add up to zero. The result is the absence of these particles
in the final states. We find here another
characteristic feature of the deconfinement transition:
it corresponds to a transition amplitude rather than to the
vacuum. This is the key to find a synthesis
between the high and the low energy scattering
experiments, described in terms of the partons and the
hadronic bound states, respectively. In other words,
as the time of a collision process is shortened the transition matrix
elements go over the ``deconfined'',
center symmetry broken phase and the elementary
constituents (partons) appear.
\underbar{6. Permanent confinement of triality~\cite{mech}:}
(i) The deconfining phase transition consists of the
dynamical breakdown of the Gauss' law and the
modification of the Hilbert space for gluons,
\begin{equation}
H=\cases{H_0&$T<T_c,$\cr H_0\oplus H_{-1}\oplus H_1&$T>T_c$,}
\end{equation}
where the subscript stands for the triality, the center charge
\footnote{The wave functional
$\Psi[\vec A(\vec x)]\in H_\ell$ changes by the phase factor
$e^{2i\pi\ell n/3}$ when the global center gauge transformation
$e^{2i\pi n/3}$ is performed on $\vec A(\vec x)$. The multi-valued
nature of the wave functional is to keep track of the global center
gauge transformations, the elements of the fundamental group of the
gluonic configuration space which are represented in a trivial manner
on the gluon field.}.
Such a description of the phase transition is the resolution
of the puzzle mentioned in point 1.
(ii) The triality is permanently confined at any temperature.
The deconfined quark seen in the numerical simulation
is actually a composite particle containing a quark and its
vacuum polarization cloud. The latter has a multi-valued
wave functional in such a manner that the total (quark plus
gluon) wave functional is single valued. The triality charge
of the quark is screened by the unusual gluon state.
(iii) The color-magnetic monopoles relate the rotations
in the external and the color spaces. These monopoles
acquire a half-integer spin in the gluonic states with
multi-valued wave functional, a manner similar to the
generation of the spin for skyrmions. The
unusual gluonic screening cloud is the sum of
states with odd and even number of monopoles.
These components correspond to fermionic and bosonic
exchange statistics. Thus the state of a deconfined quark is
the sum of components with bosonic (odd number of monopoles)
and fermionic (even number of monopoles) properties. The breakdown
of the center symmetry leads to the mixing of the fermi and
bose statistics for the deconfined quarks.
\underbar{7. Triality-canonical ensemble:} The transition between the
canonical and the grand-canonical ensembles requires smooth enough
dependence on the density. Due to the confinement mechanism the formal energy density
diverges for non-integer baryon numbers, or non-vanishing triality
charges (point 4.). It turns out that the triality-canonical ensemble predicts
different center domain structure at the deconfining phase transition
than the usual grand-canonical ensemble~\cite{canon}. This may happen
because the center symmetry is broken
spontaneously by the quark-anti quark see for $T<T_c$ and dynamically
by the kinetic energy for $T>T_c$ in the canonical ensemble. This
furthermore means that the
formal center symmetry is preserved in the presence of dynamical quarks
and the results mentioned in this talk remain valid in the triality-canonical
ensemble with dynamical quarks.
\section{Density matrix for the A-B system and for gluons}
\underbar{A-B system:}
Consider a charged particle moving on the unit circle in periodic gauge
where the wave function is periodic, $\psi(\phi+2\pi)=\psi(\phi)$.
The hamiltonian is $H=(-i\partial_\phi-\Theta/2\pi)^2/2$, where
$\Theta=2\pi A_\phi$ stands for the magnetic flux of the circle.
The eigenstates and the eigenvalues are $\psi_n(\phi)=e^{in\phi}$,
and $E_n=(n-\Theta/2\pi)^2/2$, respectively. The density matrix
is given by
$\rho(\alpha,\beta)=Z^{-1}\sum_ne^{in(\alpha-\beta)-(n-\Theta/2\pi)^2/2T}$,
where $Z$ is the partition function, $Z=\sum_ne^{-(n-\Theta/2\pi)^2/2T}$.
Notice that the probability density $p(\phi)=\rho(\phi,\phi)$ is
real non-negative, as it should be. The periodicity of the
wave functions gives $\rho(\alpha,\alpha+2\pi)=\rho(\alpha,\alpha)$.
Let us go into an aperiodic gauge by performing the
transformation $\psi(\phi)\to e^{-i\phi\Theta/2\pi}\psi(\phi)$.
The hamiltonian is simpler, $H\to-\partial_\phi^2/2$, but has the
same spectrum as before because the wave functions are multi-valued,
$\psi(\phi+2\pi)=e^{-i\Theta}\psi(\phi)$. In particular,
the eigenvectors are
$\psi_n(\phi)=e^{i\phi(n-\Theta/2\pi)}$. The density matrix transforms as
$\rho(\alpha,\beta)\to e^{-i(\alpha-\beta)\Theta/2\pi}\rho(\alpha,\beta)$,
and becomes multi-valued, as well,
$\rho(\alpha,\alpha+2\pi)=e^{i\Theta}\rho(\alpha,\alpha)$ which
makes the construction of the probability density non-trivial.
In fact, the choice of different Riemann-sheets for the two
coordinate variables yields complex probability and partition
function. But notice that the complex factor is the same
for each contribution,
\begin{equation}
Z_{compl}=\int d\phi\rho(\phi,\phi+2\pi)=e^{i\Theta}
\int d\phi\rho(\phi,\phi),
\end{equation}
and the imaginary part of the entropy is an overall constant
which does not influence the thermalization and
thermodynamics can be applied.
\underbar{QCD:}
A similar argument can easily be constructed for gluons
yielding the following results: (i) The multi-valued
nature of the gluonic wave functional of a deconfined quark
is shown by the possible non-positive or complex expectation
value of the order parameter $\langle\omega\rangle$, the
partition function of a quark. (ii) The density matrix
for gluons and a deconfined quark is
multi-valued as it happens for the A-B system in the aperiodic gauge.
The change of the Riemann-sheet,
$\rho(\alpha,\alpha)\to\rho(\alpha,\alpha+2\pi)$,
corresponds to the center transformation. (iii) Thus the
complex part of the free energy and the entropy of
a deconfined quark is a simple kinematical constant which agrees for
each contribution to the partition function and does not
influence the thermalization and the applicability of the
rules of thermodynamics. (iv) The complex part of the deconfined quark
entropy may lead to observable effects in the triality-canonical
ensemble~\cite{mich} which is more realistic than the grand-canonical one.
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\section{Introduction and Notation}\label{sec:IntrNotMain}
The arithmetic-geometric-harmonic mean inequality together with minimum and maximum (which can be seen as the extreme means) states in the two-argument case
\begin{equation}\label{eq:means_of_numbers}
\min\{a,b\}\leq\left(\frac{a^{-1}+b^{-1}}{2}\right)^{-1}\leq \sqrt{ab}\leq \frac{a+b}{2}\leq\max\{a,b\}
\end{equation}
for any $a, b > 0$, with equality in any of the inequalities if and only if $a=b$ (see \cite{HLP,Sch}).
For any $X\subset\mathbb R^n$ let $\mathrm{conv}(X)$ denote the \cemph{convex hull}, i.e.,~the smallest convex set containing $X$. A \cemph{segment} is the convex hull of $\{x,y\} \subset \mathbb R^n$, which we abbreviate by $[x,y]$.
For any $X,Y \subset\mathbb R^n$, $\rho \in \mathbb R$ let $X+Y =\{x+y:x\in X,y\in Y\}$ be the \cemph{Minkowski sum} of $X$ and $Y$, and $\rho X= \{ \rho x: x \in X\}$ the \cemph{$\rho$-dilatation} of $X$. We abbreviate $(-1)X$ by $-X$. The family of all \cemph{convex bodies} (full-dimensional compact convex sets) is denoted by $\mathcal K^n$ and for any $C\in\mathcal K^n$ we write $C^\circ=\{a\in\mathbb R^n: a^T x \leq 1,\, x\in C\}$ for the \cemph{polar} of $C$.
All the means above can be generalized for convex sets. One may identify means of numbers by means of segments via associating $a, b > 0$ with $[-a,a]$ and $[-b,b]$. Thus, e.g., the arithmetic mean of $a$ and $b$ is identified with $[-\frac{1}{2} \left( a+b \right), \frac{1}{2} \left( a+b \right) ] = \frac{1}{2} \left( [-a,a]+[-b,b] \right)$. In general, the \cemph{arithmetic mean} of $K,C \in \mathcal K^n$ is defined by $\frac{1}{2} (K+C)$, the \cemph{minimum} by $K\cap C$, and the \cemph{maximum} by $\mathrm{conv}(K\cup C)$.
Since polarity can be regarded as the higher-dimensional counterpart of the inversion operation $x\rightarrow 1/x$ (cf.~\cite{MR}), the \cemph{harmonic mean} of $K$ and $C$ is defined by $\left( \frac{1}{2}(K^\circ+C^\circ) \right)^{\circ}$.
The geometric mean has been extended in several ways (cf.~\cite{BLYZ} or \cite{MR}). It would need a separate, more involved treatment. Here we focus only on the four other means. The study of means of convex bodies has been started by Firey in the 1960's \cite{F,F2,F3}, but there also exist several recent papers (see, e.g.,~\cite{MR,MR2,MMR}).
Perphaps the most essential result of Firey is the extension of the harmonic-arithmetic mean inequality from positive numbers to convex bodies with 0 in their interior in \cite{F} (see \cite{MR} for a nice and short proof). Moreover, Firey's inequality may again be extended involving minimum and maximum.
\begin{proposition} \label{prop:means_of_sets}
Let $C,K \in\mathcal K^n$ with $0$ in their interior. Then
\begin{equation}\label{eq:means_of_sets}
K\cap C\subset \left(\frac{K^\circ+C^\circ}{2}\right)^{\circ}\subset\frac{K+C}{2}\subset\mathrm{conv}(K\cup C),
\end{equation}
with equality between any of the means if and only if $K=C$.
\end{proposition}
In the following we analyze sharpness of the set-containment inequalities with respect to optimal containment (instead of equality of sets):
For any $C, K\in\mathcal K^n$ we say $K$ is \cemph{optimally contained} in $C$ and denote it by $K\subset^{opt}C$, if $K\subset C$ and $K\not \subset \rho C+t$ for any $\rho \in [0,1)$ and $t\in\mathbb R^n$. For $C_1, \dots, C_k \in\mathcal K^n$ we say $C_1 \subset \ldots \subset C_k$ is \cemph{left-to-right optimal} if $C_1 \subset^{opt} C_k$.
The starting point of our investigation is the following generalization of
\cite[Theorem 3]{BDG} for arbitrary convex sets with 0 in their interior.
\begin{thm}\label{thm:Charact_Opt_Means_KC}
Let $C,K \in \mathcal K^n$ with $0 \in \mathrm{int}(K \cap C)$.
Then
\[K \cap C \subset^{opt} \mathrm{conv}(K \cup C) \iff \left(\frac12 (K^\circ+C^\circ) \right)^{\circ}\subset^{opt} \frac12 (K+C).\]
\end{thm}
Note that Theorem \ref{thm:Charact_Opt_Means_KC} implies that left-to-right optimality in \eqref{eq:means_of_sets} depend solely on the optimal containment of the harmonic in the arithmetic mean.
If $C=-C+t$ for some $t \in \mathbb R^n$, we say $C$ is \cemph{symmetric}, and if $C=-C$, we say $C$ is \cemph{$0$-symmetric}.
The family of 0-symmetric convex bodies is denoted by $\mathcal K^n_0$.
A special focus in our study lies on optimal containments of means of $C$ and $-C$ of a convex body $C$, which are all symmetrizations of $C$.
Symmetrizations are frequently used in convex geometry, e.g.,~as extreme cases of a variety of geometric inequalities. Consider, e.g., the Bohnenblust inequality \cite{Bo}, which bounds the ratio of the circumradius and the diameter of convex bodies in arbitrary normed spaces. The equality case in this inequality is reached in normed spaces with $S \cap (-S)$ or $ \frac{1}{2} (S-S)$ as their unit balls \cite{BrK}, where $S$ denotes an $n$-simplex with center of gravity in 0. These means also appear in characterizations of spaces for which $C$ is complete or reduced
\cite[Prop.~3.5 -- 3.10]{BGJM}.
We provide more motivation on considering optimal containments between different symmetrizations of $C$ in the Appendix.
A major part of this paper is devoted to a better understanding of the optimal containments between those symmetrizations depending on the asymmetry of the initial body. We naturally require all symmetrizations of an already symmetric $C$ to coincide with $C$. This is always true for the arithmetic mean $\frac12(C-C)$, but $0$ needs to be the center of symmetry for the other three considered means. This indicates the need of fixing a meaningful center for every convex body. The most common choice of an asymmetry measure and a corresponding center are the \cemph{Minkowski asymmetry} of $C \in \mathcal K^n$, which is defined by
\[s(C):=\inf \{ \rho >0: C-c \subset \rho (C-c), c \in \mathbb R^n \},\] and the (not necessarily unique) \cemph{Minkowski center} of $C$, which is any $c \in \mathbb R^n$ fulfilling $C-c \subset s(C)(c-C)$ \cite{Gr, BG}. If $0$ is a Minkowski center, we say $C$ is \cemph{Minkowski centered}.
Note that $s(C)\in[1,n]$, with $s(C)=1$ if and only if $C$ is symmetric, and $s(C)=n$ if and only if $C$ is an $n$-dimensional simplex \cite{Gr}. Moreover, the Minkowski asymmetry $s:\mathcal K^n\rightarrow[1,n]$ is continuous w.r.t.~the Hausdorff metric (see \cite{Gr}, \cite{Sch} for some basic properties) and invariant under non-singular affine transformations.
We believe that the Minkowski asymmetry is most suitable for studying optimal containments and consequently focus on Minkowski centered convex sets.
The classical norm relations $\|x\|_\infty \leq \|x\|_2\leq \|x\|_1$ with $x\in\mathbb R^n$ can be naturally reversed
by the inequalities $\|x\|_1 \leq \sqrt{n}\|x\|_2 \leq n \|x\|_\infty$, which both transfer to left-to-right optimal containments between the corresponding unit ball of these $\ell_p$-spaces. Similarly, we consider the norms induced by the means of $K$ and $C$. Doing so, \eqref{eq:means_of_sets} can be read as follows:
\begin{equation}\label{eq:normrelations}
\|x\|_{\mathrm{conv}(K\cup C)} \leq \|x\|_{\frac{K+C}{2}} \leq \|x\|_{\left(\frac{K^\circ+C^\circ}{2}\right)^{\circ}} \leq \|x\|_{K\cap C}.
\end{equation}
In order to reverse this chain of inequalities, we need to provide a chain of (optimal) inclusions, which is reverse to \eqref{eq:means_of_sets}. This is not possible for general convex bodies, since the scaling factors of the reverse inclusions cannot be bounded in general. However, assuming Minkowski centeredness of the considered body, this problem can be fixed.
\begin{thm}\label{thm:reverse_inclusions}
Let
$C\in\mathcal K^n$ be Minkowski centered.
Then
\begin{enumerate}[(i)]
\item $\mathrm{conv}(C\cup(-C))\subset^{opt} s(C) (C\cap(-C))$,
\item $\mathrm{conv}(C\cup(-C))\subset^{opt} \frac{2s(C)}{s(C)+1} \frac{C-C}{2}$,
\item $\left(\frac{C^\circ-C^\circ}{2}\right)^\circ \subset^{opt} \frac{2s(C)}{s(C)+1}(C\cap(-C))$,
\item $\frac{C-C}{2}\subset^{opt} \frac{s(C)+1}{2} (C\cap(-C))$, and
\item $\mathrm{conv}(C\cup(-C))\subset^{opt} \frac{s(C)+1}{2}\left(\frac{C^\circ-C^\circ}{2}\right)^\circ$.
\item $\frac{C-C}{2}\subset \frac{s(C)+1}{2}\left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, and for all $s \in [n]$ there exists a Minkowski centered $C \in \mathcal K^n$ with $s(C)=s$, such that the containment is optimal.
\end{enumerate}
\end{thm}
After the proof of Theorem \ref{thm:reverse_inclusions} we will also provide an example that shows that the containment in Part (vi) above may not be optimal and derive a lower bound for the minimal dilatation factor needed for this covering.
As a consequence of Theorem \ref{thm:reverse_inclusions}, we derive the following left-to-right optimal containment chains.
\begin{cor} \label{rem:left-to-right-opt}
Let
$C \in \mathcal K^n$ be Minkowski centered
Then the following containment chains are both left-to-right optimal:
\begin{enumerate}[(i)]
\item $\mathrm{conv}(C \cup (-C)) \subset \frac{s(C)+1}{2} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ \subset s(C) (C \cap (-C))$, and
\item $\mathrm{conv}(C \cup (-C)) \subset \frac{2s(C)}{s(C)+1}\frac{C-C}{2} \subset s(C) (C \cap (-C))$.
\end{enumerate}
Moreover, for the following containment chains always apply:
\begin{enumerate}[(i)]
\item[(iii)] $\frac{C-C}{2} \subset \mathrm{conv}(C \cup (-C)) \subset \frac{s(C)+1}{2} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, and
\item[(iv)] $\frac{C-C}{2} \subset \frac{s(C)+1}{2} C \cap (-C) \subset \frac{s(C)+1}{2} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, and
\end{enumerate}
for every $s \in [n]$ there exist $C \in \mathcal K^n$ with $s(C)=s$, such that these chains are left-to-right optimal.
\end{cor}
Based on this corollary, one obtains, e.g., that the following reverse inequality chain of \eqref{eq:normrelations} is sharp w.r.t.~$s(C)$
\begin{equation}\label{eq:LtoR}
\|x\|_{C\cap (-C)} \leq \frac{s(C)+1}{2} \|x\|_{\frac{C-C}{2}} \leq s(C) \|x\|_{\mathrm{conv}(C\cup (-C))}. \end{equation}
Some containments of symmetrizations in the forward direction are always optimal (see \cite{BDG}):
\begin{equation*}
\frac{C-C}{2} \subset^{opt} \mathrm{conv} ( C \cup (-C) ) \quad\text{ and }\quad C \cap (-C) \subset^{opt} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ.
\end{equation*}
Using Proposition \ref{thm:Charact_Opt_Means_KC}, we see that \eqref{eq:means_of_sets} may be left-to-right optimal even for non-symmetric $C$.
In particular, considering a regular Minkowski centered simplex $S \in \mathcal K^3$, the four means are a cross-polytope (minimum), a rhombic dodecahedron (harmonic mean), a cube octahedron (arithmetic mean), and a cube (maximum) and they build a left-to-right optimal chain of containments (see Figure \ref{fig:symms-of-tetrahedron}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=2.5]
\draw[thick, dred,rotate around x=12, rotate around y=10](1,1,-1)--(-1,1,-1)--(-1,1,1);
\draw[thick, dred,rotate around x=12,rotate around y=10](1,1,-1)--(1,-1,-1)--(1,-1,1)--(-1,-1,1)--(-1,1,1);
\draw[thick, dred, rotate around x=12,rotate around y=10](1,-1,-1)--(-1,-1,-1)--(-1,1,-1);
\draw[thick, dred, rotate around x=12,rotate around y=10](-1,-1,-1)--(-1,-1,1);
\draw [fill,rotate around x=12,rotate around y=10] (0,0,-1) circle [radius=0.04];
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,1,0)--(-1,0,1);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,0,1)--(-1,-1,0)--(0,-1,1);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,-1,0)--(0,-1,1)--(1,-1,0);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,-1,0)--(0,-1,-1)--(1,-1,0);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,1,0)--(-1,0,-1)--(-1,-1,0);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,0,-1)--(0,-1,-1)--(1,0,-1);
\draw[thick, niceblue,rotate around x=12,rotate around y=10](-1,0,-1)--(0,1,-1)--(1,0,-1);
\draw[thick, rotate around x=12,rotate around y=10](0,-1,0)--(-1/2,-1/2,-1/2)--(0,0,-1)--(1/2,-1/2,-1/2)--(0,-1,0);
\draw[thick, rotate around x=12,rotate around y=10](0,1,0)--(-1/2,1/2,-1/2)--(0,0,-1)--(1/2,1/2,-1/2)--(0,1,0);
\draw[thick, rotate around x=12,rotate around y=10](-1/2,-1/2,-1/2)--(-1,0,0);
\draw[thick, rotate around x=12,rotate around y=10](1/2,-1/2,-1/2)--(1,0,0);
\draw[thick, rotate around x=12,fill=yellow, fill opacity=0.7, rotate around y=10](1,0,0)--(1/2,1/2,1/2)--(0,0,1)--(1/2,-1/2,1/2)--(1,0,0);
\draw[thick, rotate around x=12,fill=gold, fill opacity=0.7, rotate around y=10](0,1,0)--(1/2,1/2,1/2)--(0,0,1)--(-1/2,1/2,1/2)--(0,1,0);
\draw[thick, rotate around x=12,fill=yellow, fill opacity=0.7, rotate around y=10](-1,0,0)--(-1/2,1/2,1/2)--(0,0,1)--(-1/2,-1/2,1/2)--(-1,0,0);
\draw[thick, rotate around x=12,fill=gold, fill opacity=0.7, rotate around y=10](0,-1,0)--(1/2,-1/2,1/2)--(0,0,1)--(-1/2,-1/2,1/2)--(0,-1,0);
\draw[thick, rotate around x=12,fill=lgold, fill opacity=0.7, rotate around y=10](0,1,0)--(1/2,1/2,1/2)--(1,0,0)--(1/2,1/2,-1/2)--(0,1,0);
\draw[thick, rotate around x=12,fill=lgold, fill opacity=0.7, rotate around y=10](0,1,0)--(-1/2,1/2,1/2)--(-1,0,0)--(-1/2,1/2,-1/2)--(0,1,0);
\draw[thick, rotate around x=12,niceblue,rotate around y=10](-1,0,1)--(0,-1,1)--(1,0,1)--(0,1,1)--(-1,0,1);
\draw[thick,rotate around x=12,niceblue,rotate around y=10](-1,1,0)--(0,1,1)--(1,1,0)--(0,1,-1)--(-1,1,0);
\draw[thick,rotate around x=12,niceblue,rotate around y=10](1,0,1)--(1,-1,0)--(1,0,-1)--(1,1,0)--(1,0,1);
\draw[thick, rotate around x=12,dred, rotate around y=10](-1,1,1)--(1,1,1)--(1,-1,1);
\draw[thick, rotate around x=12,dred,rotate around y=10](1,1,1)--(1,-1,1);
\draw[thick, dred,rotate around x=12,rotate around y=10](1,1,1)--(1,1,-1);
\draw [fill,rotate around x=12,rotate around y=10] (1,0,0) circle [radius=0.04];
\draw [fill,rotate around x=12,rotate around y=10] (0,0,1) circle [radius=0.04];
\draw [fill,rotate around x=12,rotate around y=10] (-1,0,0) circle [radius=0.04];
\draw [fill,rotate around x=12,rotate around y=10] (0,1,0) circle [radius=0.04];
\draw [fill,rotate around x=12,rotate around y=10] (0,-1,0) circle [radius=0.04];
\end{tikzpicture}
\caption{Symmetrizations of a regular simplex $S \subset \mathbb R^3$: Minimum $S \cap (-S)$ is a cross-polytope (convex hull of black points), harmonic mean $\left(\frac{S^\circ-S^\circ}{2}\right)^\circ$ is a rhombic dodecahedron (yellow), arithmetic mean $\left(\frac{S-S}{2}\right)$ is a cube octahedron (blue), and maximum $\mathrm{conv}(S \cup (-S))$ is a cube (red) .
}
\label{fig:symms-of-tetrahedron}
\end{figure}
This property remains true for the four symmetrizations of a regular Minkowski centered simplex in any odd dimension. In contrast, for a regular Minkowski centered simplex $S$ in even dimensions we show in Lemma \ref{lem:Simplex_Odd} that
\[
S \cap (-S) \subset^{opt} \frac{n}{n+1} \mathrm{conv}( S \cup (-S) )\quad\text{and}\quad
\left(\frac{S^\circ-S^\circ}{2}\right)^\circ \subset^{opt}\frac{n(n+2)}{(n+1)^2} \frac{S-S}{2}.
\]
Concerning the above, we proceed with a stability result. First we introduce several parameters which we need throughout the upcoming results.
\begin{align*}
\psi&:=\psi(n,s) := \frac{(n-s+1)(s+1)}{1-n(n-s)(n+s(n+1))} - n, \\ \mu&:=\mu(n,s) = \frac{n+1}{s+1}\left( 1- \frac{s(n+1)(n-s)}{1-n(n-s)} \right), \\
\gamma_1&:=\gamma_1(n) := \frac12(n-1+\sqrt{(n-2)n+5}), \\
\gamma_2&:=\gamma_2(n):=\frac{n^4+n^3+2n^2+\sqrt{n^8+6n^7+17n^6+28n^5+28n^4+12n^3-4n^2-12n-4}}{2(n^3+2n^2+3n+1)},\\
\gamma_3&:=\gamma_3(n):=\frac{n^4+3n^3+2n^2+1+\sqrt{n^8+6n^7+13n^6+8n^5-14n^4-22n^3+8n+1}}{2(n^3+2n^2+2n)} .
\end{align*}
One can check that $n - \frac{1}{n} < \gamma_2 < \gamma_3 < n$ and that both $\psi$ and $\mu$ become 1 in case $n=s$. Moreover, we will see that $\psi \frac{n}{n+1} > 1$ for all $s > \gamma_2$, while $\mu \psi \frac{n(n+2)}{(n+1)^2} <1$ for all $s > \gamma_3$.
\begin{thm}\label{thm:minMax_mean_improved}
Let $n$ be even and $C\in\mathcal K^n$ be Minkowski centered with $s(C)=s$.
Then
\begin{enumerate}[(i)]
\item $\displaystyle C \cap (-C) \subset \psi \, \frac{n}{n+1} \mathrm{conv}(C\cup(-C))$, if $s \ge \gamma_2(n)$, and
\item $\displaystyle \left(\frac{C^\circ+(-C)^\circ}{2}\right)^{\circ} \subset \mu \psi \, \frac{n(n+2)}{(n+1)^2}
\frac{C-C}{2}$, if $s \ge \gamma_3(n)$.
\end{enumerate}
\end{thm}
One should recognize that the factor $\mu \psi \frac{n(n+2)}{(n+1)^2}$ in Part (ii) of Theorem \ref{thm:minMax_mean_improved} becomes greater than 1 for $s < \gamma_3(n)$. However, from Part (i) of Theorem \ref{thm:minMax_mean_improved} together with Theorem \ref{thm:Charact_Opt_Means_KC} we obtain that the harmonic mean of any pair of Minkowski centered convex bodies $C$ and $-C$ cannot be optimally contained in their arithmetic mean for any $s \in [\gamma_2,\gamma_3]$.
Whenever \eqref{eq:means_of_sets} is left-to-right optimal for some Minkowski centred convex body $C$ there also exist a series of Minkowski centered convex bodies with any smaller asymmetry providing a left-to-right optimality for the full chain (see Lemma \ref{lem:Asym_Descent_Chain}). Thus we aim to determine the smallest number $\gamma(n) \in [n-1,n]$ such that for every Minkowski centered $C\in\mathcal K^n$ with $s(C)\ge \gamma(n)$ the harmonic mean of $C$ and $-C$ is not optimally contained in their arithmetic mean.
We already introduced $\gamma(n)$ in \cite{BDG} as the \cemph{asymmetry threshold of means} and it is shown there that $\gamma(2) = \frac{1+\sqrt{5}}{2}=: \varphi $ is the golden ratio, while $\gamma(n)=n$ whenever $n$ is odd.
Here we present a result on the asymmetry threshold for arbitrary even dimensions.
\begin{thm}\label{thm:gamma} Let $n$ be even.
Then
\begin{equation*}
n-1 < \gamma_1 \leq \gamma(n) \leq \gamma_2<n.
\end{equation*}
\end{thm}
One may recognize the following: it is well-known that the golden ratio, which is also $\gamma(2)$, can be obtained from solving the equation $\frac{a+b}{a} = \frac{a}{b}$ for $a>b>0$. However, one can similarily obtain the values of $\gamma_1$ in Theorem \ref{thm:gamma} from solving the equation $\frac{(n-1)a+b}{a}=\frac{a}{b}$ and therefore consider the values of $\gamma_1$ as a generalized golden ratio.
The asymmetry threshold provides us with a lower bound for the values of $s$ such that \eqref{eq:means_of_sets} cannot be left-to-right optimal. In the following we want to go one step further and determine the possible values for the contraction factors $\alpha(s)$ and $\beta(s)$ for which the minimum is optimally contained in the according contraction of the maximum and for which the harmonic mean is optimally contained in the contraction of the arithmetic mean, respectively.
\begin{thm}\label{thm:small_asym_no_improve}
Let $C \in\mathcal K^n$ be Minkowski centered with $s(C)=s$.
\begin{enumerate}[a)]
\item Let $\alpha(s) \in \mathbb R$ such that $C \cap (-C) \subset^{opt} \alpha(s) \, \mathrm{conv}(C \cup (-C))$ and $\alpha_1(s)$, $\alpha_2(s)$ be the optimal lower and upper bounds on $\alpha(s)$, respectively. Then
\begin{enumerate}[(i)]
\item $\alpha_1(s) \ge \frac{2}{s+1}$ with equality at least for $s \le 2$.
\item $\alpha_2(s) = 1$ for
$s \le \gamma_1$, $\alpha_2(s) \le \psi \frac{n}{n+1}$,
for $s > \gamma_2$
and $\alpha_2(s) \ge \frac{s}{s^2-1}$ for $s \le 2$.
\end{enumerate}
\item Let $\beta(s) \in \mathbb R$ such that $\left(\frac12 (C^\circ - C^\circ)) \right)^{\circ}\subset^{opt} \beta(s) \, \frac12 (C-C)$ and $\beta_1(s), \beta_2(s)$ be the optimal lower and upper bounds on $\beta(s)$, respectively. Then
\begin{enumerate}[(i)]
\item
$\beta_1(s) \ge \frac{4s}{(s+1)^2}$ with equality at least for $s \le 2$.
\item $\beta_2(s) = 1$ for $s \le \gamma_1$, $\beta_2(s) \le \mu \psi \frac{n(n+2)}{(n+1)^2}$ for $s > \gamma_3$ and
$\beta_2(s) \ge \max \left\{ \frac{s}{s^2-1}, \frac{4s}{(s+1)^2} \right\}$ for $s \le 2$.
\end{enumerate}
\end{enumerate}
\end{thm}
Let us denote the \cemph{canonical basis} of $\mathbb R^n$ by $e^1,\dots,e^n\in\mathbb R^n$, the \cemph{Euclidean norm} of $x\in\mathbb R^n$ by $\|x\|$, and the \cemph{Euclidean unit ball} by $\mathbb{B}_2=\{x \in\mathbb R^n : \|x\|\leq 1\}$. For any $C,K \in \mathcal K^n$ the \cemph{Euclidean distance} is denoted by $d(C,K)$ and in case $C=\{p\}$ is a singleton, we abbreviate $d(\{p\},B)$ by $d(p,B)$. For any $C,K \in\mathcal K^n$ the \cemph{Banach-Mazur distance} between $K$ and $C$ is defined by $d_{BM}(K,C)=\inf\{\rho\geq 1:t^1 + K\subset L(C)\subset t^2+\rho K,\,L\in\mathrm{GL}(n), \, t^1,t^2\in\mathbb R^n\}.$ For every $X \subset \mathbb R^n$ let $\mathrm{bd}(X)$ and $\mathrm{int}(X)$ denote the \cemph{boundary} and \cemph{interior} of $X$, respectively.
For $C\in\mathcal K^n$ and $a\in\mathbb R^n$ let $\|x\|_C=\inf\{\rho>0:x\in\rho C\}$ be the \cemph{gauge function} of $C$ in $x$ and $h_C(a)=\sup\{a^Tx : x\in C\}$
be the \cemph{support function} of $C$ in $a$. Notice that $\|\cdot\|_C$ is a norm in the classic sense if and only if $C\in\mathcal K^n_0$ and remember that $\|x\|_{C}=h_{C^\circ}(x)$ for every $C\in\mathcal K^n$ and $x\in\mathbb R^n$ (see \cite{MR}).
For any $a\in\mathbb R^n \setminus \{0\}$
and $\rho\in\mathbb R$, $H^{\le}_{a,\rho} = \{x\in\mathbb R^n: a^Tx \leq \rho\}$ denotes the \cemph{halfspace} with outer normal $a$ and right-hand side $\rho$.
We say that the halfspace $H^{\le}_{a,\rho}$ \cemph{supports} $C \in\mathcal K^n$ at $q \in C$, if $C \subset H^{\le}_{a,\rho}$ and $q \in \mathrm{bd}(H^{\le}_{a,\rho})$. For any $C\in\mathcal K^n$ and $p\in \mathrm{bd}(C)$, the \cemph{outer normal cone} of $K$ at $p$ is defined as
$N(C,p) = \{ a \in \mathbb R^n : a^Tp \geq a^Tx \text{ forall } x \in C\}$.
For every $X\subset\mathbb R^n$ let us denote by $\mathrm{pos}(X)$, and $\mathrm{aff}(X)$ the
\cemph{positive} and \cemph{affine hull} of $X$, respectively, while
the \cemph{relative interior} of $X$ is denoted by $ \mathrm{relint} (X)$.
In case $u^1,\dots,u^{n+1}\in\mathbb R^n$ are affinely independent, we say that $\mathrm{conv}(\{u^1,\dots,u^{n+1}\})$ is an \cemph{$n$-simplex}.
\section{Preliminary results and lemmas}\label{sec:prelim_lemmas}
We recall the characterization of the optimal containment under homothety in terms of the touching conditions (see \cite[Theorem 2.3]{BrK}).
\begin{proposition}\label{prop:Opt_Containment}
Let $K,C\in\mathcal K^n$ and $K\subset C$. The following are equivalent:
\begin{enumerate}[(i)]
\item $K\subset^{opt}C$.
\item There exist $k\in\{2,\dots,n+1\}$, $p^j\in K\cap \mathrm{bd}(C)$, $u^j\in N(C,p^j)$, $j=1,\dots,k$, such that
$0\in\mathrm{conv}(\{u^1,\dots,u^k\})$.
\end{enumerate}
Moreover, if $K,C\in\mathcal K^n_0$, then (i) and (ii) are also equivalent to $K\cap\mathrm{bd}(C)\neq\emptyset$.
\end{proposition}
The next lemma shows that all the considered means are affine invariant.
\begin{lemma}\label{lem:Means_Invariant}
Let $K,C \in \mathcal K^n$ and $A$ be a non-singular affine transformation. Then
\begin{equation*}
\begin{split}
A(K)\cap A(C)=A(K\cap C), \qquad \left( ((A(K))^\circ-(A(C))^\circ)/2 \right)^\circ=A\left((K^\circ- C^\circ)/2
\right)^\circ, \\
(A(K)+A(C))/2=A\left( (K+C)/2\right), \qquad \mathrm{conv}\left(A(K)\cup(A(C)\right)=A\left(\mathrm{conv}(K\cup C)\right).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
From the fact that $A(C^\circ)=((A^{-1})^T(C))^\circ$ and since $A$ is non-singular, we get
\begin{equation*}
\begin{split}
\left(\frac{(A(K))^\circ-(A(C))^\circ}{2}\right)^\circ &=\left(\frac{(A^{-1})^T(K^\circ)-(A^{-1})^T(C^\circ)}{2}\right)^\circ \\
\left(\frac{(A^{-1})^T(K^\circ-C^\circ)}{2}\right)^\circ&=\left((A^{-1})^T \left( \frac{K^\circ-C^\circ}{2}\right) \right)^\circ= A \left( \left( \frac{K^\circ-C^\circ}{2}\right)^\circ \right) .
\end{split}
\end{equation*}
The other identities are trivially true.
\end{proof}
The next result is a straightforward corollary of Lemma \ref{lem:Means_Invariant}.
\begin{cor}\label{cor:invariant}
Let $C \in \mathcal{K}^n$ be Minkowski centered, $A \in \mathbb R^{n \times n}$ a regular linear transformation and $\alpha \in \mathbb R$. Then
\[
C \cap (-C) \subset^{opt} \alpha \cdot \textrm{conv} ( C \cup (-C) )
\]
if and only if
\[
A(C) \cap A(-C) \subset^{opt} \alpha \cdot \textrm{conv} ( A(C) \cup A(-C)).
\]
\end{cor}
The following proposition is an easy corollary out of Proposition \ref{prop:Opt_Containment}. It is a (variant of a) known result which in a more general version is given in \cite[(1.1)]{GrK} and we will use it to prove Lemma \ref{lem:Regard_of_Asym}.
\begin{proposition}\label{lem:polar}
Let $ C, K \in \mathcal K^n_0$.
Then $C \subset^{opt} K$ if and only if $K^{\circ} \subset^{opt} C^{\circ}$. Moreover, the touching points of $C$ to the boundary of $K$ become the outer normals of supporting halfspaces to the touching points of $K^\circ$ to the boundary of $C^\circ$ and vice versa.
\end{proposition}
Let us mention that while the containment in Proposition \ref{lem:polar} holds for any $C,K$ with $0$ in their interior, the optimality of this containment may in general be lost even in case of Minkowski centered $C$ and $K$ (see Figure \ref{fig:polar-nonopt}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.8]
\draw [thick, dred] (-3,-0.866) -- (-3,0.866) -- (3,0.866) -- (3,-0.866)-- (-3,-0.866) ;
\draw [thick, black] (-0.5,0.866) -- (1,0) -- (-0.5,-0.866) -- (-0.5,0.866);
\draw [thick, dgreen] (-0.33,0) -- (0,1.154) -- (0.33,0) -- (0,-1.154)-- (-0.33,0);
\draw [thick, dblue] (-2,0) -- (1,1.732) -- (1,-1.732) -- (-2,0);
\draw [fill] (0,0) circle [radius=0.01];
\draw (-0.1,-0.2) node {$0$};
\end{tikzpicture}
\caption{Minkowski centered $C$ (black) and $K$ (red), s.t.~$C \subset^{opt} K$ but $K^{\circ}$ (green) is not optimal contained in $C^{\circ}$ (blue).
}
\label{fig:polar-nonopt}
\end{figure}
As mentioned in the introduction, \eqref{eq:means_of_sets} is not left-to-right optimal for regular Minkowski centered simplices in even dimensions (while it is in odd dimensions). The following lemma prepares us to prove this fact in Lemma \ref{lem:Simplex_Odd}.
\begin{lemma}\label{lemma:opt_P_Ppolar}
Let $P\in\mathcal K^n_0$ be a polytope and
$v \in \mathrm{bd}(P)$ such that $v$ is also an outer normal of a closest facet of $P$ to the origin $0$, then
\[
P^\circ\subset^{opt}\frac{1}{\|v\|^2}P.
\]
\end{lemma}
\begin{proof}
Let $0<t_1\leq \cdots\leq t_m$ and $u^1,\dots,u^m\in \S^{n-1}$ be such that $P=\{x\in\mathbb R^n:|(u^i)^Tx|\leq t_i,\,i \in [m]\}.$
Then $P^\circ=\mathrm{conv}(\{\pm u^1/t_1,\dots,u^m/t_m\})$,
$t_1\mathbb{B}_2\subset^{opt}P$ and $t_1 u^1\in t_1\mathbb{B}_2\cap \mathrm{bd}(P)$.
Since $\frac1{t_1} u^1\in P^\circ\cap \mathrm{bd}(\frac1{t_1}\mathbb{B}_2)$, we have $P^\circ\subset^{opt} \frac1{t_1} \mathbb{B}_2\subset^{opt} \frac1{t_1^2} P$
and $\frac1{t_1} u^1$ is a common touching point of $P^\circ$ and $P$. Thus by part (iii) of Proposition \ref{prop:Opt_Containment}, we have $P^\circ\subset^{opt} \frac1{t_1^2}P$. Choosing $v = t_1u^1$ finishes the proof.
\end{proof}
We recall a stability result for the Banach-Mazur distance in the near-simplex case, given in \cite[Theorem 2.1]{Sch2}.
\begin{proposition}\label{prop:Schneider}
Let $S \in \mathcal K^n$ be an $n$-simplex and $C\in\mathcal K^n$ such that $s(C)=n-\varepsilon$ and $\varepsilon\in(0,\frac 1 n)$. Then
\begin{equation}\label{eq:Schne_Stabil}
d_{BM}(C,S)\leq 1+\frac{(n+1)\varepsilon}{1-n\varepsilon}.
\end{equation}
\end{proposition}
\section{Optimality in Firey's inequality chain}
As we mentioned in the introduction, two of the containments in Proposition \ref{prop:means_of_sets} are always optimal in case of the symmetrizations.
\begin{lemma}\label{lem:Regard_of_Asym}
Let $C \in \mathcal K^n$.
Then
\begin{enumerate}[(i)]
\item $\frac{C-C}{2} \subset^{opt} \mathrm{conv} ( C \cup (-C) )$,
\item $C \cap (-C) \subset^{opt} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, if also $0 \in C$.
\end{enumerate}
\end{lemma}
\begin{proof
We start proving (i). By \eqref{eq:means_of_sets}, we have $\frac{1}{2}(C-C) \subset \mathrm{conv}(C \cup (-C))$.
Now, there exists an extreme point $x$ of $C$ and an extreme point $y$ of $-C$, s.t.~$\mathrm{conv}(\{x,y\})$ is extreme in $\mathrm{conv}(C \cup (-C))$ and thus a halfspace $H^\le$ supporting $C$ and $-C$ in that edge.
This implies $\frac{1}{2}(x+y) \in (\frac{1}{2}(C-C))\cap \mathrm{bd}(\mathrm{conv}(C\cup(-C))$, and since $\frac{C-C}{2}$ and $\mathrm{conv} ( C \cup (-C) )$ are $0$-symmetric, Part
(iii) in Proposition \ref{prop:Opt_Containment} concludes the proof (i).
Since $0 \in C$, we have $C \cap (-C) \neq \emptyset$.
Applying (i) and Proposition \ref{lem:polar} implies (ii).
\end{proof}
Now we are ready for Lemma \ref{lem:Simplex_Odd}.
\begin{lemma}\label{lem:Simplex_Odd}
Let $S$ be a Minkowski centered regular $n$-simplex. Then
\begin{enumerate}[(i)]
\item $S \cap (-S) \subset^{opt} \mathrm{conv}( S \cup (-S) )$, if $n$ is odd,
\item $S \cap (-S) \subset^{opt} \frac{n}{n+1} \mathrm{conv}( S \cup (-S) )$, if $n$ is even, and
\item $\left(\frac{S^\circ-S^\circ}{2}\right)^\circ \subset^{opt}\frac{n(n+2)}{(n+1)^2} \cdot \frac{S-S}{2}$, if $n$ is even.
\end{enumerate}
\end{lemma}
\begin{proof
In order to simplify the calculations, we assume w.l.o.g.~that $S=\mathrm{conv}(\{p^1,\dots,p^{n+1}\})$ with $p^j \in \mathbb R^n$, such that $\|p^j \|=1$, $j \in [n+1]$.
\begin{enumerate}[(i)]
\item Let $n \geq 1$ be odd and $p= \frac{2}{n+1} \left( p^1+ \cdots+ p^{\frac{n+1}{2}} \right)$. Since $\sum_{i=1}^{n+1} p^i=0$, we have $-p= \frac{2}{n+1} \left( p^{\frac{n+1}{2}+1}+ \cdots+ p^{n+1} \right) \in S$
and thus $p \in S \cap (-S)$. Define
\begin{align*}
H_1^{\leq} &:= \left\{x \in \mathbb R^n: (p^1+ \dots+ p^{\frac{n+1}{2}})^T x \leq \frac{n+1}{2n} \right\},\\
H_2^{\leq} &:= \left\{x \in \mathbb R^n: (-p^1 - \dots - p^{\frac{n+1}{2}})^T x \leq \frac{n+1}{2n} \right\}.
\end{align*}
Then we obtain for $j=1,\dots,(n+1)/2$
\[
|(p^1+\cdots+p^{\frac{n+1}{2}})^T p^j| = \left|1 - \left(\frac{n+1}{2}-1\right) \frac1n \right| = \frac{n+1}{2n},
\]
and for $j=(n+3)/2,\dots,n+1$
\[
|(p^1+\cdots+p^{\frac{n+1}{2}})^T p^j| = \left|\left(\frac{n+1}{2} (-\frac1n) \right)\right| = \frac{n+1}{2n},
\]
and therefore $S\subset H_1^{\leq}\cap H_2^{\leq}$.
Moreover,
\[
|(p^1+ \dots+ p^{\frac{n+1}{2}})^T p | = \frac{n+1}{2n},
\]
which shows that $H_1^{\le}$ and $H_2^{\le}$ support $S$ at $p$ and $-p$, respectively. Hence Part (iii) of Proposition \ref{prop:Charact_Opt_Means} is fulfilled, proving the optimal containment of $S \cap (-S)$ in $\mathrm{conv}(S \cup (-S))$.
\item We start observing that
\[
S\cap (-S) =\left\{x \in \mathbb R^n : | (p^j)^T x | \leq \frac{1}{n}, j \in [n+1] \right\}.
\]
Now, let $p=\lambda_1 p^1+\cdots+\lambda_{n+1}p^{n+1} \in S$ be a vertex of $S\cap(-S)$ for some $\lambda_j \geq 0$, $j \in [n+1]$ with $\sum_{i=1}^{n+1}\lambda_i=1$.
Since $p$ is a vertex, there must be $n$ of the constraints $| (p^j)^T x | \leq \frac{1}{n}, j \in [n+1]$ active in $p$. Hence,
we may assume w.l.o.g.~that there exists $0\leq m\leq \frac n 2$, s.t.~$(p^j)^Tp = \frac1n$, $j=1,\dots, \frac n2 + m$ and $(p^j)^Tp = -\frac1n$, $j=\frac n2+m+1,\dots,n$. Thus
\[
\frac{1}{n}= (p^j)^Tp = \lambda_j - \frac{1}{n}(\lambda_1+\cdots+\lambda_{j-1}+\lambda_{j+1}+\cdots+\lambda_{n+1}) = \lambda_j - \frac{1}{n}(1-\lambda_j)
\]
for $j=1,\dots,\frac n2 + m$ and
\[
-\frac{1}{n} = (p^j)^Tp = \lambda_j - \frac{1}{n}(\lambda_1+\cdots+\lambda_{j-1}+\lambda_{j+1}+\cdots+\lambda_{n+1})=\lambda_j-\frac{1}{n}(1-\lambda_j)
\]
for $j=\frac n2 + m+1,\dots,n$. This implies $\lambda_j = \frac 2 {n+1}$ for $j=1,\dots,\frac n2 + m$ and $\lambda_j=0$ for $j=\frac n2+m+1,\dots,n$.
Hence, $0 \le \lambda_{n+1} = 1-\sum_{i=1}^{n} \lambda_i = 1 - \frac{n+2m}{n+1} = \frac{1-2m}{n+1}$, which shows that $m=0$ and
$p=\frac{2}{n+1}p^1+\cdots+\frac{2}{n+1}p^{\frac{n}{2}}+\frac{1}{n+1}p^{n+1}$.
We obtain
\[
\|p\|^2 = \frac{1}{(n+1)^2}\left(\left( \frac{n}{2} \cdot 4 + 1 \right) + \left(\frac{n}{2} \cdot \left( \frac{n}{2} -1 \right) \cdot 4 + \frac{n}{2} \cdot 2+ \frac{n}{2} \cdot 2 \right) \left( -\frac{1}{n} \right) \right)= \frac{1}{n+1}
\]
and therefore $(n+1)p \in \mathrm{bd}((S\cap (-S))^\circ)$. Finally, since $S^\circ=-nS$, we have
\[
(S\cap (-S))^\circ = \mathrm{conv}(S^\circ \cup (-S)^\circ) = n \cdot \mathrm{conv}(S \cup (-S)),\]
implying $\frac{n+1}{n} p \in \mathrm{bd}(\mathrm{conv}( S \cup (-S) ))$ and therefore
\[
S\cap (-S) \subset^{opt} \frac{n}{n+1} \mathrm{conv}( S \cup (-S) ).
\]
\item
Notice that the faces of a Minkowski sum are Minkowski sums of their faces and
\[
v := \frac12 \left(\sum_{i=1}^{\frac n 2} \frac{p^i}{\frac n 2} - \sum_{i= \frac n 2 + 1}^{n+1} \frac{p^i}{\frac n 2 + 1}\right) \in \mathrm{bd}\left( \frac{S-S}{2}\right)
\]
is also the outer normal of one of the facets of $\frac {S-S} 2$, which are the closest to the origin.
We compute $\|v\|^2$
\[
\begin{split}
\|v\|^2 & = \frac{1}{4}\left(\frac{n}{2} \frac{1}{(\frac{n}{2})^2} + \left(\frac{n}{2} + 1\right)\frac{1}{\left(\frac{n}{2}+1\right)^2} + \frac{n}{2}\left(\frac{n}{2}-1\right) \frac{1}{(\frac{n}{2})^2} \left(-\frac{1}{n}\right) \right. \\
& \left. + \left(\frac{n}{2}+1\right)\frac{n}{2} \frac{1}{(\frac{n}{2}+1)^2}\left(-\frac{1}{n}\right)
- 2 \frac{n}{2}\left(\frac{n}{2}+1\right) \frac{1}{\frac{n}{2}} \frac{1}{\frac{n}{2}+1} \left(-\frac{1}{n}\right) \right) \\
& =\frac{1}{4}\left(\frac{2}{n}+\frac{2}{n+2}-\frac{n-2}{n^2} - \frac{1}{n+2} + \frac{2}{n}\right) = \frac{(n+1)^2}{n^2(n+2)},
\end{split}
\]
Using Lemma \ref{lemma:opt_P_Ppolar} and the identity $S^\circ=-nS$ again, we obtain
\[
\left(\frac{S^\circ-S^\circ}{2}\right)^\circ=\frac1n\left(\frac{S-S}{2}\right)^\circ\subset^{opt}\frac{1}{n\|v\|^2}\frac{S-S}{2}=\frac{n(n+2)}{(n+1)^2}\frac{S-S}{2}.
\]
\end{enumerate}
\end{proof}
As mentioned in the introduction, Theorem \ref{thm:Charact_Opt_Means_KC} states that left-to-right optimality in \eqref{eq:means_of_sets} depends only on the optimal containment of the harmonic in the arithmetic mean.
\begin{proof}[Proof of Theorem \ref{thm:Charact_Opt_Means_KC}]
The forward direction directly follows from Proposition \ref{prop:means_of_sets}. Thus we only have to show the backward direction.
Let $\left(\frac{K^\circ+C^\circ}2 \right)^\circ \subset^{opt} \frac{K+C}2$. By Proposition \ref{prop:Opt_Containment} there exist $k\in\{2,\dots,n+1\}$, $p^j\in \mathrm{bd}\left(\left(\frac{K^\circ+C^\circ}2 \right)^\circ \right) \cap \mathrm{bd}\left(\frac{K+C}2 \right)$, $u^j\in N(\frac{K+C}2,p^j)$, $j \in [k]$, such that $0\in\mathrm{conv}(\{u^1,\dots,u^k\})$. Choose any $p=p^j$, $u=u^j$ with $j \in [k]$, and $\beta \in \mathbb R$, such that $H_{u,\beta}$ is the hyperplane supporting $\frac12 (K+C)$ in $p$.
Since $p \in \mathrm{bd}\left(\left(\frac{K^\circ+C^\circ}2 \right)^\circ \right) \cap \mathrm{bd}\left(\frac{K+C}2 \right)$, we have
\[
\left\|p \right\|_{\left(\frac{K^\circ+C^\circ}2 \right)^\circ} = \left\|p \right\|_{\frac{K+C}2} =1.
\]
Now,
on the one hand using $\frac{p}{\left\|p \right\|_{K}} \in K$ and $\frac{p}{\left\|p \right\|_{C}} \in C$, we see
\[
\frac{1}{2} \left( \frac{1}{\left\|p \right\|_{K}} + \frac{1}{\left\|p \right\|_{C}} \right) p \in \frac{K+C}{2},
\]
and therefore,
\[
\left\|p \right\|_{\frac{K+C}2} \leq \left( \frac{1}{2} \left( \frac{1}{\left\|p \right\|_{K}} + \frac{1}{\left\|p \right\|_{C}} \right) \right)^{-1}.
\]
On the other hand, since $h_{C^{\circ}} = \left\| \cdot \right\|_{C}$ (see \cite{Sch}), we have
\[
\frac12 \left(\left\|p \right\|_{K}+\left\|p \right\|_{C}\right) = \frac12 \left(h_{K^{\circ}}(p)+ h_{C^{\circ}}(p)\right) = h_{\frac{K^\circ+C^\circ}2} = \left\|p \right\|_{\left(\frac{K^\circ+C^\circ}2\right)^\circ}.
\]
Applying the arithmetic-harmonic mean inequality (for numbers - restating the main argument for Proposition \ref{prop:means_of_sets}) we obtain
\[
\left\|p \right\|_{\frac{K+C}2}
\leq \left( \frac{1}{2} \left( \frac{1}{\left\|p \right\|_{K}} + \frac{1}{\left\|p \right\|_{C}} \right) \right)^{-1}
\leq \frac12 \left(\left\|p \right\|_{K} + \left\|p \right\|_{C}\right)
= \left\|p \right\|_{\left(\frac{K^\circ+C^\circ}2\right)^\circ}.
\]
However, since $p \in \mathrm{bd} \left(\left(\frac{K^\circ+C^\circ}2 \right)^\circ \right) \cap \mathrm{bd} \left(\frac{K+C}2 \right)$, it follows that \[
\left( \frac{1}{2} \left( \frac{1}{\left\|p \right\|_{K}} + \frac{1}{\left\|p \right\|_{C}} \right) \right)^{-1}
= \frac12 \left(\left\|p \right\|_{K} + \left\|p \right\|_{C}\right).\]
This means that we have equality between the harmonic and arithmetic mean of $\|p\|_K$ and $\|p\|_C$, which implies
\[
\left\|p \right\|_{K}= \left\|p \right\|_{C}= \left\|p \right\|_{\frac{K+C}2} =1
\]
and as a direct implication
\[
\left\|p \right\|_{K \cap C} = \max \{\left\|p \right\|_{K}, \left\|p \right\|_{C}\}=1 .
\]
Now it suffices to show that $H_{u,\beta}$ also supports $\mathrm{conv}(K \cup C))$ at $p$. Assume that the latter is wrong. This would imply, that there exists $q \in K \setminus C$ or $q \in C \setminus K$ such that $u^T q > \beta$, say, w.l.o.g., $q \in K \setminus C$. However, this would imply $u^T \left(\frac{p+q}2\right) > \beta$, contradicting the fact that $H_{u,\beta}$ supports $\frac{K+C}2$. Hence $H_{u,\beta}$ supports also $\mathrm{conv}(K \cup C)$ at $p$.
All together we see that $p^j\in (K \cap C) \cap \mathrm{bd} \left(\mathrm{conv}(K \cup C)) \right)$, with $u^j \in N(\mathrm{conv}(K \cup C),p^j)$, $j \in [k]$, and $0 \in \mathrm{conv}(\{u^1,\dots,u^k\})$. Using Proposition \ref{prop:Opt_Containment} we obtain the optimal containment of $K \cap C$ in $\mathrm{conv}(K \cup C)$.
\end{proof}
The following proposition (see \cite[Theorem 1.3]{BDG}) is a direct application of Theorem \ref{thm:Charact_Opt_Means_KC} to $C$ and $-C$.
\begin{proposition} \label{prop:Charact_Opt_Means}
Let $C \in \mathcal K^n$ be such that $0 \in \mathrm{int}(C)$.
Then the following are equivalent:
\begin{enumerate}[(i)]
\item $C \cap (-C) \subset^{opt} \mathrm{conv}(C \cup (-C))$,
\item $\left(\frac12 (C^\circ-C^\circ)) \right)^{\circ}\subset^{opt} \frac12 (C-C)$,
\item there exist $p, -p \in \mathrm{bd}(C)$ and parallel halfspaces $H^{\le}_{a,\rho}$ and $H^{\le}_{-a,\rho}$ supporting $C$ at
$p$ and $-p$, respectively.
\end{enumerate}
\end{proposition}
In case the containment in \eqref{eq:means_of_sets} is left-to-right optimal for some Minkowski centered $C$ and $K=-C$, there also exist Minkowski centered bodies with an arbitrary smaller asymmetry providing left-to-right optimality in the full chain.
\begin{lemma}\label{lem:Asym_Descent_Chain}
Let $C \in \mathcal K^n$ be Minkowski centered. If
\[
C \cap (-C) \subset^{opt} \mathrm{conv}( C \cup (-C) ),
\]
then for every $s\in[1,s(C)]$ there exists a Minkowski centered $C_s \in \mathcal K^n$ with $s(C_s)=s$, such that
\[
C_s \cap (-C_s) \subset^{opt} \mathrm{conv}( C_s \cup (-C_s) ).
\]
\end{lemma}
\begin{proof}
Since $C$ is Minkowski centered, we obtain from Proposition \ref{prop:Opt_Containment} that there exist $p^1, \dots, p^k \in - \frac{1}{s(C)}C \cap \mathrm{bd}(C)$ with $k\in\{2,\dots,n+1\}$ and $u^j \in N(C,p^j)$, $j \in [k]$, s.t.~$0 \in \mathrm{conv}(\{u^1, \dots, u^k\})$. By Part (iii) of Proposition \ref{prop:Charact_Opt_Means} there also exist $p,-p\in (C\cap(-C)) \cap \mathrm{bd}(\mathrm{conv}(C\cup(-C)))$. For $t \in [0,1]$ let us define
\[K_t:=\mathrm{conv}\left(\{p^1, \dots, p^k, \alpha_t p^1, \dots,\alpha_t p^k,\pm p\}\right),\] with $\alpha_t:= -((1-t)s(C)+t)$.
One may recognize, that since $\alpha_0 = -s(C)$ and $\alpha_0 p^j\in C$, $j \in [k]$, we have $K_t \subset C$.
By the fact that $\pm p \in K_t$, we have $\pm p \in (K_t \cap (-K_t)) \cap \mathrm{bd}(\mathrm{conv}(K_t\cup(-K_t))$ for all $t \in [0,1]$. Hence, $K_t$ fulfills Part (iii) of Proposition \ref{prop:Charact_Opt_Means} and therefore the optimal containment.
Moreover, $K_t \subset \alpha_t K_t$ with $\alpha_t p^j \in K_t \cap \mathrm{bd}(\alpha_t K_t)$ and $-u^j\in N(\alpha_t K_t,\alpha_t p^j)$, $j \in [k]$. Thus by Proposition \ref{prop:Opt_Containment} $K_t$ is optimally contained in $\alpha_t K_t$, which shows that $s(K_t)= - \alpha_t = (1-t)s(C) + t \in [1,s(C)]$. Choosing $C_s:=K_{\frac{s(C)-s}{s(C)-1}}$ concludes the lemma.
\end{proof}
Using Proposition \ref{prop:Schneider} we now prove Theorem \ref{thm:minMax_mean_improved}.
\begin{proof}[Proof of Theorem \ref{thm:minMax_mean_improved}] First of all let $\rho=d_{BM}(C,S)$, where $S$ is a regular Minkowski centered $n$-simplex and $\varepsilon := n - s$. Since $\gamma_3 > \gamma_2 > n - \frac{1}{n}$, we see that $C$ is under the conditions of Proposition \ref{prop:Schneider}. Hence, we may use \eqref{eq:Schne_Stabil} to obtain
\begin{equation}\label{eq:rho}
\rho \leq 1+\frac{(n+1)\varepsilon}{1-n\varepsilon}=\rho_*.
\end{equation}
Let $S=\mathrm{conv}(\{p^1,\dots,p^{n+1}\})$ be a regular Minkowski centered $n$-simplex, i.e., $\|p^i-p^j\|=const$ for all $i \neq j$, $i,j \in [n+1]$ with $\|p^i\|=n$. Moreover, let $F_i=\mathrm{conv}(\{p^j : j \neq i\})$ be the facet of $S$ opposing $p^i$ and $L_i:=\mathrm{aff}(F_i)$ with $i \in [n+1]$.
Applying a suitable regular linear transformation $L$, we may assume $c^1+S\subset L(C)\subset c^2+\rho S$ for some $c^1,c^2 \in\mathbb R^n$.
Since by Corollary \ref{cor:invariant} $L(C)\cap(-L(C))\subset \gamma \cdot \mathrm{conv}(L(C)\cup(-L(C)))$ for some $\gamma>0$ is equivalent to $C\cap(-C)\subset \gamma \cdot\mathrm{conv}(C\cup(-C))$, we can replace w.l.o.g.~$C$ by $L(C)$ and assume
\begin{equation}\label{eq:cont}
c^1+S \subset C \subset c^2+\rho S.
\end{equation}
Let $\bar \mu \le 1$ be the minimal distance from $0$ to the facets of $c^1+S$, which is attained at $c^1+L_i$ for some $i\in[n+1]$. Since $C$ is Minkowski centered and $c^1+S \subset C$, we have $z:= c^1+p^i \in C$ and therefore $\frac {-z} {s(C)} = \frac{-z}{n-\varepsilon} \in C$.
Then
\[
\begin{split}
d(z,\frac{p^i}n+L_i) + \bar \mu & = d(z,\frac {p^i}n + L_i) + d(\frac {p^i}n + L_i,c^1+L_i)\\
& =d(z,c^1+L_i) = n+1
\end{split}
\]
and we obtain
\begin{equation} \label{eq:xi1}
\xi:=d(\frac{-z}{n-\varepsilon},\frac{p^i}{n}+L_i)=\frac{d(z,\frac{p^i}n + L_i)}{n-\varepsilon} = \frac{n+1-\bar \mu}{n-\varepsilon}.
\end{equation}
Now we also have that $\frac{-z}{n-\varepsilon} \in C \subset c^2+\rho S$.
Since
\[
d(c^1+L_i,c^2+\rho L_i)\leq (n+1)\rho-(n+1) = (n+1)(\rho-1),
\]
we see that
\begin{equation} \label{eq:xi2}
\begin{split}
\xi & \leq d(c^2+\rho L_i, \frac {p^i}n + L_i)\\
&\leq d(c^2+\rho L_i,c^1+L_i)+d(c^1+L_i,\frac{p^i}n + L_i)\\
& \leq (n+1)(\rho-1)+\bar \mu.
\end{split}
\end{equation}
Combining \eqref{eq:xi1} and \eqref{eq:xi2}, we obtain
\[
\frac{n+1-\bar \mu}{n-\varepsilon} \le (n+1)(\rho-1) +\bar \mu,
\]
which is equivalent to
\begin{equation}\label{eq:mu}
\bar \mu \geq \frac{n+1}{n+1-\varepsilon}(1-(n-\varepsilon)(\rho-1))=:\mu.
\end{equation}
Since $d(0,c^1+L_j) \geq \mu$ for every $j \in [n+1]$, this directly rewrites as
\begin{equation*}\label{eq:0_in_c^1_S}
0\in c^1+(1-\mu) S.
\end{equation*}
Now since
\[
0 \in c^1+(1-\mu) S \subset c^1+S \subset c^2+\rho S
\]
it holds $d(0,c^2+\rho L_j)\geq d(0,c^1+L_j)\geq \mu$ for every $j\in[n+1]$, which rewrites as
\begin{equation*}\label{eq:0_in_c^2_S}
0\in c^2+(\rho-\mu)S.
\end{equation*}
Moreover, using
\[
d(0,c^2+\rho L_j) \leq d(c^2+(\rho-\mu)p^j, c^2 + \rho L_j) = \rho + n(\rho-\mu)
\]
for every $j\in[n+1]$, thus
\begin{equation}\label{eq:c_2_S_in_S}
c^2+\rho S \subset (\rho+n(\rho-\mu)) S.
\end{equation}
Moreover, since $d(0,c^1+L_j)\geq \mu$ for every $j\in[n+1]$, then
\begin{equation}\label{eq:S_in_c^1_S}
\mu S \subset c^1+S.
\end{equation}
\begin{enumerate}[(i)]
\item
Combining \eqref{eq:c_2_S_in_S} and \eqref{eq:S_in_c^1_S} with (ii) of Lemma \ref{lem:Simplex_Odd}
directly imply
\[
\begin{split}
C\cap(-C) & \subset (c^2+\rho S)\cap(-c^2-\rho S)\\
& \subset (\rho+n(\rho-\mu))(S \cap(-S)) \\
& \subset \frac{n}{n+1} (\rho+n(\rho-\mu))\mathrm{conv}(S\cup(-S))\\
& \subset \frac{n}{n+1} \frac{(\rho+n(\rho-\mu))}{\mu}\mathrm{conv}((c^1+S)\cup(-c^1-S))\\
& \subset\frac{n}{n+1}\frac{(\rho+n(\rho-\mu))}{\mu}\mathrm{conv}(C\cup(-C)).
\end{split}
\]
Since $\rho+n(\rho-\mu)$ is increasing in $\rho$, we obtain
\[
C\cap(-C)\subset\frac{n}{n+1}\frac{(\rho_*+n(\rho_*-\mu))}{\mu}\mathrm{conv}(C\cup(-C)).
\]
Finally, from \eqref{eq:rho} and \eqref{eq:mu} (and remembering that $\varepsilon = n -s$) we obtain
\[
C \cap (-C) \subset \psi \frac{n}{n+1} \mathrm{conv}(C\cup(-C)),\] where $\psi =\frac{(n-s+1)(s+1)}{1-n(n-s)(n+s(n+1))} - n$.
Notice that solving $\psi \frac{n}{n+1} = 1$ becomes a quadratic in $s$ equation and the unique positive root has the expression
\[
\gamma_2 = \frac{n^4+n^3+2n^2+\sqrt{n^8+6n^7+17n^6+28n^5+28n^4+12n^3-4n^2-12n-4}}{2(n^3+2n^2+3n+1)}.
\]
Using Lemma \ref{lem:Asym_Descent_Chain}
we conclude that $\psi \frac{n}{n+1} < 1$, whenever $s(C) > \gamma_2$.
\item From \eqref{eq:cont}, \eqref{eq:c_2_S_in_S}, and \eqref{eq:S_in_c^1_S} we obtain
\[\mu S\subset c^1+S \subset C \subset c^2+\rho S \subset(\rho+n(\rho-\mu))S.\]
Thus, using (iii) of Lemma \ref{lem:Simplex_Odd} this shows
\[
\begin{split}
\left(\frac{C^\circ-C^\circ}{2}\right)^\circ & \subset\left(\frac{(c_2+\rho S)^\circ-(c_2+\rho S)^\circ}{2}\right)^\circ
\subset(\rho+n(\rho-\mu))\left(\frac{S^\circ-S^\circ}{2}\right)^\circ\\
& \subset \frac{n(n+2)}{(n+1)^2} (\rho+n(\rho-\mu))\frac{S-S}{2}\\
& =\frac{n(n+2)}{(n+1)^2} (\rho+n(\rho-\mu))\frac{(c_1+S)-(c_1+S)}{2}\\
& \subset \frac{n(n+2)}{(n+1)^2}(\rho+n(\rho-\mu))\frac{C-C}{2}
\end{split}
\]
and since $\rho+n(\rho-\mu)$ is increasing in $\rho$, we obtain
\[
\left(\frac{C^\circ-C^\circ}{2}\right)^\circ \subset \frac{n(n+2)}{(n+1)^2} (\rho_*+n(\rho_*-\mu))\frac{C-C}{2}.
\]
Finally, combining \eqref{eq:rho} and \eqref{eq:mu} and remembering that $\varepsilon=n-s$, we obtain
\[
\left(\frac{C^\circ-C^\circ}{2}\right)^\circ \subset \mu \psi \frac{n(n+2)}{(n+1)^2} \frac{C-C}{2}.
\]
Notice that $\mu \psi \frac{n(n+2)}{(n+1)^2} = 1$ also becomes a quadratic equation in $s$, which has a unique positive root of expression
\[
\gamma_3 = \frac{n^4+3n^3+2n^2+1+\sqrt{n^8+6n^7+13n^6+8n^5-14n^4-22n^3+8n+1}}{2(n^3+2n^2+2n)}.
\]
By Lemma \ref{lem:Asym_Descent_Chain}
we conclude that $\mu \psi \frac{n(n+2)}{(n+1)^2} < 1$, whenever
\[
s(C)>\frac{n^4+3n^3+2n^2+1+\sqrt{n^8+6n^7+13n^6+8n^5-14n^4-22n^3+8n+1}}{2(n^3+2n^2+2n)},
\]
which itself is greater than $n- \frac1n$.
\end{enumerate}
\end{proof}
We now provide the proof of Theorem \ref{thm:gamma}.
\begin{proof}[Proof of Theorem \ref{thm:gamma}]
From Theorem \ref{thm:minMax_mean_improved} we directly obtain
$\gamma(n) \le \gamma_2$ for even $n$.
In order to obtain a lower bound on $\gamma(n)$ in even dimensions, we provide a suitable family of sets with left-to-right optimal containment in \eqref{eq:means_of_sets} and asymmetry $\gamma_1 > n-1$ by extending the construction of the Golden House in \cite{BDG}. Note that this construction also holds in odd dimensions.
Let $S$ be a Minkowski centered regular $n$-simplex such that without loss of generality $S=\mathrm{conv}(\{p^1,\dots,p^{n+1}\})$ with $p^j \in \mathbb R^n$ and $\|p^j \|=1$, $j \in [n+1]$. Note that $(p^i)^Tp^j=-1/n$ for $i \not =j$. Now we define $C=\mathrm{conv}(\{p^1,\dots,p^{n+1}\})\cap H^{\pm}$ with $H^{\pm} := \{ x \in \mathbb R^n : \pm (p^1-p^2)^Tx \leq \eta \}$, where $\eta=(p^1-p^2)^T p \in (0,1+\frac1n)$ and $p=(1-\lambda)p^1+\lambda u^2 \in \mathrm{bd}(H^+)$ for some $\lambda \in[0,1]$.
Then $\eta = 1 - \lambda - \lambda + \frac{1 - \lambda}{n} - \frac{\lambda}{n}$ and therefore
\begin{equation}\label{lemma3.6:lambda}
\lambda=\frac{1+\frac1n-\eta}{2(1+\frac1n)}.
\end{equation}
Because of the symmetry of $C$ with respect to the axis orthogonal to $\mathrm{aff}(p^1,p^2)$, there exist $\nu \in \mathbb R$ and $s \in [1,n]$ such that
\begin{equation*}
\nu(p^1+p^2)-\frac1sC \subset^{opt} C,
\end{equation*}
which can be rewritten as
\begin{equation}\label{lemma3.6:c}
-\frac1s(C-c) \subset^{opt} C-c \quad \text{for} \quad c=\frac{s}{s+1}\nu(p^1+p^2),
\end{equation}
such that $c$ is the Minkowski center of $C$.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=1.5]
\draw [thick] (0,1.61) -- (1,0)-- (1,-1)-- (-1,-1)-- (-1,0)--(0,1.61);
\draw [thick, dashed, rotate around={180:(0,0)}, scale=0.615] (0,1.61) -- (1,0)-- (1,-1)-- (-1,-1)-- (-1,0)--(0,1.61);;
\draw [thick, gray] (1,-1.2)-- (1,0.2);
\draw [thick, gray] (-1,-1.2)-- (-1,0.2);
\draw [thick, dashed, gray] (1,0)-- (1.6,-1);
\draw [thick, dashed,gray] (1.6,-1)-- (1,-1);
\draw [thick, dashed, gray] (-1,0)-- (-1.6,-1);
\draw [thick, dashed,gray] (-1.6,-1)-- (-1,-1);
\draw [fill] (0,0) circle [radius=0.01];
\draw [fill] (1,0) circle [radius=0.02];
\draw [fill] (1,-1) circle [radius=0.02];
\draw [fill] (-1,-1) circle [radius=0.02];
\draw [fill] (-1,0) circle [radius=0.02];
\draw [fill] (0,1.61) circle [radius=0.02];
\draw [fill] (0.61,0) circle [radius=0.02];
\draw [fill] (-0.61,0) circle [radius=0.02];
\draw [fill] (0,-1) circle [radius=0.02];
\draw [fill] (0.61,0.61) circle [radius=0.02];
\draw [fill] (-0.61,0.61) circle [radius=0.02];
\draw [fill] (-1.6,-1) circle [radius=0.02];
\draw [fill] (1.6,-1) circle [radius=0.02];
\draw (-0.1,-0.1) node {$c$};
\draw (1.4,0.4) node {$\mathrm{bd}(H^+)$};
\draw (-1.4,0.4) node {$\mathrm{bd}(H^-)$};
\draw (0,1.85) node {$p^3$};
\draw (-1.8,-1.1) node {$p^2$};
\draw (1.8,-1.1) node {$p^1$};
\draw (-1.5,0.8) node {$\nu (p^1+p^2)- \frac{p}{s}$};
\draw (1.2,-1.2) node {$p$};
\draw (1.2,-0.02) node {$q$};
\end{tikzpicture}
\caption{Construction from the proof of Theorem \ref{thm:gamma} for $n=2$: \\ $C$ (black), $-\frac1s C$ (dashed) and $\mathrm{bd}(H^+)$, $\mathrm{bd}(H^-)$ (gray dashed).
}
\end{figure}
Since $\nu(p^1+p^2)-\frac{p}{s}$ and $\nu(p^1+p^2)-\frac{p^3}{s}$ belong to the facets of $S$ with outer normals $-p^1$ and $-p^3$, respectively, we obtain
\[
\left( \nu(p^1+p^2)-\frac{p}{s}\right)^T(-p^1) =\frac1n
\quad\text{ and } \quad
\left( \nu(p^1+p^2)-\frac{p^3}{s} \right)^T (-p^3) =\frac1n.
\]
The latter two conditions translate into
\[
-\left(\nu-\frac{1-\lambda}{s}\right)+\left(\nu-\frac{\lambda}{s}\right)\frac1n=\frac1n
\quad\text{ and } \quad
\frac{2\nu}{n}+\frac1s = \frac1n,
\]
which can be simplified to
\begin{equation}\label{lemma3.6:mu}
s=n-2\lambda \quad \text{ and } \quad \nu=\frac{\lambda}{2\lambda-n}.
\end{equation}
Inserting \eqref{lemma3.6:lambda} we obtain
\begin{equation}\label{lemma3.6:mu1a}
s=\frac{n- \frac1n+\eta}{1+\frac1n} \quad \text{ and } \quad
\nu=\frac{1+\frac1n - \eta}{2 \left( \frac1n -n- \eta \right)}.
\end{equation}
Next, let $q=c+\xi(p^1-p^2)$, with $\xi>0$ such that $q \in \mathrm{bd}(H^+)$, which belongs to the facet of $S$ with outer normal $-p^2$.
Using $(p^1+p^2)^T(p^1-p^2)=0$, $(p^1-p^2)^T(p^1-p^2)=2 \left( 1+ \frac1n \right)$ and \eqref{lemma3.6:c}, this implies
\begin{equation} \label{eq:insert-nu}
\left(\frac{s}{s+1}\nu+\xi\right)\frac1n-\left(\frac{s}{s+1}\nu-\xi\right) = \frac1n \quad \text{and} \quad \eta = 2 \xi \left(1+\frac1n\right).
\end{equation}
Inserting $\eta$ from \eqref{eq:insert-nu} into \eqref{lemma3.6:mu1a} leads to
\[
s = n - 1 + 2\xi \quad \text{ and } \quad
\nu=\frac{1+\frac1n - 2\xi\left(1+\frac1n\right)}{2 \left( \frac1n -n- 2\xi \left(1+\frac1n\right)\right)}=\frac{1-2\xi }{1-2\xi-n}.
\]
and inserting this result for $\nu$ in \eqref{eq:insert-nu} gives us
\begin{equation*}
\left(\frac{s}{s+1}\frac{1-2\xi }{1-2\xi-n}+\xi\right)\frac1n-\left(\frac{s}{s+1} \frac{1-2\xi }{1-2\xi-n}-\xi\right) = \frac1n,
\end{equation*}
which one can solve for $\xi$ and with it for $s$ to obtain
\[
\xi=\frac{1-n+\sqrt{(n-2)n+5}}{4} \quad \text{ and } \quad s=n+2\xi-1=\frac{n-1+\sqrt{(n-2)n+5}}{2} = \gamma_1.
\]
Since condition (iii) of Proposition \ref{prop:Charact_Opt_Means} is fulfilled
for the Minkowski centered $C-c$ at the points $\pm \xi(p^1-p^2)$, \eqref{eq:means_of_sets} is optimal for $C-c$ and $c-C$,
and $s(C-c)=\gamma_1$, as desired.
Finally, by Lemma \ref{lem:Asym_Descent_Chain} we see that for
$s \le \gamma_1$ there exists a Minkowski centered $C\in\mathcal K^n$ such that $C\cap(-C)\subset^{opt}\mathrm{conv}(C\cup(-C))$, proving
$\gamma(n) \ge \gamma_1$.
\end{proof}
\section{Reverse containment}\label{sec:reverse_inclusions}
In this section we prove Theorem \ref{thm:reverse_inclusions}. While the proof of Parts (i)-(v) is straightforward, understanding (vi) needs some additional effort: on the one hand, we show that $C=S\cap(-sS)$, where $S$ is a Minkowski regular simplex, provides optimality in (vi) for each $s\in[1,n]$, while on the other hand, we find a more intriguing family of sets not fulfilling that optimality (see Example \ref{ex:notopt}).
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=3]
\draw [thick, black] (-0.11547,0.8) -- (0.11547,0.8) -- (0.75038,-0.2997)-- (0.6348,-0.5)-- (-0.6348,-0.5)-- (-0.75038,-0.2997)--(-0.11547,0.8);
\draw [thick, dashed] (0.11547,-0.8) -- (-0.11547,-0.8) -- (-0.75038,0.2997)-- (-0.6348,0.5)-- (0.6348,0.5)-- (0.75038,0.2997)--(0.11547,-0.8);
\draw [thick, dred] (0.75038,-0.2997) -- (0.6348,-0.5)-- (0.11547,-0.8) -- (-0.11547,-0.8)--(-0.6348,-0.5)-- (-0.75038,-0.2997)--(-0.75038,0.2997)-- (-0.6348,0.5)--(-0.11547,0.8) -- (0.11547,0.8)--(0.6348,0.5)-- (0.75038,0.2997)--(0.75038,-0.2997);
\draw [thick, dblue] (0.75038,0) -- (0.375135,-0.65) -- (-0.375135,-0.65)-- (-0.75038,0) -- (-0.375135,0.65)-- (0.375135,0.65)-- (0.75038,0);
\draw [thick, fill=lgold, fill opacity=0.7] (0.5774,0.225)--(0.5774,-0.225) --(0.4887,-0.3849) -- (0.089,-0.6158) -- (-0.089,-0.6158) -- (-0.4887,-0.3849) -- (-0.5774,-0.225) --(-0.5774,0.225) --(-0.4887,0.3849) --(-0.089,0.6158) --(0.089,0.6158) --(0.4887,0.3849)--(0.5774,0.225);
\draw [thick, black] (0.5774,0) -- (0.28885,-0.50035) -- (-0.28885,-0.50035) -- (-0.5774,0) -- (-0.28885,0.50035) -- (0.28885,0.50035) -- (0.5774,0);
\draw [fill] (0,0) circle [radius=0.01];
\draw [fill] (0.5774,0) circle [radius=0.02];
\draw [fill] (-0.28885,0.50035) circle [radius=0.02];
\draw [fill] (-0.28885,-0.50035) circle [radius=0.02];
\draw [fill] (-0.5774,0) circle [radius=0.02];
\draw [fill] (-0.28885,0.50035) circle [radius=0.02];
\draw [fill] (0.28885,0.50035) circle [radius=0.02];
\draw [fill] (0.28885,-0.50035) circle [radius=0.02];
\draw (-0.07,-0.07) node {$0$};
\end{tikzpicture}
\caption{$C=S \cap (-s S)$ (where $S$ is a regular triangle and $s=s(C)=1.5$, c.f.~Remark \ref{ex:partial-symm-of-simplex}) (black), $-C$ (dashed), $C \cap (-C)$ (convex hull of black points), $\mathrm{conv} (C \cup (-C))$ (red), $\frac{C-C}{2}$ (blue), and $\left(\frac{C^\circ+(-C)^\circ}{2}\right)^{\circ}$ (yellow).
}
\end{figure}
\begin{remark}\label{ex:partial-symm-of-simplex}
Let $C=S\cap(-sS)$, where $S=\mathrm{conv}(\{p^1,\dots,p^{n+1}\})$ with $\|p^i\|=1$, $i \in [n+1]$, is a regular simplex centered at $0$, and $s\in[1,n]$. Notice that
$C=S\cap(-sS)\subset s^2S\cap(-sS)=-sC$.
Let $G_i=\{x\in S:(p^i)^T x = -\frac 1 n\}$ and $F_i=S\cap(-sG_i)$ with $i\in[n+1]$, then
$G_i \subset \mathrm{bd}(C)$ and therefore $F_i\subset \mathrm{bd}(-sC)$.
Moreover, the points $\frac s n p^i$ belong to $C \cap \mathrm{bd}(-sC)$, $i \in [n+1]$, with $p^i$ being a normal vector of $-sC$ in $\frac s n p^i$. Thus by Proposition \ref{prop:Opt_Containment} we conclude that $C \subset^{opt}-sC$ and therefore that $C$ is Minkowski centered and $s(C)=s$.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:reverse_inclusions}]
We begin the proof by showing
containments in Parts (i) - (v).
For Part (ii) notice that $-C \subset sC$ directly implies $(s+1)(-C) \subset s(C-C)$, and therefore $-C \subset \frac{2s}{s+1} \frac{C-C}{2}$. Using the $0$-symmetry of $\frac {C-C}2$, we obtain $\mathrm{conv}(C\cup(-C)) \subset \frac{2s}{s+1} \frac{C-C}{2}$.
Part (iii) now follows directly from Part (ii) by using Proposition \ref{lem:polar}.
Since $-C\subset sC$, we have $\frac{C-C}{2} \subset \frac{s+1}{2}C$. From the $0$-symmetry of $\frac{C-C}{2}$, we obtain $ \frac{C-C}{2} \subset \frac{s+1}{2} (C\cap(-C))$, which yields Part (iv).
Part (v) then follows from Part (iv) by using Proposition \ref{lem:polar} again.
Finally, notice that Parts (ii) and (iv) together directly yield Part (i).
We now show that the containment in (i) is optimal for every $C$. Indeed,
let $p\in -C\cap \mathrm{bd}(sC)$. Let $H$ be a hyperplane supporting $sC$ at $p$, then $H'$, which is the translation of $H$ with
$p/s\in H'$ supports $-C$ at $p/s$, from which $p\in\mathrm{bd}(\mathrm{conv}(C\cup(-C)))$. Moreover, since $p\in s C\cap(-C)$ and $p\in\mathrm{bd}(sC)$, then $p\in \mathrm{bd}(s(C\cap(-C)))$, and thus Proposition \ref{prop:Opt_Containment} yields the result.
Since Part (i) shows optimality for every $C$, and since either joining Parts (ii),(iv) or (iii),(v) recovers (i), each of the Parts (ii)-(v) must be optimal for every $C$.
The proof of Part (vi) is a bit more subtle.
Let $\frac{C-C}{2} \subset^{opt} \alpha \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$ for some $\alpha \ge 1$.
Using the polarity of gauge and support function and the linearity of the latter with respect to Minkoswki addition, it holds
$\|z \|_{\left(\frac{C^\circ-C^\circ}{2}\right)^\circ} = h_{\frac{C^\circ - C^\circ}{2}}(z) = \frac12 \left(h_{C^\circ}(z) + h_{-C^\circ}(z)\right) = \frac12 (\|z \|_C+ \|z \|_{-C})$. From this and Part (iv) we obtain
\begin{align} \label{eq:AMinHMupper}
\alpha &= \max_{z \in \frac{C-C}{2}}
\|z \|_{\left(\frac{C^\circ-C^\circ}{2}\right)^\circ} = \max_{z \in \frac{C-C}{2}} \frac12 \left( \|z \|_C+ \|z \|_{-C} \right) \\ & \leq \max_{z \in \frac{C-C}{2}} \max\left\{\|z \|_C,\|z \|_{-C}\right\} = \max_{z \in \frac{C-C}{2}} \|z\|_{C \cap (-C)} = \frac{s+1}{2}. \nonumber
\end{align}
Now let $C=S\cap(-sS)$ for $s\in[1,n]$ as given in Remark \ref{ex:partial-symm-of-simplex} and let $q^{n+1}$, $q^n$
be the centers of the $(n-2)$-dimensional facets
of $F_{n+1}$ and $F_{n}$, respectively, which do not contain a vertex belonging to the line segment $[p^n,p^{n+1}]$.
Now let $v$ be a vertex of $F_{n+1}$ with the outer normal $p^{n+1}$. Then $v$ belongs to an edge connecting $p^{n+1}$ with
$\mathrm{conv}(\{p^1,\dots,p^{n}\})$. Thus, $v=(1-\lambda)p^i+\lambda p^{n+1}$ for some $\lambda\in[0,1]$ and $i \neq n+1$.
Since $v \in F_{n+1}$, we know by Remark \ref{ex:partial-symm-of-simplex} that
\[
\frac{s}{n}=((1-\lambda)p^i+\lambda p^{n+1})^Tp^{n+1}=(1-\lambda)\frac{-1}{n}+\lambda,
\]
i.e.~$\lambda=\frac{s+1}{n+1}$. Thus, the vertices of $F_{n+1}$ are $\frac{n-s}{n+1}p^i+\frac{s+1}{n+1}p^{n+1}$, $i \in [n]$ and
therefore
\[
\begin{split}
q^{n+1} & =\frac{1}{n-1}\sum_{i=1}^{n-1}\left(\frac{n-s}{n+1}p^i+\frac{s+1}{n+1}p^{n+1}\right) \\
& = \frac{1}{n-1}\left(\frac{(n-1)(s+1)}{n+1}p^{n+1}+\frac{n-s}{n+1}(-p^n-p^{n+1})\right) \\
& = \frac{1}{n^2-1}((ns-1)p^{n+1}-(n-s)p^n),
\end{split}
\]
where we have used that $\sum_{i=1}^{n-1}p^i=-p^n-p^{n+1}$.
For the same reasons
\[
-q^n=\frac{1}{n^2-1}((n-s)p^{n+1}-(ns-1)p^n).
\]
Now, let $z := \frac{q^{n+1}-q^n}{2} \in\frac{C-C}{2}$. Then
\[
\begin{split}
z &= \frac{1}{2(n^2-1)}\left((ns+n-s-1)p^{n+1}-(ns+n-s-1)p^n\right) =\frac{s+1}{2(n+1)}(p^{n+1}-p^n).
\end{split}
\]
Moreover, one should notice that when $s=1$
\[
q^{n+1} = -q^{n} = \frac{1}{n+1}(p^{n+1}-p^n) \in \mathrm{bd}\left(\frac{(S \cap (-S)) - (S \cap (-S))}{2}\right)=\mathrm{bd}(S \cap (-S)).
\]
Since we also have $S \cap (-S) = C \cap (-C)$, independently of our choice of $s\in[1,n]$, it follows
\[
\|z\|_{C}=\|z\|_{-C}=\|z\|_{C\cap(-C)}=\|z\|_{S \cap (-S)}=\frac{s+1}{2}\left\|\frac{1}{n+1}(p^{n+1}-p^n)\right\|_{S \cap (-S)}=\frac{s+1}{2},
\]
which shows equality in \eqref{eq:AMinHMupper} and thus that $\frac{C-C}{2} \subset^{opt} \frac{s+1}{2} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, concluding the proof.
\end{proof}
Note that only in Part (vi) of Theorem \ref{thm:reverse_inclusions} the containment may not always be optimal. Below we give two examples: the first one shows that at least in 2-space all regular $k$-gons achieve optimality, while the second one provides a construction of sets in arbitrary dimensions where the containment is not optimal.
\begin{example}
Let $C\subset\mathbb R^2$ be a Minkowski centered regular $k$-gon with odd $k$.
Then
\[
\frac{C-C}{2}\subset^{opt} \frac{s(C)+1 }{2}\left(\frac{C^\circ-C^\circ}{2}\right)^\circ.
\]
\end{example}
\begin{proof}
Let $r(C)$, $R(C)$ be the in- and circumradius of $C$ in the Euclidean distance, respectively. Assume w.l.o.g.\ that $R(C)=1$.
Since the Minkowski center of $C$ coincides with the in- and circumcenter, we can easily conclude that $s(C)=\frac{R(C)}{r(C)}$.
Now, since for any $k$-gon $r(C)=R(C) \cos \left(\frac{ \pi}{k} \right)$, it follows
\[
s:=s(C)=\frac{R(C)}{R(C) \cos \left(\frac{ \pi}{k} \right)}= \frac{1}{\cos \left(\frac{ \pi}{k} \right)}.
\]
We choose a vertex $p= \frac{u-v}{2}$ of $\frac{C-C}{2}$ , where $u$ and $v$ are vertices of $C$. Assume w.l.o.g.\ that $\|u\|=\|v\|=1$. Then since $C$ is Minkowski centered, $C^\circ =\rho (-C)$ for some $\rho >0$. Note that $R(C)=1$, thus $r(C^\circ)=1$ and $\rho=s$.
Since $C^\circ$ is again a regular $k$-gon, the distance from the origin to any edge
of $\left(\frac{C^\circ-C^\circ}{2} \right)^{\circ}$ is the same.
Using $C^\circ =s (-C)$, we have that $w=s p$ is a vertex of $\frac{C^\circ-C^\circ}{2}$. Moreover, $\{ x \in \mathbb R^n : w^T x =\|w\|^2 \}$ determines an edge of $\left(\frac{C^\circ-C^\circ}{2}\right)^\circ$ with outer-normal vector $\frac{w}{\|w \|^2}$. This implies $\frac{C^\circ-C^\circ}{2} \subset^{opt} \| w \|^2 \left(\frac{C^\circ-C^\circ}{2}\right)^\circ.$
Since $\cos \left(\frac{ \pi}{k} \right) =\frac{1}{s}$ and $R(C)=1$, we get
\begin{align*}
\| w \|^2=
\frac{ s^2}{4} \| u-v \|^2 &= \frac{ s^2}{4} \left( \| u\|^2+\|-v \|^2 +2 u^T(-v) \right) = \frac{ s^2}{4} \left( 2 R(C)^2+ 2R(C)^2 \cos \left(\frac{ \pi}{k} \right) \right) \\
&=\frac{ s^2}{2} \left( 1+ \cos \left(\frac{ \pi}{k} \right) \right)
= \frac{ s^2}{2} \left( 1+ \frac{ 1}{s} \right).
\end{align*}
Therefore,
\[
\frac{C-C}{2} = \frac{1}{s} \frac{C^\circ-C^\circ}{2} \subset^{opt} \frac{1}{s} \frac{ s^2}{2} \left( 1+ \frac{ 1}{s} \right) \left(\frac{C^\circ-C^\circ}{2}\right)^\circ=\frac{s+1}{2} \left(\frac{C-C}{2}\right)^\circ .
\]
\end{proof}
\begin{example}\label{ex:notopt}
Note that here we provide a construction only for $n=2$. For the higher dimensional case one can simply embed the construction below into the according space, keeping the Minkowski center 0. This keeps its asymmetry value and the same factor for the containment of the arithmetic mean within the harmonic mean. Thus, the construction essentially provides a family of sets in arbitrary dimensions with asymmetry $s \in (1,2)$ (only), such that the arithmetic mean is contained in the interior of the harmonic mean scaled by $\frac{s+1}2$.
Let $K=S\cap(-sS)$, where $S$ is a Minkowski centered regular triangle and $s\in(1,2)$. By $p^1,\dots,p^6$ we denote the vertices of $K$, counted in clockwise order, such that $[p^i,p^{i+1}]$ with $i=1,3,5$ are the shorter edges of $K$.
Let
\[
C=\mathrm{conv}(\{p^2, p^4, p^6, \frac{p^1+p^2}{2}, \frac{p^3+p^4}{2}, \frac{p^5+p^6}{2}, \frac{p^1-p^4}{s+1}, \frac{p^3-p^6}{s+1}, \frac{p^5-p^2}{s+1}\})
\]
(c.f.~Figure \ref{fig:ScapsS_cut}).
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=2.7]
\draw [thick, black] (0,0.8) -- (0.11547,0.8)-- (0.28885,0.50035)-- (0.5774,0)-- (0.69,-0.4)--(0.6348,-0.5)--(0.28885,-0.50035)--(-0.28885,-0.50035)-- (-0.69,-0.4)--(-0.75038,-0.2997)--(-0.5774,0)--(-0.28885,0.50035)--(0,0.8);
\draw [thick, dashed] (0,-0.8)--(-0.11547,-0.8) --(-0.28885,-0.50035)--(-0.5774,0)--(-0.69,0.4)--(-0.6348,0.5)--(-0.28885,0.50035)--(0.28885,0.50035)--(0.69,0.4)-- (0.75038,0.2997)--(0.5774,0)--(0.28885,-0.50035)--(0,-0.8);
\draw [thick, black, dotted] (-0.11547,0.8) -- (0.11547,0.8) -- (0.75038,-0.2997)-- (0.6348,-0.5)-- (-0.6348,-0.5)-- (-0.75038,-0.2997)--(-0.11547,0.8);
\draw [thick, black, dotted] (0.11547,-0.8) -- (-0.11547,-0.8) -- (-0.75038,0.2997)-- (-0.6348,0.5)-- (0.6348,0.5)-- (0.75038,0.2997)--(0.11547,-0.8);
\draw [thick, dred, dotted] (0.75038,-0.2997) -- (0.6348,-0.5)-- (0.11547,-0.8) -- (-0.11547,-0.8)--(-0.6348,-0.5)-- (-0.75038,-0.2997)--(-0.75038,0.2997)-- (-0.6348,0.5)--(-0.11547,0.8) -- (0.11547,0.8)--(0.6348,0.5)-- (0.75038,0.2997)--(0.75038,-0.2997);
\draw [thick, dred] (0.69,-0.4)--(0.6348,-0.5)-- (0,-0.8) -- (-0.11547,-0.8) --(-0.69,-0.4)--(-0.75038,-0.2997)--(-0.69,0.4)--(-0.6348,0.5)--(0,0.8)--(0.11547,0.8)--(0.69,0.4)-- (0.75038,0.2997)--(0.69,-0.4);
\draw [thick, dblue, dashed] (0.75038,0) -- (0.375135,-0.65) -- (-0.375135,-0.65)-- (-0.75038,0) -- (-0.375135,0.65)-- (0.375135,0.65)-- (0.75038,0);
\draw [thick, dblue] (0.46,-0.50035) --(0.3,-0.65)--(-0.2,-0.65)--(-0.4,-0.6) --(-0.46,-0.50035)--(-0.66,-0.15)--(-0.72,0.05) --(-0.46,0.50035)-- (-0.3,0.65)--(0.2,0.65)-- (0.4,0.6)--(0.665,0.15)-- (0.72,-0.05)--(0.46,-0.50035);
\draw [thick, fill=lgold, fill opacity=0.7] (-0.28885,0.50035)--(0.28885,0.50035)--(0.5774,0)--(0.28885,-0.50035)--(-0.28885,-0.50035)--(-0.5774,0);
\draw [thick, black] (0.5774,0) -- (0.28885,-0.50035) -- (-0.28885,-0.50035) -- (-0.5774,0) -- (-0.28885,0.50035) -- (0.28885,0.50035) -- (0.5774,0);
\draw [thick, black] (0,0) -- (0.4,0.6);
\draw [thick, black] (0,0) -- (0.2,0.65);
\draw [fill] (0,0) circle [radius=0.01];
\draw [fill] (0.4,0.6) circle [radius=0.02];
\draw [fill] (0.31,0.45) circle [radius=0.02];
\draw [fill] (0.2,0.65) circle [radius=0.02];
\draw [fill] (0.16,0.5) circle [radius=0.02];
\draw (-0.07,-0.07) node {$0$};
\draw (0.5,0.7) node {$q^1$};
\draw (0.1,0.58) node {$q^2$};
\draw (0.2,0.9) node {$p^2$};
\draw (0.7,-0.55) node {$p^4$};
\draw (-0.85,-0.27) node {$p^6$};
\draw (0.28,0.18) node {$\frac{2}{s+1} q^1$};
\draw (-0.08,0.35) node {$\frac{2}{s+1} q^2$};
\draw (0.87,-0.42) node {$\frac{p^3+p^4}{2}$};
\draw (-0.87,-0.45) node {$\frac{p^5+p^6}{2}$};
\draw (-0.07,0.93) node {$\frac{p^1+p^2}{2}$};
\end{tikzpicture}
\caption{
Construction from Example \ref{ex:notopt}, $s=1.5$: $C$ (black), $-C$ (black dashed), $K$, $-K$ (black dotted), $\mathrm{conv}(C\cup(-C))$ (red), $\mathrm{conv}(K\cup(-K))$ (red dotted),
$\frac{C-C}{2}$ (blue), $\frac{K-K}{2}$ (blue dashed), $C\cap(-C)=K\cap(-K)$ (yellow).
}
\label{fig:ScapsS_cut}
\end{figure}
Note that
\begin{align*}
\frac{K-K}{2}&= \left(\mathrm{conv}(\left\{\pm \frac{p^1-p^4}{2}, \pm \frac{p^2-p^5}{2}, \pm \frac{p^3-p^6}{2} \right\}\right) \quad \text{and} \\
K\cap(-K)&=\mathrm{conv}\left(\left\{\pm \frac{p^1-p^4}{s+1}, \pm \frac{p^2-p^5}{s+1}, \pm \frac{p^3-p^6}{s+1} \right\}\right).
\end{align*}
Moreover, $K$ and $C$ are Minkowski centered with $s(K) = s(C)=s$ and
\begin{align*}
\frac{C-C}{2}&= \mathrm{conv}\left(\left\{ \pm \frac{2p^2-p^5-p^6}{4}, \pm \frac{2p^4-p^1-p^2}{4}, \pm \frac{2p^6-p^3-p^4}{4}\right.\right.,\\
&\left.\left.\pm \frac{(s+2)p^2-p^5}{2(s+1)}, \pm \frac{(s+2)p^6-p^3}{2(s+1)}, \pm \frac{(s+2)p^4-p^1}{2(s+1)}\right\}\right).
\end{align*}
Now, assume that
\[
\frac{C-C}{2} \subset^{opt} \frac{s+1}{2}\left(\frac{C^\circ-C^\circ}{2}\right)^\circ.
\]
Then there must exist a vertex of $\frac{C-C}{2}$, which also belongs to $\mathrm{bd} \left( \frac{s+1}{2} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right)$. Since the vertices of $\frac{C-C}{2}$ have two different types, we have to consider both of them:
let w.l.o.g. $q^1=\frac{2p^2-p^5-p^6}{4}$ and $q^2=\frac{(s+2)p^2-p^5}{2(s+1)}$. Then, for both $i=1,2$,
\[
\frac{2}{s+1} q^i \in \frac{2}{s+1} \frac{K-K}{2}= K\cap(-K)=C\cap(-C)= \mathrm{conv}\left(\left\{ \pm \frac{p^2-p^5}{s+1}, \pm \frac{p^3-p^6}{s+1}, \pm \frac{p^4-p^1}{s+1}\right\}\right).
\]
Remember, that by Part (ii) of Lemma \ref{lem:Regard_of_Asym} we have $C \cap (-C) \subset^{opt} \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$. Thus we would obtain
\[
\pm \frac{p^2-p^5}{s+1}, \pm \frac{p^3-p^6}{s+1}, \pm \frac{p^4-p^1}{s+1} \in (C\cap(-C)) \cap \mathrm{bd} \left( \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right).
\]
If $\frac{2}{s+1} q^1 \in \mathrm{bd} \left( \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right)$ would be true, then
\[
\frac{p^2-p^5}{s+1}, \frac{p^3-p^6}{s+1}, \frac{2}{s+1} q^1 \in (C\cap(-C)) \cap \mathrm{bd} \left( \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right),
\]
which is only possible if the full edge $[\frac{p^2-p^5}{s+1}, \frac{p^3-p^6}{s+1}] \subset (C\cap(-C)) \cap \mathrm{bd} \left( \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right)$.
By the symmetries of $C$ we would conclude that $C\cap(-C) = \left(\frac{C^\circ-C^\circ}{2}\right)^\circ$, obtaining a contradiction to Part (iii) of Theorem \ref{thm:reverse_inclusions}. In case $\frac{2}{s+1} q^2 \in \mathrm{bd} \left( \left(\frac{C^\circ-C^\circ}{2}\right)^\circ\right)$, we obtain a similar conclusion (c.f.~Figure \ref{fig:ScapsS_cut}).
Thus, $C$ fulfills $\frac{C-C}{2} \subset \mathrm{int}(\frac{s+1}{2}\left(\frac{C^\circ-C^\circ}{2}\right)^\circ)$.
\begin{remark}
Let $\omega(s) \in [1,\frac{s+1}{2}]$ be such that $\frac{C-C}{2} \subset^{opt} \omega(s) \left(\frac{C^\circ - C^\circ}{2}\right)^\circ$, where $s=s(C)$.
Then there exists $x \in \mathbb R^n$ such that
\[
\|x\|_{\frac{C-C}{2}}=\|x\|_{\omega(s) \left(\frac{C^\circ - C^\circ}{2}\right)^\circ}.
\]
On the one hand, $\frac{x }{\|x\|_{C}} \in \mathrm{bd}(C)$ and $\frac{x }{\|x\|_{-C}} \in \mathrm{bd}( -C)$, implying $\frac{1}{2} \left( \frac{1 }{\|x\|_{K}}+\frac{1}{\|x\|_{C}} \right) x \in \frac{C-C}{2}$. And thus, we get
$\|x\|_{\frac{C-C}{2}}\leq
\left(
\frac{\frac{1}{\|x\|_C}+\frac{1}{\|x\|_{-C}}}{2} \right)^{-1}$. On the other hand, $\|x\|_{ \left(\frac{C^\circ - C^\circ}{2}\right)^\circ}=\frac{\|x\|_C+ \|x\|_{-C}}{2}$. Therefore,
\[
\omega(s)=\frac{\|x\|_{ \left(\frac{C^\circ - C^\circ}{2}\right)^\circ}}{\|x\|_{\frac{C-C}{2}}} \geq \frac{\|x\|_C+ \|x\|_{-C}}{2} \frac{\frac{1}{\|x\|_C}+\frac{1}{\|x\|_{-C}}}{2}=\frac{(\|x\|_C+ \|x\|_{-C})^2}{4 \|x\|_C \|x\|_{-C}}.
\]
Let w.l.o.g. $\|x\|_C \geq \|x\|_{-C}$ and $\rho:=\frac{\|x\|_C}{\|x\|_{-C}}$. Since $C$ is Minkowski centered,
we have $1 \leq \rho \leq s$, and thus
\begin{equation}\label{eq:LBofOmega}
\omega(s) \geq \frac{(\rho+1)^2}{4 \rho}.
\end{equation}
Note, that the right term above
attains its maximum for $\rho=s$. Since we actually get this value, whenever
$x$ is an asymmetry point of $C$, we conclude the assertion.
\end{remark}
\end{example}
\begin{remark}
Let $P \subset \mathbb R^n$ be a Minkowski centered polytope. By Proposition \ref{prop:Opt_Containment} there exist $a^i \in \mathbb R^n$ and vertices $x^i$, such that $H_{a^i,1}^{\le}$ define facets of $P$, $i \in [k+1]$ and $s(P) \leq k \leq n$,
such that $0 \in \mathrm{conv}(\{a^1,\dots,a^{k+1}\})$ and $-x^i \in H_{a^i,s(P)}$, $i \in [k+1]$.
Now, we do not only have $\frac{P-P}2 \subset^{opt} \frac{(s(P)+1)}2 (P \cap (-P))$ from Theorem \ref{thm:reverse_inclusions}, but also the fact that for any vertex $y$ of the facet $P \cap H_{a^i,1}$ of $P$, we have $\frac{y-x^i}2 \in H_{a^i,\frac{s(P)+1}2} \cap \frac{P-P}2 \cap \frac{s(P)+1}{2} (P \cap (-P))$, i.e.~$\frac{P-P}{2}$ touches $\frac{s(P)+1}{2} (P \cap (-P))$ in all the facets $\frac{s(P)+1}{2} (P \cap (-P)) \cap H_{a^i,\frac{s(P)+1}2}$ with a full facet (c.f.~\cite[Lemma 2.8]{BG}, where this fact is shown for simplices).
\end{remark}
\begin{remark}
It is well known, that $s(K) = \inf_{C \in \mathcal K_0^n} d_{BM}(K,C)$ for every $K \in \mathcal K^n$ (see, e.g., \cite{Gr}). Furthermore, in \cite[Prop. 3.1]{BrG2} it is shown that this infimum is always attained by $\frac{K-K}{2}$. In general, if $C \in \mathcal K^n_0$ and $K \in \mathcal K^n$ we see from the definition of the Banach-Mazur distance that
\[
d_{BM}(K,C) = s(K) \Longleftrightarrow \exists \, L \in GL(n), t^1,t^2 \in \mathbb R^n \text{ s.t. } -K - t^1 \subset L(C) \subset s(K)K + t^2.
\]
Since $L(C)$ is symmetric, we may symmetrize and replace the right-hand side above by
\[
\exists \, L \in GL(n), t^1,t^2 \in \mathbb R^n \text{ s.t. } \mathrm{conv}((K+t^1) \cup (-K-t^1)) \subset L(C) \subset s(K) ((K + t^2) \cap (-K-t^2)).
\]
For a Minkowski concentric $K$ we now immediately obtain that all four choices
\[
C \in\left\{K\cap(-K),\left(\frac{K^\circ-K^\circ}{2}\right)^\circ,\frac{K-K}{2},\mathrm{conv}(K\cup(-K))\right\}
\]
of symmetrizations of $K$ considered in this paper fulfill $d_{BM}(K,C) = s(K)$ and are therefore minimizers for the Banach-Mazur distance between $K$ and $\mathcal K^n_0$.
Moreover, with help of the reverse containments from Theorem \ref{thm:reverse_inclusions} we obtain some upper bounds on the Banach-Mazur distances of pairs of these symmetrizations, e.g.~
$d_{BM}(K \cap (-K),\mathrm{conv}(K \cup (-K)) \le s(K)$ or $d_{BM}(K \cap (-K), \frac{K^\circ-K^\circ}{2}) \le \frac{2s(K)}{s(K)+1}$. However, this bounds do not even have to be tight when the containments between those sets are. E.g.~is the Banach-Mazur distance of any two of the symmetrizations of a regular triangle in the plane exactly 1, as they are all regular hexagons.
\end{remark}
\section{Improving the containment factors in the forward direction}\label{sec:improving_inclusions}
Now we prove Theorem \ref{thm:small_asym_no_improve}.
\begin{proof}[Proof of Theorem \ref{thm:small_asym_no_improve}]
We start showing Part a).
\begin{enumerate}[(i)]
\item We first show that
$\alpha_1(s) \geq \frac{2}{s+1}$ independently of $n$.
By the definition of $\alpha(s)$ and Part (ii) of Theorem \ref{thm:reverse_inclusions} we have $C \cap (-C) \subset^{opt} \alpha(s) \cdot \mathrm{conv}(C \cup (-C)) \subset^{opt} \frac{\alpha(s) (s+1)}{2} \cdot \left(\frac{C^\circ -C^\circ}{2}\right)^{\circ}$, while
Part (ii) of Lemma \ref{lem:Regard_of_Asym} gives us $C \cap(-C) \subset^{opt} \left(\frac{C^\circ - C^\circ}{2}\right)^{\circ}$. Hence, $\alpha(s)$ must be always at least $\frac{2}{s+1}$.
Next we show $\alpha_1(s)=\frac{2}{s+1}$ in any dimension if $s \le 2$.
First let $n=2$ and consider $C = S \cap (-s S)$ with $s \in [1,2]$ and the regular triangle $S=\mathrm{conv} \left(\left\{p^1, p^2,p^3 \right\}\right)$.
Now choose w.l.o.g. the vertex $v$ of $C \cap (-C)$ with $v \in \mathrm{pos} \left(\left\{ p^1, p^2 \right\}\right)$ and let $\mu \ge 1$ be such that $\mu v \in \mathrm{bd}(\mathrm{conv}(C \cup (-C)))$. Let $q$ be a vertex of $\mathrm{conv}(C \cup (-C))$, such that $q \in [p^2, v]$.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=3]
\draw [thick, black] (-0.11547,0.8) -- (0.11547,0.8) -- (0.75038,-0.2997)-- (0.6348,-0.5)-- (-0.6348,-0.5)-- (-0.75038,-0.2997)--(-0.11547,0.8);
\draw [thick, dashed] (0.11547,-0.8) -- (-0.11547,-0.8) -- (-0.75038,0.2997)-- (-0.6348,0.5)-- (0.6348,0.5)-- (0.75038,0.2997)--(0.11547,-0.8);
\draw [thick, black, dotted] (-0.11547,0.8) -- (0,1)--(0.11547,0.8);
\draw [thick, black, dotted] (0.75038,-0.2997)-- (0.87,-0.5)--(0.6348,-0.5);
\draw [thick, black, dotted] (-0.75038,-0.2997)-- (-0.87,-0.5)--(-0.6348,-0.5);
\draw [thick, gray] (0.75038,0.2997)--(0.75038,-0.2997);
\draw [thick, gray] (0,0)--(0.86,-0.5);
\draw [thick, gray] (0,0)--(0.75,0);
\draw [fill] (0,0) circle [radius=0.01];
\draw [fill] (0.5774,0) circle [radius=0.02];
\draw [fill] (0.86,-0.50035) circle [radius=0.02];
\draw [fill] (0,1) circle [radius=0.02];
\draw [fill] (0.75,0) circle [radius=0.02];
\draw [fill] (0.75,-0.29) circle [radius=0.02];
\draw (-0.07,-0.07) node {$0$};
\draw (0.42,0.1) node {$v$};
\draw (0.9,0) node {$\mu v$};
\draw (0.85,-0.28) node {$q$};
\draw (0,1.1) node {$p^1$};
\draw (1,-0.55) node {$p^2$};
\end{tikzpicture}
\caption{Construction from the proof of Part (i) of Theorem \ref{thm:small_asym_no_improve}: $C=S \cap (-s S)$ ($s=s(C)=1.5$) (black), $-C$ (dashed).
}
\end{figure}
Since $\|p^2\|=1$, we have
\[
\left\|v-q \right\|=\frac{\left \|p^2-\left(\frac12 p^2 \right)\right\|-\left \|p^2-\left(\frac{s}{2}p^2 \right)\right\|}{\cos (\pi/6) }= \frac{2}{\sqrt{3}} \left( \frac s2 -\frac12\right).
\]
Thus, since $\|v\|=\frac{1}{\sqrt{3}}$, we have
\[
\frac{\left\|\mu v \right\|}{\left\|v \right\|}= 1+ \frac{\left\|\mu v -v\right\|}{\left\|v \right\|}= 1+ \frac{\left\|v-q \right\| \sin (\pi/6)}{\left\|v \right\|}=1+ \frac{ \frac{1}{2} \left\|v-q \right\| }{\frac{1}{\sqrt{3}}}= 1+\frac s2 -\frac1n,
\]
which implies
\[
\alpha(s)= \frac{1}{1+\frac s2 -\frac12} =\frac{2}{s+1}.
\]
For $n\geq 3$ we can simply embed the above $C$ keeping the Minkowski center to be still 0 into $n$-space. This keeps its asymmetry value and also the correct factor for the containment between $C \cap (-C)$ and $\mathrm{conv}(C \cup (-C))$.
\item Obviously, from Proposition \ref{prop:means_of_sets} we know that $\alpha(s) \le 1$.
We now show that for any $s \ge \varphi$
there exists a Minkowski centered $C$ with $s(C)=s$ for which $\alpha(s)=\frac{s}{s^2-1}$.
For any $s \ge \varphi$ we define
\[C=\mathrm{conv}\left(\left\{
\begin{pmatrix} \pm 1 \\ (\varphi+1)(2-s-\frac{1}{s+1})\end{pmatrix},
\begin{pmatrix} \pm 1 \\ - \frac{\varphi+1}{s+1} \end{pmatrix},
\begin{pmatrix} 0 \\ s\frac{\varphi+1}{s+1} \end{pmatrix}\right\}\right).\]
Since $-(1/s) C \subset C$ with
\[
-\frac1s \begin{pmatrix} \pm 1 \\ - \frac{\varphi+1}{s+1} \end{pmatrix} \in
\left[\begin{pmatrix} \mp 1 \\ (\varphi+1)(2-s-\frac{1}{s+1})\end{pmatrix} , \begin{pmatrix} 0 \\ s\frac{\varphi+1}{s+1} \end{pmatrix}\right]
\]
and
\[
-\frac1s \begin{pmatrix} 0 \\ s\frac{\varphi+1}{s+1} \end{pmatrix} \in
\left[\begin{pmatrix} 1 \\ - \frac{\varphi+1}{s+1} \end{pmatrix}, \begin{pmatrix} -1 \\ - \frac{\varphi+1}{s+1} \end{pmatrix}\right],
\]
we obtain from Proposition \ref{prop:Opt_Containment} that $C$ is Minkowski centered and $s(C)=s$.
Let $\alpha(s)<1$ be such that $C \cap (-C) \subset^{opt} \alpha(s) \cdot \mathrm{conv}(C \cup (-C))$. Then due to the symmetries of $C$, we have
\[
\alpha(s)=\max \left\{ \frac{\|v\|}{\|w\|}, \frac{\|p\|}{\|q\|} \right\},
\]
where $v$ and $p$ are vertices of $C \cap (-C)$, such that $v = \begin{pmatrix} \frac{s}{s^2-1} \\ 0\end{pmatrix}$ and
\[p=\left[\begin{pmatrix} 0 \\ s\frac{\varphi+1}{s+1} \end{pmatrix},\begin{pmatrix} 1 \\ (\varphi+1)(2-s-\frac{1}{s+1})\end{pmatrix}\right] \cap \left[\begin{pmatrix} 1 \\ \frac{\varphi+1}{s+1} \end{pmatrix},\begin{pmatrix} -1 \\ \frac{\varphi+1}{s+1} \end{pmatrix}\right],
\]
while $w$ and $q$ are rescalations of $v$ and $p$, respectively, belonging to $\mathrm{bd}(\mathrm{conv}(C \cup (-C))).$
Now, we see
\[w=\frac12 \left( \begin{pmatrix} 1 \\ - \frac{\varphi+1}{s+1} \end{pmatrix}+ \begin{pmatrix} 1 \\ \frac{\varphi+1}{s+1} \end{pmatrix} \right)=\begin{pmatrix} 1 \\ 0\end{pmatrix} \in \mathrm{bd}(\mathrm{conv}(C \cup (-C))),
\]
which shows $w=\frac{s^2-1}{s} v$.
Note that for some $x \in \mathbb R$ and some $\lambda \in [0,1]$, we have
\[
p=\begin{pmatrix} x \\ \frac{\varphi+1}{s+1} \end{pmatrix}=\lambda \begin{pmatrix} 0 \\ \frac{s(\varphi+1)}{s+1} \end{pmatrix} +(1-\lambda) \begin{pmatrix} 1 \\ \frac{(\varphi+1)(-s^2+s+1)}{s+1}
\end{pmatrix}.
\]
Thus, $\lambda=\frac{s}{s+1}$ and
\[
p = \frac{1}{s+1}\begin{pmatrix}1 \\ \varphi+1 \end{pmatrix}.
\]
Now let $q=\nu p \in \mathrm{bd}(\mathrm{conv}(C \cup (-C)))$ for some $\nu >1$. Then
\[
q=\lambda \begin{pmatrix} 0 \\ \frac{s(\varphi+1)}{s+1} \end{pmatrix} +(1-\lambda) \begin{pmatrix} 1 \\ \frac{\varphi+1}{s+1}
\end{pmatrix}.
\]
Thus, $\lambda=\frac12$ and
\[
q = \frac12 \begin{pmatrix} 1 \\ \varphi+1 \end{pmatrix}.
\]
Let $\alpha(s)<1$ be such that $C \cap (-C) \subset^{opt} \alpha(s) \cdot \mathrm{conv}(C \cup (-C))$. Then due to the symmetries of $C$, we have
\[
\alpha(s)=\max \left\{ \frac{\|v\|}{\|w\|}, \frac{\|p\|}{\|q\|} \right\} = \max \left\{ \frac{2}{s+1}, \frac{s}{s^2-1} \right\} = \frac{s}{s^2-1}.
\]
By the definition of $\gamma(n)$, we have $\alpha_2(S) = 1$ for $s \leq \gamma(n)$, while $\alpha_2(s) \le \psi \frac{n}{n+1}$ if $s > \gamma_2(n)$ follows from Part (i) of Theorem 1.5.
\end{enumerate}
\begin{figure}[ht]
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=3.5]
\draw[thick, discont] (0.05,0) -- (0.25,0);
\draw[thick, discont] (0,0.05) -- (0,0.25);
\draw [thick] (-0.2,0) -- (0.05,0);
\draw [thick] (0,-0.2) -- (0,0.05);
\draw[->] [thick] (0.25, 0) -- (1.7, 0) node[right] {$s$};
\draw[->] [thick] (0, 0.25) -- (0, 0.7);
\draw [thick,gray, shift={(0,-0.5)}] (1.67,1.75) -- (1.8,1.75)--(1.8,0.95)--(1.67,0.95)--(1.67,1.75);
\draw [thick,gray, shift={(0,-0.5)}] (1.45,1.05) -- (1.55,1.05)--(1.55,0.62)--(1.45,0.62)--(1.45,1.05);
\draw [thick,gray, shift={(0,-0.5)}] (1.55,0.9) -- (1.67,1.4);
\draw [thick, dred,shift={(-0.5,-0.5)}] (1,1) -- (1.61,1);
\draw [thick, dred,shift={(-0.5,-0.5)}] (1.61,1) -- (1.9914,1);
\draw[thick, dred, domain=1.9914:2, smooth, variable=\x,shift={(-0.5,-0.5)}] plot ({\x},
{(-26*\x^2+36*\x+34)/(18*\x^2-24*\x-21)});
\draw[thick, dred, domain=1.9914:2, smooth, variable=\x,shift={(-0.25,-0.6)}] plot ({\x},
{1.7*(-26*\x^2+36*\x+34)/(18*\x^2-24*\x-21)});
\draw[thick, dblue, domain=1:2, smooth, variable=\x, dblue,shift={(-0.5,-0.5)}] plot ({\x}, {2/(\x+1)});
\draw[thick, dgreen, domain=1.61:2, smooth, variable=\x,shift={(-0.5,-0.5)}] plot ({\x}, {(\x)/((\x)^2-1)});
\fill [fill=lgold, fill opacity=0.7, domain=1:1.3, variable=\x,shift={(-0.5,-0.5)}] (1,1) -- (1.3,1)-- (1.3,0.875)--(1,1);
\fill [fill=lgold, fill opacity=0.7, domain=1.3:1.61, variable=\x,shift={(-0.5,-0.5)}] (1.3,1) -- (1.61,1)-- (1.61,0.775)--(1.3,0.875);
\fill [fill=lgold, fill opacity=0.7, domain=1.3:1.61, variable=\x,shift={(-0.5,-0.5)}] (1.61,1) -- (1.8,0.79)--(1.8,0.72)-- (1.61,0.775)--(1.61,1);
\fill [fill=lgold, fill opacity=0.7, domain=1.8:2, variable=\x,shift={(-0.5,-0.5)}] (1.8,0.79) -- (2,0.67)--(1.8,0.72)-- (1.8,0.79)--(1.8,0.79);
\draw [thick, dashed,gray] (0,0.5) -- (0.5,0.5);
\draw [thick, dashed,,gray] (1.11,0)--(1.11,0.28);
\draw [thick, dashed,gray] (1.5,0) -- (1.5,0.17);
\draw [thick, dashed,gray] (0.5,0) -- (0.5,0.5);
\draw [thick, dashed,gray] (1.75,0.55) -- (1.76,1.11);
\draw [fill,shift={(-0.5,-0.5)}] (1,1) circle [radius=0.01];
\draw [fill,shift={(-0.5,-0.5)}] (1.61,1) circle [radius=0.01];
\draw [fill,shift={(-0.5,-0.5)}] (2,0.67) circle [radius=0.01];
\draw [fill,shift={(-0.5,-0.5)}] (1.9914,1) circle [radius=0.01];
\draw [fill] (1.74,1.11) circle [radius=0.01];
\draw [fill] (1.75,0.55) circle [radius=0.01];
\draw (-0.15,0.7) node {$\alpha(s)$};
\draw (-0.1,0.5) node {$1$};
\draw (0.5,-0.1) node {$1$};
\draw (1.11,-0.1) node {$\gamma(2)$};%
\draw (1.5,-0.1) node {$2$};
\end{tikzpicture}
\caption*{$n=2$: $\alpha_2(s) = 1$ for $s \leq \gamma(2)$ (red). For $s \ge \gamma(2)$ we have $\alpha_2 (s) \ge \frac{s}{s^2-1}$ (green), while $\alpha_2(s) < 1$ for $s > \gamma(2)$ and $\alpha_2 \le \frac{2}{3} \psi(2) = \frac{-26s^2+36s+34}{18s^2-24s-21}$ for $s > \gamma_2(2)$ (red). The lower bound is $\alpha_1(s) = \frac{2}{s+1}$ (blue).}
\label{fig:GH}
\begin{minipage}{.1cm}
\vspace{1em}
\end{minipage}
\end{subfigure} \hfill
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=3.5]
\draw[thick, discont] (0.05,0) -- (0.25,0);
\draw[thick, discont] (0,0.05) -- (0,0.25);
\draw [thick] (-0.2,0) -- (0.05,0);
\draw [thick] (0,-0.2) -- (0,0.05);
\draw[->] [thick] (0.25, 0) -- (1.7, 0) node[right] {$s$};
\draw[->] [thick] (0, 0.25) -- (0, 0.7);
\draw [thick, dred] (0.5,0.5) -- (1.21,0.5);
\draw [thick, dred] (1.21,0.5) -- (1.4837,0.5);
\draw [thick, dred] (1.4837,0.5) -- (1.5,0.39);
\draw [thick,gray, shift={(0,-0.5)}] (1.67,1.75) -- (1.8,1.75)--(1.8,0.95)--(1.67,0.95)--(1.67,1.75);
\draw [thick,gray, shift={(0,-0.5)}] (1.45,1.05) -- (1.55,1.05)--(1.55,0.83)--(1.45,0.83)--(1.45,1.05);
\draw [thick,gray, shift={(0,-0.5)}] (1.55,0.9) -- (1.67,1.4);
\draw[thick, dred] plot [smooth] coordinates { (1.71,1.1) (1.73,0.77) (1.75,0.59) };
\draw [thick, dashed,gray] (1.75,1.1) -- (1.75,0.59);
\draw [fill] (1.71,1.1) circle [radius=0.01];
\draw [fill] (1.75,0.59) circle [radius=0.01];
\draw[thick, dblue, domain=1:2, smooth, variable=\x,shift={(-0.5,-0.5)}] plot ({\x}, {(4*\x)/(\x+1)^2)});
\draw [thick, dashed,gray] (0,0.5) -- (0.5,0.5);
\draw [thick, dashed,gray] (1.11,0) -- (1.11,0.5);
\draw [thick, dashed,gray] (1.5,0) -- (1.5,0.39);
\draw [thick, dashed,gray] (0.5,0) -- (0.5,0.5);
\draw[thick, dgreen, domain=1.61:1.67, smooth, variable=\x,shift={(-0.5,-0.5)}] plot ({\x}, {(\x)/((\x)^2-1)});
\fill [fill=lgold, fill opacity=0.7, domain=1.61:1.67, variable=\x,shift={(-0.5,-0.5)}] (1.61,1)--(1.67,0.93) --(1.61, 0.95);
\fill [fill=lgold, fill opacity=0.7, domain=1:1.3, variable=\x,shift={(-0.5,-0.5)}] (1,1) -- (1.3,1)--(1.3,0.99)--(1,1);
\fill [fill=lgold, fill opacity=0.7, domain=1.3:1.61, variable=\x,shift={(-0.5,-0.5)}] (1.3,1) -- (1.61,1)--(1.61,0.95)--(1.3,0.99);
\fill [fill=lgold, fill opacity=0.7, domain=1.61:2, variable=\x,shift={(-0.5,-0.5)}] (1.67,0.94) --(2, 0.905)--(2, 0.9)--(1.67,0.95);
\draw [fill] (0.5,0.5) circle [radius=0.01];
\draw [fill] (1.11,0.5) circle [radius=0.01];
\draw [fill] (1.5,0.39) circle [radius=0.01];
\draw [fill] (1.4837,0.5) circle [radius=0.01];
\draw [fill] (1.17,0.438) circle [radius=0.01];
\draw (-0.15,0.7) node {$\beta(s)$};
\draw (-0.1,0.5) node {$1$};
\draw (0.5,-0.1) node {$1$};
\draw (1.11,-0.1) node {$\gamma(2)$};
\draw (1.5,-0.1) node {$2$};
\end{tikzpicture}
\caption*{$n=2$: $\beta_2(s) = 1$ for $s \leq \gamma(2)$ (red); for $s \in [\varphi,2]$ the upper bound is at least $\max \{ \frac{s}{s^2-1}, \frac{4s}{(s+1)^2} \}$ (green) and for $s \in [\gamma_2(n),2]$ also at most
$ \frac{8}{9} \mu \psi$
(red). The lower bound is
$\frac{4s}{(s+1)^2}$, which is sharp for $s \in [1,2]$ (blue).} \label{fig:GH-symm}
\begin{minipage}{.1cm}
\vspace{1em}
\end{minipage}
\end{subfigure}
\caption{Region of possible values of parameters $\alpha(s)$, $\beta(s)$ with $s \in [1,n]$, s.th. $C \cap (-C) \subset^{opt} \alpha(s) \mathrm{conv}(C \cup (-C))$, $\left(\frac12 (C^\circ - C^\circ)) \right)^{\circ}\subset^{opt} \beta(s) \, \frac12 (C-C)$ for some Minkowski centered $C \in\mathcal K^n$ with $s(C)=s$ from Theorem \ref{thm:small_asym_no_improve}.}
\end{figure}
We now proceed with Part b).
\begin{enumerate}[(i)]
\item
Let $\beta(s) \leq 1$ be such that
\[
\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} \subset^{opt} \beta(s) \frac{C-C}{2}.
\]
On the one hand, by Part (iv) of Theorem \ref{thm:reverse_inclusions} we have
\[
\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} \subset^{opt} \beta(s) \frac{C-C}{2} \subset^{opt} \beta(s) \frac{s+1}{2} C \cap (-C).
\]
On the other hand, from Part (iii) of Theorem \ref{thm:reverse_inclusions} we know that
\[
\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} \subset^{opt} \frac{2s}{s+1} C \cap (-C).
\]
Hence $ \frac{2s}{s+1} \leq \beta(s) \frac{s+1}{2}$,
which implies $\beta_1(s) \geq \frac{4s}{(s+1)^2}$.
Now, consider the hexagon
\[C=\mathrm{conv}\left(\left\{
\begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left( 1- \frac{s}{2} \right) \\ \frac{s}{2} \end{pmatrix},
\begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left( \frac{s+1}{2} \right) \\ \frac{1}{2}-\frac{3s}{4} \end{pmatrix}, \begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left( s- \frac{1}{2} \right) \\ -\frac{1}{2} \end{pmatrix} \right\}\right)
\]
with $s\in[1,2]$. For all $s\in[1,2]$ we have
\begin{align*}
- \frac1s \begin{pmatrix} \frac{\sqrt{3}}{3} \left( 1- \frac{s}{2} \right) \\ \frac{s}{2} \end{pmatrix} &= \frac{2(s^2-1)}{s(4s-1)} \begin{pmatrix} -\frac{\sqrt{3}}{3} \left( s- \frac{1}{2} \right) \\ -\frac{1}{2} \end{pmatrix} + \left( 1-\frac{2(s^2-1)}{s(4s-1)}\right) \begin{pmatrix} \frac{\sqrt{3}}{3} \left( s- \frac{1}{2} \right) \\ -\frac{1}{2} \end{pmatrix},
\end{align*}
as well as
\begin{align*}
- \frac1s \begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left( s- \frac{1}{2} \right) \\ -\frac{1}{2} \end{pmatrix} &= \frac{3s^2-2s+2}{s(5s-2)} \begin{pmatrix} \frac{\sqrt{3}}{3} \left( 1- \frac{s}{2} \right) \\ \frac{s}{2} \end{pmatrix} + \left(1-\frac{3s^2-2s+2}{s(5s-2)}\right) \begin{pmatrix} \frac{\sqrt{3}}{3} \left( \frac{s+1}{2} \right) \\ \frac{1}{2}-\frac{3s}{4} \end{pmatrix},
\end{align*}
with $\frac{2(s^2-1)}{s(4s-1)}, \frac{3s^2-2s+2}{s(5s-2)} \in [0,1]$.
Hence, $C$ is Minkowski centered with $s(C)=s$.
Since $C$ is a hexagon with 3 pairs of parallel edges, it
turns out that its arithmetic mean stays to be a hexagon:
\[
\frac{C-C}{2}=\mathrm{conv}\left(\left\{
\begin{pmatrix} \pm \frac{\sqrt{3}}{12} \left( s+1 \right) \\ \frac{s+1}{4} \end{pmatrix}, \begin{pmatrix} \pm \frac{\sqrt{3}}{12} \left( s+1 \right) \\ -\frac{s+1}{4} \end{pmatrix}
\begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left( \frac{s+1}{2} \right) \\ 0 \end{pmatrix} \right\}\right).
\]
The next step to do is to calculate $C^\circ=: \mathrm{conv}(\{q^1, \dots, q^6\})$. Since the vertices of $C$ are the outer normals of the edges of $C^\circ$ and $0 \in \mathrm{int} (C)$, we obtain the vertices of $C^\circ$ as the solution of pairs of inequalities of the form $(v^i)^T x = 1$, where the $v^i$'s are consecutive vertices of $C$.
Moreover, we make use of the fact that $C$, and therefore also $C^\circ$, is symmetric w.r.t.~the $y$-axis. Hence, it suffices to calculate four of the vertices of $C^\circ$.
Let $q^1$ fulfill the equations
\begin{align*}
(q^1)^T \begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left(1-\frac{s}{2}\right) \\ \frac{s}{2} \end{pmatrix} = 1,
\end{align*}
which obviously needs
$q^1=\begin{pmatrix} 0 \\ \frac{2}{s} \end{pmatrix}$.
For $q^2$ we demand
\begin{align*}
(q^2)^T \begin{pmatrix} \frac{\sqrt{3}}{3} \left(1-\frac{s}{2}\right) \\ \frac{s}{2} \end{pmatrix}=(q^2)^T \begin{pmatrix} \frac{\sqrt{3}}{3} \left(\frac{s+1}{2}\right)\\ \frac{1}{2}-\frac{3s}{4} \end{pmatrix} = 1.
\end{align*}
and obtain $q^2=\begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}$.
The third vertex $q^3$ should fulfill
\begin{align*}
(q^3)^T \begin{pmatrix} \frac{\sqrt{3}}{3} \left(\frac{s+1}{2}\right)\\ \frac{1}{2}-\frac{3s}{4} \end{pmatrix} =(q^3)^T \begin{pmatrix} \frac{\sqrt{3}}{3} \left(s-\frac{1}{2}\right) \\ -\frac{1}{2} \end{pmatrix}= 1.
\end{align*}
This leads to $q^3=\begin{pmatrix} \frac{\sqrt{3}}{s} \\ -\frac{1}{s} \end{pmatrix}$.
Finally, for $q^4$ we have to solve
\begin{align*}
(q^4)^T \begin{pmatrix} \pm \frac{\sqrt{3}}{3} \left(s-\frac{1}{2}\right) \\ -\frac{1}{2} \end{pmatrix} = 1,
\end{align*}
which gives $q^4=\begin{pmatrix} 0 \\ -2 \end{pmatrix}$.
Altogether, we obtain
\[
C^\circ=\mathrm{conv}\left(\left\{
\begin{pmatrix} 0 \\ \frac{2}{s} \end{pmatrix},
\begin{pmatrix} \pm \sqrt{3} \\ 1 \end{pmatrix},
\begin{pmatrix} \pm \frac{\sqrt{3}}{s} \\ -\frac{1}{s} \end{pmatrix}, \begin{pmatrix} 0 \\ -2 \end{pmatrix} \right\}\right).
\]
Since $C^\circ$ has no parallel edges, $\frac{C^\circ-C^\circ}{2}$ is a 12-gon that computes to
\[
\frac{C^\circ-C^\circ}{2} = \mathrm{conv}\left(\left\{
\begin{pmatrix} 0 \\ \pm \frac{s+1}{s} \end{pmatrix},
\begin{pmatrix} \pm \frac{\sqrt{3}}{2} \\ \pm \frac32 \end{pmatrix},
\begin{pmatrix} \pm \sqrt{3} \\ 0 \end{pmatrix}, \begin{pmatrix} \pm \sqrt{3} \frac{s+1}{2s} \\ \pm \frac{s+1}{2s} \end{pmatrix} \right\}\right).
\]
Now let $\beta(s)>1$ be such that
\[
\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} \subset^{opt} \beta(s) \frac{C-C}{2}.
\]
Note that 6 vertices of $\frac{C^\circ-C^\circ}{2}$ are rescales of the outer-normals of $\frac{C-C}{2}$, thus the 12-gon $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}$ has 6 edges, which are parallel to the corresponding edges of $\frac{C-C}{2}$, thus $v \in \mathrm{bd} (\beta(s) \frac{C - C}{2})$ for all vertices $v$ of $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}$.
So we choose the vertex $v$ that fulfills the equations
\begin{align*}
v^T \begin{pmatrix} 0 \\ \frac{s+1}{s} \end{pmatrix} =
v^T \begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac32 \end{pmatrix}= 1,
\end{align*}
which gives $v=\begin{pmatrix} \frac{2-s}{\sqrt{3}(s+1)} \\ \frac{s+1}{s} \end{pmatrix}$.
Since $v$ is contained in the edge of $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}$ with the outer-normal $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$, thus $\frac1{\beta(s)}v$ is contained in the edge of $\frac{C - C}{2}$ with the same outer-normal and
\[
\frac{1}{\beta(s)} \begin{pmatrix} \frac{2-s}{\sqrt{3}(s+1)} \\ \frac{s+1}{s} \end{pmatrix} = \lambda \begin{pmatrix} -\frac{s+1}{4 \sqrt{3}} \\ \frac{s+1}{4} \end{pmatrix} + (1-\lambda) \begin{pmatrix} \frac{s+1}{4 \sqrt{3}} \\ \frac{s+1}{4} \end{pmatrix},
\]
from which we obtain $\lambda=\frac{s-1}{s}$ and $\beta(s)=\frac{4s}{(s+1)^2}$, thus proving that $\beta_1(s) = \frac{4s}{(s+1)^2}$ for all $s \in [1,2]$.
For higher dimensions we can simply embed the above construction in a way keeping the Minkowski center 0.
\item By the definition of $\gamma(n)$, we have $\beta_2(s)=1$ for $s \leq \gamma(n)$ while $\beta_2(s) \le \mu \psi \frac{n(n+2)}{(n+1)^2}$ if $s > \gamma_3(n)$ follows from Part (ii) of Theorem 1.5.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=3]
\draw [thick] (0,0.94) -- (0.6,-0.1)-- (0.6,-0.55)--(-0.6,-0.55)--(-0.6,-0.1)--(0,0.94);
\draw [thick, rotate around={180:(0,0)}, dashed] (0,0.94) -- (0.6,-0.1)-- (0.6,-0.55)--(-0.6,-0.55)--(-0.6,-0.1)--(0,0.94);
\draw [thick, dblue] (0.6,0.225) -- (0.6,-0.225)--(0.3,-0.75)--(-0.3,-0.75)--(-0.6,-0.225) -- (-0.6,0.225)--(-0.3,0.75)--(0.3,0.75)--(0.48,0.44)--(0.6,0.225);
\draw [thick, fill=lgold, fill opacity=0.7] (0,0.7) --(0.41,0.435)--(0.54,0.05)--(0.54,-0.05)--(0.41,-0.435)--(0,-0.7)--(-0.41,-0.435)--(-0.54,-0.05)--(-0.54,0.05)--(-0.41,0.435)--(0,0.7);
\draw [thick, dred] (0,0.94) --(0.6,0.55)--(0.6,-0.55)--(0,-0.94)--(-0.6,-0.55)--(-0.6,0.55)--(0,0.94);
\draw [fill] (0,0) circle [radius=0.01];
\end{tikzpicture}
\caption{Construction from the proof of Part b) (ii) of Theorem \ref{thm:small_asym_no_improve} for
$s=1.7$: $C$ (black) and $-C$ (black dashed),
$\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}$ (yellow),
$\frac{C - C}{2}$ (blue) and $\mathrm{conv}(C\cup(-C))$ (red).
}
\label{fig:ScapsS}
\end{figure}
In order to show an upper bound on $\beta_2$,
which is strictly smaller than 1 for $s \in [\varphi,2]$, we first provide the following construction
in $\mathbb R^2$
Let $C=\mathrm{conv}\left(\left\{
\begin{pmatrix} \pm \frac{\sqrt{3}}{2} (s-1) \\ \frac32 \left( -\frac{1}{s+1}+2-s \right) \end{pmatrix},
\begin{pmatrix} \pm \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix},
\begin{pmatrix} 0 \\ \frac{3s}{2(s+1)} \end{pmatrix}\right\}\right)$, $s \in [\varphi,2]$.
Since
\begin{align*}
\begin{pmatrix} 0 \\ \frac{3}{2(s+1)} \end{pmatrix} &\in
\left[\begin{pmatrix} -\frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix}, \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix}\right] \quad \text{and} \\
\begin{pmatrix} \pm \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix} &\in
\left[\begin{pmatrix} \mp \frac{\sqrt{3}}{2} (s-1) \\ \frac32 \left( -\frac{1}{s+1}+2-s \right) \end{pmatrix}, \begin{pmatrix} 0 \\ \frac{3s}{2(s+1)} \end{pmatrix}\right],
\end{align*}
we obtain that $C$ is Minkowski centered with $s(C)=s$.
Note that $C$ has 5 edges, 2 of which are parallel. This implies that $\frac{C-C}{2}$ has 4 different pairs of parallel edges and 8 vertices. Calculating them gives
\[
\frac{C-C}{2}= \mathrm{conv} \left(\left\{
\begin{pmatrix} \pm \frac{\sqrt{3}}{2} (s-1) \\ \frac{3}{4} (2-s) \end{pmatrix}, \begin{pmatrix} \pm \frac{\sqrt{3}}{2} (s-1) \\ - \frac{3}{4} (2-s) \end{pmatrix},
\begin{pmatrix} \pm \frac{\sqrt{3}}{4} (s-1) \\ \frac{3}{4} \end{pmatrix}, \begin{pmatrix} \pm \frac{\sqrt{3}}{4} (s-1) \\ - \frac{3}{4} \end{pmatrix} \right\}\right).
\]
Next, we determine $C^\circ =: \mathrm{conv}(\{q^1, q^2, q^3, q^4, q^5\})$. Since the vertices of $C$ are the outer normals of the edges of $C^\circ$ and $0 \in \mathrm{int} (C)$, we obtain the vertices of $C^\circ$ as the solution of pairs of inequalities of the form $(v^i)^T x = 1$, such that the corresponding $v^i$ are consecutive vertices of $C$. Since $C$ is symmetric w.r.t.~the $y$-axis, so is $C^\circ$. Hence it suffices to calculate $q^1, q^2, q^3$.
Let $q^1$ be such that
\begin{align*}
(q^1)^T \begin{pmatrix} 0 \\ \frac{3s}{2(s+1)} \end{pmatrix} =
(q^1)^T \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ \frac32 \left( -\frac{1}{s+1}+2-s \right) \end{pmatrix}= 1.
\end{align*}
This gives $q^1=\begin{pmatrix} \frac{2}{\sqrt{3}} \left( \frac{s+1}{s} \right) \\ \frac{2}{3} \left( \frac{s+1}{s} \right) \end{pmatrix}$. Now, assume $q^2$ be such that
\begin{align*}
(q^2)^T \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix} =
(q^2)^T \begin{pmatrix} \frac{-\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix} = 1.
\end{align*}
We obtain $q^2=\begin{pmatrix} 0 \\ -\frac{2}{3} \left( s+1 \right) \end{pmatrix}$. For $q^3$ we assume
\begin{align*}
(q^3)^T \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ \frac32 \left( -\frac{1}{s+1}+2-s \right) \end{pmatrix} =
(q^3)^T \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{2(s+1)} \end{pmatrix}= 1.
\end{align*}
Then $q^3=\begin{pmatrix} \frac{2}{\sqrt{3}(s-1)} \\ 0 \end{pmatrix}$.
Thus,
\[
C^\circ= \mathrm{conv}\left(\left\{ \begin{pmatrix} \pm \frac{2}{\sqrt{3}} \left( \frac{s+1}{s} \right) \\ \frac{2}{3} \left( \frac{s+1}{s} \right) \end{pmatrix},
\begin{pmatrix} 0 \\ -\frac{2}{3} \left( s+1 \right) \end{pmatrix},
\begin{pmatrix} \pm \frac{2}{\sqrt{3}(s-1)} \\ 0 \end{pmatrix}
\right\}\right).
\]
Note that $C^\circ$ has 5 edges, none of which are parallel, thus $\frac{C^\circ-C^\circ}{2}$ must have 5 different pairs of parallel edges and 10 vertices. We obtain
\begin{align*}
\frac{C^\circ-C^\circ}{2} = \mathrm{conv}\left(\left\{ \begin{pmatrix} \pm \frac{1}{\sqrt{3}}\left( \frac{s+1}{s} \right) \\ \frac{1}{3} \frac{(s+1)^2}{s} \end{pmatrix}, \begin{pmatrix} \pm \frac{1}{\sqrt{3}}\left( \frac{s+1}{s} \right) \\ - \frac{1}{3} \frac{(s+1)^2}{s} \end{pmatrix}, \begin{pmatrix} \pm \frac{2}{\sqrt{3}}\left( \frac{s+1}{s} \right) \\ 0 \end{pmatrix} \right.\right. , \\
\left.\left.\begin{pmatrix} \pm \frac{1}{\sqrt{3}} \frac{s^2+s-1}{s(s-1)} \\ \frac{1}{3} \frac{s+1}{s} \end{pmatrix}, \begin{pmatrix} \pm \frac{1}{\sqrt{3}} \frac{s^2+s-1}{s(s-1)} \\ - \frac{1}{3} \frac{s+1}{s} \end{pmatrix} \right\}\right).
\end{align*}
The last set to be calculated is $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} =: \mathrm{conv}\left(\left\{v^1, \dots, v^{10}\right\}\right)$.
Let $v^1$ be such that
\begin{align*}
(v^1)^T \begin{pmatrix} \frac{1}{\sqrt{3}} \frac{s+1}{s} \\ \frac{1}{3} \frac{(s+1)^2}{s}\end{pmatrix}= 1 \quad\text{and}\quad
(v^1)^T \begin{pmatrix} \frac{1}{\sqrt{3}} \frac{s^2+s-1}{s(s-1)} \\ \frac{1}{3} \frac{s+1}{s} \end{pmatrix}= 1.
\end{align*}
Thus, $v^1=\begin{pmatrix} \frac{s-1}{s} \\ \frac{(3-\sqrt{3})s^2+\sqrt{3}}{s(s+1)^2} \end{pmatrix}$. Let $v^2$ be such that
\begin{align*}
(v^2)^T \begin{pmatrix} \frac{2}{\sqrt{3}} \frac{s+1}{s} \\ 0 \end{pmatrix}= 1 \quad\text{and}\quad
(v^2)^T \begin{pmatrix} \frac{1}{\sqrt{3}} \frac{s^2+s-1}{s(s-1)} \\ \frac{1}{3} \frac{s+1}{s} \end{pmatrix}= 1.
\end{align*}
Thus, $v^2=\begin{pmatrix} \frac{\sqrt{3}}{2} \frac{s}{s+1} \\
\frac{1}{2} \frac{s^2-s-1}{s^2-1}
\end{pmatrix}$.
Let $v^3$ be such that
\begin{align*}
(v^3)^T \begin{pmatrix} \frac{1}{\sqrt{3}}\left( \frac{s+1}{s} \right) \\ \frac{1}{3} \frac{(s+1)^2}{s} \end{pmatrix}= 1 \quad\text{and}\quad
(v^3)^T \begin{pmatrix} -\frac{1}{\sqrt{3}}\left( \frac{s+1}{s} \right) \\ \frac{1}{3} \frac{(s+1)^2}{s} \end{pmatrix}= 1.
\end{align*}
Thus, $v^3=\begin{pmatrix} 0 \\
\frac{3s}{(s+1)^2} \end{pmatrix}$. Finally, we conclude from the symmetries of $\frac{C^\circ - C^\circ}{2}$ that
\[
\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}= \mathrm{conv}\left(\left\{\begin{pmatrix} \pm \frac{s-1}{s} \\ \pm \frac{(3-\sqrt{3})s^2+\sqrt{3}}{s(s+1)^2} \end{pmatrix}, \begin{pmatrix} \pm \frac{\sqrt{3}}{2} \frac{s}{s+1} \\ \pm
\frac{1}{2} \frac{s^2-s-1}{s^2-1}
\end{pmatrix}, \begin{pmatrix} 0 \\
\pm \frac{3s}{(s+1)^2} \end{pmatrix}\right\}\right).
\]
The next thing we do is to compute scaling factors $\mu_1,\mu_2,\mu_3$ with respect to the different types of vertices of $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ}$ mapping them to the boundary of $\frac{C-C}{2}$. Using the geometry of the two sets the calculations below suffice.
For $\mu_1$ we have to solve
\[
\mu_1 \begin{pmatrix} \frac{s-1}{s} \\ \frac{(3-\sqrt{3})s^2+\sqrt{3}}{s(s+1)^2} \end{pmatrix} = \lambda \begin{pmatrix} \frac{\sqrt{3}}{4} (s-1) \\ \frac{3}{4} \end{pmatrix} + (1-\lambda) \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ \frac{3}{4} (2-s) \end{pmatrix}.
\]
with $\lambda \in [0,1]$ and obtain
\[
\lambda=\frac{s^2+2s(\sqrt{3}-1)-3}{s^2+\sqrt{3}s-1} \quad \text{and} \quad
\mu_1=\frac{ \frac{\sqrt{3}}{4} s(s+1)^2 }{s^2+\sqrt{3} s-1}.
\]
For $\mu_2>1$ we obtain that either the equations
\begin{align*}
\mu_2 \begin{pmatrix} \frac{\sqrt{3}}{2}\frac{s}{s+1} \\
\frac{1}{2} \frac{s^2-s-1}{s^2-1}
\end{pmatrix} &= \lambda \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ \frac{3}{4} (2-s) \end{pmatrix}+(1-\lambda) \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ -\frac{3}{4} (2-s) \end{pmatrix}
\intertext{or the equations}
\mu_2 \begin{pmatrix} \frac{\sqrt{3}}{2} \frac{s}{s+1} \\ \frac{1}{2} \frac{s^2-s-1}{s^2-1} \end{pmatrix} &= \lambda
\begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\
\frac{3}{4}
\end{pmatrix}+(1-\lambda) \begin{pmatrix} \frac{\sqrt{3}}{2} (s-1) \\ \frac{3}{4} (2-s) \end{pmatrix}
\end{align*}
have to be fulfilled for some $\lambda \in [0,1]$.
It turns out that for $s \le \frac{4+\sqrt{26}}{5}$ holds the first system from which we obtain $\lambda= \frac{-s^2+4s-2}{6s(2-s)}$ and $\mu_2=\frac{s^2-1}{s}$.
In case $\frac{4+\sqrt{26}}{5} \le s \le 2$ the second system is the right one, which gives us
$\lambda= \frac{5s^2-8s-2}{-2s^2+8s-1}$ and $\mu_2=\frac{3(3s-7)}{2s^2-8s+1}$.
For the last factor, $\mu_3>1$, we have to solve
\[
\mu_3 \begin{pmatrix} 0 \\ \frac{3s}{(s+1)^2} \end{pmatrix} =\lambda \begin{pmatrix} -\frac{\sqrt{3}}{4} (s-1) \\ \frac{3}{4} \end{pmatrix}+ (1-\lambda) \begin{pmatrix} \frac{\sqrt{3}}{4} (s-1) \\ \frac{3}{4} \end{pmatrix}
\]
for some $\lambda \in [0,1]$, which leads to
$\lambda= \frac12$ and $\mu_3 = \frac{(s+1)^2}{4s}$.
Finally, let $\beta(s) \leq 1$ be such that $\left(\frac{C^\circ - C^\circ}{2} \right)^{\circ} \subset^{opt} \beta(s) \cdot \frac{C-C}{2}$. Then, due to the symmetries of $C$ and since $\mu_1 > \max\{\mu_2,\mu_3\}$,
we obtain
\begin{equation*}
\beta(s) = \max \left\{ \frac{1}{\mu_1}, \frac{1}{\mu_2}, \frac{1}{\mu_3} \right\} = \max \left\{ \frac{1}{\mu_2}, \frac{1}{\mu_3} \right\} =
\begin{cases}
\frac{s}{s^2-1} & \text{if} \quad s \in [\frac{1+\sqrt{5}}{2},\frac53],\\
\frac{4s}{(s+1)^2}, & \text{if} \quad s \in [\frac53,2].
\end{cases}
\end{equation*}
\end{enumerate}
Here one should note that we already know from (i) that $\beta_2 \ge \beta_1 \ge \frac{4s}{(s+1)^2}$, i.e.~the $\frac{4s}{(s+1)^2}$ part for $s \in [\frac53,2]$ of the above calculation provides essentially no new information.
\end{proof}
Let us remark that for any asymmetry value $s \in [1,2]$ and any factor $\alpha(s) \in [\alpha_1(s),\alpha_2(s)]$ or $\beta(s) \in [\beta_1(s),\beta_2(s)]$ from Theorem \ref{thm:small_asym_no_improve}, respectively, there exists a Minkowski centered set
$C$ with asymmetry $s(C)=s$ such that $C \cap (-C) \subset^{opt} \alpha(s) \mathrm{conv}(C \cup (-C))$ or $(\frac{C^\circ - C^\circ}{2})^\circ \subset^{opt} \beta(s) \frac{C-C}{2}$, respectively.
Finally, we would also like to mention that similar to the upper bounds on $\alpha(s)$ and $\beta(s)$ for $s(C)$ close to $n$ from
Theorem \ref{thm:small_asym_no_improve} (namely,
$\alpha_2(s) \le \psi \frac{n}{n+1}$ and $\beta_2(s) \le \mu \psi \frac{n(n+2)}{(n+1)^2}$)
one may use the ideas from the proof of Theorem \ref{thm:minMax_mean_improved} to derive also lower bounds on $\alpha(s)$ and $\beta(s)$ for $s(C)$ close to $n$, i.e., $\alpha_1(s) \ge f_1(s)$ and $\beta_1(s) \ge \mu f_2(s)$ for some continuous functions $f_1$, $f_2$ fulfilling $f_1(n)=\frac{n}{n+1}$ and $f_2(n)=\frac{n(n+1)}{(n+1)^2}$.
\section*{Appendix} \label{sec:app}
The \cemph{circumradius} of $K$ w.r.t.~the gauge body $C$ is defined as
\[R(K,C) = \min \{\rho \ge 0 : K \subset \rho C +t, \ t \in \mathbb R^n\}.\]
Surprisingly, the definition of a diameter with respect to a (possibly) non-symmetric gauge body $C \in \mathcal K^n$ (with $0 \in \mathrm{int}(C)$) is not unified. While in \cite{Le}
it is defined as
\[
D_{\max}(K,C) = \max_{x,y \in K} \|x-y\|_C,
\]
which we call the \cemph{maximal diameter}, and which at first view is the most natural generalization of a diameter for non-symmetric gauges; others (see c.f.~\cite{DGK}) preferred and partly argued to choose the following definition
of the \cemph{diameter} of $K$ w.r.t.~$C$:
\[
D(K,C) = 2 \max_{x,y \in K} R(\{x,y\},C).
\]
The latter definition allows to see the diameter as a best 2-point approximation of the circumradius of the whole set $K$. Another advantage of it is that it is translation invariant in both arguments. In contrast, for the maximal diameter
choosing $C$ with $0$ close to the boundary of $C$, the circumradius-diameter ratio may get arbitrarily small.
However, the choice of a definition should always fit its desired properties. For instance, if choosing an asymmetric gauge body is motivated by the desire to measure the distance from $x$ to $y$ different than that from $y$ to $x$, the latter should possibly be reflected in the length measurements (instead of measuring the length of the segment $[x,y]$ independently of its direction).
Thus there may be applications where we would prefer to measure the distance from $x$ to $y$ by $\|x-y\|_C$, which then would lead us to the maximal diameter.
And, this is part of our motivation for the investigation above, one can see that
\[
D_{\max}(K,C) = D(K, C \cap (-C)), \quad \text{while} \quad D(K,C) = D\left(K, \frac{C-C}2\right).
\]
Moreover, if $C$ is Minkowski centered, the results above show us, that we can bound those diameters in terms of the other and therefore also the circumradius-diameter ratio for the maximal diameter.
Finally, it is easy to see that there are also well motivated definitions of lengths of segments or directional breadths w.r.t.~a given gauge $C$ that lead to diameters that depend on the harmonic mean $\left(\frac{C^\circ - C^\circ}2\right)^\circ$ or the maximum $\mathrm{conv}(C \cup (-C))$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,399 |
Q: simple induction related question where we are doing it for k ∈ R suppose we want to prove by induction that
$$1^2+
2^2+
3^2+
4^2+ ....... + n^2= \frac{n}{6}(n+1)(2n+1)$$
it is very easy where we assume $k$ and $k+1$.
suppose we want to prove $$3^k>2^k, \forall k\in\mathbb R^+$$
then while doing induction (if we are allowed to), can we assume for any general $k$ and instead of proving it for $k+1$, are we allowed to do it for limit as $h$ tends to zero $k+h$ ?
A: The proof technique of induction is usually only used for natural numbers (0 inclusive). So I think you would be better off with a deductive proof rather than doing it by induction.
Check out this post for more about why.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,925 |
Фрэнсис «Фрэнк» Мерфи (, 21 мая 1947, Дублин, Ирландия — 5 января 2017, там же) — ирландский легкоатлет, серебряный призёр чемпионата Европы по лёгкой атлетике в Афинах (1969).
Спортивная карьера
В школьные годы становился чемпионом Ирландии по кроссу среди юниоров; также увлекался гэльским футболом. Выступал за спортивный клуб Clonliffe Harriers. Получил стипендию Джамбо Эллиотта Университета Вилланова в Филадельфии, штат Пенсильвания. В составе университетской команды дважды выигрывал эстафету в помещении и трижды — титул командные соревнования по кроссу
в первенстве Национальной ассоциации студенческого спорта (NCAA).
Являлся 13-кратным чемпионом Ирландии; трижды — на 800 м (1970, 1972, 1976), пятикратным — на 1500 м (1967—1971) и пятикратным — в беге на милю (1966—1967, 1969, 1971 и 1972). Был первым ирландцем, пробежавшим 800 м за 1:48 мин., 1500 м за 3:40 мин. и 5000 м за 14:00 мин.
На чемпионате Европы по легкой атлетике в Афинах (1969) завоевал серебряную медаль на дистанции 1500 м. На континентальном первенстве по легкой атлетике в помещении в Вене (1970) также был вторым, установив национальный рекорд. Выступал на летних Олимпийских играх в Мехико (1968) и в Мюнхене (1972) в той же дисциплине.
Дальнейшая карьера
После завершения обучения в области делового администрирования в США он вернулся в Ирландию и работал в различных коммерческих компаниях. За заслуги в спорте в 2014 г. он был введен в Ирландский зал легкоатлетической славы.
Ссылки
https://www.athleticsireland.ie/index.php/news/frank-murphy-hall-of-fame-induction-2014
https://www.irishtimes.com/sport/other-sports/irish-olympic-track-runner-frank-murphy-dies-aged-69-1.2927004
Бегуны на средние дистанции Ирландии
Легкоатлеты на летних Олимпийских играх 1968 года
Легкоатлеты на летних Олимпийских играх 1972 года | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 761 |
\subsection*{Abstract}
Mixed-criticality systems combine real-time components of different
levels of criticality, i.e.\ severity of failure, on the same
processor, in order to obtain good resource utilisation. They must
guarantee deadlines of highly-critical tasks at the expense of
lower-criticality ones in the case of overload. Present operating
systems provide inadequate support for this kind of system, which is
of growing importance in avionics and other verticals. We
present an approach that provides the required asymmetric integrity
and its implementation in the high-assurance seL4 microkernel.
\section{Introduction}\label{s:intro}
Traditionally, critical real-time systems use dedicated
microcontrollers for each function. With increasing functionality and
complexity of cyber-physical and other real-time systems, this is
creating space, weight and power (SWaP) problems, which force
consolidation onto a smaller number of more powerful processors. For
example, top-end cars reached 100 processors a few years ago
\citep{Hergenhan_Heiser_08}; with the robust packaging and wiring required for
vehicle electronics, the SWaP problem is obvious, and a driver for the
adoption of multitasking OSes \citep{AUTOSAR:AUTOSAR_Spec_42}.
The potential for consolidation is limited unless it is possible
to safely co-host functions of different \emph{criticality}, where
criticality is a well-established notion that represents the severity
of failure \citep{DO178B}. Certification standards require that safe operation of a
particular component must not depend on any less-critical components
\citep{ARINC653}.
Such \emph{mixed-criticality systems} (MCS) are becoming the norm in
avionics, but presently in a very restricted form: the system is
orthogonally portioned spatially and temporally, and partitions are
scheduled round-robin with fixed time slices \citep{ARINC653}. This
limits integration and cross-partition communication, and implies long
interrupt latencies and poor resource utilisation. The simple
partitioning approach will not meet the requirements of future
mixed-criticality systems \citep{Barhorst_BBHPSSSSU_09}.
Fundamental to good resource utilisation in MCS is the ability to
over-commit safely: The system's core \emph{integrity property} is
that deadlines of the highest criticality tasks must be guaranteed,
meaning that there is always time to let such tasks execute their full \emph{worst-case
execution time} (WCET). This may be orders of magnitude larger than the
typical execution time, and computation of safe WCET bounds for
non-trivial software tends to be highly pessimistic \citep{Wilhelm_EEHTWBFHMMPPSS_08}. This
means that most of the time the highly-critical components leave plenty of slack, which
should be available to less critical components, but must be available
to the critical component when needed.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{uav}
\caption{Highly simplified autonomous aerial vehicle architecture red is most critical,
blue least.}
\label{f:aav}
\end{figure}
Such a system needs support for downgrading timeliness guarantees selectively,
least critical ones first. In general, this cannot be achieved by
simply giving the most critical tasks the highest priority. Consider
the simplified architecture of an \emph{autonomous aerial vehicle} (AAV)
in \autoref{f:aav}. The most critical component is the low-level
flight control, which keeps the vehicle stable and moving towards a
waypoint. It executes every 100\,ms and normally takes about 10\,ms but
has a WCET of 70\,ms. Next critical are the mission plan, sensor
filtering and C\&C components, which have execution rates of between 1
and 10\,Hz, normally run for a combined 200\,ms every second but have a combined
WCET of 500\,ms per second. The CAN bus, which connects a video camera
and various
sensors of secondary importance, can deliver packets every 12.5\,\(\mu\)s
and does not buffer.
If the critical components are given higher
priority than the CAN driver, it will drop many
packets even during normal operation, despite the system
having sufficient headroom to run everything.
The standard realtime (RT) scheduling approach is \emph{rate-monotonic
priority assignment} (RMPA) \citep{Liu_Layland_73}, which gives
highest priority to tasks with the shortest periods. RMPA is easy to
analyse and known to be optimal for fixed priorities; it is highly desirable to retain it
for MCS.
A further complication is that components of different criticality
must be able to communicate, and access shared data
\citep{Burns_Baruah_13}. For example, the
AAV's mission plan defines the waypoints to be used by the flight
control, including some fail-safe return-home path in case the AAV
loses ground-station connectivity. It is updated by the ground station via the command and
control (C\&C) component,
and amended by the sensor filtering component for obstacle avoidance;
the latter component receives input from various sensors, including
camera and other sensor input via the CAN bus.
Such communication, including concurrency control between
components accessing the same data, must be possible while
guaranteeing critical deadlines.
In summary, an OS for mixed-criticality systems must:
\begin{compactitem}
\item provide high-assurance spatial and temporal isolation, to allow
critical components to be assured independently of less critical
ones;
\item decouple criticality from priority, to ensure critical,
low-rate threads meet their deadlines;
\item provide mechanisms that allow analysing the timeliness of
critical tasks, even if they communicate with less critical ones;
\item have well-understood temporal behaviour, especially
bounded and known WCET for all operations;
\item be highly assured for correct operation.
\end{compactitem}
No such OS exists to date.
We present the design and implementation of such an OS, based on the
seL4 microkernel for single-core systems. seL4 is an attractive starting point, as it is a
high-assurance OS kernel that has been
comprehensively verified \citep{Klein_AEMSKH_14}, and is the first and
still only protected-mode OS in the literature with a complete and sound WCET analysis
\citep{Blackham_SCRH_11}.
We do \emph{not} claim to have invented new scheduling models or
theory. In fact, the system we present in \autoref{s:design} is, as
scheduling theory goes, known as \emph{static mixed criticality}
\citep{Baruah_BD_11}. Our claims are about practical systems, specifically:
\begin{compactenum}
\item the design of a low-overhead temporal resource management model
that is based on a small number of simple, policy-free mechanisms,
suitable for a high-assurance implementation,
matches the above requirements of MCS but also supports a wide
range of other uses (\autoref{s:design});
\item its implementation in the seL4 microkernel in a way that retains
seL4's general-purpose
nature and verifiability\footnote{We have not
formally re-verified the modified kernel, and only claim that our
modifications are moderate in terms of kernel changes and no more
difficult to verify than the baseline kernel.} (\autoref{s:impl});
\item an evaluation that demonstrates that the modifications do not
unduly impact seL4's performance, and support low-overhead
implementations of different real-time and best-effort scheduling
models (\autoref{s:eval}).
\end{compactenum}
\section{Background and Related Work\label{s:background}}
In the rest of this paper, and this section specifically, we talk
about general real-time concepts as well as OS
abstraction. Specifically there are two related concepts relating to
the execution model. We will use the term \emph{task} in the sense
established in the RT community, namely \emph{a set of related jobs
which jointly provide some system function}, where a job is \emph{a
unit of work that is scheduled and executed by the system}
\citep{Liu:rts}. We use the term (kernel-scheduled) \emph{thread} to refer to the
execution abstraction familiar to the OS community. The term ``job'',
which we will not use further, corresponds to a unit of work that is conducted
by a thread, while ``task'' maps onto a thread, plus code and data.
In short, we will use ``task'' when referring to general RT issues,
and ``thread'' when talking about a specific OS concept. In practice,
the terms are largely interchangeable.
\subsection{Scheduling models vs.\ mixed criticality}\label{s:mc-sched}
RT scheduling generally assumes \emph{periodic tasks}, which maps well
onto typical control systems, where different activities execute
periodically albeit with different periods. Non-periodic
(``sporadic'', i.e.\ interrupt-driven) tasks are incorporated in such a model by requiring
a defined \emph{minimum arrival time}, corresponding to a maximum
interrupt rate, which is used as the task's period for the
schedulability analysis. RT tasks have a \emph{deadline} by which a
computation must be finished. The general assumption is that deadlines
are implicit, meaning the deadline is the end of the period.
As discussed in the introduction, the ability to overload, while
guaranteeing critical deadlines, is core to the notion of
MCS. Classical RT scheduling approaches have a notion of (fixed or
dynamic) priority as the sole determinant of access to CPU time, with
equal-priority tasks (if permitted) being (preemptively or non-preemptively)
scheduled FIFO. If the system is overloaded, this means that
the lowest-priority deadlines are missed. When using RMPA, this
victimises the tasks with the lowest rates. In effect, criticality
equals rate in RMPA.
The main alternative to RMPA is \emph{earliest deadline first} (EDF)
scheduling. This is a dynamic priority scheme, which at any time
schedules the task with the closest deadline. Unlike fixed-priority
schemes, such as RMPA, EDF is optimal on a uniprocessor in that it can schedule
any task set, as long as the total utilisation does not exceed
100\%. However, the dynamic prioritising implies that under overload,
EDF drops deadlines of all tasks \citep{Buttazzo_05}, meaning that there is no
concept of task criticality at all.
MCS require control over which deadlines will miss in the case of
overload: those of the tasks with low criticality (called \crit{low}
tasks from now on), while guaranteeing deadlines of \crit{high} tasks,
\emph{irrespective of scheduling priority}. This requires a mechanism for
limiting CPU time of high-priority tasks.
An established way of providing isolation is through \emph{scheduling
reservations} \citep{Mercer_ST_93, Oikawa_Rajkumar_98}, where a
reservation \emph{guarantees} a certain share of the CPU to a periodic task. Such schemes are
popular in soft RT systems, e.g.\ multimedia, and some allow slack
time to be used by best-effort tasks \citep{Brandt_BLB_03}.
Scheduling reservations can be implemented as \emph{sporadic servers}
for RMPA \citep{Sprunt_SL_89} and with
\emph{constant bandwidth servers} (CBS) \citep{Abeni_Buttazzo_04}
on EDF. \label{s:reservations}
Reservations present a guarantee by the kernel that the reserved
bandwidth is available. This means that they do not support
over-committing. Also, the kernel must perform a schedulability
analysis as \emph{admission control} whenever a reservation is
created. Schedulability tests can be complicated and frequently
constitute a trade-off between cost of the test and achievable
utilisation.
Recently the concept of a \emph{mode switch} was introduced to support
mixed criticality \citep{Burns_Davis_14}: when the system is unable to
meet its deadline, it enters a high-criticality mode, where the
priority of \crit{high} tasks is boosted above all \crit{low} tasks.
To achieve this, \crit{high} tasks are assigned multiple reservations, one per criticaly level.
In a two-criticality system, \crit{high} tasks have a pessimistic WCET
and an optimistic \emph{worst-observed execution time} (WOET).
\crit{low} tasks have just one estimate.
When the system is in \crit{low} mode, \crit{high} tasks run according to their WOET.
If all reservations in this mode are schedulable, temporal isolation is guaranteed.
However, if a \crit{high} task exceeds its WOET, the system switches to \crit{high} mode, degrading \crit{low} tasks and assuring assymetric protection between \crit{high} and \crit{low} threads without falsely correlating rate and urgency.
\subsection{Support for sharing and communication}\label{s:sharing}
As indicated in the introduction, integrity of critical components
must be assured even when tasks communicate and share. In our AAV
example of \autoref{f:aav}, the mission plan component encapsulates
waypoints. The \crit{high} flight-control component must be able to access a
consistent view of the flight plan, despite the \crit{lower} C\&C and other
components performing updates.
Encapsulating the shared data and the code that accesses and modifies
it into a single-threaded \emph{resource server}
\citep{Brandenburg_14} is a simple and effective way to achieve the
necessary transaction semantics. Obviously, this server has the criticality level of its most
critical client, but must also act on behalf of a \crit{low} client. This
creates a temporary \emph{criticality inversion} where the \crit{low} client
blocks the \crit{high} one. This is an unavoidable consequence of
sharing, and the design must ensure that it does not
cause the \crit{high} task to miss deadlines.
\begin{figure}[ht]
\centering
\setlength{\unitlength}{1mm}
\begin{picture}(50,25)(-5,-5)
\thicklines
\put(-5,0){\vector(1,0){50}}
\put(7,-4){Priority inversion bound}
\put(0,-5){\vector(0,1){25}}
\put(-4.5,2){\rotatebox{90}{Complexity}}
\put(2,15){OPCP}
\put(12,3){IPCP}
\put(25,11){PIP}
\put(35,1.5){NCP}
\end{picture}
\caption{Comparison of real-time locking protocols based on
implementation complexity and priority inversion bound.}
\label{f:locking}
\end{figure}
There are multiple ways to achieve mutual exclusion in fixed-priority
RT systems \citep{Sha_RL_90}, the most common being non-preemptive critical sections (NCP), the priority inheritance protocol (PIP), and the immediate\footnote{Also known as highest lockers protocol and \texttt{PRIO\_PROTECT} in POSIX.} and original priority ceiling protocols (IPCP and OPCP). For RT systems, the most important factor for mutual exclusion is the bound on priority inversion, where a low priority task blocks a high one. For efficient systems, the concern is execution cache performance, for secure systems
the concern is the avoidance of channels.
\autoref{f:locking} shows the four protocols in terms of complexity
and priority inversion, none is a silver bullet.
NCP is simplest yet has the longest blocking time, IPCP requires the
priorities of all lockers to be known \emph{a priori}, PIP has high
implementation complexity and risks deadlock if resource ordering is
not used. OPCP is even more complex, and requires global state to be
maintained across all locks in the system, which is not acceptable for
seL4 as it introduces covert channels and is incompatible with seL4's
decentralised user-level resource management. We will show in
\autoref{s:sc} how IPCP can be easily implemented without the kernel
requiring knowledge about critical sections.
As a mechanism for supporting sharing,
Fiasco~\citep{Steinberg_04:dipl} introduced the idea of scheduling contexts, separate to
execution contexts (threads). Scheduling contexts encapsulate priority, scheduling parameters and accounting detail,
and pass between threads over IPC. \citet{Steinberg_BK_10} extended this with bandwidth
inheritance~\citep{Lamastra_LA_01, Lipari_LA_04, Faggioli_LC_10} over IPC. This is
equivalent to PIP combined with reservations.
When an IPC from client $B$ arrives at a server $S$, who is serving a client $A$ with an expired budget, $B$ budget is used to complete $A$'s request such that $B$ does not have to wait for $A$'s reservation to be replenished.
This kernel-implemented policy, also referred to as \emph{helping},
prevents the server from choosing alternatives which might be more
appropriate in a particular situation, such as aborting $A$'s request.
The Fiasco design of scheduling contexts \citep{Lackorzynski_WVH_12} is tied to the traditional L4 model of
sending IPC messages directly to threads, a model which
has been abandoned in modern L4 kernels (including Fiasco and seL4) as it introduces covert
channels~\citep{Shapiro_03}. It is not supported on the later,
capability-based Fiasco.OC kernel.
\textrm{Composite}\xspace~\citep{Parmer_West_08} completely frees the kernel from any scheduling policy by providing
mechanisms for hierarchical user-level scheduling. It reduces overhead-related capacity loss
by configuration buffers shared between user-level and the kernel.
Some capacity loss remains as timer interrupts must be delivered down
the scheduling hierarchy. This approach does not suit seL4, as the
required reasoning about concurrent access (by kernel and user-level)
to those buffers would drastically
increase verification overhead \citep{Klein_AEMSKH_14}.
Unlike all L4 microkernels, \textrm{Composite}\xspace implements a migrating thread
model~\citep{Ford_Lepreau_94}. This implies that access to shared resources
does not block, thus avoiding priority inversion, although at the cost
of requiring all server code to be re-entrant, which is fairly
heavy-handed policy for a microkernel.
Also, it only shifts the problem, as mutual exclusion is
still needed, including a way of limiting priority inversions. Given
the challenges of getting concurrent code right, it should be
minimised in high-assurance systems.
Linux introduced an implementation of the POSIX
\texttt{SCHED\_DEADLINE} in 3.14, which implements EDF with CBS for
temporal isolation. However RT tasks in Linux are higher priority than
all other tasks in the system, and cannot be over-committed (although
cgroups allow limiting the RT class to a certain share of the CPU).
Quest-V~\citep{Li_WCM_14} and PikeOS~\citep{Kaiser_Wagner_07} are both separation kernels for multicore systems that dedicate
cores to different criticalities.
AUTOBEST~\citep{Zuepke_BL_15} is another separation kernel where the authors demostrate
implementations of AUTOSAR and ARINC653 in separate partitions.
\iffalse
\bbb{This discussion seems to gloss over the vast literature specifically on MC scheduling.
For example, the model advanced in this paper --- fixed priority +
budget monitoring --- has been studied and analyzed in detail in
\citep{Baruah_BD_11}. What I gather seL4 implements is called "static
mixed criticality" (SMC) by Sanjoy and friends.}
\FIXME{Mention UNC's MC\(^2\) framework and EDF-VD?}
\bbb{You should probably also review EDF-VD, which has featured in many recent papers, is provably optimal wrt some metrics, and which I think is generally understood to be the best MC scheduler from a theory point of view today. (Well, Sanjoy had another optimal something at the last RTSS, don't know about it in detail.)
In terms of practical approaches, UNC's MC\(^2\) framework is probably the most well-regarded. It is inspired by avionics criticality levels A-F; I would consider it relevant in the context of this paper as well, especially because they also have an in-kernel prototype.}
\fi
\subsection{seL4}\label{s:sel4}
seL4 is a high-performance OS microkernel with an unprecedented degree
of assurance: it features formal proofs of implementation correctness
down to the binary, proofs of spatial isolation properties
(enforcement of confidentiality, availability and integrity) and a
complete and sound analysis of worst-case execution times on ARMv6 processors
\citep{Klein_AEMSKH_14}. This assurance makes seL4 an appealing
candidate OS for critical systems.
seL4 is designed to be a general-purpose platform,
supporting a wide range of use cases. This is a reason why it has a
strong emphasis on performance, as many of the envisioned deployment
scenarios are performance-sensitive (e.g.\ mobile devices). Formal
verification is a strong motivator for generality: the cost of
assurance is best amortised if all use cases are supported by the
same, unmodified kernel \citep{Heiser_Elphinstone_16}. As the
maintainers commit to re-verify any changes to the mainline kernel,
they are only interested in changes that make the kernel more general,
not more specialised.
\subsubsection{seL4 overview}
In line with the microkernel minimality principle
\citep{Liedtke_95}, seL4 only provides a small number of policy-free
mechanisms. Specifically it provides for threads, represented as
\emph{thread control blocks} (TCBs), \emph{address spaces}, which are thin wrappers
around hardware page tables, and \emph{frame} objects, which represent physical
memory that can be used to populate address spaces by \emph{mapping}.
It further provides port-like \emph{endpoint} objects for synchronous
(rendezvous-style) communication and \emph{notification} objects,
which are essentially arrays of binary semaphores.
Like other security-oriented systems, seL4 uses capabilities
\citep{Dennis_VanHorn_66} for controlling access to all spatial
resources and providing complete mediation similar to KeyKOS
\citep{Bromberger_FFHLS_92} and EROS \citep{Shapiro_SF_99}. Besides
its assurance story, seL4's most characteristic aspect is its
isolation-oriented approach to memory management, which is made
policy-free by fully delegating it to user level.
Specifically, the kernel never allocates memory. After booting, seL4
hands all rights to any unused memory to the first user process in the
form of capabilities to \emph{Untyped} memory. The only operation
supported on Untyped is to \emph{retype} into some other object type
(TCB, page tables, frames etc), or to \emph{revoke} of an earlier retype. That way
user-level managers have full responsibility for any memory
management. For example, the initial process can partition Untyped
memory into several disjoint pools, and set up secondary resource
managers in each partition. The partitions are then totally isolated,
unless the initial process also provides access to some shared
resources (e.g.\ frames or endpoints) to support communication.
Like any kernel operation (other than the \code{yield()} syscall which
simply forfeits the remainder of the present time slice), IPC and
notifications are authorised by capabilities: a thread needs an
endpoint capability in order to send or receive messages, and a
notification capability for signalling or collecting notifications.
Similar to other L4 kernels, the kernel not only supports basic
\code{send()} and \code{receive()} operations, but also two
versions of a send followed by a receive in one atomic syscall. First
there is the RPC-like \code{call()}, which is typically used by a
client invoking a server. When invoking \code{call()} on an
endpoint, the kernel creates a single-use \emph{reply capability},
which refers to a virtual, temporary \emph{reply endpoint}. The kernel
delivers the reply capability to the receiver listening on the endpoint, and
makes the sender wait on the reply endpoint.
The second combined call is \code{reply\_receive()}, which sends a
message to the (implicitly supplied) reply endpoint and then makes the
invoker wait on a new request on the endpoint specified in the
syscall. Once used in the reply, the reply endpoint and capability are
removed.
\subsubsection{Scheduling}\label{s:sel4sched}
Management of time is comparatively under-developed in seL4. It presently
implements the same simplistic scheduling model used in most L4
kernels for 20 years: priority-based round robin. The only
controllable parameters are a thread's priority and time slice. This
is not sufficient for supporting MCS, as indicated by the examples
given in the introduction.
On IPC, seL4 uses a \emph{direct process switch}
\citep{Liedtke_93} where possible, to avoid the cost of invoking the scheduler: the
IPC switches context from sender to receiver, but with the receiver
running on the sender's time slice, until it replies or is
preempted. When rescheduled after preemption, the server will execute
on its own time slice (and after replying to the client, the latter
may execute on the server's time slice). The IPC paths which do not
require scheduler invocation are implemented by separate,
highly-optimised \emph{fast-path} code.
This form of time-slice donation \citep{Steinberg_BK_10}
has been criticised as inappropriate for RT systems
\citep{Ruocco_08}, as time is not accounted
properly. Consequently, the Fiasco L4 kernel allows the sender to
specify whether donation is permitted.
However, even without time-slice donation, traditional L4 scheduling
is problematic. Consider a typical scenario of two clients, \(A\),
\(B\), invoking server \(S\).
Both clients have the same priority, which is lower than the server's,
and the same time slice length, so they ought to get equal amounts of
time. Assume client \(A\) requests long-running operations from \(S\),
while \(B\)'s requests are short. The server's time is not accounted against the
clients, and \(A\) gets a much higher share of the system than \(B\).
Furthermore, if the scheduler is invoked on each IPC, \(A\) and \(B\)
will alternate execution after each server invocation, making it very
difficult to reason about the progress of individual
tasks. Alternatively, if \(A\) continues executing after the invocation of \(S\)
returns, then \(A\) can effectively deny \(B\)'s service by invoking \(S\)
in a tight loop.
In summary, the L4 model of managing time is unsatisfactory no matter how it is
implemented. The bandwidth-inheritance approach taken in some
kernels \citep{Steinberg_BK_10} is not a good solution either for
the reasons explained in \autoref{s:sharing}: on the one hand there is
the general issue of complexity and poor priority-inversion bound of
inheritance. On the other hand, inheritance offers no policy flexibility on managing overruns in servers.
Additionally, while Fiasco's implementation of bandwidth inheritance
allows for bounded priority inversion,
it violates temporal isolation: \(A\) is allowed to consume \(B\)'s
budget.
\subsection{Summary}
We want a model that is simple enough to be suitable for seL4,
provides temporal isolation, and provides freedom in the
implementation of policies for dealing with isolation violations. At
the same time, it must continue to support all existing or anticipated
use cases of the kernel.
\iffalse
\footnote{For security-oriented
temporal isolation the scheduler is configurable with
multiple non-preemtible scheduling domains, which are scheduled for a fixed time
slice. These domains are unsuitable for real-time use due to the
large algorithmic capacity loss and the high interrupt latencies.}
\fi
\section{Scheduling Model}\label{s:design}
We now present a scheduling model for seL4 which satisfies all
requirements for MCS stated in \autoref{s:intro}. It is based on a
small number of abstractions, namely
\begin{compactitem}
\item periodic threads with hard CPU bandwidth limits
\item scheduling contexts
\item timeout exceptions
\item notion of criticality in addition to priority \& explicit mode switches.
\end{compactitem}
Our model matches the approach known as \emph{static mixed
criticality} in scheduling theory \citep{Baruah_BD_11}, which
provides appropriate tools for analysis.
\newlength{\Unit}\setlength{\Unit}{1em}
\newcommand{\Bx}[1]{\raisebox{0.7ex}{\fbox{\rule{#1\Unit}{0pt}\rule{-0.7em}{0pt}}}}
\newcommand{\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}}{\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}}
\newcommand{\rule{\Unit}{0pt}}{\rule{\Unit}{0pt}}
\newcommand{\Wp}{}
\begin{figure*}[t]\centering
\subfloat[Two periodic RT threads plus one best-effort thread running in slack time.]{
\begin{tabular}{@{}rrrrrc@{}}
\bf P& \bf T&\bf B& \bf U& \bf u& \bf Schedule \\
3 & 5 & 1 & 0.2 & 0.2 & \Bx{1}\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk \\
2 & 10 & 5 & 0.5 & 0.5 & \rule{\Unit}{0pt}\Bx{4}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\rule{\Unit}{0pt}\Bx{2}\Wp\\
1 & 20 &20 & 1.0 & 0.3 & \rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\Bx{3}\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\\
\end{tabular}
}
\hspace{5ex}
\subfloat[Three full-budget threads scheduled as in traditional L4.]{
\begin{tabular}{@{}rrrrrc@{}}
\bf P& \bf T&\bf B& \bf U& \bf u& \bf Schedule \\
2 & 1 & 1 & 1.0 & 0.5 & \Bx{1}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\\
2 & 1 & 1 & 1.0 & 0.5 & \rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\Bx{1}\rule{\Unit}{0pt}\Bx{1}\\
1 & 1 & 2 & 1.0 & 0.0 & \rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\rule{\Unit}{0pt}\rule{-1pt}{0pt}\rule{1pt}{0.2ex}\Tk\\
\end{tabular}
}
\caption{Examples of thread schedules. P=priority, T=period,
B=budget, U=max.\ utilisation, u=actual utilisation.}
\label{f:budget-ex}
\end{figure*}
\subsection{Execution-time limits: Budgets}
A key observation from \autoref{s:mc-sched} is that pure
priority-based scheduling cannot satisfy the requirements of MCS, and
we need a mechanism for temporal isolation. To achieve this we
introduce the notion of a \emph{budget}, which is a \emph{hard limit} on the
time a thread can consume during a \emph{period}. The ratio of budget
over period is the limit of CPU bandwidth a thread can consume.
Budgets are similar to the reservations introduced in
\autoref{s:reservations}, except that the kernel makes \emph{no guarantee
that any bandwidth is achieved}, only that the limit is not
exceeded. This makes admission control a user-level responsibility,
avoiding any policy in the kernel about whether admission should be
determined on- or off-line, should be static or dynamic, or should be
hierarchical of flattened \citep{Lackorzynski_WVH_12}. In particular,
the system designer may decide to trust a particular task not to use
its budget (except in emergencies) and perform the schedulability
analysis based on that knowledge.
\emph{Despite providing weaker guarantees, budgets are a more powerful
concept than reservations.} Specifically, if a set of reservations is
schedulable, i.e.\ admission control succeeds, then budgets will
produce the same schedule, i.e.\ they behave like reservations. If,
however, the total is not schedulable, but the sum of all budgets
above some threshold priority \(p\) is, then all budgets of tasks whose
priority exceeds \(p\) still behave like reservations, but nothing of
priority \(\leq p\) is guaranteed any CPU time.
This property allows us to safely overload a system with predictable
outcomes and without the kernel performing any admission control. We
will see later how this example of \emph{less is more} allows us to support MCS.
Specifically, we replace the kernel's notion of a \emph{time slice} by
two new attributes: \emph{period} and \emph{budget}. The budget is
less than or equal to the period and the ratio specifies the maximum
share (utilisation) of the CPU the thread can possibly get. This is essentially the
model of sporadic servers introduced by \citet{Sprunt_SL_89}, except
that we use budgets instead of reservations.
The operation of the seL4
scheduler changes only slightly: it still picks the highest-priority
runnable thread, using round-robin within a priority. The difference
is that when the kernel schedules a thread, it sets a timer to enforce
the budget, and a thread whose budget is expired is no longer
runnable. The period specifies when the thread's budget is
replenished, thus making it runnable again. \autoref{f:budget-ex} shows
some examples.
Similarly to the budget not guaranteeing any time, the
period does not guarantee that a thread is actually scheduled
periodically (which depends on the priorities, periods, and budgets of
all other threads with the same or higher priority). Note also that
this model exactly emulates the existing seL4 scheduler when all
budgets are \emph{full}: if every thread has a budget that is equal to its
period, the period has the same semantics as the time slice used to
have.
\subsection{Scheduling contexts}\label{s:sc}
In order to provide better control over the time resource, we
introduce a \emph{scheduling context} (SC) object that grants access
to time. Instead of a time slice, a thread (in its TCB) holds a
scheduling context capability (scCap), without such a valid scCap, the thread is not
runnable.
The SC consists of the period, budget pair introduced above and thus
represents the maximum bandwidth a thread may consume.
The semantics of SCs are equivalent to
hard reservations in Linux/RK~\citep{Rajkumar_JMO_98}, in that once the budget is exhausted,
no thread can run on that SC until it is replenished, however they differ in two ways.
First, we only allow one thread per SC at a time, but SCs can be passed between threads
via IPC for cooperative scheduling.
This allows for a minimal, single level scheduler in the kernel.
Second, the kernel does not conduct an admission test. Our SCs differ
from those of NOVA \citep{Steinberg_BK_10} in that priority remains
a thread attribute instead of being associated with an SC, and we
allow only one SC per thread.
SCs are like other seL4 objects, in that any thread that can allocate memory
can create them. However, setting the budget requires
special privilege, as creating budgets amounts to control over the
right to consume CPU time. It must be authorised by a capability.
We use an approach that is analogous to managing interrupt sources in
seL4. Specifically, there is a per-core virtual scheduling-control
object, represented by the \code{sched\_control} capability. This capability
must be presented when setting the budget of an SC. The
kernel creates this capability at boot time and hands it to the initial task
as part of the startup protocol.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\columnwidth]{server}
\caption{Resource server implementation as a ``passive'' thread
without a scheduling context.}
\label{f:passive}
\end{figure}
SCs provide a clean solution to the shared-server accounting dilemma
outlined in \autoref{s:sel4sched}. We allow a (server) thread without
an scCap to wait on an endpoint, we call this a \emph{passive
server}. If a client sends a message to this
endpoint, the IPC will transfer the client's scheduling context to the
server, which then executes on the client's \emph{borrowed} budget. The SC returns to
the client when the server replies to the client request. An example
is given in \autoref{f:passive}, where an SC-less resource server has
two clients, each holding an SC (indicated by the clock dial
representing a CPU bandwidth bound).
For security, the sender must agree to the donation, the IPC will fail if
the receiver has no SC but the sender is unwilling to lend its own. No
SC transfer takes place if the receiver has its own SC (active
server).
A passive server can trivially implement the immediate priority
ceiling protocol introduced in \autoref{s:sharing}, by setting its
priority to the ceiling of priorities of all clients. As any client
needs a send capability on the server's endpoint, usermode managers
can control access to the server, and thus enforce the priority
ceiling. We discuss in \autoref{s:timeout} how we deal with the server running
out of budget.
\iffalse
Both cases enforce temporal isolation, either completely
(active server) or partially but controlled and correctly accounted
(passive server). We will
\fi
\begin{figure}[htb]\small
\hspace*{0.5em}%
\begin{minipage}{0.5\columnwidth}
\begin{verbatim}
notification_t ntfn;
sched_context s_sc;
tcb_t s_tcb;
void init() {
// bind SC to TCB
bind(s_sc, s_tcb);
// create server thrd
start_thread(s_tcb);
// block and allow
// server to run
wait(ntfn);
// server initialised
// convert to passive
unbind(s_sc);
}
\end{verbatim}
\end{minipage}
\begin{minipage}{0.4\columnwidth}
\begin{verbatim}
notification_t ntfn;
endpoint_t ep;
void server() {
// run on init SC
initialise();
// signal & block
signal_recv(ntfn, ep);
while (true) {
// run on client SC
process_request();
// reply & block
reply_recv(ep);
}
}
\end{verbatim}
\end{minipage}
\caption{Passive server initialisation.}
\label{f:init}
\end{figure}
Passive servers must be initialised with an initialisation SC
and then communicate to the initial task when they are done,
such that the SC can be removed. We support this with
a new system call \texttt{signal\_receive()}, which combines
signalling a notification with an IPC receive. \autoref{f:init} shows
how initialisation works in principle.
Call-reply\&wait IPC with SC transfer avoids invoking the scheduler or
updating accounting data during IPC, and thus retains the low overhead
of the direct process switch optimisation. It has in fact many of the properties of a
migrating thread model \citep{Ford_Lepreau_94}, specifically it
avoids having multiple schedulable entities for what is logically a
single-threaded operation. The advantage over migrating threads is
that the kernel does not have to provide stacks on the fly, and thus is free
of policy decisions such as determining stack sizes, charging for
memory, whether to cache stacks. Instead, our model requires explicit
user-level management of stacks through thread objects.
\FIXME{Mention that scheduling theory just works, with budget=WCET.}
\subsection{Managing thread execution}
A periodic thread needs to suspend itself when it has finished
processing for the current period. It does so by calling
\code{yield()} on its own SC.\footnote{Authorising \code{yield()} with
an SC capability removes seL4's
previous anomaly of having a syscall that requires no capability to
execute. \code{yield()} can also be called on another thread's SC,
cancelling that thread's current budget. However, we do not claim
that there is a good use case for this.} An event-triggered (sporadic)
thread instead waits on its IRQ notification. Of course, even if the notification
is signalled (by an IRQ or another thread), the sporadic thread will
only execute if it has budget.
\iffalse
\autoref{f:trigger}
shows the templates for both kinds of threads.
\begin{figure}[htb]\small
\hspace*{0.5em}%
\begin{minipage}{0.5\columnwidth}
\begin{verbatim}
sched_context_t my_sc;
void periodic() {
initialise_p();
while (1) {
yield(my_sc);
process_p();
}
}
\end{verbatim}
\end{minipage}
\begin{minipage}{0.4\columnwidth}
\begin{verbatim}
notification_t my_irq;
void sporadic() {
initialise_s();
while (1) {
wait(my_irq);
process_s();
}
}
\end{verbatim}
\end{minipage}
\caption{Typical structure of periodic and sporadic threads.}
\label{f:trigger}
\end{figure}
\fi
Sometimes explicit changes of a thread's priority are needed, e.g.\
when implementing IPCP without encapsulating the critical section into
a separate server. In order to change a thread \(B\)'s priority,
thread \(A\) must hold a capability to \(B\)'s TCB. In order to
prevent arbitrary priority changes, we re-introduce the concept of a
\emph{maximum controlled priority} (MCP) that was used in early L4
versions \citep{Liedtke_96:rm}.
Specifically, \(A\) cannot \emph{raise}
any thread's priority, including its own, higher than \(A\)'s
MCP. Note that this does not stop \(A\) from having a priority higher
than its MCP, but some other thread must have set it up.
We add two further operations on scCaps in order to allow fine-tuning
scheduling decisions. The first, \code{consume}, obtains the total
time accounted the designated scheduling context since the last such
enquiry.
The second, \code{yieldto()}, allows user level to manipulate
the kernel's scheduling queues. When invoked on a scCap, \textbf{and} the
designated SC is presently associated with a thread whose priority
does not exceed the callers MCP, \textbf{and} the thread has budget
available in its present period, \textbf{then} that thread is moved to the head
of the ready queue of its priority. This ensures that it is the next
thread to be scheduled if no higher-priority threads are
runnable. Invoking \code{yieldto()} implicitly invokes \code{consume},
i.e. it returns and resets the time accumulated on the SC.
\iffalse
\begin{table}[t]\centering
\subfloat[All critical deadlines will be met.]{\label{f:simple-a}
\begin{tabular}{@{}lrrrrrrc}
&\bf C & \bf P& \bf T&\bf B& \bf U\\
\(T_2\) & 2 & 2 & 100 & 40 & 0.40 \\
\(T_1\) & 1 & 1 &1000 & 200 & 0.20 \\
\(T_0\) & 0 & 3 & 10 & 4 & 0.40 \\
\end{tabular}
}
\subfloat[Medium-critical deadlines may miss.]{\label{f:simple-b}
\begin{tabular}{@{}lrrrrrrc}
&\bf C & \bf P& \bf T&\bf B& \bf U\\
\(T_2\) & 2 & 2 & 100 & 40 & 0.40 \\
\(T_1\) & 1 & 1 &1000 & 400 & 0.40 \\
\(T_0\) & 0 & 3 & 10 & 4 & 0.40 \\
\end{tabular}
}
\caption{Parameters of a simple sample system. C=criticality, P=priority, T=period,
B=budget, U=utilisation.}
\label{f:simple}
\end{table}
\fi
\subsection{Budget overrun}\label{s:timeout}
We provide \emph{timeout exceptions} in order to detect budget
overrun, analogous to seL4's treatment of other exceptions. An
seL4 thread already has an exception
endpoint. If an exception is triggered, the kernel sends a message to
the appropriate endpoint on the faulting thread's behalf. An exception
handler waiting on the endpoint will then receive the message and take
appropriate action. By replying to the exception message, it unblocks
the faulting thread (possibly after adjusting its instruction pointer
to skip an emulated instruction). In practice, many threads share the
same exception endpoint (and thus handler).
We extend this model by adding a timeout-exception endpoint: the
kernel sends a message to that endpoint when the thread exceeds its
budget, and the handler can take appropriate action, which may include
adjusting the faulting thread's budget. If the handler increases the
budget and then replies to the fault message, the thread will continue
to run on the remainder of the enlarged budget.
A thread without a timeout-exception endpoint is simply rate limited.
\iffalse
Obviously, handling such a timeout exception is not cheap, as it
requires multiple system calls to handle: sending a message to the
handler, and running the handler, plus whatever syscalls that one
needs to deal with the situation. It is thus not meant for routine
cases, such as a best-effort thread exhausting its present budget, but
for truly exceptional situations.
One such use is for the passive resource server of
\autoref{f:passive}. Assume that Client\(_2\) (\(C_2\)) is of higher criticality
than C\(_1\). Using budgets we can protect C\(_2\)'s execution from
direct interference by C\(_1\). However, this is not sufficient, if
the server runs out of budget while executing on C\(_1\)'s SC. In such
a case, C\(_2\) would be blocked until C\(_1\)'s budget is
replenished, allowing the single-threaded server to complete the
request and being able to handler C\(_2\)'s request.
\fi
Timeout exceptions allow recovering from priority/criticality
inversions, as possible in the passive resource server of
\autoref{f:passive}. If the server's borrowed SC runs out of budget,
its timeout handler can implement appropriate policy, such as letting
the server complete the request on an emergency budget, forcing a
reset or roll-back, possibly
coupled with taking some additional safety precautions prejudicial to
C\(_1\), such as suspending C\(_1\) or affecting a criticality mode
switch. This is in contrast to kernel-implemented helping schemes,
which implement a specific policy.
\iffalse
\autorefsub{f:simple}{f:simple-a}. It consists of three tasks, the
\crit{high} \(T_2\), the \crit{medium} \(T_1\)
and a best-effort but high-rate \(T_0\). With the given parameters,
both RT tasks will meet all their deadline, despite a best-effort task
running at higher priority.\footnote{We ignore for simplicity that
RMPA cannot generally achieve 100\% utilisation.}
However, if we assume that the WCET of \(T_1\) is actually 400 time
units, its deadline cannot be guaranteed, as shown in
\autorefsub{f:simple}{f:simple-b}.
The two parameter sets can actually be part of the same
scenario. Assume that 400 is the real WCET of \(T_1\), while 200 is an
optimistic approximation. We can imagine running the system with the
optimistic parameters of \autorefsub{f:simple}{f:simple-a}, until we
encounter a situation where \(T_1\) runs out of budget. We can use
\(T_1\)'s timeout exception
\fi
\begin{table}[th]\centering
\begin{tabular}{|lrrrrr|}
\hline
&\bf C & \bf P& \bf T&\bf B & \bf U\\
\hline
\rowcolor{Lavender}
\(T_5\) & 1 & 6 & 10 & 2 & 0.20 \\
\rowcolor{Lavender}
\(T_4\) & 1 & 5 & 20 & \(2|7\) & \(0.1|0.35\) \\
\rowcolor{SkyBlue}
\(T_3\) & 0 & 4 & 25 & 5 & 0.20 \\
\rowcolor{Lavender}
\(T_2\) & 1 & 3 & 40 & 4 & 0.20 \\
\rowcolor{SkyBlue}
\(T_1\) & 0 & 2 & 60 & 6 & 0.20 \\
\(T_0\) & 0 & 1 & 100 & 100 & 0.00 \\
\hline
\end{tabular}
\caption{Parameters of a sample system, where \(T_4\) has a
\crit{low} budget of 2 and a \crit{high} budget of 7. C=criticality, P=priority, T=period,
B=budget, U=utilisation.}
\label{t:mc-params}
\end{table}
\subsection{Criticality mode switches}
Another case of budget overrun is the system shown in \autoref{t:mc-params},
consisting of three \crit{high} tasks (pink) and two
\crit{low} tasks (blue), plus \(T_0\) which runs in slack time. With
\(T_4\)'s \crit{low} budget of 2~units, this system is RMPA
schedulable --- the RMPA utilisation bound for 5 tasks is 74\% --- so all tasks
will meet their deadlines.
Now assume that \(T_4\) overruns its budget, triggering a timeout
exception. The handler can adjust its budget to the \crit{high} value
of 7~units, however, the resulting system is no longer
schedulable. Since the 4-task utilisation bound of RMPA is 75\%, not
only the \crit{low} task \(T_1\) may miss its deadlines, but also the
\crit{high} task \(T_2\).
We can repair this situation by a criticality switch that prevents \crit{low} tasks,
specifically \(T_3\) from competing with \(T_4\).
We support this by introducing an explicit
notion of a \emph{system criticality level}, as well as a new
\emph{thread criticality} attribute. When setting the criticality
system level to \(C\), we boost the priority of all threads with
criticality \(\geq C\) by a constant amount, so that they all have
priorities above any lower-criticality threads \citep{Burns_Baruah_13}. In the above example,
the timeout handler not only increases \(T4\)'s budget, but also
raises the criticality level to one.
The lowest-priority task \(T_0\) will only run if there is slack in
the system. If so, the criticality level can be reset to zero
(possibly after waiting for a few of \(T_0\)'s periods).
We control thread criticality changes similarly to priority changes: a
thread attribute \emph{maximum controlled criticality} (MCC)
determines limits how a thread can chance another thread's
criticality, just as the MCP limits priority changes. Setting the
system criticality level requires the \code{sched\_control} capability.
\section{Implementation}\label{s:impl}
\subsection{Objects and methods}
We add a new 64-byte scheduling context object type, and modify global
state by eight words plus the number of criticalities.
In TCB objects we replace the \texttt{timeslice}) by the scCap, add a
timeout handler capability,
criticality, MCP, and a number of bookkeeping fields, a total of nine
extra fields. As TCB objects must be powers of two in size, this has
no effect on the size of a TCB object.
We add three methods on TCBs. SCs have 5 methods, the new
\code{sched\_control} has two. There are also three new methods for manipulating
reply capabilities: the ability to set your reply slot, save another threads reply capability,
and the ability to swap your reply capability with one saved earlier.
This extra flexibilty with reply capabilities allows for more efficient
user-level scheduling via IPC, and allows the
timeout handler to access the faulter's reply capability, so it can
unblock the client on the server's behalf. Also new is \code{nbsend\_wait}.
\iffalse
\autoref{t:methods} shows the eight methods (object invocations) added to the kernel, and additionally we remove the
\textbf{yield} system call and add \textbf{nbsend\_wait}. Finally one domain related method is removed, as well as the domain object, since the domain scheduler can be emulated at user-level.
\FIXME{Is it worth mentioning why it can't be send\_wait?}
\fi
\iffalse
\begin{table}[t]\centering
\begin{tabular}{|c|p{0.35\columnwidth}|p{0.35\columnwidth}|}\hline
\textbf{Object} & \textbf{Field} & \textbf{Purpose} \\\hline
\multirow{12}{*}{TCB}
& \texttt{word\_t mcp} & Max. controlled priority. \\\cline{2-3}
& \texttt{word\_t criticality} & Crit. of this TCB.\\\cline{2-3}
& \texttt{word\_t mcc} & Max. controlled criticality.\\\cline{2-3}
& \texttt{sched\_context\_t *sched\_context} & TCBs's current SC.\\\cline{2-3}
& \texttt{sched\_context\_t *yield\_to} & SC that yielded to this SC.\\\cline{2-3}
& \texttt{sched\_context\_t *home\_sc} & SC this TCB is bound to.\\\cline{2-3}
& \texttt{tcb\_t *call\_stack\_prev} & Call stack pointer.\\\cline{2-3}
& \texttt{tcb\_t *call\_stack\_next} & Call stack pointer.\\\cline{2-3}
& \texttt{tcb\_t *crit\_prev} & Crit. queue pointer.\\\cline{2-3}
& \texttt{tcb\_t *crit\_next} & Crit. queue pointer.\\\hline
\multirow{10}{*}{SC}
&\texttt{uint64\_t budget} & Budget in ticks.\\\cline{2-3}
&\texttt{uint64\_t period} & Period in ticks.\\\cline{2-3}
&\texttt{uint64\_t remaining} & Remaining budget.\\\cline{2-3}
&\texttt{uint64\_t next} & Next period.\\\cline{2-3}
&\texttt{tcb\_t *tcb} & TCB executing on SC.\\\cline{2-3}
&\texttt{tcb\_t *sc} & TCB this SC is bound to.\\\cline{2-3}
&\texttt{word\_t data} & ID set by user.\\\cline{2-3}
&\texttt{uint64\_t consumed} & Ticks TCB has consumed.\\\cline{2-3}
&\texttt{tcb\_t* yieldFrom} & TCB that yielded to this SC.\\\hline
\multirow{7}{*}{Global}
& \texttt{word\_t crit} & Current system crit. level. \\\cline{2-3}
& \texttt{tcb\_t *release\_head}& Head of the release queue. \\\cline{2-3}
& \texttt{bool\_t reprogram} & true if timer reprogram required.\\\cline{2-3}
& \texttt{uint64\_t curr\_time} & Timestamp read at kernel entry.\\\cline{2-3}
& \texttt{uint64\_t consumed} & Difference between previous timestamp value \& current.\\\cline{2-3}
& \texttt{sched\_context\_t *curr\_SC} &Currently running scheduling context.\\\cline{2-3}
& \texttt{crit\_queues[]} & Queues of TCBs per criticality.\\\hline
\end{tabular}
\caption{Fields added to TCB and SC objects, and global kernel state.}
\label{t:fields}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{|c|l|p{2cm}|}\hline
\textbf{Object} & \textbf{Method} & \textbf{Params.} \\\hline
\multirow{3}{*}{\texttt{TCB}}
&\texttt{set\_criticality} & \texttt{crit.} \\\cline{2-3}
&\texttt{set\_MCC} & \texttt{mcc}\\\cline{2-3}
&\texttt{set\_MCP} & \texttt{mcp}\\\hline
\multirow{2}{*}{\texttt{sched\_ctrl}}
&\texttt{configure} & \texttt{budget}, \texttt{period}, \texttt{data} \\\cline{2-3}
&\texttt{set\_criticality} & \texttt{crit.} \\\hline
\multirow{5}{*}{\texttt{SC}}
&\texttt{yield} & \textit{none} \\\cline{2-3}
&\texttt{yieldTo} & \textit{none} \\\cline{2-3}
&\texttt{bind} & \texttt{capability} \\\cline{2-3}
&\texttt{unbind\_object} & \texttt{capability} \\\cline{2-3}
&\texttt{unbind} & \textit{none} \\\hline
\multirow{3}{*}{\texttt{CNode}}
&\texttt{save\_tcb\_caller} & \texttt{tcb}, \texttt{cslot} \\\cline{2-3}
&\texttt{set\_caller} & \texttt{slot} \\\cline{2-3}
&\texttt{swap\_caller} & \textit{dest, src} \\\hline
\end{tabular}
\caption{Methods added to the kernel}
\label{t:methods}
\end{table}
\fi
\subsection{Scheduling algorithm}\label{s:sched}
Baseline seL4 has a ready queue, which satisfies the invariant that it
contains all runnable threads except the one presently executing
\citep{Blackham_SH_12}. It is implemented as a priority-indexed array
of queues. A two-level bitfield of occupied priorities ensures O(1)
access.
The main change required to the existing seL4 scheduler is the
addition of a \emph{release queue}. A thread whose budget expired
before its period is up is removed from the ready queue and inserted
into the release queue. This retains the existing invariant for the
ready queue, while the release queue is characterised as holding all
threads that would be runnable but are presently lacking budget. The
queue is ordered by the time of the threads' next budget refresh,
i.e. the time their next period is up.
Whenever the kernel schedules a thread, it sets the timer to fire when
the thread's SC's remaining budget is due to expire, or for the next
wake-up time for the head of the release priority queue (whichever is first). If an SC switch
occurs, because the timer fires or the thread blocks without an SC
transfer, the consumed time is subtracted from the SC's budget and
added to the accumulated time.
On kernel entry (except on the fastpath, which never leads to an SC
change or scheduler invocation) the kernel updates the current
timestamp and stores the time since the last entry. It then checks
whether the thread has sufficient budget to complete the kernel
operation. If not, the kernel pretends the timer has already fired,
resets the budget and adds the thread to the release queue.
This adds a new
invariant that any thread in the scheduling queues must have enough budget to exit the kernel.
This makes the scheduler precision equal to the kernel's WCET, which for
seL4 is known (unlike any other protected-mode OS we are aware of).
Threads are only charged if the scheduling context changes, in order to avoid
reprogramming the timer which can be expensive on many platforms.
Else, the timestamp update is rolled back by subtracting the
stored consumed value from the timestamp.
\subsection{Criticality}
\newcommand{N_\mathit{crit}}{N_\mathit{crit}}
A core integrity requirement of MCS is that the timeliness of
\crit{high} tasks is unaffected by low tasks. This includes the mode
switch: its cost must not depend on the number of \crit{low} tasks in
the system. We implement criticality as follows.
The kernel supports base priorities in the range \([0,2^{N_p}-1]\),
where \(N_p\) is a kernel build option. The base priority is the thread's actual
priority at system criticality level zero.
The number of criticality levels, \(N_\mathit{crit}\), is also a build option. Typically,
it is a small number, e.g. \ \citet{DO178B} specifies five
levels. We require that \(N_\mathit{crit} \times 2^{N_p} \leq 1024\).
For each criticality level the kernel maintains a queue of threads, threads
that are explicitly suspended (as opposed to out of budget or blocked
in IPC) are not in any criticality queue.
When system criticality changes from $C$ to $C'$, the kernel iterates through
the criticality queues from $C'$ to $N_\mathit{crit}-1$. For each thread in those
queues, the kernel changes the present priority \(P\) to $P_0 \land
(C' \ll 8)$. This ensures that the
priority of all \crit{high} threads is above those of all \crit{low}
ones (with respect to $C'$).
The per-priority ready queues are doubly-linked lists of TCBs, so
moving a thread from one queue to another is a constant-time
operation. Hence, the total time for the priority adjustments is
proportional to the number of threads at criticality $C'$ or higher.
If during a criticality increase the kernel detects any threads that are running on a borrowed
scheduling context (comparing \texttt{tcb->sc->home} to \texttt{tcb}),
and the SC's owner is \crit{low} (\texttt{tcb->sc->home->crit} $\leq
C'$), it generates timeout exception for that thread. This allows a
server to abort any operation on behalf of a \crit{low} thread. If the
thread running on an SC borrowed from a \crit{low} thread has no
timeout hander, it will complete normally. In this case, the
worst-case blocking time is the worst-case server request time, plus
the cost of the mode switch.
\begin{table}[h!]\centering
\begin{tabular}{|l|l|l|}\hline
& Co-operative & Preemptive \\\hline
Shared SC & IPC & Timer notifications \\\hline
SC per TCB & Signals & Timeout execeptions \\\hline
\end{tabular}
\caption{Mechanisms for user-level scheduling}
\label{t:ul-sched-mech}
\end{table}
\subsection{User-level scheduling}
The kernel provides fixed-priority scheduling with budgets. This is a
particular (although quite flexible) policy. Fortunately, our
mechanisms allow us to implement very general policies, as indicated
in \autoref{t:ul-sched-mech}.
For example, cooperative scheduling with an arbitrary policy can be
implemented with a shared SC, where the threads cooperate via IPC, or
per-thread SCs, where synchronisation is via notifications (although
it is unclear why one would want the latter). Pseudocode for both
variants is shown in \autoref{f:coop}.
Similarly, arbitrary preemptive scheduling policies can be
implemented, \autoref{f:preempt} shows pseudocode for schedulers with
shared or per-thread SCs. The shared-SC case uses one SC for all
threads, and a separate one for the timer.
\begin{figure}[htb]\small
\hspace*{0.5em}%
\begin{minipage}{0.5\columnwidth}
\begin{verbatim}
void coop_sched_s() {
reply_recv(ep);
p = t;
t = pick_thread(p);
swap_caller(t, p);
}
void coop_yield_s() {
//yield
call(ep);
}
\end{verbatim}
\end{minipage}
\begin{minipage}{0.4\columnwidth}
\begin{verbatim}
void coop_sched_m() {
t = pick_thread(t);
signal(t->ntfn);
yieldTo(t->sc);
}
void coop_yield_m() {
//yield
wait(ntfn);
}
\end{verbatim}
\end{minipage}
\caption{User-level cooperative scheduler and thread yield function
using a shared SC (left) and per-thread SCs (right).}
\label{f:coop}
\end{figure}
\begin{figure}[thb]\small
\hspace*{0.5em}%
\begin{minipage}{0.5\columnwidth}
\begin{verbatim}
void pr_schd_s(prev) {
// wait for timer
wait(timer);
t = pick_thread();
// change sc over
swap_sc(t, prev);
program_timer();
ack_irq();
}
\end{verbatim}
\end{minipage}
\begin{minipage}{0.4\columnwidth}
\begin{verbatim}
void pr_schd_m() {
// wait for timeout
recv(ep);
t = pick_thread();
// place at head
// of prio queue
yield_to(t);
}
\end{verbatim}
\end{minipage}
\caption{User-level preemptive scheduler with shared (left) and
per-thread (right) SCs.}
\label{f:preempt}
\end{figure}
\FIXME{At some point we need to argue the case for keeping a scheduler in the kernel at all - COMPOSITE doesnt have one at all}
\section{Evaluation}\label{s:eval}
We conducted our evaluation on two machines, both configured to use one core:
\begin{compactitem}
\item \textbf{Sabre:} 1\,GHz ARM Cortex~A9 system on chip on
a Freescale i.MX6 SABRE Lite development board.
\item \textbf{Haswell:} 3.1\,GHz Haswell E1220v3 processor in a server
machine running in 32-bit mode (64-bit seL4 is in development).
\end{compactitem}
\subsection{Microbenchmarks}
\begin{figure}[t]\centering
\begin{tabular}{|c|l|l|l|l|}\hline
\textbf{Arch} & \textbf{Operation} & \textbf{Baseline} & \textbf{RT} & \textbf{Diff} \\ \hline
\multirow{5}{*}{Sabre}
& \texttt{call()} & 279 & 282 & +1\% \\ \cline{2-5}
& \texttt{replyrecv()} & 291 & 311 & +7\% \\ \cline{2-5}
& IRQ latency & 467 & 578 & +24\% \\\cline{2-5}
& \texttt{signal()} & 107 & 111 & +4\% \\\cline{2-5}
& Schedule & 875 & 1242 & +42\% \\\hline
\multirow{5}{*}{Haswell}
& \texttt{call()} & 412 & 415 & +1\% \\ \cline{2-5}
& \texttt{replyrecv()} & 414 & 426 & +3\% \\ \cline{2-5}
& IRQ latency & 952 & 1448 & +44\% \\\cline{2-5}
& \texttt{signal()} & 383 & 387 & +1\% \\\cline{2-5}
& Schedule & 972 & 1532 & +58\% \\\hline
\end{tabular}
\caption{Microbenchmarks of seL4 baseline vs. RT kernels, standard
deviations are negligible.}
\label{t:micro}
\end{figure}
\subsubsection{Kernel microbenchmarks}
\autoref{t:micro} shows the cost of the (performance-wise) most
important kernel operations of our present implementation compare to
the baseline seL4 kernel. Latency of the main IPC send+receive
operations increases by three cycles (call) and by 12--20 cycles
(reply\&wait). These are the result of extra checks on the fastpath to
accommodate scheduling contexts and ordering IPC, but the increase in cost is clearly
negligible. The same can be said for signalling a notification.
The actual cost of the model can be seen in the IRQ and scheduler
latency. Part of that is due to the need to reprogram the timer to
enforce the budget, which is needed on every scheduler invocation, but
also on an IRQ, as this normally unblocks a waiting handler. We
measure the cost of reprogramming the timer to be 55~cycles on the
Sabre, but about 200~cycles on the Haswell.
The rest of the increase is the result of the significant extra code
from dealing with scheduling contexts. Note that scheduling is
considered an expensive operation in seL4, and happens much less
frequently than IPC.
\subsubsection{Mode switch}
\begin{table}[h]
\centering
\iffalse
\begin{tabular}{|l|l|}\hline
\textbf{Criticality} & \textbf{No. tasks} \\\hline
3 & 4 \\\hline
2 & 8 \\\hline
1 & 16 \\\hline
0 & 32 \\\hline
\textbf{Total} & 64 \\\hline
\end{tabular}
\caption{Number of tasks for each criticality level (3 is highest) in mode switch microbenchmark.}
\fi
\begin{tabular}{|c|c|r|r|r|r|}\hline
\textbf{Criti-} &\textbf{Threads} & \multicolumn{2}{c|}{\textbf{ARM}} & \multicolumn{2}{c|}{\textbf{x86}} \\
\textbf{cality} &\textbf{boosted} & \textbf{up} & \textbf{down} & \textbf{up} & \textbf{down} \\\hline
3 & 4 & 1.4$\mu$s & 1.7$\mu$s & 0.4$\mu$s & 0.5$\mu$s \\\hline
2 & 12 & 2.4$\mu$s & 2.4$\mu$s & 0.5$\mu$s & 0.6$\mu$s \\\hline
1 & 28 & 4.3$\mu$s & 3.7$\mu$s & 0.8$\mu$s & 0.7$\mu$s \\\hline
\end{tabular}
\caption{Results of switching from criticality level 0 to the
criticality listed in column 1. Column 2 shows the number of tasks
that need boosting. Standard deviations are no more than 2\%.}
\label{t:mode-switch}
\end{table}
To evaluate the cost of changing the system criticality level,
configure the kernel with 256 priorities and 4 criticality levels
(0--3). We then set up a system with 60 threads, of which 32, 16, 8
and 4 have criticality 0, 1, 2 and 3
respectively.
\autoref{t:mode-switch} shows the cost of switching
criticality level between zero and one of the other levels. For each
data point we took 10,000 measurements with a primed cache.
\autoref{t:mode-switch} shows the cost of switching criticality level
between zero and one of the other levels. As the table shows, when
switching to level three, the three threads at that level need to be
boosted, while a switch to level one requires boosting all 28 threads
of criticality greater than zero.
The results show that a mode switch is fairly fast, around
1,500~cycles on both platforms as long as the affected number of
threads is small (which is to be assumed for \crit{high} threads), and
cost is roughly linear in the number of threads to be boosted. This
is important, as the schedulability analysis must allow for that
cost. However, the numbers shown in \autoref{t:mode-switch} are
hot-cache (best-case) numbers, while the criticality analysis must be
based on WCET.
\autoref{t:mode-switch} shows the number of threads for each criticality and the results of the microbenchmark.
\iffalse
\subsubsection{WCET}
\FIXME{Shouldn't change much}
\fi
\subsection{Case studies}
\subsubsection{Linux CFS}
As an example of a complex dynamic-priority scheduling policy
implemented at user level, we implement a version of Linux' so-called
\emph{completely fair scheduler} (CFS). The implementation uses a
red-black tree and calls to \code{consumed()} to adjust the
weights. The scheduler runs one seL4 priority above its clients.
\autoref{f:cfs} shows the scheduling cost for two scenarios, shared
and per-client SDC. Cost is measured by taking a
time stamp in the client, which then calls \code{yield()}, with
another time stamp taken right after (in the next client thread).
We also show the cost of the same operations under Linux, which takes
about 50--60\% of the time. However, this turns out to be mostly the
syscall cost, as the Linux \code{yield()} bypasses the scheduler. So,
our user-level implementation looks quite competitive.
\begin{figure}[t]
\centering
\includegraphics{cfs-x86}
\caption{Execution time of yield operation measured from user-level
for CFS compared with Linux yield on Haswell. Standard deviations
are 2--3\%.}
\label{f:cfs}
\end{figure}
\iffalse
CFS works pretty much as above in the psuedocode with a red-black tree and calls to consumed to adjust the weights. Points measured from client threads before and after yield, so includes full scheduling cost.
Linux only dispatching - not hitting scheduler, much higher cost.
\fi
\subsubsection{EDF scheduler}
\iffalse
EDF more complex - algorithm basically
while (1)
thread = head(deadline queue);
wake_threads();
timeout = next_wakeup();
set_timeout();
if (thread != NULL) {
if reply_cap_saved:
reply_recv();
else
swap_reply()
reply_recv();
else
recv()
switch (badge)
if timer
ack
else
save_reply()
remove(thread, release_queue)
insert(thread, release_queue)
EDF is all over the place as it has to
wake up up to N tasks
ack irq
program irq
reply_recv or recv (if no threads) or yeildTO (for preempted threads)
EDF threads call when they are done, but it's preemptive (which is why have o tyield to as we maynot have a reply cap).
In all cases the schedulers run at +1 to their schedulees
EDF measured from when sched wakes up and then goes to sleep again (so doesn't include call).
EDF using separate scheduling contexts - ran out of time.
Both EDF and CFS use efficient red-black tree to schedule (via time vs wait).
\fi
As a second scheduling policy we implement EDF at user level, this
time only the scenario with a shared SC. The results are shown in
\autoref{f:edf}. The standard deviations are very big, especially on
the Haswell platform. This is not unexpected, as the amount of work
EDF has to do on each scheduling operation is very sensitive to the
present state of the deadline and release queues.
The scheduler may have to release threads, reprogram the timer for the next release,
ack the previous interrupt and IPC the next thread, or resume a preempted thread with
\texttt{yieldto}.
We used the \emph{randfixedsum}~\citep{Emberson_SD_10} algorithm to generate 10 EDF
task sets for each of the 10 data points, with periods between 10--100ms. Each task
set ran 1000 times, for 10,000 runs for each data point.
A better metric in this case is the minimum scheduler time, shown in
the figure as ``Sabre min'', ``Haswell min''. It is reasonably stable
around 2\,\(\mu\)s for the Sabre, and 0.5--0.9\,\(\mu\)s for the
Haswell platform. This is an excellent result:
\citet{Cerqueria_Brandenburg_13} measured the latencies of various
in-kernel Linux schedulers on a Xeon~X7550 platform and found the
minimum to be around 1.5\,\(\mu\)s for all schedulers. While
comparions across different hardware must be taken with a grain of
salt, the fact that latencies of our user-level implementation is a factor four
less indicates that our performance is competitive.
\begin{figure}[t]
\centering
\includegraphics{edf}
\caption{Execution time of EDF on Sabre and Haswell.}
\label{f:edf}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics{redis-arch}
\caption{Network server setup, explanation in text.}
\label{f:redis-arch}
\end{figure}
\subsubsection{Network server}
In order to demonstrate temporal isolation, we use a network
benchmark, specifically the Yahoo! Cloud Serving Benchmarks (YCSB)
\citep{Cooper_STRS_10}. We run this against a server using the Redis
key-value store \citep{redis:url}.
The server setup is shown in \autoref{f:redis-arch}. Dashed arrows
show synchronisation operations through notifications (semaphores)
indicated by flags, with coloured, broken single arrows indicating the
direction of the signal. The OS server, which contains the IP stack and is
implemented as a passive server, presents a POSIX interface, which is
implemented by an RPC protocol through an endpoint. The (active) Redis
server invokes the OS server (coloured, solid single arrow), which
then runs on Redis' scheduling context. Redis and the OS share a
buffer for passing bulk data (black, solid double arrows). The OS also
shares a buffer with the Ethernet driver, which uses a second
notification (red) for signalling completion to the OS. That
notification is ``bound'', meaning the signals are delivered to the
waiting OS as an IPC apparently coming from the endpoint.
\FIXME{12\% overhead between RT kernel and baseline}
\begin{table}[th]
\centering
\begin{tabular}{|l| c | c | c |}
\hline
\bf Thread& \bf Prio & \bf Period & \bf Budget \\
\hline
Hog & 254 & 1\,ms & variable \\
Driver & 253 & 2\,ms & 2\,ms \\
Redis & 252 & 1\,s & 1\,s \\
OS & 252 & -\,- & -\,- \\
\hline
\end{tabular}
\caption{Scheduling parameters of network server setup.}
\label{t:redis-param}
\end{table}
Not shown is a
separate CPU hog thread, which does not communicate with this setup,
but is competing for CPU time. The hog runs at highes priority (254)
with a 1\,ms period. The Ethernet driver runs at priority 253
We use the budget of the hog to control the amount of time left over
for the server configuration. \autoref{f:redis} shows the bandwidth
achieved by the YCSB-A work load as a function of the available CPU
bandwidth (i.e.\ the complement of the bandwidth granted to the hog
thread). The figure also shows the total CPU idle time.
\begin{figure}[htb]
\centering
\includegraphics{redisworkloada}
\caption{Throughput of Redis YCSB workload A vs available bandwidth,
also showing idle time.}
\label{f:redis}
\end{figure}
The graph shows that the server is CPU limited (very low idle time)
and consequently throughput scales linearly with available CPU
bandwidth.
\subsubsection{Server rollback}
As an example of a shared server running out of budget, we implement
the scenario of \autoref{f:passive} of a passive server with two
clients. The server is providing an encryption service using AES-256
using a block size of 16~bytes. The server alternates between two
buffers, of which one always contains consistent state, the other is
dirty during processing.
When the server runs out of budget, its timeout fault hander gets
invoked. It rolls the server back to the last consistent state and
makes it ready for the next client.
We measure rollback time, from the time the fault handler is
invoked, until the server is ready for the next request. Given the
small amount of rollback state, this measures the baseline overhead,
for servers with more state, the handling that state would
have to be added.
We run this on the Sabre and find a mean rollback time of 12 $mu$s,
with a 31\% standard deviation on 12 runs with a cold cache. The
individual times fluctuated between 9 and 24\,$mu$s.
\iffalse
256 AES server doing encryption, has temporal exception handler. Passive. 2 clients, not enough budget.
2 choices: can roll back (abort client request) or report back how much done.
Only 4 words of state. Use 2 state structures, swap pointer shared between timeout fault handler and
server over once each block is finished.
Blocks size 16.
Implemented for arm. State is small as we work in chunks.
Times for rollback (measured from point timeout fault raised to server being restored:
min: 9.3 $mu$s
max: 23.7 $mu$s
mean: 12.2 $mu$s
stddev: 31\% $mu$s
Only 12 results, not hot cache.
\fi
\iffalse
\subsubsection{Quadcopter}
\FIXME{Won't get there I think.}
\fi
\section{Conclusions and Future Work}\label{s:concl}
Mixed criticality systems are gaining traction in avionics and the
automotive sector, due to the SWaP issues created by mushrooming
functionality. In order to get the full benefit of MCS, we need an OS
supporting strong, but asymmetric temporal isolation. Inherent in the
notion of MCS is also a requirement for high assurance.
While there is a wealth of theory about MCS, little of it is
implemented in more than a proof-of-concept, certainly not in a
high-assurance OS. We have identified a model for temporal resource
management that lends itself to efficient and policy-free
implementation in a high-performance and high-assurance OS. We have
implemented this in seL4, and have demonstrated that the base model
supports the efficient implementation of a range of different
scheduling policies, and allows efficient handling of various
emergencies.
Presently the main limitation of the work is the restriction to a
single core, which is the target of future work, as is the formal
verification of the real-time seL4 kernel.
\ifAnon\else
\section*{Acknowledgements}
The authors gratefully acknowledge Hesham Almatary's help in producing
the web-server benchmarks, and Bj\"{o}rn B. Brandenburg and Leonid
Ryzhyk for providing feedback on an early draft. \fi
\balance
{ \sloppy
\label{p:last}
\bibliographystyle{plainnat}
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} | 2,072 |
var JsonTransform = DS.Transform.extend({
deserialize: function (serialized) {
return JSON.parse(serialized);
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serialize: function(deserialized) {
return JSON.stringify(deserialized);
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export default JsonTransform;
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Social & Society
About the Role of Nonconventional Solutions during Extraordinary Times: A Case for Achieving a Tangible Result.
by Georgia Today
in Analysis, Newspaper, Politics
Reading Time: 13min read
The Quadrilateral Security Dialogue, in which the United States, Japan, Australia, and India participate aims to maintain a kind of "status quo" in the Pacific by maximum containment and restraining Chinese expansion. Source: foreignbrief.com
The revision of modern security systems is causing immediate fundamental changes to international relations. This close relationship is very natural, because any kind of global or regional order is inconceivable between the main participants in these relations, the states, without the formation and regulation of physical security in one way or another. It is also noteworthy that our own security models are evolving and often transforming, which in turn is the result of the changing circumstances in world politics.
One such powerful conjunctural change is related to the geopolitical self-determination of the countries on the dividing line of the Russian Federation and Euro-Atlantic unity and the ensuing cataclysms of this process as a result of the collapse of the Soviet Union and the end of the Cold War. It must be said that in this broad picture, it is not only these "buffer countries" that are facing the challenges of self-determination. Georgia's main strategic partner, as well as our other partners, have faced a number of unresolved challenges and unanswered questions. In recent years, the main issue has emerged: How to resolve conflicts between the two major geopolitical camps. What will be the fate of the "buffer zone" and what kind of security configuration will it consider, including our country?
An interest or a value?
Finding the answer to this question alone cannot be the subject of a theoretical exercise. Finding the right answer is possible only through continuous communication between the Georgian side and its Western allies, and, at the same time, requires continuous monitoring and evaluation of the accents and priorities of international relations. One of the priorities for us is to promote interest in the equation of "interest and values" in contrast to values.
It must be said that the balancing of states around these two categories in the global or regional context is not new and, in fact, begins with the emergence of the state on a national basis. However, the starting point of the recent history of the "struggle" of interest and value with exceptional force and clarity is the beginning of the formation of the modern state after the Second World War. At the same time, the uncertainty and unresolved situation caused by the end of the Cold War further deepened the imbalance in the two-part equation: The dominance of interest in relation to value became apparent, and in some cases virtually irreversible.
We've discussed the reasons and nature of such imbalances many times before. However, in summary, we would like to say that all these reasons are centralized around the main issue – the declining ability of the modern formation of the state to cope with accumulated socio-economic and socio-political challenges. As a result of this almost chronic incapability, we've experienced a breakdown of internal and external unity, which soon manifested itself in the form of radicalization of thoughts, incompatibility of positions, and extremist tendencies. Moreover, instead of resolving these acute crises in the domestic processes, one country or another shifted its focus to the external arena and in this way attempted to stop the further disintegration of the state and society. In any case, the shift of attention has not gone unnoticed: Modern international relations have undergone a considerable devolution, because, instead of consolidating around values, the driving factor in these relations has only been the satisfaction of one's own national interest at the expense of others. Hence, the principle of "zero sum" – which means that a particular entity receives the maximum benefit at the expense of neglecting others – has undergone a new renaissance.
As a result, more or less successfully disguised selfishness in world politics has finally emerged and has become a declared line of international conduct of the state. In this rather peculiar process, we have come to a reality in which the category of values has become a small and ritual appendage of egoism, and the geopolitical teaching of "realism" has become much more realistic. All this, in turn, necessitated the search for new models of coexistence in the field of security, and the alliances or unions formed according to the specifics of a particular region acquired a much narrower thematic relationship in order to bring more interests together. In a nutshell, time has created the inevitability of experiments with the existing useless approaches to finally say farewell and find a new solution.
More flexibility, more essence, more focus…
Western global security during the Cold War was largely based on the axis of the North Atlantic Treaty Organization (NATO). This system continues to operate today; however, with two essential questions that remain unanswered in the current context. First, what is the Alliance's updated functional burden, and second, what is the Alliance's principled approach to possible new geographical expansion. Discussions on both issues are in an active phase, and their urgency is evidenced by the recent NATO summit and the 2030 Strategic Vision document adopted there, which, it must be said, still does not adequately cover current issues for Georgia.
Together with the NATO Multilateral Collective Alliance and in parallel to it, our main strategic ally – the United States – years ago established a second, no less interesting system, which provided thematic-geographical military and security cooperation. A visible example of this approach is the hub-and-spoke model set by the bilateral agreements established with South Korea, Australia, the Philippines, Japan, and Thailand. It must be said that this type of system found its practical purpose in the implementation of George Kennan's "line of containment" against the Soviet Union during the Cold War.
However, as already mentioned, over time the nature of the relationship changes and the rules and constructions of security behavior change with it. One such major change is the compact multilateral party alliances that have emerged among narrow-format bilateral agreements and the large alliances (in modern terms – clusters) that offer participating countries along with more explicit functional tasks, improved mobility and optimal utilization of resources to achieve a goal. Not so rarely, such alliances are referred to as a "Small NATO", although, to some extent, this comparison is still incorrect.
In any case, the above-mentioned process is of practical interest for the Georgian government and specialized and analytical circles. At the same time, studying it and modeling it in our region may help us to overcome the shortcomings of long-established security models or approaches – in some cases, anachronisms – in maximizing the actual support needed for Georgian statehood, not with words but by actions.
In fact, the essence of the mentioned "cluster" (so-called "Small NATO") system is the matching and overlap of the actual and non-declarative mutual interests of the participating countries. The amplitude of such coherent aspirations can range from selectively highlighted collaborative areas to a variety of tasks. Hence, it is considered (without idealization here as well) that actually effective security is provided by the "clusters" in which the participating countries naturally unite; and in the post-unification period, the set goals are followed with a rationale understandable to them individually and collectively.
However, to better illustrate the brief description here, we will move on to one more specific precedent.
Already an example, already existing experience
One such interesting thematic association has been formed in the Pacific Ocean: the Quadrilateral Security Dialogue (QUAD), in which the United States, Japan, Australia, and India participate. This project is significant to us as it offers very practical material to discuss the topic covered in this article.
The cooperation of the abovementioned countries can be explained for several reasons, however, the most important of which is to maintain a kind of "status quo" in the Pacific (and partly in the Indian Ocean) by maximum containment and restraining Chinese expansion. Clearly, this is easier said than done, and among the difficulties is not only the growing Chinese factor in the region, but also the conceptual or tactical differences on a number of issues and approaches between the countries participating in the Quadrilateral Security Dialogue. In general, the process of forming the Dialogue and its further development, as well as the study of related problems, is obviously equally valuable for a better understanding of the model. But, though we do realize that discussion around this topic in detail will take us far, we want to take a step back and highlight the main point again and emphasize it.
The importance of the Dialogue's format has increased especially recently, which has been led by China's systematic efforts to rearrange the regional order based on its own goals, to impede free naval movement in the region, and to violate the sovereign rights of a number of countries. It is noteworthy that according to the Dialogue, Beijing's efforts to "rewrite" a number of international norms to realize their geopolitical views and ambitions have acquired an alarming character. It is quite understandable that development of the process in this way threatens regional stability and the balance of power, and the prevention of this threat and the further radicalization of events is the main challenge for the countries participating; the steps taken to deepen cooperation within the framework of the Dialogue serve the same purpose.
To cut to the chase, a "Small NATO" operating in the Pacific as a "quadrilateral security format" is a response to its revivalist democracies to the policy of containment in the region based on a number of principles enshrined in the Cold War-era policy. At the same time, it should be noted that the format is not a formal alliance and is not accompanied by a NATO Article 5 analogy – there is no formal commitment to mutual military assistance.
It should be noted that in addition to cooperation in the field of security and defense, the area of interests provided by the Dialogue includes coordination of many other topics, be it migration, energy, terrorism, etc. Although, there is no doubt that the coincidence of geopolitical interests in the geographical area, which is recognized as a "core area of interest" by the United States, is a major motivator for the participants of the Dialogue. The importance of the region for the United States has not only been reflected in a number of US policy documents, but has also been set as a special coordinator for the Indo-Pacific region under US national security during President Biden's administration. It is noteworthy that the growing attention to the region by leading EU states has also been expressed through very specific actions. For example, France first unveiled a new strategy on the issue, and then appointed an Ambassador Extraordinary and Plenipotentiary for the Indo-Pacific, and Germany began developing an emergency strategy for the region in 2020.
This brief excursion was necessary to move on to the main point and discuss Georgia's security variations in terms of modeling the unified security of the Black Sea. It is also worthy of special reservation that our country continues and develops a course of integration with NATO, which was and still is a national priority. At the same time, it is precisely the already familiar problem of the Alliance's eastward expansion that makes it necessary to consider additional or alternative ways towards Western political and security integration: Again, given that the modern trend requires the Georgian state to make more decisions, as well as have more flexibility and adaptability in terms of actions. It is necessary not only for Georgia, but also for our Western partners to get rid of stereotypes and clichés in the process of thinking, to have courage and adequacy and to measure achievements with concrete, tangible results.
"Small NATO" of the Black Sea?
Today, the issue of the regional security of the Black Sea is increasingly discussed and reviewed at meetings of various formats and levels. However, despite such a tendency, the gap between word and deed is still noticeable. Yes, relations with NATO are developing. Yes, it is filled with new elements. Yes, new security line projects and initiatives are being added with the participation of our key strategic partners. All this is true, however, the rapidly changing situation in the region requires much more – a more in-depth and fully tailored solution to the issues.
In order to create more guarantees for our country, membership in the Alliance should envisage clear deadlines and quick procedures. In this respect, however, the picture is still unsatisfactory. We would also consider the transition to a strategic contractual alliance with our key strategic partner in the field of defense and security as a kind of "alternate" option. The likelihood of this option, as well as its "experimentality" given the Black Sea regional context, have been discussed in previous publications as far as possible. This time we will try to discuss the "Small NATO" of the Black Sea – the cluster of the regional security of the Black Sea – and we shall mark its regional specification for further discussion. We believe that the Black Sea precedent of the abovementioned Quadrilateral Security Dialogue deserves a detailed study by the buffer states located between the two geopolitical camps on the Black Sea, especially for those countries that have made their own foreign policy choices.
Firstly, further structuring relations between geopolitically related countries on the Black Sea would be one very concrete and practical step. In the case of Georgia in particular (and, obviously, not only) to reduce harmful and destructive external influences. As a result, a format similar to the Dialogue for the Black Sea would mitigate the pressure on the buffer states under which they get exposed because of the realization of the foreign vector that is directed against the sovereignty and territorial integrity of these countries.
At the same time, no less important is the establishment of a similar structure of relations, which would significantly highlight Western interests in the region and completely nullify the urgency of a well-known question: "Where is the West in the region?" As opposed to the Baltic countries, in the context of NATO's fragmented representation in the Black Sea, this reorganization of relations would establish the groundwork for the conceptualization of American Eurasian policy and make Western statements on real strategic interests in the region much more credible.
It is noteworthy that in this way, Georgia's main strategic partner would create a higher quality security system in the Greater Black Sea Region (macro-region of the Black and Caspian Seas). Through this system, the interests of politically different players would be brought together on a more logical basis: The stability of the countries participating in the format equates to the stability of the region. In the wake of this great regional task, the proposed format of cooperation, under the guidance of a policy of collective restraint in the Black Sea area, will serve as a relatively more effective deterrent to Russian revisionism, as well as counteracting the harmful effects of other state or non-state actors on the region.
We are well aware of the fact that the harmonized coordination of the countries participating in the "Small NATO" format of the Black Sea is not always easy: There are internal political complications, as well as social and economic differences, between countries or heterogeneity in the perception of human rights and freedoms. However, a strong factor for overcoming and amalgamating all this is the substantial concomitancy of the participants' foreign policy vectors. Virtually, participating countries will partner where they have their common interests mostly concentrated, in particular, such as common security risks and challenges. Establishing a reliable communications system for the exchange of classified information is one of the essential components towards the way of minimizing these risks.
We note that such a cooperation proposed in the Black Sea format, such as the Quadrilateral Security Dialogue, would be devoid of bureaucratic formalities and strict treaty provisions. Moreover, any country participating in the format would continue its path of security integration with NATO or the United States, as well as the implementation of commitments or programs already made in these two areas. Consequently, the "Small NATO" of the Black Sea is equally permissible as a unity of such interdependent treaties, the main binding and unifying factor of which will be the intention to strengthen and enforce certain rules of conduct in the region.
Cooperation in our region is not only about security and there is a reason for that. The Black Sea itself and the Greater Black Sea region as a whole are important to maintain a stable balance of power in the Eurasian space, though participants in the above format should focus not just on security issues.
To this end, we have repeatedly mentioned the Black Sea Declaration in the recent past, the signing and enactment of which would qualitatively contribute to the common Western political, economic, or security space in the region. Moreover, such a document would be additional proof that the interest of Georgia's main strategic partner in the region has not slowed down at all; that by enacting the Declaration, the West expresses its firm readiness not to recognize the exclusive influence of others on the buffer countries of the Black Sea; not to mention that the Declaration would give practical means to building a strong democratic and economic order in such countries.
The Black Sea Declaration would also receive the significance of a "soft impact" and would neutralize the possible "militarization" of the collaborative format discussed in this article. In particular, it would also involve attracting additional investment resources for the implementation of regional projects, as well as infrastructure, environmental, and energy projects. The declaration would also specifically signify the unrestricted traffic necessary for trade unions and the expediency of free trade agreements or blocks. In order to intensify regional cooperation and Western participation in it, the declarant countries would discuss current geopolitical (pseudo-ethnic) conflicts in the region, cyber security, illegal migration and joint measures to combat international terrorism.
As a result, along with the Black Sea analogue of the Quadrilateral Security Dialogue discussed in the introductory part of this article, determining the fate of the Eurasian space in our most important region would lay the groundwork for the Association of Black Sea Countries. This would be another powerful mechanism for resilience and regional sustainability within countries.
Additionally – about unconventionality
In the present world's foreign or domestic policy, turning to excessive "classicism" can be a reason to miss a real result. In contrast, non-standard decisions, as well as the uncommon actions resulting from them – a kind of unconventionality – can serve as a solid statement for achievement. Getting rid of fruitless stereotypical approaches in the field of security and trying new ones should be considered as a call for this.
It is desirable that the discussion of a specific model in this article to be considered as another demonstration that the talks of the present and the future of the Georgian state should be focused on interests with a realistic content. We would like also to point out that by no means do we want anyone to get the impression that we are undermining the significance of values (in the light of interest) and want to write them off. Of course not. But it should also be noted that in this highly unpredictable world it is inadmissible to overshadow practical results with pointless theorizing, while political realism is swallowed up by a dizzying and meaningless cacophony. On the steep ascent of the historical development of a complex region, a small nation facing both internal and external challenges has neither sufficient resources nor luxury. Maturity and vigilance dictated by a healthy Georgian egocentric standard should make Georgia's state realism the norm of our actions.
The fact is that thinking about development is inconceivable without a reliable mechanism of national security. In the medium-term, the regional format of the Black Sea mentioned in this letter (for example, in the absence of a further delay in NATO expansion to the east or in the absence of real bilateral defense ties) may be considered as one such mechanism. Its practical embodiment becomes a clear message that our region is in the sphere of interest of Western civilization and is a natural part of its main "geopolitical geography". It is also noteworthy that for the West and the United States one of the most essential components of their Eurasian policy emerges through the "Small NATO" of the Black Sea. In addition, this is done by experimenting with a model that serves realistic tasks with relatively low cost and low risk (for example, no formal alliance and no additional infrastructure or physical representation in the region). At the same time, despite the formalized structure, the proposed association maintains the very minimum required for its effectiveness, as it is based on the natural concurrence of interests, the voluntary participation of the subjects, and a recognition of balance. At the same time, the relative compactness gives it an additional value just as the development of modern regional diplomacy within this elastic geopolitical geometry.
In a nutshell, there is a lot of thought and work to be done to structure regional security. We've said and would like to repeat once again a simple truth: This time the whirlpool needs a bold rogue and sound political effort both regionally and nationally. I think we have the most difficulty with the latter one, because due to the trifles of the Georgian political culture and way of life, we do not spend enough time on big and urgent tasks. Or at least, in fact, we have already lost a significant part of the ability and sense to feel and understand national issues…
Analysis by Victor Kipiani, Geocase Chairmain
Tags: Black SeaGeocaseNATOQUADVictor Kipiani
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2000-2021 © Georgia Today | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 439 |
A father who immigrated to the US from Tunisia has been found dead in a car alongside his two young daughters, aged nine and 12, in what police suspect is a murder-suicide.
Sara, 9, and Sophia, 12, were unresponsive in the back seat of their father Hamdy Rouin's car in West Sacramento, California when he was found dead in the driver's seat around 10pm on New Year's Eve.
The two girls, identified by Muslim community members, were rushed to the hospital but did not survive.
Rouin, 46, was pronounced dead on the scene, and his identity was confirmed by coroners.
Following a bitter custody battle with his American ex-wife, Rouin faced a court date on Tuesday for allegedly violating a protective order, according to court documents reported by the Sacramento Bee.
Police have not yet released the causes of death, but said that the girls and the father did not have visible gunshot or knife wounds.
Some witnesses said it appeared that there had been an attempt to set the car on fire near the entrance to Casa Mobile Park, where the girls lived with their mother.
M.A. Azeez, the senior imam of the Roseville Tarbiya Institute, told the Sacramento Bee that he had counseled the father, a Tunisian immigrant who struggled in the US, and his ex-wife, a convert to Islam.
Rouin's wife Amy Hunter filed for divorce in December 2014, the couple had gone through a protracted custody battle, the imam said.
'I remember visibly his constant frustration that he is not understood,' Azeez said.
'I heard that many times,' Azeez said.
After the divorce was finalized in December 2016, the father did not get to live with the children or get the custody he was seeking.
The court issued at least four restraining orders during the divorce battle, records show.
In August of 2017, Rouin was arrested for violating one of the restraining orders as well as for contempt of court, according to court records.
He was scheduled to appear on those charges in Yolo County Superior Court on Tuesday at 9.30am.
Police said the mother, identified by friends as Amy Hunter, was cooperating with investigators, and was not present at the time of the deaths.
Meanwhile, the community mourned the loss of young Sara, who wanted to be a mathematician for NASA, and Sophia, who wanted to be a scientist and travel the world.
Both girls were 'passionate about feeding the homeless', according to a fundraising appeal to help pay for funeral expenses.
'Sophia was contemplative and insightful. Often, you could find her deep in thought, as if taking a cue from the meaning of her name,' the Tarbiya Institute wrote on Facebook.
'Sara was the personification of the saying 'big things come in small packages'. Her smile could light up the room and her energy was always infectious,' the group said. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,748 |
\section{Introduction}
Throughout, ring means an associative ring with $1$. Let $K$ be an
arbitrary ring (not necessarily commutative). The {\em Grassmann
algebra} (the {\em exterior algebra}) $\Lambda_n = \Lambda_n (K)= K\lfloor
x_1, \ldots , x_n\rfloor$ is generated freely over $K$ by elements
$x_1, \ldots , x_n$ that satisfy the defining relations:
$$ x_1^2=\cdots = x_n^2=0 \;\; {\rm and}\;\; x_ix_j=-x_jx_i\;\;
{\rm for \; all} \;\; i\neq j.$$ The Grassmann algebra $\Lambda_n=
\oplus_{i\in \mathbb{N}}\Lambda_{n,i}$ is an $\mathbb{N}$-graded algebra
($\Lambda_{n,i}\Lambda_{n,j} \subseteq \Lambda_{n,i+j}$ for all $i,j\geq 0$)
where $\Lambda_{n,i} := \oplus_{|\alpha | =i}Kx^\alpha$, $x^\alpha :=
x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, and $|\alpha |:=
\alpha_1+\cdots +\alpha_n$.
{\bf Derivations of the Grassmann algebras}. Let $ {\rm Der }_K(\Lambda_n)$,
$ {\rm Der }_K(\Lambda_n)^{ev}$, $ {\rm Der }_K(\Lambda_n)^{od}$ and ${\rm IDer}_K(\Lambda_n)$ be
the set of all, even, odd and inner derivations of $\Lambda_n(K)$
respectively. Note that ${\rm IDer}_K(\Lambda_n)= \{ {\rm ad } (a) \, | \, a\in
\Lambda_n\}$ where $ {\rm ad } (a) (x):= ax-xa$. Let $\L_n^{ev}$ and $\L_n^{od}$ be
the set of even and odd elements of $\Lambda_n$. Let
$\partial _1:=\frac{\partial }{\partial x_1}, \ldots , \partial _n:=\frac{\partial }{\partial
x_n}$ be partial skew $K$-derivations of $\Lambda_n$ ($\partial _i(x_j) =
\delta_{ij}$, the Kronecker delta, and $\partial _i( a_ja_k) = \partial _i
(a_j) a_k+(-1)^ja_j\partial _i(a_k)$ for all $a_i\in \Lambda_{n,i}$ and
$a_j\in \Lambda_{n,j}$).
\begin{itemize}
\item (Theorem \ref{13Sep06}) {\em Suppose that $K$ is a
commutative ring with $\frac{1}{2}\in K$. Then}
\begin{enumerate}
\item $ {\rm Der }_K(\Lambda_n) = \Der_K(\L_n)^{ev} \oplus \Der_K (\L_n)^{od}$. \item $\Der_K(\L_n)^{ev} =
\oplus_{i=1}^n \L_n^{od} \partial _i$. \item $\Der_K (\L_n)^{od} = {\rm IDer}_K(\Lambda_n)$.
\item $ {\rm Der }_K(\Lambda_n)/ {\rm IDer}_K(\Lambda_n)\simeq \Der_K(\L_n)^{ev}$.
\end{enumerate}
\end{itemize}
So, each derivation $\delta \in {\rm Der }_K(\Lambda_n)$ is a unique sum $\delta =
\delta^{ev} +\delta^{od}$ of an even and odd derivation. When $K$ is a
field of characteristic $\neq 2$ this fact was proved by Djokovic,
\cite{Djokovic78}. For an even $n$,
let $ \L_n'^{od} := \L_n^{od} $. For an odd $n$, let $ \L_n'^{od}$ be the
$K$-submodule of $\L_n^{od}$ generated by all `monomials' $x^\alpha$
but $\theta := x_1\cdots x_n$, i.e. $\L_n^{od} = \L_n'^{od} \oplus K\theta$.
The next result gives explicitly derivations $\delta^{ev}$
and $\delta^{od}$ via the elements $\delta(x_1), \ldots , \delta (x_n)$.
\begin{itemize}
\item (Corollary \ref{c13Sep06}) {\em Let $K$ be a commutative
ring with $\frac{1}{2}\in K$, $\delta$ be a $K$-derivation of
$\Lambda_n(K)$, and, for each $i=1, \ldots , n$, $\delta (x_i) = u^{ev}_i +
u^{od}_i$ for unique elements $u^{ev} \in \L_n^{ev} $ and $u^{od} \in
\L_n^{od}$. Then}
\begin{enumerate}
\item $\d^{ev} =\sum_{i=1}^n u^{od}_i\partial _i$, {\em and} \item $\d^{od} =
-\frac{1}{2} {\rm ad } (a)$ {\em where the {\em unique} element $a\in
\L_n'^{od}$ is given by the formula}
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{ev}_{i+1})+\partial _1 (u^{ev}_1).$$
\end{enumerate}
\end{itemize}
The next results describes differential ideals of $\Lambda_n$ (i.e.
which are stable under all derivations).
\begin{itemize}
\item (Proposition \ref{m19Sep06}) {\em Let $K$ be a commutative
ring with $\frac{1}{2}\in K$, ${\cal F}_n(K):= \{ I: I_0\subseteq I_1
\subseteq \cdots \subseteq I_n\, | \, I_i\; {\rm are\; ideals\;
of}\; K\}$ be the set of $n$-flags of ideals of $K$, ${\rm DI} (\Lambda_n)$
be the set of all differentiable ideals of $\Lambda_n(K)$. Then the map
$$ {\cal F}_n (K)\rightarrow {\rm DI} (\Lambda_n) , \; I\mapsto \widehat{I} :=
\oplus_{i=0}^n\oplus_{|\alpha | = i} I_ix^\alpha, $$ is a
bijection. In particular, $\mathfrak{m}^i$, $0\leq i\leq n+1$, are
differential ideals of $\Lambda_n$; these are the only differential
ideals of $\Lambda_n$ if $K$ is a field of characteristic} $\neq 2$.
\item (Theorem \ref{26Sep06}) {\em Let $K$ be a reduced
commutative ring with $\frac{1}{2}\in K$, $n\geq 1$. Then
1. $ {\rm Der }_K(\Lambda_n)$ is a faithful ${\rm Aut}_K(\Lambda_n)$-module iff $n\geq
2$.
2. The ${\rm Aut}_K(\Lambda_n)$-module $ {\rm Der }_K(\Lambda_n)$ is not simple. }
\end{itemize}
{\bf Skew derivations of the Grassmann algebras}. Let
${\rm SDer}_K(\Lambda_n)$, ${\rm SDer}_K(\Lambda_n)^{ev}$, ${\rm SDer}_K(\Lambda_n)^{od}$ and
${\rm ISDer}_K(\Lambda_n)$ be the set of all, even, odd and inner skew
derivations of $\Lambda_n(K)$ respectively. ${\rm ISDer}_K(\Lambda_n)=\{ {\rm sad} (a)
\, | \, a\in \Lambda_n\}$ and ${\rm sad} (a) (a_i) := aa_i-(-1)^ia_ia$
($a_i\in \Lambda_{n,i}$) is the inner skew derivation determined by the
element $a$. For an odd $n$, let $ \L_n'^{ev} := \L_n^{ev} $. For an even
$n$, let $ \L_n'^{ev}$ be the $K$-submodule of $\L_n^{ev}$ generated by
all `monomials' $x^\alpha$ but $\theta := x_1\cdots x_n$, i.e. $\L_n^{ev}
= \L_n'^{ev} \oplus K\theta$.
\begin{itemize}
\item (Theorem \ref{a13Sep06}) {\em Suppose that $K$ is a
commutative ring with $\frac{1}{2}\in K$. Then}
\begin{enumerate}
\item ${\rm SDer}_K(\Lambda_n) = \SDer_K(\L_n)^{ev} \oplus \SDer_K (\L_n)^{od}$. \item $\SDer_K (\L_n)^{od} =
\oplus_{i=1}^n \L_n^{ev}\partial _i$. \item $\SDer_K(\L_n)^{ev} = {\rm ISDer}_K(\Lambda_n)$.
\item ${\rm SDer}_K(\Lambda_n)/ {\rm ISDer}_K(\Lambda_n)\simeq \SDer_K (\L_n)^{od}$.
\end{enumerate}
\end{itemize}
So, any skew $K$-derivation $\delta$ of $\Lambda_n$ is a unique sum $\delta =
\d^{ev} +\d^{od}$ of an even and odd
skew derivation, and $\d^{ev} := \frac{1}{2}{\rm sad} (a)$ for a unique
element $a\in \L_n'^{ev}$. The next corollary describes explicitly the
skew derivations $\d^{ev}$ and $\d^{od}$.
\begin{itemize}
\item (Corollary \ref{ca13Sep06}) {\em Let $K$ be a commutative
ring with $\frac{1}{2}\in K$, $\delta$ be a skew $K$-derivation of
$\Lambda_n(K)$, and, for each $i=1, \ldots , n$, $\delta (x_i) = u^{ev}_i +
u^{od}_i$ for unique elements $u^{ev}_i\in \L_n^{ev} $ and $u^{od}_i\in
\L_n^{od} $. Then}
\begin{enumerate}
\item $\d^{od} =\sum_{i=1}^n u^{ev}_i\partial _i$, {\em and} \item $\d^{ev} =
\frac{1}{2}{\rm sad} (a)$ {\em where the unique element $a\in \L_n'^{ev}$
is given by the formula}
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{od}_{i+1})+\partial _1 (u^{od}_1).$$
\end{enumerate}
\end{itemize}
{\bf The action of ${\rm Aut}_K(\Lambda_n)$ on the set of generic normal
non-units of $\Lambda_n$}.
Let ${\cal N}$ be the set of all the normal elements of the Grassmann
algebra $\Lambda_n = \Lambda_n(K)$, ${\cal U}$ be the set of all units of
$\Lambda_n$, and $G:={\rm Aut}_K(\Lambda_n)$ be the group of $K$-automorphisms of
$\Lambda_n$. Then ${\cal U} \subseteq {\cal N}$. The set ${\cal N}$ is a disjoint
union of its $G$-invariant subsets,
$${\cal N} = \cup_{i=0}^n{\cal N}_i, \;\; {\cal N}_i:= \{ a\in {\cal N} \, | \, a=
a_i+\cdots , 0\neq a_i\in \Lambda_{n,i}\}.$$ Clearly, ${\cal N}_0= {\cal U}$. The
next result shows that `generic' normal non-unit elements of
$\Lambda_n$ (i.e. the set ${\cal N}_1$) form a single $G$-orbit if $n$ is
even, and two $G$-orbits if $n$ is odd.
\begin{itemize} \item (Theorem \ref{5Nov06}) {\em Let $K$ be a field of
characteristic $\neq 2$ and $\Lambda_n = \Lambda_n(K)$. Then}
\begin{enumerate} \item ${\cal N}_1= Gx_1$ {\em if $n$ is
even.}\item ${\cal N}_1= Gx_1\cup G(x_1+x_2\cdots x_n)$ {\em is the
disjoint union of two orbits if $n$ is odd.}
\end{enumerate}
\end{itemize}
The stabilizers of the elements $x_1$ and $x_1+x_2\cdots x_n$ are
found (Lemma \ref{c5Nov06} and Lemma \ref{s10Nov06}).
\section{Derivations of the Grassmann
rings}\label{DRGA
In this section, the results on derivations from the Introduction
are proved. First, we recall some facts on Grassmann algebra (more
details the reader can find in \cite{BourbakiAlgCh1-3}).
{\bf The Grassmann algebra and its gradings}. Let $K$ be an {\em
arbitrary} ring (not necessarily commutative). The {\em Grassmann
algebra} (the {\em exterior algebra}) $\Lambda_n = \Lambda_n (K)= K\lfloor
x_1, \ldots , x_n\rfloor$ is generated freely over $K$ by elements
$x_1, \ldots , x_n$ that satisfy the defining relations:
$$ x_1^2=\cdots = x_n^2=0 \;\; {\rm and}\;\; x_ix_j=-x_jx_i\;\;
{\rm for \; all} \;\; i\neq j.$$ Let ${\cal B}_n$ be the set of all
subsets of the set of indices $\{ 1, \ldots , n\}$. We may
identify the set ${\cal B}_n$ with the direct product $\{ 0,1\}^n$ of
$n$ copies of the two-element set $\{ 0, 1\}$ by the rule $\{ i_1,
\ldots , i_k\} \mapsto (0, \ldots, 1, \ldots , 1, \ldots , 0)$
where $1$'s are on $i_1, \ldots , i_k$ places and $0$'s elsewhere.
So, the set $\{ 0, 1\}^n$ is the set of all the characteristic
functions on the set $\{ 1, \ldots , n\}$.
$$ \Lambda_n = \bigoplus_{\alpha \in {\cal B}_n} Kx^\alpha = \bigoplus_{\alpha \in {\cal B}_n} x^\alpha K,
\;\; x^\alpha :=x_1^{\alpha_1} \cdots x_n^{\alpha_n}, $$
where $\alpha = (\alpha_1, \ldots , \alpha_n)\in \{ 0, 1\}^n =
{\cal B}_n$. Note that the order in the product $x^\alpha$ is fixed.
So, $\Lambda_n$ is a free left and right $K$-module of rank $2^n$. Note
that $(x_i):= x_i\Lambda_n = \Lambda_nx_i$ is an ideal of $\Lambda_n$. Each
element $a\in \Lambda_n$ is a unique sum $a= \sum a_\alpha x^\alpha$,
$a_\alpha \in K$. One can view each element $a$ of $\Lambda_n$ as a
`function' $a= a(x_1, \ldots , x_n)$ in the non-commutative
variables $x_i$. The $K$-algebra epimorphism
\begin{eqnarray*}
\Lambda_n &\rightarrow & \Lambda_n / (x_{i_1}, \ldots , x_{i_k})\simeq K\lfloor x_1, \ldots
, \widehat{x_{i_1}}, \ldots , \widehat{x_{i_k}}, \ldots ,
x_n\rfloor, \\
a&\mapsto & a|_{x_{i_1}=0, \ldots , x_{i_k}=0} :=a+ (x_{i_1},
\ldots , x_{i_k}),
\end{eqnarray*}
may be seen as the operation of
taking value of the function $a(x_1, \ldots , x_n)$ at the point
$x_{i_1} = \cdots = x_{i_k}=0$ where here and later the hat over a
symbol means that it is missed.
For each $\alpha \in {\cal B}_n$, let $ | \alpha | := \alpha_1 +\cdots
+ \alpha_n$. The ring $\Lambda_n=\oplus_{i=0}^n \Lambda_{n,i}$ is a
$\mathbb{Z}$-{\em graded} ring ($\Lambda_{n,i}\Lambda_{n,j}\subseteq \Lambda_{n,i+j}$ for
all $i,j$) where $\Lambda_{n,i}:= \oplus_{|\alpha | =i}Kx^\alpha$. The
ideal $\mathfrak{m} := \oplus_{i\geq 1} \Lambda_{n,i}$ of $\Lambda_n$ is called the
{\em augmentation} ideal. Clearly, $K\simeq \Lambda_n/ \mathfrak{m}$, $\mathfrak{m}^n=
Kx_1\cdots x_n$ and $\mathfrak{m}^{n+1}=0$. We say that an element $\alpha$
of ${\cal B}_n$ is {\em even} (resp. {\em odd}) if the set $\alpha$
contains even (resp. odd) number of elements. By definition, the
empty set is even. Let $\mathbb{Z}_2:= \mathbb{Z} / 2\mathbb{Z} = \{\overline{0}, \overline{1} \}$. The ring
$\Lambda_n = \L_{n,\overline{0}} \oplus \L_{n,\overline{1}}$ is a $\mathbb{Z}_2$-{\em graded} ring where
$\L_{n,\overline{0}} := \L_n^{ev} :=\oplus_{\alpha \; {\rm is \; even}}Kx^\alpha$ is
the subring of even elements of $\Lambda_n$ and $\L_{n,\overline{1}} := \L_n^{od}
:=\oplus_{\alpha \; {\rm is \; odd}}Kx^\alpha$ is the
$\L_n^{ev}$-module of odd elements of $\Lambda_n$. The ring $\Lambda_n$ has the
$\mathfrak{m}$-{\em adic} filtration $\{ \mathfrak{m}^i\}_{i\geq 0}$. The even
subring $\L_n^{ev}$ has the induced $\mathfrak{m}$-adic filtration $\{ \Lambda_{n,
\geq i}^{ev} := \L_n^{ev} \cap \mathfrak{m}^i \}$. The $\L_n^{ev}$-module $\L_n^{od}$
has the induced $\mathfrak{m}$-adic filtration $\{ \Lambda_{n, \geq i}^{od} :=
\L_n^{od} \cap \mathfrak{m}^i \}$.
The
$K$-linear map $ a\mapsto \overline{a}$ from $\Lambda_n$ to itself which is
given by the rule
$$\overline{a} :=
\begin{cases}
a,& \text{if $a\in \L_{n,\overline{0}}$},\\
-a,& \text{if $a\in \L_{n,\overline{1}}$},
\end{cases}$$
is a ring automorphism such that $\overline{\overline{a}}= a$ for all
$a\in \Lambda_n$. For all $a\in \Lambda_n$ and $i=1, \ldots , n$,
\begin{equation}\label{xiaib}
x_ia= \overline{a} x_i \;\; {\rm and}\;\; ax_i= x_i\overline{a} .
\end{equation}
So, each element $x_i$ of $\Lambda_n$ is a {\em normal} element, i.e.
the two-sided ideal $(x_i)$ generated by the element $x_i$
coincides with both left and right ideals generated by $x_i$:
$(x_i) = \Lambda_n x_i= x_i\Lambda_n$.
For an arbitrary $\mathbb{Z}$-graded ring $A= \oplus_{i\in \mathbb{Z}} A_i$, an
additive map $\delta :A\rightarrow A$ is called a {\em left skew derivation}
if
\begin{equation}\label{dsd}
\delta (a_ia_j) = \delta (a_i) a_j+(-1)^ia_i\delta (a_j)\;\; {\rm for \;
all}\;\; a_i\in A_i, \; a_j\in A_j.
\end{equation}
In this paper, a skew derivation means a {\em left} skew
derivation. Clearly, $1\in {\rm ker } (\delta )$ ($ \delta (1) = \delta (1\cdot 1) =
2\delta (1)$ and so $\delta (1)=0$). The restriction of the left skew
derivation $\delta$ to the even subring $A^{ev}:= \oplus_{i\in 2\mathbb{Z}}
A_i$ of $A$ is an ordinary derivation. Recall that an additive
subgroup $B$ of $A$ is called a homogeneous subgroup if $B=
\oplus_{i\in \mathbb{Z} } B\cap A_i$.
{\it Definition}. For the ring $\Lambda_n (K)$, consider the set of
{\em left} skew $K$-derivations:
$$ \partial _1:= \frac{\partial }{\partial x_1}, \ldots , \partial _n:=
\frac{\partial }{\partial x_n}$$ given by the rule $\partial _i (x_j)= \delta_{ij}$,
the Kronecker delta. Informally, these skew $K$-derivations will
be called (left) {\em partial skew derivatives}.
{\it Example}. $\partial _i (x_1 \cdots x_i \cdots x_k)= (-1)^{i-1}
x_1\cdots x_{i-1} x_{i+1} \cdots x_k$.
If the ring $K$ is commutative, $2\in K$ is regular (i.e. $2\lambda
=0$ in $K$ implies $\lambda =0$), and $n\geq 2$, then the centre of
$\Lambda_n(K)$ is equal to
$$ Z(\Lambda_n)=\begin{cases}
\L_{n,\overline{0}} , & \text{if $n$ is even},\\
\L_{n,\overline{0}}\oplus Kx_1\cdots x_n, & \text{if $n$ is odd }.
\end{cases}$$
Let $K$ be a commutative ring. For an even $n$, let $ \L_n'^{od} :=
\L_n^{od} $. For an odd $n$, let $ \L_n'^{od}$ be the $K$-submodule of
$\L_n^{od}$ generated by all `monomials' $x^\alpha$ but $\theta :=
x_1\cdots x_n$, i.e. $\L_n^{od} = \L_n'^{od} \oplus K\theta$.
Similarly, for an odd $n$, let $
\L_n'^{ev} := \L_n^{ev} $. For an even $n$, let $ \L_n'^{ev}$ be the
$K$-submodule of $\L_n^{ev}$ generated by all `monomials' $x^\alpha$
but $\theta := x_1\cdots x_n$, i.e. $\L_n^{ev} = \L_n'^{ev} \oplus K\theta$.
For any $n$,
\begin{equation}\label{Lndd}
\L_n^{od} = \L_n'^{od} \oplus \L_n^{od}\cap Z(\Lambda_n ).
\end{equation}
So, one can naturally identify $\L_n^{od} / \L_n^{od} \cap Z(\Lambda_n )$ with
$\L_n'^{od}$.
Consider the sets of {\em even} and {\em odd} $K$-derivations of
$\Lambda_n(K)$:
\begin{eqnarray*}
\Der_K(\L_n)^{ev} & :=& \{ \delta \in {\rm Der }_K(\Lambda_n )\, | \, \delta (\L_{n,\overline{i}} ) \subseteq \L_{n,\overline{i}} , \; \overline{i} \in \mathbb{Z}_2 \},\\
\Der_K (\L_n)^{od} & :=& \{ \delta \in {\rm Der }_K(\Lambda_n )\, | \, \delta (\L_{n,\overline{i}} ) \subseteq \Lambda_{n, \overline{i}+\overline{1}} , \; \overline{i} \in \mathbb{Z}_2
\}.
\end{eqnarray*}
So, even derivations are precisely the derivations that respect
$\mathbb{Z}_2$-grading of $\Lambda_n$, and the odd derivations are precisely
the derivations that reverse it. The set of odd and even
derivations are left $\L_{n,\overline{0}}$-modules. For each element $a\in \Lambda_n$,
one can attach the $K$-derivation of $\Lambda_n$ $ {\rm ad } (a) : b\mapsto
[a,b]:= ab-ba$, so-called, the {\em inner} derivation determined
by $a$. The set of all inner derivations is denoted by
${\rm IDer}_K(\Lambda_n)$, and the map
$$\Lambda_n/ Z(\Lambda_n)\rightarrow {\rm IDer}_K(\Lambda_n), \;\; a+Z(\Lambda_n) \mapsto {\rm ad } (a),
$$
is an isomorphism of left $Z(\Lambda_n)$-modules. The next theorem describes
explicitly the sets of all/inner/even and odd derivations.
\begin{theorem}\label{13Sep06
Suppose that $K$ is a commutative ring with $\frac{1}{2}\in K$.
Then
\begin{enumerate}
\item $ {\rm Der }_K(\Lambda_n) = \Der_K(\L_n)^{ev} \oplus \Der_K (\L_n)^{od}$. \item $\Der_K(\L_n)^{ev} =
\oplus_{i=1}^n \L_n^{od} \partial _i$. \item $\Der_K (\L_n)^{od} = {\rm IDer}_K(\Lambda_n)$ and
the map
$$ {\rm ad } : \L_n'^{od} = \L_n^{od} / \L_n^{od} \cap Z(\Lambda_n )\rightarrow {\rm IDer}_K(\Lambda_n),
\;\; a\mapsto {\rm ad } (a), $$ is the $Z(\Lambda_n)$-module isomorphism.
\item $ {\rm Der }_K(\Lambda_n)/ {\rm IDer}_K(\Lambda_n)\simeq \Der_K(\L_n)^{ev}$.
\end{enumerate}
\end{theorem}
{\it Proof}. Since $\Der_K(\L_n)^{ev} \cap \Der_K (\L_n)^{od} = 0$, one has the inclusion
\begin{equation}\label{1ri}
{\rm Der }_K(\Lambda_n)\supseteq \Der_K(\L_n)^{ev} \oplus \Der_K (\L_n)^{od} .
\end{equation}
Clearly,
\begin{equation}\label{2ri}
\Der_K(\L_n)^{ev} \supseteq \sum_{i=1}^n \L_n^{od} \partial _i =\bigoplus_{i=1}^n
\L_n^{od} \partial _i ,
\end{equation}
${\rm IDer}_K(\Lambda_n) \simeq \Lambda_n / Z(\Lambda_n) = (\L_n^{ev} \oplus \L_n^{od} ) /
Z(\Lambda_n)\simeq \L_n^{od} / \L_n^{od} \cap Z(\Lambda_n)\simeq \L_n'^{od}$ since
$\L_n^{ev} \subseteq Z(\Lambda_n)$. For each $a\in \L_n^{od}$, $ {\rm ad } (a) \in
\Der_K (\L_n)^{od}$, hence
\begin{equation}\label{3ri}
\Der_K (\L_n)^{od} \supseteq {\rm IDer}_K(\Lambda_n).
\end{equation}
Note that statement 4 follows from statements 1 and 3. Now, it is
obvious that in order to finish the proof of the theorem it
suffices to show that
{\it Claim}. $ {\rm Der }_K(\Lambda_n) \subseteq \sum_{i=1}^n \L_n^{od} \partial _i
+{\rm IDer}_K(\Lambda_n)$.
Indeed, suppose that the inclusion of the claim holds then, by
(\ref{2ri}) and (\ref{3ri}),
$$ {\rm Der }_K(\Lambda_n) \subseteq \sum_{i=1}^n \L_n^{od} \partial _i
+{\rm IDer}_K(\Lambda_n)\subseteq \Der_K(\L_n)^{ev} \oplus \Der_K (\L_n)^{od},$$ hence statement 1
is true by (\ref{1ri}). Statement 1 together with inclusions
(\ref{2ri}) and (\ref{3ri}) implies statements 2 and 3.
{\it Proof of the Claim}. Let $\delta$ be a $K$-derivation of $\Lambda_n$.
We have to represent the derivation $\delta $ as a sum
$$ \delta = \sum_{i=1}^n a_i\partial _i + {\rm ad } (a), \;\; a_i\in \L_n^{od}, \;\;
a\in \Lambda_n.$$ The proof of the claim is constructive.
According to the decomposition $\Lambda_n= \L_n^{ev} \oplus \L_n^{od}$ each
element $u$ of $\Lambda_n$ is a unique sum
\begin{equation}\label{u=ueo}
u= u^{ev} + u^{od}
\end{equation}
of its even and odd components ($u^{ev} \in \L_n^{ev}$
and $u^{od} \in \L_n^{od}$). For each $i$, let $u_i:= \delta (x_i) =
u^{ev}_i+u^{od}_i$,
$$\partial := \sum_{i=1}^n u^{od}_i\partial _i \;\; {\rm and}\;\; \delta':= \delta -
\partial .$$ Note that $\partial \in \sum_{i=1}^n \L_n^{od} \partial _i$, hence
changing $\delta$ for $\delta'$, if necessary, one may assume that all
the elements $u_i$ are even. So, it suffices to show that $\delta =
{\rm ad } (a)$ for some $a$. We produce such an $a$ in several steps.
{\it Step 1}. Let us prove that, {\em for each $i=1, \ldots , n$,
$u_i= v_ix_i$ for some element } $v_i\in K\lfloor x_1, \ldots ,
\widehat{x_i}, \ldots , x_n\rfloor^{od}$. Note that $0= \delta (0)= \delta
(x_i^2) = u_ix_i + x_iu_i= 2u_ix_i$, and so $u_ix_i=0$ (since
$\frac{1}{2}\in K$). This means that $u_i= v_ix_i$ for some
element $v_i\in K\lfloor x_1, \ldots , \widehat{x_i}, \ldots ,
x_n\rfloor^{od}$ since $u_i$ is even. For $n=1$, it gives $u_1=0$
since $K^{od}=0$, and we are done. So, let $n\geq 2$.
{\it Step 2}. We claim that, {\em for each pair} $i\neq j$,
\begin{equation}\label{ast2}
v_i|_{x_j=0}=v_j|_{x_i=0}.
\end{equation}
Evaluating the derivation $\delta$ at the element $0= x_ix_j+x_jx_i$
and taking into account that all the elements $u_i$ are even
(hence central) we obtain
$$ 0= 2( u_ix_j+u_jx_i) = 2(v_ix_ix_j+v_jx_jx_i) = 2( v_i-v_j)
x_ix_j.$$ This means that $v_i-v_j\in (x_i, x_j)$ since
$\frac{1}{2}\in K$, or, equivalently, $v_i|_{x_i=0,
x_j=0}=v_j|_{x_i=0, x_j=0}$. By Step 1, this equality can be
written as (\ref{ast2}).
{\it Step 3}. Note that $\delta (x_1)=v_1x_1= {\rm ad } (\frac{1}{2}v_1)
(x_1)$ since $v_1$ is odd, i.e. $(\delta - {\rm ad }
(\frac{1}{2}v_1))(x_1)=0$. So, changing $\delta$ for $\delta - {\rm ad }
(\frac{1}{2}v_1)$ one can assume that $\delta (x_1)=0$, i.e. $v_1=0$.
Then, by (\ref{ast2}), $v_i|_{x_1=0}=0$ for all $i=2, \ldots , n$,
and so $v_i\in (x_1 x_i)$ for all $i=2, \ldots , n$. Summarizing,
we can say that by adding to $\delta$ a well chosen inner derivation
one can assume that $\delta (x_1)=0$ and $\delta (x_i) \in (x_1 x_i)$ for
all $i\geq 2$. This statement serves as the base of the induction
in the proof of the next statement. For each $k$ such that $1\leq
k\leq n$, by adding to $\delta$ a certain inner derivation we can
assume that
\begin{equation}\label{kind}
\delta (x_1)=\cdots = \delta (x_k)=0, \;\; \delta (x_i) \in (x_1 \cdots
x_kx_i), \;\; k<i\leq n .
\end{equation}
So, assuming that (\ref{kind}) holds for $k$ we must prove the
same statement but for $k+1$. Note that $v_{k+1} x_{k+1} = \delta (
x_{k+1})\in (x_1\cdots x_k)$, hence $v_{k+1} = x_1\cdots x_kv$ for
some $v\in K\lfloor x_{k+2} , \ldots , x_n\rfloor$. Consider the
derivation $\delta':= \delta - {\rm ad } (\frac{1}{2}v_{k+1})$. For each $i=1,
\ldots , k$, $\delta'(x_i)= \delta (x_i) =0$ as $v_{k+1} \in ( x_1\cdots
x_k)$; and $\delta'( x_{k+1}) = v_{k+1} x_{k+1}- [\frac{1}{2}v_{k+1},
x_{k+1}]= v_{k+1} x_{k+1} - v_{k+1} x_{k+1}=0$. These prove the
first part of (\ref{kind}) for $k+1$, namely, that
$$ \delta' (x_1)=\cdots = \delta' (x_{k+1})=0.$$
So, changing $\delta$ for $\delta'$ one can assume that
$$ \delta (x_1)=\cdots = \delta (x_{k+1})=0.$$
These conditions imply that $v_1=\cdots = v_{k+1}=0$. If $n=k+1$,
we are done. So, let $k+1<n$. Then, by (\ref{ast2}), for each
$i>k+1$, $v_i \in \cap_{j=1}^{k+1} (x_j)= (x_1\cdots x_{k+1})$,
hence $\delta (x_i) = v_ix_i \in ( x_1\cdots x_{k+1} x_i)$. By
induction, (\ref{kind}) is true for all $k$. In particular, for
$k=n$ one has $\delta =0$. This means that $\delta$ is an inner
derivation, as required. $\Box $
The ring $K[x]/(x^2)$ of dual
numbers is the Grassmann ring $\Lambda_1$.
\begin{corollary}\label{cc13Sep06
Suppose that $K$ is a commutative ring with $\frac{1}{2}\in K$.
Then $ {\rm Der }_K(K[x]/(x^2))= {\rm Der }_K(K[x]/(x^2))^{ev} = Kx\frac{d}{dx}$
and $ {\rm Der }_K(K[x]/(x^2))^{od}=0$ where $\frac{d}{dx}$ is the skew
$K$-derivation of $K[x]/(x^2)$.
\end{corollary}
A Lie algebra $({\cal G} , [\cdot , \cdot ])$ over $K$ is positively
graded if ${\cal G} = \oplus_{i\geq 0} {\cal G}_i$ is a direct sum of
$K$-submodules such that $[{\cal G}_i, {\cal G}_j]\subseteq {\cal G}_{i+j}$ for
all $i,j\geq 0$.
$( {\rm Der }_K(\Lambda_n), [\cdot , \cdot ])$ is a Lie algebra over $K$ where
$[\delta , \partial ] := \delta \partial - \partial \delta$. By Theorem \ref{13Sep06}, the
Lie algebra $ {\rm Der }_K(\Lambda_n)=\oplus_{i\geq 0}D_i$ is a positively
graded Lie algebra where $$D_i:= \{ \delta \in {\rm Der }_K(\Lambda_n)\, | \, \delta
(\Lambda_{n,j})\subseteq \Lambda_{n,j+i}, \; j\geq 0\}.$$ Clearly, $D_i=0$,
$i\geq n$. For each even natural number $i$ such that $0\leq i\leq
n-1$,
\begin{equation}\label{Diev}
D_i=\oplus_{j=1}^n\Lambda_{n,i+1}\partial _j.
\end{equation}
For each odd natural number $i$ such that $1\leq i\leq n-1$,
\begin{equation}\label{Diod}
D_i=\{ {\rm ad } (a) \, | \, a\in \Lambda_{n,i}\} \simeq \Lambda_{n, i},\;\; {\rm ad }
(a) \mapsto a.
\end{equation}
The zero component $D_0= \oplus_{i,j=0}^nKx_i\partial _j$ of
$ {\rm Der }_K(\Lambda_n)$ is a Lie subalgebra of $ {\rm Der }_K(\Lambda_n)$ which is
canonically isomorphic to the Lie algebra ${\rm gl}_n(K):=
\oplus_{i,j=1}^nKE_{ij}$ via $D_0\rightarrow {\rm gl}_n(K)$, $x_i\partial _j\mapsto
E_{ij}$, where $E_{ij}$ are the matrix units. By the very
definition, $D_+:= \oplus_{i\geq 1}D_i$ is a nilpotent ideal of
the Lie algebra $ {\rm Der }_k(\Lambda_n)$ such that $ {\rm Der }_K(\Lambda_n)= D_0\oplus
D_+\simeq {\rm gl}_n(K)\oplus D_+$ and $ {\rm Der }_K(\Lambda_n)/D_+\simeq
{\rm gl}_n(K)$. So, if $K$ is a field of characteristic zero then $D_+$
is the radical of the Lie algebra $ {\rm Der }_K(\Lambda_n)$.
The Lie algebra $ {\rm Der }_K(\Lambda_n)= \Der_K(\L_n)^{ev} \oplus \Der_K (\L_n)^{od}$ is a
$\mathbb{Z}_2$-graded Lie algebra.
By Theorem \ref{13Sep06}, any $K$-derivation $\delta$ of $\Lambda_n$ is a
unique sum $\delta = \d^{ev} +\d^{od}$ of even and odd derivations, and
$\d^{od} := -\frac{1}{2} {\rm ad } (a)$ for a {\em unique} element $a\in
\L_n'^{od}$. In order to find the element $a$ (Corollary
\ref{c13Sep06}), we need two theorems which are interesting on
their own right. Theorem \ref{14Sep06} gives a unique (sort of
`triangular') canonical presentation of any element of $\Lambda_n$.
This presentation is important in dealing with derivations and
skew derivations. The element $a$ in $\d^{od} = -\frac{1}{2} {\rm ad } (a)$
is given in this form (Corollary \ref{c13Sep06}). In order to find the element
$a$ we need to find solutions to the system of equations (Theorem
\ref{s14Sep06}). This system is a kind of Poincar\'{e} Lemma for
the (noncommutative) Grassmann algebra $\Lambda_n$.
\begin{theorem}\label{14Sep06
\cite{jacgras} Let $K$ be an arbitrary (not necessarily
commutative) ring. Then
\begin{enumerate}
\item the Grassmann ring $\Lambda_n(K)$ is a direct sum of right
$K$-modules
\begin{eqnarray*}
\Lambda_n(K)&=& x_1\cdots x_nK \oplus x_1\cdots x_{n-1}K \oplus
x_1\cdots x_{n-2}K\lfloor x_n\rfloor \oplus\cdots \\
&\cdots &\oplus x_1\cdots x_iK\lfloor x_{i+2}\ldots , x_n\rfloor
\oplus\cdots \oplus x_1 K\lfloor x_3\ldots , x_n\rfloor\oplus
K\lfloor x_2\ldots , x_n\rfloor .
\end{eqnarray*}
\item So, each element $a\in \Lambda_n(K)$ is a unique sum
$$ a= x_1\cdots x_na_n+ x_1\cdots x_{n-1}b_n+\sum_{i=1}^{n-2}
x_1\cdots x_ib_{i+1} + b_1$$ where $a_n, b_n\in K$, $b_i\in
K\lfloor x_{i+1}\ldots , x_n\rfloor$, $1\leq i\leq n-1$.
Moreover,
\begin{eqnarray*}
a_n&=& \partial _n\partial _{n-1}\cdots \partial _1(a), \\
b_{i+1}&=&\partial _i\partial _{i-1}\cdots \partial _1(1-x_{i+1}\partial _{i+1})(a), \; 1\leq i\leq n-1, \\
b_1&=&(1-x_1\partial _1)(a).
\end{eqnarray*}
So, $$ a= x_1\cdots x_n\partial _n\partial _{n-1}\cdots
\partial _1(a)+\sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots
\partial _1(1-x_{i+1}\partial _{i+1})(a)+(1-x_1\partial _1)(a).$$
\end{enumerate}
\end{theorem}
By Theorem \ref{14Sep06}, the identity map ${\rm id}_{\Lambda_n}
:\Lambda_n\rightarrow \Lambda_n$ is equal to
\begin{equation}\label{idLn}
{\rm id}_{\Lambda_n}=x_1\cdots x_n\partial _n\partial _{n-1}\cdots \partial _1+
\sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots
\partial _1(1-x_{i+1}\partial _{i+1})+(1-x_1\partial _1).
\end{equation}
\begin{theorem}\label{s14Sep06
\cite{jacgras} Let $K$ be an arbitrary ring, $u_1, \ldots , u_n\in
\Lambda_n(K)$, and $ a\in \Lambda_n(K)$ be an unknown. Then the system of
equations
$$\begin{cases}
x_1a=u_1 \\
x_2a=u_2 \\
\;\;\;\; \;\;\;\vdots \\
x_na=u_n
\end{cases}
$$
has a solution in $\Lambda_n$ iff the following two conditions hold
\begin{enumerate}
\item $u_1\in (x_1), \ldots , u_n\in (x_n)$, and \item
$x_iu_j=-x_ju_i$ for all $i\neq j$.
\end{enumerate}
In this case,
\begin{equation}\label{alsol}
a= x_1\cdots x_na_n+\sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots
\partial _1\partial _{i+1}(u_{i+1})+\partial _1(u_1), \;\; a_n \in K,
\end{equation}
are all the solutions.
\end{theorem}
The next corollary describes explicitly $\d^{ev}$ and $\d^{od}$ in $\delta =
\d^{ev} + \d^{od}$.
\begin{corollary}\label{c13Sep06
Let $K$ be a commutative ring with $\frac{1}{2}\in K$, $\delta$ be a
$K$-derivation of $\Lambda_n(K)$, and, for each $i=1, \ldots , n$, $\delta
(x_i) = u^{ev}_i + u^{od}_i$ for unique elements $u^{ev} \in \L_n^{ev} $
and $u^{od} \in \L_n^{od}$. Then
\begin{enumerate}
\item $\d^{ev} =\sum_{i=1}^n u^{od}_i\partial _i$, and \item $\d^{od} =
-\frac{1}{2} {\rm ad } (a)$ where the {\em unique} element $a\in \L_n'^{od}$
is given by the formula
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{ev}_{i+1})+\partial _1 (u^{ev}_1).$$
\end{enumerate}
\end{corollary}
{\it Proof}. 1. This statement has been proved already in the
proof of Theorem \ref{13Sep06}.
2. For each $i=1, \ldots , n$, on the one hand $\d^{od} (x_i) = (\delta -
\d^{ev} )(x_i) = u^{ev}_i$; on the other, $ \d^{ev} (x_i) =
-\frac{1}{2}(ax_i-x_ia) = \frac{1}{2}2x_ia= x_ia$. So, the element
$a$ is a solution to the system of equations
$$\begin{cases}
x_1a=u^{ev}_1 \\
x_2a=u^{ev}_2 \\
\;\;\;\; \;\;\;\vdots \\
x_na=u^{ev}_n.
\end{cases}
$$
By Theorem \ref{s14Sep06} and the fact that $a\in \L_n'^{od}$ (i.e.
$a_n=0$), we have
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{ev}_{i+1})+\partial _1 (u^{ev}_1).\;\;\; \Box$$
Let $K$ be a field. Let $V$ be a finite dimensional vector space
over $K$ and $a\in {\rm End }_K(V)$, a $K$-linear map on $V$. The vector
space $V$ is the $K[t]$-module where $t\cdot v := av$; $V$ is the
$K[a]$-module for short. The linear map $a$ is called {\em
semi-simple} (resp. {\em nilpotent}) if the $K[a]$-module $V$ is
semi-simple (resp. $a^k=0$ for some $k\geq 1$). It is well-known
that $a$ is a unique sum $a= a_s+a_n$ where $a_s$ is a semi-simple
map, $a_n$ is a nilpotent map, and $a_s, a_n\in K[a]:= \sum_{i\geq
0} Ka^i$ (in particular, the maps $a$, $a_s$, and $a_n$ commute).
If $V$ is a finite dimensional $K$-algebra and $a$ is a
$K$-derivation of the algebra $V$ then the maps $a_s$ and $a_n$
are also $K$-derivations.
The subsets of $ {\rm Der }_K(V)$ of all semi-simple derivations
$ {\rm Der }_K(V)_s$ and all nilpotent derivations $ {\rm Der }_K(V)_n$ do not
meet, i.e. $ {\rm Der }_K(V)_s\cap {\rm Der }_K(V)_n=0$. In general, the sets
of semi-simple and nilpotent derivations are {\em not} vector
spaces, though they are closed under scalar multiplication.
Let $K$ be a commutative ring with $\frac{1}{2}\in K$. The
following $K$-derivations of $\Lambda_n= \Lambda_n(K)$
$$ h_1:= x_1\partial _1, \ldots , h_n:= x_n\partial _n,$$
commute, $h_1^2=h_1, \ldots , h_n^2=h_n$, and $h_i(x^\alpha ) =
\alpha_ix^\alpha$ for all $i$ and $\alpha$. Since $h_i(x_j)=
\delta_{ij}x_j$, the maps $h_1, \ldots , h_n$ are linearly
independent. So, $h_1, \ldots , h_n$ are {\em commuting,
semi-simple, $K$-linearly independent, idempotent $K$-derivations}
of the algebra $\Lambda_n$. For each $i=1, \ldots , n$, $\Lambda_n=
K_i\oplus x_iK_i$ where $K_i:= {\rm ker } (\partial _i) = K\lfloor x_1,
\ldots , \widehat{x_i}, \ldots , x_n\rfloor$, and $h_i: \Lambda_n\rightarrow
\Lambda_n$ is the projection onto $x_iK_i$.
Let $H$ be the subalgebra of the endomorphism algebra
$ {\rm End }_K(\Lambda_n)$ generated by the elements $h_1, \ldots , h_n$. As
an abstract algebra $H\simeq K[H_1, \ldots , H_n]/ (H_1^2, \ldots
, H_n^2)$. The algebra $\Lambda_n=\oplus_{\alpha \in {\cal B}_n} Kx^\alpha$
is a semi-simple $H$-module where each isotypic component is
simple: $Kx^\alpha$ is the simple $H$-module, and $Kx^\alpha\simeq
Kx^\beta$ as $H$-modules iff $\alpha = \beta$. Let
\begin{eqnarray*}
\Der_K(\L_n)^{ev}_s:= {\rm Der }_K(\Lambda_n)^{ev}\cap {\rm Der }(\Lambda_n)_s &
\Der_K(\L_n)^{ev}_n := {\rm Der }_K(\Lambda_n)^{ev}\cap {\rm Der }(\Lambda_n)_n,\\
\Der_K(\L_n)^{od}_s:= {\rm Der }_K(\Lambda_n)^{od}\cap {\rm Der }(\Lambda_n)_s, & \Der_K(\L_n)^{od}_n
:= {\rm Der }_K(\Lambda_n)^{od}\cap {\rm Der }(\Lambda_n)_n.
\end{eqnarray*}
{\it Definition}. An ideal $\mathfrak{a}$ of $\Lambda_n$ is called a {\em
differential} ideal (or a $ {\rm Der }_K(\Lambda_n)$-{\em invariant} ideal) if
$\delta (\mathfrak{a} ) \subseteq \mathfrak{a}$ for all $\delta \in {\rm Der }_K(\Lambda_n)$.
The next proposition describes all the differential ideals of
$\Lambda_n(K)$.
\begin{proposition}\label{m19Sep06
Let $K$ be a commutative ring with $\frac{1}{2}\in K$, ${\cal F}_n(K):=
\{ I: I_0\subseteq I_1 \subseteq \cdots \subseteq I_n\, | \, I_i\;
{\rm are\; ideals\; of}\; K\}$ be the set of $n$-flags of ideals
of $K$, ${\rm DI} (\Lambda_n)$ be the set of all differentiable ideals of
$\Lambda_n(K)$. Then the map
$$ {\cal F}_n (K)\rightarrow {\rm DI} (\Lambda_n) , \; I\mapsto \widehat{I} :=
\oplus_{i=0}^n\oplus_{|\alpha | = i} I_ix^\alpha, $$ is a
bijection. In particular, $\mathfrak{m}^i$, $0\leq i\leq n+1$, are
differential ideals of $\Lambda_n$; these are the only differential
ideals of $\Lambda_n$ if $K$ is a field of characteristic $\neq 2$.
\end{proposition}
{\it Proof}. Recall that $ {\rm Der }_K(\Lambda_n) =\oplus_{i\geq 0} D_i$. By
(\ref{Diev}) and (\ref{Diod}), the map $I\mapsto \widehat{I}$ is
well-defined and injective, by the very definition. It remains to
show that each differential idea, say $\mathfrak{a}$, of $\Lambda_n(K)$ is equal
to $\widehat{I}$ for some $I\in {\cal F}_n(K)$. Since $\sum_{i=1}^n
Kh_i\subseteq {\rm Der }_K(\Lambda_n)$ and $ \Lambda_n =\oplus_{\alpha \in {\cal B}_n}
Kx^\alpha$ is the direct sum of non-isomorphic simple $H$-modules,
we have
$$\mathfrak{a} = \oplus_{\alpha \in
{\cal B}_n}(\mathfrak{a} \cap Kx^\alpha )= \oplus_{\alpha \in {\cal B}_n} \mathfrak{a}_\alpha
x^\alpha$$ where $\mathfrak{a}_\alpha$ is an ideal of $K$ such that
$\mathfrak{a}_\alpha x^\alpha = \mathfrak{a} \cap Kx^\alpha$. For each $i$ such that
$0\leq i \leq n$, $\oplus_{|\alpha |=i}\mathfrak{a}_\alpha x^\alpha$ is a
$D_0$-module where $D_0=\oplus_{i,j=1}^n Kx_i\partial _j$, hence all
the ideals $\mathfrak{a}_\alpha$ coincide where $|\alpha | = i$. Let $I_i$
be their common value. Since $\mathfrak{a}$ is an ideal of $\Lambda_n$, $\{
I:I_0\subseteq \cdots \subseteq I_n\} \in {\cal F}_n(K)$, and so $\mathfrak{a} =
\widehat{I}$, as required. $\Box$
{\it Definition}. A ring $R$ is called a {\em differentiably
simple} ring if it is a simple left $ {\rm Der } (R)$-module.
So, the algebra $\Lambda_n$ is {\em not} differentiably simple if
$n\geq 1$ where $K$ is a commutative ring with $\frac{1}{2}\in K$.
Let $K$ be a {\em reduced} commutative ring with $\frac{1}{2}\in
K$. Let ${\cal S}$ be the set of all $n$-tuples $(s_1, \ldots , s_n)$
where $s_1, \ldots , s_n$ are commuting, idempotent
$K$-derivations (i.e. $s_i^2=s_i$) of $\Lambda_n$ such that the
following conditions hold: $\cap_{i=1}^n {\rm ker } (s_i) = K$; all the
$K$-modules ${\cal K}_i:= \mathfrak{m} \cap {\rm ker } (s_i-1) \cap \cap_{j\neq i}
{\rm ker } (s_j)$ are free of rank $1$ over $K$, i.e. ${\cal K}_i =
Kx_i'\simeq K$ for some element $x_i'\in \mathfrak{m}$ $K$; $\Lambda_n = {\rm ker }
(s_1)+{\cal K}_1 {\rm ker } (s_1)$; and, for each $i=1, \ldots , n-1$,
\begin{equation}\label{K1i}
K_{1, \ldots , i}= K_{1, \ldots , i+1}+{\cal K}_{i+1} K_{1, \ldots ,
i+1}
\end{equation}
where $K_{1, \ldots , i}:= \cap_{j=1}^i {\rm ker } (s_j)$.
Clearly, $(h_1, \ldots, h_n)\in {\cal S}$ (see (\ref{CKxi}) below). For
each $(s_1, \ldots, s_n)\in {\cal S}$, the maps $s_1, \ldots , s_n$ are
$K$-linearly independent ($\sum \mu_i s_i=0$ $\Rightarrow$
$0=(\sum \mu_is_i)({\cal K}_i)= \mu_iKx_i'$ $\Rightarrow$ $\mu_i=0$).
Let $G:= {\rm Aut}_K(\Lambda_n)$ be the group of $K$-algebra automorphisms
of the Grassmann algebra $\Lambda_n$. For $\sigma \in G$, let $x_i':= \sigma
(x_i)$. Then $x_i'^2=\sigma(x_i^2)= \sigma (0)=0$. If $ \lambda_i\equiv
x_i'\mod \mathfrak{m}$ for some $\lambda_i\in K$ then $\lambda_i^2=0$, hence $\lambda_i=0$
since $K$ is reduced. Therefore, $\sigma (\mathfrak{m} ) = \mathfrak{m}$, and so
\begin{equation}\label{smi=mi}
\sigma (\mathfrak{m}^i)=\mathfrak{m}^i \;\; {\rm for \; all}\;\; i\geq 1.
\end{equation}
By (\ref{smi=mi}), the group $G$
acts on the set ${\cal S}$ by conjugation (i.e. by changing
generators): $\sigma \cdot (s_1, \ldots, s_n):= (\sigma s_1\sigma^{-1},
\ldots, \sigma s_n\sigma^{-1})$ We prove shortly that the group $G$ act
transitively on the set ${\cal S}$ (Corollary \ref{a19Sep06}.(2)) and
the stabilizer ${\rm St} (h_1, \ldots , h_n)$ of the element $(h_1,
\ldots , h_n)$ is equal to the `$n$-{\em dimensional algebraic
torus}' (where $K^*$ is the group of units of $K$)
$$ \mathbb{T}^n:= \{ \sigma_\lambda \, | \, \lambda \in K^{*n}, \sigma_\lambda ( x_i) =
\lambda_i x_i, 1\leq i\leq n\} \simeq K^{*n} \;\;\;\; ({\rm Lemma} \;
\ref{t19Sep06}).$$ Therefore,
\begin{equation}\label{SGhTn}
{\cal S} = G\cdot (h_1, \ldots , h_n) \simeq G/ \mathbb{T}^n.
\end{equation}
Note that ${\rm St} (h_1, \ldots , h_n) = \{ \sigma \in G \, | \, \sigma h_i=
h_i\sigma,1\leq i \leq n\}$.
\begin{lemma}\label{t19Sep06
Let $K$ be a reduced commutative ring with $\frac{1}{2}\in K$.
Then ${\rm St} (h_1, \ldots , h_n) = \mathbb{T}^n$.
\end{lemma}
{\it Proof}. Clearly, the torus is a subgroup of the stabilizer.
We have to show that each element $\sigma$ of the stabilizer belongs
to the torus. Since the automorphism $\sigma$ commutes with all the
$h_i$, the automorphism $\sigma$ respects eigenspaces of $h_i$ (i.e.
$ {\rm ker } (h_i)$ and $ {\rm ker } (h_i-1)$) and their intersections. In
particular, for each $ i=1, \ldots , n$, the vector space
\begin{equation}\label{CKxi}
{\rm ker } (h_1) \cap \cdots \cap {\rm ker } (h_{i-1}) \cap {\rm ker } (h_i-1)\cap
{\rm ker } (h_{i+1}) \cap \cdots \cap {\rm ker } (h_n) = Kx_i
\end{equation}
is $\sigma$-invariant, i.e. $\sigma (x_i)= \lambda_ix_i$ for some $\lambda_i\in
K^*$, and so $\sigma \in \mathbb{T}^n$. $\Box $
Let ${\cal A}$ be the set of all the $n$-tuples $(x_1', \ldots , x_n')$
of canonical generators for the $K$-algebra $\Lambda_n(K)$ ($x_i'^2=0$
and $x_i'x_j'=-x_j'x_i'$). Clearly, all $x_i'\in \mathfrak{m}$ since $K$ is
reduced. The group $G$ acts on the set ${\cal A}$ in the obvious way:
$\sigma (x_1', \ldots , x_n')= (\sigma (x_1'), \ldots , \sigma (x_n'))$. The
action is transitive and the stabilizer of each point is trivial
(by the very definition of the group $G$).
\begin{lemma}\label{e19Sep06
Let $K$ be a reduced commutative ring with $\frac{1}{2}\in K$,
$(s_1, \ldots , s_n)\in {\cal S}$, and ${\cal K}_i = Kx_i'$, $1\leq i \leq
n$. Then the element $(x_1', \ldots , x_n')$ belongs to the set
${\cal A}$, and $s_1=x_1'\frac{\partial }{\partial x_1'}, \ldots
,s_n=x_1'\frac{\partial }{\partial x_n'}$.
\end{lemma}
{\it Proof}. By Proposition \ref{m19Sep06}, $\delta (\mathfrak{m} )
\subseteq \mathfrak{m}$ for any $K$-derivation $\delta$ of $\Lambda_n$. In
particular, $s_1(\mathfrak{m} ) \subseteq \mathfrak{m} , \ldots ,s_n(\mathfrak{m} ) \subseteq
\mathfrak{m}$. Using $n-1$ times (\ref{K1i}), we have
\begin{eqnarray*}
\Lambda_n&=& K_1+x_1'K_1=( K_{1,2}+ x_2'K_{1,2})+ x_1'( K_{1,2}+ x_2'K_{1,2})\\
&=&K_{1,2}+x_2'K_{1,2}+x_1'K_{1,2}+x_1'x_2'K_{1,2}=\cdots \\
&=& \sum_{\alpha \in {\cal B}_n} x'^\alpha K_{1, \ldots , n} =
\sum_{\alpha \in {\cal B}_n} x'^\alpha K = \sum_{\alpha \in {\cal B}_n}
Kx'^\alpha
\end{eqnarray*}
since $ K_{1, \ldots , n}=K$. Since $\Lambda_n(K)$ is a free module of
rank $2^n$ over the commutative ring $K$ with identity,
$x'^\alpha \neq 0$ for all $\alpha$, and the sums above are the
direct sums, i.e. $\Lambda_n= \oplus_{\alpha \in {\cal B}_n} Kx'^\alpha$.
For each $i$ and $\alpha$, $s_i( x'^\alpha)= \alpha_ix'^\alpha$
and $ s_i(y_{\alpha , \nu })=\alpha_iy_{\alpha , \nu }$ where
$y_{\alpha , \nu }:= x_{\nu (1)}'^{\alpha_{\nu (1)}}\cdots x_{\nu
(n)}'^{\alpha_{\nu (n)}}$ and $\nu \in S_n$ ($S_n$ is the
symmetric group). Therefore, $Kx'^\alpha := Kx_1'^{\alpha_1}\cdots
x_n'^{\alpha_n}= K x_{\nu (1)}'^{\alpha_{\nu (1)}}\cdots x_{\nu
(n)}'^{\alpha_{\nu (n)}}$ for any permutation $\nu \in S_n$ (since
the sums above are direct). In particular, for each $i\neq j$,
$x_i'x_j'= \lambda x_j'x_i'$ for some $\lambda = \lambda_{ij}\in K$. We claim
that $\lambda =-1$. For, note that $\mathfrak{m} = (x_1', \ldots , x_n')$ and so
the set $\overline{x}_1':= x_1'+\mathfrak{m}^2, \ldots , \overline{x}_n':= x_n'+\mathfrak{m}^2$ is a
basis for the vector space $\mathfrak{m} / \mathfrak{m}^2$ over $K$. In $\mathfrak{m}^2/
\mathfrak{m}^3$, on the one hand, $\overline{x}_i'\overline{x}_j' = - \overline{x}_j' \overline{x}_i'\neq 0$;
on the other, by taking the equation $x_i'x_j'= \lambda x_j'x_i'$
modulo $\mathfrak{m}^3$, we have $\overline{x}_i'\overline{x}_j' = \lambda \overline{x}_j' \overline{x}_i'$; hence
$\lambda=-1$, as required.
For each $i$, $x_i'^2\in {\rm ker } (s_i)$ (since $s_i(x_i'^2) =
2x_i'^2$ and 2 is not an eigenvalue for the idempotent derivation
$s_i$ as $\frac{1}{2}\in K$), hence
$x_i'^2\in \mathfrak{m} \cap \cap_{i=1}^n {\rm ker } (s_i)= \mathfrak{m}
\cap K=0$, i.e. $x_i'^2=0$. This proves that the elements $x_1',
\ldots x_n'$ are {\em canonical} generators for the algebra
$\Lambda_n$. Now, it is obvious that $s_1=x_1'\frac{\partial }{\partial x_1'},
\ldots ,s_n=x_n'\frac{\partial }{\partial x_n'}$. $\Box $
If a group ${\cal G}$ acts on a set $X$ we say that $X$ is a ${\cal G}$-{\em
set}. Let $Y$ be a ${\cal G}$-set. A map $f:X\rightarrow Y$ is called a
${\cal G}$-map if $f(gx) = gf(x)$ for all $x\in X$ and $g\in {\cal G}$. A
${\cal G}$-{\em isomorphism} is a ${\cal G}$-map which is a bijection. The
torus $\mathbb{T}^n$ acts on the set ${\cal A}$ by the rule $(\lambda_i)
(x_i') := (\lambda_ix_i')$. Let $ {\cal A} /\mathbb{T}^n$ be the set of all
$\mathbb{T}^n$-orbits.
\begin{corollary}\label{a19Sep06
Let $K$ be a reduced commutative ring with $\frac{1}{2}\in K$.
Then
\begin{enumerate}
\item The map ${\cal S} \rightarrow {\cal A} /\mathbb{T}^n$, $ (s_1, \ldots, s_n)
\mapsto \mathbb{T}^n (x'_1, \ldots , x'_n)$, is a $G$-isomorphism
with the inverse $\mathbb{T}^n(x_1', \ldots , x_n')\mapsto
(x_1'\frac{\partial }{\partial x_1'}, \ldots ,x_n'\frac{\partial }{\partial x_n'})$.
\item In particular, $G$ acts transitively on the set ${\cal S}$.
\end{enumerate}
\end{corollary}
{\it Proof}. 1. This follows directly from Lemma \ref{e19Sep06}.
2. The group $G$ acts transitively on the set ${\cal A}$, hence it does
on the set ${\cal S}$, by statement 1. $\Box $
For $n=1$, Corollary \ref{a19Sep06} gives ${\cal S} = \{ x_1\partial _1\}$
since $G=\mathbb{T}$.
Let ${\cal G}$ be a group and $M$ be a ${\cal G}$-module. We say that $M$ is
a {\em faithful} ${\cal G}$-module (or the group ${\cal G}$ acts {\em
faithfully} on $M$) if the map ${\cal G} \rightarrow {\rm End} (M)$ is
injective.
\begin{theorem}\label{26Sep06
Let $K$ be a reduced commutative ring with $\frac{1}{2}\in K$,
$n\geq 1$. Then
\begin{enumerate}
\item $ {\rm Der }_K(\Lambda_n)$ is a faithful $G$-module iff $n\geq 2$.\item
The $G$-module $ {\rm Der }_K(\Lambda_n)$ is not simple.
\end{enumerate}
\end{theorem}
{\it Proof}. 1. For $n=1$, $ {\rm Der }_K(\Lambda_1) = Kx_1\partial _1$ (Corollary
\ref{cc13Sep06}) and $G=\mathbb{T}={\rm St} (x_1\partial _1)$.
Therefore, $ {\rm Der }_K(\Lambda_1)$ is not a faithful $G$-module. So, let
$n\geq 2$.
Suppose that an element $\sigma \in G$ acts trivially on
$ {\rm Der }_K(\Lambda_n)$, i.e. $\sigma \delta \sigma^{-1}=\delta$ for all $\delta \in
{\rm Der }_K(\Lambda_n)$. We have to show that $\sigma = e$, the identity element
of $G$. By Lemma \ref{t19Sep06}, $\sigma = \sigma_\lambda \in \mathbb{T}^n$ for
some $\lambda \in K^{*n}$. For each $i=1, \ldots , n$, $ {\rm ad } ( x_i) =
\sigma {\rm ad } (x_i)\sigma^{-1} = {\rm ad } (\sigma (x_i))= \lambda_i {\rm ad } (x_i)$, hence
$\lambda_i=1$ (choose $j$ such that $j\neq i$; then $0=(\lambda_i-1) {\rm ad }
(x_i)(x_j) = 2(\lambda_i-1) x_ix_j$, and so $\lambda_i-1=0$), i.e. $\sigma =e$.
2. If $n\geq 1$, then the $G$-module $ {\rm Der }_K(\Lambda_n)= D_0 \oplus
D_+$
contains the proper
submodule $D_+$, and so $ {\rm Der }_K(\Lambda_n)$ is not a simple $G$-module.
$\Box $
\section{Skew derivations of the Grassmann
rings}\label{SKDGA
Let $K$ be a commutative ring. Recall that the Grassmann
$K$-algebra $\Lambda_n = \L_{n,\overline{0}} \oplus \L_{n,\overline{1}}$ is a $\mathbb{Z}_2$-graded algebra,
$ \L_{n,\overline{0}} = \L_n^{ev}$ and $\L_{n,\overline{1}} = \L_n^{od}$. Each element $a$ of $\Lambda_n$ is
a unique sum $a= a_{\overline{0}} + a_{\overline{1}}$ with $a_{\overline{0}}\in \L_{n,\overline{0}}$ and $a_{\overline{1}}\in \L_{n,\overline{1}}$.
We also use the alternative notation: $a= a^{ev}+a^{od}$ where
$a^{ev}:=a_{\overline{0}}$ and $a^{od}:= a_{\overline{1}} $.
Recall that a $K$-linear map $\delta : \Lambda_n\rightarrow \Lambda_n$ is called a {\em
(left) skew $K$-derivation}, if for any $b_s\in \Lambda_{n,s}$ and $
b_t\in \Lambda_{n,t}$ (where $s,t\in \mathbb{Z}_2$),
$$ \delta ( b_sb_t) = \delta (b_s) b_t+ (-1)^s b_s\delta (b_t).$$
The set of all skew derivations ${\rm SDer}_K(\Lambda_n)$ is a left
$Z(\Lambda_n)$-module and a left $\L_n^{ev}$-module since $\L_n^{ev} \subseteq
Z(\Lambda_n)$.
Consider the sets of even and odd skew $K$-derivations of
$\Lambda_n(K)$:
\begin{eqnarray*}
\SDer_K(\L_n)^{ev} & :=& \{ \delta \in {\rm SDer}_K(\Lambda_n )\, | \, \delta (\L_{n,\overline{i}} ) \subseteq \L_{n,\overline{i}} , \; \overline{i} \in \mathbb{Z}_2 \},\\
\SDer_K (\L_n)^{od} & :=& \{ \delta \in {\rm SDer}_K(\Lambda_n )\, | \, \delta (\L_{n,\overline{i}} ) \subseteq \Lambda_{n, \overline{i}+\overline{1}} , \; \overline{i} \in \mathbb{Z}_2
\}.
\end{eqnarray*}
So, even skew derivations are precisely the skew derivations that
respect $\mathbb{Z}_2$-grading of $\Lambda_n$, and the odd skew derivations are
precisely the skew derivations that reverse it. The set of odd and
even skew derivations are left $\L_{n,\overline{0}}$-modules. For each element
$a\in \Lambda_n$, one can attach, so-called, the {\em inner} skew
$K$-derivation of $\Lambda_n$: ${\rm sad} (a) : b_s\mapsto ab_s-(-1)^sb_sa$,
where $b_s\in \Lambda_{n, s}$, $s\in \mathbb{Z}_2 $. The set of all inner
skew derivations is denoted by ${\rm ISDer}_K(\Lambda_n)$.
The kernel of the $K$-linear map ${\rm sad} : \Lambda_m\rightarrow {\rm ISDer}_K(\Lambda_n)$,
$a\mapsto {\rm sad} (a)$, is equal to $ {\rm ker } ({\rm sad} )= \L_n^{od}
+Kx_1\cdots x_n$, and
$$ \Lambda_n/ {\rm ker } ({\rm sad} )= \L_n^{ev} \oplus \L_n^{od}/ (\L_n^{od}
+Kx_1\cdots x_n)\simeq \L_n^{ev} / \L_n^{ev} \cap Kx_1\cdots x_n\simeq
\L_n'^{ev}.$$
By the Homomorphism Theorem, the map
\begin{equation}\label{Ln1s}
\L_n'^{ev} \rightarrow {\rm ISDer}_K(\Lambda_n), \;\; a\mapsto {\rm sad} (a),
\end{equation}
is a bijection and ${\rm ISDer}_K(\Lambda_n)= \{ {\rm sad} (a) \, | \,
a\in \L_n'^{ev}\}$.
The next theorem describes
explicitly the sets of all/inner/even and odd skew derivations.
\begin{theorem}\label{a13Sep06
Suppose that $K$ is a commutative ring with $\frac{1}{2}\in K$.
Then
\begin{enumerate}
\item ${\rm SDer}_K(\Lambda_n) = \SDer_K(\L_n)^{ev} \oplus \SDer_K (\L_n)^{od}$. \item $\SDer_K (\L_n)^{od} =
\oplus_{i=1}^n \L_n^{ev}\partial _i$. \item $\SDer_K(\L_n)^{ev} = {\rm ISDer}_K(\Lambda_n)= \{
{\rm sad} (a) \, | \, \in \L_n'^{ev} \}$, and the map $ {\rm sad} : \L_n'^{ev} \rightarrow
{\rm ISDer}_K(\Lambda_n)$, $a\mapsto {\rm sad} (a)$, is a bijection. \item
${\rm SDer}_K(\Lambda_n)/ {\rm ISDer}_K(\Lambda_n)\simeq \SDer_K (\L_n)^{od}$.
\end{enumerate}
\end{theorem}
{\it Proof}. By (\ref{Ln1s}), the map in statement 3 is a
bijection and ${\rm ISDer}_K(\Lambda_n)= \{ {\rm sad} (a) \, | \, a\in \L_n'^{ev}\}$.
Since $\SDer_K(\L_n)^{ev} \cap \SDer_K (\L_n)^{od} = 0$, one has the inclusion
\begin{equation}\label{s1ri}
{\rm SDer}_K(\Lambda_n)\supseteq \SDer_K(\L_n)^{ev} \oplus \SDer_K (\L_n)^{od} .
\end{equation}
Clearly,
\begin{equation}\label{s2ri}
\SDer_K (\L_n)^{od} \supseteq \sum_{i=1}^n \L_n^{ev} \partial _i =\bigoplus_{i=1}^n
\L_n^{ev} \partial _i .
\end{equation}
For each $a\in \L_n'^{ev}$, ${\rm sad} (a) \in
\SDer_K(\L_n)^{ev}$, hence
\begin{equation}\label{s3ri}
\SDer_K(\L_n)^{ev} \supseteq {\rm ISDer}_K(\Lambda_n).
\end{equation}
Note that statement 4 follows from statements 1 and 3. Now, it is
obvious that in order to finish the proof of the theorem it
suffices to prove the next claim.
{\it Claim}. ${\rm SDer}_K(\Lambda_n) \subseteq \sum_{i=1}^n \L_n^{ev} \partial _i
+{\rm ISDer}_K(\Lambda_n)$.
Indeed, suppose that the inclusion of the claim holds then
$$ {\rm SDer}_K(\Lambda_n) \subseteq \sum_{i=1}^n \L_n^{ev} \partial _i
+{\rm ISDer}_K(\Lambda_n)\subseteq \SDer_K(\L_n)^{ev} \oplus \SDer_K (\L_n)^{od},$$ hence statement
1 is true by (\ref{s1ri}). Statement 1 together with inclusions
(\ref{s2ri}) and (\ref{s3ri}) implies statements 2 and 3.
{\it Proof of the Claim}. Let $\delta$ be a skew $K$-derivation of
$\Lambda_n$. We have to represent the derivation $\delta $ as a sum
$$ \delta = \sum_{i=1}^n a_i\partial _i + {\rm sad} (a), \;\; a_i\in \L_n^{ev} , \;\;
a\in \L_n'^{od}.$$ The proof of the claim is constructive.
By (\ref{u=ueo}), for each $i=1, \ldots , n$, let
\begin{equation}\label{1Lnz}
u_i:= \delta (x_i)= u^{ev}_i + u^{od}_i, \;\;\; {\rm where} \;\; u^{ev}_i\in
\L_n^{ev}, \;\; u^{od}_i\in \L_n^{od} .
\end{equation}
Let $\partial := \sum_{i=1}^n u^{ev}_i\partial _i$ and $ \delta':= \delta - \partial $.
Then $\delta' (x_i)= u^{od}_i$. Note that $\partial \in \sum_{i=1}^n \L_n^{ev}
\partial _i$. Then changing $\delta$ for $\delta'$, if necessary, one may
assume that all $u_i$ belong to $\L_n^{od}$. Now, it suffices to
show that $\delta = {\rm sad} (a)$ for some $a\in \L_n^{ev}$. We construct such
an $a$ in several steps.
{\it Step 1}. Let us prove that, {\em for each $i=1, \ldots , n$,
$u_i= v_ix_i$ for some element } $v_i\in K\lfloor x_1, \ldots ,
\widehat{x_i}, \ldots , x_n\rfloor^{ev}$. Note that $0= \delta (0)= \delta
(x_i^2) = u_ix_i - x_iu_i=2u_ix_i$ since $u_i$ is odd, and so
$u_ix_i=0$ since $\frac{1}{2}\in K$. It follows that $u_i=
v_ix_i$ for some element $v_i\in K\lfloor x_1, \ldots ,
\widehat{x_i}, \ldots , x_n\rfloor^{ev}$. If $n=1$ then $v_i\in
K$, and so $\delta = {\rm sad} (\frac{1}{2}v_i)$, and we are done. So, let
$n\geq 2$.
{\it Step 2}. We claim that, {\em for each pair} $i\neq j$,
\begin{equation}\label{sast2}
v_i|_{x_j=0}=v_j|_{x_i=0}.
\end{equation}
Evaluating the skew derivation $\delta$ at the element $0=
x_ix_j+x_jx_i$ and taking into account that all the elements $u_i$
are odd we get
\begin{eqnarray*}
0&=& u_ix_j-x_iu_j+u_jx_i-x_ju_i= 2(u_ix_j+u_jx_i) \\
&=& 2(v_ix_ix_j+v_jx_jx_i)= 2(v_i-v_j) x_ix_j.
\end{eqnarray*}
This means that $v_i-v_j\in (x_i, x_j)$ since $\frac{1}{2}\in K$,
or, equivalently, $v_i|_{x_i=0, x_j=0}=v_j|_{x_i=0, x_j=0}$. By
Step 1, this equality can be written as (\ref{sast2}).
{\it Step 3}. Note that $\delta (x_1)=v_1x_1= {\rm sad} (\frac{1}{2}v_1)
(x_1)$ since $v_1$ is even, and so $(\delta -{\rm sad}
(\frac{1}{2}v_1))(x_1)=0$. Changing $\delta$ for $\delta -{\rm sad}
(\frac{1}{2}v_1)$ one can assume that $\delta (x_1)=0$, i.e. $v_1=0$.
Then, by (\ref{sast2}), $v_i|_{x_1=0}=0$ for all $i=2, \ldots ,
n$, and so $v_i\in (x_1)$ for all $i=2, \ldots , n$.
The idea of the proof is to continue in this way killing the
elements $v_i$. Namely, we are going to prove by induction on
$k$ that, for each $k$ such that $1\leq k\leq n$, by adding to
$\delta$ a well chosen inner skew derivation we have
\begin{equation}\label{skind}
\delta (x_1)=\cdots = \delta (x_k)=0, \;\; \delta (x_i) \in (x_1 \cdots
x_kx_i), \;\; k<i\leq n .
\end{equation}
The case $k=1$ has just been established. Suppose that
(\ref{skind}) holds for $k$, we have to prove the same statement
for $k+1$. Note that $v_{k+1} x_{k+1} = \delta ( x_{k+1})\in
(x_1\cdots x_kx_{k+1})$ (see Steps 1 and 2), hence $v_{k+1} =
x_1\cdots x_kv$ for some $v\in K\lfloor x_{k+2} , \ldots ,
x_n\rfloor$. Consider the derivation $\delta':= \delta -{\rm sad}
(\frac{1}{2}v_{k+1})$. For each $i=1, \ldots , k$, $\delta'(x_i)= \delta
(x_i) =0$ as $v_{k+1} \in ( x_1\cdots x_k)$; and
\begin{eqnarray*}
\delta'( x_{k+1}) &=& v_{k+1} x_{k+1}- \frac{1}{2}(v_{k+1}
x_{k+1}-(-1)x_{k+1}v_{k+1})\\
&=&v_{k+1} x_{k+1}- \frac{1}{2}(v_{k+1} x_{k+1}+v_{k+1}x_{k+1})=0.
\end{eqnarray*}
So, we have proved the first part of (\ref{skind}) for $k+1$,
namely, that
$$ \delta' (x_1)=\cdots = \delta' (x_{k+1})=0.$$
So, changing $\delta$ for $\delta'$ one can assume that
$$ \delta (x_1)=\cdots = \delta (x_{k+1})=0.$$
These conditions imply that $v_1=\cdots = v_{k+1}=0$ (by Step 1).
If $n=k+1$, we are done. So, let $k+1<n$. Then, by (\ref{sast2}),
for each $i>k+1$, $v_i \in \cap_{j=1}^{k+1} (x_j)= (x_1\cdots
x_{k+1})$, hence $\delta (x_i) = v_ix_i \in ( x_1\cdots x_{k+1} x_i)$,
and we are done.
By induction, (\ref{skind})
is true for all $k$. In particular, for $k=n$ one has $\delta =0$.
This means that $\delta$ is an inner skew derivation, as required.
$\Box $
By Theorem \ref{a13Sep06}, any skew $K$-derivation $\delta$ of $\Lambda_n$
is a unique sum $\delta = \d^{ev} +\d^{od}$ of {\em even} and {\em odd}
skew derivations, and $\d^{ev} := \frac{1}{2}{\rm sad} (a)$ for a {\em unique}
element $a\in \L_n'^{ev}$. The next corollary describes explicitly the
skew derivations $\d^{ev}$ and $\d^{od}$.
\begin{corollary}\label{ca13Sep06
Let $K$ be a commutative ring with $\frac{1}{2}\in K$, $\delta$ be a
skew $K$-derivation of $\Lambda_n(K)$, and, for each $i=1, \ldots ,
n$, $\delta (x_i) = u^{ev}_i + u^{od}_i$ for unique elements $u^{ev}_i\in
\L_n^{ev} $ and $u^{od}_i\in \L_n^{od} $. Then
\begin{enumerate}
\item $\d^{od} =\sum_{i=1}^n u^{ev}_i\partial _i$, and \item $\d^{ev} =
\frac{1}{2}{\rm sad} (a)$ where the {\em unique} element $a\in \L_n'^{ev}$
is given by the formula
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{od}_{i+1})+\partial _1 (u^{od}_1).$$
\end{enumerate}
\end{corollary}
{\it Proof}. 1. This statement has been proved already in the
proof of Theorem \ref{a13Sep06}.
2. For each $i=1, \ldots , n$, on the one hand $\d^{ev} (x_i) = (\delta -
\d^{od} )(x_i) = u^{od}_i$; on the other, $ \d^{ev} (x_i) =
\frac{1}{2}{\rm sad} (a) (x_i)=\frac{1}{2}2x_ia= x_ia$. So, the element
$a$ is a solution to the system of equations
$$\begin{cases}
x_1a=u^{od}_1 \\
x_2a=u^{od}_2 \\
\;\;\;\; \;\;\;\vdots \\
x_na=u^{od}_n.
\end{cases}
$$
By Theorem \ref{s14Sep06} and the fact that $a\in \L_n'^{ev}$ (i.e.
$a_n=0$), we have
$$ a= \sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots \partial _1\partial _{i+1}
(u^{od}_{i+1})+\partial _1 (u^{od}_1).\;\;\; \Box$$
{\it Definition}. An ideal $\mathfrak{a}$ of $\Lambda_n$ is called a {\em skew
differential} ideal if $\delta (\mathfrak{a} ) \subseteq \mathfrak{a}$ for all $\delta \in
{\rm SDer}_K(\Lambda_n)$.
\begin{lemma}\label{16Mar7
Let $K$ be a commutative ring with $\frac{1}{2}\in K$, ${\cal I} (K)$
be the set of ideals of the ring $K$, ${\rm SDI} (\Lambda_n)$ be the set of
all skew differential ideals of $\Lambda_n$. Then the map
$$ {\cal I} (K) \rightarrow {\rm SDI} (\Lambda_n), \;\; I\mapsto I\Lambda_n,$$
is a bijection. In particular, if $K$ is a field of characteristic
$\neq 2$, then $\Lambda_n$ is a skew differentiably simple algebra,
i.e. $0$ and $\Lambda_n$ are the only skew differential ideals of
$\Lambda_n$.
\end{lemma}
{\it Proof}. The map $I\mapsto I\Lambda_n$ is well-defined an
injective. It remains to prove that it is surjective. Let $\mathfrak{a}$ be
a skew differential ideal of $\Lambda_n$. First, let us show that
$$\mathfrak{a} = \oplus_{\alpha \in {\cal B}_n} (\mathfrak{a} \cap Kx^\alpha ) ,$$ i.e. if $a=
\sum_{\alpha \in {\cal B}_n} \lambda_\alpha x^\alpha\in \mathfrak{a}$, $\lambda_\alpha\in
K$, then all $\lambda_\alpha x^\alpha\in \mathfrak{a}$. The case $a=0$ is
trivial. So, let $a\neq 0$ and $i:= \max \{ |\alpha | \, | \,
\lambda_\alpha \neq 0\}$. We use induction on $i$. The case $i=0$ is
obvious. So, let $i>0$. Then, $\lambda_\alpha = \partial ^\alpha (a)\in \mathfrak{a}$
for each $\alpha $ such that $|\alpha | = i$ where $\partial ^\alpha :=
\partial _n^{\alpha_n} \partial _{n-1}^{\alpha_{n-1}}\cdots
\partial _1^{\alpha_1}$. Applying induction to the element
$a-\sum_{|\alpha | =i}\lambda_\alpha x^\alpha\in \mathfrak{a}$, we get the
result. So, $\mathfrak{a} = \oplus_{\alpha \in {\cal B}_n} \mathfrak{a}_\alpha x^\alpha$
for some ideals $\mathfrak{a}_\alpha $ of $K$. Let $I:= \mathfrak{a}_0$. On the one
hand, $I\Lambda_n\subseteq \mathfrak{a}$, and so $I\subseteq \mathfrak{a}_\alpha$ for all
$\alpha\in {\cal B}_n$. On the other, $\mathfrak{a}_\alpha =
\partial ^\alpha(\mathfrak{a}_\alpha x^\alpha ) \subseteq I$, hence $\mathfrak{a} =
I\Lambda_n$. So, the map $I\mapsto I\Lambda_n$ is a surjection. $\Box $
\section{Normal elements of the Grassmann algebras}\label{NEGA
In this section, it is proved that the set of `generic' normal
non-units forms no more that two orbits under the action of the
group $G:= {\rm Aut}_K(\Lambda_n )$ (Theorem \ref{5Nov06}). The
stabilizers of elements from each orbit are found (Lemma
\ref{c5Nov06} and Lemma \ref{s10Nov06}).
In this section, $K$ is a {\em reduced commutative} ring with
$\frac{1}{2}\in K$ and $n\geq 2$. Recall than an element $r$ of a
ring $R$ is called a {\em normal} element if $rR=Rr$. Each unit is
a normal element.
Recall that $G:= {\rm Aut}_K(\Lambda_n)$ is the group of $K$-automorphisms
of $\Lambda_n$. Consider some of its subgroups:
\begin{itemize}
\item $\Omega := \{ \omega _{1+a}\, | \, a\in \Lambda_n'^{od} \}$, where $\omega _u :
\Lambda_n\rightarrow \Lambda_n$, $ x\mapsto uxu^{-1}$, is an inner automorphism.
\item $\Gamma := \{ \gamma_b \, | \, \gamma_b(x_i) = x_i+b_i,
\; b_i \in \L_n^{od} \cap \mathfrak{m}^3, i=1, \ldots , n\}$, $ b=(b_1, \ldots
, b_n)$, \item ${\rm GL}_n(K)^{op}:= \{ \sigma_A\, | \, \sigma_A(x_i)=
\sum_{j=1}^n a_{ij}x_j, \; A=(a_{ij})\in {\rm GL}_n(K)\}$.
\end{itemize}
For each $a\in \L_n^{od}$ and $x\in \Lambda_n$, $\omega _{1+a} (x) = x+[a,x]$
(Lemma 2.8.(3), \cite{jacgras}). Note that
$G= (\Omega \rtimes \Gamma ) \rtimes {\rm GL}_n(K)^{op}$ (Theorem 2.14, \cite{jacgras}). So, each element $\sigma
\in G$ has the unique presentation as the product $\sigma = \omega _{1+a}
\gamma_b\sigma_A $ where $\omega _{1+a} \in \Omega$ ($a\in \Lambda_n'^{od}$), $\gamma_b \in
\Gamma$, $\sigma_A\in {\rm GL}_n(K)^{op}$ where $\Lambda_n'^{od} := \oplus_i
\Lambda_{n,i}$ and $i$ runs through all odd natural numbers such that
$1\leq i\leq n-1$. For more information on the group $G$, the
reader is refereed to \cite{Berezin67} and \cite{Djokovic78} where
$K$ is a field, and to \cite{jacgras} where $K$ is a commutative
ring.
\begin{theorem}\label{M30Sep06
\cite{jacgras} Let $K$ be a reduced commutative ring with
$\frac{1}{2}\in K$. Then each element $\sigma \in G$ is a unique
product $\sigma = \omega _{1+a} \gamma_b\sigma_A$ where $a\in \Lambda_n'^{od}$ and
\begin{enumerate}
\item $\sigma (x) = Ax +\cdots $ (i.e. $\sigma (x) \equiv Ax\mod \mathfrak{m}$) for
some $A\in {\rm GL}_n(K)$, \item $b= A^{-1} \sigma(x)^{od}-x$, and \item
$a= -\frac{1}{2}\gamma_b (\sum_{i=1}^{n-1} x_1\cdots x_i\partial _i\cdots
\partial _1\partial _{i+1} (a_{i+1}') +\partial _1(a_1'))$ where $a_i':= (A^{-1}
\gamma_b^{-1} (\sigma (x)^{ev}))_i$, the $i$'th component of the
column-vector $A^{-1} \gamma_b^{-1} (\sigma (x)^{ev})$.
\end{enumerate}
\end{theorem}
{\it Remark}. In the above theorem the following abbreviations are
used
$x=\begin{pmatrix}
x_1\\
\vdots \\
x_n \\
\end{pmatrix}$, $b=\begin{pmatrix}
b_1\\
\vdots \\
b_n \\
\end{pmatrix}$, $\sigma (x)=\begin{pmatrix}
\sigma (x_1)\\
\vdots \\
\sigma (x_n) \\
\end{pmatrix}$, $\sigma (x)^{ev}=\begin{pmatrix}
\sigma (x_1)^{ev}\\
\vdots \\
\sigma (x_n)^{ev} \\
\end{pmatrix}$, $\sigma (x)^{od}=\begin{pmatrix}
\sigma (x_1)^{od}\\
\vdots \\
\sigma (x_n)^{od} \\
\end{pmatrix}$, any element $u\in \Lambda_n$ is a unique sum $u=
u^{ev}+u^{od}$ of its even and odd components. Note that the
inversion formula for $\gamma_b^{-1}$ is given in \cite{jacgras}.
Let ${\cal N}$ be the set of all normal elements of the Grassmann
algebra $\Lambda_n = \Lambda_n(K)$ and let ${\cal U}$ be the set of all units of
$\Lambda_n$. Then ${\cal U} \subseteq {\cal N}$. By (\ref{smi=mi}), the set ${\cal N}$
is a disjoint union of its $G$-invariant subsets,
$${\cal N} = \cup_{i=0}^n{\cal N}_i, \;\; {\cal N}_i:= \{ a\in {\cal N} \, | \, a=
a_i+\cdots , 0\neq a_i\in \Lambda_{n,i}\}.$$ Clearly, ${\cal N}_0= {\cal U}$.
Similarly, by (\ref{smi=mi}), the set ${\cal U}$ is a disjoint union of
its $G$-invariant subsets ${\cal U}_i$,
$${\cal U} = \cup_{i=0}^n{\cal U}_i, \;\; {\cal U}_i:= \{ a\in {\cal U} \, | \, a=a_0+
a_i+\cdots , a_0\in K^*, 0\neq a_i\in \Lambda_{n,i}\}.$$
\begin{lemma}\label{k5Nov06
$\L_n^{ev} \cup \L_n^{od} \subseteq {\cal N}$.
\end{lemma}
{\it Proof}. It is obvious. $\Box $
The next result shows that `generic' normal non-unit elements of
$\Lambda_n$ (i.e. the set ${\cal N}_1$) form a single $G$-orbit if $n$ is
even, and two $G$-orbits if $n$ is odd.
\begin{theorem}\label{5Nov06
Let $K$ be a field of characteristic $\neq 2$ and $\Lambda_n =
\Lambda_n(K)$. Then\begin{enumerate} \item ${\cal N}_1= Gx_1$ if $n$ is
even.\item ${\cal N}_1= Gx_1\cup G(x_1+x_2\cdots x_n)$ is the disjoint
union of two orbits if $n$ is odd.
\end{enumerate}
\end{theorem}
{\it Proof}. The elements $x_1$ and $y:=x_1+x_2\cdots x_n$ are
normal. First, let us prove that if $n$ is odd then the orbits
$Gx_1$ and $Gy$ are distinct. Suppose that they coincide, i.e.
$y=\sigma (x_1)$ for some automorphism $\sigma \in G$, we seek a
contradiction. By Theorem \ref{M30Sep06}, $\sigma = \omega _{1+a}\gamma_b\sigma_A$.
By taking the equality $\sigma (x_1) = y$ modulo the ideal $\mathfrak{m}^2$, we
have $\sigma_A(x_1)= x_1$, hence $$x_1+x_2\cdots x_n= y= \sigma (x_1) =
\omega _{1+a} \gamma_b(x_1) = \omega _{1+a} (x_1+b_1) = x_1+b_1+[a, x_1+b_1].$$
Equating the odd parts of both ends of the equalities above gives
$x_1= x_1+b_1$, hence $b_1=0$ and $x_2\cdots x_n= [a,x_1]\in
(x_1)$, a contradiction. Therefore, the orbits $Gx_1$ and $Gy$ are
distinct.
It remains to prove that ${\cal N}_1\subseteq Gx_1$ and ${\cal N}_1\subseteq
Gx_1\cup Gy$ in the first and the second case respectively.
Let $a= a_1+a_2+\cdots \in {\cal N}_1$ where all $a_i\in \Lambda_{n,i}$ and
$0\neq a_1\in \Lambda_{n, 1}$. Up to the action of the group
${\rm GL}_n(K)^{op}$, one can assume that $a_1=x_1$. The automorphism
$\gamma : x_1\mapsto x_1+a_3+a_5+\cdots$, $x_i\mapsto x_i$, $i\geq 2$,
is an element of the group $\Gamma$. Now, $a= \gamma (x_1) +a^{ev}$ where
$a^{ev} := a_2+a_4+\cdots $ is the even part of the element $a$,
and so $\gamma^{-1} (a) = x_1+\gamma^{-1} (a^{ev})$. Note that $\gamma^{-1}
(a^{ev})$ is an even element of the set $\L_n^{ev} \cap \mathfrak{m}^2$.
Therefore, up to the action of the group $\Gamma$, one can assume that
$a= x_1+a^{ev}$ where $a^{ev}$ is an even element of $\mathfrak{m}^2$.
Since $\frac{1}{2}\in K$, the element $a^{ev}$ is the unique sum
$a^{ev}= 2\alpha x_1+\beta$ where $\alpha$ and $\beta$ are
respectively odd and even elements of the Grassmann algebra
$K\lfloor x_2, \ldots , x_n\rfloor$ and $\alpha, \beta \in \mathfrak{m}$.
Applying the inner automorphism $\omega _{1-\alpha}=
(\omega _{1+\alpha})^{-1}$ to the equality
$$ a= x_1+2\alpha x_1+\beta = x_1+ [ \alpha , x_1]+\beta =
\omega _{1+\alpha}(x_1) +\beta = \omega _{1+\alpha}(x_1+\beta )$$ we have
the equality $\omega _{1-\alpha}(a) = x_1+\beta$. So, up to the action
of the group $\Omega$, one can assume that $a^{ev} = \beta \in
K\lfloor x_2, \ldots , x_n\rfloor^{ev}_{\geq 2}$. If $\beta =0$
then we are done. So, let $\beta \neq 0$.
{\em Case 1. $\beta x_i=0$ for all} $i=2, \ldots , n$, i.e. $\beta
= \lambda x_2, \ldots x_n$, $\lambda \in K$, hence $n$ must be odd since
$\beta$ is even. In this case, $a= \sigma_A(y)$ where
$\sigma_A(x_2):=\lambda^{-1} x_2$ and $ \sigma_A(x_i)= x_i$ for all $i\neq 2$,
and we are done.
{\em Case 2. $\beta x_i\neq 0$ for some $i\geq 2$}. We aim to show
that this case is impossible, we seek a contradiction. Let $\beta
= a_{2m} +\cdots$ where $a_{2m}\in K\lfloor x_2, \ldots ,
x_n\rfloor_{2m}$, $a_{2m}x_i\neq 0$, and the three dots mean
higher terms with respect to the $\mathbb{Z}$-grading of the Grassmann
algebra $\Lambda_n$. The element $a= x_1+a_{2m}+\cdots$ is normal, and
so $ax_i= ba$ for some element $b= b_0+b_1+\cdots \in \Lambda_n$ where
$b_i\in \Lambda_{n,i}$. In more detail,
$$(x_1+a_{2m}+\cdots ) x_i= (b_0+b_1+\cdots )(x_1+a_{2m}+\cdots
).$$ Clearly, $b_0=0$. Equating the homogeneous components of
degrees $1,\ldots , 2m+1$ of both sides of the equality we have
the system of equations:
$$\begin{cases}
b_1x_1= x_1x_i,\\
b_2x_1=0, \\
\;\;\;\; \;\;\;\vdots \\
b_{2m-1}x_1=0,\\
b_1a_{2m} +b_{2m}x_1= a_{2m} x_i.
\end{cases}
$$
The first equation gives $b_1= -x_i+\lambda x_1$ for some $\lambda \in K$.
By taking the last equation modulo the ideal $(x_1)$ of $\Lambda_n$ we
have the following equality in the Grassmann algebra $K\lfloor
x_2, \ldots , x_n\rfloor$: $-x_ia_{2m}= a_{2m} x_i$, hence
$2a_{2m} x_i=0$. Dividing by 2, we have the equality
$a_{2m}x_i=0$, which contradicts to the assumption that
$a_{2m}x_i\neq 0$. $\Box $
Let ${\rm Stab} (x_1):= \{ \sigma \in G\, | \, \sigma (x_1)=x_1\}$ be the {\em
stabilizer} of the element $x_1$ in $G$ and ${\cal N}_1$ be the set of
normal elements as in Theorem \ref{5Nov06}. By Theorem
\ref{5Nov06}.(1), the map
\begin{equation}\label{GStx1}
G/ {\rm Stab} (x_1) \rightarrow {\cal N}_1, \;\; \sigma {\rm Stab} (x_1) \mapsto \sigma (x_1),
\end{equation}
is a bijection where $K$ is a field of characteristic $\neq 2$.
The next lemma describes the stabilizer ${\rm Stab} (x_1)$.
\begin{lemma}\label{c5Nov06
Let $K$ be a
reduced commutative ring with $\frac{1}{2}\in K$, $\Omega_{x_1}:= \Omega
\cap {\rm Stab} (x_1) = \{ \omega _{1+a} \, | \, a\in (x_1)\cap \L_n^{od} \}$,
$\Gamma_{x_1} := \Gamma \cap {\rm Stab} (x_1)=\{ \gamma_b\, | \, b= (0, b_2, \ldots
, b_n) \in \Lambda_{n, \geq 3}^{od} \}$, and $ {\rm GL}_n(K)^{op}_{x_1} :=
{\rm GL}_n(K)^{op}\cap {\rm Stab} (x_1)=\{ \sigma_A\, | \, \sigma_A(x_1) = x_1, A\in
{\rm GL}_n(K)\}$. Then
$$ {\rm Stab} (x_1) = \Omega_{x_1} \Gamma_{x_1} {\rm GL}_n(K)^{op}_{x_1}= (\Omega_{x_1}\rtimes
\Gamma_{x_1})\rtimes {\rm GL}_n(K)^{op}_{x_1}.$$
\end{lemma}
{\it Proof}. The last equality follows from the equality $G= (\Omega
\rtimes \Gamma )\rtimes {\rm GL}_n(K)^{op}$ (Theorem 2.14, \cite{jacgras})
provided the equality before is true. Let ${\cal M} := \Omega_{x_1}
\Gamma_{x_1} {\rm GL}_n(K)^{op}_{x_1}$. Then ${\rm Stab} (x_1) \supseteq {\cal M}$.
It remains to prove the reverse inclusion. Let $\sigma \in {\rm Stab}
(x_1)$. By Theorem \ref{M30Sep06}, $\sigma = \omega _{1+a} \gamma_b\sigma_A$. Since
$\sigma (x_1) \equiv \sigma_A(x_1)\mod \mathfrak{m}^2$ and $\sigma (x_1) = x_1$ we must
have $\sigma_A\in {\rm GL}_n(K)^{op}_{x_1}$. Now,
$$ x_1= \sigma (x_1) = \omega _{1+a}(x_1+ b_1) = x_1+b_1+[a, x_1+b_1]=
x_1+b_1+ 2a(x_1+b_1),$$and equating the odd parts of the elements
at both ends of the equalities we see that $x_1= x_1+b_1$, hence
$b_1=0$, and so $\gamma_b\in \Gamma_{x_1}$. Putting $b_1=0$ in the
equalities above gives $x_1= x_1+2ax_1$, hence $0=
x_1^{ev}=(x_1+2ax_1)^{ev} = 2ax_1$, and so $a\in (x_1)$. This
means that $\sigma \in \Omega_{x_1}$, as required. $\Box $
\begin{corollary}\label{d5Nov06
Let $K$ be a
reduced commutative ring with $\frac{1}{2}\in K$, and $I$ be a
non-empty subset of $\{ 1, \ldots , n\}$. Then
$$ \cap_{i\in I} {\rm Stab} (x_i) = (\cap_{i\in I}\Omega_{x_i})\cdot (\cap_{i\in I} \Gamma_{x_i})\cdot ( \cap_{i\in I}{\rm GL}_n(K)^{op}_{x_i})
= (\cap_{i\in I}\Omega_{x_i})\rtimes (\cap_{i\in I}\Gamma_{x_i})\rtimes (
\cap_{i\in I}{\rm GL}_n(K)^{op}_{x_i}).$$
\end{corollary}
{\it Proof}. This follows from Lemma \ref{c5Nov06} (and uniqueness
of the decomposition $ \sigma = \omega _{1+a} \gamma_b\sigma_A$). $\Box $
If $K$ is a field of characteristic $\neq 2$ and $n$ is an odd
number then, by Theorem \ref{5Nov06}.(2), the map (where
$y:=x_1+x_2\cdots x_n$)
\begin{equation}\label{GSty}
G/{\rm Stab} (x_1) \cup G/ {\rm Stab} (y) \rightarrow {\cal N} , \;\; \sigma {\rm Stab}
(x_1)\mapsto \sigma (x_1), \;\ \tau {\rm Stab} (y)\mapsto \tau (y),
\end{equation}
is a bijection. The next lemma describes the stabilizer ${\rm Stab} (
x_1+x_2\cdots x_n)$.
\begin{lemma}\label{s10Nov06
Let $K$ be a
reduced commutative ring with $\frac{1}{2}\in K$, and $n\geq 3$
be an odd number. Then ${\rm Stab} (x_1+x_2\cdots x_n)= \{
\omega _{1+\frac{1}{2}\partial _1\gamma_b\sigma_A(x_2\cdots x_n)+x_1c} \gamma_b \sigma_A\,\,
|\,\, \gamma_b\in \Gamma_{x_1},\;\; \sigma_A\in {\rm GL}_n(K)^{op}_{x_1}$, $
(1-x_1\partial _1)\gamma_b \sigma_A(x_2\cdots x_n) = x_2\cdots x_n, \; c\in
K\lfloor x_2, \ldots , x_n\rfloor^{ev}\}$.
\end{lemma}
{\it Proof}. Let $y:=x_1+x_2\cdots x_n$ and $\sigma \in {\rm Stab} (y)$.
Note that $\sigma = \omega _{1+a} \gamma_b\sigma_A$. Since $x_1\equiv y \equiv \sigma
(y) \equiv \sigma (x_1)\mod \mathfrak{m}^2$, we must have $\sigma_A\in
{\rm GL}_n(K)^{op}_{x_1}$. Then
\begin{eqnarray*}
x_1+x_2\cdots x_n&=&y=\sigma (y) = \omega _{1+a}\gamma_b(x_1+\sigma_A(x_2\cdots
x_n)) = \omega _{1+a} (x_1+b_1+\gamma_b\sigma_A(x_2\cdots x_n))\\
& =& x_1+b_1+ \gamma_b \sigma_A( x_2\cdots x_n) + [ a, x_1+b_1].
\end{eqnarray*}
Equating the odd parts of the beginning and the end of the series
of equalities above we obtain $x_1= x_1+b_1$, hence $b_1=0$, i.e.
$\gamma_b\in \Gamma_{x_1}$, and then
\begin{equation}\label{2x1a}
\gamma_b\sigma_A(x_2\cdots x_n) - x_2\cdots x_n = 2x_1a.
\end{equation}
Each element $u\in \Lambda_n$ is a unique sum $u= x_1\alpha + \beta$
for unique elements $\alpha , \beta \in K\lfloor x_2, \ldots ,
x_n\rfloor$. Clearly, $\alpha = \partial _1(u)$ and $\beta =
(1-x_1\partial _1) (u)$. The odd element $a$ is a unique sum $a=
x_1c+d$ for some elements $c\in K\lfloor x_2, \ldots ,
x_n\rfloor^{ev}$ and $ d\in K\lfloor x_2, \ldots ,
x_n\rfloor^{od}_{\geq 1}$. The equation (\ref{2x1a}) can be
written as follows $\gamma_b\sigma_A (x_2\cdots x_n) - x_2\cdots x_n =
2x_1d$. This equality is equivalent to two equalities $d=
\frac{1}{2}\partial _1(\gamma_b\sigma_A (x_2\cdots x_n)- x_2\cdots x_n)=
\frac{1}{2}\partial _1\gamma_b\sigma_A(x_2\cdots x_n)$ and $0=(1-x_1\partial _1)
(\gamma_b\sigma_A (x_2\cdots x_n)- x_2\cdots x_n)= (1-x_1\partial _1)\gamma_b\sigma_A
(x_2\cdots x_n)- x_2\cdots x_n$. This finishes the proof of the
lemma. $\Box$
Let ${\cal U}_1'$ be the image of the injection $K^*\times {\cal N}_1\rightarrow
{\cal U}_1$, $ (\lambda , u ) \mapsto \lambda (1+u)$. The next corollary follows
from Theorem \ref{5Nov06}.
\begin{corollary}\label{u11Nov06
Let $K$ be a field of characteristic $\neq 2$, and $\Lambda_n =
\Lambda_n(K)$. Then
\begin{enumerate}
\item ${\cal U}_1'= \cup_{\lambda \in K^*} G\cdot \lambda (1+x_1)$ is the
disjoint union of orbits if $n$ is even. \item ${\cal U}_1'= \cup_{\lambda
\in K^*} (G\cdot \lambda (1+x_1)\cup G\cdot \lambda (1+x_1+x_2\cdots x_n))$
is the disjoint union of orbits if $n$ is odd.
\end{enumerate}
\end{corollary}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,688 |
1902 Encyclopedia > Festivals
FESTIVALS. A festival or feast is a day or series of days specially and publicly set apart for religious observ-ances. Whether its occurrence be casual or periodic, whether its ritual be grave or gay, carnal as the orgies of Baal and Astarte or spiritual as the worship of a Puritan Sabbath, it is to be regarded as a festival or " holy day " as long as it is professedly held in the name of religion.
To trace the festivals of the world through all their variations would be to trace the entire history of human religion and human civilization. Where no religion is, there can of course be no feasts; and without civilization any attempt at festival-keeping must necessarily be fitful and comparatively futile. But as religion develops, festivals develop with it, and assume their distinctive character; and an advancing civilization, at least in its earlier stages, will generally be found to increase their number, enrich their ritual, fix more precisely the time and order of their recurrence, and widen the area of their observance.
Some uncivilized tribes, such as the Juangs of Bengal, the Fuegians, and the Andamanese, have been described as having no word for God, no idea of a future state, and con sequently no religious ceremonies of any kind whatever. But such cases, doubtful at the best, are confessedly excep-tional. In the vast majority of instances observed and re-corded, the religiosity of the savage is conspicuous. Even when incapable of higher manifestations, it can at least take the form of reverence for the dead; the grave-heap can become an altar on which offerings of food for the departed may be placed, and where in acts of public and private worship the gifts of survivors may be accompanied with praises and with prayers. That the custom of ghost-pro-pitiation by some sort of sacrifice is even now very widely diffused among the lower races at least, and that there are also many curious " survivals " of such a habit to be traced among highly civilized modern nations, has been abundantly shown of late by numerous collectors of folk-lore and students of sociology; and indications of the same pheno-mena can be readily pointed out in the Big-Veda, the Zend-Avesta, and the Pentateuch, as well as in the known usages of the aucient Egyptians, Greeks, and Romans. In many cases the ceremonial observed is of the simplest; but it ever tends to become more elaborate; and above all it calls for repetition, and repetition, too, at regular intervals. Whenever this last demand has made itself felt, a calendar begins to take shape. The simplest calendar is obviously the lunar. " The Naga tribes of Assam celebrate their funeral feasts month by month, laying food and drink on the graves of the departed." But it soon comes to be combined with the solar. Thus the Karens, " while habi-tually making oblations, have also annual feasts for the dead, at which they ask the spirits to eat and drink." The natives of the Mexican valley in November lay animals, edibles, and flowers on the graves of their dead relatives and friends. The common people in China have a similar custom on the arrival of the winter solstice. The ancient Peruvians had the custom of periodically assembling the embalmed bodies of their dead emperors in the great square of the capital to be feasted in company with the people. The Athenians had their annual NeKixna or Ne/xea-era. and the Romans their Feralia and Lemuralia. The Egyptians observed their three " festivals of the seasons," twelve "festivals of the month," and twelve "festivals of the half month," in honour of their dead. The Parsees, too, were re-quired to render their afringaus (blessings which were to be recited over a meal to which an angel or the spirit of a deceased person was invited) at each of the six seasons of the year, and also on certain other days.3
In the majority of recorded instances, the religious feel-ing of the savage has been found to express itself in other forms besides that of reverence towards the dead. The oldest literatures of the world, at all events, whether Aryan or Semitic, embody a religion of a much higher type than ancestor worship. The hymns of the Rig-Veda, for example, while not without traces of the other, yet indicate chiefly a worship of the powers of nature, connected with the regular recurrence of the seasons. Thus in iv. 57 we have a hymn designed for use at the commencement of the ploughing time j and in the Aitareya-Brahmana, the earliest treatise on Hindu ceremonial, we already find a complete series of sattras or sacrificial sessions exactly following the course of the solar year. They are divided into two distinct sections, each consisting of six months of thirty days each. The sacrifices are allowed to commence only at certain lucky constellations and in certain months. So, for in-stance, as a rule, no great sacrifice can commence during the sun's southern progress. The great sacrifices generally take place in spring, in the months of April and May. [n the Parsee Scriptures the year is divided into six seasons or gahanbars of two months each, concluding with February, the season at which " great expiatory sacrifices were offered for the growth of the whole creation in the last two months of the year." We have no means of know-ing precisely what were the arrangements of the Phoenician calendar, but it is generally admitted that the worship was solar, the principal festivals taking place in spring and in autumn, Among the most characteristic celebrations of the Egyptians were those which took place at the acj>avicrgo<s or disappearance of Osiris in October or November, at the search for his remains, and their discovery about the winter solstice, and at the date of his supposed entrance into the moon at the beginning of spring. The Phrygian festivals were also arranged on the theory that the deity was asleep during the winter and awake during the summer; in the autumn they celebrated his retiring to rest, and in spring with mirth and revelry they roused him from his slumbers. The seasonal character of the Teutonic Ostern, the Celtic Beltein, and the Scandinavian Yule is obvious. Nor was the habit of observing such festivals peculiar to the Aryan or the Semitic race. The Mexicans, who were remarkable for the perfection of their calendar (see vol i. 695), had also an elaborate system of movable and immovable feasts distri-buted over the entire year; the principal festivals, however, in honour of their chiefs gods, Tezcatlipoca, Huitzilopochtli, and Tlaloc, were held in May, June, and December. Still more plainly connected with the revolutions of the seasons was the public worship of the ancient Peruvians, who, besides the ordinary feast at each new moon, observed four solar festivals annually. Of these the most important was the Yntip-Raymi(Sun-feast), which,preceded byathree days' fast, began with the summer solstice, and lasted for nine days. Its ceremonies have been often described. A similar but less important festival was held at the winter solstice. The Cusqui-Kaymi, held after seed-time, as the maize began to appear, was celebrated with sacrifices and banquets, music and dancing. A fourth great festival, called Citua, held on the first new moon after the autumnal equinox, was preceded by a strict fast and special observances intended for purposes of purification and expiation, after which the festivities lasted until the moon entered her second quarter.
Greek Festivals.—Perhaps the annual Attic festival in honour of Erechtheus alluded to in the Iliad (ii. 550) ought to be regarded as an instance of ancestor-worship ; but the seasonal character of the koprq or new-moon feast in Od., xx. 156, and of the OaXvaia or harvest-festival in II, ix. 533, is generally acknowledged. The older Homeric poems, however, give no such express indications of a fully-developed system of festivals as are to be met with in the
! so-called " Homeric" hymns, in the Works and Bays of Hesiod, in the pages of Herodotus, and so abundantly in most authors of the subsequent period; and it is manifest that the calendar of Homer or even of Herodotus must have been a much simpler matter than that of the Taren-tines, for example, came to be, of whom we are told by Strabo that their holidays were in excess of their working days. Each demos of ancient Greece during the historical period had its own local festivals (iopral 8roj.oTi.Kat), often largely attended and splendidly solemnized, the usages of which, though essentially alike, differed very considerably in details. These details have in many cases been wholly lost, and in others have reached us only in a very fragment-ary state. But with regard to the Athenian calendar, the most interesting of all, our means of information are for-tunately very copious. It included some 50 or 60 days on which all business, and especially the administration of justice, was by order of the magistrates suspended. Among these lepop,vviai were included—in Gamelion (January), the Lencea or wine-press feast in honour of Dionysus; in Anthesterion (February), the Anthesteria, also in honour of Dionysus, lasting three days (Pithoigia, Choes, and Chutroi); the Biasia in honour of Zeus, and the lesser Fleusinia; in Elaphebolion (March), the Pandia of Zeus, the Elaphe-bolia of Artemis, and the greater Dionysia ; in Munychion, the Munyehia of Artemis as the moon goddess (Motw^a) and the Belphinia of Apollo; in Thargelion (May), the Thargelia of Apollo and the Plynteria and Gallynteria of Athene; in Skirophorion (June), the Biipolia of Zeus and the Skirophoria of Athene; in Hekatombaion hecatombs were offered to Apollo the summer-god, and the Gronia of Cronus and the PanatJiencea of Athene were held ; in Metageitnion, the Metageitnia of Apollo; in Boedromion, the Boedromia of .Apollo the helper, the Nekusia or Nemeseia (the festival of the dead), and the greater Eleus-inia; in Pyanepsion, the Pyanepsia of Apollo, the Oscho-phoria of Dionysus (probably), the Clialkeia or Aihencea of Athene, the Thesmoplioria of Demeter, and the Apatiiria; in Maimakterion, the Maimakteria of Zeus; and in Posei-deon (December), the lesser Dionysia.
Of these (for the more important of which reference is made to the separate articles) some are commemorative of historical events, and one at least may perhaps be regarded as a relic of ancestor-worship; but the great majority are nature-festivals, associating themselves in the manner that has already been indicated with the phenomena of the seasons, the equinoxes and the solstices. In addition to their numerous public festivals, the Greeks held various family celebrations, also called iopral, in connexion with weddings, births, and similar domestic occurrences. The great national iravgyvpaos—Olympian, Pythian, Nemean, and Isthmian—will be found under separate headings.
Roman Festivals.—For the purpose of holding comitia and administering justice, the days of the Bomanyear were regarded as being either dies fasti or dies nefasti—the dies fasti being the days on which it was lawful for the praetors to administer justice in the public courts, while on the dies nefasti neither courts of justice nor meetings of comitia were allowed to be held. Some days were fasti during one por-tion and nefasti during another; these were called dies intercisi. For the purposes of religion a different division of the year was made; the days were treated as festi or as profesti,—-the former being consecrated to acts of public worship, such as sacrifices, banquets, and games, while the latter (whether fasti or nefasti) were not specially claimed for religious purposes. The dies festi or feriae publicae were either stativae, conceptivae, or imperativae. The stativae were such as were observed regularly, each on a definite day; the conceptivae were observed annually on days fixed by the authorities for the time being; the imperativae were publicly appointed as occasion called for them. In the Augustan age the ferios stativas were very numerous, as may be seen from what we possess of the Fasti of Ovid. The number was somewhat fluc-tuating. Festivals frequently fell into desuetude or were revived, were increased or diminished, were shortened or prolonged at the will of the emperor, or under the caprice of the popular taste. Thus Augustus restored the Com-pitalia and Lupercalia; while Marcus Antoninus in his turn found it expedient to diminish the number of holidays.
The following is an enumeration of the stated festivals as given by Ovid and contemporary writers. The first day of January was observed somewhat as is the modern New Year's Day: clients sent presents to their patrons, slaves to their masters, friends and relations to one another. On the 9th the Agonalia were held, apparently in honour of Janus. On the 11th the Carmentalia were kept as a half-holiday, but principally by women; so also on the 15th. On the 13th of February were the Faunalia, on the 15th the Lupercalia, on the 17th the Quirinalia, on the 18th the Feralia, on the 23d (at one time the last day of the Roman year) the Terminalia, on the 24th the Regifugium or Fugalia, and on the 27th the Fquiria (of Mars). On the 1st of March were the Matronalia, on the 14th a repeti-tion of the Equiria, on the 15th the festival of Anna Ber-enna, on the 17th the Liberalia or Agonalia, and from the 19th to the 23d the Quinquatria (of Minerva). On the 4th of April were the Megalesia (of Cybele), on the 12th the Cerealia, on the 21st the Palilia, on the 23d the Vinalia, on the 25th the Robigalia, and on the 28th the Floralia. The 1st of May was the festival of the Lares Praestites; on the 9th, 11th, and 13th the Lenmria were celebrated; on the 12th the Ludi Martiales, and on the 15th those of Mercury. June 5 was sacred to Semo Sancus ; the Vestalia occurred on the 9 th, the Matralia on the 11th, and the Quinquatrus Minusculce on the 13th. The Ludi Apolli-nares were on the 5th, and the Neptunalia on the 23d of July. On the 13th of August were the Nemoralia, in honour of Diana ; on the 18th the Gonsualia, on the 19th the Vinalia Riistica, and on the 23d the Vulccmalia. The Ludi Magni, in honour of Jupiter, Juno, and Minerva, began on September 4. The Meditrinalia (new wine) were on the 11th of October, the Faunalia on the 13th, and the Equiria on the 15th. The Epulum Jovis was on 13th November. The December festivals were—on the 5th Faunalia, and towards the close Opalia, Saturnalia, Larentalia.
The calendar as it stood at the Augustan age was known to contain many comparatively recent accessions, brought in under the influence of two " closely allied powers, the foreign priest and the foreign cook" (Mommsen). The Megalesia, for example, had been introduced 204 B.C. The Ludi Apollinares could not be traced further back than 208 B.C. The Floralia and Cerealia had not come in much earlier. Among the oldest feasts were undoubtedly the Lupercalia, in honour of Lupercus, the god of fertility; the Equiria, in honour of Mars; the Palilia; the great September festival; and the Saturnalia.
Among the feriae conceptivae were the very ancient feriae Latinae, held in honour of Jupiter on the Alban Mount, and attended by all the higher magistrates and the whole body of the senate. The time of their celebration greatly de-pended on the state of affairs at Rome, as the consuls were not allowed to take the field until they had held the Latinae, which were regarded as days of a sacred truce. The ferias sementivae were held in the spring, and the Ambarvalia in autumn, both in honour of Ceres. The Paganalia of each pagus, and the Compitalia of each vicus were also concep-tivae. Of feriae imperatives,—that is to say, festivals appointed by the senate, or magistrates, or higher priests to commemorate some great event or avert some threatened disaster,—the best known is the Novendiale, which used to be celebrated as often as stones fell from heaven (Livy, xxi. 62, xxv. 7, &c). In addition to all those already mentioned, there occasionally occurred ludi votivi, which were cele-brated in fulfilment of a vow; ludi funebres, sometimes given by private persons; and ludi seculares, to celebrate certain periods marked off in the Etrusco-Roman religion.
Feasts of the Jews.—By Old Testament writers a festival or feast is generally called either 2n (compare the Arabic Hadj), from 33n to rejoice, or IjfiO, from IV), to appoint. The words n2E> and K'Tl'p K"ipp are also occasionally used. In the Talmud the three principal feasts are called DW, after Exod. xxiii. 14. Of the Jewish feasts which are usually traced to a pre-Mosaic origin the most important and characteristic was the weekly Sabbath, but special im-portance was also attached from a very early date to the lunar periods. It is probable that other festivals also, of a seasonal character, were observed (see Exod. v. 1). In common with most others, the Mosaic system of annual feasts groups itself readily around the vernal and autumnal equinoxes. In Lev. xxiii., where the list is most fully given, they seem to be arranged with a conscious reference to the sacred number seven (compare Numb, xxviii.). Those belonging to the vernal equinox are three in number; a preparatory day, that of the Passover, leads up to the prin-cipal festival, that of unleavened bread, which again is followed by an after-feast, that of Pentecost (see PASSOVER, PENTECOST). Those of the autumnal equinox are four; a preparatory day on the new moon of the seventh month (the Feast of Trumpets) is followed by a great day of rest, the day of Atonement (which, however, was hardly a festival in the stricter sense of the word), by the Feast of Taber-nacles, and by a great concluding day (Lev. xxiii. 36 ; John vii. 37). If the feast of the Passover be excepted, it will be seen that all these celebrations or commemorations asso-ciate themselves more readily with natural than with his-torical events. There was also a considerable number of post-Mosaic festivals, of which the principal were that of the Dedication (described in 1 Mace. iv. 52-59; comp. John x. 22) and that of Purim, the origin of which is given in the book of Esther (ix. 20 sq.). It has probably no connexion with the Persian festival Furdigan (see ESTHER).
Earlier Christian Festivals.—While making it abund-antly manifest that Christ and His disciples observed the appointed Jewish feasts, the New Testament nowhere re-cords the formal institution of any distinctively Christian festival. But we have unambiguous evidence of the actual observance, from a very early period, of the first day of the week as a holy day (John xx. 19, 26; 1 Cor. xvL 2; Acts xx. 7 ; Rev. i. 10). Bliny in his letter to Trajan describes the Christians of Bithynia as meeting for religious purposes on a set day; that this day was Sunday is put beyond all reasonable doubt by such a passage as that in the Apology of Justin Martyr, where he says that " on Sunday (TTJ TOV r¡ktov AeyoiuVg v^fü) and the Christians living either in the city or the country met together." The Jewish element, in some churches at least, and especially in the East, vías strong enough to secure that, along with the dies dominica, the seventh day should continue to be kept holy. Thus in the Apostolic Constitutions (ii. 59) we find the Saturday specially mentioned along with the Sunday as a clay for the assembling of the church ; in v. 15 it is ordained that there shall be no fasting on Saturday, while in viii, 33 it is added that both on Saturday and Sunday slaves are to have rest from their labours. The 16th canon of the council of Laodicea almost certainly means that solemn public service was to be held on Saturday as well as on Sunday. In other quarters, however, the tendency to re-gard both days as equally sacred met with considerable re-sistance. The 36th canon of the council of Illiberis, for example, deciding that Saturday should be observed as a fast-day, was doubtless intended to enforce the distinction between Saturday and Sunday. At Milan in Ambrose's time Saturday was observed as a festival; but Pope Inno-cent is found writing to the bishop of Eugubium to urge that it should be kept as a fast. Ultimately the Christian church came to recognize but one weekly festival.
The numerous yearly festivals of the later Christian church, when historically investigated, can be traced to very small beginnings. Indeed, while it appears to be tolerably certain that Jewish Christians for the most part retained all the festivals which had been instituted under the old dispensation, it is not at all probable that either they or their Gentile brethren recognized any yearly feasts as of distinctively Christian origin or obligation. It can-not be doubted, however, that gradually, in the course of the 2d century, the universal church came to observe the anniversaries of the death and resurrection of Christ —the Traerla crravpwcnp.ov and the Tracryu avacrTácn¡j.ov, as they were respectively called (see EASTER and GOOD FRIDAY). Not long afterwards Whitsunday also came to be fixed in the usage of Christendom as a great annual festival. Even Origen (in the 8th book Against Celsus) enumerates as Christian festivals the Sunday, the irapa-o-Kevr¡, the Passover with the feast of the Resurrection, and Pentecost; under which latter term, however, he includes the whole period between Easter and Whitsuntide. About Cyprian's time we find individual Christians commemorat-ing their departed friends, and whole churches commemor-ating their martyrs ; in particular, there are traces of a local and partial observance of the feast of the Innocents. Christinas day and Epiphany were among the later introductions, the feast of the Epiphany being somewhat the earlier of the two. Both are alluded to indeed by Clemens Alexandrinus (i. 340), but only in a way which indicates that even in his time the precise date of Christ's birth was unknown, that its anniversary was not usually observed, and that the day of his baptism was kept as a festival only by the followers of Basilides (see EPIPHANY).
When we come down to the 4th century we find that, among the 50 days between Easter and Pentecost, Ascen-sion day has come into new prominence, Augustine, for example, enumerates as anniversaries celebrated by the whole church those of Christ's passion, resurrection, and ascension, along with that of the outpouring of the Holy Ghost, while he is silent with regard to Christmas and Epiphany. The general tendency of this and the following-centuries was largely to increase the festivals of the church, and by legislation to make them more fixed and uniform. Many passages, indeed, could be quoted from Chrysostom, Jerome, and Augustine to show that these fathers had not by any means forgotten that comparative freedom with regard to outward observances was one of the distinctive excellencies of Christianity as contrasted with Judaism and the various heathen systems (compare Socrates, II. E., v. 22). But there were many special circumstances which seemed to the leaders of the Church at that time to neces-sitate the permission and even legislative sanction of a large number of new feasts. The innovations of heretics some-times seemed to call for rectification by the institution of more orthodox observances; in other instances the propen-sity of rude and uneducated converts from paganism to cling to the festal rites of their forefathers proved to be invincible, so that it was seen to be necessary to seek to adapt the old usages to the new worship rather than to abolish them altogether; moreover, although the empire had become Christian, it was manifestly expedient that the old holidays should be recognized as much as possible in the new arrangements of the calendar. Constantine soon after his conversion enacted that on the dies dominica there should be no suits or trials in law; Theodosius the Great added a prohibition of all public shows on that day, and Theodosius the younger extended the prohibition to Epi-phany and the anniversaries of martyrdoms, which at that time included the festivals of St Stephen, and of St Peter and St Baul, as also that of the Maccabees. In the 21st canon of the council of Agde (506), besides Easter, Christmas, Epiphany, Ascension, and Pentecost, we find the Nativity of John the Baptist already mentioned as one of the more important festivals on which attendance at church was regarded as obligatory. To these were added, in the centuries immediately following, the feasts of the Annunciation, the Purification, and the Assumption of the Virgin; as well as those of the Circumcision, of St Michael, and of All Saints.
Festivals were in practice distinguished from ordinary days in the following ways:—all public and judicial business was suspended, as well as every kind of game or amuse-ment which might interfere with devotion; the churches were specially decorated; Christians were expected to attend public worship, attired in their best dress; love feasts were celebrated, and the rich were accustomed to show special kindness to the poor; fasting was strictly forbidden, and public prayers were said in a standing posture.
Later Practice.—In the present calendar of the Roman Catholic Church the number of feast days is very large. Each is celebrated by an appropriate office, which, accord-ing to its character, is either duplex, semi-duplex, or simplex. A duplex again may be either of the first class or of the second, or a major or a minor. The distinctions of ritual for each of these are given with great minuteness in the general rubrics of the breviary; they turn chiefly on the number of Psalms to be sung and of lessons to be read, on the manner in which the antiphons are to be given, and on similar details. The duplicia of the first class are the Nativity, the Epiphany, Easter with the three preceding and two following days, the Ascension, Whitsunday and the two following clays, Corpus Christi, the Nativity of John Baptist, Saints Beter and Paul, the Assumption of the Virgin, All Saints, and, for each church, the feast proper to its patron or title and the feast of its dedication. The duplicia of the second class are the Circumcision, the feast of the Holy Name of Jesus, of the Holy Trinity, and of the Most Precious Blood of Christ, the feasts of the Purification, Annunciation, Visitation, Nativity, and Conception of the Virgin, the Natalitia of the Twelve Apostles, the feasts of the Evangelists, of St Stephen, of the Holy Innocents, of St Joseph ana of the Patrocinium of Joseph, of St Lawrence, of the Invention of the Cross, and of the Dedication of St Michael. The Dominicas majores of the first class are the first Sunday in Advent, the first in Lent, Passion Sunday, Palm Sunday, Easter Sunday, Dominica in Alhis, Whitsunday, and Trinity Sunday; the Dominicas majores of the second class are the second, third, and fourth in Advent, Septuagésima, Sexagésima, and Quinquagesima Sundays, and the second, third, and fourth Sundays in Lent.
In the canons and decrees of the council of Trent re-peated allusions are made to the feast days, and their fitness, when properly observed, to promote piety. Those entrusted with the cure of souls are urged to see that the feasts of the church be devoutly and religiously observed, the faithful are enjoined to attend public worship on Sundays and on the greater festivals at least, and parish priests are bidden to expound to the people on such days some of the things which have been read in the office for the day. Since the council of Trent, the practice of the church with respect to the prohibition of servile work on holidays has varied considerably in different Catholic countries, and even in the same country at different times. Thus in 1577, in the diocese of Lyons, there were almost 40 annual festi-vals of a compulsory character. By the concordat of 1802 the number of such festivals was for France reduced to four, namely, Christmas day, Ascension day, the Assumption of the Virgin, and All Saints day.
The calendar of the Greek Church is even fuller than that of the Latin, especially as regards the éopral TW áyióV. Thus on the last Sunday in Advent the feast of All Saints of the Old Covenant is celebrated; while Adam and Eve, Job, Elijah, Isaiah, &c, have separate days. The dis-tinctions of ritual are analogous to those in the Western Church. In the Coptic Church there are seven great festi-vals, Christmas, Epiphany, the Annunciation, Palm Sunday, Easter Sunday, Ascension, and Whitsunday, on all of which the Copts "wear new clothes (or the best they have), feast, and give alms " (Lane). They also observe, as minor festivals, Maundy Thursday, Holy Saturday, the feast of the Apostles (11th July), and that if the Discovery of the Cross.
In common with most of the churches of the Beforma-tion, the Church of England retained a certain number of feasts besides all Sundays in the year. They are, besides Monday and Tuesday both in Easter-week and Whitsun-week, as follows :—the Circumcision, the Epiphany, the Conversion of St Paul, the Purification of the Blessed Virgin, St Matthias the Apostle, the Annunciation of the Blessed Virgin, St Mark the Evangelist, St Philip and St James (Apostles), the Ascension, St Barnabas, the Nativity of St John Baptist, St Peter the Apostle, St James the Apostle, St Bartholomew, St Matthew, St Michael and all Angels, St Luke the Evangelist, St Simon and St Jude, All Saints, St Andrew, St Thomas, Christmas, St Stephen, St John the Evangelist, the Holy Innocents. The 13th canon enjoins that all manner of persons within the Church of England shall from henceforth celebrate and keep the Lord's day, commonly called Sunday, and other holy days, according to God's holy will and pleasure, and the orders of the Church of England prescribed in that behalf, that is, in hearing the Word of God read and taught, in private and public prayers, in acknowledging their offences to God and amendment of the same, in reconciling themselves charitably to their neighbours where displeasure hath been, in oftentimes receiving the communion of the body and blood of Christ, in visiting of the poor and sick, using all godly and sober conversation. (Compare Hooker, E. P., v. 70.) In the Directory for the Public Worship of God which was drawn up by the Westminster Assembly, and accepted by the Church of Scotland in 1645, there is an appendix which declares that there is no day commanded in Scripture to be kept holy under the gospel but the Lord's day, which is the Christian Sabbath; festival days, vulgarly called holy-days, having no warrant in the Word of God, are not to be continued; nevertheless it is lawful and necessary, upon special emergent occasions, to separate a day or clays for public fasting or thanksgiving, as the several eminent and extraordinary dispensations of God's providence shall ad-minister cause and opportunity to his people.
Several attempts have been made at various times in western Europe to reorganize the festival system on some other scheme than the Christian. Thus at the time of the French Revolution, during the period of Robespierre's ascendency, it was proposed to substitute a tenth day (Decadi) for the weekly rest, and to introduce the following new festivals:— that of the Supreme Being and of Nature, of the Human Race, of the French People, of the Benefactors of Mankind, of Freedom and Equality, of the Martyrs of Freedom, of the Republic, of the Freedom of the World, of Patriotism, of Hatred of Tyrants and Traitors, of Truth, of Justice, of Modesty, of Fame and Immortality, of Friendship, of Temperance, of Heroism, of Fidelity, of Unselfishness, of Stoicism, of Love, of Conjugal Fidelity, of Filial Affection, of Childhood, of Youth, of Manhood, of Old Age, of Misfortune, of Agriculture, of Industry, of our Forefathers, of Posterity and Felicity. The proposal, however, was never fully carried out, and soon fell into oblivion.
Mahometan Festivals.—These are chiefly two—the 'Eed es-Sagheer (or minor festival) and the'Eed el-Kebeer (or great festival), sometimes called 'Eed el-Kurban. The former, which lasts for three days, immediately follows the month Bamadan, and is generally the more joyful of the two; the latter begins on the tenth of Zu-l-Heggeh (the last month of the Mahometan year), and lasts for three or four days. Besides these festivals they usually keep holy the first ten days of Moharram (the first month of the year), especially the tenth day, called Yom Ashoora; the birthday of the prophet, on the twelfth day of the third month; the birth-day of El-Hoseyn, in the fourth month ; the anniversary of the prophet's miraculous ascension into heaven, in the seventh month; and one or two other anniversaries (see vol. vii. p. 727). Friday, called the clay of El-Gumah (the assembly), is a day of public worship ; but it is not usual to abstain from public business on that day except during the time of prayer.
Hindu and Buddhist Festivals.—In modern India the leading popular festivals are the Holi, which is held in March or April and lasts for five days, and the Dasahara, which occurs in October (see Hunter's Statistical Account of Bengal). Although in its origin Buddhism was a deliberate reaction against all ceremonial, it does not now refuse to observe festivals. By Buddhists in China, for example, three days in the year are especially observed in
honour of the Buddha,—the eighth day of the second month, when he left his home; the eighth day of the fourth month, the anniversary of his birthday ; and the eighth of the twelfth, when he attained to perfection and entered Nirvana. In Siam the eighth and fifteenth days of every month are considered holy, and are observed as days for rest and worship. At Trut, the festival of the close of the year, visiting and play-going are universal.
The new year (January) is celebrated for three days; in February is another holiday; in April is a sort of Lent, ushering in the rainy season; on the last day of June presents are made of cakes of the new rice ; in August is the festival of the angel of the river, " whose forgiveness is then asked for every act by which the waters of the Meinam have been rendered impure."
See Bowring's Siam and Carne's Travels in Indo-China and the Chinese Empire. Copious details of the elaborate festival-system of the Chinese may be found in Doolittle's Social Life of the Chinese. (J. S. BLJ
2 See Spencer, Principles of Sociology, i. 170, 280, 306.
3 Haug, Parsis, 224, 225.
8 In this month the anniversaries of the battle of Marathon, and of the downfall of the thirty tyrants, were also publicly celebrated.
6 See Schoemann, Grieschische Alterthumer, ii. 439 sq.; Momrnsen, Seortologie.
" May the heavens, the waters, the firmament, he kind to us ; may
the lord of the field he gracious to us.....May the oxen (draw)
happily, the men labour happily; may the traces bind happily, wield the goad happily " (Wilson's translation, iii. 224).
- See Haug's Aitareya-br&hmanam of the Rig-Veda; Max Muller's Chips from a German Workshop, i. 115.
Visperad. See Haug, Parsis, 192; Richardson's Dissertation on the
Language, &c., of Eastern Nations, p. 184; Morier's Journey through
Persia.
* Plutarch. De Iside et Osiride; Macrobius, Saturnalia, i. 21.
Feriaa privatse, such as anniversaries of births, deaths, and the like,
were observed by separate clans, families, or individuals.
On the whole subject of Jewish festivals see Reland, Antiq. Hebr.; Knobel, Leviticus (c. 23); George, Die Jüdischen Feste; Hupfeld, De primitives fest. ap. Hebr. ratione; Ewald, Alterthümer des Volkes Israel; Dillmann in Schenkel's Bibel-lexicon, art. "Feste."
2 In the "parallel" passages, there is considerable variety in the designation and arrangement of these feasts. While Ex. xii. approxi-mates most closely to Lev. xxiii. and Num. xxviii., Ex. xxiii. has stronger affinities with Deut. xvi. The relations of these passages are largely discussed by Graf, Die Geschichtlichen Bücher des A. T., pp. 34-41, and by other recent critics.
As, at a later period (601), Gregory the Great instructed his Anglo-Saxon missionaries so to Christianize the temples, festivals, &c., of the heathen " ut dura? mentes gradibus vol passions, non autem saltihus, eleventur."
Manumission, however, was lawful on any day. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,613 |
{"url":"https:\/\/nforum.ncatlab.org\/discussion\/6343\/aufhebung\/","text":"# Start a new discussion\n\n## Not signed in\n\nWant to take part in these discussions? Sign in if you have an account, or apply for one below\n\n## Discussion Tag Cloud\n\nVanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.\n\n\u2022 CommentRowNumber1.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\nThomas Holder has been working on Aufhebung. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references, cross-linked with duality of opposites).\n\n\u2022 CommentRowNumber2.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\nFor completeness I have added an brief entry level of a topos and cross-linked a bit.\n\n\u2022 CommentRowNumber3.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\u2022 (edited Nov 25th 2014)\n\nLet me look at the most basic case. That $(\\flat \\dashv \\sharp)$ is Aufhebung for $(\\emptyset \\dashv \\ast)$ means that\n\n$\\sharp \\emptyset \\simeq \\emptyset$\n\nand that $\\sharp$ is minimal with this property. What is the statement regarding conditions under which this is the case?\n\nBy adjunction and using that the initial object in a topos is stict, it follows that $\\sharp \\emptyset \\simeq \\emptyset$ is equivalent to the statement that the only object with no global points is $\\emptyset$ itself, because\n\n$(X \\to \\sharp \\emptyset) \\simeq (\\flat X \\to \\emptyset) \\simeq (\\flat X \\simeq \\emptyset) \\,.$\n\nIs that automatic? (This is probably elementary, please bear with me.) And how to see or check that $\\sharp$ is minimal with this property?\n\n\u2022 CommentRowNumber4.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 25th 2014\n\nI guess my wording in the entry is a bit careless, as I brought in the general context of categories of being in order to motivate the metaphysical lingo for $\\empty\\dashv\\ast$ and its Aufhebung, though Lawvere proves this only in special cases e.g. in the 1991 \u2019Hegelian taco\u2019 paper. so unless, a proof for the general case can be cooked up, I am afraid I\u2019ve to row back a bit in the entry.\n\nIt maybe worth mentioning that a negative result of a (sufficiently) cohesive topos where this Aufhebungs relation fails to hold would also show the limits of the particular categorical interpretation of Hegel\u2019s logic.\n\n\u2022 CommentRowNumber5.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nLet me connect this with something more familiar to me. To say that $(i^*,i_*):E\\to F$ is an essential subtopos, with essentiality $i_!$, is almost the same as to say that $(i_!,i^*):F\\to E$ is a connected local geometric morphism, except that we don\u2019t ask $i_!$ to be left exact. Right?\n\n\u2022 CommentRowNumber6.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nWhat does it mean for two levels to be \u201cconsecutive\u201d?\n\n\u2022 CommentRowNumber7.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\u2022 (edited Nov 25th 2014)\n\nre #3:\n\nto say that in a more pronounced way, the condition $\\sharp \\emptyset \\simeq \\emptyset$ implies that the ambient $\\infty$-topos has homotopy dimension $\\leq 0$ relative to the sub-$\\infty$-topos of $\\flat$-modal objects. This is a property enjoyed by all models for which the $\\flat$-subtopos is $\\infty Grpd$ (by this proposition). So is this \u201cAufhebung\u201d a strong classicality condition on the $\\flat$-modal subtopos?\n\nre #5: I\u2019d say \u201cyes, of course\u201d, but that means I am probably missing some subtlety that you have in mind :-)\n\n\u2022 CommentRowNumber8.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nAre you sure that $\\lozenge _i$ and $\\Box _i$ are named correctly? If the are intended to suggest the \u201cpossibly\u201d and \u201cnecessarily\u201d modalities of classical modal logic, then $\\lozenge$ should be the monad and $\\Box$ the comonad.\n\n\u2022 CommentRowNumber9.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\u2022 (edited Nov 25th 2014)\n\nre #6:\n\nthis refers to the Idea section here. I just meant two levels where one includes the previous one, I have changed it now to read as follows:\n\nIf for two levels $\\mathbf{H}_{1} \\hookrightarrow \\mathbf{H}_2$ the second one includes the modal types of the idempotent comonad of the first one, and if it is minimal with this property, then Lawvere speaks of \u201cAufhebung\u201d (see there for details) of the unity of opposites exhibited by the first one.\n\n\u2022 CommentRowNumber10.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 25th 2014\n\n@#8: the notation was chosen by me probably after having looked in which way the modal operators are adjoint to replace $L\\dashv R$ of Lawvere (1989) or the $sk\\dashv cosk$ used in KRRZ11. In complies broadly with the intuition to have necessity on the \u2019right\u2019 side of being\/sheaf but feel free to choose something more suitable.\n\n\u2022 CommentRowNumber11.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nThomas, can you explain further your reasons for assigning $\\lozenge$ and $\\Box$ in that order? I don\u2019t understand why necessity would be on the right side.\n\n\u2022 CommentRowNumber12.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 25th 2014\n\nMy decision came from a short look at p.116 in the \u2019 generic figures\u2019 book of Reyes et.al. where they define $\\lozenge:=i^*\\circ i_!$ and $\\Box :=i^*\\circ i_*$ in context of an inclusion of posets; in which case then $\\lozenge\\dashv \\Box$ whereas the nlab entry defines the composition dually in reverse order. so this should probably be changed, sorry for the confusion!\n\n\u2022 CommentRowNumber13.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\u2022 (edited Nov 25th 2014)\n\nI think Mike\u2019s point is that for the notation here its not the side of the adjunction but whether we have a monad or comonad that decides what the box is. The box should be a comonad as for necessity. That is consistent with what you cite from Reyes, but not with using box for $\\sharp$.\n\n(We are lacking an $n$Lab page that says this comprehensively. Something should go at modal operator )\n\n\u2022 CommentRowNumber14.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 25th 2014\n\nI agree, my choice was hasty as I looked only for an adjunction between the modal operators without realizing that they were defined dually. Though I must admit that I found the choice esthetically pleasing - it would be good to have some suggestive symbols.\n\n\u2022 CommentRowNumber15.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\nI added a remark at modality in a section Notation. Everyone is kindly invited to expand and\/or edit.\n\n\u2022 CommentRowNumber16.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\nBack to the Aufhebung via global points:\n\nso over every infinity-cohesive site it is true that $\\sharp\\emptyset \\simeq \\emptyset$, I suppose.\n\nSince here indeed only the initial sheaf has no global points. (E.g. if a sheaf on $CartSp$ has no global points, that means it assigns the empty set to $\\mathbb{R}^0$, but that means it must assigns the empty set to each $\\mathbb{R}^n$ since there do exist maps $\\mathbb{R}^0\\to \\mathbb{R}^n$).\n\n\u2022 CommentRowNumber17.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 25th 2014\n\nWhat I am talking about is now cleaned up here. Let me know if I am hallucinating (it\u2019s late here).\n\n\u2022 CommentRowNumber18.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nOk, I\u2019ve switched $\\Box$ and $\\lozenge$ in the entry.\n\n\u2022 CommentRowNumber19.\n\u2022 CommentAuthorMike Shulman\n\u2022 CommentTimeNov 25th 2014\n\nNow I\u2019ve looked at your notation remark at modality; should we use $\\bigcirc$ rather than $\\lozenge$ at Aufhebung since it is right adjoint to $\\Box$? (What\u2019s the origin of $\\lozenge$ being left adjoint to $\\Box$ and $\\bigcirc$ being right adjoint to it? I didn\u2019t think that necessity and possibility were adjoint in either direction; are they?)\n\n\u2022 CommentRowNumber20.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nI did this just by exclusion principle: according to #12 the diamond has been used for the left adjoint, so the circle remains for the right adjoint.\n\nBut I am happy with any other convention\/tendency.\n\n\u2022 CommentRowNumber21.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nCancel what I just wrote having actually read what was written.\n\nIf that\u2019s suggesting in modality that $\\lozenge$ is less used by modal logicians for possibility, then that\u2019s wrong. As far as I\u2019m concerned it\u2019s used as much as $\\bigcirc$:\n\nAlso $\\lozenge$ is used for a modality, in particular if it is left adjoint to a $\\Box$.\n\nOn the other hand we could choose to make such a convention of distinguishing left and right adjoint monads. But let\u2019s be explicit.\n\nWhat about the differential cohesive comonads? Shouldn\u2019t we have notation to distinguish left and right adjoints there too?\n\n\u2022 CommentRowNumber22.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nHow do modal logicians choose between $\\lozenge$ and $\\bigcirc$? What does the choice indicate for them, if anything?\n\n\u2022 CommentRowNumber23.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\nAs far as I\u2019m aware it\u2019s just an arbitrary choice of notation, like $\\wedge$, $\\cdot$, or & is a choice for conjunction.\n\n\u2022 CommentRowNumber24.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nThanks. I\u2019ll rephrase the statement in the entry then. Just a moment\u2026\n\n\u2022 CommentRowNumber25.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\nIf anything I\u2019d say $\\lozenge$ is the more commonly used of the two. The SEP modal logic entry chooses it. So, all things equal, I\u2019d have it as the left adjoint monadic modality.\n\n\u2022 CommentRowNumber26.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nOkay, I have tried to edit accordingly here. Please feel invited to further edit if you see further need.\n\n\u2022 CommentRowNumber27.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nWhen adjunctions between modalities matter, there is a tendency \u2026\n\nMy knowledge of these situations is so small I couldn\u2019t speak of a tendency. I don\u2019t know of any philosophers who have raised the issue of adjunctions. Do people know about computer science, etc.? Even Hermida as mentioned here seems to have the dichotomies lined up.\n\nBut what about my final point in #21? Should one use just $\\Box$ for left\/right adjoint comodalities $Red$ and $\\flat_{inf}$? Sometimes people have used L and M for necessity\/possibility.\n\n\u2022 CommentRowNumber28.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nOkay, I have changed \u201ctendency to use\u201d to \u201csome authors use\u201d. Then I added one more example, namely\n\n\u2022 Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)\n\n(where the left part of cohesion appears in terms of adjoint modalities on p. 367).\n\nMaybe \u201csome authors\u201d is just \u201cGonzalo Reyes\u201d, though? Hermida does not seem to use the symbols from modal logic, does he?\n\nRegarding notation for differential cohesion: I may not know what candidates there are from traditional theory. Maybe none? I don\u2019t know. The situation of differential cohesion seems to have been missed, by and large.\n\n\u2022 CommentRowNumber29.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nHermida uses $\\Box$ and $\\lozenge$ in the abstract, but then angled and square parentheses around a relation symbol.\n\n\u2022 CommentRowNumber30.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nAh, right. So under this translation, does he have $\\Box$ as the right adjoint?\n\n\u2022 CommentRowNumber31.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\nIt seems that $[R]$ is always on the right.\n\n\u2022 CommentRowNumber32.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nThanks. I see it now in remark 3.3. Good, so I\u2019ll add that to the entry modality, too. So then maybe \u201ctendency to use\u201d is not that bad after all. Do we known an author who explicitly does not use that convention?\n\n\u2022 CommentRowNumber33.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nHm, on the other hand, Hermida\u2019s article has $\\Box$ be a monad, not a comonad.\n\n\u2022 CommentRowNumber34.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 26th 2014\n\nWhere? I see him say\n\nMonadic interpretation of $\\langle - \\rangle$,\n\nbut that\u2019s expected.\n\n\u2022 CommentRowNumber35.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nIn that remark 3.3 he says that his $\\langle R\\rangle$ is the composition of pullback followed by dependent sum, and that his $[R]$ is the composition of pullback followed by dependent product. That makes $[R]$ a monad.\n\n\u2022 CommentRowNumber36.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nSorry for bringing this up again, as the entry now has $\\Box$ on the left, I would suggest wispering to use $\\bigcirc$ on the right in the context of Aufhebung as this nicely suggests \u2019being\u2019 unless this interferes negatively with the convention you just set up. As this is just a tiny detail I would very much like to avoid a discussion on this and propose to drop the subject when you don\u2019t like the idea.\n\n\u2022 CommentRowNumber37.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nYes, that\u2019s exactly what we seem to have agreed on over at modality \u2013 Notation.\n\nI\u2019ll change it at Aufhebung, too.\n\nBTW, any comment on #17? We talked about it by email. It now seems to me that it does work for the \u201cstandard examples\u201d of cohesion. But let me know if I am missing something.\n\n\u2022 CommentRowNumber38.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nAssuming that you use corollary II of Yoneda lemma in the last step of the proof of prop.1 at Aufhebung I think that prop.1 is ok.\n\nAs far as I can tell the proof of prop.1 uses only the strictness of $\\empty$ and the rest is purely general using just the adjointness and the specific equivalences you assume, so this seems to be valid more generally for extensive cats with an adjunction that relates to $\\empty$ in the appropriate way without being necessarily $\\flat\\dashv\\sharp$, no !?\n\nTo see through prop.2 I\u2019d need time to acquire some knowledge on infinity-cohesive-sites. This should be easier to see for someone with better grounding in the higher categorical point of view though, I guess.\n\nIn any case, it would be nice to have this somewhat more general result for the Aufhebung of $\\empty\\dashv {\\ast}$.\n\n\u2022 CommentRowNumber39.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\u2022 (edited Nov 26th 2014)\n\nThanks for the feedback. For the argument in prop. 2 the homotopy theory is irrelevant (I should have pointed to cohesive site!), you may just as well consider just plain presheaves. It\u2019s meant to be a trivial argument: the site by assumption has a terminal object and there is a morphism from that terminal object to every other object. That implies that if a presheaf (of sets) assigns the empty set to that terminal object, it has to assign the empty set to every other object of the site, too.\n\n\u2022 CommentRowNumber40.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nI have edited the formatting of the central definition a bit in order to make the central ideas spring to the eye more vividly. Now it reads as follows:\n\nLet $i,j$ be levels, def. \\ref{Level}, of a topos $\\mathcal{A}$ we say that the level $i$ is lower than level $j$, written\n\n$\\array{ \\Box_i &\\leq& \\Box_j \\\\ \\bot && \\bot \\\\ \\bigcirc_i &\\leq& \\bigcirc_j }$\n\n(or $i\\leq j$ for short) when every i-sheaf ($\\bigcirc_i$-modal type) is also a j-sheaf and every i-skeleton ($\\Box_i$-modal type) is a j-skeleton.\n\nLet $i\\leq j$, we say that the level $j$ resolves the opposite of level $i$, written\n\n$\\array{ \\Box_i &\\ll& \\Box_j \\\\ \\bot && \\bot \\\\ \\bigcirc_i &\\ll& \\bigcirc_j }$\n\n(or just $i\\ll j$ for short) if $\\bigcirc _j\\Box_i=\\Box _i$.\n\nFinally a level $\\bar{i}$ is called the Aufhebung of level $i$\n\n$\\array{ \\Box_i &\\ll& \\Box_{\\bar i} \\\\ \\bot &\\searrow& \\bot \\\\ \\bigcirc_i &\\ll& \\bigcirc_{\\bar i} }$\n\niff it is a minimal level which resolves the oppose of level $i$, i.e. iff $i\\ll\\bar{i}$ and for any $k$ with $i\\ll k$ then it holds that $\\bar{i}\\ll k$.\n\n=\u2013\n\n\u2022 CommentRowNumber41.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 26th 2014\n\nIt seems to me that this here is proof that over a cohesive site $(\\flat \\dashv \\sharp)$ is Aufhebung of $(\\emptyset \\dashv \\ast)$. But check that I am not being stupid here:\n\n+\u2013 {: .num_prop #OverCohesiveSiteBecomingIsAufgehoben}\n\n###### Proposition\n\nLet $\\mathcal{S}$ be a cohesive site (or \u221e-cohesive site) and $\\mathbf{H} = Sh(\\mathcal{S})$ its cohesive sheaf topos with values in Set (or $\\mathbf{H} = Sh_\\infty(S)$ its cohesive (\u221e,1)-topos ).\n\nThen in $\\mathbf{H}$ we have Aufhebung, def. \\ref{Aufhebung}, of the duality of opposites of becoming $\\emptyset \\dashv \\ast$.\n\n=\u2013\n\n+\u2013 {: .proof}\n\n###### Proof\n\nBy prop. \\ref{OverCohesiveSiteBecomingIsResolved} we have that $(\\flat\\dashv \\sharp)$ resolves $(\\emptyset \\dashv \\ast)$ and so it remains to see that it is the minimal level with this property. But the subtopos of sharp-modal types is $\\simeq$ Set which is clearly a 2-valued Boolean topos. By this proposition these are the atoms in the subtopos lattice hence are minimal as subtoposes and hence also as levels.\n\n=\u2013\n\n\u2022 CommentRowNumber42.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 27th 2014\n\u2022 (edited Nov 27th 2014)\n\nDavid,\n\ncoming back to #34-#35 and also to our discussion in person on the justification of calling a $\\Box$-modality the \u201cnecessity\u201d modality:\n\nif we turn what Hermida does around, then everything makes sense to me.\n\nNamely: if we\n\n\u2022 read \u201cnecessarily\u201d as the name specifically for the \u201cfor all\u201d-operation turned into a comonad;\n\n\u2022 read \u201cpossibly\u201d as the name specifically for the \u201cthere exists\u201d-operation turned into a monad;\n\nthen first of all $\\Box$ is a comonad as desired and moreover then its interpretation as formalizing \u201cnecessity\u201d is indeed justified:\n\nfor let $X$ be a \u201ccontext\u201d which you may want to think of as the \u201ctype of all possible worlds\u201d, and if $P$ is a proposition about terms of type $X$, then\n\n\u2022 the statement that \u201cfor $x \\colon X$ it is necessarily true that $P(x)$\u201d is just another way to say in Enlish that \u201cfor all $x \\colon X$ it is true that $P(X)$\u201d;\n\n\u2022 the statement that \u201cfor $x \\colon X$ it is possibly true that $P(x)$\u201d is just another way to say in Enlish that \u201cthere exists $x \\colon X$ such that it is true that $P(X)$\u201d;\n\nSo, more formally, given the adjoint triple of dependent sum $\\dashv$ context extension $\\dashv$ dependent product\n\n$\\mathbf{H}_{\/X} \\stackrel{\\stackrel{\\sum_X}{\\longrightarrow}}{\\stackrel{\\stackrel{X^\\ast}{\\longleftarrow}}{\\underset{\\prod_X}{\\longrightarrow}}} \\mathbf{H}$\n\nthen it makes justified sense to call the induced (co-)-monads\n\n$(\\lozenge_X \\dashv \\Box_X) \\coloneqq ( X^\\ast \\underset{X}{\\sum} \\dashv X^\\ast \\underset{X}{\\prod}) \\colon \\mathbf{H}_{\/X} \\to \\mathbf{H}_{\/X}$\n\n\u201cpossibility\u201d and \u201cnecessity\u201d, respectively, because\n\n\u2022 by the established interpretation of $\\sum_X$ as \u201cthere exists $x \\colon X$\u201d we have that $\\lozenge_X P$ holds for a given $x\\in X$ precisely if it holds for at least one $x \\in X$, hence if it is possible in the context $X$ for it to hold at all;\n\n\u2022 by the established interpretation of $\\prod_X$ as \u201cfor all $x \\colon X$\u201d we have that $\\Box_X P$ holds for a given $x\\in X$ precisely if it holds for all $x \\in X$ hence if it is inevitable, hence necessary, in the context $X$ that if holds.\n\n\u2022 CommentRowNumber43.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 27th 2014\n\nSorry for the interruption, but I had to fix the definition of Aufhebung as the minimality is defined relative to the usual order of the levels in the literature. For this I reused $\\leq$ as the order of levels and tried to use predecessor-equality for the resolution relation but the local teX doesn\u2019t know my dialect.\n\n\u2022 CommentRowNumber44.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 27th 2014\n\nLet me see if I understand \u2013 you changed the notation in the first clause in the definition, but should you not also change it in the last line then? Maybe I am missing something.\n\nBy the way, regarding your email: the reason that I oriented these diagrams as I did as opposed to with levels going upwards is just because I wouldn\u2019t know how to typeset the order relation going vertically.\n\n\u2022 CommentRowNumber45.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 27th 2014\n\nUrs, re #40, I was trying to do something like that back here.\n\nI had the feeling Neel was onto something too a few comments later.\n\n\u2022 CommentRowNumber46.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 28th 2014\n\u2022 (edited Nov 28th 2014)\n\nad 44#: to me the definition appears to be as intended unless I forgot to switch notation at some place. The problem was previously it used $\\ll$ for the minimality where it should have used the essential subtopos order which I then thought best to denote with the least marked $\\leq$. I think I\u2019ll try later to use $\\sqsubset$ and $\\lhd$ instead of $\\prec$ and $\\ll$ as they have rotated versions $\\sqcup$ and $\\bigtriangledown$.\n\n\u2022 CommentRowNumber47.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 28th 2014\n\nThomas, if it is as intended, then my question would be why this is intended. Why use in the last line of the definition not the same order relation as introduced in the first line? What\u2019s the rationale for this?\n\n\u2022 CommentRowNumber48.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeNov 28th 2014\n\u2022 (edited Nov 28th 2014)\n\nI start to see your point here, the way I stated the definition is cooked up out of Lawvere 89 who defines this for graphic toposes and states everything in terms of the pretty-well behaved ideals in the underlying category, so I kept his terminology but looked for the definitions in terms of subtoposes to KRRZ11 which I\u2019ve read in the way that $\\prec$ and $\\leq$ are two different things, but actually life becomes much easier if they are the same. I was already looking for a proposition showing their compatibility in order to prove that quintessential localizations are their proper Aufhebung. I guess you are right.\n\n\u2022 CommentRowNumber49.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 28th 2014\n\u2022 (edited Nov 28th 2014)\n\nIt seems we\u2019d have the freedom to make a definition either way, after all this is to formalize something that Hegel said, and depending on how we feel about that and depending on which mathematics we would like to see developed, we may feel that different definitions are appropriate. I am just thinking that in any case there should be some kind of justification. The order relation via subcollections of modal types seems well motivated, but then switching to a different relation along the way seems to call for a reason.\n\nI\u2019ll be happy with whatever definition leads to something interesting. In any case I am presently short of examples of subtoposes which would be in relation in one of the senses under consideration, but not in the other. Maybe we should try to get hold of some examples for such a phenomenon to get a feeling for which kind of subtlety the definition should try to take care of.\n\n\u2022 CommentRowNumber50.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeDec 9th 2014\n\u2022 (edited Dec 9th 2014)\n\nThis is long overdue: I have started at differential cohesion \u2013 relation to infinitesimal cohesion to add some first notes on how differential cohesion\n\n$\\array{ \\Re &\\dashv& \u0283_{inf} &\\dashv& \\flat_{inf} \\\\ && \\vee && \\vee \\\\ && \u0283 &\\dashv& \\flat &\\dashv& \\sharp }$\n\ninduces \u201crelative\u201d shape and flat $\u0283^{rel} \\dashv \\flat^{rel}$ \u2013 such that when this extends to a level\n\n$\\array{ \\flat^{rel} &\\dashv& \\sharp^{rel} \\\\ \\vee && \\vee \\\\ \\flat &\\dashv& \\sharp \\\\ \\vee && \\vee \\\\ \\emptyset &\\dashv& \\ast }$\n\nthen this level exhibits infinitesimal cohesion.\n\nThis is the case in particular for the model of formal smooth \u221e-groupoids and all its variants (formal complex-analytic $\\infty$-groupoids, etc.).\n\nBut I think in all these cases $(\\flat^{rel} \\dashv \\sharp^{rel})$ does not provide Aufhebung for $(\\flat \\dashv \\sharp)$.\n\nThis is because: for $X$ being $\\flat$-modal hence being a discrete object, then maps $U \\to \\sharp^{rel} X$ out of any object $U$ in the site, which are equivalently maps $\\flat^{rel}U \\to X$, are maps out of the disjoint union of all formal disks in $U$ into $X$. These are again representable (we are over the site of formal smooth manifolds) and so these maps are equivalent to $X(\\flat^{rel}U)$. But $X$ is discrete and hence constant as a sheaf on the Cahiers-site, and so these are equivalent to $X(\\flat U)$ which in turns is equivalent to maps $\\flat U \\to X$ and hence to maps $U \\to \\sharp X$. So by Yoneda we conclude that $\\sharp^{rel} X \\simeq \\sharp X$ in this case, but this is in general not equivalent to $X$.\n\n\u2022 CommentRowNumber51.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeDec 9th 2014\n\u2022 (edited Dec 9th 2014)\n\nSo, do you mean that $\\flat\\dashv\\sharp$ lacks Aufhebung ? If this is the case, couldn\u2019t $\\flat^{rel}\\dashv\\sharp^{rel}$ be regarded as a sort of homotopy approximation to the Aufhebung i.e. is construction of $\\flat^{rel}\\dashv\\sharp^{rel}$ from $\\flat\\dashv\\sharp$ sufficiently canonical !?\n\n\u2022 CommentRowNumber52.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeDec 9th 2014\n\u2022 (edited Dec 9th 2014)\n\nI haven\u2019t really thought about under which conditions $(\\flat \\dashv \\sharp)$ has Aufhebung, all I meant to say here is that there is naturally this level $(\\flat^{rel} \\dashv \\sharp^{rel})$ sitting above it, but that in the standard model this level, at least, is not even a resolution of $(\\flat \\dashv \\sharp)$. There might be others that are, though.\n\nAnd regarding this being canonical: the claim is that if differential cohesion is given, then $(\\int^{rel} \\dashv \\flat^{rel})$ is canonically given. Think of it this way: differential cohesion is not really a level above cohesion, because of the \u201ccarrying\u201d of the adjoints to the left. But it canonically induces $\\flat^{rel}$ and so as a soon as that happens to have an adjoint $\\sharp^{rel}$, then thereby it induces something that is a level over cohesion.\n\nSo maybe it\u2019s good to think of this as some kind of \u201ccarrying back to the right\u201d-operation, if you wish.\n\n\u2022 CommentRowNumber53.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeDec 10th 2014\n\u2022 (edited Dec 10th 2014)\n\nUnfortunately, I am not sufficiently familiar with your example to fully understand the details but I asked because I have the understanding that $\\flat^{rel}\\dashv\\sharp^{rel}$ is actually a quality type hence its own Aufhebung !? So my idea is that in some sense it comes reasonably closest to provide Aufhebung for a level which otherwise lacks Aufhebung.\n\nIn order for this to make sense, one would probably like to demand that $\\flat^{rel}\\dashv\\sharp^{rel}$ is the smallest quality type that subsumes $\\flat\\dashv\\sharp$.\n\n\u2022 CommentRowNumber54.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeDec 10th 2014\n\u2022 (edited Dec 10th 2014)\n\nYes, right, in the given model the $(\\flat^{rel} \\dashv \\sharp^{rel})$-level exhibits what \u201cwe\u201d here had decided to call \u201cinfinitesimal cohesion\u201d, which is essentially another word for what Lawvere had called a \u201cquality type\u201d.\n\nAnd yes, I\u2019d agree that it would make much sense to regard $(\\flat^{rel} \\dashv \\sharp^{rel})$ as being the \u201cnext\u201d level after $(\\flat \\dashv \\sharp)$. After all, the sequence of inclusions of levels\n\n$\\array{ \\flat^{rel} &\\dashv& \\sharp^{rel} \\\\ \\vee && \\vee \\\\ \\flat &\\dashv & \\sharp \\\\ \\vee && \\vee \\\\ \\emptyset && \\ast }$\n\nreads in words \u201ca) the single point, b) collections of points, c) collections of points with infinitesimal thickening\u201d.\n\nAnd it seems clear in the model (though I\u2019d have to think about how to prove it) that $\\flat^{rel} \\dashv \\sharp^{rel}$ (when given by first-order infinitesimals) should be the smallest nontrivial level above flat $\\flat \\dashv \\sharp$.\n\n\u2022 CommentRowNumber55.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeDec 10th 2014\n\u2022 (edited Dec 10th 2014)\n\nAh, I should be saying this more properly (and this maybe highlights a subtlety in language that we may have not properly taken account of somewhere else in the discussion):\n\nin the topos over the site of formal smooth manifolds, the sub-topos of $\\flat^{rel}$-modal types is \u201cinfinitesimally cohesive\u201d in that restricted to it the map $\\flat \\to \\int$ is an equivalence.\n\n\u2022 CommentRowNumber56.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeJan 5th 2015\n\ncoming back to #50:\n\non the other hand, of course $\\flat_{inf}$ does provide Aufhebung of cohesion in the sense that $\\flat_{inf} \\int \\simeq \\int$.\n\nOf course this follows trivially here, since we are one step to the left and both of $\\int$ and $\\flat$ correspond to the same subcategory.\n\n\u2022 CommentRowNumber57.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeJan 6th 2015\n\u2022 (edited Jan 6th 2015)\n\ncoming back to #16:\n\nI used to think and say that in the axioms of cohesion the extra exactness condtions on the shape modality seem to break a little the ultra-elegant nicety of the rest of the axioms. There is an adjoint triple of (co-)monads, fine\u2026 and in addtition the leftmost preserves the terminal object \u2013 what kind of axiomatics is that?!\n\nBut Aufhebung now shows the pattern: that extra condition on the shape modality\n\n$\u0283 {}_\\ast \\simeq {}_\\ast$\n\nis just a dual to the \u201cAufhebung of becoming\u201d\n\n$\\sharp \\emptyset \\simeq \\emptyset\\,.$\n\nMaybe a co-Aufhebung, or something.\n\n\u2022 CommentRowNumber58.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeJan 7th 2015\n\u2022 (edited Jan 7th 2015)\n\nThat makes me want to experiment with re-thinking about a possibly neater way of defining differentially cohesive toposes.\n\nSomething like this:\n\nA differential cohesive topos is (\u2026of course\u2026) a topos $\\mathbf{H}$ equipped with two idempotent monads $\\sharp,\\Re : \\mathbf{H} \\to\\mathbf{H}$ such that there are adjoints $\\int \\dashv \\flat \\dashv \\sharp$ and $\\Re \\dashv \u0283_{inf} \\dashv \\flat_{inf}$ (\u2026but now:) and such that\n\n1. (clear:) $\u0283 {}_{\\ast}\\simeq {}_{\\ast}$ and $\\sharp \\emptyset \\simeq \\emptyset$\n\n2. (maybe:) $\\flat_{inf} \\Pi \\simeq \\Pi$ and $\\Re \\flat \\simeq \\flat$.\n\nI need to go through what I have to see what the minimum needed here is. I certainly need $\\flat_{inf} \\flat \\simeq \\flat$ for the relative infinitesimal cohesion to come out right. Also $\\Re \\ast \\simeq \\ast$, which would follow from the above.\n\nIn any case, I feel now one should think of these axioms as describing a picture of the following form (the Proce\u00df)\n\n$\\array{ &\\stackrel{}{}&& id &\\stackrel{}{\\dashv}& id \\\\ &\\stackrel{}{}&& \\vee && \\vee \\\\ && & \\Re &\\dashv & \u0283_{inf} & \\\\ &&& \\bot && \\bot \\\\ &&& \u0283_{inf} &\\dashv& \\flat_{inf} \\\\ &&& \\vee && \\vee \\\\ &&& \u0283 &\\dashv& \\flat & \\\\ &&& \\bot && \\bot \\\\ &&& \\flat &\\dashv& \\sharp & \\\\ &&& \\vee && \\vee \\\\ &&& \\emptyset &\\dashv& \\ast & \\\\ }$\n\nand the question is what an elegant minimal condition is to encode the $\\vee$-s.\n\n\u2022 CommentRowNumber59.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeApr 22nd 2015\n\u2022 (edited Apr 22nd 2015)\n\nI wrote out a more detailed proof of the statement here that the bosonic modality $\\rightsquigarrow$ preserves local diffeomorphisms.\n\nIt seems the proof needs not just Aufhebung in that\n\n$\\rightsquigarrow \\Im \\simeq \\Im$\n\nbut needs also that this is compatible with the $\\Im$-unit in that $\\rightsquigarrow$ sends the $\\Im$-unit of an object $\\stackrel{\\rightsquigarrow}{X}$ to itself, up to equivalent\n\n$\\rightsquigarrow( \\stackrel{\\rightsquigarrow}{X} \\stackrel{\\eta_{\\stackrel{\\rightsquigarrow}{X}}}{\\longrightarrow} \\Im \\stackrel{\\rightsquigarrow}{X} ) \\;\\;\\; \\simeq \\;\\;\\; ( \\stackrel{\\rightsquigarrow}{X} \\stackrel{\\eta_{\\stackrel{\\rightsquigarrow}{X}}}{\\longrightarrow} \\Im \\stackrel{\\rightsquigarrow}{X} )$\n\nThis is true in the model of super formal smooth $\\infty$-stacks, so I am just adding this condition now to the axioms. But it makes me wonder if one should add this generally to the concept of Aufhebung, or, better, if I am missing something and this condition actually follows from the weaker one.\n\n\u2022 CommentRowNumber60.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeJul 18th 2015\n\u2022 (edited Jul 18th 2015)\n\nadded to the discussion here of Aufhebung $\\sharp \\empty \\simeq \\empty$ over cohesive sites pointer to lemma 4.1 in\n\n\u2022 William Lawvere, Mat\u00edas Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)\n\nwhich obverseves this more generally when pieces-have-points.\n\n\u2022 CommentRowNumber61.\n\u2022 CommentAuthorUrs\n\u2022 CommentTimeNov 12th 2015\n\u2022 (edited Nov 12th 2015)\n\nAt MPI Bonn this Aufhebungs-announcement is flying around (full pdf by In Situ Art Society).\n\nI have take the liberty to add it to the entry Aufhebung.\n\n\u2022 CommentRowNumber62.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeNov 12th 2015\n\nThe quote from Hegel is 113 from here:\n\nCancelling, superseding, brings out and lays bare its true twofold meaning which we found contained in the negative: to supersede (aufheben) is at once to negate and to preserve.\n\n\u2022 CommentRowNumber63.\n\u2022 CommentAuthorMatt Earnshaw\n\u2022 CommentTimeJul 4th 2016\n\nthe functors L and R must actually correspond to inclusions of disjoint subcategories\n\nI take it this is not necessary in general, and is only true here since the composites $T \\circ L, T \\circ R$ are equal to the identity. I think that in general, when the composites are merely isomorphic to the identity, the subcategories have intersection given by the equalizer of the subcategory inclusions. Is this reasoning sound?\n\n\u2022 CommentRowNumber64.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeJul 5th 2016\n\nFrom a quick glance at the section you link, to the disjointness property is meant only for the particular example. In the general case, the two subcategories are usually far from disjoint, in fact, one can think of the process of Aufhebung as a gradual level-to-level augmentation of the objects in the intersection, that contains the \u2019true thoughts\u2019 where content (left inclusion) coincides with notion (right inclusion), starting from $0\\cap 1=\\emptyset$ up to $id_\\mathcal{E}\\cap id_\\mathcal{E}=\\mathcal{E}$.\n\n1. Disclaimer: This is my first post here. I'm a philosopher by training, but I'm trying to teach myself enough category theory in order to eventually participate in the project in a meaningful way.\n\nI wanted to offer a reason to reconsider the following quote from the entry:\n\n\"However critical these idealist systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself.\"\n\nI can't speak for Fichte, but there's some good evidence that this is not the case for Kant. Dorothea Achourioti and Michiel van Lambalgen have an excellent paper titled 'A Formalization of Kant's Transcendental Logic' (http:\/\/philpapers.org\/rec\/ACHAFO) that makes the case that Kant's logic is actually geometric logic. The paper is aimed at philosophers, and so goes out of its way to not use category theory (it uses inverse limits of a system of inverse sets), but I have confirmed with van Lambalgen that this was a very deliberate de-categorification for philosophical consumption. I actually think that the semantics they provide is probably way too simple, but the exegetical case for interpreting Kant's logic as geometric is very good in my view, and the connection to Grothendieck topoi provides various interesting connections between Kant's transcendental psychology and mathematics\/compsci (e.g., the idea that object synthesis is about tracking local invariants in varying heterogeneous data (intuition) attached to\/organised by some base topology (forms thereof)).\n\nIf you want a potted example, Kant's distinction between negative judgments and infinite judgements makes perfect sense from the perspective of Steve Vickers's ideas about geometric type theory. For Kant, the crucial difference between the two (corresponding to propositional vs. predicate negation) is that the latter has existential import (relation to an object) and the former does not. Vickers's idea is that the infinitary disjunctions of geometric logic can be re-interpreted using typed existential quantification, and this is essentially what provides the existential import\/objective validity of the infinite judgment from the Kantian perspective. Using a standard and overly simplistic example, judging that an extended object is non-blue excludes a determinate range of possibilities for that object, because the type of extended objects includes a colour attribute with a strictly delimited but potentially infinite range of possible variations. There's more that could be said here, but I think this indicates that Kant is more interesting for the nLab project than the above quote suggests.\n\nI'll close with one more observation about the history of logic. I'm no expert, but the history of logic between Leibniz and Boole is really not very well articulated, and it seems that there are a lot of lines of influence that really aren't properly understood. It's all too easy to represent it as a sort of logical dark age where nothing took place. I honestly don't know how well Hegel would have understood Kant's perspective on logic, given that much of what we now understand comes from the lectures he gave on the topic, which wouldn't have been widely available at the time. However, it seems that Hegel and Schelling were significantly influenced by Ploucquet, from whom they seem to have contracted the idea of the reversibility of subject and predicate. Ploucquet is one of the significant figures recognised in the tradition post-Leibniz, but his work doesn't seem to be widely studied (it's all in latin, with no English translation from what I can gather) either on its own terms or as an influence on Hegel's logical views. It's worth avoiding 'logical dark age' rhetoric, whether it is between Aristotle and Frege or Leibniz and Hegel, as there are hidden lines of research and influence that were still in the process of uncovering and reconstructing.\n\u2022 CommentRowNumber66.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeJul 6th 2016\n\nWelcome Peter, from another philosopher.\n\nI\u2019m very interested in your \u201cpotted example\u201d. The reading group I belong to will soon be reading Paul Redding on the lost subtleties of negation possible in term logic in his \u2019Analytic Philosophy and the Return of Hegelian Thought\u2019.\n\nHave you written on the Kant-Vickers connection?\n\n2. Hi David, thanks for the welcome. I still feel somewhat out of my depth here, even if I've been browsing nLab for at least a year at this point. I considered submitting something for your HoTT and philosophy workshop, but nervousness and the need to move to South Africa on short notice mitigated against it.\n\nI haven't written anything up on the Kant-Vickers connection yet. It's part of a cache of insights I've been slowly tripping over since discovering the A&L paper on transcendental logic and getting serious about understanding sheaves and Grothendieck topoi. Strangely, I got into all this by trying to figure out what was so unsatisfactory about Brandom's formal incompatibility semantics, which has some Hegelian philosophical inspiration, but ends up being horribly gerrymandered into classical rubbish. One of the inconsistencies between Brandom's philosophical inspirations and his formalism (as pointed out to me by Ken Westphal) is precisely that it ignores Kant's distinction between negative and infinite judgments, collapsing everything back into (classical) propositional negation, and thereby being completely unable to account for the sorts of concrete incompatibilities between predicates he starts from (e.g., between blue and the various predicates - green, red, yellow, etc. - that fall under non-blue). There are more complaints I could make on this front, as learning what's wrong in Brandom's project has been quite enlightening, but I'll stop there.\n\nThe overarching project that these ideas belong to is the development of what I'm calling 'computational Kantianism', reading Kant's transcendental psychology as essentially already the project of AGI, using contemporary work in logic\/maths\/compsci to make sense of Kant and using Kant to provide some overarching structure connecting this same work. I gave a talk in Dublin recently that went over some of the overarching methodological ideas of this approach, but it's not written up. I'm due to give a seminar at a Summer school in NYC directly on computational Kantianism later this month, but I'm still working from notes, trying to condense things down into something tractable. I think one can draw a useful line between Kant's insistence on the primacy of judgment as a starting point for transcendental psychology and Harper's idea of computational trinitarianism, and that this (along with certain stories one can tell about subterranean connections between Kant and constructivism in the history of mathematics: i.e., Brouwer-Heyting, topos theory, Curry-Howard, HoTT) opens up the possibilities for reflecting back and forth between Kant and contemporary work I'm proposing. The really novel thing I think can be imported back from Kant is his conception of the relationship between mathematical and empirical judgment\/cognition, which I think can be roughly understood through the duality between intuitionistic and co-intuitionistic logic (judgment) and computational data and co-data (cognition). I think this line of thinking inevitably leads you from Kant to Hegel, as the relationship between imagination and understanding needs to be supplemented with that between understanding and reason, but it's nice to start with Kant's emphasis on our (computational) finitude and build up to Hegel from there. It also has the advantage of suggesting how to extend computational trinitarianism beyond HoTT, insofar as one can project something like a co-intuitionistic dual of HoTT for empirical cognition. There's a few more things I could say about this, but I'll stop myself before I get further out of my depth!\n\nIs this reading group online? If so, I'd very be interested in tagging along. One of the things reading the work being done here has taught me is how much was lost in the transition from the Aristotelian to the Fregean logical paradigm.\n\u2022 CommentRowNumber68.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeJul 6th 2016\n\nWell you\u2019d have been very welcome to the workshop anyway even if just to attend.\n\nInteresting you mention Brandom. I jotted down a note which sounds like it may be in the same direction as your criticism. (By the way, I overlapped here with Ken Westphal for a couple of years.)\n\nWhen you have something to read on what you describe in the 3rd paragraph, I\u2019d be very interested.\n\nUnfortunately, the reading group is just a bunch of us in a room thrashing things out.\n\n3. I'll pass on to you anything that gets written up, or perhaps recordings of the talk if they appear first.\n\nThis is probably a silly question, but would the right etiquette be to edit your Brandom note if I wanted to respond to your thoughts on the topic?\n\u2022 CommentRowNumber70.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeJul 6th 2016\n\u2022 (edited Jul 6th 2016)\n\n@#65: Peter, concerning the passage you criticize though I would admit the sin of \u2019rhetoric\u2019 I would deny the charge of \u2019dark age rhetoric\u2019. The intention there is to provoke the reader with the idea that the \u2019logical lightweight\u2019 Hegel outdoes Kant when it comes to having a critical attitude to traditional logic (note the implied suggestion to view Hegel as an expansion of the Kantian project to logic) and more generally that the postKantian philosophers of the 1790 were quick to dismiss practically all preceding \u2019dogmatic\u2019 metaphysics but often took the traditional laws of formal reasoning for granted. I am probably willing now to exempt at least some of the postKantians from this charge since some of them felt indeed that the critical philosophy demanded a revision of traditional logic e.g. Salomon Maimon published a \u2019Neue Theorie des Denkens\u2019 in 1794, Jacob Siegismund Beck, a mathematician from Kant\u2019s inner circle published a \u2019Lehrbuch der Logik\u2019 in 1820 introducing transcendental concepts into traditional logic, and Fichte in 1808 lectured on \u2019transcendental logic\u2019 producing a large posthumously published text. The point is that Kant did not feel this need, the Jaesche-Logik contains the famous quote that general and pure logic is dull and short and basically a closed chapter since antiquity (or something like this), a quote that made it into the 1928 textbook of Hilbert and Ackermann who obviously did not think that chapter quite as closed neither did Leibniz before them.\n\nThis does not mean that the Jaesche-Logik is unimportant for the philosophy of logic nor that Kant\u2019s transcendental logic cannot not fruitfully confronted with geometric logic, though calling the later \u2019Kant\u2019s logic\u2019 runs into the problem that Kant admitted traditional logic as a valid form of reasoning regardless of the objective content of the concepts employed i.e. to the extent that Kant \u2019had\u2019 a logic traditional logic is a better candidate for it, in my view.\n\nThat Kant\u2019s reasoning is inherently constructive is due to his attempt to model philosophy on the reasoning with constructions in Euclid\u2019s geometry and the later is also an albeit remote source of geometric logic. Anyway, I am the last person to belittle Kant who is in fact one of the brightest stars on my philosophical firmament. In the later passages of the nLab article the continuity between Kant and the postkantian systems and Hegel is stressed. I generally find it useful to view thinkers like Kant, Fichte, Schelling and Hegel to be involved in a common project of transcendental philosophy which in my view is highly relevant to contemporary philosophy or cognitive science and deserves to be formalized by methods of modern mathematics.\n\nConcerning Ploucquet, there is a German-Latin edition of his Logic by Michael Franz available as well as an article by Redding exploring the connection between Hegel and Ploucquet called THE ROLE OF LOGIC \u201cCOMMONLY SO CALLED\u201d IN HEGEL\u2019S SCIENCE OF LOGIC presumably available from his homepage as a preprint. In the context of cognitive underpinning for sheaf theory the link to the Petitot paper at Aufhebung might be interesting as well.\n\nIn any case, feel free to edit or expand Aufhebung when you cannot stomach certain passages. Additional insights or views are always appreciated and generally encouraged by the nLab!\n\n\u2022 CommentRowNumber71.\n\u2022 CommentAuthorDavid_Corfield\n\u2022 CommentTimeJul 7th 2016\n\n@#69 Peter, regarding editing pages, the general rule is that anything on nLab can be edited, with announcement here if substantial. For others\u2019 private webs, I just correct typos.\n\nAs for my own, where that Brandom note is, I just collect together some sketchy thoughts there. I\u2019d be happy to read your thoughts there, if you could designate them as yours.\n\n\u2022 CommentRowNumber72.\n\u2022 CommentAuthorDavidRoberts\n\u2022 CommentTimeJul 7th 2016\n\n@Peter,\n\nthere is a syntax for query boxes, we don\u2019t use it much these days, with discussion being held here instead, but you could still use it. That would help separate your questions\/comments from what David C wrote.\n\n4. Thanks guys. I'll try to figure out the neatest way of commenting that separates out my comments from David's thoughts.\n\nThomas, thanks for the reference to the Redding paper, it looks great!\n\nAs for Kant\/Hegel, I appreciate that what you're doing is provocation. It is of course incredibly important to get people to consider Hegel as an actual logician, and indeed, thereby to broaden their understanding of what logic is. I simply think it is worth pushing the envelope further back and giving Kant the same treatment, and indeed, thereby allowing us to draw more interesting *logical* lines from Kant to Hegel. Here are a couple further reasons to reconsider Kant's status as a logician, responding to your points:\n\n1. One of van Lambalgen's students, Riccardo Pinosio, wrote an excellent MSc thesis reconstructing Kant's philosophy of mathematics based on the idea that transcendental logic is geometric logic (https:\/\/www.illc.uva.nl\/Research\/Publications\/Reports\/MoL-2012-13.text.pdf). It's really good, and it does an excellent job of dismissing Friedman's claim that Kant's constructivism was motivated by the inability of his logic to formulate \u2200\u2203 judgements. There's more work to be done along these lines, as the role of temporality in Kant's account of construction is still not completely explicated, but I think it's a great start.\n\n\u2022 CommentRowNumber74.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeSep 30th 2019\n\nAdded a reference to the recent article by Marmolejo-Menni on \u201clevel $\\epsilon$\u201d.\n\n\u2022 CommentRowNumber75.\n\u2022 CommentAuthorThomas Holder\n\u2022 CommentTimeDec 8th 2019","date":"2021-09-18 14:06:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 219, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8159046769142151, \"perplexity\": 1569.2503518934227}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056476.66\/warc\/CC-MAIN-20210918123546-20210918153546-00559.warc.gz\"}"} | null | null |
I grew up as a country boy and literally lived right next to a junk yard. My parents never liked it, but my brother and I were in Heaven. We had the most elaborate dirt bike trail imaginable.
Country Time was a bit slower and we had our own phrases for everything. We didn't wash clothes, we warshed them. The word color we pronounced as keller.
So you'll have to forgive me going back to olde tyme phrases in my excitement. It was recently announced that TeddyCon 2017 is now a complete and entire hotel takeover ABDL convention.
The main convention space, the hallways, the pool, the courtyard, the lobby, the breakfast area, the bar … the entire building will now be included in TeddyCon 2017. There will be no vanilla folks in the building for the convention. And this means you don't have to cover your littleness up when walking through the halls or anywhere else in the building.
It opens up so many other possibilities that hadn't been possible since the days of those Vermont Ski Lodge Conventions.
And I am losing my mind with excitement.
And the entire hotel building is ours.
This entry was posted in ZorroDaddy's Blog and tagged abdl, ABDL Convention, activities, crafts, medfet, Teddy Clinic, TeddyCon, TeddyCon 2017, TeddyCon hotel takeover. Bookmark the permalink. | {
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The stature of Mastercard's Global Diversity and Inclusion Council (GDIC) reflects the value the company attaches to diversity. It acts as a board of directors for our Global Diversity Office, providing direction to ensure our strategy is embedded throughout the organisation.
Chaired by our Chief Executive Officer, the GDIC has members drawn from all of the company's business regions: North America; Latin America/Caribbean; Europe; Asia/Pacific; and Middle East and Africa. It meets six to eight times a year to evaluate different programmes, partnerships and other proposals that are presented as potential means of enhancing shareholder value.
We leverage the unique strengths, views, and experiences of our employees through our support of Business Resource Groups (BRGs). These self-governed groups are comprised of individuals who come together based on similar interests or experiences, such as gender or ethnicity. BRG members help us to identify business programmes that address the needs of diverse consumers by providing feedback on new ideas and initiatives, partnering with specific organisations, and reaching out to their communities.
At Mastercard, we want to create an environment where the best people choose to be. We provide opportunities for our people to do purpose driven work that impacts customers, communities and their co-workers on a global scale.
Our industry expertise is enhanced by the diverse insights and perspectives of our global workforce, which is at the core of our inclusion and diversity strategy. We are dedicated to cultivating an environment where all of our employees feel valued, respected, and have the opportunity to reach their greatest potential regardless of their difference. To meet those expectations and to sustain our growth and performance, we must always challenge ourselves to provide a workplace where the best individuals can thrive.
In particular gender diversity continues to be an area of focus. As of December 31, 2017 nearly 40 percent of our global workforce was female.
While there is more work to be done, we are making progress in the percentage of women holding senior positions at our company.
While we recognise the challenges we face in improving our gender mix, we are committed to the principle of equal pay for equal work.
To support our commitment, we have a framework in place for annually examining pay practices. All roles in our organisation are reviewed and benchmarked to the external market on an annual basis. We also assess compensation decisions for potential pay disparities by gender, among other things. If disparities are found and not explained in an acceptable manner, appropriate responsive action is taken. Furthermore, we offer employees multiple channels to raise pay disparity concerns, such as the employee's manager, our Ethics Helpline, our Employee Relations team or the Law Department.
This year, to further support and enhance our process, we have also retained a third party to validate our assessments, and this will become a part of our annual process.
Following the conclusion of our assessment and validation process, globally, women at our company earn $0.991 to every $1.000 earned by men for equal performance at the same level.
As part of this year's compensation cycle, we are taking appropriate actions to help close the gap for women.
Gender equality is the foundational core of our commitment to building an inclusive, high-performing culture at the company, so we remain dedicated to maintaining practices designed to ensure there is equal pay for equal performance at the same levels.
Pay equity is just one component of our efforts to create a high-performing, diverse organisation. We are also actively engaged in initiatives that will improve our gender diversity at all levels of the company and will continue to look for new opportunities for women to build their careers across all parts of the business.
One of our objectives is to recruit from diverse candidate slates, and as a general matter, hiring managers start from a diverse slate of candidates. Last year 83% of slates globally had at least one female candidate. As a result, in 2017 40% of our global hires and 55% of our university hires were women.
In terms of career development we are focused on creating opportunities for our high potential women and expanding their skills and experiences through lateral and upward job assignments. We have created a global female leadership development program to identify our next generation of female talent and to ensure these women have the necessary skills to take on broader roles within the organisation – whether through intensive workshops, career moves and other means. We also have formal, targeted discussions to discuss any development and succession planning gaps by gender at our senior management levels.
Additionally, we are involved in a number of other external partnerships and programs designed to facilitate gender diversity. For example, our Board Chairman and other Mastercard management are members of the 30% Club and 25% of our board of directors are female.
Women in Technology: cultivates the pipeline of girls and women entering the STEM/technology fields. Finds women to join Mastercard workforce. Empowers women within their technology careers.
Girls4Tech™ program: our award-winning education program developed in conjunction with top engineers and technologists at Mastercard to teach the foundations of STEM principles to girls aged 10-13 around the world. We are committed to reaching 200,000 girls by 2020.
LaunchCode: Providing women with opportunities for entry into tech. In addition to being a talent partner for hiring, Mastercard supports LaunchCode's CoderGirl program, a weekly meet-up for aspiring female coders to receive mentoring from seasoned pros.
Our commitment to gender equality is ongoing. We continue to listen to our employees and to take actions aligned with our commitment. We are focused on growing our diverse organisation to reflect all the communities that we serve as a business. All of our employees deserve to feel valued and respected, and empowered to reach their greatest potential. We succeed as a company when we bring together our diverse workforce to innovate and develop solutions for our customers and the communities we support.
In April 2017, the UK Government introduced a requirement that both public and large private sector employers must publish an annual snapshot of what they pay their male and female employees. This has been designed to shine a spotlight on the issue and, ultimately, over time, improve gender pay disparity in the UK.
At Mastercard we aspire to ensure that men and women participate equally in all levels of our company, with the same access to compensation and career development opportunities. To achieve this, we continue to be committed to our existing initiatives, as well as to developing new approaches to improve how we recruit, retain and develop women.
Read both Mastercard UK and VocaLink's reports for 2017.
Mastercard CEO Ajay Banga shares his personal stories of equality and diversity with NYU Stern Business School graduates.
Get a closer look at how employee diversity is front and center at Mastercard. | {
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Written by <span itemprop="author" itemscope itemtype="http://schema.org/Person"><span itemprop="name">Henry Golding</span></span> · Published Oct 7, 2015
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<p><img src="/images/featured/featured-eleven-books-you-need-to-read-before-youre-21.jpg" alt="11 books you need to read before you’re 21" /></p>
<h2 id="the-outsider-by-albert-camus"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9004424?QRY=CTIBIB%3C%20IRN(8800)&QRYTEXT=The%20outsider"><cite>The outsider</cite> by Albert Camus</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9004424?QRY=CTIBIB%3C%20IRN(8800)&QRYTEXT=The%20outsider"><img src="/images/article/the-outsider.jpg" alt="The outsider by Albert Camus" /></a></p>
<p>The brutal debut novel of Albert Camus, philosopher of the absurd, tells the story of Mersault, who apathetically kills a man he recognises on a beach. His lack of motivation, remorse or hope of redemption confuses and disturbs the men who sentence him to death.</p>
<h2 id="the-sailor-who-fell-from-grace-with-the-sea-by-yukio-mishima"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9010536?QRY=CTIBIB%3C%20IRN(539104)&QRYTEXT=The%20sailor%20who%20fell%20from%20grace%20with%20the%20sea"><cite>The sailor who fell from grace with the sea</cite> by Yukio Mishima</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9010536?QRY=CTIBIB%3C%20IRN(539104)&QRYTEXT=The%20sailor%20who%20fell%20from%20grace%20with%20the%20sea"><img src="/images/article/the-sailor-who-fell-from-grace.jpg" alt="The sailor who fell from grace with the sea by Yukio Mishima" /></a></p>
<p>Noboru and his friends believe the adult world is hypocritical, illusory and sentimental. They call their savage ideology 'objectivism' and spend their spare time dissecting stray cats. Too bad that Noboru's step-dad should appear on the scene just when they start to look for bigger prey.</p>
<h2 id="a-season-in-hell-in-rimbaud-complete-by-arthur-rimbaud"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9015459?QRY=CTIBIB%3C%20IRN(640114)&QRYTEXT=Rimbaud%20complete"><cite>A season in hell (in Rimbaud complete)</cite> by Arthur Rimbaud</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9015459?QRY=CTIBIB%3C%20IRN(640114)&QRYTEXT=Rimbaud%20complete"><img src="/images/article/rimbaud-complete.jpg" alt="A season in hell (in Rimbaud complete) by Arthur Rimbaud" /></a></p>
<p>When Arthur Rimbaud was sixteen, he left home to become a poet, and in doing so became one of the wildest and most uncompromising writers of his age. His greatest work, Illuminations, was written after he had run away to London with his boyfriend Paul Verlaine. Verlaine, a jealous type, shot Rimbaud but failed to kill him. Deciding that poetry was too dangerous, Rimbaud gave up writing at the age of 21 and became a gun smuggler in Africa.</p>
<h2 id="dubliners-by-james-joyce"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9017525?QRY=CTIBIB%3C%20IRN(13293)&QRYTEXT=Dubliners"><cite>Dubliners</cite> by James Joyce</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9017525?QRY=CTIBIB%3C%20IRN(13293)&QRYTEXT=Dubliners"><img src="/images/article/dubliners.jpg" alt="Dubliners by James Joyce" /></a></p>
<p>James Joyce is known for writing some long, strange and difficult novels towards the end of his career, but Dubliners, his first work of fiction, is notable precisely because of how plain its language is. In this collection of fifteen short stories, Joyce describes a whole range of aspects of life in turn-of-the-century Dublin. Its beauty lies in its simplicity. Publishers were shocked with Joyce's warts-and-all depiction of life and insisted he take all the swear words out. He refused, and after being turned down by fifteen publishers, got his book printed after ten years of trying.</p>
<h2 id="down-and-out-in-paris-and-london-by-george-orwell"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9020043?QRY=CTIBIB%3C%20IRN(15334)&QRYTEXT=Down%20and%20out%20in%20Paris%20and%20London"><cite>Down and out in Paris and London</cite> by George Orwell</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9020043?QRY=CTIBIB%3C%20IRN(15334)&QRYTEXT=Down%20and%20out%20in%20Paris%20and%20London"><img src="/images/article/down-and-out-in-paris-and-london.jpg" alt="Down and out in Paris and London by George Orwell" /></a></p>
<p>George Orwell was a well educated, well-off young man teaching English from his flat in Paris when he fell ill and then had all his money stolen. For the next few months he worked washing dishes before falling into absolute poverty when he returned home to England. His account of dosshouses, overnight shelters, simple acts of charity and hunger demonstrates both the reality and the beauty of poverty.</p>
<h2 id="narcissus-and-goldmund-by-hermann-hesse"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9022760?QRY=CTIBIB%3C%20IRN(1961419)&QRYTEXT=Narcissus%20and%20goldmund"><cite>Narcissus and Goldmund</cite> by Hermann Hesse</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9022760?QRY=CTIBIB%3C%20IRN(1961419)&QRYTEXT=Narcissus%20and%20goldmund"><img src="/images/article/narcissus-and-goldmund.jpg" alt="Narcissus and Goldmund by Hermann Hesse" /></a></p>
<p>What are reason and sobriety without the knowledge of intoxication? Narcissus and Goldmund are two students at a monastery in the middle ages. They are the best of friends and absolutely different from each other. Narziss stays within the monastery, serious and pure, and Goldmund 'let himself be led into the night, into the forest, into the blind secret wordless, thoughtless country…' Hesse's description of his long journey around the countryside and back to the monastery is the most beautiful thing written in the twentieth century.</p>
<h2 id="a-separate-peace-by-john-knowles"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9025205?QRY=CTIBIB%3C%20IRN(473053)&QRYTEXT=A%20separate%20peace"><cite>A separate peace</cite> by John Knowles</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9025205?QRY=CTIBIB%3C%20IRN(473053)&QRYTEXT=A%20separate%20peace"><img src="/images/article/a-separate-peace.jpg" alt="A separate peace by John Knowles" /></a></p>
<p>Regarded as a modern classic in the author's home country America, this novel of adolescent idealism, imagination and confusion is sadly neglected east of the Atlantic. A group of boys are made to stay at school over the summer and there they invent sports to amuse themselves. Phineas is good-looking, charming and athletic and his devoted best friend Gene might just be getting a little bit jealous. One day, whilst up a tree, he makes a decision that could ruin his best friend's life forever.</p>
<h2 id="the-picture-of-dorian-gray-by-oscar-wilde"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9028107?QRY=CTIBIB%3C%20IRN(56972)&QRYTEXT=The%20picture%20of%20Dorian%20Gray"><cite>The picture of Dorian Gray</cite> by Oscar Wilde</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9028107?QRY=CTIBIB%3C%20IRN(56972)&QRYTEXT=The%20picture%20of%20Dorian%20Gray"><img src="/images/article/the-picture-of-dorian-gray.jpg" alt="The picture of Dorian Gray by Oscar Wilde" /></a></p>
<p>At the time of its publication, one reviewer claimed that The Picture of Dorian Gray 'will taint every young mind that comes in contact with it'. It is vulgar, unclean, poisonous, discreditable and an immoral sham. Or so they'd have us think. In reality it is an elegant story of a young man who wishes that a portrait of himself will show the effects of all the bad things he does whilst he himself remains young and good-looking. Remarkably, the uncensored version of this book remained unpublished until 2011. But it's still rather tame by modern standards.</p>
<h2 id="catcher-in-the-rye-by-j-d-salinger"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9032313?QRY=CTIBIB%3C%20IRN(564250)&QRYTEXT=The%20catcher%20in%20the%20rye"><cite>Catcher in the Rye</cite> by J D Salinger</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9032313?QRY=CTIBIB%3C%20IRN(564250)&QRYTEXT=The%20catcher%20in%20the%20rye"><img src="/images/article/catcher-in-the-rye.jpg" alt="Catcher in the Rye by J D Salinger" /></a></p>
<p>Sadly spoilt for many readers due to its inclusion on the GCSE English curriculum, Catcher in the Rye is in fact a wonderful book about trying to just be you whilst everyone else is trying too hard to fit in and be liked by everyone. Screw the phonies. Read this for fun.</p>
<h2 id="sons-and-lovers-by-d-h-lawrence"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9034974?QRY=CTIBIB%3C%20IRN(28160)&QRYTEXT=Sons%20and%20lovers"><cite>Sons and lovers</cite> by D H Lawrence</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9034974?QRY=CTIBIB%3C%20IRN(28160)&QRYTEXT=Sons%20and%20lovers"><img src="/images/article/sons-and-lovers.jpg" alt="Sons and lovers by D H Lawrence" /></a></p>
<p>These days, DHL is better known for writing Lady Chatterley's Lover, which was banned for being smutty. His other novels aren't as rude and are much better-written. T his, his third, is an extraordinary story of a talented artistic lad growing up in a mining community who comes into conflict with both his parents, in very different ways.</p>
<h2 id="the-turn-of-the-screw-by-henry-james"><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9049974?QRY=CTIBIB%3C%20IRN(144305)&QRYTEXT=The%20turn%20of%20the%20screw"><cite>The turn of the screw</cite> by Henry James</a></h2>
<p><a href="https://suffolk.spydus.co.uk/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/9049974?QRY=CTIBIB%3C%20IRN(144305)&QRYTEXT=The%20turn%20of%20the%20screw"><img src="/images/article/the-turn-of-the-screw.jpg" alt="The turn of the screw by Henry James" /></a></p>
<p>One of the best ghost stories ever written, yet completely atypical of the genre. No headless men or bumps in the night, just a load of unanswered questions.</p>
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| {
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} | 1,111 |
Shapur Mihran (), known in Armenian sources as Shapuh Mihran (Armenian: Շապուհ Միհրան), was a Sasanian nobleman from the House of Mihran. He served as the marzban of Persian Armenia briefly in 482.
Biography
Shapur belonged to the House of Mihran, one of the Seven Parthian clans; he was the son of a certain Mihran. He was a foster brother of the Sasanian shah Peroz I, who was himself married to a princess from the Mihran family. During the reign of Peroz, the Mihran family enjoyed a high status, and played an important role in Sasanian politics. Shapur, during his youth, was raised in Armenia, which made him, unlike other Sasanian nobles, act more tolerant towards Christianity.
In 475, the Mamikonian princess Shushanik was murdered by her husband Prince Varsken, who was a convert to Zoroastrianism, and related to the Mihran family. The reason for this murder was because she had refused to convert to Zoroastrianism and wanted to stay Christian. Varsken, because of his actions, was in 482 executed by Vakhtang I, king of Iberia. Peroz I sent an army to punish Vakhtang for intervening. However, Vakhtang was joined by the Armenians, and a revolt broke out in Armenia, led by Vahan I Mamikonian.
Peroz I, eager to avenge Varsken, sent his general Shapur Mihran to Iberia. To defend himself, Vakhtang appealed to the Huns and the Armenian nobles, citing solidarity between Christians. After carefully weighing the decision, Vahan Mamikonian agreed to revolt against the Sasanians. He defeated the marzban Adhur Gushnasp, and declared Sahak II Bagratuni as the new marzban. He also kept defeating several Sasanian counter-attacks.
In 482, Shapur Mihran began to become a big threat to the security of Iberia, which made Vakhtang request Armenian help. Vahan and Sahak shortly arrived to Iberia at the head of a big army, but were defeated in Akesga, where Sahak was killed. Vahan fled with the remnants of the Armenian army into the mountains, where he led guerrilla actions against the Sasanians, while Shapur managed to regain control of Armenia. However, Shapur was shortly ordered to return to the Sasanian capital of Ctesiphon. Vahan quickly used the opportunity to regain control of Armenia.
In the spring of 484, however, Shapur Mihran returned as the head of a new army and forced Vahan to flee to refuge near the Byzantine frontier, at Tao and Taron. During the same period, the Sasanian noble Zarmihr Karen from the Karenid family, was also successful in a campaign against the Armenians, and managed to capture several of them, including nobles from the Kamsarakan family. Zarmihr shortly delivered the Armenian captives to Shapur, who delivered them to Izad Gushnasp, and promised the Armenian captives to make Peroz spare them.
During the same period, several of Shapur's relatives, including his father Mihran, were summoned by Peroz to aid him in his campaigns against in Central Asia against the Hephthalites. However, the campaign ended disastrously, and all of the Sasanian army, including Peroz and Mihran, were exterminated.
After hearing about the death of Peroz I, Shapur left Caucasus and returned to Ctesiphon, in order to protect the Sasanian Empire from the Hephthalites and to elect a new king. Balash, the brother of Peroz I, was crowned as the new king of the Sasanian Empire. However, it was in reality the father of Zarmihr Karen, Sukhra who exercised real power over the Sasanian Empire. After this event, Shapur is no longer mentioned in any sources.
References
Sources
Further reading
5th-century births
Year of death unknown
Sasanian governors of Armenia
5th-century Iranian people
House of Mihran
Generals of Peroz I | {
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Q: Не могу расшифровать ключ Вот класс шифровки и расшифровки ключа
public class SecurityClass {
private static Cipher serCretss(String mode) throws NoSuchAlgorithmException, NoSuchPaddingException, InvalidKeyException {
SecretKeySpec sks = null;
SecureRandom sr = SecureRandom.getInstance("SHA1PRNG");
sr.setSeed("any data used as random seed".getBytes());
KeyGenerator kg = KeyGenerator.getInstance("AES");
kg.init(128, sr);
sks = new SecretKeySpec((kg.generateKey()).getEncoded(), "AES");
Cipher c = Cipher.getInstance("AES");
if (mode.equals("d")){
c.init(Cipher.DECRYPT_MODE, sks);
}else {
c.init(Cipher.ENCRYPT_MODE, sks);
}
return c;
}
public static byte[] doCript(String myText) {
byte[] encodedBytes = null;
try {
encodedBytes = serCretss("e").doFinal(myText.getBytes());
} catch (Exception e) {
Log.e("Crypto", "AES encryption error");
}
return encodedBytes;
}
public static byte[] decodeCript(byte[] convertbyte) {
byte[] decodedBytes = null;
try {
decodedBytes = serCretss("d").doFinal(convertbyte);
} catch (Exception e) {
Log.e("Crypto", "AES decryption error");
}
return decodedBytes;
}
}
так делаю шифровать
SecurityClass.doCript(a)
так делаю расшифровать
SecurityClass.decodeCript(a)
шифрует но при расшифровке дает null
A: При шифровании и дешифровании вы используете разные ключи.
Метод serCretss() каждый раз заново генерирует новый ключ KeyGenerator.init() - который использует рандомный генератор поданный ему на вход.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,715 |
{"url":"http:\/\/openstudy.com\/updates\/510ec0dce4b0d9aa3c47a6ce","text":"## Luigi0210 3 years ago A rectangle is bounded by the x and y axes and the graph of the line y= (-1\/2)x+3. What length and width should the rectangle have so its area is maximum?\n\n1. Luigi0210\n\n|dw:1359921377233:dw|\n\n2. anonymous\n\nbase of rectangle will be $$x$$ and height will be $$-\\frac{1}{2}x+3$$ so area is $A(x)=x(-\\frac{1}{2}x+3)=-\\frac{1}{2}x^2+3x$\\]\n\n3. anonymous\n\nmax will be at the vertex, which is at $$-\\frac{b}{2a}=3$$\n\n4. Elsa213\n\noooo a year later c: Heyo humans :D","date":"2016-05-27 22:19:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6102834939956665, \"perplexity\": 1228.423334085379}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049277091.36\/warc\/CC-MAIN-20160524002117-00006-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.usanewscourt.com\/what-is-3-8-as-a-decimal-form\/","text":"What is 3\/8 as a decimal form?\n\nWhat is 3\/8 as a decimal form?\n\nSolution\n\nIn the fraction 3\/8, 3 is the numerator and 8 is the denominator, the fraction bar implies \u201cdivided by\u201d. So, the fraction 3\/8 also means \u201c3 divided by 8\u201d or \u201c3 \u00f7 8\u201d.\n\nWhen it is calculated, 3\/8 in decimal form is:\n\n3 \u00f7 8 = 0.375\n\nTherefore, the decimal form of 8\/3 is 0.375.\n\nWhat is a decimal?\n\nA decimal is a number with two parts: a whole number and fraction part separated by a decimal point, like 0.6 or 1.1. Decimals are used to measure small amounts of money, such as the cost of an item. They can also be used to measure percentages, such as the percentage of a person\u2019s body weight that is fat.\n\nLike fractions, decimals are an important part of everyday math. You will use decimals often when you need to measure very small amounts of money or much larger amounts of money (such as when you have to pay taxes). Decimals are also used in science and engineering.\n\nEngineers often use decimals when they are designing things like bridges and buildings because decimals allow them to make calculations much faster than if they were using fractions. It might interest you to know that a number written as 1.4 means that it has four tenths (1\/10) in it. This is because a tenth is 1\/10 of something.","date":"2023-02-03 00:35:33","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8439851403236389, \"perplexity\": 619.3073815651842}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500041.2\/warc\/CC-MAIN-20230202232251-20230203022251-00833.warc.gz\"}"} | null | null |
Q: Object creation at runtime using google guice I am building an application which downloads the files from S3 and DynamoDB on behalf of my users from their aws accounts. Each user registers with their AWS account details in my system to start with.
When my application starts I read all my users information and create an S3Client or DyanmoDBClient for them depending on the one they registered with. At runtime i retrieve objects using their registered account details. The code looks something like this.
interface ReaderClient {
public Object read();
}
public class S3Client implements ReaderClient {
getInstance(account);
}
public class DynamoDBClient implements ReaderClient {
getInstance(account);
}
class ReaderClientFactory {
public static ReaderClient getReaderClient(User) {
switch(user.database) {
case S3:
return S3Client.getInstance(user.account);
break;
case Dynamo:
return DynamoDBClient.getInstance(user.account)
break
}
}
}
How do I replace this factory with Google guice? Is this possible at all?
I digged through providers and assisted inject but not able to fit wither of them for this usecase. Any help would be appreciated.
A: public class Main {
public static void main(String[]args){ReaderClientFactory factory = Guice.createInjector(new ReaderClientModule(user).getInstance(ReaderClientFactory.class));
ReaderClient client = factory.create();}}interface ReaderClient {public Object read();}class S3Client implements ReaderClient {
public Object read(){
System.out.println("S3 Client");
return new Object();
}}class DynamoDBClient implements ReaderClient {
public Object read(){
System.out.println("Dynamo DB Client");
return new Object();
}}interface ReaderClientFactory {
public ReaderClient create();}class ReaderClientModule extends AbstractModule{
User user;
@Inject
ReaderClientModule(@Assited User user){
this.user = user;
}
@Override
protected void configure(){
switch(user.database){
case S3:
install(new FactoryModuleBuilder()
.implement(ReaderClient.class, S3Client.class)
.build(ReaderClientFactory.class));
break:
case Dynamo:
install(new FactoryModuleBuilder()
.implement(ReaderClient.class, DynamoDBClient.class)
.build(ReaderClientFactory.class
break;
}} }
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 416 |
The Dordogne department (department no 24) is in the Aquitaine region in the South West of France. The Dordogne area of France covered by this guide runs from Bergerac to Rocamadour in the south, and to Perigueux and Brantôme in the north, including a great deal of interesting and beautiful things to see in between, the Dordogne valley and Vezere valley encompassing the principle attractions. The area has little public transport, so you will need a car (or a bike or walking boots) to make the most of a holiday in the region.
The Perigord Noir is the south-easterly part of the perigord and is named for the dark oak forests that cover much of the landscape.
It is in the Perigord Noir that most of the highlights of the Dordogne area can be found. So many are concentrated into this area that we would recommend splitting your visit in to two.
The first visit could encompass the lower Dordogne valley, the area between Saint Cyprien and Sarlat. Some of the many highlights of this visit will include canoeing on the Dordogne, visiting the most beautiful villages of Domme and Beynac and the magnificent castles along this stretch of the river. This visit could stretch further upstream (towards the east), along the Dordogne Valley from Sarlat to Rocamadour via Souillac. Be sure to see the caves at Gouffre de Padirac. See Perigord Noir.
The second visit would cover the Vezere Valley from Le Bugue up to Montignac. The Vezere valley is home to an astonishing number of prehistoric caves and shelters as well as many charming towns and villages. Some of the many highlights of this visit will include visiting the caves at Lascaux and Font de Gaume and exploring the charming market towns of Le Bugue and Montignac. See Perigord Noir.
The south west Perigord is known as the Perigord Pourpre for the vineyards that are common in this area.
Highlights of the Perigord Pourpre include the lovely town of Bergerac with its historic centre and the beautiful village of Monpazier. The bastide towns of Beaumont and Eymet are also well worth a visit. The Chateau de Monbazillac is at the heart of the Monbazillac wine region, reknowned for its sweet white wine. See Perigord Pourpre.
The Perigord Blanc is found in the centre of the Perigord area and is named for the white colour of the chalk soil.
Here you will find the beautiful town of Perigueux with its beautifull multi-domed cathedral.
The chateau de Neuvic and the Chateau de Montreal are both in this area. See Perigord Blanc.
A visit to the Perigord Blanc could be linked to a visit to the Perigord Vert as there are fewer sights to see in these areas than the southern parts of the Dordogne.
The Perigord Vert is the most northerly part of the Dordogne and is named for the green chestnut and oak forests common to the area.
The stunning riverside village of Brantome is a must-see in this area as is the nearby village and chateau of Bourdeilles and the beautiful village of Saint-Jean-de Cole. Nontron and its famous knives is also worth a visit as is the market town of Riberac and the Dronne valley. See and Perigord Vert.
A trip to the Perigord Vert could easily be combined with a trip to the Perigord Blanc.
The reference to the Dordogne regions above as Perigord is based on their (common) classification using the historical name for the region. You can see these regions marked on the map above - essentially Nontron in the north is the centre of Perigord Vert; Perigueux is at the heart of Perigord Blanc in the centre of the region; Bergerac and the south-west part is known as Perigord Pourpre; and the region around Sarlat is often called Perigord Noir.
On arriving, your first port of call should always be the local tourist office - most towns in the Dordogne have one - for up to date information about events for that period.
All these different regions of the Dordogne area of France, from the Dordogne Valley in the south to the border with the Limousin in the north, are covered by this website. For the next part of your grand tour you will need to visit our Lot-et-Garonne website at southofthedordogne.com, for the many places of interest just to the south of the river and into the northern Lot-et-Garonne department. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,053 |
{"url":"http:\/\/tex.stackexchange.com\/questions\/65707\/is-it-possible-to-include-tikz-picture-from-a-file","text":"# is it possible to include tikz picture from a file\n\nI have a tikz picture in a tex file\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\resizebox{.4\\textwidth}{!}{\\begin{tikzpicture}[\n....\n\n\\end{tikzpicture}}\n\\end{center}\n\\end{figure}\n\n\nThe ... portion is quite long. Is it possible to put the ... portion in a file and load it with code like \\include...?\n\n-\nYou could simply use \\input. \u2013\u00a0 Stefan Kottwitz Aug 3 '12 at 12:12\n\u2013\u00a0 Ahmed Musa Aug 3 '12 at 13:41\n\n\\documentclass{article}\n\\usepackage{filecontents}\n\n\\begin{filecontents*}{tikzcode.tex}\n\\documentclass[tikz]{standalone}\n\\usepackage{tikz}\n\\begin{document}\n\\begin{tikzpicture}\n\\fill[red] (1,2) circle (3);\n\\end{tikzpicture}\n\\end{document}\n\\end{filecontents*}\n\n\\usepackage{tikz}\n\\usepackage{standalone}\n\\begin{document}\n\\input{tikzcode.tex}\n\\end{document}\n\n-\nI almost forgot that I have answered this question. \u2013\u00a0 kiss my armpit Oct 30 '12 at 17:05","date":"2015-05-24 15:25:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9970598220825195, \"perplexity\": 3291.8343306782986}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1432207928019.82\/warc\/CC-MAIN-20150521113208-00316-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
Q: How to remove all the packages that come with gnome? I have recently switched from gnome to unity and have removed gnome with
sudo apt remove ubuntu-gnome-desktop
sudo apt remove gnome-shell
I have also purged config files like below
sudo apt purge ubuntu-gnome-desktop
Removed unwanted dependencies with
sudo apt auto-remove
But I don't know how to delete all the bloatware that comes with gnome.
When I run
sudo apt list gnome-*
It gives me a huge list which is really difficult to remove manually.
How can I remove the apps like settings, calendar music and so on without manually deleting them one by one?
| {
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To Stay and Deliver
Providing humanitarian assistance amid conflict has always been a dangerous and difficult endevour; however, over the last decade aid worker casualties tripled, reaching over 100 deaths per year. From 2005 onwards the largest numbers of violent attacks on humanitarian personnel have been concentrated in a small number of countries representing the most difficult and volatile operating environments. Attacks in some of these settings have also grown more lethal and sophisticated and the number of kidnappings has risen dramatically.
Publication Date: Feb 01, 2011
Author(s) / Contributor(s): Abby Stoddard, Adele Harmer, Jan Egeland
Region/Country: East Asia, Horn of Africa, Latin America, Middle East, Sub-Saharan Africa
Topic(s): Humanitarian Crises
The State of the Humanitarian System | Assessing Performance and Progress (2010)
o The ability to monitor and report on performance is increasingly important for any successful sector. Individuals, organisations or systems cannot improve unless their shortcomings are identified and practical and creative solutions for improvements are put forward. This report aims to provide a system-level mapping and assessment of international humanitarian assistance.
Publication Date: Jan 04, 2010
Author(s) / Contributor(s): Abby Stoddard, Paul Harvey, Adele Harmer, Victoria DiDomenico, Lauren Brander
Topic(s): Crises, Fragile States, Global Governance, Humanitarian Crises
The Aid Worker Security Database
The Aid Worker Security Database (AWSD) records major incidents of violence against aid workers, with incident reports from 1997 through the present. Initiated in 2005, to date the AWSD remains the single most comprehensive global source of this data, providing a much-needed quantitative evidence base for analysis of the changing security environment for civilian aid operations. For more detail on the AWSD click here.
Author(s) / Contributor(s): Abby Stoddard, Adele Harmer, Katherine Haver
Region/Country: Central Asia, East Asia, Horn of Africa, Latin America, Middle East, Sub-Saharan Africa, South Africa, West Africa
Principles of Protection for Migrants, Refugees, and Displaced People During COVID-19
COVID-19 and Public Support for Radical Policies
Do or Die: COVID-19 and Imprisonment in Syria
Kabila's impasse
Bringing Development to Fore of Refugee Response
The Mess Obama Left Behind in Iraq
Should the U.S. have kept Iraq's oil, as Donald Trump argues? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,105 |
ACCEPTED
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Trans. Br. mycol. Soc. 89(1): 117 (1987)
#### Original name
Spororminula Arx & Aa
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,516 |
\section{Introduction}
\label{sec:DM_intro}
\input{inputs/sec1-intro}
\section{Composite dark matter candidates}
\label{sec:DM_candidates}
\input{inputs/sec2-candidates}
\section{Direct detection of composite dark matter}
\label{sec:DM_direct}
\input{inputs/sec3-direct-arxiv}
\section{Collider implications}
\label{sec:DM_other}
\input{inputs/sec4-other}
\section{Outlook}
\label{sec:DM_outlook}
\input{inputs/sec5-outlook}
\section*{Acknowledgements}
\label{sec:acknowledgements}
We thank
Y.~Bai,
T.~Cohen,
Y.~Hochberg,
R.~Lewis,
M.~McCullough,
M.~Pospelov,
E.~Rinaldi,
F.~Sannino, and
T.~Tait
for providing helpful comments, corrections, and suggestions for
improvement on a preliminary version of this review. The authors are
supported in part by the U.~S.~Department of Energy under contract
nos. DE-SC0011640 (GDK) and DE-SC0010005 (ETN). Brookhaven
National Laboratory is supported by the DoE under contract no.~DE-SC0012704.
\subsection{Motivation}
\textbf{Dark Matter Stability}. One of the principle attractions of
composite dark matter candidates is that stability can be an automatic
consequence of the accidental global flavor symmetries of the
underlying theory. Once above the compositeness scale,
the (gauge) symmetries of the dark constituents may
restrict the dimension of the leading operators
$d$ that violate the global flavor symmetries
to be sufficiently high that even with maximal violation by
Planck-suppressed operators (e.g., $\mathcal{O}_d/M_{Pl}^{d-4}$),
the dark matter stability is far longer than the age of the universe.
This is, after all, the reason that proton stability is well understood
in the standard model, despite possible baryon number violation
by Planck-suppressed operators.
\textbf{Naturalness}. Just like QCD, once a non-Abelian theory
confines, a new scale appears through dimensional transmutation,
the dark confinement scale $\Lambda_d$. This scale is technically
natural, allowing for an effective theory description of the
infrared theory -- the dark mesons and baryons -- through operators
suppressed by this scale.
\textbf{Dark Matter Neutrality}. In theories where the
constituents transform under (part of) the standard model,
confinement can lead to color, weak, and charge-neutral
dark hadrons that provide candidates for neutral dark matter.
This typically imposes constraints on the parameters of the
underlying theory.\footnote{This is not unlike analogous constraints
on theories of elementary dark matter candidates, in which the lightest dark parity-odd particle is
required to be neutral under the standard model.}
\textbf{Suppressed interactions}. The effective theory
below the confinement scale can be expressed in terms of
higher dimensional operators involving (pairs of) dark matter
fields with standard model fields, suppressed by powers of
the dark confinement scale. This can provide a beautiful mechanism
to suppress dark matter scattering off nuclei, below the
tight experimental bounds that exist from direct detection experiments.
The coefficients
of the dark moments can in principle be computed from the underlying
ultraviolet theory. This is a major motivation for lattice
simulations that can provide superior estimates of the coefficients
of the dark moments over naive dimensional analysis power counting.
\textbf{Self-interactions}. Strongly-coupled theories naturally
have strong self-interactions among the mesons and baryons.
If the scales are arranged appropriately, these self-interactions
may be responsible for addressing the observed galactic structure
anomalies\cite{Spergel:1999mh,Vogelsberger:2012ku,Rocha:2012jg,Peter:2012jh,Zavala:2012us}.
This has provided
a strong motivation for recent consideration of strongly-coupled
self-interacting dark matter
\cite{Carlson:1992fn,Cline:2013zca,Boddy:2014yra,Hochberg:2014dra,Boddy:2014qxa,Soni:2016gzf}.
\textbf{New observables}. The rich spectrum of dark hadrons
that appear after the dark non-Abelian theory confines
provide a plethora of experimental targets. This includes
novel detection strategies such as inelastic scattering to
excited states \cite{ArkaniHamed:2008qn,Alves:2010dd,Kumar:2011iy},
dark absorption lines \cite{Profumo:2006im,Kribs:2009fy},
effects on the CMB and $N_{\rm eff}$
\cite{Garcia:2015loa,Buen-Abad:2015ova}
and a host of collider phenomenology consequences (to be discussed below). We note that although not focused on dark matter models, spectroscopy of SU$(N)$
gauge theories in the large-$N$ limit has been studied extensively on the lattice \cite{Panero:2012qx,DeGrand:2013nna,Lucini:2014bwa, Cordon:2014sda}.
\subsection{Meson dark matter I: Pion-like}
There are three broad classes of composite dark matter made from
mesons of a confining, strongly-coupled non-Abelian group:
pion-like ($m_q \ll \Lambda_d$), quarkonia-like
($m_q \gg \Lambda_d$), and an intermediate regime
($m_q \sim \Lambda_d$) or mixed regime ($m_{q_1} < \Lambda_d < m_{q_2}$).
Meson stability relies on accidental dark flavor
(or ``species'' \cite{Kilic:2009mi}) symmetries. The dark flavor
symmetries could be continuous or discrete, such as $G$-parity
\cite{Bai:2010qg}.
Several models of pion-like dark matter have been proposed
\cite{Ryttov:2008xe,Hambye:2009fg,Bai:2010qg,Lewis:2011zb,Buckley:2012ky,Frigerio:2012uc,Bhattacharya:2013kma,Hochberg:2014kqa,Hietanen:2014xca,Carmona:2015haa,Hochberg:2015vrg}.
One reason for their popularity is familiarity from QCD,
and specifically, utilizing chiral effective theory techniques
to characterize the mass spectrum and pion interactions.
In the following, we describe only a selection of models that
have been proposed.
Weakly interacting stable pions was proposed in Ref.\cite{Bai:2010qg}.
In this theory, stability is ensured through $G$-parity,
that is a modified charge conjugation operation allowed
when using real representations of the SM gauge group.
Dark fermions transform in vector-like
representations of an $SU(N)_d \times SU(2)_L$,
where the reality of $SU(2)$ representations permits
$G$-parity to be preserved in the Lagrangian. Pions transform
as $\Pi^{(J M)} \stackrel{G}{\rightarrow} (-1)^G \Pi^{(J M)}$,
and thus the lightest $G$-odd pion, $\Pi^{(1 \, 0)}$,
is a dark matter candidate. At the level of the
chiral Lagrangian, $\Pi^{(1 0)}$ does not decay through
the usual axial anomaly (unlike the standard QCD case,
$\pi^0 \rightarrow \gamma\gamma$), due to the vanishing of
the relevant isospin trace.
A different proposal to use weakly interacting pions was
proposed in \cite{Buckley:2012ky}. Dark fermions transform
in vector-like representations of $SU(2)_d \times U(1)_Y$.
The use of $SU(2)$ for the new confining group
(called ``ectocolor'' \cite{Buckley:2012ky}) has the feature that there
are five pseudo-Goldstone bosons -- three are the usual pions,
while the remaining two can be identified as ``baryon-like''
dark matter in the theory. Of course no baryons result once
$SU(2)_d$ confines, however, a global $U(1)_X$ symmetry
can be imposed to ensure the baryon-like pions are
stable with respect to the low energy effective theory.
Once again, chiral Lagrangian techniques can be used to calculate
the leading scattering cross sections and decay rates of
the pions. That the dark matter baryon-like pions are
close in mass to the pions that do not carry a conserved global
$U(1)_X$ number leads to a novel, nontrivial freezeout process
effectively driven by co-annihilation among the light pion
species \cite{Griest:1990kh,Buckley:2012ky}.
This leads to weak scale masses and pion decay
constants that can be probed by collider experiments.
More recently, a strongly-coupled theory with pions as dark matter
was proposed (``SIMP'' for strongly interacting massive particle)
\cite{Hochberg:2014kqa,Hochberg:2015vrg}.
Again, chiral Lagrangian techniques allow the estimation
of the leading pion interactions. Here, the novel feature
that occurs for a wide class of non-Abelian theories
($SU(N_c)$ or $SO(N_c)$ with $N_f \ge 3$; $Sp(N_c)$ with $N_f \ge 2$)
in the pion-like limit is that there is a 5-point interaction arising
from the Wess-Zumino-Witten action enabling a $3 \rightarrow 2$
annihilation process.
Unlike the earlier proposals, all of the dark fermions are
neutral under the SM, and so a new mediator is necessary to
connect the thermal bath of the standard model with this dark sector.
In \cite{Hochberg:2015vrg}, a global $U(1)$ flavor symmetry of the theory
is gauged (and broken explicitly) leading to a massive ``dark photon''
with assumed kinetic mixing with hypercharge. Under suitable conditions,
the $3 \rightarrow 2$ process is active and leads to the freeze out
of dark matter. Intriguingly, the relevant scales of the
new non-Abelian dark sector that lead to the correct dark matter
relic abundance is very similar to QCD, $m_\pi \sim 300$~MeV
with $f_\pi \sim \mathrm{few} \times m_\pi$. Constraints on
the dark photon mass and kinetic mixing that enable this
mechanism were presented in \cite{Hochberg:2015vrg}. Principally,
the pions cannot decay into dark photons. This can be guaranteed
if all the pions transform non-trivially under part of the
unbroken flavor symmetry. And, near degeneracy of the dark quarks
is required, so that at least five different pions participate
in the $3 \rightarrow 2$ process via the WZW term.
\subsection{Meson dark matter II: Quarkonium-like}
In the regime where there is at least one heavy dark fermion
with mass $m_q > \Lambda_d$, heavy quark effective theory
can be applied, and qualitative differences from the
pion-like theories result.
One example of this class of model is ``composite inelastic dark matter''
\cite{Alves:2009nf,Lisanti:2009am,Alves:2010dd}, where dark matter
is a meson made from one light and one heavy quark. In this theory,
the hyperfine interactions split the ground state by a small
energy that can be relevant to dark matter direct detection.
The possibility that dark matter may have its dominant interaction
with nuclei through an inelastic scattering process is
well known \cite{TuckerSmith:2001hy,TuckerSmith:2004jv,Chang:2008gd}.
Strongly-coupled composite theories provide a natural home for
small inelastic transitions, and this can lead to a rich
spectroscopy.
In the regime where all of the dark fermions are heavier than
the confinement scale leads to another class of composite
dark matter generally known as ``quirky
\footnote{The name ``quirky'' came from
a fascinating class of theories that postulate new dark
fermions that transform under part of the standard model, and also
transform under a new non-Abelian group that confines
at a scale far below the mass of the fermions.\cite{Kang:2008ea}}
dark matter'' \cite{Kribs:2009fy}.
The dark fermions were taken to be in a chiral representation
of the electroweak group, using the Higgs mechanism to give them mass.
For the specific theory of SU(2),
it was known \cite{Peskin:1980gc,Preskill:1980mz}
that confinement aligned the vacuum towards an electroweak
preserving minimum, and thus not substantially affecting electroweak
symmetry breaking. In addition, with the bound states containing
exactly two heavy dark fermions, a perturbative non-relativistic
treatment of the composite dark matter mesons is possible.
This allowed an estimate of the excited meson masses,
as well as the coefficients of the effective operators
leading to quirky dark matter scattering with nuclei.
Quirkonium production and decay were considered in
\cite{Harnik:2011mv,Fok:2011yc}.
One drawback to dark matter composed of dark mesons is the
potential difficulty in maintaining the exactness
of the global flavor quantum number that ensures that
the dark matter is (sufficiently) stable. For example,
already at dimension-5 there can be operators that violate
global flavor symmetries
\begin{eqnarray}
\frac{1}{\Lambda} \overline{\Psi} \Psi H^\dagger H \quad , \quad
\frac{1}{\Lambda} \overline{\Psi} \sigma^{\mu\nu} \Psi B_{\mu\nu} \, .
\label{eq:dim-5-ops}
\end{eqnarray}
Here $\overline{\Psi}\Psi$ is a fermion bilinear that transforms
nontrivially under the global flavor symmetry that protects against
meson decay in the effective theory.
Even with $\Lambda = M_{\rm Pl}$, these operators with order one coefficients
lead to dark meson lifetimes much shorter than the age of the
universe. Of course there are ways to suppress these interactions,
but it requires additional model-building at higher scales.
\subsection{Baryon-like dark matter}
One of the principle reasons to consider baryon-like candidates
for dark matter is robust stability, i.e., safety
from higher dimensional interactions that lead to decay on
timescales short compared with the age of the universe,
c.f. Eq.~(\ref{eq:dim-5-ops}).
For theories with fermion constituents, $SU(N_c)$
with $N_c \ge 3$, higher dimensional operators are at least
dimension-6 or higher. In these theories, dark matter is automatically
sufficiently stable, and no further ultraviolet model-building is
needed. This is a superior property of composite baryonic dark matter.
Early work on technicolor theories demonstrated the potential
of technibaryons as a dark matter candidate
\cite{Nussinov:1985xr,Chivukula:1989qb,Barr:1990ca}.
In these theories,
dark fermions transformed under a chiral representation of
$SU(2)_L \times U(1)_Y$, and so after confinement, lead
to dynamical electroweak symmetry breaking. The technibaryons
carried an accidental global quantum number, technibaryon number,
that suggested the lightest technibaryon is a natural dark matter
candidate. Early investigation in these theories revealed an
elegant mechanism to obtain the correct cosmological abundance of
dark matter through a technibaryon number asymmetry
\cite{Barr:1990ca,Barr:1991qn,Kaplan:1991ah}.
These investigations continued into aspects of direct detection
\cite{Chivukula:1992pn,Bagnasco:1993st,Pospelov:2000bq}.
More recent investigations into technibaryon dark matter and
other related candidates can be found
\cite{Dietrich:2006cm,Gudnason:2006yj,Nardi:2008ix,Foadi:2008qv,Ryttov:2008xe,Khlopov:2008ty,Sannino:2009za,Mardon:2009gw,Lewis:2011zb,Cline:2013zca,Hietanen:2013fya,Brod:2014loa,Hietanen:2014xca}.
With the discovery of the Higgs boson in 2012, technicolor
theories, at least as originally formulated, are under siege.
(Theories that lead to a Higgs-like boson, such as composite Higgs theories,
remain interesting, but also have substantial constraints from
the LHC.) This leaves open the possibility of theories containing
approximately vector-like fermions transforming under (part of) the
standard model, using the strong dynamics to set mass scales
as well as to provide a viable electroweak-neutral composite dark
matter candidate.
The LSD Collaboration has investigated both fermion and scalar
baryonic candidates for dark matter from confining SU(3) and SU(4)
dark color theories
\cite{Appelquist:2013ms,Appelquist:2015yfa,Appelquist:2015zfa}.
In both cases, dark fermions were
assumed to transform under (nearly) vector-like representations
of the electroweak group, leading to negligible corrections to
electroweak precision observables. In the dark SU(3) case,
the lightest baryon was found to be a neutron-like fermionic
dark baryon with a significant magnetic moment due to the
electrically charged dark fermion constituents.
The spectrum and the leading interactions with nuclei
were determined using lattice simulations.
Given existing bounds from Xenon experiments, the lower bound
on the mass of this fermionic dark baryon was found to be
$10$~TeV \cite{Appelquist:2013ms}. This comparatively large mass
arises due to the relatively low mass dimension of the magnetic
moment interaction (dimension-5).
In \cite{Appelquist:2015yfa,Appelquist:2015zfa}, the LSD
Collaboration proposed and investigated scalar dark baryons from an
SU(4) dark confining interaction, called ``Stealth Dark Matter''.
When combined with a dark custodial SU(2) symmetry (that leads to
equal masses for the lightest $q = \pm 1/2$ electrically charged
dark fermion constituents),
stealth dark matter was found to be remarkably safe from
direct detection experiments due to the high dimension of the leading
interaction -- electromagnetic polarizability -- of the
scalar baryon with the standard model. Dark baryons as light as
$300$~GeV were possible for an order one pion-to-vector mass ratio.
These relatively precise estimates were possible by performing
lattice simulations to determine the hadron mass spectrum
as well as the coefficients of the dominant operators
leading to direct detection.
Given that the polarizability-induced spin-independent direct detection
cross section scales as $Z^{8/3}$, heavier element experiments
(including xenon and tungsten) clearly have better sensitivity than
lighter elements (such as germanium and argon). In addition, this theory
contains a rich spectrum of mesons somewhat below the mass
of the dark baryon that are ripe for exploration at the LHC.
\subsection{Dark glueballs}
Any non-Abelian gauge sector is also expected to contain a number of glueball bound states which have no valence fermion content. In theories such as QCD where the colored fermions are light compared to the confinement scale, these glueballs are broad resonances which decay readily into lighter mesons and baryons, and often mix with neutral mesons as well. These properties make QCD glueballs rather difficult to isolate, and no conclusive experimental observation has been reported to date \cite{Crede:2008vw}.
However, if all fermions in a dark non-Abelian sector are very heavy compared to the confinement scale $\Lambda_d$, then the lightest particles in the spectrum will be the glueballs, with masses of order $\Lambda_d$. Like baryonic states, the lightest glueballs are stabilized by accidental symmetry, since as color-singlet bound states their creation operators are dimension 4, of the form ${\rm Tr} (G_{\mu \nu} G^{\mu \nu})$, or dimension 6 of the form ${\rm Tr}(G_{\mu \nu}^3)$. The leading operators in the Lagrangian which can mediate glueball decay are then of the form \cite{Faraggi:2000pv}
\begin{equation}
\mathcal{L} \supset \frac{c_H}{M^2} H^\dagger H {\rm Tr} (G_{\mu \nu} G^{\mu \nu}) + \frac{c_F}{M^4} {\rm Tr} (G_{\mu \nu} G^{\mu \nu}) {\rm Tr} (F_{\mu \nu} F^{\mu \nu}),
\end{equation}
where $F_{\mu \nu}$ is the field-strength tensor for one of the standard model gauge fields, and $M$ is the scale of new physics, e.g. some heavy fermions which carry both standard model and hidden sector charge. These operators mediate glueball decays which scale as $\Gamma \sim c_H \Lambda_d^9 / M_H^4 M^4$ and $\Gamma \sim c_F \Lambda_d^9 / M^8$ respectively, so that the glueballs can be stabilized for a wide range of $\Lambda_d$, as long as $M$ is sufficiently large: for example taking $M = M_{\rm GUT} = 10^{16}$ GeV, the glueballs will be stable on the lifetime of the universe if $\Lambda_d \lesssim 10^{5}$ GeV \cite{Faraggi:2000pv}. A detailed effective field theory description of glueball decay processes has been studied in \cite{Juknevich:2009ji,Juknevich:2009gg}.
Hidden sector glueballs are thus a natural dark matter candidate \cite{Okun:1980kw,Okun:1980mu}, with a number of interesting properties. Hidden sectors of this type can fit nicely into larger models of new physics, e.g. as part of the MSSM with anomaly-mediated supersymmetry breaking \cite{Feng:2011ik,Boddy:2014yra,Boddy:2014qxa} - in which case the dark matter consists of both glueballs and glueballinos, with abundance depending on parameter choices. Grand unification can also lead to hidden non-Abelian sectors (see e.g. \cite{Kakushadze:1996jm,Kakushadze:1997ne} for a partial classification); GUT and string-theory motivated studies of glueball dark matter are undertaken in \cite{Faraggi:2000pv, Yamanaka:2014pva}.
Collider bounds on glueball dark matter are relatively weak compared to other composite candidates, because unlike the mesonic and baryonic cases, no light charged states exist in the spectrum if all of the hidden-sector fermions are heavy. The absence of strong constraints from LEP makes it much easier to construct viable glueball dark matter models with a dark matter mass below 100 GeV. In particular, the strong self-interactions of the glueballs can provide an explanation of galactic structure anomalies \cite{Spergel:1999mh,Vogelsberger:2012ku,Rocha:2012jg,Peter:2012jh,Zavala:2012us} for glueball dark matter masses in the MeV to GeV range, depending on the specific choice of hidden gauge group \cite{Boddy:2014yra, Boddy:2014qxa,Soni:2016gzf}.
Although there have been no lattice calculations to date specifically focused on glueball dark matter, there are a number of more general results on the glueball spectrum and selected matrix elements \cite{Morningstar:1999rf,Lucini:2004my,Chen:2005mg,Loan:2006gm,Lucini:2010nv, Lucini:2014paa}. Since any hypercolor-charged particles are taken to be heavy compared to the confinement scale, from the perspective of a lattice practitioner the theory of interest is ``pure-gauge" Yang-Mills; this is an attractive theory to study, because in the absence of fermions large-scale studies can be undertaken with relatively modest computational resources.
\begin{figure}[t]
\centering
\label{fig:glue-3}
\begin{minipage}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{figs/glueballs-su3}
\end{minipage}
\hfill
\begin{minipage}[b]{0.54\textwidth}
\includegraphics[width=\textwidth]{figs/glueballs-suinf}
\end{minipage}
\caption{Glueball spectrum obtained from lattice simulations for SU$(3)$ pure gauge theory \cite{Chen:2005mg} (left) and for SU$(N)$ pure gauge theory extrapolated to the large-$N$ limit \cite{Lucini:2010nv} (right).}
\end{figure}
\subsection{Abundance}
To obtain a cosmologically significant thermal relic abundance
of dark matter, the standard approach is to allow the strongly-coupled
composites to annihilate into other, light, unstable dark sector
states that decay back into standard model particles.
The annihilation cross sections are generically large,
due to strong coupling. If the annihilation rate saturates
the unitarity bound\cite{Griest:1989wd,Blum:2014dca},
the dark matter mass is on the order of $100$~TeV\@.
If the dark sector contains many states close in mass,
the annihilation rates can be substantially modified,
and the mass of the dark matter can be smaller,
closer to the TeV scale.\cite{Buckley:2012ky},
A more recent exception
to the usual $2 \rightarrow 2$ strongly-coupled annihilation rate
is when a $3 \rightarrow 2$
process dominates the thermal freezeout of dark pions. For this
process to be operable, both a
new interaction with standard model (e.g.\ dark photons kinetically
mixed with hypercharge) is required as well as degeneracy of the
lightest flavors of dark fermions to ensure the effective
Wess-Zumino-Witten 5-point pion interaction is unsuppressed.
An asymmetric abundance of strongly-coupled dark matter
was considered long ago in the context of technibaryon
dark matter \cite{Barr:1990ca,Barr:1991qn,Kaplan:1991ah}.
That technibaryons transformed under a chiral representation
of the electroweak group implies that electroweak sphalerons
preserve the linear combination $U(1)_{B - L - D}$,
opening up the possibility that an asymmetric abundance
of dark baryons is automatically generated following
baryogenesis or darkogenesis
\cite{Barr:1991qn,Kribs:2009fy,Shelton:2010ta,Davoudiasl:2010am,Buckley:2010ui}.
Intriguingly, the residual abundance of dark baryons is
$\rho \sim m_B n_B$ where the number density is
proportional to $\exp[-m_{\rm B}/T_{\rm sph}]$,
where $T_{\rm sph}$ is the temperature
at which sphaleron interactions shut off.
If the baryon and dark baryon number densities are comparable,
the would-be overabundance of dark matter (from $m_B \gg m_{\rm nucleon}$)
is compensated by the Boltzmann suppression. Very roughly,
$m_{\rm baryon} \sim 1$-$2$~TeV
is the natural mass scale that matches the
cosmological abundance of dark matter \cite{Barr:1990ca}. A similar mass scale
can be arrived at through dynamics coupling the QCD scale to the dark confinement scale
\cite{Bai:2013xga}.
In theories with vector-like and electroweak symmetry breaking
masses, such as stealth dark matter
\cite{Appelquist:2015yfa,Appelquist:2015zfa},
it was anticipated that an asymmetric abundance was still
possible, but with further suppression by the amount of
electroweak symmetry breaking in the dark sector.
\subsection{Dark Nuclei and Dark Nucleosynthesis}
Strongly-coupled hadrons from a dark sector may combine to form stable
composites of the hadrons themselves: dark nuclei. Certainly
the standard model provides a clear proof of principle that
such nuclei exist and provide an order one fraction of the
energy density of matter. In the presence of a light mediator,
``darkleosynthesis'' was shown to be possible and
efficient for asymmetric composite dark matter from a
confining non-Abelian theory\cite{Krnjaic:2014xza}.
A quantitative exploration of dark nuclei
was performed for a dark SU(2) with two
flavors.\cite{Detmold:2014qqa,Detmold:2014kba}
Lattice simulations of this model
demonstrated stable nuclear states are possible with the lowest
lying states being bound states of the pion and vector mesons
and their baryonic partners. This suggests the possibility of
analogues of nuclei should be considered in any strongly interacting
composite model.
In \cite{Detmold:2014qqa,Detmold:2014kba}, it was shown that
for both symmetric and asymmetric origins of dark hadrons,
the early universe cosmology can be substantially altered by
dark nucleosynthesis, perhaps having most dark nucleons processed
into dark nuclei. Importantly, new signals of indirect detection
of asymmetric dark matter are possible as dark hadrons combine
into dark nuclei, emitting photons [either directly or as a result
of $U(1)_Y$ kinetically mixed with a dark U(1)]. Other exotic
phenomena may also occur, such as the ejection of asymmetric
dark nuclei from stars, thereby suppressing the accumulation
of asymmetric dark matter in these objects. It remains to be
seen how generalizable these results are to different numbers of
color, flavors, and dark fermion mass spectrum.
Finally, an intriguing possibility is that strongly-coupled
dark baryons could form very large dark nuclei, forming an
extended semi-uniform object \cite{Hardy:2014mqa,Hardy:2015boa}.
This is counter to the usual intiution from the standard model,
where only light elements form from big-bang nucleosynthesis.
It was found that dark nuclei with large dark nucleon number,
$A \gtrsim 10^8$ may be synthesized. \cite{Hardy:2014mqa,Hardy:2015boa}.
This qualitatively changes direct detection and capture rates in
astrophysical objects.
\subsection{Photon interactions}
If the composite dark matter candidate $\chi$ is neutral, but its constituents carry electromagnetic charge, then its coupling to the photon is proportional to the matrix element
\begin{equation}
\langle \chi(p') | j_{\rm EM}^\mu | \chi(p) \rangle = F(q^2) q^\mu,
\end{equation}
where $q_\mu = p_\mu + p'_\mu$, $j_{\rm EM}^\mu$ is the electromagnetic current, and $F(0) = 0$. In the limit that the momentum transfer $|q|$ is very small compared to the compositeness scale $\Lambda$, which is appropriate for dark matter direct detection, the form factor can be described in terms of effective field theory operators of increasing dimension. The leading C and P-conserving operators are \cite{Bagnasco:1993st,Pospelov:2000bq} the magnetic moment
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda} \bar{\chi} \sigma^{\mu \nu} \chi F_{\mu \nu},
\end{equation}
the charge radius
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda^2} \bar{\chi} \gamma^\nu \chi \partial^\mu F_{\mu \nu},\ \ \frac{1}{\Lambda^2} \phi^\dagger \phi v^\nu \partial^\mu F_{\mu \nu},
\end{equation}
and the electromagnetic polarizability
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda^3} \bar{\chi} \chi F_{\mu \nu} F^{\mu \nu},\ \ \frac{1}{\Lambda^3} \phi^\dagger \phi F_{\mu \nu} F^{\mu \nu},
\end{equation}
where $\chi$ represents a fermionic dark matter candidate, and $\phi$ a spin-zero bosonic candidate. If $\phi$ were a boson with non-zero spin, it would also have a magnetic moment operator. (Note that the dimensions of the operators are the same for scalar dark matter, due to the non-relativistic normalization of the fields which is most appropriate for treating dark matter direct detection \cite{Hill:2014yxa}.) Other operators at similar orders in the effective expansion, e.g. an electric dipole moment, can appear if CP violation occurs in the dark sector \cite{Pospelov:2000bq}. Phenomenological treatments of dark matter with some or all of these effective interactions have been considered in the literature, both independently and in the context of composite models \cite{Bagnasco:1993st,Pospelov:2000bq,Sigurdson:2004zp,Gardner:2008yn,Chang:2010en,Banks:2010eh,Barger:2010gv,DelNobile:2012tx,Weiner:2012cb,Pospelov:2013nea,Ovanesyan:2014fha}.
Detailed formulas for the interaction cross sections mediated by these interactions are derived in the above references. We will not reproduce them here, but it is worth observing that the scaling of the cross section with the choice of nuclear target can be dramatically different \cite{Dent:2015zpa,DelNobile:2015tza,DelNobile:2015rmp}, depending on which operator dominates. The per-nucleon interaction cross section is expected to scale as $\mu^2 (J+1)/J$, $Z^2 / A^2$, and $Z^4 / A^{8/3}$ for the magnetic moment, charge radius, and electromagnetic polarizability operators respectively, where $\mu$ is the nuclear magnetic moment, $J$ is the nuclear spin, and $Z$ and $A$ are the standard proton and atomic mass numbers. (Note that for a dark matter magnetic moment, the scaling given is for the moment-moment interaction; there is also a magnetic moment-nuclear charge interaction \cite{Banks:2010eh,Appelquist:2013ms}, which scales as $Z^2/A^2$ like the charge radius.)
The value of these prefactors for several nuclear targets currently used in direct detection experiments are tabulated in Table~\ref{tab:EM_scaling_normalized}, scaled so that the value for xenon is set to 1. Especially dramatic differences are seen for the coupling to the nuclear magnetic moment. We also note that the electromagnetic polarizability interaction in principle has a very large uncertainty; since the interaction contains two photons, scattering proceeds through a loop diagram, so this interaction may be particularly sensitive to poorly-known nuclear matrix elements involving excited states \cite{Appelquist:2015zfa}.
On the dark matter side, determination of the coefficients of these operators requires a non-perturbative calculation. We now turn to lattice calculations focused on photon direct-detection operators.
\begin{table}[ph]
\caption{Leading scaling of direct-detection interactions involving photon exchange: dark magnetic moment-nuclear moment (first column), dark magnetic moment-nuclear charge or dark charge radius (second column), and dark electromagnetic polarizability (third column), relative to the given prefactor for xenon. We average over natural isotopic abundance for each element, taking the nuclear magnetic moment $\mu$ and spin $J$ from the literature \cite{Fuller:1976xx}.}
{\begin{tabular}{c|cccc}
\hline
target & $\mu^2 (J+1)/J$ & $Z^2/A^2$ & $Z^4/A^{8/3}$ \\
\hline
Xe &1 &1 &1 \\
Si &0.06681 &1.472 &0.2766 \\
Ge &0.1130 &1.152 &0.6010 \\
Na &12.68 &1.357 &0.1798 \\
O &0.003029 &1.482 &0.1323 \\
I &17.09 &1.033 &1.018 \\
Ca &0.004658 &1.476 &0.4464 \\
W &0.009074 &0.9608 &1.442 \\
Ar &0. &1.201 &0.2949 \\
C &0.02518 &1.481 &0.0900 \\
F &32.07 &1.331 &0.1341 \\
\hline
\end{tabular}
\label{tab:EM_scaling_normalized}
}
\end{table}
Calculation of the magnetic moment and charge radius for a given dark matter candidate can be accomplished through a direct lattice calculation of the form factor $F(Q^2)$ itself. That is, the three-point correlation function
\begin{equation}
C_3^\mu(t,t') = \sum_{\vec{x}, \vec{y}} e^{-i \vec{p}' \cdot \vec{x}} e^{-i (\vec{p}-\vec{p}') \cdot \vec{y}} \langle B^\dagger(0,0) V^\mu(\vec{x}, t) B(\vec{y}, t') \rangle
\end{equation}
is computed directly, where $B$ is the composite object interpolating operator and $V^\mu$ the electromagnetic current. Calculating at several values of the discretized momentum transfer and fitting the momentum dependence allows the magnetic moment and charge radius to be determined.
Results on the lattice have been obtained for SU$(2)$ \cite{Hietanen:2013fya} and SU$(3)$ \cite{Appelquist:2013ms} gauge theories. In the former case, the calculated charge radius for the meson-like dark matter candidate is found to be roughly consistent with its predicted value from vector meson dominance, using the value of the vector-meson mass determined from the lattice as well. Fairly strong bounds are found from the Xenon100 and LUX experiments, although their model has an additional adjustable parameter $d_b$ which can suppress the charge radius interaction (with $d_b = 0$ corresponding to the restoration of an isospin-like symmetry.) For the SU$(3)$ study, strong bounds are found particularly from the magnetic moment, restricting the dark matter mass to be larger than roughly 10 TeV from Xenon100 constraints alone.
Determination of the electromagnetic polarizability can be somewhat more difficult, due to its suppression by large powers of the momentum transfer. An alternative to direct calculation of the form factor is to apply the background field method \cite{Detmold:2010ts}. In this approach, a background static electric field $\mathcal{E}$ is applied by use of appropriate boundary conditions in the lattice simulation. Measuring the ground-state energy of the dark matter candidate as a function of $|\mathcal{E}|$ allows determination of the polarizability from the quadratic Stark shift, e.g.\cite{Appelquist:2015zfa}
\begin{equation}
E_X(|\mathcal{E}|) = m_X + \left( 2C_F - \frac{\mu_X^2}{8m_X^3} \right) |\mathcal{E}|^2 + \mathcal{O}(|\mathcal{E}|^4),
\end{equation}
where $C_F$ is the polarizability and $\mu_X$ is the magnetic moment of $X$.
This approach has been used so far to study two different theories on the lattice. The LSD collaboration has calculated the polarizability in SU$(4)$ gauge theory for their ``stealth dark matter" model \cite{Appelquist:2015zfa}. In units of the SU$(4)$ baryon mass, the polarizability was found to be comparable to that of the neutron in QCD, much larger than naive dimensional analysis would indicate. The resulting direct-detection cross section in LUX diminishes rapidly with the dark matter mass and falls below the expected cosmic neutrino background, but an interesting window for direct detection remains below 1 TeV or so. The polarizability has also been studied in an SU$(2)$ gauge theory \cite{Drach:2015epq} for ``template composite dark matter", finding essentially no bound on their model from direct detection.
\begin{figure}
\label{fig:pol-su4}
\centering
\includegraphics[width=0.6\textwidth]{figs/Pol_Plot_Corrected_Band}
\caption{Direct detection cross-section prediction for xenon (purple band) for stealth dark matter interacting through electromagnetic polarizability, calculated using lattice results \cite{Appelquist:2015zfa}. The blue shaded region (top) indicates current experimental bounds from LUX \cite{Akerib:2013tjd}; the grey region (left) shows collider bounds on charged mesons in this model; the orange region (bottom) shows the anticipated irreducible cosmic neutrino background.}
\end{figure}
\subsection{Higgs interaction}
If the composite dark sector contains fundamental fermions $f$, it is natural for them to obtain some of their mass from a Yukawa coupling $y_f$ to the Higgs boson, inducing a mass of order $m_f \sim y_f v$. If this coupling is present, it will induce a Higgs coupling to any composite state, e.g. a dark baryon $B$ formed from the $f$ fields, of the form
\begin{equation}
\sum_f y_f \langle B | \bar{f} f | B \rangle.
\end{equation}
This mirrors the way in which the Higgs couplings of the proton and neutron in the standard model arise; they depend on the individual quark Yukawa couplings, and on the scalar-current matrix element, also known as the ``sigma term''.
The coupling to the Higgs need not be the only source of mass for the $f$ fermions; they may also have (technically natural) vector-like mass terms, or Yukawa couplings to other new scalar fields. In general, we can parameterize the fraction of the fermion mass which is due to the Higgs field by defining the parameter\cite{Appelquist:2014jch}
\begin{equation}
\alpha = \frac{v}{m_f} \left. \frac{\partial m_f(h)}{\partial v} \right|_{h=v}
\end{equation}
where $m_f(h) = m + yh/\sqrt{2}$, and $m$ encapsulates other sources of mass. This parameter varies from $\alpha=0$ if $y=0$ (no Higgs contribution to $m_f$), to $\alpha=1$ when $m=0$ (so the Higgs boson is the only source of mass for $f$.)
If the ratio $m_f / m_B$ is kept fixed, then the direct detection cross section for the dark baryon $B$ increases quadratically with $m_B$, leading to fairly strict bounds from current experiments when $\alpha$ is large. In particular, comparison with LUX yields the bound \cite{Appelquist:2014jch}
\begin{equation} \label{eq:alpha-bound}
\alpha \lesssim \left( \frac{370\ \rm{GeV}}{m_B} \right)^{1/2} \times \begin{cases}
0.34& m_{PS} / m_V = 0.55, \\
0.05& m_{PS} / m_V = 1,\end{cases}
\end{equation}
where $m_{PS} / m_V$ is the ratio of pseudoscalar to vector meson mass in the SU$(4)$ theory considered, which is a proxy for $m_f / m_B$. These results strongly disfavor $\alpha = 1$, i.e. a purely electroweak origin for the dark sector fermion masses is essentially ruled out.
Although this result assumes a particular dark sector model based on SU$(4)$ gauge theory, there is some evidence that the constraint $\alpha < 1$ is fairly robust. The main non-perturbative input which gives the bound Eq.~\ref{eq:alpha-bound} is the ``dark sigma term"
\begin{equation}
f_f^{(B)} \equiv \frac{\langle B | m_f \bar{f} f | B \rangle}{m_B} = \frac{m_f}{m_B} \frac{\partial m_B}{\partial m_f},
\end{equation}
applying the Feynman-Hellmann theorem to obtain the last equality. This quantity is readily determined from lattice spectroscopy of the baryon mass vs. input fermion mass. Lattice results for a number of different gauge theories \cite{DeGrand:2015lna} are shown in Fig.~\ref{fig:sigma-lat}, and indicate that for similar mass ranges, the non-perturbative value of $f_f^{(B)}$ obtained tends to be consistent across different strongly-coupled theories.
\begin{figure}
\label{fig:sigma-lat}
\centering
\includegraphics[width=0.6\textwidth]{figs/sigma-term-lattice}
\caption{Results obtained from lattice simulations\cite{DeGrand:2015lna} in various theories for the ``dark sigma term" $f_f^{(B)}$, defined in the text. The results are generally quite consistent as a function of fermion mass, even as the gauge group and fermion representation are varied.}
\end{figure}
\subsection{Photon interactions}
If the composite dark matter candidate $\chi$ is neutral, but its constituents carry electromagnetic charge, then its coupling to the photon is proportional to the matrix element
\begin{equation}
\langle \chi(p') | j_{\rm EM}^\mu | \chi(p) \rangle = F(q^2) q^\mu,
\end{equation}
where $q_\mu = p_\mu + p'_\mu$, $j_{\rm EM}^\mu$ is the electromagnetic current, and $F(0) = 0$. In the limit that the momentum transfer $|q|$ is very small compared to the compositeness scale $\Lambda$, which is appropriate for dark matter direct detection, the form factor can be described in terms of effective field theory operators of increasing dimension. The leading C and P-conserving operators are \cite{Bagnasco:1993st,Pospelov:2000bq} the magnetic moment
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda} \bar{\chi} \sigma^{\mu \nu} \chi F_{\mu \nu},
\end{equation}
the charge radius
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda^2} \bar{\chi} \gamma^\nu \chi \partial^\mu F_{\mu \nu},\ \ \frac{1}{\Lambda^2} \phi^\dagger \phi v^\nu \partial^\mu F_{\mu \nu},
\end{equation}
and the electromagnetic polarizability
\begin{equation}
\mathcal{L} \supset \frac{1}{\Lambda^3} \bar{\chi} \chi F_{\mu \nu} F^{\mu \nu},\ \ \frac{1}{\Lambda^3} \phi^\dagger \phi F_{\mu \nu} F^{\mu \nu},
\end{equation}
where $\chi$ represents a fermionic dark matter candidate, and $\phi$ a spin-zero bosonic candidate. If $\phi$ were a boson with non-zero spin, it would also have a magnetic moment operator. (Note that the dimensions of the operators are the same for scalar dark matter, due to the non-relativistic normalization of the fields which is most appropriate for treating dark matter direct detection \cite{Hill:2014yxa}.) Other operators at similar orders in the effective expansion, e.g. an electric dipole moment, can appear if CP violation occurs in the dark sector \cite{Pospelov:2000bq}. Phenomenological treatments of dark matter with some or all of these effective interactions have been considered in the literature, both independently and in the context of composite models \cite{Bagnasco:1993st,Pospelov:2000bq,Sigurdson:2004zp,Gardner:2008yn,Chang:2010en,Banks:2010eh,Barger:2010gv,DelNobile:2012tx,Weiner:2012cb,Pospelov:2013nea,Ovanesyan:2014fha}.
Detailed formulas for the interaction cross sections mediated by these interactions are derived in the above references. We will not reproduce them here, but it is worth observing that the scaling of the cross section with the choice of nuclear target can be dramatically different \cite{Dent:2015zpa,DelNobile:2015tza,DelNobile:2015rmp}, depending on which operator dominates. The per-nucleon interaction cross section is expected to scale as $\mu^2 (J+1)/J$, $Z^2 / A^2$, and $Z^4 / A^{8/3}$ for the magnetic moment, charge radius, and electromagnetic polarizability operators respectively, where $\mu$ is the nuclear magnetic moment, $J$ is the nuclear spin, and $Z$ and $A$ are the standard proton and atomic mass numbers. (Note that for a dark matter magnetic moment, the scaling given is for the moment-moment interaction; there is also a magnetic moment-nuclear charge interaction \cite{Banks:2010eh,Appelquist:2013ms}, which scales as $Z^2/A^2$ like the charge radius.)
\begin{table}[t]
\begin{tabular}{c|cccc}
\hline
target & $\mu^2 (J+1)/J$ & $Z^2/A^2$ & $Z^4/A^{8/3}$ \\\hline
Xe &1 &1 &1 \\
Si &0.06681 &1.472 &0.2766 \\
Ge &0.1130 &1.152 &0.6010 \\
Na &12.68 &1.357 &0.1798 \\
O &0.003029 &1.482 &0.1323 \\
I &17.09 &1.033 &1.018 \\
Ca &0.004658 &1.476 &0.4464 \\
W &0.009074 &0.9608 &1.442 \\
Ar &0. &1.201 &0.2949 \\
C &0.02518 &1.481 &0.0900 \\
F &32.07 &1.331 &0.1341 \\ \hline
\end{tabular}
\label{tab:EM_scaling_normalized}
\caption{Leading scaling of direct-detection interactions involving photon exchange: dark magnetic moment-nuclear moment (first column), dark magnetic moment-nuclear charge or dark charge radius (second column), and dark electromagnetic polarizability (third column), relative to the given prefactor for xenon. We average over natural isotopic abundance for each element, taking the nuclear magnetic moment $\mu$ and spin $J$ from the literature \cite{Fuller:1976xx}.}
\end{table}
The value of these prefactors for several nuclear targets currently used in direct detection experiments are tabulated in Table~\ref{tab:EM_scaling_normalized}, scaled so that the value for xenon is set to 1. Especially dramatic differences are seen for the coupling to the nuclear magnetic moment. We also note that the electromagnetic polarizability interaction in principle has a very large uncertainty; since the interaction contains two photons, scattering proceeds through a loop diagram, so this interaction may be particularly sensitive to poorly-known nuclear matrix elements involving excited states \cite{Appelquist:2015zfa}.
On the dark matter side, determination of the coefficients of these operators requires a non-perturbative calculation. We now turn to lattice calculations focused on photon direct-detection operators.
Calculation of the magnetic moment and charge radius for a given dark matter candidate can be accomplished through a direct lattice calculation of the form factor $F(Q^2)$ itself. That is, the three-point correlation function
\begin{equation}
C_3^\mu(t,t') = \sum_{\vec{x}, \vec{y}} e^{-i \vec{p}' \cdot \vec{x}} e^{-i (\vec{p}-\vec{p}') \cdot \vec{y}} \langle B^\dagger(0,0) V^\mu(\vec{x}, t) B(\vec{y}, t') \rangle
\end{equation}
is computed directly, where $B$ is the composite object interpolating operator and $V^\mu$ the electromagnetic current. Calculating at several values of the discretized momentum transfer and fitting the momentum dependence allows the magnetic moment and charge radius to be determined.
Results on the lattice have been obtained for SU$(2)$ \cite{Hietanen:2013fya} and SU$(3)$ \cite{Appelquist:2013ms} gauge theories. In the former case, the calculated charge radius for the meson-like dark matter candidate is found to be roughly consistent with its predicted value from vector meson dominance, using the value of the vector-meson mass determined from the lattice as well. Fairly strong bounds are found from the Xenon100 and LUX experiments, although their model has an additional adjustable parameter $d_b$ which can suppress the charge radius interaction (with $d_b = 0$ corresponding to the restoration of an isospin-like symmetry.) For the SU$(3)$ study, strong bounds are found particularly from the magnetic moment, restricting the dark matter mass to be larger than roughly 10 TeV from Xenon100 constraints alone.
Determination of the electromagnetic polarizability can be somewhat more difficult, due to its suppression by large powers of the momentum transfer. An alternative to direct calculation of the form factor is to apply the background field method \cite{Detmold:2010ts}. In this approach, a background static electric field $\mathcal{E}$ is applied by use of appropriate boundary conditions in the lattice simulation. Measuring the ground-state energy of the dark matter candidate as a function of $|\mathcal{E}|$ allows determination of the polarizability from the quadratic Stark shift, e.g.\cite{Appelquist:2015zfa}
\begin{equation}
E_X(|\mathcal{E}|) = m_X + \left( 2C_F - \frac{\mu_X^2}{8m_X^3} \right) |\mathcal{E}|^2 + \mathcal{O}(|\mathcal{E}|^4),
\end{equation}
where $C_F$ is the polarizability and $\mu_X$ is the magnetic moment of $X$.
This approach has been used so far to study two different theories on the lattice. The LSD collaboration has calculated the polarizability in SU$(4)$ gauge theory for their ``stealth dark matter" model \cite{Appelquist:2015zfa}. In units of the SU$(4)$ baryon mass, the polarizability was found to be comparable to that of the neutron in QCD, much larger than naive dimensional analysis would indicate. The resulting direct-detection cross section in LUX diminishes rapidly with the dark matter mass and falls below the expected cosmic neutrino background, but an interesting window for direct detection remains below 1 TeV or so. The polarizability has also been studied in an SU$(2)$ gauge theory \cite{Drach:2015epq} for ``template composite dark matter", finding essentially no bound on their model from direct detection.
\begin{figure}
\label{fig:pol-su4}
\centering
\includegraphics[width=0.6\textwidth]{figs/Pol_Plot_Corrected_Band}
\caption{Direct detection cross-section prediction for xenon (purple band) for stealth dark matter interacting through electromagnetic polarizability, calculated using lattice results \cite{Appelquist:2015zfa}. The blue shaded region (top) indicates current experimental bounds from LUX \cite{Akerib:2013tjd}; the grey region (left) shows collider bounds on charged mesons in this model; the orange region (bottom) shows the anticipated irreducible cosmic neutrino background.}
\end{figure}
\subsection{Higgs interaction}
If the composite dark sector contains fundamental fermions $f$, it is natural for them to obtain some of their mass from a Yukawa coupling $y_f$ to the Higgs boson, inducing a mass of order $m_f \sim y_f v$. If this coupling is present, it will induce a Higgs coupling to any composite state, e.g. a dark baryon $B$ formed from the $f$ fields, of the form
\begin{equation}
\sum_f y_f \langle B | \bar{f} f | B \rangle.
\end{equation}
This mirrors the way in which the Higgs couplings of the proton and neutron in the standard model arise; they depend on the individual quark Yukawa couplings, and on the scalar-current matrix element, also known as the ``sigma term''.
The coupling to the Higgs need not be the only source of mass for the $f$ fermions; they may also have (technically natural) vector-like mass terms, or Yukawa couplings to other new scalar fields. In general, we can parameterize the fraction of the fermion mass which is due to the Higgs field by defining the parameter\cite{Appelquist:2014jch}
\begin{equation}
\alpha = \frac{v}{m_f} \left. \frac{\partial m_f(h)}{\partial v} \right|_{h=v}
\end{equation}
where $m_f(h) = m + yh/\sqrt{2}$, and $m$ encapsulates other sources of mass. This parameter varies from $\alpha=0$ if $y=0$ (no Higgs contribution to $m_f$), to $\alpha=1$ when $m=0$ (so the Higgs boson is the only source of mass for $f$.)
If the ratio $m_f / m_B$ is kept fixed, then the direct detection cross section for the dark baryon $B$ increases quadratically with $m_B$, leading to fairly strict bounds from current experiments when $\alpha$ is large. In particular, comparison with LUX yields the bound \cite{Appelquist:2014jch}
\begin{equation} \label{eq:alpha-bound}
\alpha \lesssim \left( \frac{370\ \rm{GeV}}{m_B} \right)^{1/2} \times \begin{cases}
0.34& m_{PS} / m_V = 0.55, \\
0.05& m_{PS} / m_V = 1,\end{cases}
\end{equation}
where $m_{PS} / m_V$ is the ratio of pseudoscalar to vector meson mass in the SU$(4)$ theory considered, which is a proxy for $m_f / m_B$. These results strongly disfavor $\alpha = 1$, i.e. a purely electroweak origin for the dark sector fermion masses is essentially ruled out.
Although this result assumes a particular dark sector model based on SU$(4)$ gauge theory, there is some evidence that the constraint $\alpha < 1$ is fairly robust. The main non-perturbative input which gives the bound Eq.~\ref{eq:alpha-bound} is the ``dark sigma term"
\begin{equation}
f_f^{(B)} \equiv \frac{\langle B | m_f \bar{f} f | B \rangle}{m_B} = \frac{m_f}{m_B} \frac{\partial m_B}{\partial m_f},
\end{equation}
applying the Feynman-Hellmann theorem to obtain the last equality. This quantity is readily determined from lattice spectroscopy of the baryon mass vs. input fermion mass. Lattice results for a number of different gauge theories \cite{DeGrand:2015lna} are shown in Fig.~\ref{fig:sigma-lat}, and indicate that for similar mass ranges, the non-perturbative value of $f_f^{(B)}$ obtained tends to be consistent across different strongly-coupled theories.
\begin{figure}
\label{fig:sigma-lat}
\centering
\includegraphics[width=0.6\textwidth]{figs/sigma-term-lattice}
\caption{Results obtained from lattice simulations\cite{DeGrand:2015lna} in various theories for the ``dark sigma term" $f_f^{(B)}$, defined in the text. The results are generally quite consistent as a function of fermion mass, even as the gauge group and fermion representation are varied.}
\end{figure}
\subsection{Quirky signals}
A dark hidden sector with a new non-Abelian force that confines
can leave very unusual collider phenomenology.
This was recognized long ago in the context of
``hidden valley'' models.\cite{Strassler:2006im,Han:2007ae}
A striking example was emphasized in the context
of ``quirky'' models\cite{Kang:2008ea} -- a dark sector contains
dark fermions that transform under part of the standard model,
while being deep in the quarkonia regime, $\Lambda_d \ll m_q$.
Here $m_q \lesssim \sqrt{s}$ ($\sqrt{\hat{s}}$) so that
pairs of dark fermions could be produced easily by
Drell-Yan or other standard model processes.
With a dark confinement scale much smaller than the dark fermion
mass scale, the dark color strings cannot fragment, and so the
dark fermions can travel macroscopic distances while still held
together by a very weak unbreakable dark color string.
Depending on the standard
model charges of the dark fermions, this can leave highly exotic tracks
and energy deposition in detectors that is unlike anything
produced in the standard model.\cite{Kang:2008ea,Harnik:2008ax}
Once the dark fermions are sufficiently heavy, they can be
integrated out, giving effective operators between
standard model fields and the dark glue fields.
This can provide an opportunity to probe glueball
phenomenology at colliders \cite{Juknevich:2009ji}.
Additionally, models with $\Lambda_d \ll m_q \simeq \sqrt{s}$ can give
glueball dark matter, so long as $\Lambda_d$ is
sufficiently small that the glueball has a lifetime longer than
the age of the universe.
\subsection{Dark shower signals}
Qualitatively distinct phenomenology can occur in a different regime
(e.g., see\cite{ArkaniHamed:2008qn,Baumgart:2009tn})
in which $m_q \lesssim \Lambda_d \ll \sqrt{s}$ ($\sqrt{\hat{s}}$),
where the dark non-Abelian sector is expected to shower,
fragment, hadronize, and decay (for the states that have
no conserved quantum
number)\cite{Carloni:2010tw,Carloni:2011kk,Schwaller:2015gea,Cohen:2015toa}.
One possibility follows from production of dark fermions that
subsequently shower in the dark sector, followed by
decays of dark mesons back to standard model jets.
In the case where the decay lengths of the dark mesons are
macroscopic, this can give emerging jet signals
at colliders.\cite{Schwaller:2015gea}
Another possibility is that there are stable dark mesons --
dark matter -- also produced during the dark shower.
In this case, the dark matter is produced in an ordinary
QCD-like parton shower along with other light degrees
of freedom that decay hadronically. The result is a
multijet plus missing transverse momentum signature
where one of the jets is closely aligned with the direction
of the missing momentum, called a ``semi-visible''
jet\cite{Cohen:2015toa}.
\subsection{Meson production and decay}
In the regime where $\Lambda_d \sim m_q \sim \sqrt{s}$ ($\sqrt{\hat{s}}$),
meson production and decay is likely the most promising way to
probe confining, non-Abelian dark sectors.
At LEP II and the LHC, with $\sqrt{s} \sim \sqrt{\hat{s}} \sim v_{\rm EW}$,
the signals bear some resemblance to the older studies of technicolor
theories where the meson phenomenology dominates the experimental
observables.\cite{Hill:2002ap,Martin:2008cd,Barbieri:2010mn,Andersen:2011yj,Brod:2014loa}
This is not hard to understand - in general collider experiments can
much more easily produce dark mesons than dark baryons
(just like their QCD analogues). In other cases where the dark
sector mass scales are smaller, for example
$\Lambda_d \sim m_q \sim 1$~GeV (where dark matter self-interactions
could affect small-scale structure), the spectroscopy of these
theories could be probed by lower energy experiments,
such as high luminosity $b$-factories\cite{Hochberg:2015vrg}.
Composite dark sectors contain a large number of resonances.
Among the mesons, both the (pseudo)scalar and vector mesons
provide excellent opportunities for collider studies.
The scalar mesons are generally the lightest new particles
in theories with $m_q \lesssim \Lambda_d$ as demonstrated
by lattice simulations. Vector mesons also provide an
excellent probe of composite dynamics, especially when
there is some effective kinetic mixing between the photon
or electroweak gauge bosons and the new vector mesons.
In the following, we consider a study of one concrete example,
the lightest charged meson in Stealth Dark Matter.
\subsubsection{Case Study Example: Lightest Stealthy Mesons}
In Stealth Dark Matter, the lightest charged meson is a $0^{-+}$
that we denote by $\Pi^{\pm}$. In contrast to e.g.
supersymmetric extensions of the standard model where the lightest
supersymmetric particle serves as a dark matter candidate,
in composite models these charged states can be significantly
lighter than the dark matter itself. Direct searches for
charged states can therefore have better reach than generic
missing-energy searches for certain composite models.
As a first approximation, treating the $\Pi^\pm$ as point-like scalars
carrying unit electric charge, the production cross-section
from electron-positron collisions is
\begin{equation}
\sigma(e^+ e^- \rightarrow \Pi^+ \Pi^-) = \frac{\pi \alpha^2}{8E^2} \left(1 - \frac{M_{\Pi}^2}{E^2} \right)^{3/2},
\end{equation}
where $E$ is half of the center-of-mass energy of the collision,
roughly 100 GeV for LEP. This gives e.g. a cross section of 0.2 pb
with $M_\Pi = 80$ GeV; since the LEP experiments recorded
approximately 1000 pb${}^{-1}$ of integrated luminosity,
hundreds of candidate events would have been produced.
If the $\Pi^{\pm}$ are stable on collider timescales, then searches
for charged tracks can give a bound.
On the other hand, with appropriate electroweak couplings the
$\Pi^{\pm}$ can decay to standard model particles via annihilation
of its constituent fermions into a $W$ boson \cite{Appelquist:2015yfa}.
Because the initial state is spin-zero, this decay proceeds only
through the longitudinal part of the $W$, if the decay is into a
fermion doublet $ff'$, the width will be proportional to the
final-state fermion masses: with $m_f \gg m_{f'}$,
\begin{equation}
\Gamma \propto m_f^2 \left( 1 - \frac{m_f^2}{M_\Pi^2} \right)^2.
\end{equation}
Again focusing on LEP, for $M_\Pi$ of order 100 GeV, the branching of $\Pi^{\pm}$ decays is roughly 70\% into $\tau \bar{\nu}_\tau$, and 30\% into $c\bar{s}$ pairs. Combined with the large production cross-section, this leads to a robust constraint from stau searches at LEP \cite{Heister:2001nk,Heister:2003zk,Abdallah:2003xe,Achard:2003ge,Abbiendi:2004gf} that require $M_\pi \gtrsim 90$ GeV (assuming that the decay of $\Pi$ is prompt). More restrictive bounds may be obtainable by using LHC data, but would require a more detailed study.
The translation of this bound into a bound on the dark matter mass itself depends on the spectrum of the dark sector. If the composite dark matter candidate is the lightest meson, then the $\Pi^{\pm}$ will tend to be nearly degenerate with it, so that the bound applies directly to $M_{DM}$. (Exceptions are possible, for example, in models which are embedded directly into electroweak symmetry breaking, the $\Pi^{\pm}$ can become the longitudinal modes of the $W$ and $Z$ bosons \cite{Ryttov:2008xe,Belyaev:2010kp}, removing the bound.) On the other hand, if the dark matter is baryonic, then it will tend to be heavier than $\Pi^{\pm}$, so that the bound is stronger; for example in \cite{Appelquist:2015yfa}, the dark matter mass bound is $M_{DM} \gtrsim 250-320$ GeV, depending on the specific model parameters which determine $M_{DM} / M_\Pi$. For the case of glueball dark matter, any charged fermions will be much heavier than the dark matter candidate by necessity to ensure its stability, which again effectively removes the bound.
Going forward, it is important to note that this analysis is relatively simplistic. In particular, we have treated the $\Pi^{\pm}$ as point-like charged particles, but their interactions with the photon will actually be proportional to a momentum-dependent form factor $F(q^2)$. The form factor satisfies $F(0) = 1$ - that is, at zero momentum transfer the $\Pi^{\pm}$ do appear point-like - but for e.g. Drell-Yan photoproduction, the momentum transfer at the vertex becomes $q^2 = 4M_\Pi^2$ at threshold, and the form factor may be significantly different. (In QCD the form factor is larger at this value of $q^2$, growing to $|F| \sim 6$ due to the dominant effects of the $\rho$ vector meson in this channel \cite{Feng:2014gba}.)
Lattice calculation of such form factors would allow for more accurate calculations of the production of the $\Pi^{\pm}$. However, lattice simulations are carried out in Euclidean spacetime, which means that only spacelike form factors ($q^2 < 0$) are readily accessible. Lattice studies both for QCD (reviewed in \cite{Brandt:2013ffb}) and for SU$(2)$ gauge theory \cite{Hietanen:2013fya} have shown good agreement with vector-meson dominance (VMD) models in the spacelike region, although far from the vector ($\rho$) resonance itself. The more challenging direct calculation of the timelike form factor using the L\"{u}scher finite-volume method has been demonstrated recently in lattice QCD \cite{Feng:2014gba}. A similar calculation in a different strongly-coupled theory could provide direct input for collider studies, as well as giving an interesting test of vector-meson dominance away from QCD.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,204 |
module Finance.OpenSymbology.Types (
BloombergEntry (..)
, BloombergId (..)
, BloombergName (..)
, BloombergSymbol (..)
, PricingSource (..)
) where
import Data.Text (Text)
import Finance.OpenSymbology.PricingSourceAbbreviations
import Finance.OpenSymbology.PricingSourceCategories
import Finance.OpenSymbology.PricingSourceDescriptions
data BloombergId = BloombergId {bId :: Text} deriving (Eq,Ord,Show)
data BloombergName = BloombergName {bName :: Text} deriving (Eq,Ord,Show)
data BloombergSymbol = BloombergSymbol {bSymbol :: Text} deriving (Eq,Ord,Show)
data BloombergEntry =
BloombergHeader |
BloombergEntry {
beName :: BloombergName,
beSymbol :: BloombergSymbol,
bePricingSource :: Maybe Abbreviation,
beSecurityType :: Maybe Text,
beMarketSector :: Text,
beBloombergId :: BloombergId,
beBloombergCompositeId :: BloombergId,
beSourceId :: Maybe Text,
beUniqueId :: Maybe Text } deriving (Show,Eq)
data PricingSource = PricingSource Category Abbreviation Description deriving (Ord,Eq,Show,Read)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,382 |
import csv, string, sys
nomenclatural_statuses_to_keep = {}
for status in [
'',
'conservandum',
'nom. cons.',
'Nom. cons.',
'nom. cons. des.',
'nom. cons. prop.',
'protectum',
'Nomen novum',
'correctum',
'orth. cons.',
'legitimate',
'www.nearctica.com/nomina/beetle/colteneb.htm',
'Ruhberg et al., 1988',
'later usage',
]:
nomenclatural_statuses_to_keep[status] = True
not_extinct = ['1531', # Sarcopterygii
'10565', # Saurischia
'118547', # Aviculariidae
'1402700', # Trophomera
# '11919', # Didelphimorphia
# '1021564', # Cruciplacolithus
# '1530', # Actinopterygii
#'1170022', # Tipuloidea
#'1340611', # Retaria
#'1124871', # Labyrinthulomycetes [Labyrinthomorpha??]
#'102024', # Ophiurinidae - problem is Ophiurina
#'1064058', # Rhynchonelloidea genus/superfamily
#'1114655', # Tetrasphaera - different from GBIF
]
# These are the taxa with taxonomic status '' that occurred in phylesystem as of
# June 2016 (when these taxa were deleted from OTT). (from deprecated.tsv)
grandfathered = {}
def init_grandfathered(grandfathered):
for (id, name) in [
('10180190', 'Opulaster opulifolius'),
('11899420', 'Palaemonetes granulosus'),
('11704707', 'Olivioxantho denticulatus'),
('10527330', 'Chamaeleolis chameleontides'),
('11399158', 'Phyrignathus lesuerii'),
('10527966', 'Cylicia magna'),
('11444963', 'Epicrates anguilifer'),
('11078615', 'Egernia whitei'),
('10522666', 'Trogon aurantiventris'),
('10692084', 'Tauraco livingstoni'),
('10525002', 'Piculus leucolalemus'),
('10520170', 'Archaeopteryx lithographica'),
('11444785', 'Mesopropithecus pithecoides'),
('11167068', 'Zalambdalestes lechei'),
('10531957', 'Protungulatum donnae'),
('11024850', 'Megaladapis madagascariensis'),
('11078603', 'Megaladapis grandidieri'),
('11458858', 'Anthropornis nordenskjoeldi'),
('11081142', 'Gobipteryx minuta'),
('11390044', 'Pagophilus groenlandica'),
('10793056', 'Ommatophoca rossi'),
('10525092', 'Sivatherium giganteum'),
('10692824', 'Mesohippus bairdi'),
('10689467', 'Penaeus semisculcatus'),
('10543655', 'Palaemonetes atribunes'),
('10530648', 'Albunea occulatus'),
('102843', 'Hypsidoridae'),
('10697026', 'Badumna longinquus'),
('10184114', 'Cylactis pubescens'),
('11256401', 'Melanobatus leucodermis'),
('11083597', 'Squilla mikado'),
('11102182', 'Basilosaurus cetoides'),
('11103647', 'Pseudastacus pustulosa'),
('10532033', 'Hyopsodus paulus'),
('1435408', 'Auletes'),
('10532250', 'Allosaurus fragilis'),
('10537012', 'Gallimimus bullatus'),
('1178867', 'Songlingornis'),
('10532020', 'Cypselosoma australis'),
('1407317', 'Lithornis'),
('10957072', 'Velociraptor mongoliensis'),
]:
grandfathered[id] = name
init_grandfathered(grandfathered)
irmng_file_name = sys.argv[1]
profile_file_name = sys.argv[2]
taxonomy_file_name = sys.argv[3]
synonyms_file_name = sys.argv[4]
taxa = {}
synonyms = {}
roots = []
class Taxon:
def __init__(self, id, parentid, name, rank, tstatus, nstatus):
self.id = id
self.parentid = parentid
self.name = name
self.rank = rank
self.tstatus = tstatus
self.nstatus = nstatus
self.keep = False
self.extinctp = False
def read_irmng():
# 0 "TAXONID","SCIENTIFICNAME","SCIENTIFICNAMEAUTHORSHIP","GENUS",
# 4 "SPECIFICEPITHET","FAMILY","TAXONRANK","TAXONOMICSTATUS",
# 8 "NOMENCLATURALSTATUS","NAMEACCORDINGTO","ORIGINALNAMEUSAGEID",
# 11 "NAMEPUBLISHEDIN","ACCEPTEDNAMEUSAGEID","PARENTNAMEUSAGE",
# 14 "PARENTNAMEUSAGEID","TAXONREMARKS","MODIFIED","NOMENCLATURALCODE"
rows = 0
source_taxon_count = 0
source_synonym_count = 0
unallocated = 0
with open(irmng_file_name, 'rb') as csvfile:
csvreader = csv.reader(csvfile)
header = csvreader.next()
if header[5] != 'FAMILY':
print >>sys.stderr, '** Unexpected column name in header row', header[-3]
for row in csvreader:
taxonid = row[0]
longname = row[1]
auth = row[2]
rank = row[6]
tstatus = row[7] # TAXONOMICSTATUS
nstatus = row[8] # NOMENCLATURALSTATUS
syn_target_id = row[12]
parent = row[-4]
synonymp = (tstatus == 'synonym' or (syn_target_id != '' and syn_target_id != taxonid))
if synonymp:
source_synonym_count += 1
else:
source_taxon_count += 1
# Kludge to get rid of redundancies e.g. Megastoma
if tstatus == '':
for value in row:
if 'awaiting allocation' in value:
tstatus = 'lose'
unallocated += 1
break
if parent == '':
roots.append(taxonid)
# Calculate taxon name
genus = row[3]
if rank == 'species':
epithet = row[4]
name = ('%s %s')%(genus,epithet)
elif rank == 'genus':
name = genus
elif rank == 'family':
family = row[5]
name = family
elif len(auth) > 0 and longname.endswith(auth):
name = longname[0:len(longname)-len(auth)-1]
else:
name = longname
taxon = Taxon(taxonid, parent, name, rank, tstatus, nstatus)
if synonymp:
taxon.parentid = syn_target_id
synonyms[taxonid] = taxon
else:
taxa[taxonid] = taxon
rows += 1
if rows % 250000 == 0:
print >>sys.stderr, rows, taxonid, name
# FOR DEBUGGING
# break
print >>sys.stderr, 'Source: %s taxa, %s synonyms' % (source_taxon_count, source_synonym_count)
print >>sys.stderr, 'Flushing %s unallocated' % unallocated
print >>sys.stderr, 'Processed: %s taxa, %s synonyms' % (len(taxa), len(synonyms))
def fix_irmng():
# Get rid of all synonym of a synonym
# "10704","Decapoda Latreille, 1802","Latreille, 1802",,,,"order",,,,,,,"Malacostraca","1190","cf. Decapoda (Mollusca)","01-01-2012","ICZN"
loser_synonyms = {}
for syn in synonyms.itervalues():
if syn.parentid in synonyms:
loser_synonyms[syn.id] = True
print >>sys.stderr, "Indirect synonyms:", len(loser_synonyms)
for syn_id in loser_synonyms:
del synonyms[syn_id]
# Short-circuit taxon parents that are synonyms
loser_parent_count = 0
for taxon in taxa.itervalues():
if taxon.parentid in synonyms:
taxon.parentid = synonyms[taxon.parentid].parentid
loser_parent_count += 1
print >>sys.stderr, "Indirect parents:", loser_parent_count
# Decide which taxa to keep
keep_count = 0
missing_parent_count = 0
taxon_statuses_to_keep = ['accepted', 'valid', '']
for taxon in taxa.itervalues():
if taxon.keep:
True # already seen
elif (taxon.id in grandfathered or
(taxon.tstatus in taxon_statuses_to_keep and
# reduces number of kept taxa from 1685133 to 1351145
taxon.nstatus in nomenclatural_statuses_to_keep)):
scan = taxon
while not scan.keep:
if scan.id in grandfathered: print >>sys.stderr, 'Grandfathering', taxon.name
scan.keep = True
keep_count += 1
if scan.parentid == '':
break
parent = taxa.get(scan.parentid)
if parent == None:
missing_parent_count += 1
break
scan = parent
print >>sys.stderr, "Keeping %s taxa" % keep_count
print >>sys.stderr, "%s missing parents" % missing_parent_count
# Read the file that has the extinct annotations
with open(profile_file_name, 'rb') as csvfile:
csvreader = csv.reader(csvfile)
header = csvreader.next()
if header[1] != 'ISEXTINCT':
print >>sys.stderr, "** Expected to find ISEXTINCT in header row but didn't:", header[1]
for row in csvreader:
taxonid = row[0]
taxon = taxa.get(taxonid)
if taxon == None: continue
taxon.extinctp = (row[1] == 'TRUE')
if taxonid in not_extinct:
if not taxon.extinctp:
print >>sys.stderr, 'Already not extinct: %s(%s)' % (taxonid, taxon.name)
else:
print >>sys.stderr, 'Fixing extinctness of %s(%s)' % (taxonid, taxon.name)
taxon.extinctp = False
def extinctness_report():
# Report on nonextinct descended from extinct
count = 0
for taxon in taxa.itervalues():
if taxon.keep and not taxon.extinctp:
parentid = taxon.parentid
parent = taxa.get(parentid)
if parent != None and parent.extinctp:
count += 1
if taxon.rank != 'species':
print >>sys.stderr, ("Extant taxon %s(%s) with extinct parent %s(%s)"%
(taxon.id, taxon.name, parentid, parent.name))
print >>sys.stderr, 'Extant taxa with extinct parent:', count
# Write it out
# Returns True if one or more children also got written
def write_irmng():
def write_taxon(taxon, taxfile):
parentid = taxon.parentid
if parentid == '':
parentid = '0'
flags = ''
if taxon.extinctp:
flags = 'extinct'
taxfile.write('%s\t|\t%s\t|\t%s\t|\t%s\t|\t%s\t|\t\n'%(taxon.id, parentid, taxon.name, taxon.rank, flags))
with open(taxonomy_file_name, 'w') as taxfile:
print 'Writing %s'%taxonomy_file_name
taxfile.write('%s\t|\t%s\t|\t%s\t|\t%s\t|\t%s\t|\t\n'%('uid', 'parent_uid', 'name', 'rank', 'flags'))
taxfile.write('%s\t|\t%s\t|\t%s\t|\t%s\t|\t%s\t|\t\n'%('0', '', 'life', 'no rank', ''))
for taxon in taxa.itervalues():
if taxon.keep:
write_taxon(taxon, taxfile)
with open(synonyms_file_name, 'w') as synfile:
print 'Writing %s'%synonyms_file_name
synfile.write('uid\t|\tname\t|\ttype\t|\t\n')
for syn in synonyms.itervalues():
taxon = taxa.get(syn.parentid)
if taxon != None and taxon.keep and not taxon.extinctp:
status = syn.nstatus
if status == '':
status = syn.tstatus
if status == '': status = 'synonym'
synfile.write('%s\t|\t%s\t|\t%s\t|\t\n'%(syn.parentid, syn.name, status.lower()))
read_irmng()
fix_irmng()
extinctness_report()
write_irmng()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,319 |
#include "Precompiled.h"
#include "../TestChipmunk2.h"
static cpSpace* space;
typedef struct OneWayPlatform
{
cpVect n; // direction objects may pass through
cpArray* passThruList; // list of objects passing through
} OneWayPlatform;
static OneWayPlatform platformInstance;
static int
preSolve ( cpArbiter* arb, cpSpace* space, KDvoid* ignore )
{
CP_ARBITER_GET_SHAPES ( arb, a, b );
OneWayPlatform *platform = ( OneWayPlatform* ) ( a->data );
if ( cpvdot ( cpArbiterGetNormal ( arb, 0 ), platform->n ) < 0 )
{
cpArbiterIgnore ( arb );
return cpFalse;
}
return cpTrue;
}
static KDvoid update ( KDint ticks )
{
KDint steps = 1;
cpFloat dt = 1.0f / 60.0f / (cpFloat) steps;
for ( KDint i = 0; i < steps; i++ )
{
cpSpaceStep ( space, dt );
}
}
static cpSpace* init ( KDvoid )
{
cpResetShapeIdCounter ( );
space = cpSpaceNew ( );
space->iterations = 10;
space->gravity = cpv ( 0, -100 );
cpBody *body, *staticBody = space->staticBody;
cpShape* shape;
// Create segments around the edge of the screen.
shape = cpSpaceAddShape ( space, cpSegmentShapeNew ( staticBody, cpv ( -320, -240 ), cpv ( -320, 240 ), 0.0f ) );
shape->e = 1.0f; shape->u = 1.0f;
shape->layers = NOT_GRABABLE_MASK;
shape = cpSpaceAddShape ( space, cpSegmentShapeNew ( staticBody, cpv ( 320, -240 ), cpv ( 320, 240 ), 0.0f ) );
shape->e = 1.0f; shape->u = 1.0f;
shape->layers = NOT_GRABABLE_MASK;
shape = cpSpaceAddShape ( space, cpSegmentShapeNew ( staticBody, cpv ( -320, -240 ), cpv ( 320, -240 ), 0.0f ) );
shape->e = 1.0f; shape->u = 1.0f;
shape->layers = NOT_GRABABLE_MASK;
// Add our one way segment
shape = cpSpaceAddShape ( space, cpSegmentShapeNew ( staticBody, cpv ( -160,-100 ), cpv ( 160,-100 ), 10.0f ) );
shape->e = 1.0f; shape->u = 1.0f;
shape->collision_type = 1;
shape->layers = NOT_GRABABLE_MASK;
// We'll use the data pointer for the OneWayPlatform struct
platformInstance.n = cpv ( 0, 1 ); // let objects pass upwards
shape->data = &platformInstance;
// Add a ball to make things more interesting
cpFloat radius = 15.0f;
body = cpSpaceAddBody ( space, cpBodyNew ( 10.0f, cpMomentForCircle ( 10.0f, 0.0f, radius, cpvzero ) ) );
body->p = cpv ( 0, -200 );
body->v = cpv ( 0, 170 );
shape = cpSpaceAddShape ( space, cpCircleShapeNew ( body, radius, cpvzero ) );
shape->e = 0.0f; shape->u = 0.9f;
shape->collision_type = 2;
cpSpaceAddCollisionHandler ( space, 1, 2, KD_NULL, preSolve, KD_NULL, KD_NULL, KD_NULL );
return space;
}
static KDvoid destroy ( KDvoid )
{
ChipmunkDemoFreeSpaceChildren ( space );
cpSpaceFree ( space );
}
chipmunkDemo OneWay =
{
"One Way Platforms",
KD_NULL,
init,
update,
destroy,
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 856 |
LatestCongressElectionsFeaturesWhite HouseTrump AdministrationThe Future of Labor
Prosecutors Say Jeffrey Epstein Had Safe Full of Cash, Diamonds, and a Fake Passport
Rafi Schwartz
Filed to:Jeffrey Epstein
Image: Elizabeth Williams (AP)
This week, accused child rapist Jeffrey Epstein will find out whether he will be allowed to spend his pre-trial days in his plush Manhattan home or if he'll remain in jail until his day in court.
Speaking at a bail hearing in a Manhattan courtroom on Monday, prosecutors also revealed what investigators found in a locked safe at Epstein's luxury New York home: diamonds, "piles of cash," and—perhaps most damning for Epstein's argument that he should be allowed to remain free while he awaits trail—an expired passport with what appeared to be his photo but with a fake name that registers his residence as Saudi Arabia, according to the New York Times.
These are just the latest bizarre finds in Epstein's palatial Manhattan estate, which also reportedly featured a collection of fake eyes, a chess set with pieces designed to look like his staffers in their underwear, and a commissioned mural painting of Epstein in prison. Attorneys for Epstein have argued their client should be allowed to remain at the property and does not present a flight risk, despite the fact that prosecutors say he has three active U.S. passports, a jet, and his own private island dubbed "pedophile island" by locals.
In a court filing last week, Epstein's lawyers claimed that:
A spotless 14-year record of walking the straight and narrow, complemented by an exemplary 10-year history of diligent sex offender registration and reporting, is compelling proof [Epstein] was able, once the prior investigation commenced, to conform his conduct to the law's dictates.
"He didn't re-engage in this activity," Epstein's attorney Martin Weinberg said in court on Monday, per the Times. "It's not like he's an out-of-control rapist."
Epstein is accused of having "sexually exploited and abused dozens of minor girls" according to federal prosecutors. He has pleaded not guilty to all charges.
More from Splinter
Jeffrey Epstein's New York Mansion Is Reportedly Full of Psycho Shit
I Wonder Why Jeffrey Epstein Reportedly Shipped Himself a Massive Paper Shredder
Billionaire Pedophile Jeffrey Epstein Is Finally Arrested
Senior writer. When in doubt he'll have the soup. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,828 |
\section*{Key Points}
\textbf{Question:} Can an interpretable machine learning model distinguish between cEEG activities and provide meaningful explanations of each prediction?
\\
\textbf{Findings:} Interpretable neural network ProtoPMed-EEG uses case-based reasoning to distinguish six EEG activities with superior performance to the corresponding state-of-the-art black box (i.e. uninterpretable) model. Interpretable dimension reduction tools allow us to map the ictal-interictal-injury continuum to 2D, with demonstrations provided in a series of videos. \\
\textbf{Meaning:} We demonstrate the value of interpretability in machine learning for providing insight into the ictal-interictal-injury continuum. Interpretable, accurate models reduce the obstacles in the deployment of cEEG monitoring as the model's explanation can be reviewed by a trained practitioner.
\\
\section*{Structured Abstract}
IMPORTANCE:
An interpretable machine learning model can provide faithful explanations of each prediction and yet maintain higher performance than its black box counterpart. \\
OBJECTIVE:
To design an interpretable machine learning model which accurately predicts EEG protopatterns while providing an explanation of its predictions with assistance of a specialized GUI. To map the cEEG latent features to a 2D space in order to visualize the ictal-interictal-injury continuum and gain insight into its high-dimensional structure. \\
DESIGN, SETTING, AND PARTICIPANTS:
50,697 50-second cEEG samples from 2,711 ICU patients collected between July 2006 and March 2020 at Massachusetts General Hospital. Samples were labeled as one of 6 EEG activities by domain experts, with 124 different experts providing annotations.\\
MAIN OUTCOMES AND MEASURES:
Our neural network is interpretable because it uses case-based reasoning: it compares a new EEG reading to a set of learned prototypical EEG samples from the training dataset. Interpretability was measured with task-specific neighborhood agreement statistics. Discriminatory performance was evaluated with AUROC and AUPRC. \\
RESULTS:
The model achieves AUROCs of 0.87, 0.93, 0.96, 0.92, 0.93, 0.80 for classes Seizure, LPD, GPD, LRDA, GRDA, Other respectively. This performance is statistically significantly higher than that of the corresponding uninterpretable (black box) model with $p<0.0001$.
Videos of the ictal-interictal-injury continuum are provided. \\
CONCLUSION AND RELEVANCE:
Our interpretable model and GUI can act as a reference for practitioners who work with cEEG patterns. We can now better understand the relationships between different types of cEEG patterns.
In the future, this system may allow for targeted intervention and training in clinical settings. It could also be used for re-confirming or providing additional information for diagnostics. \\
\section*{Introduction}
Seizures and status epilepticus are found in 20\% of patients with severe medical and neurologic illness who undergo brain monitoring with electroencephalography (EEG) because of altered mental status \cite{pmid21204818, pmid10668693, pmid10478706}.
Every hour of seizures detected on EEG further increases the risk of permanent disability or death \cite{De_Marchis2016, pmid24595203}.
More ambiguous patterns of brain activity, consisting of periodic discharges or rhythmic activity, are even more common, and occur in nearly 40\% of patients undergoing EEG monitoring \cite{pmid26943901}.
More than two decades ago Chiappa et al hypothesized that IIIC activity lies along a spectrum, an ``ictal-interictal-injury continuum'' (IIIC) \cite{pmid8978624}:
patterns at one end of this continuum are hypothesized to cause brain injury and are difficult to distinguish from seizures on the EEG; patterns at the other bear little resemblance to seizures and are thought to cause little to no harm. Although this hypothesis has gained wide acceptance \cite{chong2005eeg, pmid33475321},
confirming it has been difficult because, until recently, the only method for quantifying IIIC patterns has been manual review of the EEG, which does not scale to large cohorts. Consequently, debates about the clinical significance of IIIC patterns and how to treat them have been going on for decades \cite{pmid31198061,pmid29139014,pmid8978624, pmid29979290, pmid19851892}.
Recent progress in machine learning (``AI''), and the availability of large EEG datasets, has recently made it possible to develop an automated algorithm (SPaRCNet) \cite{older_eeg_paper} to detect and classify IIIC patterns with a level of accuracy comparable to physician experts \cite{pmid33131680}.
Using this algorithm to comprehensively annotate nearly 2000 continuous EEG recordings, two recent studies found evidence that prolonged IIIC activity, like seizures, increases the risk of disability and death \cite{pmid34231244, arXiv:2203.04920[stat.ME]}.
However, SPaRCNet, like most AI approaches, is uninterpretable (``black box'') meaning that it cannot explain how it reaches its conclusions. In other medical applications, uninterpretable models often fail to generalize to the clinic or are later shown to depend on confounding factors, generating risks in real-world applications \cite{rudin2019stop}. Furthermore, despite high performance on specific tasks, uninterpretable models cannot provide broader medical insights to advance the field. In the case of classifying IIIC patterns, the black box model does not provide insight into the nature of the underlying ``continuum.'' That is, by grouping all variants within each broad class of rhythmic and periodic discharge patterns together, the black box model does not help us to discover which patterns within each class are harmful and merit treatment, and which are benign.
In this paper, we present novel \textit{interpretable} machine learning methods to classify seizures and rhythmic and periodic EEG patterns \cite{chen2019this,barnett2021case} and visualize their relationships within the IIIC. These tools allow us to classify EEG samples in an auditable way and to map the IIIC onto a two-dimensional `map' that reveals high-dimensional information. Specifically, we introduce an interpretable neural network, ProtoPMed-EEG, that leverages case-based reasoning. ProtoPMed-EEG is more accurate than its uninterpretable counterpart, SPaRCNet \cite{older_eeg_paper}, and provides an explanation of every prediction through comparisons to learned prototypical patterns. These explanations are of the form ``this EEG looks like that EEG,'' the \textit{TEEGLLTEEG explanation method}. Then, using a second algorithm, PaCMAP \cite{wang2021understanding}, to reduce EEG information from a high-dimensional feature space to 2D while still preserving global structure, we can inspect relationships and distances between EEG patterns, allowing us to ``map'' the IIIC into a 2D space that users can explore. The map reveals that, despite being given distinct class names, EEG patterns within the IIIC do not exist in isolated islands. Rather, each class is connected to each other class via a sequence of intermediate patterns. This provides the quantitative support for the IIIC hypothesis, by explicitly demonstrating the existence of any underlying continuum among pathological patterns of brain activity.
\section*{Methods}
\subsection*{EEG data and expert labels}
ProtoPMed-EEG was trained and tested on a large-scale EEG study \cite{iir_eeg_paper} consisting of 50,697 events from 2,711 patients hospitalized between July 2006 and March 2020 who underwent continuous EEG as part of clinical care at Massachusetts General Hospital. EEG electrodes were placed according to the International 10-20 system. The large group was intended to ensure broad coverage of all variations of IIIC events encountered in practice.
124 EEG raters from 18 centers labeled varying numbers of 10-second EEG segments in 2 stages. The first stage involved targeted annotations by small groups of independent experts. The second stage involved multiple labeling rounds by larger groups of independent experts. Raters were given a forced choice of six options: seizure (SZ), lateralized periodic discharges (LPD), generalized periodic discharges (GPD), lateralized rhythmic delta activity (LRDA), generalized rhythmic delta activity (GRDA), and ``Other'' if none of those patterns was present.
\subsection*{Model Interpretability}
In this paper, we define interpretable models as models that ``explain their predictions in a way that humans can understand'' \cite{rudin2019stop}.
Our model uses case-based reasoning to provide interpretability for the end user. Case-based reasoning is using previous examples to reason about a new case. The model learns a set of previous cases called \textit{prototypical samples}. Each prototypical sample is a 50-second EEG sample from the training set that is particularly useful for classifying new cases.
For each prediction, the model generates similarity scores between the new case and the set of prototypical samples learned during training. Each explanation is of the form ``this sample is class X because it is similar to these prototypes of class X, and not similar to prototypes of other classes." We call this method the TEEGLLTEEG explanation method because it makes explanations of the form ``this EEG looks like that EEG.''
Every prediction made by our model follows the same logic as the explanation provided by the model. This means that the model explanations have \textit{perfect fidelity} with the underlying decision-making process.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{fig/modeArch.jpg}
\caption{Model architecture. Input sample $x$ is passed through a feature extractor $f()$, followed by a prototype layer $g()$. The prototype layer calculates the angular distances between the sample feature and the prototypes. The angular distances are multiplied with class affinity to generate the logits (class scores). The softmax calculation converts the logits into prediction probabilities.}
\label{fig:arch}
\Description{Model architecture. Input sample $x$ is passed through a feature extractor $f()$, followed by a prototype layer $g()$. The prototype layer calculates the angular distances between the sample feature and the prototypes. The angular distances are multiplied with class affinity to generate the logits. The softmax calculation converts the logits into prediction probabilities.}
\end{figure}
\subsection*{Model Training and Development}
\ifx\mycmd\undefined
Our dataset consists of 50-second EEG samples with class labels indicating the EEG pattern. As in \citet{older_eeg_paper}, the patients were split into approximately equally sized training and test sets. Then, samples with higher counts of expert votes ($\geq 20$) were selected from the training set to form the \textit{prototype set} of cases that are candidates to become prototypical samples.
ProtoPMed-EEG is an adaptation of ProtoPNet, a convolutional neural network model with a prototype interpretability layer, built on recent work by \citet{LiLiuChenRudin}, \citet{chen2019this}, and \citet{barnett2021case}, incorporating angular similarity as in \citet{donnelly2022deformable}. The network architecture for ProtoPMed-EEG is shown in Figure \ref{fig:arch}. Our model consists of feature extraction layers, which are initialized by weights from \citet{older_eeg_paper}; a prototype layer, which computes the similarity (angular distance) between each learned prototype and feature extractor output for each sample; and a final linear layer, which maps the similarity to each prototype into class scores for each EEG pattern using \textit{class-connection} vectors. Each prototype is a vector of length 1275 that exists in the same \textit{latent space} as the outputs from the feature extraction layer. Each learned prototype corresponds to an actual EEG sample from the prototype set, where the output of the feature extractor on a prototypical sample is the prototype corresponding to that sample. The final linear layer is initialized such that the first 30 prototypes are single-class prototypes (that is, being similar to the prototype increases the class score for one of the EEG patterns) and the next 15 prototypes are dual-class prototypes (that is, being similar to the prototype increases the class score for two of the EEG patterns).
The model is trained with an objective function containing loss terms to encourage accuracy, a clustering structure in the latent space, and separation between prototypes.
The training was completed in 4 hours using two NVIDIA V100 GPUs.
Refer to Appendix \ref{app:model_details} for further information on model architecture and training.
\else
Our dataset consists of 50-second EEG samples $x_i$ with class labels $y_i$ indicating the EEG pattern. As in \citet{older_eeg_paper}, the patients were split into approximately equally sized training and test sets. Then, samples with higher counts of expert votes ($\geq 20$) were selected from the training set to form the \textit{prototype set} of cases that are candidates to become prototypical samples.
ProtoPMed-EEG is an adaptation of ProtoPNet, a convolutional neural network model with a prototype interpretability layer, built on a recent series of works by \citet{LiLiuChenRudin}, \citet{chen2019this}, and \citet{barnett2021case}, incorporating angular similarity as in \citet{donnelly2022deformable}. The network architecture for ProtoPMed-EEG is shown in Figure \ref{fig:arch}. Our model consists of feature extraction layers $f()$, which are initialized by weights from \citet{older_eeg_paper}; prototype layer $g()$, which computes the similarity (angular distance) between each learned prototype and feature extractor output $f(\mathbf{x})$ for each sample $\mathbf{x}$; and a final linear layer $h()$, which maps the similarity to each prototype into class scores for each EEG pattern using \textit{class-connection} vectors. The set of $m$ prototypes is $P=\{p_j\}_{j=0}^m$, where each prototype is a vector of length 1275 that exists in the same \textit{1275-dimensional latent space} as outputs $f(\mathbf{x})$. Each learned prototype corresponds to an actual EEG sample from the prototype set, where $f(\mathbf{x}_{i'})$ is the prototype corresponding to the prototypical sample $\mathbf{x}_{i'}$. Network layer $h()$ is initialized such that the first 30 prototypes are single-class prototypes (that is, being similar to the prototype increases the class score for one of the EEG patterns) and the next 15 prototypes are dual-class prototypes (that is, being similar to the prototype increases the class score for two of the EEG patterns).
The model is trained with an objective function containing loss terms to encourage accuracy, a clustering structure in the latent space and separation between prototypes.
Training is completed in 4 hours using two Nvidia V100 GPUs.
Refer to Appendix \ref{app:model_details} for further information on model architecture and training.
\fi
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{fig/ProtoEEG_layout_newscreenshot.jpg}
\caption{The graphical user interface (GUI) of the interpretable system. On the left top panel is the 2D embedding map, with each dot representing one EEG sample. Dots can be displayed with shading according to 9 different available schemes (human majority, model prediction, model uncertainty, Seizure burden etc.). A user can click on the map to select any sample of interest; the 3 nearest prototypes as displayed on the right, ranked according to similarity score (SIM). For each sample/prototype, 10 seconds of the EEG and a 10 minute spectrogram (centered on the 10-second EEG segment) are displayed with human votes and model predictions shown on top of the EEG; a pie chart is provided to visualize the class distribution according to the model or human votes, depending on the selected color scheme. For each prototype, under its pie chart, we also list the values of three terms: similarity score (SIM), class connection (AFF), and class contribution score (SCORE).}
\label{fig:GUI_screenshot}
\Description{[A] Current sample waveform (10s). [B] Current sample spectrogram (600s). [C] Expert votes for the current sample (if available). [D] Model prediction on the current sample. [E] A mapping of samples (dots) and prototypes (triangles). [F] Current sample location in mapping. [G] Nearest prototypes' locations in the map. [H] Mapping options. [I] Option to show prototypes from the nearest 3 classes instead of the nearest 3 prototypes overall. [J] Spectrogram and waveform of the nearest prototype. [K] Expert annotations on prototypes. [L] Model prediction for prototypes. [M] Similarity score between prototype and current sample. [N] Class connection of prototype with top prototype class. [O] Top class score contribution from prototype.}
\end{figure}
\subsection*{Graphical User Interface}
We also present a specialized graphical user interface (GUI) which allows users to explore the interpretable model and its predictions on the test set. A screenshot is shown in Figure \ref{fig:GUI_screenshot}.
On the bottom left of the GUI, we present the waveform [A], spectrogram [B] and the expert-annotator votes [C] of the currently selected sample. Below the expert-annotator votes we display the model prediction [D] on the currently selected sample. On the upper left of the GUI, we present the 2D representation of the embedded space that was generated using PaCMAP [E]. We can see the location of the current sample [F], as well as the location of the nearest prototypes [G] in the 2D representation. The color schema and mapping methods can be changed using drop-down menus [H]. Which prototypes are displayed can be changed using a selection box [I], where one option shows the nearest prototypes regardless of class and the other shows the nearest prototype from each of the three highest-score classes. Along the right-hand side, we display the waveform, spectrogram [J], expert-annotator votes [K] and model prediction [L] for the prototypical samples from the three displayed prototypes (in this case, the three nearest prototypes). For each of the displayed prototypes, we show the similarity score between the prototypical sample and the currently selected sample [M], the class-connection between the prototype and the predicted class (affinity) [N] and the class score added by that prototype to the predicted class [O].
\subsection*{Evaluation of model performance}
We evaluate our model's performance using area under receiver operating characteristic curve (AUROC) scores, area under precision-recall curve (AUPRC) scores and neighborhood analysis measures. For comparing AUROC scores between the black box model and ours, we use the Delong test \cite{delong1988comparing} for statistical significance. For AUPRC comparisons, we test for statistical significance using the bootstrapping method with 1000 bootstrap samples. For more detail on these statistical significance tests, refer to Appendix \ref{app:sig}. We further evaluated model performance using neighborhood analysis. Details are provided in the Results section.
\section*{Results}
\subsection*{Model Performance}
The classification performance of our interpretable model ProtoPMed-EEG statistically significantly exceeds that of its uninterpretable counterpart SPaRCNet in distinguishing Seizures, LPDs, GPDs, LRDAs, and GRDAs, as measured both by AUROC and AUPRC scores (p<0.001). Results for ROC and PRC curve analysis are shown in Figure \ref{fig:results} and Table \ref{tab:aucs} for values. These findings hold when bootstrapping by patient or by sample.
\begin{table*}
\caption{AUROC, AUPRC and neighborhood analysis of the interpretable model compared to its uninterpretable counterpart, SPaRCNet \cite{older_eeg_paper}. Each prediction problem is one-vs-all. The column name ``All'' refers to a mean for all classes weighted by the number of samples in each class. For ``Neighborhood Analysis by Votes,'' a lower score is better; for all other metrics, a higher score is better. 95\% confidence intervals are shown in square brackets.
We use the bootstrapping method described in Appendix \ref{app:boot} for AUPRC and AUROC. We use $\sigma_u=\frac{\sigma}{\sqrt{N}}$ for the neighborhood analyses. The test set size $N$ is $35740$ cEEG samples. Our results show statistically significant improvements over SPaRCNet for all comparisons, see Appendix Table \ref{tab:significance_test_percent} for AUROC and AUPRC significance test results and Appendix Table \ref{tab:significance_test} for neighborhood analysis significance test results.}
\label{tab:aucs}
\scriptsize
\begin{tabular}{llccccccc}
\toprule
& & Other & Seizure & LPD & GPD & LRDA & GRDA & All\\
\midrule
\multirow{2}{1cm}{AUROC} & Interp. & \textbf{0.80 [0.80, 0.80]} & \textbf{0.87 [0.87, 0.87]} & \textbf{0.93 [0.93, 0.93]} & \textbf{0.96 [0.96, 0.96]} & \textbf{0.92 [0.92, 0.93]} & \textbf{0.93 [0.93, 0.93]} & \textbf{0.91 [0.91, 0.91]} \\
& Uninterp. \cite{older_eeg_paper} & 0.79 [0.79, 0.79] & 0.86 [0.86, 0.86] & 0.90 [0.90, 0.90] & 0.94 [0.94, 0.94] & 0.92 [0.92, 0.92] & 0.92 [0.92, 0.92] & 0.89 [0.89, 0.89] \\
\multirow{2}{1cm}{AUPRC} & Interp. & \textbf{0.52 [0.52, 0.52]} & \textbf{0.25 [0.24, 0.25]} & \textbf{0.81 [0.81, 0.81]} & \textbf{0.92 [0.92, 0.92]} & \textbf{0.76 [0.76, 0.76]} & \textbf{0.67 [0.67, 0.67]} & \textbf{0.74 [0.74, 0.74]}\\
& Uninterp. \cite{older_eeg_paper} & 0.47 [0.47, 0.47] & 0.19 [0.19, 0.19] & 0.73 [0.73, 0.73] & 0.89 [0.89, 0.89] & 0.74 [0.74, 0.74] & 0.63 [0.63, 0.63] & 0.70 [0.70, 0.70] \\
Neighborhood & Interp. & \textbf{0.52 [0.51, 0.53]} & \textbf{0.31 [0.29, 0.34]} & \textbf{0.73 [0.72, 0.74]} & \textbf{0.87 [0.87, 0.88]} & \textbf{0.75 [0.75, 0.76]} & \textbf{0.61 [0.60, 0.62]} & \textbf{0.79 [0.79, 0.79]}\\
Analysis by Max & Uninterp. \cite{older_eeg_paper} & 0.47 [0
47, 0.48] & 0.23 [0.21, 0.25] & 0.65 [0.65, 0.66] & 0.84 [0.83, 0.84] & 0.67 [0.66, 0.68] & 0.56 [0.55, 0.57] & 0.65 [0.65, 0.65] \\
Neighborhood & Interp. & \textbf{1.61 [1.61, 1.61]} & \textbf{1.66 [1.65, 1.68]} & \textbf{1.46 [1.45, 1.46]} & \textbf{1.46 [1.46, 1.47]} & \textbf{1.58 [1.58, 1.59]} & \textbf{1.55 [1.55, 1.56]} & \textbf{1.53 [1.53, 1.53]}\\
Analysis by Votes & Uninterp. \cite{older_eeg_paper} & 1.63 [1.63, 1.63] & 1.69 [1.68, 1.71] & 1.50 [1.49, 1.50] & 1.48 [1.48, 1.48] & 1.61 [1.60, 1.61] & 1.55 [1.55, 1.56] & 1.55 [1.55, 1.56]\\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{fig/plotprroc_v5.jpg}
\caption{The receiver operating characteristic curves and precision-recall curves for ProtoPMed-EEG (solid lines) compared to its uninterpretable counterpart SPaRCNet (dashed lines).}
\label{fig:results}
\Description{The receiver operating characteristic curves and the precision-recall curves for ProtoPMed-EEG (our model) compared to its uninterpretable counterpart. The AUROCs and AUPRCs are also displayed on the chart, matching those in Table \ref{tab:aucs}.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{fig/Figure_prototypesNneighbors.png}
\caption{The nearest neighbors for a prototype of the ProtoPMed-EEG model. We show top 3 nearest test samples corresponding to the prototype in 8 cases. The full set of prototypes and their neighbors can be found in Appendix \ref{app:neighbor_analyses}.}
\Description{We show 32 EEG readouts corresponding to the 8 prototypes, and the nearest three test samples for each prototype.}
\label{fig:neighborhood}
\end{figure}
\subsection*{Neighborhood Analysis: Quantitative} As one way to evaluate the interpretability of ProtoPMed-EEG, we analyzed the neighborhood of each prototype to examine the structure of the learned latent space. In a well-trained system, the neighborhood of a $c$-class prototype will primarily contain samples from class $c$. For each sample in the test set, we calculate the percentage of the 10 nearest test set neighbors where the class with the most votes is the same as for the sample (``by max''). We also consider the neighborhood analyses ``by vote,'' where for each sample we calculate the mean cross-entropy of the vote distribution of the sample with the vote distribution of each of the 10 nearest neighbors. Here, we consider cross entropy as a discrete distribution across classes, and check whether the cross entropy of the test point matches the distribution of classes from the nearest neighbors. The interpretable model does statistically significantly better than its uninterpretable counterpart across all metrics and classes with $p<0.05$ for each comparison (see Table \ref{tab:aucs}).
\subsection*{Neighborhood Analysis: Qualitative} For the system to be understandable, we require that samples in the neighborhood around a prototype are also qualitatively similar to the prototype according to domain experts.
In Figure \ref{fig:neighborhood}, for each of eight prototypes from our model, we explore the three nearest neighbors from the test set to the prototype. In each case, the neighboring samples are similar to the prototype not only in class, but also in amplitude, peak-to-peak distance and other domain-relevant qualities. This demonstrates to domain experts our model's concept of ``similarity.'' Qualitative neighborhood analyses for all prototypes showing the six nearest neighbors from each of the training and test sets can be found in Appendix \ref{app:neighbor_analyses}.
\subsection*{Mapping the Ictal-Interictal-Injury Continuum}
In Figure \ref{fig:GUI_screenshot}, the PaCMAP coloring and distance between samples (points) are based on the model class scores for each sample. This results in a structure with outer points (arms) corresponding to single classes and reveals dense, thread-like paths mapping a gradual change between IIIC classes. This lends credence to the concept of a ``continuum'' between ictal and interictal EEG patterns. We further sampled along those paths between each pair of IIIC patterns, and produced videos which demonstrate the smooth continuum from one pattern to the other. Video links are provided in Appendix \ref{app:links}.
\section*{Discussion}
In this study, we developed an interpretable deep learning model to classify seizures and rhythmic and periodic patterns of brain activity that occur commonly in patients with severe neurologic and medical illness, and introduced a specially-designed user interface to explore the interpretable model. Each explanation follows the TEEGLLTEEG explanation method (``this EEG looks like that EEG.'') This is the first adaption of interpretable neural networks to cEEG and the first quantitative exploration of the ictal-interictal-injury continuum. The model is trained on a large and diverse set of EEG signals annotated by multiple independent experts, and evaluated against the best current state-of-the-art black box model. Our work yielded advancements in both the model performance and model interpretability. Compared to the black box baseline model, our model achieves significant improvements in pattern classification performance as measured by AUROC, AURPC, and in neighborhood agreement metrics (Table \ref{tab:aucs}). Higher neighborhood analysis scores indicate that the interpretable model learned a more consistent neighborhood in the latent space. High neighborhood consistency and agreement is especially desirable as it indicates the model is more robust and more interpretable.
While machine learning, and specifically deep learning, has been used for EEG classification tasks including seizure detection \cite{older_eeg_paper,tzallas2012automated,abbasi2019machine,craik2019deep,amin2019deep}, our interpretable model goes beyond traditional tasks, providing clinicians with the means to validate diagnoses and providing researchers verifiable evidence for smooth continuity of the IIIC, confirming the existence of an underlying continuum. Although there are past works on leveraging prototypes to provide explanations for model predictions \cite{zhang2020tapnet,huang2019deep,gee2019explaining}, the prototypes were limited to single-classes which is insufficient for mapping IIIC patterns, as evidenced by the common occurrence in our dataset of patterns on which expert opinions are divided as to the correct classification. Our introduction of dual-class prototypes enables our model to place prototypes between two classes in the latent space, providing insights into EEG patterns in the transitional states.
Due to the existence of samples between classes in the EEG pattern classifications, particularly the transitional states, human annotations often yield disagreements. Our interpretable model can provide additional support for clinicians in day-to-day ICU patient monitoring. The model provides prediction along with its neighborhood and matched prototype information to the users, allowing users to evaluate the reliability of predictions, thereby acting as an ``assistant'' in the process. Since our GUI provides expert annotation records alongside model predictions, this tool may also be helpful in training clinicians; our case-based-reasoning design would provide not only explanations for model predictions but also insights for existing expert annotations.
\subsection*{Limitations}
In our dataset, the number of seizure class samples was substantially smaller compared to other classes. We alleviated the class imbalance issue during the training process, however, a more balanced dataset would be helpful for achieving better performance. More seizure data samples could also help the model learn a better set of seizure prototypes with more robust representations. In future studies, we could also leverage the additional 10-minute spectrogram information as it is also a component in the human decision process.
\section*{Conclusions}
In conclusion, our interpretable deep learning was able to accurately classify six common clinically-relevant patterns of potentially harmful brain activity that occur commonly in ICU patients. It showed better performance compared to the previous black box approach, and provides sufficient explanation for its own predictions.
\section*{Acknowledgments}
We acknowledge support from the National Science Foundation under grants IIS-2147061 (with Amazon), HRD-2222336 and IIS-2130250, from the NIH (R01NS102190, R01NS102574, R01NS107291, RF1AG064312, RF1NS120947, R01AG073410, R01HL161253), and from NSF (2014431).
We also acknowledge Drs. Aaron F. Struck, Safoora Fatima, Aline Herlopian, Ioannis Karakis, Jonathan J. Halford, Marcus Ng, Emily L. Johnson, Brian Appavu, Rani A. Sarkis, Gamaleldin Osman, Peter W. Kaplan, Monica B. Dhakar, Lakshman Arcot Jayagopal, Zubeda Sheikh, Olha Taraschenko, Sarah Schmitt, Hiba A. Haider, Jennifer A. Kim, Christa B. Swisher, Nicolas Gaspard, Mackenzie C. Cervenka, Andres Rodriguez, Jong Woo Lee, Mohammad Tabaeizadeh, Emily J. Gilmore, Kristy Nordstrom, Ji Yeoun Yoo, Manisha Holmes, Susan T. Herman, Jennifer A. Williams, Jay Pathmanathan, Fábio A. Nascimento, Mouhsin M. Shafi, Sydney S. Cash, Daniel B. Hoch, Andrew J. Cole, Eric S. Rosenthal, Sahar F. Zafar, and Jimeng Sun, who played major roles in creating the labelled EEG dataset and SPaRCNet in the study.
\section*{Conflict of Interest Disclosures}
Dr. Westover is a co-founder of Beacon Biosignals, which played no role in this work.
\section*{Author Contribution Statement}
Idea conception and development: Alina Jade Barnett, Zhicheng Guo, Jin Jing, Brandon Westover, Cynthia Rudin. Interpretable model code: Zhicheng Guo, Alina Jade Barnett. GUI code: Jin Jing. Data preparation: Jin Jing, Zhicheng Guo, Alina Jade Barnett, Wendong Ge. Writing: Alina Jade Barnett, Zhicheng Guo, Jin Jing, Brandon Westover, Cynthia Rudin.
\FloatBarrier
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,808 |
Q: No results found in query I'm trying to modify a field for the the currentUser in Parse. I run the following code and my log keeps returning "no results found". I've logged the ObjectId for reference and verified that it's correct in my database.
ParseQuery<ParseUser> query = ParseQuery.getQuery("User");
query.getInBackground(ParseUser.getCurrentUser().getObjectId(), new GetCallback<ParseUser>() {
public void done(ParseUser object, ParseException e) {
if (e == null) {
Log.i("objectId", userObjectId);
object.put("instructorId", instructorId.getText().toString());
object.saveEventually();
} else {
Log.e("objectId", "Error: " + e.getMessage());
Log.i("objectId", userObjectId);
}
}
});
}
});
A: Your assumption that the User class has the classname User is probably the issue. Try _User.
BUT, you should really be creating your query as
ParseQuery<ParseUser> query = ParseUser.getQuery();
That way you don't even need to know the classname for the user object.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,412 |
Grimmeodendron is a plant genus of the family Euphorbiaceae first described as a genus in 1908. It is native to the West Indies.
Species
Grimmeodendron eglandulosum (A.Rich.) Urb. - Bahamas, Cuba, Hispaniola (Dominican Republic, Haiti)
Grimmeodendron jamaicense Urb. - Jamaica
References
Euphorbiaceae genera
Flora of the Caribbean
Hippomaneae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 799 |
package br.com.pucrs.io;
import br.com.pucrs.collections.GeneralTree;
import br.com.pucrs.model.ContentBook;
import java.io.IOException;
import java.nio.file.Files;
import java.nio.file.Path;
import java.util.Scanner;
public class BookReader {
private ContentBook capituloAtual;
private ContentBook secaoAtual;
private ContentBook subSecaoAtual;
private int nroCapitulos;
private int nroSecoes;
private int nroSubSecoes;
private int nroParagrafos;
public BookReader() {
capituloAtual = null;
secaoAtual = null;
subSecaoAtual = null;
nroCapitulos = 0;
nroSecoes = 0;
nroSubSecoes = 0;
nroParagrafos = 0;
}
public GeneralTree<ContentBook> readFile(Path path) throws IOException {
GeneralTree<ContentBook> tree = new GeneralTree<>();
try (Scanner scanner = new Scanner(Files.newBufferedReader(path))) {
while (scanner.hasNextLine()) {
String line = scanner.nextLine();
String tipo = line.substring(0, 2).trim();
ContentBook page = new ContentBook(line.substring(2), tipo);
switch (tipo) {
case "L":
tree.add(page, null);
break;
case "C":
addChapter(tree, page);
break;
case "S":
addSection(tree, page);
break;
case "SS":
addSubSection(tree, page);
break;
case "P":
addParagraph(tree, page);
break;
default:
//não faz nada
break;
}
}
} catch (IOException e) {
throw new IOException(e.getMessage(), e);
}
return tree;
}
public void clear() {
capituloAtual = null;
secaoAtual = null;
subSecaoAtual = null;
nroCapitulos = 0;
nroSecoes = 0;
nroSubSecoes = 0;
nroParagrafos = 0;
}
private void addSubSection(GeneralTree<ContentBook> tree, ContentBook page) {
subSecaoAtual = page;
tree.add(subSecaoAtual, secaoAtual);
nroSubSecoes++;
}
private void addParagraph(GeneralTree<ContentBook> tree, ContentBook page) {
if (subSecaoAtual != null) {
tree.add(page, subSecaoAtual);
} else if (secaoAtual != null) {
tree.add(page, secaoAtual);
} else {
tree.add(page, capituloAtual);
}
nroParagrafos++;
}
private void addSection(GeneralTree<ContentBook> tree, ContentBook page) {
subSecaoAtual = null;
secaoAtual = page;
tree.add(secaoAtual, capituloAtual);
nroSecoes++;
}
private void addChapter(GeneralTree<ContentBook> tree, ContentBook line) {
if (capituloAtual != null) {
secaoAtual = null;
subSecaoAtual = null;
}
capituloAtual = line;
tree.add(line, tree.getRoot());
nroCapitulos++;
}
public int getNroCapitulos() {
return nroCapitulos;
}
public int getNroSecoes() {
return nroSecoes;
}
public int getNroSubSecoes() {
return nroSubSecoes;
}
public int getNroParagrafos() {
return nroParagrafos;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,535 |
\section{INTRODUCTION}
Coarse-grained (CG) molecular dynamics simulations are computationally efficient and can simulate long time scale processes that are not accessible to all-atom simulations \cite{Saunders2013Coarse-GrainingBiology,Kmiecik2016Coarse-GrainedApplications,Nielsen2004CoarseMaterials,dePablo2011Coarse-GrainedNanocomposites}. They are widely used for understanding dynamical processes in physics, chemistry and biology\cite{Souza2021MartiniDynamics,Dignon2018SequenceModel,BEST2017,Joseph2021Physics-drivenAccuracy,Hyeon2011CapturingModels,Denesyuk2013Coarse-GrainedThermodynamics,Farr2021NucleosomeInteractions,Grime2016Coarse-grainedSelf-assemblyb,Mansbach2017Coarse-GrainedPeptides,Shmilovich2020DiscoverySimulation,Latham2021,Latham2019ImprovingData,Giulio2021}. The accuracy of these simulations depends on how well the force fields can describe the interactions among various components of the system under investigation. Therefore, algorithms and methodologies that can produce high-quality coarse-grained force fields (CGFF), or CG potential energy, are of key interest.
Numerous approaches have been introduced for systematically parameterizing CGFFs.\cite{Noid2013Perspective:Systems,Gkeka2020MachineSystems}.
Top-down approaches often rely on a set of experimental structural or thermodynamic properties to fine-tune CGFFs and ensure the physical relevance of CG simulations
\cite{Noid2013Perspective:Systems,Latham2022UnifyingProteins,Souza2021MartiniDynamics,Nielsen2004CoarseMaterials,Shelley2001ASimulations,Davtyan2012AWSEM-MD:Biasing,Wu2018AWSEM-IDP:Proteins,Darre2015SIRAH:Electrostatics}. On the other hand, bottom-up approaches learn CGFFs from an ensemble of atomistic configurations collected using simulations performed at finer resolution, typically with all-atom force fields \cite{Ercolessi1994InteratomicMethod,Izvekov2004EffectiveForce-matching,Zhang2018DeePCG:Networks,Zhang2018DeepMechanics}.
From the configurational ensemble, various physical quantities and correlation functions can be computed to serve as targets for recreation with CGFF. \cite{Tschop1998SimulationPolycarbonates,Louis2000CanColloids,Davtyan2012AWSEM-MD:Biasing,Wu2018AWSEM-IDP:Proteins,Savelyev2009MolecularDNA,Akkermans2000AMelts,Ruhle2009VersatileApplications,Noid2013SystematicProtocols,Toth2007EffectivePotentials,Li2010Characterizing-Hairpin,Hansen_2005,Liwo2001Cumulant-basedField,Clark2012ThermodynamicLiquids,Mirzoev2013MagiC:Modeling}
In addition, CGFFs can also be optimized to enforce the statistical consistency between their corresponding Boltzmann distributions and the reference configurational distribution with variational methods.\cite{Izvekov2005ASystems,Shell2008TheProblems,Noid2008TheModels,Noid2008TheModelsb}. The consistency is achieved when the CG potential energy matches the potential of mean force dictated by the all-atom force field and the mapping that connects atomistic and CG configurations.
Existing variational methods optimize force field parameters by formulating and solving regression problems or maximizing the likelihood of observing the reference configurations. The force matching method\cite{Izvekov2005ASystems,Ercolessi1994InteratomicMethod} and its generalization \cite{Mullinax2009GeneralizedSystems,Kohler2022Force-matchingForces,Mullinax2009GeneralizedSystems} belong to the former category and aim to minimize the difference between forces for CG coordinates calculated from the CGFF and target values estimated from all-atom simulations. A perfect match in forces ensures that the CG energy function reproduces the potential of mean force. On the other hand, the relative entropy method\cite{Shell2008TheProblems}, or equivalently maximum likelihood \cite{Noid2013SystematicProtocols}, directly optimizes the CG energy function by minimizing the relative entropy and maximizing the overlap between the CG Boltzmann distribution and the configurational distribution from all-atom simulations.
The relative entropy is minimized when the CG energy function reproduces the potential of mean force, and the CG Boltzmann distribution assigns high probabilities to configurations from all-atom simulations.
While existing force field parameterization methods have found great success in many applications, they are not without limitations. For example, the relative entropy method needs to run simulations to sample from trial CG potentials in every optimization step and can be computationally expensive. While the force matching method can learn the CG potential directly without iterative sampling, it often requires extra atomic force information, and the quality of the resulting potential can be sensitive to the accumulation of errors through the integration of the estimated force.
Here we developed a new variational method called potential contrasting for learning CGFFs, and applied it for multi-scale coarse-graining of protein folding. Potential contrasting generalizes the noise contrastive estimation method \cite{Gutmann2010Noise-contrastiveModels} to formulate force field parameterization into a classification problem. Input for the method is a target ensemble of protein conformations from all-atom simulations, and no atomic force information is required. When applied to the peptide Trp-cage, we found that potential contrasting can produce force fields that accurately reproduce the all-atom conformational ensemble and capture the complex folding landscape. The method also revealed the importance of including many-body potentials in CG models to describe protein biophysics with a reduced degree of freedom and implicit solvation. In addition, we showed that potential contrasting is computationally efficient and trivially parallelizable, enabling the parameterization of transferable force fields using large datasets collected from multiple proteins.
\section{METHODS}
Potential constrasting combines a machine learning method called noise contrastive estimation\cite{Gutmann2010Noise-contrastiveModels} (NCE) with molecular simulation techniques. In this section, we first introduce NCE using the M\"uller potential\cite{Muller1979LocationProcedure} as an example. Then we present how the NCE method is generalized and used in potential constrasting to learn CGFFs for protein folding.
\subsection{Noise contrastive estimation}
\begin{figure}[t]
\includegraphics[width=\textwidth]{./figure_1.eps}
\caption{
Noise contrastive learning accurately reproduces the M\"uller potential from sampled data only.
(a) Illustration of the noise contrastive estimation method. Representative data samples from Monte Carlo sampling of the M\"uller potential and noise samples from a noise distribution $q(\bm{x})$ are shown in blue and orange, respectively. The target data distribution is parameterized using a potential energy function $u_p(\bm{x};\bm{\theta})$, i.e., $p(\bm{x}; \bm{\theta}) \propto \exp(-\beta u_p(\bm{x}; \bm{\theta}))$, where $\bm{\theta}$ is a set of parameters. $\bm{\theta}$ is optimized in a logistic regression to classify the data and noise samples.
(b) Contour plot of the M\"uller potential. Energy is shown in the units of $k_BT$.
(c) Contour plot of the potential energy function $u_p(\bm{x}; \bm{\theta}^*)$ learned using noise contrastive estimation. Energy is shown in the units of $k_BT$.
}
\label{fig:mp}
\end{figure}
The NCE method\cite{Gutmann2010Noise-contrastiveModels} learns a probabilistic model on observed data. It is especially useful for learning unnormalized statistical models where the probability density function is only specified up to a normalization constant. It is evident that NCE is connected to bottom-up force field optimization, which aims to parameterize an energy function or an unnormalized Boltzmann distribution from data produced by all-atom simulations.
Here we use the M\"uller potential as an example to show how NCE helps to learn energy functions. Given a set of data (Figure \ref{fig:mp}a) drawn from the M\"uller potential with Markov chain Monte Carlo sampling, NCE aims to approximate their probability distribution with $p(\bm{x};\bm{\theta})$ defined as $\log p(\bm{x}; \bm{\theta}) = - \beta [u_p(\bm{x}; \bm{\theta}) - F_p]$, where $u_p(\bm{x}; \bm{\theta})$ is the potential energy parameterized with $\bm{\theta}$ and $F_p$ is the free energy. To optimize the parameters $\bm{\theta}$, NCE performs a logistic regression to discriminate the $N_p$ data samples $\{\bm{x}^i_p\}_{i=1}^{N_p}$ from $N_q$ noise samples $\{\bm{x}^i_q\}_{i=1}^{N_q}$ (Figure \ref{fig:mp}a) that are drawn from a noise distribution $q(\bm{x})$. Specifically, we assign binary labels of $y = 1$ and $y = 0$ to data and noise samples, respectively. NCE parameterizes the energy function by maximizing the following averaged log-likelihood of labels:
\begin{equation}
\label{eqn:log-likelihood}
\ell(\bm{\theta}, F_p) = \frac{1}{N_p} \Big[ \sum_{i=1}^{N_p}\log P(y=1|\bm{x}_p^i) + \sum_{i=1}^{N_q}\log P(y=0|\bm{x}_q^i) \Big],
\end{equation}
with
\begin{equation}
\label{eqn:posterior}
P(y = 1 | \bm{x}) = \frac{p(\bm{x}; \bm{\theta})}{p(\bm{x};\bm{\theta}) + \nu q(\bm{x})}\ \mathrm{and} \ P(y = 0 | \bm{x}) = \frac{\nu q(\bm{x})}{p(\bm{x};\bm{\theta}) + \nu q(\bm{x})},
\end{equation}
where $\nu = P(y=0)/P(y=1) = N_q/N_p$.
By definition, maximizing the above objective function forces the probability function $p(\bm{x};\bm{\theta})$ to assign high values to data samples (the first term) and low values to noise samples (the second term). In that regard, NCE is similar to the standard maximum likelihood estimation \cite{Myung2003TutorialEstimation.}, which assigns high probability on training data. Previous works\cite{Gutmann2010Noise-contrastiveModels} have proven that the solution $\bm{\theta}^*$ for optimizing $\ell(\bm{\theta}, F_p)$ behaves like the maximum likelihood estimator for large noise sample sizes and $p(\bm{x};\bm{\theta}^*)$ converges to the true data distribution. The advantage of NCE over maximum likelihood estimation is that the free energy $F_p$ is treated as a free parameter, and the optimization avoids the computationally expensive procedure for evaluating $F_p$ rigorously. In addition, a nice property of $\ell(\bm{\theta}, F_p)$ is that it is a concave function and has a unique maximum point if the potential energy function $u_p(\bm{x};\bm{\theta})$ is linear to $\bm{\theta}$.
Treating $F_p$ as an independent variable, while being advantageous, also introduces a dependence of NCE's performance on the noise distribution because the noise sample size is always limited in practice. If $p(\bm{x}; \bm{\theta})$ is a normalized density with conserved probability mass, as in the maximum likelihood optimization, increasing its value on data samples would implicitly decrease its value on regions outside the data. Such a balance of probability density is not guaranteed in NCE since $p(\bm{x}; \bm{\theta})$ is not strictly normalized due to the approximate treatment of $F_p$. The use of a noise distribution remedies this issue by allowing an explicit probability minimization for the region covered by noise samples. While a comprehensive theory is still missing on designing optimal noise distributions\cite{Chehab2022TheThink}, we find that a useful guiding principle is to design the noise distribution such that it covers the phase space occupied by and surrounding the data samples. Without significant overlap between data and noise samples, the objective function, $\ell(\bm{\theta}, F_p)$, can be trivially optimized by assigning high probability on data samples and low probability on noise samples without forcing $p(\bm{x}; \bm{\theta})$ to capture the distributional structure within the data samples. In such cases, both terms in the objective function approach the constant zero, and the gradient on $\bm{\theta}$ vanishes, hindering the optimization.
We parameterized the potential energy function $u_p(\bm{x}; \bm{\theta})$ using a two dimensional cubic spline\cite{Hastie2009ThePrediction} with 169 spline coefficients. The noise distribution $q(\bm{x})$ was chosen as the uniform distribution. $500,000$ samples were generated for both data and noise. We learned the parameters $\bm{\theta}$ by maximizing the NCE objective function $\ell(\bm{\theta}, F)$ (Eq. \ref{eqn:log-likelihood}) using the L-BFGS algorithm\cite{Zhu1997AlgorithmOptimization} (In practice, we minimize the negative of the NCE objective function). As shown in Figure \ref{fig:mp}c, $u_p(\bm{x}; \bm{\theta}^*)$ closely matches the underlying M\"uller potential (Figure \ref{fig:mp}b), supporting the effectiveness of NCE for learning potential energy functions.
\subsection{Potential contrasting for learning force fields}
\begin{figure}[t]
\includegraphics[width=1.0\textwidth]{./figure_2.eps}
\caption{
Workflow of the potential contrasting method for learning coarse-grained force fields for the Trp-cage protein.
The functional form of the potential energy function is chosen as $u_p(\bm{x}; \bm{\theta})$, where $\bm{\theta}$ is the set of parameters that need to be learned.
The ensemble of conformations from all-atom simulations are converted into a coarse-grained ensemble using a predefined CG mapping as data samples.
Here we map each amino acid into one coarse-grained particle at the $\mathrm{C}_\alpha$ position.
Based on the data samples, a noise potential $u_q(\bm{x})$ is designed and used to generate an ensemble of noise conformations and optimize the parameters $\bm{\theta}$ with potential contrasting. }
\label{fig:pc}
\end{figure}
We find that the current formulation of NCE, although theoretically sound, is not effective for learning molecular force fields in practice. Therefore, we developed a new method named potential contrasting by generalizing NCE and introducing a customized way of defining the noise distribution. We present details of the method with applications to protein molecules in mind, for which the development of CGFFs is of great significance but has been challenging \cite{Kmiecik2016Coarse-GrainedApplications, Latham2022UnifyingProteins}. However, potential contrasting is general and can be applied to other types of molecules.
\textbf{Generalizing NCE to unnormalized noise distributions.}
Current formulation of NCE\cite{Gutmann2010Noise-contrastiveModels} requires specifying noise distributions for which the normalized probability density can be determined at ease. This requirement restricts the choice of noise distributions because the normalization constant is difficult to compute for many probabilistic functions, including Boltzmann distributions defined by complex potentials. Here we propose that this requirement is not necessary, and generalize NCE to use noise distributions specified with a potential energy function $u_q(\bm{x})$. Specifically, we set $q(\bm{x}) = e^{-\beta[u_q(\bm{x}) - F_q]}$, where $F_q$ is the free energy. Similarly to $F_p$, we treat $F_q$ as an extra parameter in the optimization instead of computing its value explicitly. As a result, the logistic regression objective function in Eq.~\ref{eqn:log-likelihood} becomes
\begin{align}
\label{eqn:glog-likelihood_0}
\ell(\bm{\theta}, F_p, F_q) &= \frac{1}{N_p} \Big[ \sum_{i=1}^{N_p}\log \frac{1}{1 + \nu e^{-\beta[u_q(\bm{x}^i_p) - u_p(\bm{x}^i_p; \bm{\theta}) + F_p - F_q]}} + \sum_{i=1}^{N_q}\log \frac{1}{1 + \nu^{-1} e^{-\beta[u_p(\bm{x}^i_q; \bm{\theta}) - u_q(\bm{x}^i_q) + F_q - F_p]}} \Big].
\end{align}
Because the value of $\ell(\bm{\theta}, F_p, F_q)$ in Eq. \ref{eqn:glog-likelihood_0} only depends on $\bm{\theta}$ and the difference between $F_p$ and $F_q$, we merge the two free energy into one free parameter $\Delta F = F_p - F_q$, i.e.,
\begin{align}
\label{eqn:glog-likelihood}
\ell(\bm{\theta}, \Delta F) = \frac{1}{N_p} \Big[ \sum_{i=1}^{N_p}\log \frac{1}{1 + \nu e^{-\beta[u_q(\bm{x}^i_p) - u_p(\bm{x}^i_p; \bm{\theta}) + \Delta F]}} + \sum_{i=1}^{N_q}\log \frac{1}{1 + \nu^{-1} e^{-\beta[u_p(\bm{x}^i_q; \bm{\theta}) - u_q(\bm{x}^i_q) - \Delta F]}} \Big].
\end{align}
Potential contrasting uses $\ell(\bm{\theta}, \Delta F)$ as the objective function and optimizes the parameters $\bm{\theta}$ by maximizing $\ell(\bm{\theta}, \Delta F)$. $\bm{\theta}^*$ is used to represent optimized parameters.
\textbf{Defining the noise distribution for learning CGFFs of protein folding.}
As mentioned before, the performance of NCE depends critically on the noise distribution, which should produce samples with sufficient overlap with the training data. For low dimensional systems, a feasible choice for the noise is the uniform distribution used in the M\"uller potential example. For complex systems such as protein molecules, uniform distributions suffer the dimensionality curse to cover the relevant phase space. Our generalization to unnormalized Boltzmann distributions significantly broadens the choices of noise distributions to facilitate producing complex molecular structures that resemble data samples. We further propose an umbrella sampling procedure to design noise potential energy functions and enhancer overlap between noise and data samples.
We design the noise potential energy function such that the noise samples contain both folded and unfolded structures to match the configurational ensemble from all-atom simulations. For a given protein, we start with an energy function that includes terms for bonds, angles and dihedral angles defined as
\begin{equation}
\label{eqn:bonded}
u_\mathrm{bonded}(\bm{x}) = \sum_{i=1}^{L-1} \frac{1}{2}k_i (b_i - b_i^\circ)^2 + \sum_{i=1}^{L-2} S_\mathrm{angle}(a_i; \bm{c}^a_i) + \sum_{i=1}^{L-3} S_\mathrm{dihedral}(d_i; \bm{c}^d_i),
\end{equation}
where $L$ is the number of residues in the protein, and $b_i, a_i$ and $d_i$ represent the $i$th bond, angle and dihedral angle. A quadratic function is used for energies on bonds. $k_i$ and $b_i^\circ$ are the force constant and the equilibrium value for the $i$th bond. Cubic spline functions, $S_\mathrm{angle}$ and $S_\mathrm{dihedral}$, are used for energies on angles and dihedral angles. $\bm{c}^a_i$ and $\bm{c}^d_i$ are spline coefficients for the $i$th angle and dihedral angle, respectively. Using data samples, we fit each bonded energy term in $u_{\mathrm{bonded}}(\bm{x})$ independently such that it will reproduce the marginal distribution of the corresponding degree of freedom from the data samples. To generate both folded and unfolded structures for noise samples, we further carried out umbrella sampling simulations \cite{Torrie1977NonphysicalSampling,Tiwary2016} with $u_\mathrm{bonded}(\bm{x})$ by biasing the root-mean-squared-deviation (RMSD) from the folded structure towards different values.
We combined configurations sampled from all $M$ umbrella simulations together to construct the noise ensemble. The probability distribution of the generalized ensemble can be described as $p_\mathrm{gm}(\bm{x}) \propto \sum_{i=1}^M \exp(- \beta [u_i(\bm{x}) + v_i])$ \cite{PhysRevLett.63.1195,Kumar1992THEMethodb,Shirts2008StatisticallyStates,Bennett1976EfficientDatab,Ding2019FastEquations}, where $u_i(\bm{x})$ is the energy function used in the $i$th umbrella simulation that includes both $u_\mathrm{bonded}(\bm{x})$ and the bias function on the RMSD. $v_i$ are adjustable energies that need to be fitted and added to the potential energy $u_i(\bm{x})$ so that the relative free energies of the $M$ states match the relative populations of structures sampled from these states. Correspondingly, the noise potential function can be computed as $u_q(\bm{x}) = - \beta^{-1} \log \sum_{i=1}^M \exp(- \beta [u_i(\bm{x}) + v_i])$. More details on the procedure are included in the Supporting Information.
\textbf{Extending potential contrasting to multiple proteins.}
With the developments outlined above, potential contrasting can be used to parameterize CG energy functions for a specific protein by optimizing $\ell(\bm{\theta}, \Delta F)$ defined in Eq.~\ref{eqn:glog-likelihood}. It can be further generalized to learn CG potential functions with transferable parameters. Suppose that we can produce data and noise samples for a collection of proteins, the objective function to ensure that the CGFF reproduces the target configurational distribution for each protein can be defined as
\begin{align}
\label{eqn:glog-likelihood-tot}
\ell_\mathrm{tot}(\bm{\theta}, \{\Delta F_k\}_{k=1}^K) = \sum_{k=1}^K \frac{1}{N^k_p} \Big[ &\sum_{i=1}^{N^k_p}\log \frac{1}{1 + \nu_k e^{-\beta[u^k_q(\bm{x}^{ki}_p) - u_p(\bm{x}^{ki}_p; \bm{\theta}) + \Delta F_k]}} \nonumber \\
+ &\sum_{i=1}^{N^k_q}\log \frac{1}{1 + \nu_k^{-1} e^{-\beta[u_p(\bm{x}^{ki}_q; \bm{\theta}) - u^k_q(\bm{x}^{ki}_q) - \Delta F_k]}} \Big].
\end{align}
The above expression is a sum of potential contrasting objective functions (Eq. \ref{eqn:glog-likelihood}) introduced for each individual protein. $\{\bm{x}_p^{ki}: i = 1, \cdot\cdot\cdot N_p^k\}$ and $\{\bm{x}_q^{ki}: i = 1, \cdot\cdot\cdot, N_q^k\}$ represent the data and noise samples for the $k$th protein, with $N_p^k$ and $N_q^k$ corresponding to the respective sample sizes, and $\nu_k = N_q^k/N_p^k$. While the same energy function $u_p(\bm{x}^{ki}_p; \bm{\theta})$ with transferable parameters $\bm{\theta}$ is used, different noise potential energy functions, $ u^k_q(\bm{x}^{ki}_q)$, can be introduced for individual proteins. The aggregated objective function maintains the property of being concave if the CG energy function is linear to $\bm{\theta}$. We note that the objective function can be generalized straightforwardly if the CGFF introduces non-transferable parameters across proteins, as detailed in the Supporting Information.
\section{RESULTS}
Potential contrasting is a general-purpose method for force field parameterization. We focus on its application to protein folding and show that it can be used to optimize CGFFs for a specific protein and a collection of proteins. Given a sufficiently flexible functional form, the force field produced by potential contrasting can accurately reproduce the configurational distribution of all-atom simulations. We further demonstrate its efficiency by simultaneously optimizing over 12 proteins to derive CG potential functions with transferable parameters.
\subsection{Coarse-grained force field for the Trp-cage protein}
We applied potential contrasting to learn CGFFs for a 20 amino acids long peptide, Trp-cage. As detailed in the \emph{Methods Section}, potential contrasting parameterizes the force field by maximizing its effectiveness in differentiating data samples from noise samples. We use as data samples a total of $N_p = 1,044,000$ conformations from a 208-$\mu$s long molecular dynamics simulation with explicit solvents performed in Ref.~\citenum{Kresten2011HowFold}. This fully atomistic simulation captures multiple folding and unfolding events for the peptide. We generated $N_q = 1,044,000$ noise samples (Figure S3) that include both folded and disordered configurations and computed the noise potential $u_q(\bm{x})$ using the umbrella sampling procedure described in the \emph{Methods Section}. In the following, we use potential contrasting to learn three CGFFs with different flexibility and complexity. For simplicity, we only use $\mathrm{C}_\alpha$ atoms to represent protein conformations and define energies, but potential contrasting can be easily generalized to more refined structural models.
\begin{figure}[t!]
\includegraphics[width=0.95\textwidth]{./figure_3.eps}
\caption{ Parameterizing CGFFs for the Trp-cage protein using potential contrasting and all-atom simulations.
(a-c) Distributions of RMSD with respect to the folded structure for conformations sampled from the all-atom simulation (orange) and CG simulations with learned CG potentials that differ in the representation of the non-bonded interactions (Eq.~7-9).
(d) Free energy profiles along the RMSD with respect to the folded structure for conformations sampled from the all-atom simulation and CG simulations with the three different learned potentials.
(e-h) Free energy surfaces over the first two tICA coordinates for the all-atom simulation (h) and CG simulations with the three different learned potentials.
The three meta-stable states in h are labels as 1, 2, and 3, with the corresponding representative structures shown in part j, k, and l.
(i) The many-body potential $u_\mathrm{ss}^\mathrm{mb}(\bm{x}; \bm{\phi}^*)$ as a function of the RMSD with respect to the folded $\alpha$-helix structure.
}
\label{fig:trp-cage}
\end{figure}
\textbf{CGFF with bonded terms and pairwise non-bonded interactions.}
We first learned a CGFF, $u_p^\mathrm{pair}(\bm{x};\bm{\theta})$, that includes bonded terms and pairwise non-bonded terms defined as
\begin{align}
\label{eqn:pair}
u_p^\mathrm{pair}(\bm{x}; \bm{\theta}) = & u_\mathrm{bond}(\bm{x}) + u_\mathrm{angle}(\bm{x}) + u_\mathrm{dihedral}(\bm{x}) + u_\mathrm{elec}(\bm{x}) + u_\mathrm{contact}(\bm{x}) \nonumber \\
= &\sum_{i=1}^{L-1} \frac{1}{2} k_i (b_i - b_i^\circ)^2 + \sum_{i=1}^{L-2} S_\mathrm{angle}(a_i; \bm{c}^a_i) + \sum_{i=1}^{L-3} S_\mathrm{dihedral}(d_i; \bm{c}^d_i) + \nonumber \\
& \sum_{i=1}^{L-4} \sum_{j=i+4}^L \frac{q_i q_j}{4 \pi \epsilon r_{ij}} \exp(-r_{ij}/\lambda_D) + \sum_{i=1}^{L-4} \sum_{j=i+4}^L S_\mathrm{contact}(r_{ij}; \bm{c}_{ij}).
\end{align}
The bond, angle, and dihedral terms are similarly defined as in Eq.~\ref{eqn:bonded}. Non-bonded terms include electrostatics $u_\mathrm{elec}(\bm{x})$ and a contact energy term $u_\mathrm{contact}(\bm{x})$, both of which act between pairs of CG particles that are separated by four or more bonds. The electrostatic interaction is modeled using the Debye-H\"uckel theory, where $q_i$ is the net charge of the $i$th residue, $\lambda_D$ is the Debye screening length, and $r_{ij}$ is the distance between residues $i$ and $j$. The non-bonded contact energy is defined with cubic spline functions $S_\mathrm{contact}(r_{ij}; \bm{c}_{ij})$ and $\bm{c}_{ij}$ are spline basis coefficients (Figure S1). Because bond energies are much stronger than others, the parameters $b_i^\circ$ and $k_i$ were directly fitted based on the mean and the variance of the $i$th bond's distribution in the data samples. Therefore, the parameter $\bm{\theta}$ only includes spline basis coefficients, i.e., $\bm{\theta}=\{\bm{c}_i^a, \bm{c}_i^d, \bm{c}_{ij}\}$. To prevent overfitting, regularization terms on the potential energy $u_p(x; \bm{\theta})$ are added in the optimization to control their smoothness. Details on regularization terms are included in the Supporting Information. Since the energy function depends on the parameters $\bm{\theta}$ linearly, potential contrasting is guaranteed to produce a unique solution $\bm{\theta}^*$.
We carried out molecular dynamics simulations (see the Supporting Information for details) with the learned CGFF $u_p^\mathrm{pair}(\bm{x}; \bm{\theta}^*)$ to evaluate the resulting structural ensemble. Similar to that from the all-atom simulation, the distribution of RMSD with respect to the folded structure for CGFF is bimodal (Figure \ref{fig:trp-cage}a). Therefore, the learned CG potential function $u_p^\mathrm{pair}(\bm{x}; \bm{\theta}^*)$ captures both folded and unfolded structures. However, a significant discrepancy exists between the two distributions. The CG simulation produced fewer folded structures, and the two maximums of the corresponding RMSD distribution do not exactly match that of the all-atom result. The discrepancy is more clear if we convert the RMSD distribution histogram into free energy surfaces (Figure \ref{fig:trp-cage}d). Deviations can also be seen when comparing the free energy surface over the first two components of the time-independent component analysis\cite{Molgedey1994SeparationCorrelations,Naritomi2011SlowMotions,Scherer2015PyEMMAModelsb} (tICA), which describe the slowest processes observed in the simulation. The all-atom surface has three meta-stable states: one folded state and two different unfolded states that cannot be differentiated using RMSD alone (Figure \ref{fig:trp-cage}h). Although the CG simulation samples all three meta-stable states (Figure \ref{fig:trp-cage}e), it produces a smaller population of the folded state and does not capture the cooperative transitions between folded and unfolded structures (Figure S4).
\textbf{Adding many-body interactions parameterized using neural networks.}
The discrepancy between the CG and the all-atom simulations could be caused by the pair-wise potential being too restrictive and cannot capture many-body interactions that might arise due to coarse-graining. Next, we learned a more flexible energy function that includes an extra term parameterized using a feed-forward neural network with parameters $\bm{\phi}$,
i.e.,
\begin{equation}
\label{eqn:mb_nn}
u_p^\mathrm{nn}(\bm{x}; \bm{\theta}, \bm{\phi}) = u_p^\mathrm{pair}(\bm{x}; \bm{\theta}) + u^\mathrm{nn}_\mathrm{mb}(\bm{x}; \bm{\phi}).
\end{equation}
The additional energy term, $u^\mathrm{nn}_\mathrm{mb}(\bm{x}; \bm{\phi})$, is invariant to translations and rotations and takes angles, dihedral angles, and pairwise distances as inputs (Figure S2). It can represent complex interactions involving multiple residues because the neural network is fully connected to couple different degrees of freedom\cite{Wang2020}.
A CG simulation performed with the learned potential function $u_p^\mathrm{nn}(\bm{x}; \bm{\theta}^*, \bm{\phi}^*)$ now indeed matches the all-atom results well. The maximums of the RMSD distribution are much better placed (Figure \ref{fig:trp-cage}c), suggesting that the CG simulation accurately predicts the folded structure. Importantly, the CG simulation reproduces the relative population of the folded structure and the unfolded ensemble and the free energy barrier between them (Figure \ref{fig:trp-cage}d and S4). Similarly, the free energy surface of the first two tICA coordinates (Figure \ref{fig:trp-cage}g and \ref{fig:trp-cage}h) agrees well with the all-atom one. Therefore, despite only using only $\alpha$-carbons, the CGFF captures the complex folding landscape of the peptide determined from atomistic explicit solvent simulations.
\textbf{Adding secondary structure inspired many-body potentials.}
Although parameterizing the many-body energy term using a neural network improves the accuracy of the resulting force field, it has a few disadvantages. For instance, the potential function $u_p^\mathrm{nn}(\bm{x}; \bm{\theta}, \bm{\phi})$ is not linear to $\bm{\phi}$, and the optimized parameters depends on initial conditions. Moreover, it is difficult to interpret the many-body energy in simple physical terms. To avoid these issues, we learned a CG potential function with a secondary structure based many-body energy term. Secondary structure biases are frequently incorporated into coarse-grained models as fragment memories for improved quality of structural predictions\cite{Davtyan2012AWSEM-MD:Biasing, Latham2021,Rohl2004ProteinRosetta}. They help account for cooperative effects arising from water molecules involving many residues that are challenging to describe with pair-wise potentials.
Specifically, the secondary structure based many-body energy term is defined as
\begin{equation}
\label{eqn:mb_ss}
u_p^\mathrm{ss}(\bm{x}; \bm{\theta}, \bm{\phi}) = u_p^\mathrm{pair}(\bm{x}; \bm{\theta}) + u^\mathrm{ss}_\mathrm{mb}(\bm{x}; \bm{\phi}).
\end{equation}
It is parameterized using cubic spline functions as
$u^\mathrm{ss}_\mathrm{mb}(\bm{x}; \bm{\phi}) = S_\mathrm{ss}(\mathrm{rmsd\_ss}(\bm{x}, \bm{x}_\circ); \bm{c}^\mathrm{ss})$. Here $\mathrm{rmsd\_ss}(\bm{x}, \bm{x}_\circ)$ is the RMSD calculated on the $\alpha$-helix (residue 3 to residue 15) between a given structure $\bm{x}$ and the folded structure $\bm{x}_\circ$. The parameter $\bm{\phi}$ includes all the spline basis coefficients $\bm{c}^\mathrm{ss}$.
This design of the energy function in Eq.~\ref{eqn:mb_ss} further ensures linear dependence on parameters and a unique solution for force field optimization.
The CG simulation results using the learned potential $u_p^\mathrm{ss}(\bm{x}; \bm{\theta}^*, \bm{\phi}^*)$ are shown in Figure \ref{fig:trp-cage}b, \ref{fig:trp-cage}d , \ref{fig:trp-cage}f, and S4. Although the many-body energy term is restricted within the $\alpha$-helix, the CG simulation correctly reproduces the relative populations of folded and unfolded states and the free energy barrier. Its performance is almost as good as the potential with a neural network based many-body term defined over the whole protein. The learned many-body potential function $u^\mathrm{ss}_\mathrm{mb}(\bm{x}; \bm{\phi}^*)$ along the $\alpha$-helix RMSD is shown in Figure \ref{fig:trp-cage}i. It has a deep well near 0 nm and quickly approaches zero when the RMSD is larger than 0.3 nm. Therefore, the potential only plays a significant role in stabilizing the folded structure when the $\alpha-$helix is already close to the native state. Its impact is minimal when the $\alpha$-helix adopts unfolded configurations.
\subsection{Efficient optimization of transferable force fields with data from multiple proteins}
\begin{figure}[ht!]
\includegraphics[width=0.8\textwidth]{./figure_4.eps}
\caption{
Comparison between all-atom simulations and CG simulations performed with the learned transferable force field.
(a) For each of the 12 proteins, we show the folded structure (red) from the all-atom simulation, the structure (blue) from the CG simulation that has the lowest RMSD with respect to the folded structure, and the C$_\alpha$-RMSD (over all residues) between the two structures.
The two plots on the right of structures are distributions of RMSD to the folded structure and distributions of Rg (radius of gyration) for conformations sampled from all-atom simulations (orange) and CG simulations (blue).
(b) Trajectories of Rg and RMSD with respect to the folded structure for the all-atom simulation (orange) and the CG simulation (blue) of the Protein B. Although the data from all-atom simulations and CG simulations are plotted in the same figure, their time scales are different. Similar plots for other proteins are included in the Supporting Information.
}
\label{fig:all_hist}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=0.6\textwidth]{./figure_5.eps}
\caption{
Learned transferable contact potential energy functions between representative pairs of amino acids.
Similar plots for other pairs of amino acids are included in the Supplementary Information.
}
\label{fig:lj_contact}
\end{figure}
The above results suggest that potential contrasting is a powerful tool to parameterize flexible CGFFs for specific proteins and capture their complex folding landscapes. Next, we show that the method also allows efficient optimization of transferable force fields using all-atom simulations of 12 fast-folding proteins performed in Ref. \citenum{Kresten2011HowFold}.
The transferable force field for the $k$th protein is defined using Eq.~\ref{eqn:mb_ss} as
\begin{align}
\label{eqn:transferable}
u_p^k(\bm{x}_k) = u_\mathrm{bond}(\bm{x}_k) + u_\mathrm{angle}(\bm{x}_k) + u_\mathrm{dihedral}(\bm{x}_k) + u_\mathrm{contact}(\bm{x}_k) + u_\mathrm{elec}(\bm{x}_k) + u^\mathrm{ss}_\mathrm{mb}(\bm{x}_k; \bm{\phi}_k).
\end{align}
As a proof of principle, we only shared parameters for pair-wise non-bonded interactions and allowed protein-specific non-transferable parameters for both the bonded term and the many-body term.
The pair-wise contact potential is now defined as
\begin{equation}
\label{eqn:contact_transferable}
u_\mathrm{contact}(\bm{x}) = \sum_{i=1}^{L-4} \sum_{j=i+4}^L S_\mathrm{contact}(r_{ij}; \bm{c}^\mathrm{contact}_{IJ}).
\end{equation}
While $S_\mathrm{contact}(r; \bm{c}^\mathrm{contact}_{IJ})$ shares the same functional form as that in Eq.~\ref{eqn:pair}, its parameters now only depend on residue types $I$ and $J$. Because $\bm{c}^\mathrm{contact}_{IJ}$ are made to depend on residue types alone, they are transferable among proteins. Our choice of limiting the force field's transferability is due to the well-known challenges of predicting secondary structures in CG models \cite{Kmiecik2016Coarse-GrainedApplications}. While potential contrasting allows efficient optimization of all parameters across proteins, the accuracy of the resulting CGFF may be poor. Allowing protein-specific potentials alleviates the challenges in describing secondary structures using CG models with only one particle per residue.
Both transferable parameters and non-transferable parameters were learned by optimizing the aggregated objective function defined in Eq.~\ref{eqn:glog-likelihood-tot}. For each of the 12 proteins, we used evenly spaced 250,000 conformations from the corresponding all-atom simulation as data samples. Using the umbrella sampling procedure described in the \emph{Methods Section}, we generated the same number of noise samples and computed the noise potentials $u_q^k(\bm{x})$. Because the energy function $u_p^k(\bm{x})$ is linear to all parameters, optimizing the aggregated objective function (Eq.~\ref{eqn:glog-likelihood-tot}) converges to a unique solution. In addition, because the aggregated objective function is a weighted sum of objective functions for individual proteins, its computing and optimization can be easily parallelized among proteins. Using 12 Nvidia Volta V100 GPUs, each assigned to calculate the potential contrasting objective function of one protein, we can optimize the aggregated objective function (Figure S5) and learn all parameters in 30 minutes.
CG simulations using the learned potential functions are compared to all-atom simulations (Figure \ref{fig:all_hist} and S6) in terms of the radius of gyration (Rg) and the RMSD from the folded structures. Structures close to the native state are sampled in the CG simulations for all proteins (Figure \ref{fig:all_hist}). The lowest RMSD for configurations sampled in CG simulations range from 0.2 \AA\ to 5.3 \AA\ and are less than 4 \AA\ for 10 out of 12 proteins. Because the CG potential functions (Eq.~\ref{eqn:transferable}) are restricted to share transferable non-bonded interactions, their performances at reproducing all-atom simulations are compromised compared to the potential function $u_p^\mathrm{ss}$ that is specific to the Trp-cage protein and has no transferable parameters. Nonetheless, the CG simulations capture folding and unfolding transitions for all but the NTL9 proteins (Figure \ref{fig:all_hist}). The learned transferable contact potential energy functions between pairs of amino acids are shown in Figure \ref{fig:lj_contact} and S7. Although we parameterize these non-bonded contact potentials using cubic splines and do not restrict them to specific mathematical expressions, they all converge to functions that resemble the Lennard-Jones potential widely used in all-atom and CG force fields.
\section{CONCLUSION and DISCUSSION}
By generalizing noise contrastive estimation with unnormalized noise distributions, we developed a new method, potential contrasting, for learning force fields from reference molecular configurations. Potential contrasting combines the advantages of existing variational methods such as force matching and relative entropy minimization. As with the force matching method, it is computationally efficient and does not need sampling during force field optimization. Like the relative entropy method, potential contrasting does not require force information. We showed that the method is effective and succeeds in producing CG energy functions that accurately reproduce configurational distributions obtained from all-atom simulations. In addition, potential contrasting can be trivially parallelized for efficient learning of transferable CGFFs using simulation data of multiple systems. With its efficacy and efficiency, potential contrasting is well-positioned to systematically learn transferable CGFFs based on all-atom force fields, addressing one of the significant challenges in coarse-grained modeling.
Although we focused in this study on using potential contrasting to learn CGFFs, the method is general. It can be applied to learning various types of force fields. For instance, potential contrasting can be readily applied to parameterize implicit solvent models using all-atom simulations with explicit water molecules. With further development, it could also be used to improve existing all-atom force fields by incorporating information from quantum mechanical calculations or experimental data. Such applications and development will be investigated in future studies.
Our use of unnormalized noise distributions produced with umbrella sampling is essential for parameterizing accurate CGFFs. Unnormalized noise distributions defined with molecular energy functions allow the generation of noise samples that resemble the configurations produced from all-atom simulations. Therefore, significant overlap in the phase space between noise and data samples can be achieved. Such overlap can be difficult to ensure with arbitrary noise distributions since all-atom simulations only sample limited regions of phase space with low energy. We note that, upon training molecular simulation data, probabilistic models parameterized with normalizing flows\cite{Rezende2015VariationalFlowsc,Dinh2016DensityNVP,Papamakarios2019NormalizingInferenced} have been shown to produce realistic and stable molecular conformations \cite{Gao2020FlowModels,Noe2019BoltzmannLearningb,Ding2020ComputingModels,Ding2021DeepBAR:Computationb,Wirnsberger2020,Kohler2022Force-matchingForces}. These models have indeed been proposed to serve as noise distributions for contrastive learning to guarantee overlap with data samples \cite{Gutmann2010Noise-contrastiveModels,Gao2020FlowModels}. However, we found that using flow-based models as noise distributions produced CGFFs with sub-par quality. Similar findings have been reached in other recent studies as well \cite{Chehab2022TheThink}. By optimizing the overlap with data samples, flow-based models may hinder the minimization of probability for regions outside the data. Further research is needed to design optimal noise distributions in NCE.
\begin{acknowledgement}
This work was supported by the National Institutes of Health (R35GM133580).
\end{acknowledgement}
\begin{suppinfo}
Detailed procedure for generating noise samples for learning CGFF of protein folding,
learning CG potential functions with both transferable and non-transferable parameters,
cubic splines for flexible potential energy parameterization,
parameter optimization and regularization,
molecular dynamics simulations with the CGFFs,
many-body interactions parameterized using neural network,
(Figure S1) cubic spline basis,
(Figure S2) the neural network used for parameterizing the many-body energy term,
(Figure S3) distributions of RMSD with respect to the folded structure of the Trp-cage protein for the noise samples generated using umbrella sampling,
(Figure S4) trajectories of RMSD with respect to the folded structure of the Trp-cage protein for conformations from the all-atom simulation and the CG simulations,
(Figure S5) convergence of the aggregated loss function with weight decay during the optimization
of the CG potential functions that have both transferable and non-transferable parameters,
(Figure S6) trajectories of raidus of gyration and RMSD with respect to the folded structure from the
all-atom simulation and the CG simulation of all 12 proteins,
(Figure S7) learned transferable contact potential energy functions between pairs of amino
acids.
(Table S1) setup used in umbrella sampling used for generating noise samples.
\end{suppinfo}
\clearpage
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,181 |
Daniel Orton (Oklahoma City, Oklahoma, 6 augustus 1990) is een Amerikaans basketbalspeler.
Carrière
Orton speelde collegebasketbal voor de Kentucky Wildcats van 2009 tot 2010. Hij stelde zich in 2010 kandidaat voor de NBA-draft en werd gekozen als 29e in de eerste ronde door de Orlando Magic. Bij de Magic speelde hij ook bij hun opleidingsploeg New Mexico Thunderbirds in 2010 maar door een blessure zat het seizoen er al snel op. In zijn eerste seizoen kwam hij niet aan spelen toe, in zijn tweede seizoen speelde hij 16 wedstrijden en twee als starter. Hij tekende daarna een contract bij de Oklahoma City Thunder waar hij dertien wedstrijden speelde en speelde ook voor hun opleidingsploeg Tulsa 66ers. In het seizoen 2013/14 speelde hij 22 wedstrijden voor de Philadelphia 76ers. In 2014 tekende hij een contract bij de Maine Red Claws en speelde het seizoen bij hen uit. Hij speelde tijdens de Summer League bij de Washington Wizards maar kreeg geen plaats in hun selectie voor het reguliere seizoen.
Daarop tekende hij een contract bij het Chinese Shanxi Zhongyu maar verloor zijn plaats aan Jeremy Tyler en speelde niet voor hen. Hij tekende dan bij de Sichuan Blue Whales waar hij het seizoen 2014/15 doorbracht tot op 1 februari toen hij zijn contract ontbond. Op 4 februari tekende hij bij het Filipijnse Purefoods Star Hotshots, op 11 februari noemde hij Manny Pacquiao: "A joke". Hij werd door de Filipijnse competitie geschorst voor zijn opmerking. Hij tekende op 11 maart voor de D-League ploeg Grand Rapids Drive maar vertrok er al weer de 26e. Op 31 maart tekende hij een contract bij de Idaho Stampede maar dat werd de volgende dag weer ontbonden. In augustus 2015 tekende hij een contract bij het Griekse AEK Athene BC maar speelde geen wedstrijd voor hen. In november tekende hij een contract bij de Santa Cruz Warriors waar hij in januari weer vertrok. In de volgende jaren speelde Orton bij tal van ploegen in Azië.
NBA Statistieken
Regulier seizoen
Play-offs
Amerikaans basketballer | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,021 |
L'aéroport de Birsa Munda , également appelé aéroport de Ranchi, est l'aéroport desservant la ville de Ranchi, la capitale de l'État indien de Jharkhand. Il porte le nom du combattant de la liberté tribal indien Birsa Munda et est actuellement géré par la Airports Authority of India. L'aéroport est situé à Hinoo, à environ du centre-ville. L'ensemble de l'aéroport s'étend sur 1568 acres. L'aéroport est utilisé par plus de de passagers par an et est le le plus fréquenté d'Inde.
Compagnies et destinations
Édité le 21/01/2020
Situation
Statistiques
Notes et références
Ranchi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,030 |
Q: Confused About Play2 Form Submission (login/register) This might seem very basic, but I am new to JPA/Ebean and Play Framework (not so new with Java btw).
I'm trying to make two forms. One to log in and one to register. I'm using both the book Play for Java MEAP (Early Access), the official website documentation, and the sample app "ZenTask" codes. I must say Play is strong, robust, but the documentation is so poor. I get this must have something to do with most Play users are seasoned Java Web developers, but still!
I created a model called User
@Entity
@Table(name="account")
public class User extends Model {
@Id
@Constraints.Required
@Formats.NonEmpty
public String email;
@Constraints.Required
public String displayName;
@Constraints.Required
public String password;
public static Model.Finder<String, User> find = new Model.Finder(String.class, User.class);
public static User authenticate(String email, String password) {
return find.where()
.eq("email", email)
.eq("password", password)
.findUnique();
}
}
Then I have this controller Application
public class Application extends Controller {
public static class Login {
@Constraints.Required
public String email;
public String password;
public String validate() {
if (User.authenticate(email,password)==null) {
return "Invalid email or password.";
}
return null;
}
}
public static class Register {
@Constraints.Required
public String email;
public String password;
public String cfmPassword;
public String displayName;
public String validate() {
if (cfmPassword.equals(password)) {
return "Passwords typed in does not match.";
} else if (displayName.contains(" ")) {
return "Display name cannot contain space";
}
return null;
}
}
}
I don't even understand why am I creating two nested classes, but it seems like a requirement to create forms in Play? So I made two. This is the code I use to render the page (it's inside Controller Application)
public static Result index() {
return ok(index.render(form(Login.class), form(Register.class)));
}
OK, now here comes the most frustrated part. First, I don't know if I have truly authenticated the user when logged in: (codes in Application)
/**
* Handle login form submission.
*/
public static Result authenticate() {
Form<Login> loginForm = form(Login.class).bindFromRequest();
if (loginForm.hasErrors()) {
return badRequest(index.render(loginForm, form(Register.class)));
} else {
session("email", loginForm.get().email);
return redirect(controllers.routes.Wall.index());
}
}
Second, how can I say a user's information when he/she registers. I copied these lines from the Java for Play book, but it doesn't work (my IDE says it's wrong..type mismatch)
public static Result register() {
Form<Register> RegisterForm = form(Register.class).bindFromRequest();
if (RegisterForm.hasErrors()) {
return badRequest(index.render(form(Login.class), RegisterForm));
} else {
User user = RegisterForm.get();
session("email", RegisterForm.get().email);
return redirect(controllers.routes.Wall.initiate());
}
}
I must have done something really really wrong here..can anyone enlighten me a bit?
A: OK...sorry. I have solved my problem. Here is how I handled the type mismatch problem. It turns out that I should have used User class from the model directly instead of creating new nested classes such as "Login" and "Register."
I add these lines into my register() method under my Application controller.
User user = new User();
user.email = RegisterForm.get().email;
user.displayName = RegisterForm.get().displayName;
user.password = RegisterForm.get().password;
user.save();
My question now is: "is validate() method a reserved method in Play framework that will be called by default from Form.hasErrors()? If I use my User class to construct my form and invoke .hasErrors(), since I don't have validate() method in User class, what will Play framework invoke to check error then? Is it better off for me to use a nested class or should I use classes from models directly?"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,438 |
Chignik Lake és una concentració de població designada pel cens dels Estats Units a l'estat d'Alaska. Segons el cens del 2000 tenia una població de 145 habitants.
Demografia
Segons el cens del 2000, Chignik Lake tenia 145 habitants, 40 habitatges, i 34 famílies La densitat de població era de 4,6 habitants/km².
Per edats la població es repartia de la següent manera: un 44,8% tenia menys de 18 anys, un 8,3% entre 18 i 24, un 24,8% entre 25 i 44, un 17,9% de 45 a 60 i un 4,1% 65 anys o més.
L'edat mediana era de 21 anys. Per cada 100 dones hi havia 93,3 homes. Per cada 100 dones de 18 o més anys hi havia 116,2 homes.
La renda mediana per habitatge era de 41.458 $ i la renda mediana per família de 40.938 $. Els homes tenien una renda mediana de 0 $ mentre que les dones 38.750 $. La renda per capita de la població era de 13.842 $. Aproximadament el 21,2% de les famílies i el 22% de la població estaven per davall del llindar de pobresa.
Poblacions més properes
El següent diagrama mostra les poblacions més properes.
Referències
Concentracions de població designades pel cens d'Alaska
Borough de Lake and Peninsula | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,275 |
## DEDICATION
_To the_ shokunin _of Japan, pursuers of perfection,
for showing us the true meaning of devotion_
## CONTENTS
_(Laura Pérez)_
1. Dedication
2. Foreword: In Correspondence with Bourdain
3. **TOKYO**
4. Plus: Know Before You Go
5. Food Groups
6. In the Raw
7. One Night at a Love Hotel
8. **OSAKA**
9. Plus: Operation Izakaya
10. Wagyu 101
11. The Knife Makers of Sakai
12. **KYOTO**
13. Plus: The Art of Gift Giving
14. Japan's Greatest Food Journeys
15. Gaijin Glossary
16. **FUKUOKA**
17. Plus: The Ramen Matrix
18. Dream Machines
19. Fear Not!
20. **HIROSHIMA**
21. Plus: The Evolution
22. The Eight Wonders of the Japanese Convenience Store
23. Deep Fried
24. **HOKKAIDO**
25. Plus: Amazing Shit in the Middle of Nowhere
26. One Night with the Salarymen
27. On a Stick
28. **NOTO**
29. Plus: One Night with the Geisha
30. The Beauty of Bento
31. Acknowledgments
32. About the Team Behind _Rice, Noodle, Fish_
33. About Roads & Kingdoms
34. Credits
35. Copyright
36. About the Publisher
# Guide
1. Cover
2. Contents
3. chapter 1
1. iii
2. iv
3. v
4. viii
5. ix
6. x
7. xi
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9. xiii
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## _Foreword_
## IN CORRESPONDENCE WITH BOURDAIN:
## How this book was born
Dear Tony,
I'm writing you from a laundromat attached to an old teahouse down a dark alley in Kyoto. I've spent the past month eating my way south from Hokkaido—from the _uni_ shrines of Hakodate to the _okonomiyaki_ dens of Osaka. I've been invited to dine with the Sugimoto clan tonight, the oldest family in Kyoto, in their 300-year-old home with their 600-year-old recipes, and I need something decent to wear. So while five weeks' worth of memories dissolve in the spin cycle, let me tell you about this idea I have.
If _Parts Unknown_ and its many imitators have taught us anything, it's that we're living in the Golden Age of Gastrotourism. The same people who once traveled to Rome to stare at statues now go to twirl bucatini on their forks and filter balls of burrata onto their Instagram accounts. You've helped inspire a generation of food-obsessed pilgrims, the same people we try to reach every day at Roads & Kingdoms: the ones who want to be smarter, eat better, travel deeper. We've given them ice cream crawls in Mogadishu, the chili sauce wars of the Caucasus, the burger kings of Karachi.
But it feels like there's something even bigger out there to tap into, a more complete way to capture the seismic shift that takes place inside of us as we first eat our way through a country. And Japan, where a tangle of undressed noodles can feel like a seminal life moment, is the perfect place to start. I'm imagining a book that attempts to make sense of the many wondrous, beautiful, confounding things the outsider experiences here—both at the table and beyond.
I don't have any clear answers yet, but I know you share my affection for this country and I thought this might be something you'd want to be a part of. Give it some thought and let me know what you think. I'll be here, watching the laundry spin.
Cheers,
Matt
***
Dear Matt,
That's pretty much where I'd like to be right now, preparing to go out to dinner in a 300-year-old home—in Kyoto. I stayed in a magnificent old _ryokan_ there once, so old there were sword slashes in the ceiling beams. Evidence, I was told, of samurai-related violence.
As you know, Japan hooked me. It was the first Asian country I ever visited. I was alone, clueless, horribly, cripplingly jet-lagged (back when I still suffered from such things), and on an ill-fated mission to consult on a French restaurant project. I'd wake up in Roppongi early in the morning to the shrieks of those giant crows and wander the streets, trying to summon the courage to enter a noodle shop. I will never forget the sense of deep satisfaction I felt when I finally managed to order breakfast for myself.
Tokyo was so dense, so crowded with . . . stuff, so complicated, tempting, delicious, and seemingly unknowable: layer upon layer of maddeningly interesting izakayas in one building alone. One city block a life's work of exploration. It was a glorious and lasting derangement of the senses that first trip, and I've never been the same since.
I became selfish that first time in Tokyo in ways I had never been. Previously, when viewing something incredible, impressive, strikingly beautiful, or interesting, my first instinct was to share. Who might I share this with? How might I best relate this experience?
In Tokyo, alone and traumatized in the best possible ways by this new universe of possibilities, I just said "fuck it" to that voice. This was for me. There was no sharing. I wanted more—whatever it took—and I resolved, consciously or not, I think, to burn down the whole world if necessary to get more of this.
In Japan you are confronted constantly, almost violently, with how much you don't know. I liked that feeling. I liked that steep, virtually impossible learning curve. I liked, it turned out, that feeling of being a stranger in a strange yet wonderful land, not understanding the language, lost. Every little thing was a discovery.
Things kind of worked out. I found a way to ensure many more trips to Japan, television being a small price to pay for the privilege. I know now exactly what you mean when you speak of the joys of undressed noodles. I yearn for the smoke and sizzle of many parts of pampered chickens in an old-school yakitori joint, the clean smell of the fish market at four in the morning (cigarettes and seawater), _chankonabe_ , grilled fish collars in Golden Gai, the glory of the Japanese bathroom. They may work punishingly, insanely hard in Japan. But they have relaxation down to a science. To spend a weekend at a traditional _ryokan_ , marinating in an outdoor _onsen_ , is a life-changing thing. There's no going back. Not all the way back anyway.
I don't know if you know this but I've found that if you sat at a table with eight or nine of the worlds best chefs—from France, Brazil, America, wherever—and you asked them where they'd choose if they had to eat in one, and only one country, for the rest of their lives, they would ALL of them pick Japan without hesitation. We both know why.
I have no doubt that you would make that case brilliantly in the book to come, but I'm going to need more details if I want to convince my cruel masters at HarperCollins. How do you see this playing out on the page?
Best,
Tony
***
Hi Tony,
I know what you mean when you say you've never been the same. I'm supposed to be on a honeymoon with my Catalan wife, but every time a piece of _uni_ nigiri or _shirako_ tempura is placed before me, I feel like I'm cheating on her. I try to shift the focus back to my bride, but then I look over and see her eyes glazed with that same new Japan sheen, and I know that there will forever be a line in our lives: Before Japan, After Japan.
I could see how you would want to keep this to yourself. Something so intense and intimate—it's hard to share without feeling like you're somehow butchering the translation. Judging by the episodes you've logged from Japan, though, you got over that feeling, no doubt for much the same reason that I'm getting over it: we tell stories for a living, and these stories are the best I've found anywhere.
I'm in Noto now, a windswept peninsula on the west coast known as the Kingdom of Fermentation. Breakfast this morning was a piece of mackerel cured in salt and chilies for 12 years (my body is still buzzing from the umami). Chikako Fukushita is the daughter of Noto's preeminent pickle masters: her father has been honored by the governor for his fish sauce, her mother is the sole keeper of over 300 recipes that represent the family's—and Noto's—legacy. They never had a son, so it has fallen to Chikako to catalog every last recipe before they pass away.
The plan is to stay here as long as it takes to find stories like these—deep, experiential narratives that tell us something about this country that only the food and its creators can. On the horizon: a Guatemalan immigrant turned _okonomiyaki_ master in Hiroshima, a rebel band of sea urchin fishermen in Hokkaido, and a ramen blogger from Fukuoka who eats 400 bowls of _tonkotsu_ a year.
I've talked with my Roads & Kingdoms partners about this idea and they're all in. Beyond the high-protein narratives, we see a series of lighter side stories, photo essays, and illustrated decoders illuminating the most interesting corners of Japanese culture. Doug Hughmanick built our website and would be perfect for designing big, beautiful spreads about the glories of the Japanese convenience store or how to navigate a love hotel. Nathan Thornburgh, whom you already know from his days at _Time_ magazine, is an intense and uncompromising editor, ready to make whatever I write stronger.
Good thing, because despite the beauty of these stories, there is infinite potential to make an ass of myself. I'm a novice here. I speak no Japanese. I claim no special understanding of this dense culture and hold no key to unlock the country's many closed doors. I went to a very famous sushi restaurant in Tokyo last week, a place that destroyed me the first time I ate there. I came back with a translator and a suit jacket, waited for two hours until the last guests trickled out, then asked the chef if I might arrange an interview. His jaw dropped, his face contorted. "Why would you come here?" he said. "Next time, please go through the embassy."
I spent the next 24 hours steaming, appalled by the suggestion of involving diplomats to talk about rice and fish and somehow offended that he didn't want to share his story. But deep down there's something almost noble about his reaction: with only six seats and a loyal local clientele, his only objective now is to protect what he has.
There is no escaping my place as the most outside of outsiders here, so I might as well embrace it. There will be plenty of expertise proffered along the way, just not from me—from the chefs and artisans and families who have this cuisine in their DNA, and who have opened up many doors as I've begun to eat my way through this country.
So the big question is, just who is this book for? People already on their way to Japan? People parked in the armchair with no immediate plans to hit the road? The burrata Instagrammers? You're the book guy these days, and no doubt the suits at HarperCollins will want to know. Any guidance you might have will be rewarded with a fugu sake (a blowfish tail set alight and dropped into a glass of rice wine—a group of salarymen hazed me with this hellbroth last night) next time we cross paths.
Cheers,
Matt
***
Matt,
Thinking about the smell of hinoki wood. You know the smell. One of those deep water tubs that comes up to your chin. Scaldingly hot water, washcloth on head. Maybe a bottle of sake close at hand.
Do you have any tattoos? This is one of the peculiarities of Japan I find both charming and annoying at the same time. Every time I hit a public _onsen_ —or a hotel pool for that matter—as soon as the shirt comes off, some very uncomfortable-looking attendant comes running over with a rash guard to cover me up. Apparently, it's a non-insulting way to keep yakuza out. I wish they would just go with a sign saying "NO GANGSTERS" instead of busting my balls but what can you do? I put the damn rash guard on.
What do you read when in Japan? In Vietnam, it's _The Quiet American_ every time. Often, I find it's fiction that better describes a place—the atmospherics, the soul. Graham Greene, being such a terrific traveling companion, it's too bad he never set a novel in Japan. Lowry in Mexico. Orwell in Myanmar. Theroux in Singapore. But Japan? I'm at a loss.
I usually end up watching DVDs that capture better (or more easily) the hallucinatory aspects of Tokyo or Osaka. I've described experiencing Japanese nightlife as like living inside a pinball machine—or dropping acid for the first time—inside yet always outside.
Making an ass of yourself in Japan is an inevitability. Fortunately, we gaijin seem to get cut a lot of slack. I recall with embarrassment being treated to an elaborate kaiseki meal and the elderly geishas who were there to entertain bursting into peals of laughter as I tucked unknowingly into a bowl with my chopsticks, blissfully unaware that it was the condiment not the entree. It's a minefield of potential offense. I'm quite certain that at all times that every single thing about me is somehow "wrong," from my posture, the way I hold my chopsticks, bow, pour my drink, sit, cross my legs—and so on. But I don't care. Japan is just too awesome to not just forge on.
I don't know why you would call that delightful burning fugu tail drink a hellbroth. I love that shit.
And as I sit here and reflect on "who this book is for and what its appeal might be" I no longer care. The more layers you can peel back, the better. The deeper you dive into all those things that make Japan so fascinating and so pleasurable to us, the better for humanity.
Roads & Kingdoms has for some time now been doing the best travel journalism out there. It's not just WHAT it is—but what it is NOT. You're cutting right to the good shit. A person could easily miss what you had the good sense to celebrate. There is enormous value in that.
This is, after all, the beginning of what I fully expect to be a long and fruitful relationship. An unholy alliance between you—Roads & Kingdoms—and whatever it is that I do for a living.
Readers will either read the book and immediately book tickets to Japan to explore for themselves. They will return changed. Unable to look at the world in the same way ever again. Or . . . they might refine and adjust whatever sadly misguided plans they might have had in favor of destinations described here.
Or they might sit in their chairs and dream of a faraway place where the culture is very old, the food extraordinary and refined beyond imagining, and where there are many beautiful things that feel good.
And someday, if given the opportunity to see this place for themselves, hopefully, they will leap.
The world needs Roads & Kingdoms. It needs this book. Let's give it to them.
Best,
Tony
## _Chapter One_
## TOKYO
If you listen carefully, you will hear the sounds of Japan cooking. But these are not the sounds of a typical kitchen, even a great one, at work—at least not the ones you may be used to hearing. It's not an expediter on a line asking when his rib steak will be ready. It's not the gurgle of a deep fryer violently crisping a thatch of potatoes. It's not the sound of a sauce being scraped across the plate with the back of a spoon, or the pinch of tweezers art-directing another foraged herb into position.
It is the sound of a terry-cloth towel rubbed against the grain of hardwood, scrubbing for hours each night to remove the gentle stain of fish oils accumulated on the hinoki counter over the course of a sushi service. It is the gentle rustle of fingers gliding over green coffee beans, like wind in the trees, in search of imperfections before roasting. It is the whoosh of a handmade fan used to tame a _binchotan_ fire. The dull thump of polished wood against the soft flesh of tomato. The muted cadence of a long, thin knife working its way across the flesh of a conger eel.
These are the sounds of Japan cooking. And everything that you will put in your mouth begins with one of these sounds, barely audible, that rises up and amplifies and takes on a force of its own. In the most perfect moment, when you least expect it, these little whispers will build into a great sonic boom, and all you can do is close your eyes and let it wash over you.
If all of this feels precious, that's because it is. One of your first revelations in Japan, especially while eating, will be just how much the details matter: the angle of the maple leaf garnishing your plate, the mood of the chef when frying your asparagus, the bloodline of the farmer who grew that radish. The fact that you—and everyone else, including experienced Japanese diners—will miss most of these details doesn't matter; there is the underlying belief that nearly imperceptible improvements are made in the quality of the food by the most subtle actions of its creators. The tempura batter tastes better when stirred with chopsticks from the Meiji era; the dashi is purer when simmered by a cook with a clear mind and a light heart.
But not everything is so subtle. There are succulent loins of fatty pork fried in scales of thin bread crumbs and served with bowls of thickened Worcestershire and dabs of fiery mustard. Giant pots of curry, dark and brooding as a sudden summer storm, where apples and onions and huge hunks of meat are simmered into submission over hours. Or days. There is _okonomiyaki_ , the great geologic mass of carbs and cabbage and pork fat that would feel more at home on a stoner's coffee table than a Japanese tatami mat.
And, of course, there is ramen, the loudest of all Japanese foods, a soundtrack of thwacks, sizzles, drips, and slurps that undermines everything you thought you knew about this country and its culture. Is that cook chopping leeks to the bass of a hip-hop track? Why yes, yes he is.
No country on this planet inspires wonder like Japan does. Everywhere you turn, you will find a reason to be astounded.
It starts on the airplane, twenty thousand feet above Tokyo. I remember my first approach to Narita, when the plane knifed through the clouds and suddenly there it was, the biggest city in the history of the earth, pixilated in a billion yellow dots below me. In the early 1600s, when the shogun Tokugawa Ieyasu decided to build his castle here, Tokyo was nothing but a tiny fishing village. By 1800 it was already the largest city in the world, with more than a million people calling Japan's new capital home. Over the years, it would shake, shatter, splinter, and burn again and again. And still it stands and stretches on to infinity.
I first came to Tokyo in the fall of 2008 with no plans, no reservations, not the slightest clue about the transformation that awaited me. With six thousand miles separating me from sleep, I stumbled down into the subway at dawn and emerged on the outskirts of the Tsukiji market just as the sun broke across Tokyo Bay. Inside the market, I saw the entire ocean on display: swollen-bellied salmon, dark disks of abalone, vast armies of exotic crustaceans, conger eels so shiny and new they looked to be napping in their Styrofoam boxes. I stumbled onward to a tuna auction, where a man in a trader's cap worked his way through a hundred silver carcasses scattered across the cement floor, using a system of rapid hand motions and guttural noises unintelligible to all but a select group of tuna savants. When the auction ended, I followed one of the bodies back to its buyer's stall, where a man and his son used band saw, katana blade, cleaver, and fillet knife to work the massive fish down into sellable components: sinewy tail meat for the cheap izakaya, ruby loins for hotel restaurants, blocks of marbled belly for the high-end sushi temples.
By 8:00 a.m. I was starving. First, a sushi feast, a twelve-piece procession of Tsukiji's finest—fat-frizzled bluefin, chewy surf clam, a custardy slab of Hokkaido _uni_ —washed down with frosty glasses of Kirin. Then a bowl of warm soba from the outer market, crowned at the last second with a golden nest of vegetable tempura. By the time the sun had climbed directly above me, I stood before a wall of skyscrapers, smiling stupidly, uncomfortably full but hungrier than ever.
The largest city on earth, as seen from Roppongi Hills
_(JapanExterna)_
If you've never been before, you will do what we all do when we first come to Japan: you will blink and rub your eyes like a cartoon character, you will lose yourself in the human churn of Shibuya and Shinjuku, you will bear witness to the fantastic collision of past and future as you move from neon jungle to ancient temple and back into Tomorrowland. You will marvel at the plastic food, the bullet trains, the omnipresent vending machines. You will take pictures of toilets. Your e-mails back home will be filthy with exclamation points.
You will feel completely and wonderfully overwhelmed by the stimuli, and there will be moments when you don't know what to do. Which way to turn. Which person to ask. Which dish to eat.
It's the last one that gets me every time. What to eat? You've crossed a dozen time zones to get here and you want to make every meal count. Do you start at an izakaya, a Japanese pub, and eat raw fish and grilled chicken parts and fried tofu, all washed down with a river of cold sake? Do you seek out the familiar nourishment of noodles—ramen, udon, soba—and let the warmth and beauty of this cuisine slip gloriously past your lips? Or maybe you wade into the vast unknown, throw yourself entirely into the world of unfamiliar flavors: a bowl of salt-roasted eel, a mound of sticky fermented soybeans, a nine-course kaiseki feast.
You would be ill-advised to take this decision lightly. Make no mistake about it: Tokyo is the greatest feast on earth. Not New York. Not Paris. Not Bangkok. All of these cities offer sprawling, beautiful food cultures worthy of a lifetime of exploration, but none can compare with the depth and breadth of deliciousness proffered by Tokyo's culinary legions.
First of all, it's the size. New York City has some 30,000 restaurants; Tokyo, 300,000. (Take a moment to let that sink in, please.) Whereas most of the world confines their restaurants to street level, a ten-story building in Japan might have two or three restaurants on every floor, towers of deliciousness stretched toward the heavens like Babel.
But Tokyo's preeminence as the world's most exciting dining destination isn't a quantity thing: it's a quality one. There are a dozen factors that make Japanese food so special—ingredient obsession, technical precision, thousands of years of meticulous refinement—but chief among them is one simple concept: specialization. In the Western world, where miso-braised short ribs share menu space with white truffle pizza and sea bass ceviche, restaurants cast massive nets to try to catch as many fish as possible, but in Japan, the secret to success is choosing one thing and doing it really fucking well. Forever. There are people who dedicate their entire lives to grilling beef intestines, slicing blowfish, kneading buckwheat into tangles of chewy noodles—microdisciplines with infinite room for improvement.
The concept of _shokunin_ , an artisan deeply and singularly dedicated to his or her craft, is at the core of Japanese culture. Japan's most famous _shokunin_ these days is Jiro Ono, immortalized in the documentary _Jiro Dreams of Sushi_ , but you will encounter his level of relentless focus across the entire food industry. Behind closed doors. Down dark alleyways. Up small stairwells. Hiding in every corner of this city and country: the eighty-year-old tempura man who has spent the past six decades discovering the subtle differences yielded by temperature and motion. The twelfth-generation _unagi_ sage who uses metal skewers like an acupuncturist uses needles, teasing the muscles of wild eel into new territories. The young man who has grown old at his father's side, measuring his age in kitchen lessons. Any moment now, it will be his turn to be the master, and when he does, he'll know exactly what to do.
"The _shokunin_ has a social obligation to work his or her best for the general welfare of the people," says Japanese sculptor Tasio Odate. "This obligation is both spiritual and material, in that no matter what it is, the _shokunin_ 's responsibility is to fulfill the requirement."
Tokyo is the city of ten thousand _shokunin_. If you come to Japan to eat, you come for them.
At first I didn't get this. I ate nothing but ramen and udon and tempura from any place that looked legit—and I was deeply satisfied doing it. But then a friend, Shinji Nohara, a culinary guide who makes a living out of turning first-timers into lifelong Japanophiles, took me to a small coffee shop where an old man named Katsuji Daibo had spent four decades converting muddy water into a religious experience: sifting bean-by-bean through pounds of coffee every morning, hand-roasting each batch for thirty minutes over a low flame, executing a drip-by-drip pour-over that felt like watching life move backward—a painstaking process that produced the city's richest, most expensive, most labor-intensive cup of coffee.
By the time I emerged from Daibo, Tokyo and Japan and the entire food world had changed for me. I had a new lens through which to view this country and a new reason to keep coming back: to eat the noodles and conveyor sushi and pork-belly pancakes, yes, but also to take the time to experience the true masters of Tokyo, the _shokunin_ , the ones who bless this city with their quiet pursuit of perfection.
米 麺 魚
Ginza is the heart of Tokyo's sushi culture, making it the center of Japan's sushi culture, making it the greatest neighborhood in the world for eating fish. Walk these gilded streets for a few blocks and you'll soon figure out why: this is one of Japan's wealthiest zip codes, home to extravagant department stores and a battery of international luxury brands housed in beautiful buildings created by famous architects. A perfect fit for the world's most expensive cuisine.
_(Michael Magers, lead photographer)_
Sushi as we know it today was bred in these blocks. Japanese cooks had been cycling through various permutations of _narezushi_ , fish fermented with cooked rice, since the eighth century, but it wasn't until the early 1800s, as Edo (Tokyo's original name) was taking shape as Japan's new capital, that the familiar nigiri formulation emerged. Wooden _yatai_ , street food stands, dotted this area, serving urban dwellers the best of the day's catch from Tokyo Bay. Cooks shaped warm mounds of rice by hand, covered them with a slice of fresh fish, and served individual pieces directly to hungry customers. To mimic the puckering flavor of the fermented fish of yore, sidewalk chefs added vinegar to the rice; to kill off potential toxins, they rubbed the fish with a dab of grated horseradish; to season it, a few drops of soy. Modern sushi— _edomaezushi_ —was born.
Today, in an eight-block radius you will find the finest sushi bars on earth, a concentrated cluster of polished countertops with claim to sixteen Michelin stars among them. The mighty Jiro Ono operates here, Zeus among the sushi gods of Japan, serving his twenty-minute, $350 feast to a rotating cast of curious foreigners and Japanese heavyweights. So too does Takashi Saito, the young Jedi master with the longest waiting list in town, along with many, many others.
On the third floor of an unassuming office building, one of these _shokunin_ , the one whom some have dubbed the soul of Tokyo sushi culture, stands behind a beautiful two-tiered hickory countertop, rubbing a mint-green root of fresh wasabi against a sharkskin grater, preparing for his first guests. He's young by sushi standards, forty or so, and built like a defensive back, with thick arms, shaved head, and heavy eyes that do most of the talking.
I first met Koji Sawada in 2011, when I took a seat at his counter and slowly felt the walls of my food world crumble. To call it a revelation would be to undersell the experience: the meal I had at Sawada was a full-scale transformation, a piece-by-piece poem to starch and sea, not perfect exactly, but a clear indication that a pathway to perfection existed, a stairway to heaven, and that Sawada was climbing it, one step at a time.
This wasn't my first sushi epiphany. On my maiden voyage to Tokyo, I went alone to eat lunch at Mizutani, a Michelin three-star Ginza institution run by Jiro's most famous disciple. There, in a basement restaurant, armed with only three words of Japanese and the one wrinkled button-up I could dig out of my backpack, I learned about the intimacy and artistry of a true sushi experience. (The elegance of the meal came to a crashing halt when I was told Mizutani didn't take credit cards—all too common in Japan's best restaurants—and the chef personally escorted me to the post office to take out 25,000 yen to cover lunch.)
But Sawada is something else entirely—a former trucker who turned to sushi relatively late in life but with all the manic energy and determination of a man possessed by a single idea: to create the best Edo-style sushi experience in the beating heart of sushi's birthplace.
That means starting each day at 6:00 a.m. at Tsukiji, buying each individual piece of fish from the purveyor who knows the species best. That means investing years in developing a system to serve rice at its ideal temperature and texture the moment the customers settle into their seats. That means constructing an elaborate and expensive refrigeration system cooled not by electricity but by giant blocks of ice. That means serving only six people for lunch and six for dinner. That means ending each night with a terry-cloth towel, scrubbing the hinoki countertop until his arms are sore and his head is slick with sweat and all trace of the fish oils accumulated during service vanish, a cleaning session that marches past midnight, completing an eighteen-hour day that he and his wife repeat six days a week. When I ask Sawada why he doesn't hire someone to clean after dinner service so that he might rest for a bit, he squints his eyes, cocks his head, and points toward the entrance. "You see the name on that door? It says Sawada. I'm Sawada. She's Sawada. Nobody else."
Sawada could probably wake up at 9:00 a.m., get his fish delivered to his door, use a standard refrigerator for cooling his ingredients, have his counters scrubbed by a young apprentice after dinner, and still serve some of Tokyo's most breathtaking sushi. But he doesn't. Because in Japan, it's not about the end; it's about the means.
"It comes down to _kimochi_ ," says Sawada. "'Feeling.' That's the difference between a good sushi chef and a great one. From start to finish, it's all about feeling. I want you to have the best possible sushi. That's why I go to the market at six a.m. That's why I'm still cleaning at midnight."
_Kimochi_ is a part of all _shokunin_ , says Sawada, but especially part of the sushi chef. "Your feelings come out in the sushi. There's no fire. We make it with our hands. You eat it with yours."
With the first bite—and every bite that follows—I realize something I've always been told but never believed: sushi is not about fish; it's about rice. Nigiri—ninety-five times out of a hundred, what Japanese eat when they eat sushi—comprises two components: _shari_ , the seasoned rice that forms the base, and _neta_ , the slice of fish that rests on top. Anyone can find great _neta_ at Tsukiji, the reasoning goes, but only a _shokunin_ can master _shari_. "Sushi is eighty percent rice," says Sawada.
Tales of struggle and sacrifice are told of young cooks who toil for years learning the tiny details of proper rice cookery: washing off the excess starch in successive changes of water, calculating the perfect ratio of dry to wet, learning how to properly fan the rice and season it with precise slashes of a wooden spoon. An extraordinary amount of thought goes into Sawada's rice, from the temperature ("It should be as warm as my skin") to the timing ("Rice is at its peak sixty minutes after cooking it"), to the source, which he changes with the rising global temperatures ("The best rice used to come from Niigata, but now it's coming from Hokkaido").
Sawada's _shari_ buzzes with a gentle current of acidity, a divisive move among the sushi cognoscenti of Tokyo, many of whom believe rice should be less assertive (so few and focused are the variables in this discipline that a couple extra drops of vinegar added to a mountain of rice constitutes a controversy). But Sawada's _neta_ is rich and flush with umami, and the rice's subtle vinegar edge keeps your palate primed for the long road ahead.
From the salty bite of gizzard shad to the supple sweetness of horse mackerel to the crunch and brine of ark shell clam, Sawada guides you through the full spectrum of ocean taste and texture. A giant prawn split into two pieces delivers dessert levels of sweetness. Saltwater eel is equal parts crunchy skin and tender flesh. Smoked bonito, in all its concentrated, fire-kissed intensity, will keep you awake at night.
Behind Sawada, his wife works heating stones, steaming shrimp, wordlessly anticipating everything he will need to continue his thesis. "We move together. She makes me better."
Contrary to popular belief, sushi isn't about freshness; it's about timing. Not just having your rice the proper temperature, but also having your fish the perfect age. Serve fish too soon out of the water and the muscles will be tight and the flavor underdeveloped. Wait too long and the protein turns to mush.
Before refrigeration, fish was either served immediately or marinated in vinegar, but over the years sushi chefs have come to understand that carefully aging fish can bring out its best qualities. The concept, Sawada explains, is the same principle behind aged meat: by removing the water and converting the protein into amino acids, you intensify the flavor of the fish—in particular, the natural umami, the most prized taste in Japan. Tuna tastes most like tuna not when it's still dripping with the essence of the ocean but when it's been allowed to mature for days or even weeks. Every fish has its optimal age: Sawada ages most white fish for two days, scallops and ark shell clams for one week, fattier fish for even longer.
To demonstrate, he offers a flight of tuna: _maguro_ , a lean ruby cut a few days out of the water, followed by a lightly marbled, week-old slice of _chutoro_ , preceded by an extravagantly fatty _otoro_ that floods the brain with a warm rush of endorphins. "Twelve days' aging," he says as he watches me struggle to control my emotions. "Tons of umami." The tuna tutorial concludes with a chunk of _otoro_ cooked directly over a hot stone, leaving the outside black and smoky and the inside just warm enough to let loose a tide of fat—a bite I would cross the Pacific for again and again.
Sushi at this level is the finest form of culinary alchemy: cooked rice, raw fish, unmitigated bliss. Along the way, progressions of little mysteries are unveiled: the way an angled knife stroke can relax a thousand tight muscle fibers, how cupping the fish with a warm palm can release just the right amount of natural oils, how the quantity and density of the rice base must be matched to each piece of fish. All these revelations can be tasted on the spot, six feet from where they were born. It is performance art of the highest caliber, one of the few great meals in the world I might not want to share with anyone else, lest they distract me from the theater at hand.
So why does he keep it so small? Why does this man limit himself to just twelve customers a day? When I ask him if he has plans to expand, he tries his best not to look offended. "If anything, we would like to get smaller so we could give more of our attention to our guests. We started as an eight-seat counter, but that was too much, so we downsized to six." Now his greatest ambition is to remove another two chairs.
Lunch ends with a single gooseberry: bright orange, leaves pulled back like a ponytail, no bigger than a marble. I think it's some kind of Japanese thing I don't quite grasp, and maybe it is, but then I crunch down on the berry and the skin pops and releases a flood of sour-sweet juice and I realize that somewhere out there in a field far removed from Tokyo, a farmer with a soul like Sawada is putting everything he's got into these berries.
米 麺 魚
Not far from Sawada, past an eight-story Gucci building, a billion-dollar department store, a 7-Eleven, and a handful of vending machines, there is a small, quiet café where you can drink a cup of coffee from 1954.
Japan may claim one of the world's great tea cultures, but it's no stranger to the coffee bean. Coffee arrived in the country in the eighteenth century, piled high in the bellies of Dutch trading ships. It went relatively unnoticed by most Japanese until, in the early twentieth century, the Brazilian government began sending free coffee beans to Tokyo shop owners. By the 1930s you could find three thousand _kissaten_ (called _kissa_ for short), traditional Japanese coffee shops, offering Tokyoites a current of caffeine and a respite from city life.
The war put coffee's ascendance on pause. Beans were in short supply for the Japanese, so _kissa_ owners roasted soybeans instead, doing little to win over new clientele. At the same time, the Nazis developed a taste for coffee from the Far East and bought up everything they could from Indonesia and Sumatra. When shipments to Western Europe were cut off during the later years of the war, the coffee was rerouted through Japan, where it was to be sent by railroad from China to Germany. When fighting between the Nazis and Russians compromised the train routes, the beans sat all but forgotten in warehouses in Tokyo.
Ichiro Sekiguchi, a sound engineer from Tokyo, was serving in the war at the time and learned that the German-purchased beans were being stored on the outskirts of the city. When the war ended, he decided to go into the coffee business, using what he could of the beans left to languish as the Axis powers met defeat. By the time he opened Café de l'Ambre in Ginza in 1948, he was brewing five-year-old beans from Sumatra. What was born out of necessity turned into a groundbreaking technique. "The coffee had a rich, full taste, like good wine."
Today l'Ambre offers a wide selection of global vintages: '93 Brazil, '76 Mexico, and, the oldest, a Colombian bean from 1954. "Coffee beans are breathing," says Ichiro. "They evolve and develop different flavors over time." Ichiro's disciples have spread across Tokyo over the decades, infusing the city with a heady dose of cotton-filtered aged coffee, but at 101 years old, Ichiro still shows up to work every day to toast his ancient beans on a roaster he helped design himself decades back. The classic _kissa_ , the old beans, and the man himself stand as a stubborn rebuke to the wave of chain coffee outlets, convenience stores, and vending machines that sprang up during Japan's boom years and today make up the vast majority of the coffee market.
I settle onto a stool at the long countertop and choose a cup of Cuban coffee from 1974. A middle-aged barista in a striped turtleneck spends ten minutes dribbling hot water in concentric circles through a vintage Japanese sock filter. The coffee is like nothing I've tasted before, with a round, vegetal quality and only the faintest hint of acidity.
It is not Tokyo's finest cup of coffee. Until recently, that honor belonged to Daibo-san. But after thirty-eight years of slow filtration, Daibo retired at the end of 2013.
Tokyo's current coffee king is no doubt one of the new-wave wizards at work in Ebisu or Setagaya or Yoyogi, the hip corners of the city, where modern equipment, fresh beans, and time-aged technique combine to make powerful, balanced brews. But Ichiro trades in something more than technical precision—he offers a taste of the past, a reminder that there's always another way to do things.
On my way out, I see Ichiro sitting in his office, hands on his knees, a picture of him as a younger man hanging on the wall over his shoulder. He looks worried. "My supplies are down," he tells me. "I used to have five tons of coffee aging in my storage, but now I'm down to less than a ton." The soul of a _shokunin_ : a 101-year-old man worried about inventory.
"What's your secret, Ichiro-san?" I ask. "Coffee, of course. I drink at least five cups a day."
米 麺 魚
Yoshiteru Ikegawa knew that he wanted to cook chicken before he left first grade.
"At home we ate yakitori, but like most people in Japan we did it over gas in a small kitchen. But I'll never forget the smell of charcoal when my parents first took me to a real yakitori."
Despite the early revelation, Ikegawa didn't do what most budding _shokunin_ do: he didn't begin to slaughter chickens as a prepubescent boy; he didn't study their musculature and temperament in obscure texts found in dark library corners; he didn't even apprentice under a well-known yakitori chef—at least not at first. Instead, he did what a billion Japanese men had done before him: he became a salaryman. He put on a suit, took the train to work, drank with his colleagues, and remained loyal to his boss. But this wasn't a dream deferred; on the contrary, it was part of his master plan.
"Most _shokunin_ spend their entire lives in kitchens, never learning how to work directly with people," says Ikegawa. "I knew early on that dealing with the customer is one of the most important parts of being a master, so I started with that."
When he felt the business world had taught him the subtleties of customer service—above all, he says, how to give people what they want without their having to ask for it—he left the suit behind and took up an apprenticeship at Toriyoshi, an elegant yakitori bar in Naka-Meguro, where he trained for seven years, studying the bible of the flame-grilled bird. In 2007 he opened Torishiki next to Meguro Station, a lovely low-lit restaurant with a U-shaped bar centered around a small iron grill. His wife glides around the room in a kimono, dispensing drinks and good vibes to happy guests. The master himself stands at attention behind the fire, the spitting image of a _shokunin_ : chiseled facial features, warrior stance, a rolled white bandanna tied tight around his clean-shaven head.
Yakitori, like all great food in Japan, is both perfectly simple and infinitely complex. In its most literal state, yakitori is chicken on a stick grilled over an open flame—conceptually only a step removed from caveman cuisine. It's drinking food, a companion to beer and sake found on the menus of izakaya and clusters of back-alley street stalls that cater to hungry salarymen on their way to the last train.
But yakitori cleans up nicely, too, and room for refinement in the hands of discerning Japanese chefs is infinite. The lack of variables puts all that much more pressure and scrutiny on the few factors each individual can control: the source and intensity of the flame; the provenance of the chicken; the butchering, seasoning, and, above all, careful cooking of its flesh. This isn't a matter of running a skewer through a chicken breast and cooking until firm; there are a thousand defining details that must be managed if you take yakitori as seriously as Ikegawa does.
Yoshiteru Ikegawa, yakitori master, ready for service
_(Matt Goulding)_
One of the predominant trends in the world of high-end yakitori in Tokyo today is the full anatomy experience. At places like Toritama in Shirokane, owner Shiro Izawa butchers his chickens into thirty-six distinct pieces, a forceful biology lesson for anyone who has dismissed chicken as one-dimensional. For the diner, the question isn't whether you want a skewer of small intestine, but what part of the small intestine you would like: the duodenum or the ileum?
Ikegawa doesn't subscribe to the full-anatomy theory. He doesn't divide the thigh into inner, outer, and middle pieces to challenge your understanding of a single muscle. On a given night, he offers a tasting menu that spans about a dozen cuts of chicken, the same ones you'll find at most respectable yakitori joints in Tokyo: breast, thigh, wing, organs. Seasonal vegetables find their way to the flame, as does the occasional piece of duck or pork, but chicken is the star of Ikegawa's dissertation, a spellbinding treatise on the world of tastes and textures found within a single animal.
In the procession of pieces, which Ikegawa changes based on his read of each guest, you find crunch and chew, fat and cartilage, soft, timid tenderness and bursts of outrageous savory intensity. He starts me with the breast, barely touched by the flame, pink in the center, green on top from a smear of wasabi; a single bite buries a lifetime of salmonella hysteria. A quick-cooked skewer of liver balances the soft, melting fattiness of foie with a gentle mineral bite. The _tsukune_ , a string of one-bite orbs made from finely chopped thigh meat, arrives blistered on the outside, studded with pieces of cartilage that give the meatballs a magnificent chew. _Chochin_ , the grilled uterus, comes with a proto-egg attached to the skewer like a rising sun. The combination of snappy meat and molten yolk is the stuff taste memories are made of.
What separates Ikegawa from other serious yakitori, what has earned him a Michelin star and keeps his reservation book filled six months in advance, is the amount of care he puts into every last piece of flesh that meets his fire. He tinkers with each skewer as if it's the last piece of meat he'll ever cook, twisting, brushing, dipping, timing, tweaking—employing tight bits of motion to tease out the purest expression of each piece.
Ikegawa embodies the qualities that all _shokunin_ share: unwavering focus, economy of motion, disarming humility, and a studied silence that never betrays the inner orchestra his life's work inspires. At any given time, he handles up to a dozen skewers sizzling over _binchotan_ , an expensive hardwood charcoal that burns hot for hours, taming the fire with a wooden fan he keeps tucked into the back of his belt. When each piece is ready, he dips it into a large glass jar filled with a sweet lacquer of soy, mirin, and sugar, the indispensable _tare_ that sits patiently next to all yakitori grills in Japan. His was a gift from his master, a twenty-five-year-old taste of Toriyoshi that Ikegawa feeds each day like a baker feeds an ancient starter.
After the procession of parts comes Torishiki's _onigiri_ , a hulking triangle of rice basted with lavish amounts of rendered chicken fat and grilled until it looks like a lump of gold. I decide to take it back to the hotel to paw it in private, where it will be one of the great rice experiences of my life.
Before the bill, a cup of broth, a simmering distillation of everything that came before it. Chicken tea for the soul, a last respite before night falls over Tokyo.
米 麺 魚
The nighttime starts with a drinkable geography lesson: lemon from Ehime, golden ginger from Kochi Prefecture, and rice shochu from Kumamoto. For garnish, a light dusting of _sansho_ peppercorn—then served atop a bar made from a giant cross section of five-hundred-year-old Hokkaido oak. The acidity hits first, then the warm sting of shochu, followed by a one-two punch of spice—the scratchy, throat-tickling heat of ginger, the aromatic, tongue-numbing tingle of _sansho_. The young bartender in a crisp white tuxedo looks on and nods, as if to say this is just the beginning.
Gen Yamamoto was born in a small town in Mie Prefecture and came to Tokyo to learn about beverages. He trained in Aoyama Dining Bar in Tokyo before moving to New York for eight years, working as the cocktail man behind a few of Manhattan's best Japanese restaurants. He learned a lot in those years—from serving huge crowds to constructing seasonal menus to speaking fluent English—but all along, he was distilling an idea for a cocktail bar the likes of which the world had never seen. In 2013 Gen returned to Japan and began to build out his dream. "I wanted to try something that you can only do in a city like Tokyo."
Gen Yamamoto pours the next round.
_(Matt Goulding)_
Japan has a rich cocktail culture, one studiously built around classic drinks executed with precise technique. Mixologists invest a lifetime in learning how to perfect the hard shake, the gentle stir, the crystalline sphere carved from a giant block of ice. You will find textbook Gibsons and definitive Manhattans in drinking dens from Sapporo to Kagoshima, but you won't find many bartenders in this country pushing the limits of freewheeling cocktail creation.
Gen holds the same respect for refined technique as his colleagues, but he sees an untapped resource in Japan's cocktail culture: the country's bounty of vegetables, citrus, roots, and herbs. "We have amazing local citrus with soft flavors, but they don't mix well with gimlets so you don't see them in other bars. I don't make gimlets." Instead, he draws on the full reach of Japanese climate and topography to build a menu that changes almost daily. Today he's working with papaya and passion fruit from Okinawa, tomato and wasabi from Shizuoka, corn from Miyazaki—almost all of which he sources directly from the farmers themselves. At other moments of the year, you might find Hokkaido squash, Kanagawa carrots, Nagano quince, and a host of rare roots and esoteric herbs with seasons as fleeting as a full moon.
You can order à la carte, but it's the six-course tasting menu that best showcases Gen's vision. As with a great kaiseki feast, there is an arc to the stories he tells with his drinks, drawing heavily on the rituals of the Japanese tea ceremony, where technique, aesthetics, and an unwavering focus on microseasonality combine to create a vivid narrative.
To bring these liquid tales to life, Gen eschews the highfalutin tinctures and technologies favored by many Western bartenders—no volatile distillations, no blowtorched garnishes, no advanced equipment to speak of. He works with three primary tools: a strainer, a stirrer, and a long wooden muddler. It's the latter that allows him to transform seasonal produce into magic potions, using the blunt face to work fruits and vegetables down into smooth purees that form the base of most drinks. He uses no measurements, choosing instead to build the cocktails slowly, doing little half-stirs with his metal stirring spoon and tasting constantly as he creates.
No music, no wall decorations, nothing to take your attention away from the drinks and their methodical creation. I watch as Gen reduces lipstick-red tomatoes to a fleshy mass with the muddler, then strains the pulp and mixes it with rye vodka from Lithuania, using a tiny spoon and quick, short strokes to emulsify the tomato and vodka. He tastes, adjusts with a splash of vodka, stirs, tastes again, adjusts, adds a pinch of salt, stirs, tastes. He pours the drink into a clear glass cone, then cracks open a passion fruit and spoons the seeds onto the layer of tomato foam that has risen to the rim of the glass, dark orbs hovering delicately on the surface. He sprays a piece of black slate with water, arranges a few loose flowers at one edge and the cocktail at the other.
"Thank you for waiting."
The flavors bloom like a sunrise in my mouth, evolving with each sip: rich and mellow at the top, sweet and acidic in the middle, thinner and stronger with a heavy kick of rye as the drink disappears. It's not simply a tale of taste, but of texture and temperature.
As I journey through the glass, he sets about making the next drink—muddled kiwi from Wakayama, high-proof sake, and a splash of milk. In the foam of kiwi and dairy that settles on the top of the drink he suspends a spoonful of minced fennel, which punctuates each sip with crunchy licorice bursts.
Despite the fireworks in the glass, there is nothing loud or flashy about this man: no waxed mustache or sudden movements. With his shaved head and soft features and quiet voice, he could be a monk—if you swapped the white tux for saffron robes.
For the final course, he layers twelve-year-old Yamazaki whisky with muddled sweet potato and shavings of dark chocolate: a cocktail whose sweet, smoky, bitter brilliance I'll try and fail to convey a thousand times to anyone who will listen.
米 麺 魚
Not everything is so beautiful in Tokyo. Not every meal ends with a warm ball of rice in your pocket or a sweet potato cocktail in your belly. There are 35 million strong in these streets, after all, and only so many can fit into the sacred shrines scattered throughout the cityscape.
The moments I'm not pursuing the city's _shokunin_ I spend mostly on foot, losing myself in the minimalism of Omotesando, the maximalism of Shibuya, the J-pop gyre of Harajuku. Late one night I take a train to Shinjuku, the busiest station on earth, with nearly 4 million bodies traversing its tracks each day. More than home to a frenzied train station, Shinjuku is the heart of Tokyo's entertainment district.
Seventy years ago the neighborhood was all rubble, a smoldering heap of war regrets. Prostitution flourished, and, naturally enough, so did drinking and revelry as ramshackle bars popped up east of the station in the late 1940s. In the years after reconstruction, many of Japan's largest businesses set up shop here, and soon the bulk of the city's skyscrapers sprouted from Shinjuku, creating a dual identity—modern economic might by day, throwback pleasure center by night—that persists today.
I walk under the railroad tracks and into a labyrinth of narrow corridors called Memory Lane, better known as Piss Alley, named for the unsavory smell that once filled these confined quarters before bathrooms joined the party. Today the smell is mainly of yakitori, the lion's share of the shoebox spaces dedicated to chicken parts and cold beer. This is the foil to Torishiki: loud, cramped, drunk—with little subtlety but just enough soul.
In Kabukicho, Tokyo's red-light district, three-story pachinko parlors hum with the sound of retirement checks. Steamy restaurants dispense cheap, instant sustenance—ramen, burgers, dumplings. Yakuza toughs in cheap suits roam the blocks, the not so invisible hand behind most of the night economy.
I pass hostess bars where men with briefcases pay young girls to laugh at their jokes, host bars where middle-aged women pay boy-band look-alikes to tell them they're pretty. It all feels like a twisted simulation, a paper-thin world where people pay top dollar for the promise of a payout, the scent of a woman, the scratching of an itch.
They call this mix of nocturnal carousing _mizu shobai_ , the "water trade," a business built on the back of corporate expense accounts during Japan's rapid ascendancy to economic dominance. Companies may not have footed the bill for the worst secrets that lurk behind these doors, but they paid for the booze and bonhomie that loosened the ties, cemented the deals, and fed the darker sides of those who helped build New Tokyo overnight.
Shinjuku's Kabukicho, Japan's largest "red-light" district
_(Matt Goulding)_
Those darker sides feed strange industries and sad secrets in this part of town. The saddest secret is no secret at all: the Japanese have less sex than people of any other country on the planet. The women call men _soshuku danshi_ , herbivores who graze on leaves and pass on flesh, more interested in a virtual relationship than the real thing. The death of romance, some say, is the bane of birthrates, an economic and social crisis bubbling below the surface.
On the edge of Kabukicho, a line of people stretches around the block, all waiting to gain entrance into the Robot Restaurant, home to Tokyo's mad $100 million spectacle. Inside, bikini-clad-women straddle neon tanks, and robots dance and sing in a psychedelic futurama that will take years of the spectators' lives to fully process.
Beyond the lights and the noise, a refuge of Shinjuku's past: Golden Gai, a dense concentration of two hundred–odd bars organized down a series of dimly lit alleys. The spaces are tiny, the prices are high, the bars' motifs as narrow as the alleys they live in: medical gear, horse racing, exploitation films. I try to walk into one, but the owner sees my face and crosses his two pointer fingers into an X.
I find a more welcoming crowd at Bar Plastic Model—a toy-box love letter to 1980s plastic regalia. I order a glass of Nikka whisky while the guy next to me fumbles with a Rubik's Cube. He surrenders after a few minutes and strikes up a conversation in broken English. "You like Japanese food?" he asks. I drop $15, half for the cover charge, and merge back into the drunken alleyway traffic.
The whisky weighs on my eyelids, but the bright surgical lights of the Lawson pull me in like a tractor beam. From the outside, it looks like the convenience stores back home, but inside exists a very different world—one with a sake section and platters of raw fish and skewers of exotic vegetables simmered in dashi. The young woman behind the counter greets me with more cheer than can be expected at this or any hour. She works with a palpable sense of purpose, disarming surly customers with her smile, meticulously tending to a fryer full of chicken, all the while watching my cautious movements around her store. When I pause in front of the sake, searching for a nightcap, she comes from behind the counter, grabs a small bottle with a silver label, and hands it to me. " _Oishii!_ " she says, then goes back to bronzing the skin of her fried chicken.
A convenience-store _shokunin_? A liquor-fueled fever dream? Another lovely paradox? There is beauty to be found in the snack aisle, far from the tiny restaurants with buckling waitlists. It's not always as romantic as it sounds— _karoshi_ , death from overwork, is a real thing here—but in the long, strange trip ahead, when the train conductor crisply bows to an empty passenger car or the hotel cleaning lady origamis my towel into a perfect swan or the Lawson clerk fries chicken like a Southerner and picks sake like a sommelier for no other reason than because it's the job she has chosen, Tokyo will seem so much bigger than the world's largest city.
## _Vital Intel_
## KNOW BEFORE YOU GO
### **"Yes" goes a long way.**
_Hai_ , "yes" or "okay" in Japanese, is the most valuable word in the dictionary, a single high-pitched syllable you can finesse into something resembling a conversation. As with _vale_ in Spain, tone and inflection can bend the word into a dozen different meanings—from "Yes, I'm a huge fan of this strange and beautiful country" to "Of course I'd like you to soak me in unfiltered sake." Besides, you wouldn't want to say no to the Japanese, would you? Didn't think so.
### **It's not _that_ expensive.**
Legions of potential visitors pass on a trip to Japan because of the misguided belief that the country is unbearably pricey. Compared to Thailand or Central America, it's not cheap; put next to the UK, Switzerland, or any northern European country, Japan looks like a bargain. What is expensive: cab rides, _ryokan_ and high-end hotel chains, drinking in nice bars, formal sushi meals, and Japanese beef. What isn't expensive: public transportation, business hotels, drinking in izakaya, conveyor sushi, and beautiful bowls of noodles. You can't survive on $22 a day, but you can sleep and eat pretty well in the big cities for $100.
### **English is scarce.**
Not solar-eclipse scarce but pretty close. Few people in the world speak less English than the Japanese, which means you'll need to sharpen your body language skills, learn a few key phrases, and bring a willingness to laugh at yourself in the long stream of slightly embarrassing situations that will inevitably follow you around the country. Also a smart move: memorize ten or fifteen food words you can use when you get to a restaurant and can't read a single symbol in one of Japan's three alphabets. (See the "Gaijin Glossary" in chapter 3 for further guidance.)
### **Japan is a cash society.**
It may be surprising to learn that the country that invented the bullet train and robot strippers still relies on hard currency, but places from five-star _ryokan_ to top-tier sushi restaurants refuse to take credit cards, which means you'll need to carry a thick wad of yen around at all times. Very few Japanese ATMs work with foreign cards; instead use the machines in post offices and 7-Elevens, the two most reliable ways to get cash.
### **Subtlety is king.**
Japan is a society of deep-seated traditions and formalities that can puzzle the outsider, but getting it right can really make a difference for you and your hosts. Some basics to remember: Personal contact is mostly avoided in Japan, so mind your body and be prepared to bow rather than shake hands (a gentle bow for friends and family, a deeper dip from the waist for business relations or people of importance). Be punctual; tardiness isn't tolerated. And in general, avoid anything to cause undue attention to yourself or those around you; though you'll never blend in, Japanese value subtlety over aggressive individuality.
### **Buy your ticket to freedom.**
Purchased outside the country (through a travel agency or at international airports), a Japan Rail Pass allows for unlimited travel for up to three weeks on all but a few special trains in Japan. Not only will it save you money and time (a pass costs around $300 for unlimited travel in a week and doesn't require reservations on most trains) but it also turns the country into a traveler's buffet, allowing you to improvise your daily destinations based on your various appetites for culture, climate, and regional cuisine. Go anywhere and everywhere, just never board a train without a bento box and a beverage (more on that subject on The Beauty of Bento in chapter 7).
### **Smaller is better.**
It doesn't matter what you're eating: eel, sushi, noodles, sweets, cocktails. Small establishments are where _shokunin_ do their work. It may be intimidating to walk into a six-seat bar, but this is where you will find the good stuff—a place where the chef and the staff (most likely husband and wife) are unwaveringly dedicated to their craft. The most exclusive places require an invitation or a Japanese guest to accompany you, but the country is bursting with warm, intimate establishments dying for a chance to blow your mind.
### **Don't ask; just do.**
Hoping to wander a secret section of the fish market? Want to change seats on a train? Ask and you will invariably be defeated by a series of extended deliberations and bureaucratic consultations—a reflection of the highly structured reality of daily life in Japan. As long as it's not offensive and not illegal, you're better off doing first and feigning innocence later. It might not be the most elegant alibi, but nobody expects gaijin to know what they're doing in Japan.
## _Six Styles_
## FOOD GROUPS
_(Matt Goulding)_
MENRUI
_noodles_
SOBA
Japan's most elegant noodle, made from ground buckwheat.
UDON
Thick white noodles, served hot or cold.
SOMEN
Thin wheat noodles, normally served cold for dipping.
RAMEN
Made with wheat flour and alkaline salt to help retain its chew in hot broth (see more on The Ramen Matrix in chapter 4).
_(Matt Goulding)_
AGEMONO
_fried_
TEMPURA
Seafood and vegetables battered in flour, egg, and water (see more in Deep Fried in chapter 5).
KARAAGE
Bite-sized pieces of fried chicken, shrimp, or fish. Classic drinking food.
KATSU
Pork, chicken, or beef cutlets breaded in panko bread crumbs and fried crisp.
KOROKKE
Crispy breaded croquettes made from mashed potato or mincemeat.
_(Matt Goulding)_
NABEMONO
_stews_
SHABU-SHABU
Tableside hot pot of beef, vegetables, and tofu cooked in dashi.
ODEN
Meat, egg, fish cake, and a variety of vegetables slow-simmered in dashi.
SUKIYAKI
Meat and vegetables cooked in soy-spiked dashi and dipped in raw egg yolk.
MOTSUNABE
A popular _nabe_ of stewed beef offal and cabbage cooked in dashi.
_(Matt Goulding)_
YAKIMONO
_grilled_
YAKISOBA
Wheat noodles mixed with meat and vegetables and fried on a griddle.
OKONOMIYAKI
Cabbage pancake laced with meat or seafood and topped with a flurry of condiments.
YAKINIKU
Thin slices of meat cooked over a charcoal grill or on a griddle.
YAKITORI
Skewers of chicken and vegetables grilled over a charcoal fire (see more in On a Stick in chapter 6).
_(Matt Goulding)_
SUSHI
_raw_
SASHIMI
Raw slices of fish, seafood, chicken, or beef.
TATAKI
Torched tuna (and other fish), blackened on the outside, raw in the center.
NIGIRIZUSHI
Single pieces of raw fish pressed over seasoned rice (see more in In the Raw in this chapter).
MAKIZUSHI
Rolled sushi of rice and fish or vegetables, wrapped in dried seaweed (nori).
_(Matt Goulding)_
GOHAN
_rice_
ONIGIRI
Pressed rice triangle wrapped in seaweed, often filled with fish or vegetables.
DON
Rice bowls topped with various types of raw or cooked fish and meat.
MOCHI
A soft, sticky rice cake often filled with sweetened beans.
CHAZUKE
Soup made with steamed rice and tea, a classic comfort food.
## _Sushi_
## [寿司
**IN THE RAW**](nav.xhtml#rhh3)
The most famous and revered of Japan's culinary pillars, sushi comes with a set of rules and rituals that confound most outsiders. In a country where table manners matter, it's easy to look like a jackass when eating raw fish. Here's how to do it right. (Hand and sushi modeling by the great Takashi Saito, one of Tokyo's top sushi _shokunin,_ photographed by Sander Jackson Siswojo.)
_(Sander Jackson Siswojo)_
_(Sander Jackson Siswojo)_
## THE RULES OF SUSHI
### USE YOUR HANDS
Eat sashimi with chopsticks, but high-end nigiri is delicate, and all but the finest motor skills will test the sushi's integrity. Hands serve as more elegant and perfectly acceptable tools at a sushi bar, as long as they're clean.
### RESPECT THE RICE
It's the star of this show, and soaking it in soy sauce would compromise a technique that takes most sushi masters years to perfect. Instead, roll the nigiri over and gently dip the edge of the fish in soy sauce without saturating the rice.
### HAVE IT THEIR WAY
True sushi masters serve their pieces how they want them eaten—already seasoned with wasabi and soy. Keep it clean: no ginger (it's there to clean the palate between pieces), no wasabi in your soy sauce, and eat the nigiri in one bite. Always one bite.
### KEEP PACE
Great sushi isn't a social outing; it's a communion between you and the chef behind the counter. Part of that means eating nigiri as soon as it's made, at the peak of its deliciousness. Holster your smartphone and save the long conversations for the bar afterward.
_(Sander Jackson Siswojo)_
赤身
AKAMI (lean tuna)
_(Sander Jackson Siswojo)_
鯵
AJI (jack mackerel)
_(Sander Jackson Siswojo)_
中とろ
CHUTORO (medium fatty tuna)
_(Sander Jackson Siswojo)_
烏賊
IKA (squid)
_(Sander Jackson Siswojo)_
大とろ
OTORO (fatty tuna)
_(Sander Jackson Siswojo)_
小鰭
KOHADA (gizzard shad)
_(Sander Jackson Siswojo)_
車海老
KURUMA EBI (prawn)
_(Sander Jackson Siswojo)_
穴子
ANAGO (eel)
_(Sander Jackson Siswojo)_
鰈
KAREI (flatfish)
_(Sander Jackson Siswojo)_
鰹
KATSUO (skipjack tuna)
_(Sander Jackson Siswojo)_
蛤
HAMAGURI (surf clam)
_(Sander Jackson Siswojo)_
卵焼き
TAMAGOYAKI (omelet)
## _One Night at a_
## LOVE HOTEL
_(Matt Goulding)_
## _Chapter Two_
## OSAKA
This is how it happens: You sit down at a long counter in a restaurant that feels like someone's kitchen. You are alone, a bit nervous, uncertain whether to order a drink or ask the man behind the bar how his day went. He senses your apprehension (after all, he doesn't see your kind around here too often), disappears for a second, then deposits an armful of half-drunk wine bottles before you. Close your eyes and point, he seems to be saying. You abide.
Eventually, the restaurant fills out. A couple in the corner play footsy beneath the gentle heat of the warming griddle. A party of four, the men with spiked hair, the women with skirts and thick-rimmed glasses, slide into the counter seats next to you.
With the first glass of wine, the stilted silence prevails. A plate of warm buffalo mozzarella appears, speckled with pink peppercorns, and something about that combination of tang and spice, cream and crunch, tells you that tonight will be different from the others you've spent in Japan.
With the second glass of wine, your neighbors look over and offer a _kanpai_. Another plate arrives, this one a few pieces of seared octopus, the purple tentacles curled like crawling vines around a warm mound of barely mashed potatoes.
With the third glass of wine you begin to test your Japanese. _Watashi wa Matt-o desu. California kara kimashita._ Even the stone-faced salaryman eating pasta by himself in the corner cracks a smile. Glasses are emptied in your honor.
By the time you move on to sake, you feel the sweat above your eyebrows. At first you figure it's the spirited drinking and the aggressive round of selfies that has taken over the small restaurant, but then you see the old woman behind the counter testing the griddle, showering it with little drops of water that hiss on contact. She lays down a few strips of pork belly, then a ladleful of batter that she lovingly crisps in the sheen of rendered pork fat. She flips it, dresses it with a thick, dark sauce and shaved bonito flakes, which move like flamenco hands as they hit the hot surface, then slides it across the griddle toward you and smiles.
By the time you ask for the bill, the couple has their family album open on the bar and the group of four has nuzzled their stools so close you can smell the pinot gris on their words. You make plans to eat _hakozushi_ with one, an off-duty chef; another wants to show you a secret bar that serves only grilled offal. (Back at the hotel, four friend requests await you on Facebook.)
When you leave, the entire restaurant stands to escort you out the door. The man shakes your hand vigorously. The woman hesitates, then wraps her arms around you. You stand there for a second, unsure of how to thank them for such a beautiful evening. Finally, you bow as low and as slowly as possible and step reluctantly away. As you reach the corner, you turn around one last time, just to make sure, and there they are, the entire restaurant, waiting calmly for you to disappear into the night.
米 麺 魚
A well-worn Japanese proverb has it that Tokyoites spend all their money on footwear, Kyotoites on kimonos and formal attire. But Osakans save their funds for food and drink. There's a word for this Osakan propensity, _kuidaore_ : to eat until you drop.
Unfortunately, most visitors to Japan will never have the chance to eat themselves stupid in Osaka, because most visitors from the United States and Western Europe don't come to Osaka. They go instead to Tokyo, to bask in the full bulk and breadth of the Japanese urban phenomenon. They travel by train to Kyoto, to tour temples and gardens and capture geisha with their zoom lenses. As well they should. Osaka can't compete with Tokyo's size or stature, and it doesn't have the ancient culture and spellbinding beauty of Kyoto. Travel literature does little to stoke outsider interest. _Lonely Planet_ warns readers that "Osaka is not an attractive city"; other guidebooks have similarly grim tidings to share with readers.
But Osaka doesn't seem to mind. After 1,500 years of wildly yo-yoing fortunes, the city has developed something of a thick skin. In 645 Emperor Kotoku bestowed upon Osaka, then known as Naniwa, the honor of being Japan's first capital, only to abruptly move the government seat to Asuka just ten years later. In 744 Osaka was once again the capital of Japan, but this stint was even shorter: by 745 Nara had taken up the mantle as Japan's political center.
Come the sixteenth century, Osaka was back on center stage. In 1590 Hideyoshi Toyotomi, considered Japan's second great unifier, completed construction of Osaka Castle, the largest and grandest castle in the country. His son and successor, Hideyori, chose Osaka as his base, but his rival Ieyasu Tokugawa had other ideas. He laid siege to the castle in 1615, driving Hideyori and his mother to suicide, and established as Japan's capital Edo, which lives on today as modern-day Tokyo.
Osaka remained an important commercial center but suffered a series of blows over the next three centuries, including a peasant uprising in 1855 that razed a quarter of the city, and the brunt of the American attack on the Kansai region during World War II. The Americans may have spared nearby Kyoto, but they hit Osaka with the full force of their firebombing campaign, taking aim at the city's railways and extensive industrial complex. Two thousand tons of bombs and ten thousand lives later, the city was reduced to a skeleton of its former self. The rebuild was hasty and in some ways haphazard, robbing Osaka of the pockets of old-world charm that contrast so brilliantly with the modernity in most of Japan's largest cities.
One thing that never changed in millennia of misfortune: Osaka's place as the eating center of Japan. Osaka earned the moniker "the nation's kitchen" as early as the fifteenth century, when its privileged position on the Osaka Bay created a rich, thriving merchant class with the means to eat well (even today _mokarimakka_ , "Are you making money?" is a standard greeting in local dialect). Rice, seaweed, and other staples arrived from all parts of Japan, sent by feudal lords to sell in Osaka's massive system of commodity markets. The city also served as an entry point for Chinese, Korean, and other foreign ships bearing important edible cargo, deepening Osaka's place as a breeding ground for new tastes and big appetites. The table was set for a feast that continues today.
I've barely finished my bento from Tokyo Station when I step off the train in Osaka's Tenma district. Yuko Suzuki, a friend who works with high-end food producers in her adopted city, meets me at the station, and we plunge directly into Tenma's narrow, snaking streets in search of sustenance. Yuko was born to undertake these types of high-calorie missions; not only can she direct you to the city's best place for duck soba but she can also tell you where the restaurant buys its duck and how it grinds its buckwheat.
Our first stop is Tsugie, a smoky pocket square of a bar built around a large charcoal grill covered in unfamiliar anatomy. Tsugie serves _horumonyaki_ , an Osaka specialty focusing on offal and other off-cuts left behind by most restaurants (and cities with less discerning palates). "In Kyoto, they'd throw this stuff away," Yuko says as we settle into a corner of the bar, "but in Osaka it's the star of the meal."
Grilling odd cow parts at Tsugie, one of Osaka's many offal dispensaries
_(Michael Magers, lead photographer)_
Yuko tells me about the breed of young Osakan restaurateurs, the kind who have bucked the austere traditions of Japanese restaurant culture to focus on more pressing priorities, namely fun and deliciousness. That means open kitchens, louder music, more banter, less staff, bigger flavors, cheaper prices. Tsugie's owner, Takeshi Yamakawa, could be a poster boy for this ethos: after he takes our order, he bones out a few pounds of short ribs, switches the jazz for metal, fans a bed of charcoal, pours two perfect draft beers, and generally looks like he's having the greatest night of his life.
We start strong with chunks of the cow's third stomach, raw and slippery, slicked in sesame oil and green onion. Later come rosy pieces of flank steak and short rib, soft strips of raw heart with a yuzu chili paste, and gorgeous wedges of grilled tongue dripping with ginger-spiked soy sauce.
There are no seats at Tsugie; all of this is eaten standing up at the bar, washed down with generous amounts of _biru_ and sake. _Tachinomi_ , literally "drinking while standing," is big in Japan these days, signs of a shifting dining constituency that values good food at low prices over the formalities that dominate _ryotei_ , traditional Japanese restaurants. It may not have originated in Osaka, but wander the streets of Tenma after dark and you'll find a well-lubricated mix of salarymen, hipsters, and young couples _tachinomi_ -ing like they invented the form.
Later, we deepen our investigation into the drinking-while-standing phenomenon at Mashika, an Italian izakaya in a hip pocket of Nishi-Ku. The Italian-Japanese coalition is hardly new territory in this pasta-loving country, but Mashika is a different kind of mash-up. To start with, the space isn't really a restaurant at all. During the day, grandma sells cigarettes out of the small space. When the sun goes down, grandson fires up the burners as a crowd of thirtysomething Osakans drink Spritz and fill up on charcuterie, sashimi, and funky hybrids like spaghetti sauced with grated daikon and crowned with a wedge of ocean-sweet saury tataki. The menu follows no particular rules at all. Nobody seems to notice.
Eventually we are joined by Yumiko Nakamoto, editor of _Amakara-techo_ , Osaka's largest food magazine. She takes us for a highball break at the spiffy Samboa, where tuxedoed barmen manage to turn a whisky soda into a transfixing ten-minute preparation.
"You see?" Yumiko says, pointing to the tux with the long silver spoon in his hand. "All the details matter. People are just so passionate about food in Osaka. People want to sit at counters, talk to their neighbors, talk to the chef."
Yumiko, like most people I meet here, is not shy about her Osaka love. She also isn't shy about the rivalry with their Kansai neighbor.
"The French, the Chinese, and the Kyotoites have one thing in common: they all think they're number one. We're not worried about where we rank."
The last stop of the opening Osaka salvo is at Tenpei, another shoebox joint with a six-seat counter and two wooden booths. The menu, all three words of it, hangs from the wall: Gyoza. Pickles. Beer. Gyoza is Japan's take on the Chinese pork dumpling, shrunk down and refined in the same way the Japanese like to shrink and refine most everything. Still, textbook drinking food.
We order the entire menu four times over and take a seat in the booth.
Emiko Urakami, who opened Tenpei in 1952, claims to have invented the one-bite gyoza, now the accepted style across the city. "I made them like that because my hands are small," she says, rolling her palms over to show us the proof. "You see this line here; it means I'm going to be rich." Whether or not she invented the form, she certainly mastered it: the mahogany gyoza skin glistens with griddle fat and shatters with the gentlest bite, giving way to a tide of warm pork juice. The table grows silent as we devour the dumplings.
_(Michael Magers, lead photographer)_
Emiko is plump and ornery and wears the decades of gyoza making like vets wear war on their bodies. "I've been doing this for sixty-two years, feeding lines around the block," she says.
She must really love dumplings, I suggest rather lamely.
"Not at all. I don't really like gyoza."
Exhausted from the march down memory lane, she takes a seat with us at the table and watches intently while four of us try to make 150 dumplings disappear. Every time I put my chopsticks down, she touches my arm and motions for me to keep going. "Eat! Eat!" Nobody at the table wants to let her down, least of all me, but I don't know how to tell her that this is the fifth stop of the night, that my belly is filled with cow stomach and daikon spaghetti and the warm, mysterious pleasures of a city I'm falling hard for, so instead I just keep eating.
米 麺 魚
On my first morning in Osaka, I see the strangest thing: a man jaywalking. In fact, not just one man, but men and women and young students, crossing side streets and central avenues under the red glow of a stoplight. This might be typical urban behavior in most corners of the world, but jaywalking in Japan is snow-leopard rare. Even in Tokyo, in the small, fuzzy hours of the morning, call girls and yakuza toughs will patiently wait at the city's tiniest, least-traveled intersection for the green light to tell them when to go.
出る杭は打たれる—"The nail that sticks up will be hammered down"—goes one popular Japanese proverb. Japan is a paragon of order and civil obedience. There are parts of Japanese society so polished, they make Switzerland and Scandinavia look sloppy by comparison. In most regards, this is a glorious virtue: trains are scheduled down to the nanosecond, streets are so clean you can see your reflection in the pavement, and crime—especially violent crime—is all but nonexistent. But for those of us who like our punctual trains with a dash of disorder, Japan can feel stifling.
Osaka understands this. Its guiding proverb has a decidedly more defiant tone: 十人十色: "Ten persons, ten colors." Here the ubiquitous and homogenous spiffiness (some might say stiffness) that defines so much of Japan gives way to a more diverse and familiar tapestry of rich and poor, clean and dirty, highbrow and low. Even by Japanese standards, this is a massive city, with more than 19 million inhabitants in greater Osaka. The guidebooks aren't wrong; Osaka is not a textbook beautiful city. Not a seamless stretch of civilization, but a patchwork of skyscrapers and smokestacks, Gucci and ghettos, that better approximates life as most of us know it.
With all of this in mind, it's not surprising that Osaka is a center of casual food culture. Its two most famous foods, _okonomiyaki_ (a thick, savory pancake stuffed with all manners of flora and fauna) and _takoyaki_ (a golf-ball-sized fritter with a single chewy nugget of octopus deposited at its molten core), are the kind of carby, fatty, belly-padding drinking food that can sustain a city with Osaka's voracious appetite for mischief. You'll find plenty of both all over the city, but especially at the street food stalls that dot the electric streets of Dotonbori, Osaka's central entertainment district, a high-voltage maze of karaoke bars, gentlemen's clubs, and cheap calories, all punctuated by packs of giant metal animals—dragons, crabs, blowfish—that keep watch over the frothing masses. I've ended more than a few Suntory-soaked Osaka nights there at the elbow of a _takoyaki_ cook, that selfless citizen who works tirelessly browning octopus-stuffed batter so that individuals like me might sleep a bit better at night.
But nothing in Osaka is as precious as stomach real estate, and one would be wise to save plenty of space for the city's less-obvious delights. To find the best of Osaka's street eats, you must venture farther afield. Shinsekai is an area known for its rough edges, a once-glorious neighborhood modeled after New York and Paris that became a bastion of seediness and criminal activity in the lean postwar years. But beyond the hucksters and the hustlers, the pachinko parlors and the prostitutes, you will find what you need: _kushikatsu_ , tiny skewers of deep-fried meat and vegetables that were invented on these streets. Osaka is lousy with restaurants that have taken the _kushi_ concept, cleaned it up, and marked up the price 500 percent, but that's not what you're after. You want your stick fix here in Shinsekai, where _kushi_ shops line the streets, and with them groups of old couples and construction workers looking for grease. People will proclaim one establishment's superiority over another, but when it comes to fried meat, it's best not to overthink it.
We find a space at the bar of one of three dozen nearly identical establishments and rattle off an order that surprises even the double-fisting plumbers to our left. A meat marathon ensues. Two old guys behind the counter run through the paces, the first doing the three-step shuffle (flour, egg wash, panko bread crumbs), the second working the fryer like a North Carolina line cook. On the counter, a sign lays out the only true rule of the _kushikatsu_ : no double dipping. Drop your stick into the thick Worcestershire-like sauce once and be done with it.
The sticks come flying out of the grease, golden and glistening. They taste exactly as they should, of salt and crunch and a general meaty savoriness. A wedge of raw cabbage is offered as a breath mint of sorts, but in this bar at least, vegetables are a lonely bunch.
The longer we linger, the more the sharp edges of Shinsekai soften. After another round of _kushikatsu_ , the oily residue that hangs over this part of town feels like aromatherapy; the electric patter of pachinko begins to sound like Kenny G. By the time we wander back toward downtown Osaka, I'm convinced the most dangerous move you can make in this neighborhood is the double dip.
But you won't find Osaka's most quintessential eating experience under the menacing glare of a local thug in Shinsekai or next to a giant mechanized crab in Dotonbori. You'll find it on an unsuspecting street a few hundred meters from Kyobashi Station, at what looks more like a garage sale or a homeless enclave than a dining establishment. Nothing about Toyo makes sense: the kitchen is housed in the back of a pickup truck, the tables are made from stacks of yellow Asahi crates, and the hours are as erratic as the decor. But come most days after 4:00 p.m. and you will find a line of young Osakans clutching briefcases and fingering iPhones, eager to take in the Toyo _tachinomi_ experience.
Look alive! You will never find a better perch from which to take in the dramatic transformation of the postwork Japanese. It takes place every evening between approximately five and six in cities across Japan, as salarymen and women emerge from gleaming steel structures that hold them captive during daylight hours and beeline it to the closest izakaya to eat and drink away the sting of the workday. The same people who stood so quietly, so tensely in line behind you, soon grow animated. Ties are loosened, hair let down, and _kanpai_ s ring out in spirited choruses as rank and order dissolve with each passing sip. From soba to miso to raw-tuna red, the most aggressive transformers wear the stages of devolution on their faces. You want to be near this; this is the Japan that runs antithetical to the one you have constructed in your head. This is the beauty of Japan: it builds a set of beliefs and perceptions during the day, only to destroy them once the sun goes down. Rigid? Reserved? Formal? Find a table, fill it with food and beer and new friends, and watch as all those stiff postures slacken.
Fueling this metamorphosis is Toyo-san, chef and owner of this beautiful mess, a short, muscular man in his late sixties with a shiny bald head and wildfire in his eyes. He holds forth at the stovetop with a towel wrapped around his neck like a prizefighter, a lit cigarette dangling from his lips and a full-blast blowtorch in his hand. Toyo trades in extremes. Half the food that he sends out is raw: ruby cubes of tuna dressed with a heaping mound of fresh wasabi; sea grapes the size of ball bearings that pop like caviar against the roof of your mouth; glistening beads of salmon roe meant to be stuffed into crispy sheets of nori.
The other half gets the blowtorch treatment. Tuna is transformed into a sort of tataki stir-fry, toasted, glazed with ponzu, and tossed with a thicket of spring onions. Fish heads are blitzed under the flame until the cheeks singe and the skin screams and the eyes melt into a glorious stew meant to be extracted with chopsticks. Even sea urchin, those soft orange tongues of ocean umami, with a sweetness so subtle that cooking it is considered heretical in most culinary circles, gets blasted like a crème brûlée by Toyo and his ring of fire.
From spanking raw to burning inferno and back again, he cooks like a man possessed by some gnawing gastronomic schizophrenia. Every so often he looks up and gives wide-eyed onlookers an enthusiastic thumbs-up, but mostly he keeps to his food and his flame, laughing softly to himself at something we'll never understand. In some corners of Japan's culinary world, where restaurants have roofs and ingredients come with responsibilities, he might be crucified for his blatant disregard for convention and basic decorum, but in Osaka, where eating is a sport and rules are made to be blowtorched, Toyo-san is a hero.
Toyo-san and his flaming tuna, icons of Osaka
_(Michael Magers, lead photographer)_
米 麺 魚
That's not to say Osaka doesn't dress up. After all, this is a city with more boutiques than Paris and more Michelin stars than New York. But even the high-end stuff in Osaka exudes the warm, inviting, you're-here-to-have-fun-not-whisper-to-your-waiter vibe that you find at more everyday establishments.
At the heart of this ethos is _kappo_ , counter-style dining wherein the line between chef and guest is all but dissolved entirely. Chefs talk about the menu, take orders, cook inches from your face, and reach across the counter to serve you dinner. If this sounds familiar, it's because many of the best restaurants in the world right now—Momofuku Ko in New York, L'Atelier de Joel Robuchon in Paris—model themselves after Osaka _kappo_.
You'll find the counter philosophy expressed in a variety of styles across Osaka. Kigawa is the city's _kappo_ nerve center, the birthplace of modern _kappo_ and still the breeding ground for many of the city's best young chefs. The menu offers a hundred different dishes, all heavily tied to the seasons, all built with the best of Osakan raw materials. At Kahala, a favorite of brand-name Western chefs, Yoshifumi Mori serves an eight-course showcase of expensive, obscure local ingredients that concludes with a five-layer mille-feuille of rare beef and fresh wasabi. And at Yamagata, the chef turns his counter intelligence into a treatise on Kansai beef, a _horumon_ -inspired showcase of the entire sacred cow: heart sashimi with charred edamame, grilled tongue coated in mushroom miso, and a four-ounce square of tenderloin sauced with barrel-aged soy and fresh _sansho_ peppercorns.
But my favorite _kappo_ , one of the purest expressions of Osaka-style counter eating, is found down a narrow alleyway just a few blocks removed from the madness of Dotonbori. When you walk into Wayoyuzen Nakamura, the first thing you'll see is Nakamura-san himself standing firmly behind the counter, smiling broadly and bowing as you take your seat. He'll talk to you, ask you about your day, probe the dimensions of your hunger, discuss at length your hopes and fears.
"I can tell right away by looking at you what you want to eat," he says. "I can tell you how many brothers and sisters you have."
After divining my favorite color (blue) and my astrological sign (Aquarius), Nakamura pulls out an ivory stalk of _takenoko_ , fresh young bamboo ubiquitous in Japan during the spring. "This came in this morning from Kagumi. It's so sweet that you can eat it raw." He peels off the outer layer, cuts a thin slice, and passes it across the counter.
Then he goes to work. First, he scores an inch-thick bamboo steak with a ferocious _santoku_ blade. Then he sears it in a dry sauté pan until the flesh softens and the natural sugars form a dark crust on the surface. While the bamboo cooks, he places two sacks of _shirako_ , cod milt, under the broiler. ("Milt," by the way, is a euphemism for sperm. Cod sperm is everywhere in Japan in the winter and early spring, and despite the challenges its name might create for some, it's one of the most delicious things you can eat.)
Nakamura brings it all together on a Meiji-era ceramic plate: caramelized bamboo brushed with soy, broiled cod milt topped with miso made from foraged mountain vegetables, and, for good measure, two lightly boiled fava beans. An edible postcard of spring. I take a bite, drop my chopsticks, and look up to find Nakamura staring right at me.
"See, I told you I know what you want to eat."
The rest of the dinner unfolds in a similar fashion: a little counter banter, a little product display, then back to the burners to transform my tastes and his ingredients into a cohesive unit. The hits keep coming: a staggering plate of sashimi filled with charbroiled tuna, surgically scored squid, thick circles of scallop, and tiny white shrimp blanketed in sea urchin: a lesson in the power of perfect product. A sparkling crab dashi topped with yuzu flowers: a meditation on the power of restraint. Warm mochi infused with cherry blossoms and topped with a crispy plank of broiled eel: a seasonal invention so delicious it defies explanation.
Nakamura watches me eat. He watches everyone eat. Not in a creepy surveillance way, but in a sweetly innocent, I-really-hope-he-likes-this way. Soon you get the feeling that this guy has a body double or two floating around the restaurant, because despite the lavish attention he showers on all of his customers, and despite the fact that he's personally responsible for cooking at least half the plates that cross the counter, he does nothing all night but smile and look unreasonably relaxed behind the bar.
"We don't hide behind kitchen doors," says Nakamura. "This is what makes Osaka food so special, the relationship between the chef and the guest."
In a _kappo_ setup, there are no secrets: you know the shrimp soup is laced with both the brains and the roe of the crustacean, plus a jigger of cognac; you see the nail that goes through the brain of the wriggling eel just before it's filleted; you learn that 40 degrees is the perfect angle to transform a pristine fillet of fish into a pile of perfect sashimi. (An enterprising home cook brings home a doggie bag of pro moves after dinner at Nakamura.) If you're going to spend $100 on a meal, this is how you want to spend it, on a dinner that educates and entertains as much as it satisfies.
As I'm paying the bill, an older gentleman with an electric-blue tie sparks up a conversation with the chef. "What's good right now? You have anything you're really excited about?" Nakamura reaches down into one of his coolers and pulls out a massive wedge of beef so intensely frosted with fat that only the sparest trace of protein is visible.
"A-five Omi beef." A hush falls over the restaurant; Omi beef, ludicrously fatty and fabulously expensive, may be Japan's finest Wagyu.
The man bites, and Nakamura gets to work on his dish. He sears the beef, simmers wedges of golden carrots, whisks a fragrant sauce made with butter and vanilla. It's the first time the beef has made an appearance all night, but by the time Nakamura flips the steak, three more orders come in. Suddenly, the entire restaurant is happily working its way through these heartbreaking steaks, and I'm left staring at my bill.
"Are you sure you want to leave?" Nakamura asks, and before I can say anything, he cuts another steak.
米 麺 魚
Of course, there are things to do in Osaka that don't involve fried meat and torched tuna and foie gras masquerading as beef. That is to say, there are nonedible activities here. You can, for example, take in Aleutian otters, Panamanian porcupine fish, and a whale shark the size of a small school bus at the Kaiyukan Aquarium, home to one of the largest collections of sea creatures in the world. You can visit one of Osaka's many awesome and offbeat museums: see rural life transposed onto urban at the Museum of Japanese Farmhouses, witness the world's largest collection of sake drinking vessels at the Museum of National Ceramics, or personalize your own Cup Noodles at the Museum of Instant Ramen. Or spend a day soaking your bones at Spa World, the Epcot Center of _onsen_ , where revelers can travel through space and time to bathe in Caprian grottos, Greek medicinal baths, and the Trevi Fountain.
You can give your wallet a workout for the ages. Up near Umeda, you'll find one of Japan's greatest concentrations of department store awesomeness, including a thirteen-story Hankyu that could occupy the better part of a lifetime to fully explore. (CliffsNotes: Head straight to the basement and you will learn more about the beauty of Japanese food in an hour than in a week of restaurant eating.) Stroll the tree-lined lanes of Midosuji, Osaka's largest and leafiest boulevard, and dream dreams of Armani and Dior. (Or do what I do: dream of all the Omi beef and cod milt you can buy with the money you're not spending on clothing, purses, and other inedible extravagances.) If sprawling commercial centers and polished boulevards aren't your thing, try an afternoon at Tachibana-dori, Orange Street, retail fantasy for the hipster set, comprising a thousand meters of antique shops, boutiques, and pour-over dispensaries, all of which looks to be curated expressly for your Instagram feed.
The blue-lit backstreets of Dotonbori
_(Michael Magers, lead photographer)_
But ultimately, if you've come to Osaka, you've come to eat, drink, and soak up as much of the bonhomie as possible. And the best way to do that is through a good old-fashioned crawl, in search of the soul of _kuidaore_ , a slow, prodding, improvised evening of binge eating, drinking, and socializing that pushes you, your companions, and the city itself to the breaking point.
On my last night in Osaka, Yuko Suzuki rejoins the rabble, determined to lead us to parts unknown across the nation's kitchen.
The crawl starts where most good crawls end: in a dank basement filled with sake. Shimada Shoten is primarily a sake distributor, with a storefront stocked with a selection of Japan's finest _nihonshu_ (the owners tell me they have personally visited over 250 breweries to build out their list), but drop down a secret staircase and you land in the tasting room, with transport barrels and half-drunk bottles scattered everywhere. A group of men who look like they haven't seen daylight all week herald our arrival with a chorus of grunts.
Shimada operates on the honor system. Choose your glass from a stable of beautiful ceramic sake vessels, pick your poison from a series of refrigerators, and at the end of the night, tally up all the damage. Let's go.
We warm up with a sparkling sake from Hiroshima, then move on to a _junmai daiginjo_ from Ishikawa Prefecture, one of Japan's best sake-producing regions. You can taste its greatness, a cool shower of stone fruit and spring flowers. One refrigerator houses _koshu_ , aged sake, and we take our chances with a twelve-year-old bottle from Kyoto. Aged sake makes up only a fraction of a percent of Japan's total sake production, and remains a controversial beverage, given the vast range of quality found in the end product. This particular _koshu_ is as dark and musty as the room we're drinking it in.
We need a landing pad for all this rice wine, so we order the only food they serve in this joint: chunky miso from Wakayama, purple piles of pickled plums, and a strangely delicious cream cheese spiked with sake that pairs perfectly with nearly everything we pour.
_Nihonshu_ sneaks up on you. It goes down gently, floral and cold, coating your throat in the most positively medicinal of ways. There is no recoil, no heartburn, no palpable reminder that what you're drinking is an intoxicant—just gentle sweetness and the earthy whisper of fermentation. The beauty and size of most sake glasses—scarcely larger than a shot glass—adds to the apparent innocence of it all. But once you get started on a proper sake session, with you pouring for your partners and your partners pouring for you and nobody allowing a glass to ever approach empty, it takes on a momentum of its own.
Sake is produced in all but one of Japan's forty-seven prefectures (Kagoshima reserves its distilling ambitions for potato shochu), and the early evening unspools into a liquid road trip. Nagano, Akita, Nara, Sendai, Okayama: we race our way around Japan, testing the harvest from every corner of the country, probing the borders with our tiny glasses, savoring the nuances of climate and topography: the snowmelt from the mountains above Niigata; the pristine waters that flow from the Katsura River outside Nara; the long, sunny days of Okinawa. A proper sake tasting will whisk you around Japan faster than the Shinkansen.
Somewhere in this liquid fantasy my notes degenerate into a series of miso stains and sake splotches and islands of isolated adjectives, which grow increasingly abstract and aggressive as the night inches forward:
_Roasted asparagus . . . strawberry fields . . . liquid fireballs!_
You could lose yourself quickly down the Shimada rabbit hole, which is probably why it closes at 7:00 p.m. sharp. The owner hustles us out with a broomstick, and we scatter like drunk rodents under the white glare of the streetlights.
"If we're going to make it to midnight, we'll need some real food," says Yuko, ever a beacon of wisdom in our hazy Osaka adventures. "I have an idea."
Osaka is home to a rich, closed-door dining scene, not just formal _ichigen-san okotowari_ (invitation-only restaurants found everywhere in Japan) but clandestine spots in private homes and apartments scattered across the city. Madame X (she asked me not to use her real name, to protect her establishment) greets us at the door and ushers us into her apartment, a beautiful sunken space bathed in warm lights, with an open semipro kitchen and a bar with stools overlooking the action. Rendered chicken fat pops and crackles. A wok sizzles with blistered vegetables. Outkast bumps in the background.
We crowd into a nook around a chest-high table with a view of the residential street below. Two men in their late thirties, charming, good-looking dudes still suited up from a long Friday, join the group. They both work for United Airlines, and it is clear from the way they make their wine vanish that this was a week they'd like to forget. "Wait, what are you doing in Osaka?" one asks me, a mixture of merlot and disbelief on his breath.
Madame X returns to flood our table with a selection from tonight's menu: fried tofu floating in dashi and covered with dancing bonito flakes, spring vegetables simmered in dashi and sake, and the house specialty: pizza coverd in _shirasu_ , tiny whitefish. The conversation, as it inevitably does in the presence of this foreign journalist, turns to Osaka.
Osakan fun facts ascertained during this stage of the evening:
■ Osakans are hilarious. More than 50 percent of Japan's professional comedians come from Osaka. (Kyotoites, despite living just twenty minutes away, are a decidedly unfunny species, I'm assured by everyone at the table.)
■ Everyone thinks Tokyo has Japan's best sushi, but they're wrong. Osaka does, because Osaka gets the best fish. Serious eaters from Tokyo take the train down just for dinner.
■ Osaka sake is really great, because Osaka water is really great. (At which point we switch from French wine to Osaka sake . . . Point taken.)
■ Osakans love foreigners, even if foreigners haven't fully embraced Osaka. "Please tell people to come." I'm on it, dudes.
■ Osakans once dumped a statue of KFC magnate Colonel Harland Sanders in the Dotonbori River to celebrate a victory by the Hanshin Tigers in the 1985 Japan Championship Series. The Tigers then went on an eighteen-year losing streak, giving birth to the Curse of the Colonel theory and inciting city officials to dredge the river in search of the shipwrecked birdman. He was eventually found in 2009, though his left hand and eyeglasses remain lost to the canal; only when those are found, Osakans speculate, will the curse be fully lifted.
■ Osaka is really awesome.
From Madame X's private parlor we slip our way back into the world of legal establishments. Yuji Kawabata is a well-known restaurateur with six popular izakayas clustered around the Namba area. He's also an artist, a ceramics collector, a deep thinker, and a celebrated drinker. Soul mate material.
His restaurant is closed by the time we arrive, but he ushers us upstairs to a table, opens a massive bottle of sake, and instructs his kitchen to give us everything they've got. Out comes everything: piles of blistered _shishito_ peppers, golden fried sandwiches of taro root stuffed with minced pork, bowls of dashi-braised daikon, a tower of yakitori, including my favorite, _tsukune_ , a charcoal-kissed chicken meatball rich with fat and cartilage, meant to be dipped in raw egg yolk. My chopsticks cannot move fast enough.
The Osaka night begins to buckle at the knees.
_(Michael Magers, lead photographer)_
As we work our way through a second bottle of sake, Yuji presents me with two of his favorite pieces from his ceramics collection, a violet sake pourer from a young Osakan artist and a pimply pink bowl from southern Kyushu. I do what I've been told to do with all gifts: refuse once politely, then accept with exaggerated displays of gratitude.
_Not a city, a sensation . . . lights grow, night flows. . . . Osaka decides, we can't say no._
At 2:00 a.m. the airline execs call for a nightcap. As we walk up the stairs to Teppan-Yaro, a bar not far from Yuji's restaurant, I realize that I have been here before, six months ago on my maiden Osaka voyage. That night ended in a blur of whisky shots and air guitars. Somewhere in my parting words was a promise to return soon.
We open the door and the room explodes. A team of line cooks working the griddle raises spatulas in a spirited salute to our posse. The owner, skinny with long hair and the faintest whisper of a mustache, comes from behind the counter and pulls me in close to his chest. "You came back!" Music is cranked. Drinks are proffered. The night begins to buckle at its knees.
_Clickety-clack . . . Whisky Pete is back . . . #mayocoma! . . . one-eyed purple people eater._
The Stones bleed through the speakers and the shots ring out and the men work the _teppan_ with manic fury. It's unclear if anyone has ordered food, but they keep cooking: _clickety-clack, clickety-clack._
The drink of the house is a purple potion made with vodka and juice and crushed unicorn horns. A decree has gone out across the bar to drown me in this shit.
I'm not used to this kind of treatment. The tourist is a fragile species in Japan, treated with guarded respect and kept at arm's length. Japanese are unfailingly polite, and most will go to absurd lengths to give you directions or greet you warmly as you enter their establishments. But even then, you are destined to sit on the sidelines of this society and watch it unfold from the outside. Outside that tiny yakitori joint exhaling charcoal smoke and good times. Outside the incredible sushi bar that serves only Japanese-speaking customers. Outside the animated conversation taking place on the stools next to you. This is a dense culture, steeped in a history, a code, and a language that most will never comprehend, and so we stare through the window and wonder what it must be like to understand.
But Osaka leaves the door ajar, if only a crack. Walk into a bar with an open mind and a wide smile, and someone might buy you a drink and ask you what you're up to tomorrow. It might not always be true, doors are walls here too, but everywhere you will see those little slivers of light, and when you see the light, the only thing to do is step into it.
At the end of the night, when our stomachs are stretched to the snapping point and all I can see is purple, the owner turns down the music, quiets the crowd, and makes an announcement in Japanese. Naturally, I understand none of it, but cheer along with the crowd as he punctuates his sentences with hand chops and fist pumps. Until suddenly everyone is staring at me with glasses raised. The owner comes from behind the bar and presents me with a white bandanna, the same one that he and his team of cooks are wearing.
This isn't a polite gift for an enthusiastic foreigner. It's a key to a door I thought was locked forever. And, for this one night in Osaka, it is mine.
## _Vital Intel_
## [揚げ物
**OPERATION IZAKAYA**](nav.xhtml#rhh5)
It's your first night in Japan. All is a mess of incomprehensible signs and inscrutable commuters. Then you find an izakaya—Japan's ubiquitous, open-hearted bastion of small plates and big drinks. In this, the most accessible and democratic of all Japanese institutions, you can have it all. Follow these steps, and your first night in Japan might be your best.
_(Michael Magers, lead photographer)_
_(Matt Goulding)_
**START WITH SAKE**
Izakaya means "to stay in a sake shop," and rice wine should propel your tavern experience. The most important rule of sake: keep your drinking partners well lubricated, but never serve yourself (that's what partners are for). Start with a midrange _junmai_ , a pure rice wine.
_(Matt Goulding)_
**GO RAW**
Next to first-class sushi bars, you won't find better raw fish in all of Japan. Izakaya sashimi plates typically deliver a mix of three to five different types of seasonal seafood, such as scallop, yellowtail, squid, or tilefish. A perfect match for sake.
_(Matt Goulding)_
**BRING THE FIRE**
Yakitori appears on most menus, but even better is whole grilled fish. Excavating the tastiest bits of a fish head with your chopsticks is izakaya eating at its best.
_(Matt Goulding)_
**SCALE MOUNT SAKE**
Remember, you're here to drink. Now that you've warmed up, move on to a _junmai daiginjo_ from Niigata, Japan's greatest sake-producing region. _Daiginjo_ means at least 50 percent of the rice has been polished, giving it a more delicate, complex flavor.
_(Michael Magers, lead photographer)_
**TACKLE THE TEPPAN**
Griddle-cooked staples like _yakisoba_ and _okonomiyaki_ make it onto most izakaya menus, but it's crispy gyoza, Japan's juicy pork dumplings (best when lashed with chili oil), that offer the best match for your blooming buzz.
_(Matt Goulding)_
**SWITCH YOUR POISON**
Now that you've warmed up on sake, time to wade into deeper waters. Shochu is the distilled drink of choice in Kyushu, packing twice the punch of a typical sake. Try a sweet potato–based _imo_ shochu. (No shochu? Make it a highball—a salaryman favorite.)
_(Michael Magers, lead photographer)_
**GREASE UP**
Fried food makes the perfect booze sponge. _Karaage_ (fried chicken) and _agedashi_ tofu (fried tofu) are the most ubiquitous, but crispy oysters and _satsumaage_ , fried fish cakes from Kyushu (a perfect match for the shochu!) offer a chance to break new culinary ground.
_(Michael Magers, lead photographer)_
**BE BRAVE**
Finish with something from the inevitable section of izakaya oddities: fermented squid guts, cod sperm, fried testicles. There's no better time than right now.
## _This Is the Beef_
## [和牛
**WAGYU 101**](nav.xhtml#rhh6)
_(Matt Goulding)_
### **DON'T CALL IT KOBE**
Kobe is what your local gastropub calls its sliders, not what the Japanese call their high-fat beef. Kobe is a city famous for the quality of its Wagyu (the proper name for Japanese beef), but it represents less than 1 percent of all Japanese beef. Lavishly marbled Wagyu comes from nearly all of Japan's forty-seven prefectures. Want to sound smart? Look for Matsusaka, Omi, or Mishima Wagyu, among the most revered in Japan.
### **THERE'S NO BEER IN THAT BEEF**
Rumors that Japanese cows get fat on beer, sake, and massages turn out to be greatly exaggerated. Historically, some small part of the Wagyu industry advocated beer or sake to stimulate appetite in the warmer months, while others massaged cows for better fat distribution, but the practice is limited to a tiny percentage of the overall Wagyu game. Most cows live on a diet rich in grains and move very little—two secrets to the intense intramuscular marbling.
### **IT EATS LIKE BUTTER**
Wagyu is ranked on a well-defined scale of letters and numbers based on the quality of the beef and the intensity of the marbling. A5, the highest ranking for Wagyu, indicates meat so densely marbled that the red protein is tough to spot. The best Japanese beef eats like European butter, which is a neat trick for protein, but those who love the intense minerality of, say, a grass-fed sirloin may be left wondering, Where's the beef?
### **IT COSTS A FORTUNE**
Whether it's worth it depends on how deep your pockets and your love of beef fat are. Dedicated Wagyu restaurants charge up to $200 for a basic steak dinner. Get your fix with a few bites at a high-end izakaya, or try a Wagyu _sando_ —lightly fried beef stuffed between soft bread. You'll find better value with F1 beef, a mix of Wagyu and Angus that delivers much of the extravagant richness of the high-class stuff without the price tag.
## _The_
## KNIFE MAKERS OF SAKAI
The blade makes its way from hand to hand, from anvil to grinder to whetstone. The four artisans in this famed knife town south of Osaka create some of the finest edges in a country that still reveres a balanced blade. American photographer Michael Magers crossed Japan in search of _shokunin_ , Japan's fading class of master craftsmen, and came to Sakai for these intimate portraits of the men of steel.
_(Michael Magers, lead photographer)_
_(Michael Magers, lead photographer)_
### **YOSHIKAZU IKEDA**
_Forger_ **|** 鍛冶
_(Michael Magers, lead photographer)_
### **SHUNICHI TAHARA**
_Sharpener_ **|** 刃付
_(Michael Magers, lead photographer)_
### **KOICHI MORIMOTO**
_Honer_ **|** 刃付
_(Michael Magers, lead photographer)_
### **SUSUMU WAKAI**
_Setter_ **|** 問屋
### **SANTOKU**
三徳包丁
_(Michael Magers, lead photographer)_
### **UTILITY BLADE**
In a culture with knives for every micro task, _santoku_ is the closest thing to an all-purpose chef's knife. _San_ = "three" and _toku_ = "virtue" or "character." The three virtues are meat, fish, and vegetables.
### **HIGH-CARBON STEEL**
Sakai blades are made by forging together soft ferrite and high-carbon steel at 1,000˚C, a delicate process that few blacksmiths have mastered, and can fetch up to $3,000 per knife.
### **SINGLE-BEVELED EDGE**
The edge first takes shape on a sharpening stone made from whale bones. The exact angle depends on what the blade will be used for: there are configurations for everything from root vegetables to octopus.
### **FORGED IN SAKAI**
A wealthy port town, Sakai was once home to Japan's finest sword makers. Today a small group of dedicated _shokunin_ continue the tradition by making the most coveted knives in the culinary world.
## _Chapter Three_
## KYOTO
If you blink, you might miss it. You might miss the wet floor at the threshold, symbolically cleansing you before the meal begins. You might overlook the flower arrangement in the corner, a spare expression of the passing season. You might miss the scroll on the wall drawn with a single unbroken line, signaling the infinite continuity of nature. You might not detect the gentle current of young ginger rippling through the dashi, the extra sheet of Hokkaido kelp in the soup, the mochi that is made to look like a cherry blossom at midnight.
You might miss the water.
"I believe water is the most important ingredient in Japanese cuisine," says Toshiro Ogata, chef of the eponymous two-star kaiseki restaurant in the heart of Kyoto. "I always think about different ways to showcase water."
It paints an extraordinary picture, a chef kept awake at night by the most common substance on earth, an ingredient distinguished mainly by its absence of flavor. It would be easy to dismiss as precious lip service, an affectation delivered by a man expected to obsess over the details everyone else overlooks, if not for the fact that every week he drives into the mountains outside Kyoto and comes back with the best water nature provides.
If that's not proof enough of his obsession, there are the first three courses he serves me: a bowl of rice from Niigata Prefecture, steamed seconds before I sit down, shiny with a sheen of warm starch, presented with nothing more than a pod of lightly grilled fava beans; _ichiban_ dashi, a stock of seaweed and dried tuna, twenty minutes old, served in an ink-black lacquer bowl; and finally, that same infant dashi, the same inky lacquer, this time with a pearl-white cross section of simmered onion floating in the center, a world of texture in its rings.
As I sip from the bowl cradled in my palms, I watch the line of liquid vanish against the shiny black surface—a moment of peace and mystery. Three courses, three expressions of water, collectively the most audacious and confounding start to any meal I've ever eaten.
Ogata-san is forty-seven years old, but cooks and speaks with a wisdom that suggests he's been on this rock for a few extra orbits. With each new course, he offers up little bites of the ethos that drives his cooking, the tastes and the words playing off each other like a kaiseki echo chamber.
Ark shell, a bulging, bright orange clam peeking out of its dark shell, barely cooked, dusted with seaweed salt.
_"To add things is easy; to take them away is the challenge."_
Bamboo, cut into wedges, boiled in mountain water and served in a wide, shallow bowl with nothing but the cooking liquid.
_"How can we make the ingredient taste more like itself? With heat, with water, with knifework."_
Tempura: a single large clam, cloaked in a pale, soft batter with more chew than crunch. The clam snaps under gentle pressure, releasing a warm ocean of umami.
_"I want to send a message to the guest: this is the best possible way to cook this ingredient."_
A meaty fillet of eel wrapped around a thumb of burdock root, glazed with soy and mirin, grilled until crispy: a three-bite explosion that leaves you desperate for more.
_"The meal must go up and down, following strong flavors with subtle flavors, setting the right tone for the diner."_
And it does, rising and falling, ebbing and flowing, until the last frothy drop of matcha is gone, signaling the end of the meal. Ten dishes, thirty ingredients, the breadth of kaiseki: boiled, raw, steamed, fried, grilled, all served in their proper order, all part of a poem Ogata pens to this city and to this season. A beautiful piece that I'm not sure I fully understand.
There is no questioning the quality of his ingredients, the scope of his skill, the depth of his dedication, but this is a cuisine so minimalist that it sometimes seems to not exist at all. "Western food is about addition," says Ogata. "Japanese food is about subtraction."
The wet stone, the lonely scroll, the midnight mochi: these are the tiny details that make kaiseki Japan's most elegant and extraordinary and befuddling branch of cuisine. Beautiful and austere, ancient and earnest, never has a cuisine better matched a city. And never has there been a city as mystifying as Kyoto.
米 麺 魚
Over 30 million people a year come to soak up the Kyoto experience, to visit the more than two thousand temples, marvel at the Zen restraint of hundreds of rock gardens, lose themselves in the shadows of towering bamboo forests. UNESCO must appropriate a budget solely for lavishing Kyoto with awards and designations, because over the years the UN has blessed seventeen buildings with its coveted heritage award. Add in the creaky teahouses, the tiny, mystery-filled streets of the Gion, the kimono-clad women, the sword-making men, and you see why Kyoto is considered the cultural heart of Japan, what Pico Iyer, the British essayist who came to the city in 1992 and never left, calls "a citywide shrine to Japaneseness."
Ogata's boiled bamboo in bamboo broth
_(Matt Goulding)_
No shrine to Japaneseness would be complete without its own dedicated cuisine, and _kyo-ryori_ , the food of Kyoto, proves to be every bit as decorated as the rest of the city's disciplines. Kyoto claims seven restaurants with three Michelin stars and another twenty-two with two stars making it the most Michelin-dense city on the planet. If the Michelin man doesn't impress you, consider this: in November 2013 Kyoto led a successful campaign to have Japanese cuisine enshrined with a UNESCO Cultural Heritage award, one of only a handful of cuisines in the world to be honored with the distinction.
When I first came to Kyoto, it wasn't for gardens or geisha or UNESCO-blessed shrines; it was for kaiseki. For years I had marveled at it from afar, studied the format and history, read about the quiet practitioners turning dining into an all-encompassing feast for the senses. I had seen its fingerprints all over fine dining in the rest of the world—from the mixture of minimalism and naturalism that defines modern haute cuisine to the entire concept of an interconnected tasting menu designed to tell a story larger than the sum of its tastes.
It was the fall, a beautiful time for eating in Japan, when wild mushrooms cover the forest floors and tiny sweet fish swim upstream. In four days I ate five kaiseki meals, a procession of lunches and dinners made with the best imaginable ingredients handled with tremendous precision and served in exquisite settings. There were moments of striking beauty and astounding taste, but those were ultimately overshadowed by the confusion, consternation, and, worst of all, boredom I often felt in these restaurants. Every meal contained the same plate of sashimi, the same vegetable tempura, the same stilted, slightly tense service. I began to feel that kaiseki was a movie whose plot I already knew. In five meals I had five _dobin mushi_ , a teakettle filled with conger eel and matsutake mushroom stems, along with a fragrant strip of _sudachi_ lime zest, meant to be drunk first, then eaten. The first time, it was an eyes-in-the-back-of-the-head revelation; the fifth time, it felt soulless.
By the time I boarded the Shinkansen back to Tokyo—wallet empty, belly full of mushroom tea—I felt as if I might never need to eat another kaiseki meal in my life.
Was I missing something? Was I bringing my own baggage to the dinner table, hampered by being a foreigner, or did other Japanese find kaiseki so inaccessible? Was a chef who built a menu around water totally fucking insane, or was I the crazy one? The more I thought about it, the more I came to feel for kaiseki the same way I felt for Kyoto: breathtakingly beautiful but encased in amber, more a fossil than a living, breathing creature.
I took my concerns to the people who knew best, the serious eaters of Japan, speaking with friends in Kyoto, Tokyo, and other parts of the country to get their read on kaiseki. I quickly learned that kaiseki is like a Rorschach test for foodies: some see the epitome of elegance and refinement; others see a boring, overpriced cuisine in need of a shake-up. A small group of dedicated chowhounds I consulted spoke reverentially of the importance of kaiseki, its history and impact on Japan, not unlike the way certain Americans discuss the importance of, say, the Constitution. The other, much larger group all offered variations on the same theme: kaiseki is for old, rich people.
I found myself stranded on an island in between, respectful of its beauty and refinement but wary of its rigidity. I continued to feel this way about Kyoto and its famous cuisine until, one day in the fall of 2013, I met Ken Yokoyama, and everything I thought I knew was turned upside down.
米 麺 魚
Whatever you need, Ken Yokoyama has you covered. Need tickets to that Kabuki show that sold out weeks ago? He knows a guy. Want to eat in that tiny Michelin-starred restaurant all your food-obsessed friends talk about? He'll make a call. Hoping to catch a glimpse of that austere rock garden without all the camera-clutching tourists? He'll do his best, which is always more than enough.
In a city where most doors are locked, Ken carries a skeleton key. As the general manager of the Hyatt Regency Kyoto, it's his job to open doors, but he does so with a subtlety and humility that belie his position. I first meet Ken as a guest of the hotel, which I choose based solely on his reputation, which has spread to certain corners of the country. The accolades, if anything, prove to be understated.
Every morning when I see Ken working the breakfast crowd, I think of the scene in _Casino_ in which Robert De Niro is meeting with an exec in the hotel restaurant and both are eating blueberry muffins, only his partner's is exploding with fruit, and De Niro's has only one or two sad little blueberries. So he marches back to the kitchen and tells the dumbfounded chef that he wants every last muffin to have the same amount of blueberries. _The same amount of blueberries._ That's Ken, only not just with muffins, with everything: the petals on the flower arrangements that decorate the lobby, the warm sesame tofu served in the _robata-ya_ downstairs, the hand-signed welcome note waiting on your pillow.
The Hyatt was once the indisputable hotel king of Kyoto, "the only non- _ryokan_ game in town," as a travel agent friend told me when I first visited. But in late 2013 the Ritz-Carlton set up shop across the Kamo River, opening an aggressively gorgeous hotel rumored to cost $300 million to build. They poached a few of Ken's lieutenants and no doubt went after Ken himself (even if Ken won't admit it), but he seems unfazed. If anything, it makes Yokoyama-san push even harder.
I first recognize the full scope of Ken's reach one fall afternoon a day after we meet. Over a cup of coffee we talk kaiseki, and Ken gently probes the depth of my understanding. Between business dinners, government meetings, and VIP guest treatment, Ken eats kaiseki twice a week, as much as any man in this city, and he brings an anthropologist's eye to the discipline.
After offering up a few quick history lessons, he asks me where I intend to eat. I tell him that Sojiki Nakahigashi, an intimate, modern kaiseki considered Kyoto's most difficult reservation, is at the top of my wish list. "Yes, that would be at the top of my list, too, but as you know, it's not easy even for locals to get in." A few hours after our talk, my phone rings; it's Ken. "Nakahigashi-san will met you tomorrow at five p.m. for a chat. Then at six p.m. he will cook you dinner."
Hasao Nakahigashi grew up in a Kyoto _ryokan_ , helping his parents with the cooking that forms a fundamental part of the traditional inn experience in Japan. Later he trained at Hyotei, the grandfather of all kaiseki in Kyoto, in operation since the early seventeenth century, before opening his own restaurant close to the Ginkaku-ji temple in 1992.
Nakahigashi is in his early fifties, with soft features and a gentle smile that exudes a sort of old-soul tranquillity. He wears not a chef's jacket but the white lab coat and tie favored by Kyoto's kaiseki masters. Every morning, before he puts on his coat and tie, Nakahigashi treks to the outskirts of Kyoto to pick vegetables and wild herbs from the hillsides and riverbanks. "The most important part of my cuisine is a strong sense of the season," he says. "If you had to use one word to describe Japanese cuisine, it's _nature_."
The restaurant has a floor made of small, smooth stones and a long polished cherrywood bar overlooking the open kitchen. The centerpiece of the room is a hulking orange _kamado_ , a traditional wood-fired rice cooker, the same that has been used for centuries to prepare Japan's sacred grain. As diners arrive, Nakahigashi loads up the rice and feeds the fire.
The meal begins the way all kaiseki meals begin, with _hassun_ , a mixed plate of small bites—fish and vegetables, usually—used to set the tone for the feast to come. In a bowl of pine needles and fallen leaves he hides smoky slices of bonito topped with slow-cooked seaweed, ginkgo nuts grilled until just tender, a summer roll packed with foraged herbs, and juicy wedges of persimmon dressed with ground sesame and _sansho_ flowers. Autumn resonates in every bite.
While the rice simmers away, the meal marches forward: sashimi decorated with a thicket of mountain vegetables and wildflowers; a thick slab of Kyoto-style mackerel sushi, fermented for a year, with the big, heady funk of a washed cheese; mountain fruit blanketed in white miso and speckled with black sesame and bee larvae. His skills with vegetables bear the mark of a man willing to hunt them down every day at dawn. "The vegetables tell me what to do," he says. "When I pick up a daikon, it says, 'Please bake me, please simmer me.'"
As the meal progresses deeper and deeper into the Kyoto wilderness, the anticipation for Nakahigashi's famous rice grows palpable among his patrons.
"Rice is sacred to the Japanese people," he says. "We eat it at every meal, yet we never get tired of it." He points out that the word for rice in Japanese, _gohan_ , is the same as the word for meal.
When he finally lifts the lid of the first rice cooker, releasing a dramatic gasp of starchy steam, the entire restaurant looks ready to wave their white napkins in exuberant applause.
The rice is served with a single anchovy painstakingly smoked over a charcoal fire. Below the rice, a nest of lightly grilled matsutake mushrooms; on top, an orange slice of compressed fish roe. Together, an intense wave of umami to fortify the tender grains of rice.
Next comes _okoge_ , the crispy rice from the bottom of the pan, served with crunchy flakes of sea salt and oil made from the outside kernel of the rice, spiked with spicy _sansho_ pepper. For the finale, an island of crisp rice with wild herbs and broth from the cooked rice, a moving rendition of _chazuke_ , Japanese rice-and-tea soup. It's a husk-to-heart exposé on rice, striking in both its simplicity and its soul-warming deliciousness—the standard by which all rice I ever eat will be judged.
米 麺 魚
Before you rush off to drop $300 on an eight-course kaiseki dinner, take this simple questionnaire:
■ When I eat out, I like to do so in a quiet, contemplative way.
■ I care a lot about the bowl my soup is served in.
■ I prefer subtle flavors to aggressive ones.
■ I am a big fan of negative space.
■ For me, eating is a form of meditation.
■ I am capable of feeling wonder at a single perfect ingredient.
■ I like to wander alone in the woods.
If you've answered yes to all the statements, congratulations, you have kaiseki in your DNA! Head directly to a very old, very serious restaurant and embrace the subtlety. A handful of affirmatives and you should book a table during your time in Kyoto and give it a shot. All nos? Save your money for sushi.
Regardless of your feelings on the form, the importance of kaiseki on Japanese culture cannot be overstated. Kaiseki draws on a diverse wellspring of influences, all of them intimately associated with the long, storied history of Kyoto itself.
In AD 789 the capital of Japan moved from Nara to Kyoto, where it remained uninterrupted until 1869. For more than a millennium, all life flowed through Kyoto—all political maneuvering, all artistic creation, all of the country's best ingredients for consumption by the elite. This is where Ieyasu and Hideyoshi ruled; this is where Kabuki and geisha were born. Modern Kyotoites' sense of pride ruffles the feathers of many other Japanese, but for the people of Kyoto, the city's long-spanning reign serves as empirical evidence of its superiority: _When Tokyo has been capital of Japan for more than a thousand years, then we'll talk._
This sense of confidence stretches to all sides of Kyoto culture, but it's felt especially strongly in _kyo-ryori_ , traditional Kyoto cuisine. You'll hear that the dashi is more elegant, the tofu more refined, the vegetables more dense with the flavors of Japanese terroir. If you listen closely, you will hear elaborate tales of ancient family traditions and heroic battles and extravagant courtly precedents that gave birth to Kyoto's status as one of the great eating cities of the world. Kyotoites may allow that Tokyo has a deeper overall food scene, if only for its size, but the belief of anybody born and bred on the food of the ancient capital is that Kyoto is ground zero for Japanese cuisine. And nowhere are its tenets and techniques better displayed than through kaiseki.
By the end of Kyoto's thousand-year run, kaiseki had emerged as the dominant cuisine for the local elite, with four principal branches reflecting the city's multilayered history: court cooking, a regal cuisine soaked in pageantry developed around the imperial presence during the early years of Kyoto's reign as capital; _shojin ryori_ , vegan cooking, the humble yet elaborate meals formed around Buddhist-temple dining ( _kaiseki_ literally means "a stone in the stomach," a reference to fasting Buddhist monks who used warm rocks to ward off hunger during the long, hard winters of meditation); _obanzai_ , traditional home cooking that showcases the bounty of Kyoto, in particular its vegetables; and, most important, _chakaiseki_ , the cuisine of the tea ceremony, the fountain through which nearly all traditional Japanese culture flows.
Tea came to Japan from China around the same time Kyoto was settling into its role as the country's new capital. Back then, it was consumed primarily as a form of medicine, but in time it developed into an important social ritual, one that grew more elaborate and grandiose as the years went on.
Sen no Rikyu would change all that. Born in 1522 in the nearby port town of Sakai to a wealthy merchant father, he studied tea from an early age and rose quickly in the ranks of Kyoto's tea luminaries, going on to serve as tea master under Japan's two most powerful warlords at the end of the sixteenth century. Nobunaga Oda was one of Japan's fiercest and cruelest rulers, a man best known for his blatant and bloody disregard for history, tradition, and Japanese formalities. And yet he was a lover of the tea ceremony, the most formal and traditional of all Japanese endeavors, which he used as a civilized way to discuss politics. When Nobunaga was betrayed and killed in 1582, Rikyu's talents with tea were employed for a new boss: Hideyoshi Toyotomi, a former servant to Nobunaga who would go on to be one of Japan's three great unifiers.
During his time serving Hideyoshi, Rikyu began to reshape the dynamics of the tea ceremony. Building on the Japanese idea of _wabi-sabi_ , appreciation for the imperfect and the inconstant, Rikyu worked to bring the tea ceremony back down to earth, to replace the ostentatious public ceremonies held by the Kyoto elite with private, reflective experiences designed to tease out in participants a deeper appreciation for the finer points of the moment—the shadows in a rock garden, the brushstrokes on a scroll, the gentle bitterness of the tea itself.
Rikyu achieved this by stripping away all nonessentials from the ceremony. He traded lavish halls for wooden huts, golden kettles for iron pots, elaborate ceramics for simple wooden cups. Free from fancy distractions, participants could achieve the deeper meditative state the tea ceremony was supposed to evoke.
Most students will spend decades studying and still not become official tea masters.
_(Michael Magers, lead photographer)_
In 1591 Rikyu's relationship with Hideyoshi turned sour. Some scholars speculate Hideyoshi grew resentful of Rikyu's growing influence in Kyoto; others point to a statue Rikyu erected of himself in Hideyoshi's compound as the source of the ruler's ire. Whatever the rift may have been, Hideyoshi ordered the grand tea master to commit seppuku, death by his own blade, which Rikyu did after serving up one last cup of tea to students and friends.
Four hundred years later, Rikyu is not only viewed as the father of the modern tea ceremony but he is also by extension one of the chief architects of kaiseki.
It has been said by more than a few smart people that to truly understand Japanese culture, you must first understand the Japanese tea ceremony. Packed into this single event you can experience the purest expression of the cornerstones of Japanese culture: flower and garden arrangements, calligraphy and scrolling, architecture and dress. And, of course, cuisine. Food became a part of the tea ceremony as a way to line the stomach before drinking strong beverages, but what started out as a light snack gradually grew to a meticulous multicourse feast.
Today kaiseki exists as a stand-alone experience, separate from the elaborate four-hour tea ceremonies that still take place in many corners of Japan, but appropriating many of the same aesthetic anchors—scrolls, flowers, rock gardens—to carry on Rikyu's enduring vision: an experience of gentle nourishment, a meditation on imperfection, a communion between man and nature.
米 麺 魚
The morning after the Nakahigashi dinner, I find Ken waiting in the lobby. "Well?"
"A beautiful meal from a beautiful man," I tell him, before running through a few of the highlights. But he senses something in my voice, some distant reservation that even I don't register.
"But?"
"No buts. No, no. How could there be buts when there's a six-month waiting list? The rice was lovely. The guy is obviously a genius."
He arches his eyebrows, cocks his head slightly.
"Okay, maybe there is something. I'm not really sure what it is, but it just feels like I'm missing something, like maybe I don't have all the pieces of the puzzle. Or maybe I'm just ill suited to kaiseki."
We both stand there in the lobby in silence. Finally Ken speaks up.
"What are you doing tonight?"
"Why?"
"Be here at ten p.m. There's something I want to show you."
A few ticks after ten, we're in a taxi heading west across Kyoto. Eventually we come to a river framed by a dark mass of mountains behind it. The taxi pulls up to a freestanding two-story wooden building, what looks like someone's riverside residence.
A family of four greets us at the door, bowing as we approach. Ken carries a gift from the Hyatt, little cakes and sweets from their pastry kitchen in impeccable packaging. "My favorite," says the older man, as he bows to accept the gift and welcome his late-night visitors.
As we step inside the house, I realize that it's actually a restaurant, but it doesn't look like any of the kaiseki places I've eaten in before: small and creaky, with a handful of tables and a long counter—more an izakaya than a sanctuary for quiet reflection. The room smells of grilled fish and sesame oil, but there are no customers to be found. I see two sets of chopsticks, two sake glasses, and two stalks of bamboo set at the bar. We take a seat, and the old man and his son join two other cooks behind the counter. Packed inside the bamboo is a sorbet made from _shiso_ , an herb with a flavor somewhere between mint and basil, a bracing shotgun start. Ken gives a nod, and the procession begins.
We start with a next-generation miso soup: Kyoto's famous sweet white miso whisked with dashi made from lobster shells, with large chunks of tender claw meat and wilted spinach bobbing on the soup's surface.
The son takes a cube of topflight Wagyu off the grill, charred on the outside, rare in the center, and swaddles it with green onions and a scoop of melting sea urchin—a surf-and-turf to end all others.
The father lays down a gorgeous ceramic plate with a poem painted on its surface. "From the sixteenth century," he tells us, then goes about constructing the dish with his son, piece by piece: First, a chunk of tilefish wrapped around a grilled matsutake mushroom stem. Then a thick triangle of grilled mushroom cap, plus another grilled stem the size of a D-sized battery, topped with mushroom miso. A pickled ginger shoot, a few tender soybeans, and the crowning touch, the tilefish skin, separated from its body and fried into a rippled wave of crunch.
The rice course arrives in a small bamboo steamer. The young chef works quickly. He slices curtains of tuna belly from a massive, fat-streaked block, dips it briefly in house-made soy sauce, then lays it on the rice. Over the top he spoons a sauce of seaweed and crushed sesame seeds just as the tuna fat begins to melt into the grains below.
A round of tempura comes next: a harvest moon of creamy pumpkin, a gold nugget of blowfish capped with a translucent daikon sauce, and finally a soft, custardy chunk of salmon liver, intensely fatty with a bitter edge, a flavor that I've never tasted before.
The last savory course comes in a large ice block carved into the shape of a bowl. Inside, a nest of soba noodles tinted green with powdered matcha floating in a dashi charged with citrus and topped with a false quail egg, the white fashioned from grated daikon. The chefs cheer as I lift the block to my lips.
It happens fast, ten courses in just over an hour, and it unspools so quickly that there's no time for talking or processing everything they serve us, but by the time we emerge from the restaurant under a bright bank of Kansai stars, I know that I've just eaten one of the great meals of my life.
米 麺 魚
If anyone could be expected to carry the torch for classic Kyoto cuisine, it's Shunichi Matsuno. He was born in Gion, the ancient geisha district of Kyoto and the spiritual center of kaiseki. His dad ran a private teahouse, one of the most exclusive institutions in a city built on exclusive institutions. Down the street, one aunt owned a famous soba shop and another a grilled eel restaurant, two sturdy pillars of _kyo-ryori_.
And yet, when he graduated from university, Shunichi wanted to be a salaryman—a wage warrior far away from the smoke and steam of the kitchen. But the cooking gene was strong in Shunichi, and when the business suit began to chafe, he decided to continue the family legacy, albeit with his own restaurant safely removed from the rest of the Matsuno clan.
"The Gion was filled with drunk people treating women badly. Lots of prostitution. I knew I needed to get away from the center of Kyoto." So he came to Arashiyama, six miles due west, and bought a house along the Oi River with a sweeping view of the area's guidebook beauty.
"Being next to the river and the mountains, we hear the water from the kitchen, we see the leaves change from our window."
Tempura Matsu was a true tempura restaurant for only three years. The fry business was slow, so Shunichi began to experiment with other dishes to serve alongside the tempura. It was those dishes that customers loved and came back for. Gradually the menu grew in scope and ambition, incorporating the structure of traditional kaiseki but without being bound to its strict tenets.
He ran the restaurant with his wife, Toyomi, and when his daughter, Mariko, and his son, Toshio, were born, as is tradition in Japan, they eventually became a part of the business.
I learn all this one morning in the back of Shunichi's station wagon on the way to Kyoto's central market. Toshio is riding shotgun, arguing with his dad over the quickest backstreets to take through the city. It's been nine months since the midnight meal with Ken at the Matsu counter, and barely a day has passed when I haven't thought about that crunchy tilefish skin, that miso lobster, that icy soba finale. After nearly a dozen kaiseki meals and a world of ambivalence, I felt like I finally had a breakthrough, something unequivocally worthy of the towering fame of Kyoto's cuisine, and I needed to know—and taste—more.
The benchmark for innovation in Western cuisine is high these days. Ever since Ferran and Albert Adrià of Spain's El Bulli blew the doors off the traditional French model of dining that dominated high-end restaurants for decades, unleashing on the world a palette of foams, gels, powders, and spheres to paint with, the modern kitchen has become the seat of a creative arms race. Centrifuges and thermal circulators share counter space with mortars and pestles, young chefs use liquid nitrogen like old chefs use freezers, and restaurants collaborate with physicists, chemists, even perfumists, in the search for the next big discovery. Ambition announces itself with a megaphone at these places, above all on the plate, where a tableau of strange tastes and textures paint precious—and sometimes delicious—pictures.
But in Japan, creativity takes a back seat to tradition. Chefs remain more dedicated to perfecting the old than uncovering the new. Here innovation means adding a few extra grams of _katsuobushi_ to your dashi, buying your tuna on Tuesday and serving it on Thursday, driving to the mountain to get your water. By this measure, what I tasted at Tempura Matsu was radical, if not downright heretical.
Shunichi and Toshio Matsuno, in the kitchen at Tempura Matsu
_(Michael Magers, lead photographer)_
Most visitors to Kyoto will wind up in the Nishiki Market, the spellbinding sprawl of pickle purveyors, tofu artisans, and prepared-food specialists that runs horizontally through five blocks of downtown. But Kyoto's legions of chefs do most of their shopping at the less beautiful but more functional wholesale market, a cavernous collection of bulk seafood, meat, and vegetable dealers. As with most of Japan's commercial markets, you need a special ID just to survey the goods.
The menu at Tempura Matsu is a constantly evolving animal, with dishes rotating on and off daily, if not hourly. Every night after service, Toshio and Shunichi draw up the next day's menu, but final decisions aren't made until they've done their market run and tasted as many of the menu's protagonists as possible. "Too many chefs in Kyoto cook by the calendar, not by taste," says Shunichi. "We would never write a monthly menu because if the product isn't good, you still have to use it, and that doesn't make sense."
Instead, the two of them bound from one stand to the next, tasting everything in their path, making adjustments as they go. Shunichi is a large man, half bald, with a light frost of white stubble and a round face that looks borrowed from a manga character. His default facial expression is a smile that could melt an ice bowl of soba, punctuated by the occasional furrowed brow reserved for contemplative moments. Dressed in sweatpants, a bubbly jacket vest, and long-sleeved black shirt, he could be an emcee a few rhymes past his prime.
The shopping starts with Shunichi's tuna man. When the fishmonger sees us coming, he pulls out a large katana blade and saws thin slices of meat off the tail, which he dresses with soy and passes our way. "These ones don't have a lot of fat because they're jumping so much right now," says Shunichi. "They swim around the world to get away from their wives, and then they end up in my kitchen. Ha! Ha! Ha!"
He tells the man to bring out the _otoro_ , the prized belly meat, which next to the lean tail meat looks like a winter snowstorm. Shunichi, convinced, peels off 10,000-yen notes from a thick wad he carries in his sweatpants.
We stop at a sea urchin vendor a few stands down who lines up a selection of Hokkaido and Kansai _uni_ for us to taste. "You see? The Kansai _uni_ isn't sweet enough yet," Dad says to his son. "We'll use it mostly later in the season, when it improves, but for now we'll stick with Hokkaido."
As we move through the market, gathering the building blocks for today's menu, Shunichi offers a running commentary on everything we pass. "See these eels? They're caught one by one in a net. The difference in taste is unbelievable. . . . Hokkaido asparagus is famous, but these are too fat to be delicious. . . . You know what these are? Dried sea cucumber ovaries. The most expensive ingredient in the world."
Purveyors offer us tea, ply us with samples, pull Shunichi aside to show him a special product they've saved just for him. He happily bellows out his opinions on any and all market constituents, including the vendors themselves, but when it comes time to discuss prices, he goes quiet. He'll grab a vendor by the arm, usher him to the side and whisper, often using a little piece of cardboard to cover his mouth so others can't read his lips.
He takes an enormous amount of pride in the product he procures and the prices he pays, a result, he says, of the relationships he's cultivated over four decades at the market and the fact that he always carries cash. "A meal that could cost forty thousand yen anywhere else in Kyoto costs only fifteen thousand at our place. I know how to get value."
But while purveyors all vie for his attention, not everyone loves his bargaining tactics. "They call my dad the little devil in this market," says Toshio. He leans in to add, "Sometimes I think the same thing."
"What did you say?" Shunichi asks, inspecting a large snapping turtle crawling across the market floor.
"Nothing, Dad," he says, shooting me a little wink. "How does the turtle look?"
"Delicious." After a bit of negotiating, he settles on a 3.7-kilogram turtle, not something on the shopping list today, but he likes the rim of yellow fat he spots beneath the shell.
After a breakfast of ramen and fried rice, we make the morning's last purchase just outside the market maze. Shunichi has arranged to meet one of Kyoto's top sushi chefs, who moonlights as a high-end wild boar dealer. The chef emerges from between two delivery trucks with a white plastic bag in his hand and quickly passes it off to Shunichi, as if it were a brick of Colombia's finest.
"Three-year-old virgin boars are the most delicious," he says, peeking into the bag. "The fat is perfect."
It's late April in Kyoto, not long after the last of the cherry blossoms have vanished, which means _takenoko_ (bamboo) season is in full bloom. Seasonality is a dominant tenet of Japanese cooking across the country, but in Kyoto, it dictates nearly every calorie consumed across the city. At this very moment, a thousand prep chefs are peeling back the fibrous layers of bamboo bulbs.
The Matsunos buy their bamboo from Yoshiaki Yamashita, a farmer whose family has been farming bamboo on the outskirts of Arashiyama for centuries. "He's Japan's number one bamboo farmer. He's the emperor's _takenoko_ supplier. Very high-class stuff."
The key to great bamboo, Yamashita tells me, is space. Bamboo trees can reproduce for six years, but their roots need room to spread, and the sun needs room to bake the forest floor. More than a farmer, Yamashita is a constant gardener, pruning branches, keeping the trees to a height of six meters, using rice husk to sow nutrients back into the soil.
The best bamboo is found deep underground, safely away from sunlight, turning the harvest into something resembling a truffle hunt. We walk carefully and quietly through the forest, looking for little cracks in the earth that indicate a baby bamboo trying to make its way to the surface. When we spot cracks, Yamashita comes by with a small pick and gently works the soil until he reaches the bulb.
Most bamboo you see is ruddy brown or purple, but Yamashita's _takenoko_ comes out lily white, tender, and sweet enough to eat like an apple.
"You have to cook it right away, otherwise you begin to lose the flavor," says Shunichi. He pulls his cell phone from his sweatpants and calls the restaurant.
"Tell them to start boiling the water. We're coming back."
米 麺 魚
For a place filled with so many outsiders, Kyoto is the ultimate insider's town. Everywhere you turn you find reminders of the line that exists between you and them: not just the restaurant menus dense with Japan's three alphabets, but the hidden pathways, the dancing curtains, the portals to a world that your imagination will work fiercely to construct as you wander through the shadows of Kyoto's oldest streets.
To understand just how deep the divide runs, consider the case of Ken. He is the ultimate insider, as deeply connected to this city, its culture, and its most august citizens as anyone you'll meet, yet he will never be anything but a guest in this city, a Yokohama-born transplant with Kyoto in his heart but not in his blood. Even if he had been born here, it would make no difference; Kyotoites count their history not in years or decades but in centuries.
Robert Yellin, an American expat who runs one of Kyoto's greatest ceramics galleries, tells it like this: "You aren't officially a Kyotoite until you're seventh generation. If you're sixth generation and your family has been here for two hundred years, you're still an outsider."
The veiled light of a _ryokan_ entrance
_(Laura Pérez)_
I count my Kyoto history in hours, a ship anchored in port for the night. I find myself constantly fighting the urge to abandon caution and good manners and breach the curtains into Kyoto's higher dimension. Reason and decorum save me the embarrassment, though; instead, late at night, I'll wander the smallest streets of the Gion in hope that one of those doors will suddenly slide open, an arm will reach out, like a Hollywood hand plunged into the frigid sea to save a sinking body, and pull me into the wondrous universe inside.
I have only a vague notion of what goes down in the house of the geisha. I imagine streams of sake poured from ancient ceramic sculptures by hands specifically designed for its dispensing. Long, electric conversations confronting the mysteries of our existence. Beautiful plates of food packed with textures and flavors unknown to the outside world. Busy hands, sweaty brows. Intellect and innuendo. When I close my eyes really tight, I see the last candle of the night casting an orange glow against the gossamer veil of a rice-paper door.
But I have no way of verifying any of these suspicions. Unless your family tree begins with a Tokugawa _bafuku_ , or you have befriended the daughter of a Grand Master of Tea, your imagination will do most of the feasting in this town.
It's not all mirages, though. One afternoon, I sit in on a master tea class. Five students—two men, three women, all but one over sixty—spend hours practicing to make a smokeless charcoal fire in a hole in the tatami floor. Later, one by one they whip hot water and matcha powder into a frothy emerald cup of tea with _chasen_ , bamboo tea whisks. An older woman in a purple kimono serves me one; I turn the cup three times to honor its creator, as I was taught before, then drink it down in one long gulp. Thick, vegetal, astringent—nearly a meal on its own. There are three stages one must pass before reaching the status of master; the woman who makes my tea has spent two decades in the class and remains mired in the first stage. "I know I still have much to learn before I can move on to the next level," she says, eyes closed, head bowed.
One early evening, with the sun dipping just below the crest of the mountains that loom over the city, I meet with Yoshihiro Murata, the head chef of Kikunoi, one of Kyoto's most venerable kaiseki institutions. Murata ranks among Japan's best-known chefs, the man behind the successful UNESCO bid to honor Japanese cuisine. We meet in a private room upstairs at his restaurant, just me and him and five people in suits from various government branches. Upon learning that I am from the United States, he offers up a small lesson on global cuisine: "Western cuisines are based on fat," he says, "but Japanese gets its flavor from umami, which has zero calories. That's why we live longer than everyone else."
Another day, Ken takes me to meet Setsuko Sugimoto, matriarch of one of Kyoto's ancient clans, a family that traces its roots back seventeen generations, to when the city was the center of Japan. Her home is among the oldest in Kyoto, so closely protected by the city that to rearrange a piece of furniture requires approval from multiple government offices. She serves us a traditional _obanzai_ dinner, Kyoto-style home cooking: _chazuke_ , steamed rice-and-tea soup, and a salad of tofu scraps speckled with dried fish. "We're starting to lose these traditions," she says, ladling the soup from an ancient wood-fired stovetop.
But inevitably, most of the moments that aren't spent at the kaiseki counter are spent wandering—past the shops where _wagashi_ artisans shape sweetened beans into works of edible art, through the temples and shrines that dot the winding Philosopher's Path, across the canal and into the evening glow of Shirakawa Dori, a street whose beauty leaves me breathless every time I walk it.
I dream strange dreams when I am in Kyoto. One night, I am called upon by Obama to broker a trade negotiation between Japan and the United States. The next, I iron a suit jacket that stays forever wrinkled. An anxiety lies awake in me that no flower arrangement or seasonal scroll or dimly lit path can uproot. I'm not sure if it comes from the doors I can't open or the people who guard their thresholds. Maybe it's the shoes that never slip off my feet, the density of Japanese words in my mouth, the chopsticks that feel like tree branches. I cycle through metaphors, looking for one that makes sense of what I'm feeling: Kyoto is a Christmas feast, and I'm stuck at the kids' table. Or maybe Kyoto is a poem of immense but impenetrable meaning.
And yet, even after all the doubt and restlessness, all the unsettling stimuli, when I see a young _maiko_ , a geisha-in-training, emerge from behind the curtains and fill a quiet street with her clomping wooden shoes, the whole world stops. My knees buckle and my palms sweat and in that second where everything grows wonderfully fuzzy I remember that this is where the story begins. That they are the music makers, and we are the dreamers of dreams.
米 麺 魚
By the time we get back to Tempura Matsu, the morning market haul is in various stages of undress. In Kyoto's more renowned restaurants, ingredient transformation is a delicate act—a bit of knifework, a gentle boil, a brushstroke of soy. But at Tempura Matsu, transformation takes on a more aggressive tone.
The 3.7-kilogram snapping turtle is alive no more; its shell bobs just above the water line of a simmering stock, flavoring a dashi made with leeks, ginger, sake, and mirin. The wild boar braises in a bath of white miso studded with mountain herbs and wedges of daikon. The bamboo goes directly from the trunk of the car into a massive pot of boiling water, the first step in a multipart process that will transform the tender bulbs into five separate dishes for the day's menu.
The restaurant employs five cooks who collectively have spent more than a century working in the Matsu kitchen. Kazuhiro Nakagawa, the youngest of the crew, handles the rice, the most straightforward but in some ways most stressful job on the line—rice must always be perfect in Japan. Hirofumi Oyagi works the tempura station, gently stirring batter with chopsticks, floating little drops into the hot oil to take its temperature. The same way an owner and his dog grow to resemble one another, Hirofumi and his pot have nearly become one over the decades, his eyes and hair cast-iron black, his face moist and craggy from forty-two years in front of the fryer.
If you sat on a stool and watched Takashi Shingu long enough, you would eventually unlock all the secrets of Japanese cooking. He skewers Wagyu and salmon and sacks of cod sperm and begins to grill them slowly over a charcoal fire. He turns pufferfish and squid into perfect dominos for sashimi. He nails a still-slithering eel to the cutting board, skins it with one swift motion, fillets it with one more, and has it cooking in a bamboo steamer splashed with sake before its muscles have stopped moving. He's been at Matsu since he was in high school, nearly four decades on the line, and he moves through the day's cooking like a man who has never wanted to do anything else.
Grilling, steaming, stewing, slicing, frying: the wheels of kaiseki turn with incredible ease and fluency at Tempura Matsu. With all the pieces in place, father and son both disappear upstairs to prepare for service. When Toshio comes back down, he's changed from his market clothes into a crisp white chef's jacket with a tiny Matsu kanji monogrammed on the left breast. Toshio's thirty years old but looks a decade younger, with boy-band good looks and a tommy-gun laugh that makes everything sound like the funniest moment of his life.
Toshio trained under Alain Ducasse, arguably the greatest French chef cooking today. "I taught Ducasse how to cook on hot stones," he says with a sort of sweet seriousness that dispatches any doubts you might have about the claim. "Now he's using it in all of his restaurants, but because he's so famous, people think we're the ones copying him." When Toshio looks over at me, he seems concerned that my pen isn't moving. "Please make sure to write that down."
Toshio also spent time in the kitchen at Kitcho, Kyoto's most renowned kaiseki temple, where dinner starts at $400 and dishes read like an edible history of Japan's ancient capital. You won't find many like him in the kitchens of this city: a young, hypertalented chef with one eye fixed on Kyoto and the other scanning the horizons of global cuisine.
Of course, Toshio's true master has always been his father. It was his father who taught him to how to bone a fish, how to fry a vegetable, how to fit two opposing flavors together. Normally the son of the owner, regardless of résumé, would be relegated to a supporting role in any Japanese kitchen—especially in Kyoto. But the father-son dynamic at Matsu is like none I've seen anywhere else in this country, one built on open collaboration, constant feedback, and a deep respect for each other's talents.
"When customers like a dish, they often ask who created it," says Shunichi. "But we always do it together. The base comes from one of us, but the final is a collaboration."
"Don't listen to him!" says Toshio, lining up plates for sashimi. "I came up with a dish yesterday and my dad said it was no good, but then the customers really liked it and he said he did, too. It happens all the time." He says it with a playful smile, but you can see by the way he works the kitchen, by the way men who were cooking here before he was born follow his orders with exacting discipline, that Toshio is ready to push Kyoto cuisine forward.
_(Michael Magers, lead photographer)_
_(Michael Magers, lead photographer)_
"Kyoto is a place trying to hold on to its past," says Shunichi. "Many of the young chefs training at the important restaurants in the city will go on to open the exact same kind of kaiseki place. Of course we're always chasing perfection, but not at the cost of new ideas. We don't change tradition; we build on top of it."
That's where Toshio comes in, says his father. "He's the future of this place, so I need to empower him with the ability to do what he needs to do to adapt. Times change. We have different backgrounds and we combine them and that's what makes Tempura Matsu what it is."
At 11:30 a.m., the guests begin to arrive. First, a couple from Hong Kong with their six-year-old son settle in at the countertop; then a group of four businessmen in suits are led to a private table. Later, a single woman from Tokyo and two young men from Singapore, back for their third visit, fill out the countertop.
Father and son roam freely through the kitchen, Shunichi tasting sauces and plating sashimi, Toshio buzzing from one station to the next, whisking, slicing, skewering, creating dishes on the fly. Mom and daughter work on the other side of the counter, handling reservations, recommending sake, delivering dishes to customers seated in the restaurant's quiet second floor, removed from the immediate action of the kitchen.
As I learned that first night with Ken, the Matsu service philosophy revolves around guest interaction and kitchen spectacle. "Anyone can make delicious food. It's about pleasure, having fun," says Shunichi. "It's different when you can look a customer in the eyes, when you can see her smile." Throughout lunch, they make jokes and ask questions and do a good bit of the cooking and plating directly on the countertop, to the wild delight of the diners.
But as service wears on, Dad begins to tire. He goes from plating and tasting and teasing to standing in the center of the kitchen, arms folded, watching over his restaurant. Ten years ago he suffered a heart attack, losing the use of his right hand in the kitchen in the process. You'll never hear him complain, but occasionally, in a quiet moment, he speaks obliquely about the mounting health problems. When he says Toshio is the future of the restaurant, it's not a platitude; it's a forecast.
"Maybe I won't continue the way we're going," says Toshio, when I ask him about the future of Tempura Matsu. "I don't see myself copying my father. I don't think that's the right thing to do."
From a strict skills standpoint, Toshio is more than ready to blaze his own trail. I've seen few chefs with his touch, his versatility, his innate ability to create delicious, meaningful food. But this is his father's kitchen, his father's meticulous creation. Shunichi spent a lifetime ignoring the rules of Kyoto to build a bridge to the future, and now his son fears that he might have to cross it alone.
米 麺 魚
One morning I rent a bike and ride west, away from the old city. I ride with a backpack full of salty snacks and green tea, the Golden Pavilion of Kinkaku-ji my nominal destination. I don't pedal for long before discovering that most of this city lives and breathes in the vast space between Karasuma Station and Arashiyama, shopping for flat-screens, drinking canned coffee from vending machines, catching buses and trains like the rest of Japan.
Halfway to the pavilion, I hit a red light on a busy street corner. While I wait, life passes me by: cars honk, schoolgirls in plaid skirts eat French fries from a takeaway bag, a man passes out flyers for a cell phone sale down the street. For three light cycles, I just stand there, eyes glazed, wondering what to make of this strange city before me.
It's not the Kyoto we come for, and if you want your vision of this city to be as unblemished as the Gion's baby-smooth streets, it's best to stay east of the Kamo River. Most visitors—gaijin and Japanese alike—would rather not spoil the view, but head west on foot or by bike or taxi and you will see the "other" Kyoto, a Kyoto without brooding red bulbs, rake-groomed gardens, powder-white layers of face paint.
This is a Kyoto that feeds itself the same way most of Japan feeds itself: with big steamy bowls of shoyu ramen, grilled skewers of chicken hearts, egg salad sandwiches from Lawson. For this Kyoto, traditional kaiseki is a source of pride, not a source of sustenance.
If places like Kikunoi and Ogata exist to nourish Old Kyoto (and the affluent visitors who come to immerse themselves in its unflinching antiquity), GiroGiro represents the tastes of a more modern city—young, hip, more concerned about value and good times than adherence to historical standards.
GiroGiro is a kaiseki restaurant, but only in the most liberal definition of the term. It's true, the third course they serve is sashimi, the sixth tempura, the last rice and miso soup, all in accordance with kaiseki code, but the similarities between it and the earnest institutions that have turned Kyoto into the Milky Way of Michelin stars stop there. The list of differences runs long.
To start with, there's the noise. Conversations echo off the ceilings, laughter bounces off the walls, the collective commotion a stark contrast to the pin-drop silence observed at most kaiseki restaurants.
The cooks: bright shocks of spiky hair, tattoos, loud and animated and quite possibly drunk—literally the polar opposite of almost every other cook you'll see in Japan.
The crowd looks scarcely different from the cooks: young, nicely toasted, aggressively hip. There seems to be more facial hair in this one room than in the rest of the country combined.
The price: at 3,500 yen, dinner at GiroGiro costs roughly one-tenth what you might pay at top-tier kaiseki in Kyoto, opening up the reservation book to an entirely different set of demographics.
If Nakahigashi is classical and Tempura Matsu jazz, this is punk rock kaiseki, done with attitude, volume, and a disdain for rules and expectations. There will be no scrolls to consider, no flower arrangements to admire, no ancient pottery to appreciate. As for the food, well, it's more manipulated, more dressed up, than anything you'd find in traditional kaiseki.
"In many ways, the more expensive the food, the more simple it's going to be, and that's hard to get as a beginner," says Shota Okuda, who has been cooking at GiroGiro for six years. "We have to develop techniques to work with the ingredients we can afford." Many of the resulting creations display a generous vision and a deft touch, like a bowl of rice spiked with sesame seeds and daikon, pressed fish roe and fresh strawberries, a tightrope balance of sweet and savory. Or the crispy fish cake covered in a sauce of whipped tofu and juicy segments of grapefruit—a dish with the type of contrasting tastes and textures you wish more kaiseki chefs would employ.
Other times, the limitations of a $35 kaiseki dinner are more noticeable. The final course of steamed rice and grilled _anago_ , served with mushy, flaccid eel, made me long for the crispy soy-shellacked _anago_ at Ogata.
But nobody here seems to mind. Young couples hold hands under the countertop. An apple-cheeked diner plies the staff with shots of whisky. A waitress, victim of the patron's generosity, stumbles delivering desserts to the last customers.
"Kyoto is ultimately a very young town," says Okuda, "and the university students can't go to traditional kaiseki, but they can come here with a date."
GiroGiro isn't the only restaurant pushing the kaiseki envelope. Jimbocho Den takes the traditional format and infuses it with whimsy and wizardry, serving dishes like Wagyu stained with beet blood and Dentucky Fried Chicken in a mock KFC box. At Ryugin, chef Seiji Yamamoto combines a _shokunin_ 's ingredient obsession with highfalutin techniques borrowed from modernist kitchens in the West. Both are exceptional places to eat, but both are in Tokyo, far from the standards and strictures of Kyoto.
Compared to these, GiroGiro, with its wobbling waitresses and Sid Vicious line cooks, is an especially aggressive addition to the kaiseki canon. Perhaps the most audacious part of GiroGiro is its location, along the Shirakawa Canal, a few blocks from the Gion, one of the oldest and loveliest parts of Kyoto. The past literally pushing up against the future, the conundrum of kaiseki, of Kyoto, and in many ways of all of Japan, captured in a single restaurant with a funny name.
Is this the future of kaiseki? Half of Kyoto shudders at the thought. The other half is lining up for a reservation.
米 麺 魚
As the last of the lunch customers make their way out the door, Matsuno-san seats me at the counter and says something in Japanese to his son that could mean only one thing: keep it coming.
First, a sizzling stone, the same one Toshio introduced to Ducasse years back. Today it's filled with rice and ginger juice and baby firefly squid, which crackle wildly as he tosses it all like a scalding salad and pushes it over to me. The squid guts coat the rice like an ocean risotto, give it body and funk, while the heat from the stone crisps the grains like a perfect bibimbap.
By now the other cooks have all stopped working; even Shunichi has stepped out of the kitchen to talk with his wife. It's just Toshio, and you can tell by the way he wriggles his shoulders and glides between stations that he lives for this moment.
Next comes _chawan mushi_ , a delicate egg custard studded with wild mountain vegetables and surrounded by flowers from the bamboo forest. A dish as old as Kyoto itself.
Toshio creates a new dish while his father looks on.
_(Matt Goulding)_
Toshio plucks two sacs of cod milt from the grill, slides them off the skewer into a squat clay box filled with bubbling miso. He comes back a second later with a scoop of _konawata_ , pickled sea cucumber organs. A dish as new as the spring flowers blooming just outside the window.
One by one, the market stars reappear on the plate.
A black-and-gold-lacquered bowl: Toshio pulls off the top to reveal thin slices of three-year-old virgin wild boar braised into sweet, savory submission with Kyoto white miso and chunks of root vegetables.
_Uni_ —Hokkaido and Kansai—the first atop a wedge of taro root dusted with rice flour and lightly fried, the other resting gently on a fried shiso leaf. Two bites, two urchins, an echo of the lesson in the market this morning.
Shunichi comes back to the kitchen and shuffles up behind his son, watching his moves closely.
He's on to sashimi now, fanning and curling slices of snapper and fugu into white roses on his cutting board. Before Toshio can plate the slices, Shunichi reaches over and calmly replaces the serving plate his son has chosen with an Edo-era ceramic rectangle more to his liking.
Three pieces of tempura—shrimp, eggplant, new onion—emerge hissing and golden from the black iron pot in the corner, and Toshio arranges them on small plates with wedges of Japanese lime. Before the tempura goes out, Shunichi sneaks in a few extra granules of salt while Toshio's not looking.
By now Dad is shadowing his son's every move. As Toshio waves a thin plank of sea cucumber eggs over the charcoal fire, his dad leans gently over his shoulder. "Be careful. You don't want to cook it. You just want to release its aroma."
Toshio places a fried silverfish spine on a craggy ceramic plate, tucks grated yuzu and _sansho_ flowers into its ribs, then lays a sliver of the dried eggs over the top. The bones shatter like a potato chip, and the sea cucumber detonates in my mouth.
A golden light bleeds through the window, framing the spirals of steam rising from a copper teakettle. Outside, the river sparkles at the foot of the mountains. At this hour, Tempura Matsu looks magnificent.
It's way past lunchtime. Normally the entire family would be upstairs, eating lunch together, a moment of peace, the day's first, before the dinner rush. But not today.
Toshio uses tongs to pluck a burning log of _binchotan_ charcoal from the fire, sets it on an inverted Japanese roof tile filled with sand, and places it all before me. "This is an idea I just came up with," he says with a mischievous little smile.
He pinches two slices of densely marbled Japanese beef between chopsticks and lays them directly over the _binchotan_ , a cloud of smoke rising on contact.
Shunichi inches in tight, eyebrows raised in an expression somewhere between surprise and doubt. He doesn't do or say anything, though. He doesn't grab the salt. He doesn't look for a new plate. He just stands there, close enough to breathe on his son's neck, watching him cook a dish that neither of them has ever tasted.
## [お土産
**THE ART OF GIFT GIVING**](nav.xhtml#rhh8)
**OMIYAGE**
お土産
The word for gift giving, a touchstone of Japanese culture, is _omiyage_ , which literally translates to "product of the earth," a local food from the place you were coming from. But while food (or drink) usually makes the best gift, there's more to _omiyage_ than that.
**TSUMARANAI**
つまらない
Westerners are often tempted to hype their gifts. The Japanese, not so much. When giving a gift, use the customary line: _tsumaranai mono desu ga_ : "It's nothing, actually, but please accept it." They in turn may decline it a couple times, but you should persist.
**MEIWAKU**
迷惑
One job of the gift giver? Avoiding _meiwaku_ —annoyance. Don't give a heavy bottle at the beginning of a night out. And don't get something needlessly expensive: custom requires a return gift worth about half the value, so your rich gift costs them.
**MEIBUTSU**
名物
The best gift of all may be the most classic: _meibutsu_ , the most famous foods of any given region of Japan. Hairy crab from Hokkaido, grapes from Yamanashi, and so on. It's not a unique choice, but creativity isn't the goal here. Besides, _meibutsu_ are delicious.
**TAKKYUBIN**
宅急便
Not ready to pack a hairy crab in your suitcase? Grapes don't travel well, either. Fortunately there is the ingenious _takkyubin_ transport system, which will express-ship local products anywhere in Japan for reasonable rates.
## **_Japan's_**
## **GREATEST FOOD JOURNEYS**
In a country shaped by its regional specialties, the travel dilemma isn't where to go, but what to eat. These are the answers you're looking for.
_(Matt Goulding)_
### **UDON IN TAKAMATSU**
Perhaps no city in Japan is better known for one dish than Takamatsu and its udon. Hundreds of restaurants dedicate themselves to Sanuki-style udon—thick al dente noodles afloat in dashi and topped with everything from raw egg to tempura to braised beef. Not sure where to go? Wave down one of the taxis marked with bowls of udon, and they'll deliver you to the city's finest noodle dispensaries.
_(Matt Goulding)_
### **SOBA IN NAGANO**
Soba culture gets deeper and more delicious the higher you climb, and mountainous Nagano produces some of the country's finest buckwheat noodles. Here you can have your soba cold and naked, hot and swimming in dashi, topped with wild duck and tinged black with charcoal. Both Kusabue and Fujiki-an have been in the soba game for a few centuries—worthy places for your noodle indoctrination.
_(Matt Goulding)_
### **SEAFOOD IN HAKODATE**
Hakodate at the southern tip of Hokkaido offers one of the finest displays of seafood you'll find anywhere on the planet. The morning markets teem with wild salmon, hairy crabs, giant sea scallops, and golden mountains of sea urchin roe. The best way to enjoy it all? A breakfast _donburi_ , bowls of steamed rice topped with any or all of Hokkaido's finest raw materials.
_(Michael Magers, lead photographer)_
### **STREET FOOD IN OSAKA**
Osaka's reputation as a center for good times and cheap food is well earned: the city abounds with casual eateries, lively bars, and street stands dispensing quick bites with potent flavors. _Kuiadore_ , Osakan dialect for eating yourself stupid, is a founding principle in Japan's most freewheeling city, and it should be the primary objective while in Japan's second city. One could survive very happily on _takoyaki_ , _okonomiyaki_ , cold beer, and the good vibes of the Osakan citizenry for weeks at a time.
_(Matt Goulding)_
### **PORK AND SHOCHU IN KAGOSHIMA**
There's something deeply lovable about this southern Kyushu city: maybe it's the spewing Sakurajima __volcano, the sprawling sea views, the seedy entertainment district. Probably it's the abundance of Japan's best shochu (with over a hundred distilleries in the city) and _kurobuta_ cuisine—shabu-shabu, tender braises, and ramen all made with black-footed Berkshire pork. Combine both in as many of the city's excellent izakayas as possible.
_(Matt Goulding)_
### **YATAI IN FUKUOKA**
Fukuoka is the last bastion of Japan's _yatai_ culture—a robust world of street food stalls that recalls a day when much of Japan's best food came from wooden stands. You'll find _yatai_ specializing in everything from classic cocktails to French country cuisine to regional Italian cooking. Above all, you'll find the Big Three: yakitori, _oden_ , and _tonkotsu_ ramen. With spacing tight and alcohol aplenty, _yatai_ are a good way to make friends fast.
_(Matt Goulding)_
## _Vital Intel_
## GAIJIN GLOSSARY
### **OISHII 美味しい**
_Delicious_.
If there is one word that will bring visitor and host together, this is it. Said with a slight twinkle in the eye, it can melt all the barriers of language and culture into a warm broth of love for one's fellow man.
### **SUMIMASEN すみません**
_Excuse me_.
Personal space in Japan is highly valued and yet nearly impossible to defend. _Sumimasen_ and its expat-impatient variety, _excuse-me-masen_ , are the Purell of jostling: a word you can just lather on any situation to defuse and disinfect.
### **DOZO どうぞ**
_Please, go ahead_.
Like _vale_ in Spain or _doch_ in Germany, _dozo_ in Japan is a multitool of a word. It adds politeness—not an undervalued commodity in Japan—to any situation, whether you're letting someone pass in front of you or handing over a present.
### **DOKO どこ**
_Where?_
It's not just that most people don't speak English; most street signs and place names are not in the __Romaji alphabet, and guidebook and Internet addresses routinely fail. _Doko_ is your friend.
### **TABEMASU 食べます**
_To eat._
You did come to Japan to eat, yes? Say this word (remember, the _u_ is silent in Japanese) with a question mark at the end, and you will immediately be led to Japanese food.
### **OMAKASE お任せ**
_I leave it up to you._
It's the equivalent of putting yourself in the chef's hands—most common at high-end sushi bars but also used in many top restaurants. Say it when you want to be taken on a boundless gastronomic adventure, or when you have no idea how to order à la carte.
### **ITADAKIMASU 美味しい**
_I receive this food._
Use this and _gochiso sama deshita_ to bookend mealtime, and you will win hearts everywhere you go. This is essentially a small blessing to be intoned just before you begin eating, aimed at those who prepared the food for you.
### **GOCHISO SAMA DESHITA ご馳走様でした**
_It was quite a feast._
After you finish eating, say this incantation to thank and praise the cook. When you return for lunch the next day, they'll give you a hero's welcome.
## _Chapter Four_
## FUKUOKA
Toshiyuki Kamimura eats four hundred bowls of ramen a year. That's a bowl every day for lunch or dinner, plus one for breakfast about once a week. For that weekly breakfast bowl he usually goes to Ganso Nagahama out toward the ocean, a legendary spot located in what looks like an auto-parts warehouse that stays open twenty hours a day. "Sometimes I can't wait until lunch," says Kamimura, who consumes his ramen with a sense of urgency, conveying thick ropes of noodles into his mouth and sliding them down his throat like a duck, barely pausing to chew, "so I eat with the taxi drivers getting off the late shift."
His first memories of eating ramen come from his childhood in Kagoshima, the city at the southern tip of Kyushu famous for its fat-strewn pigs and potato-based liquor. Back then, Kamimura's parents would have ramen delivered from a local restaurant as a treat for the family. Even with the distance of time and the warm mist of nostalgia, Kamimura can't help but put a critical spin on those infant ramen moments. "By the time it got home, the broth was cold and the noodles were compromised. It wasn't impressive ramen."
He moved to Fukuoka, the capital of Kyushu, when he was seventeen in order to study photography at Fukuoka University. It was in that first year living on his own that Kamimura had his ramen epiphany. The transformative bowl came from Ichiran, now a popular national chain of middling quality but back then a gateway to a new life: "It was a whole different experience. I had no idea ramen could be so good."
In the twenty years since, he has gone from being a passionate consumer to one of Japan's most important ramen bloggers. When it comes to food writing, the Japanese are avid consumers of data, and the nascent ramen blogging industry specializes in chronicling every aspect of Japan's chief noodle obsession. On his website, Junction 9 (named for a local intersection with a concentration of killer ramen), Kamimura reviews hundreds of shops across Kyushu, offering detailed analysis on broth strength, noodle type, and topping cohesion. He's also a frequent contributor to _Ramen Walker_ , the most prominent of Japan's dozen or so ramen magazines, among other publications, and appears regularly on television, offering his take on the pressing ramen issues of the day.
Ramen bloggers aren't just passive observers of the noodle soup phenomenon: they create trends, drive or deflate business, and generally analyze ramen creation, consumption, and culture down to a microbial level. In some cases they eventually find themselves on the other side of the counter, stirring the soup and kneading the noodles. They, as much as the bandanna-wearing chefs and the legions of slurping salarymen, are the heart of modern ramen culture.
To be a ramen writer of Kamimura's stature, you need to live in a ramen town, and there is unquestionably no town in Japan more dedicated to ramen than Fukuoka. This city of 1.5 million along the northern coast of Kyushu, the southernmost of Japan's four main islands, is home to two thousand ramen shops, representing Japan's densest concentration of noodle-soup emporiums. While bowls of ramen are like snowflakes in Japan, Fukuoka is known as the cradle of _tonkotsu_ , a pork-bone broth made milky white by the deposits of fat and collagen extracted during days of aggressive boiling. It is not simply a specialty of the city; it is the city, a distillation of all its qualities and calluses.
Indeed, tell any Japanese that you've been to Fukuoka and invariably the first question will be: "How was the _tonkotsu_?"
Ramen, despite its reputation as a cheap fast food, is a complex pillar of modern Japanese society, one loaded with political, cultural, and culinary importance that stretches far beyond the circumference of the bowl. And all those big ideas start here in Fukuoka, ground zero for the ramen craze, a dizzying galaxy of bone-broth dispensaries that can be overwhelming for the noodle novice.
I'm not a novice, not exactly. Like most Westerners, my ramen history begins with a brick of dried noodles and a silver spice packet, a three-for-a-dollar subsistence plan that propelled me through the lean college years. Later came the real thing, first in the early ramen boom of New York, later in noodle crawls around Tokyo that opened my mind to how sophisticated and staggeringly delicious the best bowls could be.
But this kind of ramen world, one where every block houses a bowl that could make your knees buckle, is brand-new territory, and objective is nothing if not ambitious, naive, and slightly hazardous to my health. I'm stalking the million-footed beast: not just a bowl that will make my stomach dance, but an experience that will help me better understand how a bowl of noodle soup from China came to define Japanese food culture in the twenty-first century. Any local can take you to a handful of her favorite shops, but it takes the discerning eye of a ramen blogger to understand the details. That is why I've enlisted Kamimura-san to be my ramen guru, my noodle-soup interpreter, a spirit guide in a journey to better understand the bowl behind the city, and the city behind the bowl.
米 麺 魚
In the broadest sense, a bowl of ramen comprises four principal constituents: _tare_ (a seasoning base), broth, noodles, and toppings. (Of course, ramen wonks like Kamimura could nitpick these parts into dozens of subcategories.)
Let's start from the top of the bowl and work our way down. In theory, toppings can include almost anything, but 95 percent of the ramen you consume in Japan will be topped with _chashu_ , Chinese-style roasted pork. In a perfect world, that means luscious slices of marinated belly or shoulder, carefully basted over a low temperature until the fat has rendered and the meat collapses with a hard stare. Beyond the pork, the only other sure bet in a bowl of ramen is _negi_ , thinly sliced green onion, little islands of allium sting in a sea of richness. Pickled bamboo shoots ( _menma_ ), sheets of nori, bean sprouts, fish cake, raw garlic, and soy-soaked eggs are common constituents, but of course there is a whole world of outlier ingredients that make it into more esoteric bowls, which we'll get into later.
While shape and size will vary depending on region and style, ramen noodles all share one thing in common: alkaline salts. Called _kansui_ in Japanese, alkaline salts are what give the noodles a yellow tint and allow them to stand up to the blistering heat of the soup without degrading into a gummy mass. In fact, in the sprawling ecosystem of noodle soups, it may be the alkaline noodle alone that unites the ramen universe. "If it doesn't have _kansui_ , it's not ramen," Kamimura says.
Noodles and toppings are paramount in the ramen formula, but the broth is undoubtedly the soul of the bowl, there to unite the disparate tastes and textures at work in the dish. This is where a ramen chef makes his name. Broth can be made from an encyclopedia of flora and fauna: chicken, pork, fish, mushrooms, root vegetables, herbs, spices. Ramen broth isn't about nuance; it's about impact, which is why making most soup involves high heat, long cooking times, and giant heaps of chicken bones, pork bones, or both.
Ramen is one of the few foods in Japan that comes with no rule book.
_(Matt Goulding)_
_Tare_ is the flavor base that anchors each bowl, that special potion—usually just an ounce or two of concentrated liquid—that bends ramen into one camp or another. In Sapporo, _tare_ is made with miso. In Tokyo, soy sauce takes the lead. At enterprising ramen joints, you'll find _tare_ made with up to two dozen ingredients, an apothecary's stash of dried fish and fungus and esoteric add-ons. The objective of _tare_ is essentially the core objective of Japanese food itself: to pack as much umami as possible into every bite.
With all these variables in play, the potential combinations are limitless, but in Fukuoka, the single-minded dedication to _tonkotsu_ is so relentless that all other ramen is beside the point. Kyushu has long been the center of Japan's pork industry, and no dish better expresses the potential of the pig better than _tonkotsu_. To make sure I fully understand the Fukuoka- _tonkotsu_ connection, Kamimura starts me off at one of his favorite shops, Ramen-Ya Mototsugi.
Watching Kamimura review a ramen shop is like watching a detective work a crime scene. He starts with the _noren_ , the cloth awning that invariably hangs from a shop's entrance. "If it's greasy ramen," he says, reaching up and rubbing the yellowing drapes with a nod of approval, "it will look like a dirty shirt."
Next, he inhales deeply. _Tonkotsu_ is legendary for its fragrance, which, when emanating from the most intense shops, can assault your olfactory system from a three-block radius. It's a barnyard smell, pure sweaty-foot funk, and it's everywhere in Fukuoka, a misty aroma that hangs over the city the way fog clings to the hills of San Francisco.
"When I walk into a place and smell the broth, I can imagine how it was made," says the ramen whisperer as we slip into the shop. I draw in a deep breath, and my head swims with the memory of pigs passed.
After he orders, Kamimura turns his attention to the noodles. Are they cooked in individual baskets for easy timing, or are they dropped coil by coil into an open pot of boiling water? Most cooks go the basket route these days, but Kamimura prefers the purity of a free boil. "I respect the talent it takes to cook it all together—it takes real touch and intuition."
All the while, he's watching for little precursors of quality: the way the ramen cook shakes the water from the noodles after they're done cooking so as not to dilute the precious broth; the careful hand-slicing of a roll of _chashu_ so that it melts on contact with the hot soup; the judicious layering of _negi_ and nori and other garnishes to elongate the textural juxtaposition.
"I work hard to gather all the information necessary to make my judgments. If you don't poke your head into the kitchen, you never know," Kamimura says.
Our first bowl of ramen arrives. It's a muscular rendition, the spitting definition of Hakata-style _tonkotsu_ : pale, thin, straight noodles, thick ivory broth, two slices of _chashu_ , and little else in the way of toppings. Sesame seeds, ground white pepper, and electric pink pickled ginger are the holy trinity of table condiments in Fukuoka, but Kamimura isn't much for accessories. He wastes no time in cracking his wooden chopsticks and breaching the surface, but I wade in more cautiously.
Most Japanese food is a collective experience: the sushi chef feeds you piece by piece, the yakitori arrives in a great heap for divvying up, and the shabu-shabu bubbles away between you and your dining partners. But not ramen. With ramen it is just you and the bowl—the most intense and intimate of all food experiences in Japan. You may belly up to the bar with friends or colleagues in tow, but once your bowl arrives, all talking ceases as you turn your attention entirely to the task of conveying noodles and soup from bowl to mouth. No conversing, no pausing, no "How is the soup working out for you?" from the waitstaff. You bow your head, let the steam wash over you, and don't look up again until you can see the bottom of the bowl.
Inexperienced eaters will require some practice before they learn to handle the volcanic temperatures of a proper bowl of ramen. Waiting for it to cool, though, will prove an unnerving experience for both you and the chef. The only way forward is to abandon Western decorum and embrace the slurp, the calculated introduction of air that cools the noodles upon entry. A ramen shop in full feast mode sounds like a car vacuum suctioned against your front seat. It will take a few scaldings and a few stained shirts, but until you learn to properly slurp, expect to be lapped by grandpas whose bowls are dry before you've had the chance to slip the first noodles past your lips.
When we finish our bowls—Kamimura in three minutes, me in twelve—beads of sweat have gathered above my brow. I look up, almost surprised and slightly embarrassed to find I'm not alone in the shop.
"Next stop," he says, and we step outside, swallowed by the bright lights of a Fukuoka night, in search of another bowl.
米 麺 魚
Kyushu, as the southernmost of Japan's four main islands, has always been a gateway to the outside world. In fact, for much of Japan's modern history, it was the only way in and out.
When the shogun Tokugawa Iemitsu closed Japan's borders in 1635, ushering in two centuries of virtual isolation, Nagasaki on Kyushu's west coast remained the only open port in the country. It became a tiny door through which cultural artifacts from the outside world could enter. Portuguese missionaries brought tempura and Christianity. The Koreans introduced a rich ceramics culture. And the Chinese arrived with their noodle soups, including _champon_ , a Nagasaki specialty of pork, seafood, and egg noodles that some believe to be a precursor to Japanese ramen.
At the same time, Kyushu maintained a wild, rebellious edge. For much of the seventeenth and eighteenth centuries, it became a clubhouse for buccaneers and misfits, a refuge where pirates could take advantage of Japan's lack of centralized power and public order to loot and pillage. By the nineteenth century, much of this rogue energy had coalesced into one of Japan's mightiest military factions. It was in here, in Satsuma in 1877, shortly after the inception of the Meiji era, that the last samurai took a final stand against imperial Japan.
Kyushu later became a home of industry—steel and iron, mostly—and as such took on an outsize role in World War II. The southern island became a favorite target for American attacks, starting with the 1944 bombing of Yahata. In fact, Yawata Steel Works in Fukuoka Prefecture was the original target for the second atomic bomb, but because of cloud cover, Nagasaki was razed instead.
Like the rest of Japan, Kyushu recovered quickly, rebuilding and expanding industry in the postwar years. Since then the region has been working hard to position itself as a top destination for domestic and foreign tourists. The arrival of the Shinkansen in 2004 has made the southern island more accessible than ever: board a bullet train in Kyoto, and you'll barely have time to crack a bento box and down a Kirin before you pull into Hakata Station.
Despite the ease of access, just 3 percent of American tourists in Japan ever make it to Kyushu, something that will feel like a gross oversight to anyone who has spent time in the region. This is a land for coastal cruising and mountain bounding, for hot mud baths and cold potato liquor. Above all, it's a place to eat. In Kagoshima in the south, the list of local specialties (black-footed pigs, fried fish cakes, tiny, sweet sardines) is exceeded only by the world-class shochu on offer at every bar and restaurant. In Miyazaki, on the southwestern coast where Japan's surfing community chases the country's best break, chicken is king, from blackened, charcoal-coated thighs to the scourge of Western hygienic sensibilities, chicken sashimi.
Fukuoka dusk reflected in the Naka River; nightlife here is among the best in Japan.
_(Matt Goulding)_
But Fukuoka is the center of island life—gastronomically and otherwise. The island's capital was originally divided into two urban centers: Fukuoka for the well-to-do to the west of the Nakagawa River, and Hakata for the common folk settled in the east. The two were officially merged in 1889, but the two names are still used by locals and urban planners (who named the airport after the former and the train station after the latter).
It's a city with a broad yet gentle appeal—not a love-at-first-sight destination, but a slow-burn kinda place. _Monocle_ named it the tenth most livable city in the world in its 2014 survey, a fact you're likely to hear repeated more than a few times while in town. Indeed, on paper, it stacks up favorably to any city you know: great weather, lovely coastline, plenty of parks and open spaces, fantastic food, electric nightlife. Unlike other parts of Japan, which can strike visitors as wondrous, fantastical places, spend a few days in Fukuoka and you might find yourself saying, "I could see myself living here . . ."
Here with the high-skirts and hustlers working the corners of Naksu. Here with the hipsters and the bookworms buying jean jackets and sipping matcha lattes on the narrow streets of Daimyo. Here with the tuna-cheeked businessmen bellying up to the _yatai_ , the local street food vendors, for one last round and a bowl of something warm before heading home. Fukuoka has an edge, a certain samurai resistance about it, and nowhere is the spirit of nonconformity more apparent than in its street-food scene.
Fukuoka is the last bastion of _yatai_ culture in Japan, a reminder of a past when all of Japan's most famous foods—sushi, soba, skewers—could be found at these pushcart street stands. While _yatai_ have effectively been banned across Japan, you will still find them all around Fukuoka, gathered in clusters along the river and in pockets of city nightlife centers like Tenjin and Nagahama. They take shape every evening at dusk and disappear every morning at dawn, and in the hours between they serve everything from _oden_ and yakitori to craft cocktails and escargot.
Kamimura takes me to his favorite _yatai_ , a collection of covered stands next to a famous shrine where the owner works the crowd in a full kimono and headband. Most _yatai_ seat eight people hip to hip, but this is Fukuoka's largest, a tented stand that could house your high school algebra class. It's early by _yatai_ standards, but the group of young suits next to us is already soaked in shochu and offers up spirited _kanpai_ s when our own drinks arrive. After skewers of grilled chicken parts and a few more rounds of shochu, they pay their bill and move on, but it's clear that a few more _yatai_ stops await before the night is through. Kamimura looks almost wistful as he watches them go.
" _Yatai_ life is slowly dying in Fukuoka. There used to be three hundred _yatai_. Now there are a hundred and fifty." _Yatai_ have been fighting for survival for the past two decades as business owners have decried their low-rent competitive advantage and local residents have railed against the bad behavior—the noise, the smell, the public urinating—of _yatai_ customers, many of them tourists, most of them drunk.
After a few rounds of shochu and some skewers of cheese grilled directly over charcoal, the ramen arrives—a small bowl dotted with bamboo and nori and a single thin slice of roast pork. Kamimura seems to sense my disappointment.
"There are very few _yatai_ that serve good ramen. They have limited space, so they use soup and noodles made by someone else. Others have very limited hours so it's impossible to have the same quality as a restaurant. And because it's for after drinking, the style tends to be light. But you have to respect _yatai_ for their history and for still being one of the most popular places to eat ramen here."
Kamimura is an enthusiast, a man who in private will tell you that a noodle should have been a millimeter wider but publicly, on his website and in magazines, will always try to find the positive side behind every bowl. Rather than calling a bowl small or overpriced, he says, "It's perfect for a snack." Instead of calling a broth overly fatty and without nuance, he'll say "It's best for hardcore _tonkotsu_ lovers."
That's not to say Kamimura's ramen writing isn't deeply informative; the man will lay out details about ingredients and techniques with encyclopedic exactitude. But the overall tone is always one of respect and enthusiasm for the craft. "No matter what happens in my life, ramen has always been there for me."
This is our fifth bowl of ramen over the past eight hours, and I've reached my limit, but Kamimura shows no sign of slowing down. He looks over at me and eyes the small puddle of pork broth and tiny tangle of noodles before me. "You going to finish that?" It's not a clever technique to inspire me to soldier on; it's a legitimate desire to leave no soup unslurped. For every bowl I eat, Kamimura eats two—not for research (he's been to all of these places dozens of times), not to avoid waste (all nonramen food that makes its way to the table is essentially ignored by him), and certainly not because he's hungry (by my back-of-the-napkin math, he is ingesting north of 5,000 calories' worth of ramen a day during our time together). No, Kamimura does it for the same reason he reviews packaged ramen at home and feeds his baby boy pork broth and makes his wife pull over every time they drive past an unknown shop: because his dedication to ramen is boundless. He doesn't love ramen like you love pizza or like I love _The Sopranos_ ; he loves ramen like Antony loved Cleopatra.
In Japanese, you would call Kamimura an _otaku_ , one with a deep, abiding dedication to a single topic. A nerd. _Otaku_ commonly describes manga fanatics and video game savants. But just like the chefs he admires, Kamimura is a craftsman, and his commitment to ramen writing approaches _shokunin_ status, a dedication so all-consuming that everything else in his life is a footnote.
Sometimes, he says, that love may go too deep. Eating more than a bowl of _tonkotsu_ a day will wear on a man's body, and Kamimura is no exception. "I've gained ten kilos in the last three years. I'm afraid my blood is more fat than blood now. My doctor is concerned."
But the torment goes beyond the physical; the knowledge that somewhere, in some corner of Fukuoka or Kyushu beyond, lurks a bowl of unknown provenance and deliciousness is enough to keep a man like Kamimura up until the small hours of the night. When we part each day after our ramen adventures, there's a hint of sadness in his demeanor, as if he thought our hunt would go on forever. When he waves good night, it's not with an open palm but with two fingers pressed together like chopsticks, which he shovels toward his mouth.
米 麺 魚
The earliest footprints of ramen in Japan can be found around the turn of the century, as Chinese migrants in areas like Yokohama, Hakodate, and Nagasaki, the first ports opened to the outside world after hundreds of years of isolationism, began selling the soup to construction workers. Back then it was called _shina soba_ , "Chinese noodles," and was sold mostly from street carts and, oddly enough, Western-style restaurants. The dish was a humble convergence of noodles and a light salt-based broth, but also a sign of Japan's shifting eating habits, one that signaled an increasing appetite for wheat and meat.
Kamimura Toshiyuki working on one of his four hundred annual bowls of ramen.
_(Matt Goulding)_
Whatever it might have been before the war, the events that took place between 1937 and 1945 would put ramen culture on an entirely different trajectory. Strict food rationing meant _shina soba_ all but disappeared during World War II. When the atomic dust finally settled, the Americans moved in and began to reshape Japanese eating habits in profound ways.
Japan had long struggled to feed its own citizens, given its small land mass and high population density. But with the country pockmarked by fire bombings and much of the young male population lost to the war, the Japanese became deeply reliant on American supplies as they fought to ward off starvation. Chief among the imports: American wheat and lard, the basis for a bowl of ramen.
In his excellent book _The Untold History of Ramen_ , George Solt points out that these two ingredients, along with garlic, became the basis for what the Japanese called "stamina food," belly-filling staples like gyoza, _okonomiyaki_ , and ramen that became lifelines in the scavenger years following the war. Rice harvests were largely compromised by the war, so American flour became the building block for postwar recovery, and eventually the reindustrialization of Japan.
Some scholars, including Solt, argue that the shift from rice to wheat consumption during these years was a carefully crafted political objective undertaken by the Americans and supported by the Japanese government. It also became a powerful weapon for the United States' quest to contain the spread of communism across the Far East. Internal memos between the chief architects of the postwar world—Truman, Eisenhower and MacArthur—discussed American wheat shipments down to the last ton.
Propaganda abounded. "Eating Rice Makes You Stupid," read one flyer put out by a consortium of wheat producers. Another popular leaflet, circulated by the Civil Information and Education Section, showed a muscle-bound American foisting a tray of buttered bread loaves:
_Protein is a body builder. Wheat flour contains 50% more protein than rice. America is spending $250 million for your food. Learn to use it properly to get the full benefit._
A sketchy nutrition lesson and an even sketchier claim of American altruism (Japan was forced to pay the Americans back for the food aid they provided), but because this was a vulnerable and humbled Japan, the message caught on. Between 1956 and 1974, U.S. wheat exports to Japan nearly tripled.
On August 25, 1958, Momofuku Ando, a Taiwanese-born owner of a small salt company, released the first package of instant ramen noodles, a triumph of industrial food science that would redefine ramen for generations of busy moms, hungry bachelors, and desperate stoners. It would also represent the first taste of ramen most of the world beyond Japan would ever experience, a gateway to an ever-expanding world of noodle soups. (Today 100 billion servings of instant noodles are consumed annually worldwide.)
By the 1960s Japan had passed from postwar fallout into a period of rapid reindustrialization, and the workforce turned to ramen for fuel. As cities like Tokyo and Osaka began to rebuild and expand, small ramen shops sprouted across the cityscapes to feed the growing body of construction workers at the heart of Japan's unprecedented growth. In a matter of three decades Japan went from a broken nation to one of the world's greatest economic powers, a turnaround of staggering speed and remarkable scope. Behind every step forward was a bowl of ramen feeding the fires of industry.
The 1980s marked ramen's arrival into a whole new social stratosphere. Ramen was no longer a simple staple; it became a craft food, an object of obsession, a means of expression for legions of new cooks. Whereas most Japanese food is bound by tradition and a set of unspoken rules, ramen fans embraced innovation and experimentation. Microtrends—crinkled noodles, burned garlic oil, double broths—took shape overnight. The culture of queuing, now an honored pastime in Japan, was bred into acceptance in the boiling years of ramen ascendance.
Everyone wanted a piece of the action. Salarymen, disenchanted by the soulless demands of New Japan and its economic might, traded their briefcases for stockpots and began to boil their way back into a more rewarding life. (So common is this phenomenon that it has its own name: _datsu-sara_ , "salaryman escapee.") Young cooks took up the profession in droves, brandishing bandannas, self-branded tees, and a swagger that spoke of a new era of Japanese identity.
By the time Hideto Kawahara was twenty, ramen's transformation from a humble Chinese noodle soup to a Japanese cultural juggernaut was complete. But it still had yet to hit its apex. Hideto's father was a ramen man; in 1963 in Fukuoka he opened Daruma, a small shop serving a thick, dark bowl of _tonkotsu_ to a loyal local clientele. Ramen was one of the few corners of the culinary world where young cooks and entrepreneurs could make an immediate impact, but by the time he was old enough to cook, Hideto—a competitive breakdancer, a hat-to-the-side b-boy popping and locking his way across Japan—was more interested in break beats than pork bones.
But Hideto couldn't dance forever, so at twenty-eight years he gave up the floor spins and the helicopters and waded into the simmering waters of the ramen world. But he didn't do what sons had been doing for a thousand years in Japan: he didn't learn from his father. "My father told me he didn't want me to imitate his ramen. He wanted me to develop my own."
Instead, Hideto spent five years training down the street from his dad's shop, and then branched off to start his own, which quickly grew into a popular local chain in Fukuoka. By the time he opened in Tokyo's Asakusa neighborhood in 2001, he had a camera crew following his every move for a documentary TV show. "That was a rough time in my life. I was going through a divorce and I had this huge opening. So much pressure." When the store finally opened, there were three-hour waits for Hideto's ramen.
Today Hideto is forty-eight years old. He still wears his hat to the side, still rocks the gold chain, still looks like he could drop into a 720-degree headspin at any moment, but he's now ramen royalty, owner of seventeen shops across the globe, including ramen counters in New York, Hong Kong, Singapore, and Cambodia. He's just one part of a faction of Fukuoka-based chains that have together reshaped ramen on a global scale in the new millennium.
For the better part of thirty years, sushi was Japan's primary culinary export. But come the mid-aughts, when sushi bars had infiltrated cities across the globe and spicy tuna rolls could be found in every supermarket from Milwaukee to Melbourne, a new taste of Japan found its way to Los Angeles and New York. David Chang and his Momofuku Noodle Bar in New York's East Village was an early and influential player in the ramen game, but it wasn't until Fukuoka's most famous export, Ippudo, opened a few blocks west, on Fourth Avenue, in the winter of 2006 that ramen hit full fever pitch. Now you can find ramen shops in Midwestern malls and roving food trucks, and even your weird aunt Agnes can't stop talking about those strange and delicious Japanese noodles she had last spring.
The Japan represented by sushi is a very different country from the one represented by ramen. The former was a hushed, refined, serious country of fine taste and even finer economic means, but ramen represents a less intimidating, less exotic Japan, one dominated by bright lights, bold flavors, and the electric pulse of youth-driven pop culture.
Fukuoka, more than any other city in Japan, is responsible for ramen's rocket-ship trajectory, and the ensuing shift in Japan's cultural identity abroad. Between Hide-Chan, Ichiran, and Ippudo—three of the biggest ramen chains in the world—they've brought the soup to corners of the globe that still thought ramen meant a bag of dried noodles and a dehydrated spice packet. But while Ichiran and Ippudo are purveyors of classic _tonkotsu_ , undoubtedly the defining ramen of the modern era, Hideto has a decidedly different belief about ramen and its mutability.
"There are no boundaries for ramen, no rules," he says. "It's all freestyle."
As we talk at his original Hide-Chan location in the Kego area of Fukuoka, a new bowl arrives on the table, a prototype for his borderless ramen philosophy. A coffee filter is filled with _katsuobushi_ , smoked skipjack tuna flakes, and balanced over a bowl with a pair of chopsticks. Hideto pours chicken stock through the filter, which soaks up the _katsuobushi_ and emerges into the bowl as clear as a consommé. He adds rice noodles and saw-tooth coriander then slides it over to me.
Compared with other Hide-Chan creations, though, this one shows remarkable restraint. While I sip the soup, Hideto pulls out his cell phone and plays a video of him layering hot pork cheeks and cold noodles into a hollowed-out porcelain skull, then dumping a cocktail shaker filled with chili oil, shrimp oil, truffle oil, and dashi over the top. Other creations include spicy arrabiata ramen with pancetta and roasted tomatoes, foie gras ramen with orange jam and blueberry miso, and black ramen made with bamboo ash dipped into a mix of miso and onions caramelized for forty-five days.
"It's important to make the right ramen for the right place. If I do what I do here in New York, it doesn't work," he says. "They want less salt and less fat in New York. Gluten-free noodles. New Yorkers are tough."
Suddenly Hideto jumps up from the table and announces that he needs to go. He leaves me with a bowl of industrial-strength _tonkotsu_ —pig heads viciously boiled in sixty-liter iron vats for forty-eight hours—and a rundown of his itinerary for the next week: first to Singapore, then to Phnom Penh for the opening of his first Cambodian shop, then to New York to roll out a new line of dry ramen dishes, back to Hong Kong to scout new locations, then home to Fukuoka for thirty-six hours before repeating the loop. The world, he says, is hungry for ramen.
米 麺 魚
Hideki Irie doesn't look like a typical _tonkotsu_ ramen cook. He walks into his restaurant in a shiny black bubble jacket, sunglasses perched on his head, a sparkly watch on each wrist. Even in his uniform, with the black collar popped like a Michigan frat boy, he manages to exude a sense of attitude that feels completely foreign in this country. But ramen may be the one corner of Japanese food culture where swagger is an acceptable ingredient, and Irie projects it with gusto.
We've met at his shop Mengekijo Genei, which eschews the typical ramen curtains in favor of a thick wooden door and trades a traditional countertop setup for stadium seating, each stool positioned for optimal intake of the kitchen action below. Cooks in the center toast garlic and shrimp oil in sizzling woks while a young kid behind a glass wall on the left feeds yellow balls of dough into a pasta machine to make the night's noodles.
Before turning to ramen, Irie was a private investigator, a job that he dismisses today with a single shake of the head. "I wasn't happy doing it. I would walk around with this horrible look on my face." One day during his sleuthing years, he visited a ramen shop in his hometown of Kumamoto owned by a friend. Something clicked when he saw the simplicity of it all: hot soup, happy people. "My friend told me, 'It's the most rewarding job in the world.'"
Hideki Irie, the Ramen Chemist, with his finely tuned bowl of _tonkotsu_
_(Matt Goulding)_
He left the investigating behind and took up a job behind the counter at Tenyo Ramen, where he spent five years learning the ins and outs of the craft. He discovered early on what he didn't like about ramen: he didn't like shortcuts; he didn't like cheap ingredients; he didn't like monosodium glutamate. The last point remains one of heated debate in the ramen community. In some kitchens, tubs of MSG sit openly on the counter like salt and pepper, ready to be spooned generously into each bowl before being passed across the counter. But many of the young modern ramen chefs have made it a mission to find maximum flavor without MSG.
Proponents say MSG is a natural flavor enhancer, a crystalline source of umami that has been openly harnessed for its savory powers for generations. Detractors claim it's unsafe, a catalyst for rogue headaches and strange neural reactions—or, at the very least, a dubious substitute for finesse in the kitchen. No matter what your reasons may be for keeping it out of your restaurant, one thing is certain: not using MSG puts you at a distinct disadvantage in a crowded, powder-happy market like Fukuoka. For a place to survive and thrive, it must find other ways to harness flavor.
This became Irie's obsession. He started out by learning to brew his own soy sauce. "Almost all chefs buy soy in the store, but the product is lousy. If I could develop my own soy, nobody could copy my recipe." The resulting potion took a year of research to master and costs $200 a liter to make—which, Irie says, is worth every yen. "Joel Robuchon wanted to buy it from me, and I told him no," he says, speaking of the French chef dubbed by the Michelin guides as "the greatest chef of the century." "I don't want Robuchon copying my ramen."
With the super soy calibrated, he set about tinkering with different combinations of umami-rich products until he found the perfect mix for his _tare_ : kelp, shiitakes, bonito, oysters, sardines, mackerel, dried scallops, and dried abalone.
"I'm a ramen chemist," he says, talking about the time he spent three days straight in the library studying the science of taste. "I can engineer any flavor. I could make you a bowl of _tonkotsu_ without using pork."
Irie is part of a generation of enterprising ramen chefs intent on pushing the soul of this traditional comfort food to its most sophisticated and refined expression. After listening to him talk about his top-secret _tare_ , his $200-a-liter homemade soy sauce, his years spent studying MSG, you get the sense that the 800 yen he charges for a bowl may represent one of the greatest bargains in the entire food world. And maybe it does.
But I'm not so sure Kamimura is convinced. It's clear that he respects Irie's talent and his desire to innovate, but Kamimura is a _tonkotsu_ purist, a man who would rather pay 400 yen for a bowl of pork bones and store-bought soy sauce than twice that much for a bowl refined down to its last milliliter. Ramen should be made by blue-collar cooks, not white-collar chefs. Most of the men in Fukuoka might agree, judging by the crowds I see gathered around places like Ganso and Shin Shin, classic joints serving throwback bowls for throwback prices.
But that's clearly not the audience Irie is aiming for. _Tonkotsu_ has always been an almost all-male sport, but look around Genei and you see a different clientele entirely: couples, single women, families—signs of a shifting culture.
"There are two methods to develop a ramen shop in Fukuoka," says Kamimura. "The first is to provide a single taste and dedicate yourself just to that taste. The second way is to offer a variety of flavors and changing menus. At least Hide-Chan and Genei keep their classic ramen while they experiment with new flavors. It has to be that way, because that brings in a wider variety of clientele."
Irie serves me three ramens, including a bowl made with a rich dashi and head-on shrimp and another studded with spicy ground pork and wilted spinach and lashed with chili oil. Both are exceptionally delicious, sophisticated creations, but it's his interpretation of _tonkotsu_ that leaves me muttering softly to myself. The noodles are firm and chewy, the roast pork is striped with soft deposits of warm fat, and the toppings—white curls of shredded spring onion, chewy strips of bamboo, a perfect square of toasted seaweed—are skillfully applied. Here it is the combination of _tare_ , the culmination of years of careful tinkering, and broth, made from whole pig heads and knots of ginger, that defies the laws of _tonkotsu_ : a soup with the savory, meaty intensity of a broth made from a thousand pigs that's light enough to leave you wanting more. And more. And more.
"I have no doubt that I make the best bowl of ramen in Japan," Irie says. Fighting words, to be sure, but the man may have a point.
米 麺 魚
_Tonkotsu_ , like many of the world's great dishes, was born out of a happy accident. The idea of replacing traditional chicken bones with pork bones was already in practice in Kurume in the early 1930s, adapted from the Chinese in nearby Nagasaki. As the story has it, one night an old cook at a _yatai_ left the soup on the stove too long, turning the broth thick and cloudy with melted marrow and porky intensity. It caught on quickly, spreading from _yatai_ to _yatai_ , and soon double-boiled pork-bone soup became the official ramen of Kyushu.
At the Kurume train station, twenty-five miles south of Fukuoka, a miniature bronze replica of the original _yatai_ stands as a reminder to all of where one of Japan's most famous dishes comes from. Kamimura takes me by the statue to pay our respects to ramen history, but he talks grimly about Kurume's ramen scene. He speaks of a ramen town where nobody gets along, where factional beefs and claims to history cloud the already cloudy soup, a town where the shop that invented _tonkotsu_ can't even make a decent bowl anymore. (Which is why we're genuflecting to the statue instead of the still-operational original _yatai_.)
But the trip isn't merely a historical pilgrimage; Kurume still claims a few of Kamimura's favorite shops. We start at Rai Fuku Ken, a tiny shop next to the train station that has been serving _tonkotsu_ since shortly after it was invented down the street. The owner, Akira Yoshino, is a second-generation shop owner and the current president of the Kurume _Tonkotsu_ Ramen Association. Round-faced and rosy-cheeked, with a black bandanna tied tightly across his forehead, Yoshino views himself as a guardian of the true _tonkotsu_.
"I'm proud to know that ramen has spread to places like New York and Europe," he says, "but Kurume people like Kurume ramen, and the style that people around the world know as _tonkotsu_ is not the original _tonkotsu_. We care only about keeping the soul of Kurume ramen alive."
His is a Goldilocks bowl: medium body, golden in color, made from all parts of the pig cooked over twenty-four hours with nothing but water from the Chikobe River nearby. It asserts itself, coats your throat on the way down, but it doesn't stick to your ribs the way the most intense bowls do.
It's the next stop, though, that I've been waiting for. Kamimura has been whispering all week of a sacred twenty-four-hour ramen spot located on a two-lane highway in Kurume where truckers go for the taste of true ramen. The shop is massive by ramen standards, big enough to fit a few trucks along with those drivers, and in the midafternoon a loose assortment of castaways and road warriors sit slurping their noodles. Near the entrance a thick, sweaty cauldron boils so aggressively that a haze of pork fat hangs over the kitchen like waterfall mist.
The same stock has been simmering at Maruboshi since 1955.
_(Matt Goulding)_
While few are audacious enough to claim ramen is healthy, _tonkotsu_ enthusiasts love to point out that the collagen in pork bones is great for the skin. "Look at their faces!" says Kamimura. "They're almost seventy years old and not a wrinkle! That's the collagen. Where there is _tonkotsu_ , there is rarely a wrinkle."
He's right: the woman wears a faded purple bandanna and sad, sunken eyes, but even then she doesn't look a day over fifty. She's stirring a massive metal cauldron of broth, and I ask her how long it's been simmering for.
"Sixty years," she says flatly.
This isn't hyperbole, not exactly. Kurume treats _tonkotsu_ like a French country baker treats a sourdough starter—feeding it, regenerating, keeping some small fraction of the original soup alive in perpetuity. Old bones out, new bones in, but the base never changes. The mother of all ramen.
Maruboshi Ramen opened in 1958, and you can taste every one of those years in the simple bowl they serve. There is no fancy _tare_ , no double broth, no secret spice or unexpected toppings: just pork bones, noodles, and three generations of constant simmering.
The flavor is pig in its purest form, a milky white broth with no aromatics or condiments to mitigate the purity of its porcine essence. Up until now, Kamimura has worked his way through bowls of ramen with the methodical persistence of a librarian cataloging books, but something in him changes with the first slurp of Maruboshi's bowl. His eyes light up, he wiggles his shoulders, and a childish smile breaks out across his face. "What do you think? What do you think?"
For Kamimura, it's not just a strength thing—it's a soul thing. He respects craftsmen like Hideto and Irie, but their calculated compositions don't move him the same way that a straight bowl of bone broth does. It takes time to draw out the soul of ramen—some say hours; others, like Kamimura, say lifetimes.
When the owners spot Kamimura, they hurry over to our booth, offering paper cups of coffee to go with our mystic soup. Kamimura mentions that he's been reviewing more instant ramen than ever lately, and the woman disappears and comes back with a cardboard box stacked with sixteen individual packets of Maruboshi's take-home product. But his attention isn't with the owners or the packaged noodles or the steaming cups of coffee. No, it's aimed squarely at me. He catches my eyes, then looks down at my unfinished bowl, then back up at me. I know what he wants, and after twenty-eight bowls over the course of five days, I'm more than happy to give it to him, but first, he needs to ask.
"You going to finish that?"
## _Taxonomy_
## [ラーメン
**THE RAMEN MATRIX**](nav.xhtml#rhh11)
_(Matt Goulding)_
Japan is a land of a million bowls of ramen. With over 200,000 shops and a world of microtrends and funky innovations, ramen is Japan's most personalized and boundless staple. Behind the specialty bowls, though, there are at least twenty-two accepted regional styles of ramen that bring order to the complex noodle ecosystem. Here, in part, are the most famous of Japan's regional ramen species.
**HAKODATE SHIO**
As one of Japan's first ports open to the outside world, Hakodate has a long ramen history. Light and clear like consommé, _shio_ (salt) ramen is the closest reflection of the original Chinese ramen.
_Where to eat: Ebisuken (Hakodate), Afuri (Tokyo)_
_(Michael Magers, lead photographer)_
**SAPPORO MISO**
The thickest and richest of Japan's regional ramens, designed to get people through Hokkaido's Siberian winters. Red miso and wok-fried _chashu_ and vegetables make up the soul of the bowl. Butter and corn, two Hokkaido staples, are optional.
_Where to eat: Menya Saimi (Sapporo), Hanamichi (Tokyo)_
_(Michael Magers, lead photographer)_
**HAKATA TONKOTSU**
The king of regional ramen, made exclusively with pork bones boiled for up to forty-eight hours, creating a milky white broth thick with melted marrow and collagen. Served with straight, thin noodles.
_Where to eat: Mengekijo Genei (Fukuoka and Tokyo)_
_(Matt Goulding)_
**KAGOSHIMA HYBRID**
Kagoshima ramen cooks cut a _tonkotsu_ base with chicken and vegetables for a lighter version of Hakata _tonkotsu_. Noodles are flat, broad, and cooked soft, and the _chashu_ , made from local _kurobuta_ pig, is Japan's best.
_Where to eat: Ramen Kokinta, Tontoro_
_(Matt Goulding)_
**TOKYO SHOYU**
Chicken-based broth spiked with a generous amount of shoyu (soy sauce) and often a current of _nibosh_ (dried sardine). Expect curly yellow noodles, _menma_ (pickled bamboo), seaweed, and a soy-soaked egg. Along with _tonkotsu_ , the most common style of ramen in Japan.
_Where to eat: En, Taishoken_
_(Michael Magers, lead photographer)_
**TOKYO TSUKEMEN**
Thick room-temperature noodles slicked with warm pork fat and served with _chashu_ and a concentrated broth for dipping. One of the most popular ramen trends of the past decade, perfect for steamy summer afternoons.
_Where to eat: Rokurinsha_
_(Michael Magers, lead photographer)_
**ASAHIKAWA SURF-AND-TURF**
Blending the best from the two extremes of Japan: pork _tonkotsu_ from Kyushu mixed with the best seafood from Asahikawa's northern Hokkaido backyard to create a complex broth of land and sea.
_Where to eat: Santouka (throughout Japan)_
_(Michael Magers, lead photographer)_
## **DREAM MACHINES**
Our favorite picks from Japan's ubiquitous army of vending machines
_(Ioanna Morelli)_
_(Ioanna Morelli)_
### **BOSS COFFEE**
Vending-machine coffee can be sickly sweet, but Black Boss delivers the caffeine high without the sugar crash. Red buttons on the machine mean hot coffee; blue means cold.
_(Ioanna Morelli)_
### **POCARI SWEAT**
Even more appetizing than its name is the salty-sweet rush of electrolytes it delivers. Pocari is for the morning after a long night of _chuhai_ and Yebisu.
_(Ioanna Morelli)_
### **CHU-HI**
A version of the highball made with shochu instead of whisky, _Chu-hi_ contains twice as much alcohol as most beer. Best consumed as a pre-karaoke aperitif.
_(Ioanna Morelli)_
### **YEBISU**
Lesser known but the best of Japan's major beers, malty and smooth. When you spot this rare beast in the vending wilderness, rustle up some change.
## **FEAR NOT!**
_Conquering Japan's greatest cultural challenges_
_(Nathan Thornburgh)_
### **NATTO**
Soft, slimy, with a fermented tang, _natto_ is everything Westerners don't want in food. But the dish—made from soaked, steamed, and funkified soybeans—is served at breakfast through much of Japan. (It is also a common way to test a gaijin's appetite for real Japanese food.) How to handle? First, spike it with hot mustard and soy-based _tare_ , then just close your eyes and think of England.
_(Matt Goulding)_
### **WAGASHI**
There are two genres of desserts in Japan. _Yogashi_ are European-style cakes and pastries (universally excellent throughout Japan); _wagashi_ are traditional Japanese sweets. The charms of _wagashi_ can be elusive if you're not used to sweet rice, adzuki beans, and sticky textures for dessert. The trick is to embrace the subtlety: no blast of sugar means richer, earthier flavors.
_(Matt Goulding)_
### **RYOKAN**
You may regret having chosen a _ryokan_ , a traditional Japanese inn, right around the time the innkeeper wakes everyone up for breakfast at 7:00 a.m. on Sunday. But don't rue: spending a day or two in a _ryokan_ is the quickest route to understanding the futon-sleeping, robe-wearing, big-breakfasting, hot-tub-loving Japanese soul.
### **ONSEN**
Sitting naked in a communal bath might not sound like a relaxing time, but when you slip into the therapeutic waters of a hot spring, all worries will evaporate. Prepare yourself: First, strip down. No bathing suits, no underwear. Cover tattoos (often banned because of their connection to yakuza). Scrub yourself head to toe. Now you're ready to soak it all up.
## _Chapter Five_
## HIROSHIMA
It starts with a _thwack_ , the sharp crack of hard plastic against a hot metal surface. When the ladle rolls over, it deposits a pale-yellow puddle of batter onto the griddle. A gentle sizzle, as the back of the ladle spackles a mixture of eggs, flour, water, and milk across the silver surface. A crepe takes shape.
Next comes cabbage, chopped thin—but not too thin—and stacked six inches high, lightly packed so hot air can flow freely and wilt the mountain down to a molehill. Crowning the cabbage comes a flurry of tastes and textures: ivory bean sprouts, golden pebbles of fried tempura batter, a few shakes of salt, and, for an extra umami punch, a drift of dried bonito powder. Finally, three strips of streaky pork belly, just enough to umbrella the cabbage in fat, plus a bit more batter to hold the whole thing together. With two metal spatulas and a gentle rocking of the wrists, the mass is inverted. The pork fat melts on contact, and the cabbage shrinks in the steam trapped under the crepe.
Then things get serious. Thin wheat soba noodles, still dripping with hot water, hit the _teppan_ , dancing like garden hoses across its hot surface, absorbing the heat of the griddle until they crisp into a bird's nest to house the cabbage and crepe. An egg with two orange yolks sizzles beside the soba, waiting for its place on top of this magnificent heap.
Everything comes together: cabbage and crepe at the base, bean sprouts and pork belly in the center, soba and fried egg parked on top, a geologic construction of carbs and crunch, protein and chew, all framed with the black and white of thickened Worcestershire and a zigzag of mayonnaise.
This is _okonomiyaki_ , the second most famous thing that ever happened to Hiroshima.
米 麺 魚
Fernando Lopez makes an unlikely candidate for one of Hiroshima's greatest _okonomiyaki_ chefs. He was born in Guatemala City in 1963. His father worked for Guatemala's health services, spraying DDT to combat the plague of malaria that gripped Central America in the 1960s. He spent a lot of his time on the road, often in the beds of other women. "He wasn't a good man," says Lopez.
His father had Mayan blood, with dark skin to match his dark hair. His mother was fair, with wavy hair and a sweet smile. When little Fernando was born with light skin, blue eyes, and curly hair, his father refused to believe the boy was his, and so Lopez was raised mainly by his grandmother, separate from his four brothers and two sisters.
Even when his father did finally accept Fernando as his own, it wasn't an easy relationship. At fifteen, Lopez decided to stand between the man he barely knew and a beating aimed for his mother, and years of abuse and philandering came to a head. His father left, never to come back, and the boy who'd grown up alone was left to absorb the blame for chasing away the man of the house.
Lopez survived, working hard to overcome the early challenges life had posed for him. He studied accounting for a year in college, and eventually took a job managing the books for a popular Italian restaurant in Guatemala City. He soon discovered that the managers were skimming off the top, along with other criminal activity, and he fretted over what to do with this sensitive information. When one of his coworkers turned up dead in a ditch, he knew it was time to leave Guatemala, possibly for good.
Fernando Lopez in the early moments of _okonomiyaki_ prep
_(Matt Goulding)_
He landed in New Orleans on a visa sponsored by an uncle who had lived in the States for years. He planned to stay for three months to study English, but instead took a job busing tables at an Italian restaurant. The chef had a temper issue, and one day the entire kitchen staff walked out on him. He recruited Lopez to help out in the kitchen, but the young Guatemalan knew nothing about cooking. "He fired me every fifteen minutes. It was a mess."
Soon after, while working as a dishwasher at the Fairmont, he met Andre LeDoux, a well-traveled hotel chef who would become his kitchen mentor—the first of a series of teacher-student relationships that would shape Lopez as a cook and a man. LeDoux made him a deal: Lopez would teach him Spanish, and he would teach Lopez how to cook. When LeDoux became chef of the French Quarter institute Arnaud's, he took Lopez with him, and Lopez's real education began in earnest. "At first you're a slave, you're everyone's bitch, and they can do whatever they want with you. But that's how you learn." He moved from station to station, mastering the classics of the French Creole canon: shucking and roasting oysters, making roux for gumbo, sautéing frog legs in garlic butter. "There were twelve of us feeding six hundred people a night. People walked out on him. They couldn't take the stress. But I loved it."
When LeDoux left Arnaud's to run the kitchen at the Sheraton Surfrider in Honolulu, Lopez followed him across the Pacific. The Sheraton's kitchen staff was on strike, so Lopez entered as a scab, stuffed in a van and slipped into the kitchen under the cover of darkness. He cooked nonstop for forty-three days and nights, until the strike broke and Lopez was left without a place in the kitchen. He took a job as a valet at a hotel where, one night, a young Japanese woman in a beat-up Toyota Corolla with a bad paint job pulled into the parking lot and changed his life. "Nobody else wanted to park the car because it was so beat up, they thought they wouldn't get a tip." He didn't get a tip, but he got a date out of it.
Makiko Yonezawa was from Hiroshima. Her family owned a _ryokan_ back home, and she had come to Hawaii six years earlier to study the hotel industry. They connected right away, but Lopez's timing wasn't great: Makiko returned to Japan a few months after they started dating to help with the family business. Lopez soon followed with a surprise visit, a grand gesture of young love that didn't sit well with Makiko's parents. They didn't like the idea of their daughter dating a foreigner, but her father pulled Lopez aside before he returned to Hawaii and told him that if they remained together for a year, then they could talk seriously.
Fernando and Makiko married in 1992, in a small civil ceremony in Hawaii. For their honeymoon, they took Amtrak around the United States, looking for a place to build a life together. They loved Chicago, Denver, and Seattle, but the cold and the rain scared them off. In Phoenix they fell hard for the spice-charged food of the Southwest, and they hatched a plan to open a Tex-Mex restaurant together in Hawaii. But things didn't go exactly as planned. Real estate was outrageously expensive in Honolulu, and neither of them qualified for the kind of loan they would need to build a business. So in 1995, with dwindling prospects in the States, they made the move to the Far East, to southern Japan, transporting their dream of opening a Southwestern restaurant to the heart of Hiroshima.
米 麺 魚
People around town tell me to look for the giant wooden egg. "The Giant Wooden Egg!" they say, raising their voices and stretching their wingspans out to mimic its shape—a brown oblong structure eight stories high, inside which I would find the secrets of Hiroshima's most sacred food.
The egg in question is home to Otafuku, Hiroshima's famous sauce maker, which doubles as the de facto museum to Hiroshima-style _okonomiyaki_. Maybe it's the exposed ribs, the empty spaces, the nearly naked aspect of the looming wooden structure, but the building looks less like an egg and more like the skeletal remains of the Atomic Bomb Dome, which stands as a memorial to the nuclear attack on Hiroshima. It seems like a grim architectural echo for the global headquarters of a company best known for its sticky-sweet _okonomiyaki_ topping.
That echo, however, turns out to be intentional. Otafuku ties its sauce intimately to the city it comes from, and also to the defining horror that destroyed old Hiroshima and remade everything that followed. It was in the wake of that horror that _okonomiyaki_ took shape.
_Issen yoshoku_ , "one-coin Western food," gained popularity in the early 1900s as a cheap after-school snack for kids, a crepe rolled with onions and bean sprouts and often sold in candy shops. In the years immediately following the war, as the survivors of the bomb tried to stave off starvation, the snack became a vital part of Hiroshima's revival.
In a matter of seconds, the bomb leveled every eatery in the city center, in essence wiping Hiroshima's restaurant culture clean. With nothing else to work with, loose pieces of sheet metal, the bones of buildings lost to the bomb, became street _teppan_ s, makeshift griddles heated from below with coal from the shipyard and used to cook whatever scraps of food could be thrown together: a few shreds of cabbage, loose vegetable bits, an egg or a touch of protein for the most fortunate. As American forces arrived in Japan with surplus wheat supplies, cooks in Hiroshima used flour and water to stretch and bind the dish.
The Otafuku tour begins the _okonomiyaki_ story a few years later, after the dust had settled, after the desperation had ebbed. On the main floor of the museum, the first stop is a reconstructed _okonomiyaki_ ya-san from the 1950s. Like many of the early wave of _okonomiyaki_ shops, it was connected to a home, perhaps with a small convenience store for daytime commerce, selling gum and cigarettes. More than anything, the ad hoc diners were a way for war widows to earn some money. The reconstructed space has the plastic feel of demonstration food, punctuated by a few original accents: metal _hera_ (spatulas) from the period, a small black-and-white television with old newsreels, a menu board offering _okonomiyaki_ with egg for 15 yen and without for 10.
As Japan recovered from the postwar depression, _okonomiyaki_ became the cornerstone of Hiroshima's nascent restaurant culture. And with new variables—noodles, protein, fishy powders—added to the equation, it became an increasingly fungible concept. Half a century later it still defies easy description. _Okonomi_ means "whatever you like," _yaki_ means "grill," but smashed together they do little to paint a clear picture. Invariably, writers, cooks, and _oko_ officials revert to analogies: some call it a cabbage crepe; others a savory pancake or an omelet. Guidebooks, unhelpfully, refer to it as Japanese pizza, though _okonomiyaki_ looks and tastes nothing like pizza. Otafuku, for its part, does little to clarify the situation, comparing _okonomiyaki_ in turn to Turkish pide, Indian chapati, and Mexican tacos.
There are two overarching categories of _okonomiyaki_ : Hiroshima style, with a layer of noodles and a heavy cabbage presence, and Osaka or Kansai style, made with a base of eggs, flour, dashi, and grated _nagaimo_ , sticky mountain yam. More than the ingredients themselves, the difference lies in the structure: whereas _okonomiyaki_ in Hiroshima is carefully layered, a savory circle with five or six distinct layers, the ingredients in Osaka-style _okonomiyaki_ are mixed together before cooking. The latter is so simple to cook that many restaurants let you do it yourself on tableside _teppan_ s. Hiroshima-style _okonomiyaki_ , on the otherhand, is complicated enough that even the cooks who dedicate their lives to its construction still don't get it right most of the time. (Some people consider _monjayaki_ , a runny mass of meat and vegetables popularized in Tokyo's Tsukishima district, to be part of the _okonomiyaki_ family, but if so, it's no more than a distant cousin.)
A wall of sticky-sweet _okonomiyaki_ sauce on display at Otafuku's headquarters
_(Nathan Thornburgh)_
Otafuku entered the picture in 1938 as a rice vinegar manufacturer. Their original factory near Yokogawa Station burned down in the nuclear attack, but in 1946 they started making vinegar again. In 1950 Otafuku began production of Worcestershire sauce, but local cooks complained that it was too spicy and too thin, that it didn't cling to _okonomiyaki_ , which was becoming the nutritional staple of Hiroshima life. So Otafuku used fruit—originally orange and peach, later Middle Eastern dates—to thicken and sweeten the sauce, and added the now-iconic Otafuku label with the six virtues that the chubby-cheeked lady of Otafuku, a traditional character from Japanese folklore, is supposed to represent, including a little nose for modesty, big ears for good listening, and a large forehead for wisdom.
Today Otafuku is the primary engine behind Hiroshima's massive _okonomiyaki_ industry, and as such, they invest no small amount of time and energy in making sure the city is checkered with successful vendors dispensing dark rivers of its saccharine sauce. That means connecting business owners with cabbage and pork purveyors to keep the _teppan_ s humming. That means schooling potential entrepreneurs in the economics of restaurant management. That means helping train the next wave of _okonomiyaki_ masters: disgruntled salarymen, ambitious home cooks, even the occasional Guatemalan immigrant.
米 麺 魚
Lopez and his wife were determined to bring the flavors of Phoenix and Santa Fe and El Paso to the people of Hiroshima. The only problem was that no one in Japan had ever heard of Southwestern food.
After Lopez presented his plan to a local builder, the contractor told Lopez bluntly, "I don't build restaurants that fail."
Lopez and his wife shuffled through ideas—pizzeria, bistro, sandwich shop—but nothing felt right. Eventually the conversation turned where conversations in Hiroshima normally turn when the subject of food comes up: _okonomiyaki_. "Why don't you open an _okonomiyaki_ restaurant?" friends and family started to ask.
Why not open an _okonomiyaki_ shop? Let's consider the reasons: Because Lopez was born seven thousand miles away, in one of the roughest cities on the planet. Because he didn't look Japanese, speak Japanese, or cook Japanese. Because _okonomiyaki_ isn't just a pile of cabbage and noodles and pork belly, but a hallowed food in Hiroshima, stacked with layers and layers of history and culture that he couldn't pretend to be a part of. Because even though they might accept an Italian cooking pasta and a Frenchman baking baguettes, they would never accept a Guatemalan making _okonomiyaki_.
But friends and family insisted it was a good idea—"Everybody knows and loves _okonomiyaki_ ," they would say, still confounded by the idea of fajitas—and Lopez, with few decent alternatives, agreed to attend a business workshop put on by Otafuku. By the time he emerged three days later, head full of inventory lists and _teppan_ technology, he was convinced enough to give it a run.
Otafuku provided the framework for running a business, but he still needed to learn how to cook _okonomiyaki_ , so he sought out an apprenticeship. Lopez knew a guy who knew a guy working at Hassho, one of Hiroshima's greatest _okonomiyaki_ restaurants, where every night a line filled with hungry locals and guidebook-clutching tourists snakes around the block of Hiroshima's neon Yagenbori entertainment district. He was in.
The master-apprentice relationship, in many ways, is still the beating heart of Japanese food culture, an age-old tradition that supersedes stages and cooking school as the primary engine of culinary education. Unsurprisingly, apprenticeships tend to be formal endeavors, and each style of cooking comes with its own set of rules and expectations. Serious tempura students can expect to spend five years filtering oil, stirring batter, and looking over their master's shoulder before they're deemed ready to fry. In the sushi world, the apprentice might begin with a year of washing dishes, another few years cleaning and cooking rice, and eventually dedicate a decade to quietly observing the master slice and serve fish before being released into the wild to test his skills. I once met a fifty-five-year-old man in a Matsumoto _karaage_ restaurant who had been apprenticing under his father for twenty-seven years. After three decades, the dad didn't let the son fry the chicken.
By these standards, the _okonomiyaki_ apprenticeship is relatively relaxed. Lopez spent just three months working at Hassho, learning quickly the dozens of steps that go into constructing Hiroshima's most sacred staple. "I had an advantage that most of these guys don't have: I was a professional cook. I picked it up pretty fast."
In ninety days, Hassho's owner, Ogawa Hiroki, passed along to Lopez an arsenal of tiny tricks and vital techniques it had taken him a lifetime to accumulate. Lopez learned that bean sprouts in May behave differently from bean sprouts in October. He learned that fresh noodles, cooked to order, make an _okonomiyaki_ superior to one made with the prepackaged, precooked soba everyone else uses. He learned that touch and finesse are the most vital items in an _okonomiyaki_ cook's toolkit, because every _okonomiyaki_ behaves differently.
When Lopez had metabolized the meaty lessons of _okonomiyaki_ , Hiroki didn't just pat him on the back and wish him good luck. He took an early and spirited role in assuring that Lopez would succeed on his own. He helped design the layout of the restaurant; he made sure the _teppan_ was three centimeters thick and had overlapping burners to better hold in the heat, just as he had designed it himself so many years ago; he connected Lopez with all the right purveyors, including the guy with the gorgeous eggs with double yolks that his regulars so adored.
When a new _okonomiyaki_ restaurant opens in Hiroshima, an elaborate flower arrangement adorns the front of the shop, a gift from the master to the apprentice as the latter tries to win over a new clientele. It's both a sign of respect and an easy way to establish the bona fides of the new business owner. (It's also a subtle but looming reminder to the apprentice that he better keep his shit together and not bring dishonor to the master.) When Okonomiyaki Lopez opened in the spring of 2000, Hiroki sent an elaborate $200 arrangement, a sign with his shop's logo, and a metal stand to hold it all out in front for the public to see.
But business was slow. To start with, _okonomiyaki_ joints are everywhere in this city, two thousand in total across greater Hiroshima, and it's not easy to set yourself apart from the competition. It doesn't help that Okonomiyaki Lopez is located on a quiet street in Yokogawacho, the working-class neighborhood where Makiko's family once owned its _ryokan_. This is the kind of area where small neighborhood restaurants rule, and Lopez didn't fit the profile of your Tuesday-night cook. "People would sit there and watch me with huge eyes, trying to figure out who this guy was making their _okonomiyaki_."
Less than 2 percent of Japan's 126 million citizens are immigrants, making it one of the most homogenous countries on the planet (a 2012 study in the _Journal of Economic Literature_ placed it third to last in terms of ethnic diversity, with only North and South Korea ranking lower). Chinese and Koreans, many of whom have lived here for generations, account for more than half of whatever diversity there is, meaning very few Westerners call Japan home. Part of this stems from Japan's historic aversion to non-Japanese—from the sealed borders of the Tokugawa shogunate to the forced assimilation of the Ainu in Hokkaido. Modern immigration laws, among the most draconian in the world, and a deep dedication to a belief in Japanese superiority on the part of today's most conservative leaders, have done little to make Japan a more inclusive society.
The Japanese are heroically hospitable when it comes to foreign visitors, but for immigrants the welcome mat can be harder to find. Even if you do make it here, adapt to the culture, commit a thousand kanji characters to memory, denounce your birth country, and feel deep down in your soul that you are as Japanese as pickled fish and electronic toilets, you will always be an outsider.
Being from Guatemala, which at last count had just 145 citizens calling Japan home, means you're more outside than most. "A lot of people think Guatemala is a coffee brand. 'Oh, you're from the coffee brand!'" says Lopez. "Japanese people forget about Central America. They think Mexico is attached to South America."
Knowing they were up against a formidable headwind, Lopez and Makiko worked hard to make inroads in the neighborhood. So did Hiroki, who created special cards announcing Okonomiyaki Lopez that he distributed around Yokogawacho. He instructed Lopez—who was studying Japanese in night school and by now beginning to grasp some of the many social formalities that dominate basic interactions in Japan—to follow up with free samples of his _okonomiyaki_ and to solicit feedback from potential customers.
"Many said I could do better," says Lopez. "I mean, if you ask them their opinion, they're going to tell you."
In those early days, Lopez and Makiko cooked side by side. She was pregnant with their first child, but she had trained in kitchens before and proved a talented _okonomiyaki_ cook. Plus, since she was born and raised in the neighborhood, her mere presence behind the counter gave Lopez a sparkle of authenticity.
The combination of the _oko_ offensive and the husband-and-wife dynamics worked to slowly win over the neighborhood. The biggest breakthrough, though, came from the most unlikely source of all: Guatemala. A customer from the neighborhood came in one afternoon while Lopez was making salsa for a staff meal. He saw a pile of chopped jalapeños and asked Lopez to throw a few in with his _okonomiyaki_. Lopez tried to dissuade the man, told him that jalapeños are spicy and wouldn't match well with the _okonomiyaki_ , but the customer insisted. He loved it, and came back every day for weeks, ordering the same thing, until finally another customer saw the off-menu alteration and came along for the ride. Soon the spicy supplement became a Lopez staple, and he was forced to add it to the regular menu.
Today the jalapeño _okonomiyaki_ remains the most popular item at Okonomiyaki __Lopez, much to the owner's chagrin.
"Jalapeños don't belong in _okonomiyaki_."
米 麺 魚
I eat a lot of _okonomiyaki_ when I stay in Hiroshima, which is to say, I survive on _okonomiyaki_ alone for many days at a time. I eat it in tiny shops down tiny alleys without names on the door. I eat it in the famous places with long lines and dense clouds of savory steam fogging up the windows. I eat it in Okonomi-mura, a four-story building dedicated entirely to _okonomiyaki_ , with twenty-six vendors wilting their way through vast sierras of cabbage. (I'm reminded constantly during my time in Hiroshima that Okonomi-mura is the most popular food theme park in all of Japan.) I eat it with the salarymen at noon and the hustlers at midnight; I eat it with pork and beef, shrimp and scallops, oysters and squid.
There are over 2,000 _okonomiyaki_ shops in the greater Hiroshima area.
_(Matt Goulding)_
My main takeaway, from a strict culinary perspective, is this: if handled improperly, made in a hurry, or constructed from subpar ingredients, Hiroshima-style _okonomiyaki_ is little more than prosaic drinking food— _yakisoba_ made vertically instead of horizontally.
Made with care, constructed with a deft hand, put together with finesse and talent and a few shakes of soul, it is a glorious amalgamation, so vastly superior to Osaka's version as to not even warrant a comparison. But nothing about _okonomiyaki_ feels particularly Japanese—not the flavors, not the format, and certainly not the bulk. Which, ultimately, might explain its popularity: after a breakfast of _natto_ and a lunch of grilled mackerel and steamed rice, there's nothing like tucking into a 1,500-calorie disk of destiny to remind you of the primal joys of eating. (Unsurprisingly, it's a dish that wins the hearts and stomachs of Western visitors almost instantly.)
Friday lunch at Okonomiyaki Lopez is one of the busiest shifts of the week, a time for a final splurge for the professional set before the weekend begins. I've been watching Lopez make _okonomiyaki_ all week now, and occasionally I've grabbed a spatula and made a mess of his _teppan_ , but the pace of today's business allows little time for gaijin high jinks; I take up a chair at the end of the bar and watch the great feast unfold.
Two female pharmacists in lab coats are the first to arrive, followed by a pair of older salarymen with impeccable suits and polished briefcases. Then a mother and her young son. By 11:20 every seat is taken, and the _teppan_ crackles with the sound of sizzling pork belly and wilting cabbage.
The restaurant is small, even by _okonomiyaki_ standards, with a narrow prep kitchen, a sixteen-seat counter, and a U-shaped _teppan_ that stretches nearly the entire width of the shop. From appearances alone, it's tough to tell where the fantasy of Lopez Southwest ends and the reality of Okonomiyaki Lopez begins. The chairs are covered in poncho patterns, the shop logo shines bright with yellow, red, and green, and the menu contains a few tastes of a dream deferred, including a Guatemalan tongue stew and chicken fajitas, which sit warming in green and red enamel pots at the edge of the _teppan_.
It's clear a lot has changed since the years of goosing the neighborhood with free samples. Lopez looks comfortable behind his stainless steel perch, with a white flower-studded bandanna wrapped tight around his head and a denim Otafuku apron covering his chest that reads: "Eat _Okonomiyaki_ All Together a Happy Happy Home!!"
Lopez makes his _okonomiyaki_ with a mixture of repetitive precision and intense personal interest. As soon as a customer walks in the door, before she can even sit down, he drops a ladle of batter onto the griddle and begins to build. The precise layering of ingredients, the way he cups the cabbage between two spatulas, the little beads of water he splashes on the _teppan_ to take and adjust its temperature: all point to a man who knows that the difference between commodity and craft is razor thin.
Unlike many of the _okonomiyaki_ cooks I see around town, who look as if they changed their suits and ties for aprons and bandannas in a phone booth, Lopez works the griddle like a guy who has filleted a few fish, reduced a few sauces, ruined a few soufflés in his life. For someone with his rolling-stone résumé, you might think a single savory concoction would be a death sentence, but he exudes a deep sense of calm behind the sizzle and the steam.
"People ask if I ever get bored of making the same thing. Are you kidding me? They have no idea what goes on in my head just to make this one _okonomiyaki_."
To illustrate his point, Lopez gives me a primer on cabbage. Cabbage evolves throughout the course of the year, coming from different prefectures across Japan—from the wintry mountains of Nagano to the dry flats of Fukuoka—and as the seasons change, so too does the cabbage's behavior on the _teppan_. In spring, it wilts fast and burns quickly, in the fall it retains liquid and requires a longer, slower cook. "It took me a full year just to figure out how to manage my cabbage."
Multiply that by noodles, eggs, crepes, proteins, and the capricious nature of the griddle, and you begin to understand why he doesn't seem eager to add items to his menu or build more restaurants or do anything else besides make _okonomiyaki_ exactly where he's been making it for fifteen years. That might be the most Japanese thing about Lopez: his ability to accept tiny details like a vegetable's water content and griddle heat distribution as challenges worthy of a life's dedication.
Behind Lopez, tracking his every move, are two apprentices. Futoshi Mitsumura, thirty-one, left behind a moderately successful stint as a punk rock drummer in Tokyo to return to Hiroshima and learn to cook the soul food of his hometown. He's been here for one year, and still does most of his work behind the scenes, boiling noodles, chopping cabbage, refilling bottles of Otafuku sauce.
Hidenori Takemoto, thirty, could be the poster child for the salaryman convert, an uninspired mechanic at Toyota who found his true muse in the leafy layers of this Hiroshima specialty. "At Toyota, I did what I was told and there was no praise for a job well done. With _okonomiyaki_ , I get immediate response." He's been working behind Lopez for over a year now, and he shadows his master with quiet confidence, cracking eggs, flipping crepes, splashing noodles with helpful doses of hot water. He already has a space picked out for his restaurant, where he will bring Lopez-style _okonomiyaki_ , jalapeños and all, to the people of Shikoku.
The line of people waiting for a seat at the counter continues to grow, until a small group—a pair of parking attendants, a young guy with huge headphones and a bubble jacket—forms outside. Makiko suddenly appears, apron-clad and spatula-ready, and takes her place beside Lopez at the _teppan_. She still works the griddle but mostly during the restaurant's busiest moments (the Lopez family—husband and wife, two boys, in-laws—all live in a house attached to the restaurant). She shakes spices and fries eggs and efficiently begins to finish the _okonomiyaki_ her husband starts, then slides them across the griddle to waiting customers. _Okonomiyaki_ , in the best places, at least, is eaten with a _hera_ , a thin metal spatula, directly off the _teppan_ —a dish, as Lopez likes to say, that continues to evolve down to the last bite.
These days Okonomiyaki Lopez shows up on the top-ten lists of many local experts, including a perennial slot as one of Hiroshima's best _okonomiyaki_ shops on Tablelog, Japan's massive restaurant review website. But a certain contingent of Japan's food cognoscenti still have a hard time believing that _okonomiyaki_ could come from a Guatemalan. Lopez remembers a few years back when a local journalist wrote a book dedicated to Hiroshima-style _okonomiyaki_ and its many purveyors. He ate at Okonomiyaki Lopez a few times, and politely returned months later to give Lopez a finished copy of the book. Only, Lopez wasn't listed with the other _okonomiyaki_ shops; he was written up in the "Other" section. (Two of his students, however, had made the real shop list.)
"Some people say I've Westernized _okonomiyaki_ just because I'm Western," Lopez says, with the affectless delivery of someone who appears constitutionally incapable of getting worked up over anything, a walking Venn diagram of Latin American humility and Japanese restraint. He can talk openly about the most extraordinary things—an abusive father, a transcontinental romance, the challenges of being an immigrant in Japan—with the same shoulder shrugs and steady monotone he saves for discussions about vegetables. It's tough to say if this temperament came with his Japanese citizenship or if he's been carrying it around with him since he left Guatemala, but it plays well at the _teppan_. With the right set of eyes, you might even mistake Lopez for a local.
Today's customers look comfortable at Okonomiyaki Lopez. They drink beer and take pictures and talk up the man behind the griddle—a sharp contrast to the studied silence of many Japanese restaurants. He chats with old women and young couples as they place their orders, asking regulars about family members, telling stories about mutual friends.
"Out there in the streets of Hiroshima, you don't talk with people. You live in your own world," he says. "But here, you pull up a stool, watch the cooking, and you get to know your neighbor."
米 麺 魚
One afternoon, as I sit scraping my way through a Lopez jalapeño _okonomiyaki_ at the restaurant counter, an old woman takes a seat next to me and places a large to-go order. She looks surprised to see a foreigner in the restaurant and tells me as much in near-perfect English. We get to talking about the types of things strangers talk about until she, unprompted, tells me that she is a _hibakusha_ , a bomb survivor.
"I was two years old when it happened. We lived a kilometer and a half from the center. Some people survived the initial blast in this neighborhood, but the heat was so intense that it burned for three days and many eventually died. My three older brothers died in the rubble when our building collapsed. My mother and I were the only ones from my family to survive."
We both sit quietly, staring at the little waves of heat rising off the surface of the _teppan_. After a few minutes, she breaks the silence.
"I've spent my whole life thinking about how amazing it is that in the same apartment four people died and two people lived. Life is full of mysteries."
Is it possible to write about Hiroshima without writing about the splitting of atoms? Is it possible to walk its streets and visit its markets and eat in its restaurants without thinking about oblivion? I think about it constantly, wonder why I can't get past it, wonder if I lived here if I would ever get past it. Even as I type these words, I feel a current of guilt coursing through my digits, as if I owe it to the people of Hiroshima to leave it alone, to let them get on with living.
The earth this city is built on was ready to move on before its surface had cooled. They say that after the bomb dropped and nearly blasted Hiroshima out of existence, the grass and flowers grew back almost immediately. Not months or years after the bodies had been burned and the radiation dissipated; by August 12, 1945, just a week after the _Enola Gay_ gave birth to the nuclear age, the city was blanketed in green. "Weeds already hid the ashes, and wild flowers were in bloom among the city's bones," John Hersey wrote in _Hiroshima_ , his wrenching minute-by-minute account of the aftermath of the first atomic bomb. "The bomb had not only left the underground organs of plants intact; it had stimulated them."
A modern city was transposed onto the ruined one with remarkable speed: skyscrapers were erected, a new system of streets and avenues laid out, and a sprawling memorial dedicated to peace took shape along the water. When Emperor Hirohito came in 1947 to visit the orphans of Hiroshima, he didn't find a city mourning; he found a city rising.
"This was no beaten people who welcomed the Emperor to their city," Allen Raymond, a correspondent for the _Herald Tribune_ , wrote at the time. "I have seen most of the war-damaged sections of the world, and one could not find a healthier, stronger, more cheerful population anywhere than that of Hiroshima. The city is simply crawling with new life and energy." American correspondents were known for dispensing self-serving boosterism in the wake of the war, but so many I meet in Hiroshima tell me various versions of the same story: _We were looking forward, not backward._
_(Nathan Thornburgh)_
Every morning I walk from the city center to Okonomiyaki Lopez, crossing the wide boulevards designed by the Americans, cutting through generous parks, where local women hunt wild vegetables in the bushes, getting swallowed by the shadows of buildings electric with the energy of an animated workforce. Men slurp noodles; women sell shoes; kids ride bikes: Hiroshima is nothing but a city being a city.
But every night I walk home along the Motoyasu River, seven murky fingers that splinter the city into a small archipelago, and all I can see is the past. The looming mountains, where people fled that first morning to higher ground, away from the smoldering remains. The T-shaped Aioi Bridge, the original target for Little Boy, until the bomb drifted west and detonated above a hospital instead. The river, where survivors trapped in the center submerged themselves to escape the incendiary temperatures of the burning city. The river, where the skeleton of the Atomic Bomb Dome casts a pale light on the water, a spectral reminder of Hiroshima's haunted past.
Once known as the Industrial Promotional Hall, the building was located just 160 meters from where the bomb detonated. Everyone inside was killed instantly, but besides a few scars across its facade, the structure survived intact. Many people, frightened by the eerie bones of the building in the city center, wanted to see it flattened and forgotten, but the government elected to keep it, and now it shimmers across the water like an optical illusion—a reminder of either the senselessness or the resilience of man, depending on how you squint your eyes.
Every night I think: How can the walls possibly be so smooth? How can those windows be so square? How can that dome up top still be so round? I think: After all it's been through, how does it still have the strength to stand? It begins to follow me into my dreams, just one of the many ghosts that chase me around the city.
Those ghosts show up in the form of impromptu tales told at the Lopez counter. Most are stories of impossible survival. One old man, between bites of a squid _okonomiyaki_ , recounts to the entire restaurant how he walked behind a building just as the bomb blew, unknowingly saving himself from its incinerating temperatures. Another day, with a full counter of diners around us, Makiko tells me the story of her mother, ten years old and working in a factory, building plane parts for the war, who survived when two pieces of machinery collapsed onto each other, creating a protective A-frame above her tiny body. "My mom always says, 'No wonder we lost the war, we had little girls building the weapons.'"
I try to take all of this in, to think of something appropriate to say, but nothing comes out. Being American, with a grandfather who stormed the shores of Okinawa and whose cohort likely celebrated the news of the bombing, makes it only more complicated. The emotions swirl and take shape inside you, one after the next, a tarmac procession of loaded cargo waiting to take off: guilt, regret, rationalization, anger, acceptance, ambivalence. My internal chaos contrasts sharply with the extraordinary sense of calm transmitted by everyone I meet, especially the gentle _hibakusha_ at my side, sharing her story, patiently waiting for her dinner.
I can't help but try to connect the dots—the smiling old woman with the vanished family, the stone monuments to peace, the people who gather around this improbable postwar food—and when I do, this is all I can see: a city of origami artists taking the scraps they've been given and bending them into something beautiful.
A few minutes later Lopez hands her a bag stuffed full of food—three pork _okonomiyaki_ to go—and her face lights up like Christmas Eve.
"His _okonomiyaki_ is very good," she says, then shuffles off into the night with her bag of goodies.
I fight off a few tears and look up at Lopez. He shakes his head. "She always orders _okonomiyaki_ with udon. I can't get her to try it with soba."
米 麺 魚
Ever since its owner developed tendinitis in his shoulder back in 2008, Okonomiyaki Lopez has been closed on Saturdays, a reality that doesn't sit well with the parents-in-law. "In Japan, when you're young you're supposed to work hard all the time. My mother-in-law's friends in the neighborhood ask her why we take Saturdays off." He says this with the subtle grin of a man who long ago stopped worrying about the opinions of his in-laws.
Behind the smile, Lopez is nervous, pacing slowly in front of the shuttered shop. He has been meaning to drop in on one of his apprentices for months now, ever since he opened his shop behind Hiroshima Station. He sent flowers, of course, along with a bright Okonomiyaki Lopez shop sign, but today would be the first time tasting the student's work. To add to the pressure, Lopez has invited along Hiroki, his master, to help assess the quality of the Lopez school of _okonomiyaki_.
Hiroki picks us up in front of the shop in his van, and master and student embrace like old friends. "You look good," says Lopez. "I've been worried about you." The reunion is spoiled in part by a bit of troubling news Hiroki has just received: a former student of his suddenly died last week, and now Hiroki, as cosigner on the restaurant lease, is expected to inherit the shop. The bank delivered the news earlier this week.
Hiroki is seventy-one years old, and clearly in no shape to be running another man's _okonomiyaki_ shop. He has spent the past few years in a two-front battle against liver and colon cancer, and after three operations and rounds of chemo, his body is starting to give out on him. But his dedication to his students, he says, takes precedence. "The bank told me I either have to pay or go back to work. So I'm going back to work."
When the crowds descend, Lopez's wife, Makiko, joins him at the _teppan._
_(Matt Goulding)_
Hiroki was born in Nagasaki a year after the bomb and moved to Hiroshima in 1968. He was working as a bartender in the early 1970s when an _okonomiyaki_ place opened upstairs and the owner offered to train him. Ten years later he opened his own branch and began, little by little, to make the changes to the ingredients and techniques and cooking implements that have come to define one of Hiroshima's most famous and influential strains of _okonomiyaki_.
Since then, he's trained over fifty students in the art of Hassho-style _okonomiyaki_ , an open-door, open-book philosophy that runs counter to the guardedness you find in many corners of the culinary world. "I have no secrets. I want people to do well."
Okonomiyaki Masaru shares more than a few things in common with Okonomiyaki Lopez: the long U-shaped _teppan_ , the bright colors and Latin music, the crowds that descend upon the place as soon as the sun goes down. We arrive unannounced, and Hiraoka Masaru looks dumbstruck when he sees Lopez and Hiroki walk through the door. He greets us nervously, then retreats to the _teppan_ to tend to his cabbage.
Hiroki watches him work, quietly, carefully, throwing off tiny nods of approval as he analyzes the methodical construction of Masaru's _okonomiyaki_ : the oval shape of the crepe, the freshly boiled noodles, still dripping with water, the double-yolk eggs, the rising heat off the surface of the _teppan_.
Hiroki grows silent for quite some time, looks lost in the midst of the _teppan_ , as if he's staring into a lava lamp of his life. Is he thinking of the fifty young men who have chosen him as their guide? The foulmouthed kid from Hokkaido, the one who got rich fast in Tokyo, the Guatemalan who surprised everyone? Or is he thinking about the one who just slipped away, and the painful path ahead of him?
Masaru is not his student, which makes the familiarity of his moves all the more meaningful. Why the thick _teppan_? Why fresh noodles? That's the way master did it. Why two yolks? Why? Why? Because that's how master taught me—the simple answer to the most important food questions of Japan.
Three generations, three branches of an _okonomiyaki_ discipline responsible for feeding Hiroshima the food it craves. To Masaru's right, chopping cabbage, is a fourth branch, his own disciple, who will spread the gospel in some unknown direction. He'll call it not Hassho style or Lopez style but Masaru style to his customers and to his own students one day, yet the fountain of his inspiration is seated right next to me, cancer-riddled, hard of hearing, watching the little waves of his legacy ripple across Hiroshima.
"I haven't changed anything. This is exactly as Lopez-san taught me," says Masaru, wiping off a trail of sweat inching down his forehead. "My goal is to reach his level, to make it just like his. I'm not there yet, but my customers will tell me when I am." With this last part Lopez blushes just a bit. With this last part, Hiroki snaps out of his silence, mumbles his approval, and blushes a bit too.
Another order comes in, and Masaru rushes back to the other side of the _teppan_ and gets to work. He spackles the crepe with the back of the ladle, packs the cabbage lightly, lets the noodles dance across the hot surface, paints it with a generous stroke of Otafuku sauce. And when everything is ready, stacked high and bubbling, double yolk dripping down the side, he grabs a handful of jalapeños and scatters them over the _okonomiyaki_.
"It's our bestseller."
## _Food History_
## THE EVOLUTION
**600**
Rice arrives from China, beginning a long regional trade relationship in which the Chinese and Koreans export vital cultural cornerstones (tea, Buddhism, ceramics, various culinary staples) and the Japanese reward them with a mixture of respect and resentment.
**1543**
Portuguese sailors shipwreck off the coast of Kyushu, bringing with them the blueprints for tempura and Christianity. The former is widely embraced; the latter is eventually banned and its practitioners summarily executed by the powerful ruler Toyotomi Hideyoshi.
**1873**
Emperor Meiji is seen publicly consuming beef, thus ending a 1,200-year ban on meat consumption (just one of Buddhism's many marks on Japanese cuisine). The rise of Westernized and meat-centric cuisine—yakitori, _tonkatsu_ , _yakiniku_ , and ramen—soon follows.
**1945**
Americans begin a seven-year occupation of Japan. They bring with them boatloads of surplus wheat, convincing a starving nation of its nutritional superiority. (In return for cheap wheat, Japan agrees to purchase American arms.) Ramen, udon, and _okonomiyaki_ culture flourish.
**1975**
Akira Okazaki, a Japan Airlines executive, successfully air-delivers bluefin tuna from Nova Scotia to Tokyo, ensuring decades of Japanese sushi superiority but at a steep cost to the world's ocean life. Today's fish markets in Japan are an edible atlas of the twenty-first century.
**2011**
Bread consumption surpasses rice consumption for the first time in Japanese history. While traditionalists lament the rise of wheat, Japanese cooks continue to one-up the world in the art of pizza, pastry, and baking as the national waistline inches ever so slightly outward.
## _A Beacon in the Night_
## **THE 8 WONDERS OF THE JAPANESE CONVENIENCE STORE**
Located on every block in urban areas (and every other block in rural ones), the Japanese convenience store is much more than a ubiquitous repository of junk food and cheap buzzes. It sells sushi and soba, manga and medicine, single-malt whisky and next-day hangover cures. Many Japanese swear allegiance to one of the Big Three _conbini_ —7-Eleven, Lawson, or Family Mart—but all share a common ethos of maximum utility, minimal hassle, and food that's better than it needs to be. There are many things to love about _conbini_ (and a few things not to), but these are the most heroic features of the Japanese convenience store.
_(Michael Magers, lead photographer)_
_(Matt Goulding)_
**ONIGIRI**
One of Japan's most popular snacks looms large on the shelves of _conbini_ —endless triangles of packed rice wrapped in shiny sheaths of crackly seaweed. Try it stuffed with _umeboshi_ (pickled plum) or tuna and mayo.
_(Matt Goulding)_
**KARAAGE**
Fried food has a strong presence in _conbini_ , but chicken—spicy nuggets, patties, thighs, and drumsticks—is the standout. Lawson has a deservedly strong reputation for its _karaage_ : salty, unreasonably juicy, and as delicious cold as it is hot.
_(Matt Goulding)_
**ODEN**
Come winter, _oden_ dominates the _conbini_ landscape: vegetables, meat, tofu, and eggs simmered gently in dashi. The Japanese go crazy for this stuff, and when you feel the chill in your bones, you will too.
_(Matt Goulding)_
**YOGASHI**
Pillow soft and lightly sweetened, _yogashi_ (Western-style desserts) make for a heroic breakfast or late-night binge (try anything made with green tea). Family Mart's line of high-concept pastries is especially impressive.
_(Matt Goulding)_
**ICED COFFEE**
Nearly as ubiquitous as vending machine coffee, and marginally better. It tends to be super sweet, so best to look for ones with "double" or "espresso" in the name, or custom blend your own hot or cold caffeine fix with the slick coffee machines found at all the big _conbini_ these days.
_(Matt Goulding)_
**BOOZE**
The place to stock up for a street beverage or a hotel stash. Dedicated sake sections, sprawling beer cases, wine, and whisky give the informed drinker a formidable lot to select from. _Chu-his_ and pocket Suntory bottles are two standouts.
_(Matt Goulding)_
**SANDOS**
The math doesn't work out—squishy bread, industrial fillings—but what emerges out of those plastic wrappers is glorious. Egg sandwiches from 7-Eleven and Lawson are little miracles of creamy golden yolks and umami-rich kewpie mayonnaise.
_(Matt Goulding)_
**EVERYTHING ELSE**
The bathrooms are sparkling by U.S. convenience-store standards, the employees are comically cheery, and 7-Elevens remain one of the only places where foreign ATM cards work. You can also pay bills and buy plane and concert tickets while you snack on your egg _sando_.
## _Agemono_
## [揚げ物
**DEEP FRIED**](nav.xhtml#rhh16)
_(Matt Goulding)_
### ELEVATING THE ART OF FRYING
The Japanese may boast the longest life spans on earth, but people here love grease as much as the rest of the world. From convenience-store _korokke_ to Michelin-starred tempura temples, nobody fries better than the Japanese.
_(Matt Goulding)_
### **KARAAGE**
**Chicken thighs marinated in soy, garlic, and ginger, then floured and fried. Also made with shrimp, octopus, and other sea creatures.**
_(Matt Goulding)_
### **KOROKKE**
**Filled with everything from mashed potatoes and mincemeat to curry and cream of crab. Like a Spanish croquette but executed with Japanese precision.**
_(Michael Magers, lead photographer)_
### **KUSHIKATSU**
**Fried meat on a stick eaten elbow to elbow at a bar and washed down with rivers of cold beer: What's not to love? Osaka invented the form, but you'll find it everywhere.**
_(Matt Goulding)_
### **TONKATSU**
**Panko-breaded pork loins fried to a greaseless crisp, served with hot mustard, sweet Worcestershire, steamed rice, and shredded cabbage. The best is made with _kurobuta_ (black foot) pork.**
_(Matt Goulding)_
### **TEMPURA**
**_Shokunin_ dedicate entire lives to tempura, turning battering and frying into a high art form. For the full experience, go to a tempura-only restaurant and order the _omakase_ —the chef's tasting menu.**
### THE POWER OF PANKO
Japanese chefs use panko bread crumbs—large, flat flakes that create a shattering, greaseless crust—on _tonkatsu_ , _korokke_ , and other golden-brown gems.
_(Michael Magers, lead photographer)_
### **DEEP-FRIED DEPACHIKA**
**Japanese department stores (called _depachika_ )—wondrous centers of gastronomic greatness—trade in the entire spectrum of fried specialties. A fine place for _korokke_ , _katsu_ , or tempura. (Be on high alert for free samples of each.) **
## _Chapter Six_
## HOKKAIDO
I wake up on top of the sheets of my cheap hotel bed, fully clothed, smelling of whisky and lamb. Not lamb, actually, but grilled mutton, possibly a few days or weeks past its prime. I struggle to bring the details of last night into focus. If I squint hard enough, I see a pocket bottle of Suntory, an old woman with a pile of raw onions, a smoky bar with karaoke and cheap wine.
But then I find this e-mail, sent to me sometime during the last night's stupor:
_Dear Matt,_
_I have arranged for the complimentary tickets on the SL (Steam Locomotive) Niseko, which travels from Sapporo to Niseko on the weekends during autumn only. This is designed to let people enjoy the nostalgia of travel from days gone by, enhanced with dramatic scenery and with a variety of different specialty products available in the dining car as you go through the different regions en route to Niseko._
The e-mail is from Paul Haggart, the sole representative of Niseko Tourism, who insists that I need to come to his tiny mountain community to appreciate the full pastoral majesty of Hokkaido. Paul informs me that the tickets are on hold at the information counter at JR Sapporo Station, arranged for by one Mr. Yoshitaka Ito from JR Plaza in Tokyo. The train people will be expecting me.
In the harsh glare of the Hokkaido morning light—not to mention the throbbing weight of an all-world hangover—all of this sounds like too much effort. But I have made promises, and nowhere are broken promises more perilous than in Japan, so I roll out of bed, stuff my clothes into my suitcase, and wobble my way toward the station. After days of grilled mutton and bad decisions in Sapporo, maybe a bit of mountain air will do me good.
The SL Niseko is a reptile of a train, muscular and elegant, black as a starless night, spewing thick plumes of smoke from her nose—a seasonal beast ready to slither her way through the jumbled topography of this island. A conductor in a throwback uniform stands guard at the front, his posture so stiff it could slice a soft tomato. All around, people snap photos and film videos and generally lose their shit over the old-world elegance and enduring mechanical mastery of the steam locomotive Niseko.
I fight past the crowds, who look genuinely confused and disappointed that this disheveled gaijin has in his greasy, lamb-stained palm the golden ticket. Inside, the train cars sparkle with the "nostalgia of travel from days gone by." In true Japanese fashion, the interior looks to be lifted directly from 1856, with all the tiny details Hollywood-ready: the polished oak paneling, the meticulous ironwork, the authentically uncomfortable wooden seats.
The train pulls out of Sapporo with a few proud whistles and winds its way southeast along the Sea of Japan. It's barely 8:00 a.m., but my train mates waste little time in breaking out the picnic material. But this isn't standard Japanese picnic fare: not a grain of rice or a pickled plum in sight. Instead, they fill the varnished wooden tables with thick slices of crusty bread, wedges of weeping cheese, batons of hard salamis, and slices of cured ham. To drink, bottles of local white wine, covered in condensation, and high-alcohol microbrews rich in hops and local iconography.
From the coastline we begin our slow, dramatic ascent into the mountains of Hokkaido. The colors bleed from broccoli to banana to butternut to beet as we climb, inching ever closer to the heart of autumn. My neighbors, an increasingly jovial group of thirtysomethings with a few words of English to spare, pass me a glass of wine and a plate of cheese, and I begin to feel the fog dissipate.
We stop at a small train station in the foothills outside of Ginzan, and my entire car suddenly empties. A husband-and-wife team has set up a small stand on the train platform, selling warm apple hand pies made with layers of flaky pastry and apples from their orchard just outside of town. I buy one, take a bite, then immediately buy three more.
Back on the train, young uniformed women flood the cars with samples of Hokkaido ice cream. The group behind me breaks out in song, a ballad, I'm later told, dedicated to the beauty of the season. Everywhere we go, from the golden fields of empty cornstalks to the dense forest thickets to the rushing rivers that carve up this land like the fat of a Wagyu steak, groups of camouflaged photographers lie in wait, tripods and shutter releases ready, hoping to capture the perfect photo of the SL Niseko steaming its way through the hills of Hokkaido.
As I sit there, sipping my wine and snacking on cheese, soaking up the cornucopia of autumn views and the bonhomie of my train mates, one troubling question bounces around in my brain: When did I leave Japan?
米 麺 魚
Hokkaido is roughly the size and shape of Maine, a land of towering mountains, lush valleys, and rugged, lonely coastlines. Imagine Switzerland, if Switzerland were an island in the Sea of Japan instead of a landlocked country in Europe. Separated from Honshu by the Tsugaru Strait, Hokkaido is large and sparsely populated, making up 25 percent of Japan's landmass but just 5 percent of its population. Host to the 1972 Winter Olympics, the island is known to outsiders primarily as a place to ski, its prodigious snowfall legendary as some of the world's lightest and driest powder.
Locals call Mount Yotei "Hokkaido's Fuji," for obvious reasons.
_(Matt Goulding)_
I did not come to ski. I first came to Hokkaido for two reasons: miso ramen and _uni_ , the island's most famous foods and two items on my short list for Last Supper constituents. The only thing they share in common, besides a home, is the intense fits of joy they deliver: the former made from an unholy mix of pork-bone broth, thick miso paste, and wok-crisped pork belly (with the optional addition of a slab of melting Hokkaido butter), the latter arguably the sexiest food on earth, yolk-orange tongues of raw sea urchin roe with a habit-forming blend of fat and umami, sweetness and brine. Fall for _uni_ at your own peril; like heroin and high-stakes poker, it's an expensive addiction that's tough to kick.
But my dead-simple plan—to binge on both and catch the first flight back to Tokyo—has been upended by a steam locomotive and Whole Foods foliage, and suddenly Hokkaido seems much bigger than an urchin and a bowl of soup. No one told me about the rolling farmlands, the Fuji-like volcanoes, the stunning national parks, one stacked on top of another. Nobody said there would be wine. And cheese. And bread.
Few understand my sudden itch for exploration better than Ioanna Watanabe. Ioanna came to Niseko in 2004 with plans to spend a few days snowboarding, a few more drinking and eating, before continuing her tour of the Far East. Only she fell in love with the island and its underappreciated virtues, including Hisashi Watanabe, a young Japanese man from Saitama working the ski patrol in the backcountry, and never left.
Today she and Hisashi own one of Hokkaido's hippest cocktail dispensaries, Gyu Bar, a low-lit drinking cave in Hirafu at the foot of the area's biggest ski resort. The two make a formidable team: Ioanna the resident whisky expert, Hisashi the dapper suspender-clad cocktail king. For four months of the year, Gyu Bar and every other establishment within sniffing distance of a ski slope hums with packs of Aussie boarders and Hong Kong powderhounds and the occasional Tokyoite.
But when the snow goes, so do most of the people, which is exactly why I'm here now: to focus on what really counts without the distractions of the winter-clad hordes. I meet Ioanna by chance at a wine shop in Niseko shortly after the locomotive delivers me to the mountains. When she hears about my SL Niseko revelations, she offers to take me around to experience what she calls "the mind-blowing Hokkaido."
Yes, you can come to Sapporo, drink the namesake beer and slurp ramen and enjoy one of Japan's largest and friskiest entertainment districts, take the train to Otaru for a quick _uni_ feast, then head back to Honshu, but to truly experience Hokkaido, to understand what this island is all about, you'll need to venture out beyond the handful of urban pockets and into the wild. Do that for a few days and you'll realize that, more than anything, Hokkaido is a collection of amazing shit in the middle of nowhere.
This is the kind of place where you buy your eggs on the honor system from a friend's mailbox, where supermarkets sell produce with the face of the farmer on the package so you know exactly who grew your daikon, where your neighbor raises ostriches because he spent his honeymoon in Australia and thought they looked cool and, fuck it, why not?
We spend a week crisscrossing the southern part of Hokkaido in Ioanna's well-worn Honda CRV, eating and drinking in a way that upends my understanding of Japanese food culture. Ioanna is Canadian by birth but deep down as Japanese as fermented soybeans, able to understand and decode both sides of the cultural divide with preternatural ease and grace. I learn many things from Ioanna during our time together: when and how to bow in a variety of social scenarios, the exact combination of sounds to offer up after a delicious meal, the virtues of convenience-store fried chicken.
Ten minutes outside of Hirafu, we find Del Sole, a small cabin tucked into the woods with a world-class pizza operation inside. Kenji Tsugimoto, the owner, built brick by brick the oven that ejects puffy-rimmed, blistered-bottom pies that could rival the finest pizzas of Naples. He serves just five tables at lunch and five more at dinner. "Any more, and I wouldn't be able to make the pizza I want to make."
Signs of Hokkaido's muscular dairy industry tattoo the terrain everywhere: packs of Holsteins chew cud unblinkingly in the sunlight, ice cream shops proffer hyperseason flavors to hungry leaf gazers, and giant silos offer advice to the calcium deficient: "Drink Hokkaido Milk!" Even better than drinking the island's milk is drinking its yogurt, which you can do at Milk Kobo, a converted red barn with cows and tractors and generous views of Mount Yotei, which locals call Ezo Fuji. Kobo sells all manner of dairy products, but you're here for the drinkable yogurt, which has a light current of sweetness and a deep lactic tang, a product so good that the second it hits my lips, I give up water for the week.
The Nikka distillery, one of Japan's oldest and largest whisky makers, rises out of the coastal flats of Yoichi like a high-proof oasis for thirsty island itinerants. Inside, the fires of distillation burn red-hot: like the great SL Niseko, Nikka still runs on coal. Whisky is Ioanna's wheelhouse, and she peppers the self-guided tour with fun facts about the virtues of barrel-aging and the vision of Nikka founder Masataka Taketsuru. In 1918 he traveled to Scotland to learn the secrets of brown liquor from its oldest and wisest practitioners. He returned to Japan two years later with a Scottish wife and a blueprint that would form the basis of Japan's entire whisky industry. He chose Hokkaido as his home base because it was the place that reminded him most of Scotland.
My favorite of these far-flung places, though, is a few miles from the Niseko train station, housed in a steep brown A-frame that looks more like an Austrian ski chalet than a soba shrine. Tatsuru Rai first came to Hokkaido in 1962, when as a high school freshman he rode his bike all the way from Tokyo. He fell in love with the island's rural charms, and four years later, after saving up enough money, he returned, making the thousand-kilometer trip on foot this time. He worked in a hotel at first but wanted to open his own restaurant. There was no soba in the area at the time, despite the abundance of buckwheat grown in Hokkaido, so he rolled up his sleeves and got to work.
Tatsuru built Raku-ichi himself, fashioning a twelve-seat hinoki bar into a quiet viewing area for the performance that unfolds in the kitchen. He makes every order of soba by hand, working in small batches so that by the time you've eaten, you'll have witnessed the extraordinary transformation of grain and water into noodle. It takes him eight minutes from start to finish, a process so lovely and intimate that you blush every time he looks up from his work area.
He starts with 100 percent local buckwheat—a grain stubborn enough that most soba masters cut their dough with wheat flour to make it easier to work with. Once the water is added and the dough shaped into a smooth, seamless ball, he works it with a wooden dowel, using his forearms and his palms to make the mass thinner and thinner. With each pass of the dowel, he pats the dough with his right hand, a quick, seamless motion that acts as a metronome for the elaborate rolling process. The thud of the dowel, the slap of the hand, the rustle of the buckwheat against the board: it starts soft, grows louder and faster, like the building of a great jazz performance. He rolls, slaps, rotates, rolls, slaps, rotates, rolls, slaps, rotates—over and over until the crude circle is shaped into a sharp rectangle. With a twelve-inch soba blade and a wooden board to guide him, he transforms the rectangle into thousands of dark brown strands. No wasted motion, no alien movements, not a scrap of dough lost to inexactitude or impatience.
Tatsuru Rai turns buckwheat and water into performance art.
_(Michael Magers, lead photographer)_
Nobody talks, as if too much breath might break the magical bond of buckwheat and water. (When Tatsuru traveled to Copenhagen to make his noodles in front of a crowd of food-industry luminaries, three hundred of the world's greatest chefs sat slack-jawed in silence as he did nothing more or less than what he does ten times a day in his tiny Hokkaido restaurant.)
The noodles are served by Tatsuru's wife, Midori, a lovely, soft-spoken hostess who wraps herself in gorgeous, expensive kimonos. The soba comes two ways: _seiro_ , afloat in a dark, hot dashi spiked with slices of duck breast, or _kake_ , cold and naked, to be dipped into a concentrated version of that same broth. Even if it's -50˚F outside and you've lost all sensation in your toes, eat these noodles cold, the elegant chew and earthy taste of the buckwheat uncompromised by the heat of the dashi.
"The process is everything," Tatsuru says, in what could be a four-word definition of Japan.
The young man next to me, a spiky-haired pop star from Sapporo, nods his head in agreement. "Once you eat here, it's hard to go back," he says, in what could be a nine-word definition of Hokkaido.
米 麺 魚
The story of Hokkaido is not a lovely one. It is a history of neglect and repression, displacement and discrimination, outcasts and vagabonds. Some have likened Hokkaido to the Wild West, and the parallels are easy enough to draw: the government malfeasance, the band of misfits and clansmen that came here to operate outside of the law, the world of shit forced upon the native population.
For most of its written history, Hokkaido was known as Ezo, an island occupied by the Ainu, believed to be descendants of the ancient Jomon people, with a nomadic streak and a deep dedication to their spirituality. The Ainu had little contact with the Japanese until 1590, when Hideyoshi Toyotomi granted the Matsumae clan, a group of roaming samurai that settled in southern Hokkaido, exclusive trading rights with the "barbarians from the north."
The Ainu had things the rest of Japan wanted—fish, seaweed, furs—and in turn they took what their home couldn't provide: rice, sake, and tools. But the Matsumae clan did more than just trade with the Ainu: they restricted their movements within their own lands, prohibited them from trading with outside groups, and enforced their exclusive relationship with brutal force, gutting the indigenous culture and killing Ainu leaders over minor disputes.
Even with the increased trading between the Ainu and the Japanese, Ezo remained a land apart, one not formally recognized by Japan until the Meiji Restoration was in full swing. In 1869 the new imperial government christened the island Hokkaido and began to actively encourage settlement, primarily as a buffer against Russia, which was quickly encroaching on Japanese territory from the north.
As Hokkaido became more important to the Japanese government, so too did suppressing the Ainu, whose culture they viewed as a threat to Honshu homogeneity. Ainu language was banned, religious practices snuffed out, and the people themselves forcibly assimilated into the Japanese way of life. The Ainu survived in pockets scattered around southern Hokkaido, but their home was no longer theirs alone. (Only in 2008 did the Japanese government formally recognize the Ainu as "an indigenous people with a distinct language, religion and culture." Around 25,000 Ainu live in Hokkaido today, using a mixture of tourism income and government funds to restore many of the traditions and practices they suffered the loss of over the years.)
Like the pack of thieves and scoundrels that protect the Wall in _Game of_ _Thrones_ , the earliest Japanese settlers were people from the margins of society: ex-criminals, forgotten sons, failed families. In the north they saw a chance to trade in their messy pasts for clean canvases. And the new Hokkaido government, for its part, was all too happy to provide them that opportunity.
After World War II, many of the Japanese who had occupied Manchuria repatriated to Hokkaido, adding to the motley mix of new faces looking for a fresh start in Japan's northern reaches. In 1971 the Japanese government decided it was time to finally connect Hokkaido to the rest of the country, and they began construction on an ambitious tunnel project that would reshape the island forever.
The Seikan Tunnel is the world's deepest and longest tunnel, an under-water expanse that takes twenty-two minutes traveling at 140 kilometers an hour to pass through. At the other end of the abyss is Hakodate, the gateway to Hokkaido and, for many years, to the rest of Japan. Hakodate was one of two ports to open to the outside world after Commodore Matthew Perry forced Japan to end its closed-door policy in 1854, a first stop for American and Russian ships winding their way down the country. It was once the most important city in Hokkaido—before the rise of Sapporo, before the Great Hakodate Fire of 1934—and signs of its former greatness still linger around town: the generous port and its polished warehouses, the cable cars that climb past the brick Orthodox churches in the hillside Motomachi district, the five-pointed star of Goryokaku, the European-style fort at the southern end of the city. From atop Mount Hakodate at night, you can take in the sparkle of Hakodate's hourglass body, and the bright lights of the squid boats bobbing in the water below.
The clearest signs of Hakodate's current greatness, though, can be found clustered around its central train station, in the morning market, where blocks and blocks of pristine seafood explode onto the sidewalks like an edible aquarium, showcasing the might of the Japanese fishing industry.
Hokkaido is ground zero for the world's high-end sushi culture. The cold waters off the island have long been home to Japan's A-list of seafood: hairy crab, salmon, scallops, squid, and, of course, _uni_. The word "Hokkaido" attached to any of these creatures commands a premium at market, one that the finest sushi chefs around the world are all too happy to pay.
Most of the Hokkaido haul is shipped off to the Tsukiji market in Tokyo, where it's auctioned and scattered piece by piece around Japan and the big cities of the world. But the island keeps a small portion of the good stuff for itself, most of which seems to be concentrated in a two-hundred-meter stretch in Hakodate.
Everything here glistens with that sparkly sea essence, and nearly everything is meant to be consumed in the moment. Live sea urchins, piled high in hillocks of purple spikes, are split with scissors and scraped out raw with chopsticks. Scallops are blowtorched in their shells until their edges char and their sweet liquor concentrates. Somewhere, surely, a young fishmonger will spoon salmon roe directly into your mouth for the right price.
This is Japan, after all, where freshness cannot be faked because everybody knows the difference between yesterday's scallop and today's. But sometimes, in this quest for deliciousness, lines are crossed. In the center of the morning market sits a giant tank of live squid and a handful of fishing poles. I pay my 500 yen and drop a line in. A group of Chinese tourists surround the tank, cheering me on in Mandarin as I try my best to hook one of the squirmy cephalopods. When I finally pull a squid out of the tank, it blasts a jet stream of water onto the crowd, which drives them wild. The squid is air-dropped immediately onto a cutting board where a man with a long blade and a stern face turns the dancing creature into a plate of sashimi before the muscles have a chance to stop wriggling. The body is sweet and supple, but the legs, still busily in search of their final resting state, don't go down without a fight.
The many wonders of Hokkaido's waters on display in Hakodate's morning market
_(Michael Magers, lead photographer)_
Like so much in Japan, it's equal parts cute, impressive, and unsettling. There's a reason that markets like these aren't frequented by locals; they prefer their squid without a crowd of wealthy Shanghainese urging them on. The real game, as I soon discover, is _donburi_. _Donburi_ , often shortened to _don_ , means "bowl," and the name encapsulates a vast array of rice bowls topped with delicious stuff: _oyakodon_ (chicken and egg), _unadon_ (grilled eel), _tendon_ (tempura). As nice as meat and tempura and eel can be, the _donburi_ of yours and mine and every sensible person's dreams is topped with a rainbow bounty of raw fish. Warm rice, cool fish, a dab of wasabi, a splash of soy—sushi, without the pageantry and without the price tag.
At Kikuyo Shokudo Honten you will find more than three dozen varieties of seafood _don_ s, including a kaleidoscopic combination of _uni_ , salmon, _ikura_ (salmon roe), quail eggs, and avocado. I opt for what I've come to call the Hokkaido Superhero's Special: scallops, salmon roe, hairy crab, and _uni_. It's ridiculous hyperbole to call a simple plate of food life changing, but as the tiny briny eggs pop and the sweet scallops dissolve and the _uni_ melts like ocean Velveeta, I feel some tectonic shift taking place just below my surface.
Over the next few days, I eat nothing but _donburi_. At 7:00 a.m., when the sun still sleeps with the fishes. At 2:00 p.m., as the local workforce is mustering up the strength to see the day through. At 11:00 p.m., with the staff looking on nervously, trying to determine if I might finally be full. If I had to travel to just one part of Japan to eat one type of food, it would be seafood _donburi_ in Hakodate. Truth.
If _uni_ is your objective, you can do no better than Uniya Murakami, a fifth-generation family business with unparalleled dedication to the noble urchin, which it serves in dozens of guises: lightly cured in soy sauce, folded into the soft curds of an omelet, clinging to udon noodles like a Far Eastern carbonara. All of this, of course, is a distraction from what really counts: two dozen tongues of _uni_ , an umbrella of orange with a green wasabi top, draped over warm rice, the _donburi_ to end all others.
If there is anywhere more famous for _uni_ than Hakodate, it's Otaru, a small, postcard-pretty harbor town on the west coast of Hokkaido, thirty minutes by train from Sapporo. They say the waters were once so rich in Otaru that you could catch fish with your bare hands. It was a wealthy town, the wealthiest in all of Hokkaido, built on the back of a gangbuster _nishin_ industry—tiny herring fished in abundance and processed into fertilizer. Herring mansions, fancy nineteenth-century processing centers that doubled as residences for their wealthy owners, still dot the hillsides around Otaru, but it's been many years since they've seen any action.
A picturesque canal cuts through the center of town, and on either side you'll find dozens of sprawling sushi venues offering more or less the same set 2,000-yen menu to the packs of day-trippers. But beyond the rows of restaurants, past the covered shopping arcade, down a back alley of tiny wooden huts, Sushiya Ko-Dai stands as a firm rebuke to the cookie-cutter sushi culture that dominates so much of Japan.
Technically it's a _yatai_ , a street stall, but it could be mistaken for a closet or a can of sardines. People stand at the bar, pressed against each other, pointing through the glass of the most jumbled fish case you'll ever find in Japan. Presiding over this lovely mess is twenty-eight-year-old Sanada Kodai, a warm, talkative host with a perma-smile and a penchant for self-deprecation.
"I always wanted to be a hairstylist," says Kodai, running his hand over his cue-ball scalp and laughing. "But then I thought, which job would be cooler when I'm older? I figured cutting fish would be cooler than cutting hair, so here I am."
As he slices and sculpts and passes each piece across the case to a customer, the chatter never stops. "I wanted to open a fun place, an alternative to conveyor sushi for young people. The best for me is when the counter is full of doctors and a high school student walks in and starts ordering."
Just then a group of three young Tokyoites open the sliding glass door and pull back the curtain.
"We tried to reserve," one of them says, seeing the tight space.
"We don't do reservations," says Kodai.
"Well, we're here now."
"Great, but you'll have to wait."
The place may be tiny and the mood relaxed, but the sushi itself is serious stuff. I put myself in Kodai's hands and he walks me, piece by piece, through the greatest of Hokkaido's bounty: mackerel, marinated in soy for twenty minutes ("In Tokyo they have to marinate their _saba_ for three hours"); salmon, streaked with huge deposits of fat, better for keeping the fish warm in these cold waters; a slice of scallop so tender that it seems to vanish before I have time to chew; and a generous pile of hairy crab crowning a warm, loose mound of rice, the kind of genre-defining bite that follows you places.
The Tokyo crew, who finally find a space at the counter, are visibly moved by the experience. "I wish we had a place like this back home," one of them tells me. Kodai beams like a lighthouse.
We finish with a fat piece of _uni_ that trembles like flan, so soft and sweet it could double as dessert. It's a powerful ending to one of the best sushi experiences I've had in Japan, and it cost a fifth of what the big places in Tokyo run.
"Look, I have no staff and I work in this tiny space. That's why I can afford to use the best products."
"So is most of this fish from Otaru?" I ask.
"No. Not exactly. Things are complicated here."
I press him on the complicated part.
"Tomorrow I'll take you to see the fishermen. You'll see."
Today's fishermen live in considerably more humble settings than the herring hunters of Otaru past. Most are clustered in a series of huts and small wooden houses just north of the town center. I meet Kodai there at dawn, and we knock gently on the door of what looks like someone's garage.
An imposing figure in a dark V-neck sweater answers the door. "What do you want? Don't you know I'm famous?"
I struggle to find an appropriate response. "Why are you famous?"
"Because I'm crazy."
"What do you mean crazy? Crazy at night?"
"No, at night I'm a gentleman. I'm crazy on the water—the craziest guy on the water. Today's the only day I won't go out. I sent my sons instead."
Masao-san looks like a villain from a Jean-Claude Van Damme movie: brick-house build, facial scars, handsome in a slightly menacing way. He is the unofficial leader of the fishermen of Otaru, a pack of seventy-five or so men whose families have worked these waters for generations. Masao's place, a messy shed with a few motorboats parked out back, feels more like a safe house than a fish shack, a place where he and his posse of bandits can lie low while the heat dies down.
Masao moves slowly, except when he's smoking, which is always. "I'm sorry I don't have much to offer you," he says. He reaches into a freezer and pulls out a shrink-wrapped octopus tentacle the size of a small human arm. He tears it open and cuts the tentacle into thick coins with a pocketknife, smears a wad of wasabi on a small plate, pours soy sauce onto another one, and puts it all over a stack of old newspapers.
"Breakfast is served."
Behind Masao sits a giant cooler, which I assume is stuffed full of fish, but when he opens it the only thing inside is Boss coffee—hundreds of tiny black cans emblazoned with the pipe-smoking Boss man (played by Tommy Lee Jones in real life).
Hokkaido's seafood remains the finest in Japan, but overfishing threatens its future.
_(Matt Goulding)_
As we sit chewing on frozen octopus and sipping canned coffee and bathing in cigarette smoke, Masao explains that this is the most important hour of the day, when the fishermen come back to base with their catch. Historically, Otaru has yielded a rich and diverse catch that evolves throughout the year—salmon in the summer, herring in the fall, octopus and _uni_ in the spring—but in recent years, they've been lucky to catch enough to live on.
"Every year there's less coming out. Even when I was young the herring supply was way down. This year we're seeing a third of what we had last year."
While we eat, a neighbor fisherman—short, with long hair and a rubber apron—walks in. "Zero. Zero, zero," he says, opening the cooler and grabbing a Boss. "I got enough for dinner, but that's it."
"The summers are hotter, and it's impacted our fish supplies," says Masao. "There aren't enough good bacteria. Less kombu, less places for fish to lay eggs. The balance is off."
There are many things to admire about Japanese food culture, but resource management isn't one of them. It's no secret that the Japanese are voracious consumers of life aquatic, eating fifty-five kilograms for every man, woman, and child—more than three times the global average. In the wake of World War II, with protein sources scarce, it was a matter of national policy to catch as much fish as possible, which has left fishermen with little to catch these days.
Cries from conservationists looking to impose some level of sustainability on Japan's fish habits have largely been ignored by all, including consumers. At the heart of these complicated issues—the whale hunting, the tuna desolation, the systematic emptying of the seas—is a simple argument that stops all other arguments in their tracks: it's our tradition. It's true, Japan for millennia has been a nation that survives largely off seafood, but a dense population combined with the rise of convenience-store and conveyor sushi has stretched its dining habits to the limits.
It's a loaded debate, a cultural minefield for a foreigner, but one can't help but get the sense that if the Japanese preserved ecosystems as carefully as they preserve tradition, the future of the fishing industry might not look so grim. Masao seems stranded somewhere between the two sides: he respects the tradition and is desperate to make a living, but he sees the need to adapt to the limitations of today.
Another fisherman enters and dramatically drops two live shrimp on the table. "Today's catch, your majesty." Masao lights another cigarette.
"We overfished. We should have made changes earlier. The older people would just take whatever they could take. The ones without sons were the worst. And now we're paying for it."
On cue, Masao's younger son walks in empty-handed. "I had one, it was right there, but it got away." He grabs a Boss and lights a cigarette.
Finally Masao's older son—heavyset, with dark red hair and a patchy beard—enters through the back door. He's carrying a plastic sack, which he opens and shakes in front of us. An orange octopus with two-foot-long tentacles drops to the floor and slithers its way across the cement.
"I was afraid of what Dad might do if I came back empty-handed." He gives the octopus a good kick, then looks up and notices me for the first time. "Who brought the gaijin here?" We all have a good laugh.
Soon the entire room is smoking and eating octopus and drinking cans of Boss coffee.
"Are there fishermen in America?" the older brother asks.
"Yeah," the younger one responds. "They look really cool."
A discussion about king crab ensues, and I explain that my brother used to work on the Alaskan crab boats and made some really good coin doing it. Suddenly the fishermen are ready for a relocation.
"If we're all going to America, I'm coming too," says the younger brother.
"You should all get insurance," says Masao, lighting another cigarette. "I'll stay here and collect when you die."
米 麺 魚
In the early years of the Meiji Restoration, the new imperial government of Japan implemented a series of measures that would forever change the country and its culture. Closed off to the outside world for 180 years, the leaders of new Japan looked to modernize the country overnight, and to do so called upon foreign experts to help bring the country up to speed.
And so it was that William S. Clark, a Massachusetts-born son of a country doctor, found himself in Sapporo, charged with establishing Hokkaido's first agricultural college. Clark was a powerful academic, with a doctorate in mineralogy from Germany and a high-ranking post at Amherst as a professor of chemistry, zoology, and botany. A vigorous supporter of the Union cause, he took a leave from his teaching to lead a regiment in the Civil War. His bravery earned him accolades and a legion of loyal troops; in one immortalized moment during the Battle of New Bern, he mounted a Confederate canon like a metal steed, allowing his men to advance and overtake the enemy battery. Between the war heroics, the deep education, and a prodigious growth of facial hair, Clark would have made a strong candidate for Most Interesting Man of the Nineteenth Century.
He landed in Hokkaido in the spring of 1876 and got down to business, building out the Sapporo Agricultural College in a month flat. He introduced new crops to Hokkaido, along with lessons on Western agricultural techniques, animal husbandry, and Christianity. He became a trusted adviser to Hokkaido governor Kuroda Kiyotaka, offering counsel on everything from fisheries management to architecture to the textiles industry.
Clark was called back to the States after just eight months in Hokkaido, and a pack of his students rode with him to the outskirts of Sapporo to see him off. In a good-bye that was destined to inspire generations of Japanese, he turned back to the pack of young Hokkaidoans and offered his final words: "Boys, be ambitious!"
Today statues of Clark can be found all around Hokkaido, and his parting words, emblazoned on government buildings and appropriated by makers of manga and J-pop, maintain an outsize place in Japanese culture.
The message wasn't lost on the people of Hokkaido. Since the days of Clark's dramatic send-off, they've worked to prove that Hokkaido is as much an idea as it is an island. It's the idea of a Japan apart, the world beyond Honshu, Kanto, and Kyushu, the island that operates on its own time, plays by its own rules. Even after hundreds of years of gentle growth, Hokkaido remains uncharted territory, the place you come to start new, to reinvent yourself and make a footprint in a way that would be impossible down below, where conformity is the unwritten rule that governs so much of society. For those who need to breathe deeply, to live beyond the white noise of the urban experiment; for those with a few jagged skeletons stuffed in their closets; for those who won't be tethered to the totems of history and tradition that cast impossible shadows across the rest of Japan, there will always be Hokkaido: once and forever the new frontier.
Takahiko Soga knows the frontier spirit better than most. He was born in the mountains around Nagano, son to a first-generation winemaker. He inherited the family winery, along with his brother, but quickly realized he needed his own space to make the wine he wanted. "The first thing that I learned was having a winery with your brother isn't a good idea. We needed some distance between us, so I came to Hokkaido."
Domaine Takahiko is a ten-acre winery situated on the rolling hills outside of Yoichi, about three miles from the coast. At the crest of the hill, Hokkaido flags flap in the gentle breeze that sweeps in off the sea.
Japanese wine consumption has shot up since the 1980s, when a growing appetite for foreign culture and the blossoming of expense accounts introduced the country to the virtues of French burgundies and Italian Barolos. The domestic wine industry took off around the same time, centered around Yamanashi Prefecture, close to where Soga was born, and spreading from the northern reaches to the bottom of Kyushu. Hokkaido is proving one of the country's most promising regions for wine production, not just because of the terrain and the weather but because producers have the space—physical, psychic—to experiment with their fruit.
Not everyone is playing the right way, though, according to Soga. He says that too many Japanese wineries blindly imitate California and France without considering the soil, the conditions, or the type of food that will be served with their wines. "We should be making wines to pair with Japanese food, and umami and dark, intense wines don't pair well."
Instead, Soga has embraced natural wines, a lighter, more esoteric style better suited to Japanese terrain and Japanese palates. He experiments with up to a hundred different types of wild yeasts, prefers long periods of fermentation, and above all wants his wines to express the earth they come from. "I want you to be able to taste Hokkaido in every bottle," he says.
As we sit talking and tasting in his bodega, taking in the damp funk of deep fermentation, it's easy to see what he means. Soga, like most of the serious terroir evangelists, thinks of wine as more art than science, a craft that requires soul and touch and deep-seated dedication. His mind is filled with big ideas about wine, and he reaches constantly for metaphors and analogies to drive his points home. In the first hour we meet, he likens wine to burgers, pickles, seaweed, miso, onion rings, bowls of ramen. "If you use the best seaweed in your _tonkotsu_ broth, you'll never taste it because _tonkotsu_ is so intense. But with _shio_ ramen, you taste every ingredient. I like my wine like _shio_ ramen."
He wants to be sure I understand that Domaine Takahiko is not a company and he is not a businessman. I take the bait and ask him how many people he employs. Soga brings his two hands together to make a large zero.
"If I was going to do it, I wanted to be a part of every step of the process." He tends to the wines, picks the grapes, smashes, inoculates, ferments, ages, and bottles them. He even designs the labels for everything he produces. It's hard to imagine Soga gets a lot of shut-eye, but this level of dedication is not without its rewards: everything he makes sells out months before it's ready to drink, making him the island's cult winemaker of choice.
"If you're a _shokunin_ , Hokkaido is a good place to be."
Saito Narumitsu would probably agree. He dedicates his life to making small batches of high-quality cheeses, which he sells out of a small wooden stand on the side of a two-lane highway between Niseko and Otaru. He learned the craft during a five-year apprenticeship at Kyodo Gakusha, one of Hokkaido's most famous cheese producers, which operates as an incubator for the island's rapidly expanding cheese culture.
In 2007 he opened Tokari Ranch, a business he runs with his wife, whom he met studying cheese at Kyodo, and his brother, who raises the cows.
Like many of Hokkaido's young entrepreneurs, Saito isn't from the island; he moved from Niigata a dozen years ago, when life stalled on Honshu and he saw an opportunity to try something new. "In Honshu, a hundred years is nothing. But here we have a much shorter history, so instead of tradition, we have room to develop our own culture."
There is one Hokkaido tradition, though, that he does follow. "This was Ainu land a thousand years ago, and we wanted to respect that." He honors the Ainu roots in the names of his creations. Retara, which means "white" in Ainu, is a soft, fresh cheese similar to a ricotta or a fromage blanc. Another—a firm, nutty cheese with a grassy finish—has a name that means "waking of the springtime."
My Ainu is rusty, so I can't help but refer to each cheese by its apparent European inspiration—Gruyère, Camembert, scamorza—much to Saito's (understandable) consternation.
"Sure, there are strong French, Italian, and American influences, but the ingredients are from Hokkaido and we are from Hokkaido, so this is Hokkaido cheese. It's not world-class, not yet, but our cheese is getting better every year. With time, we will get there."
Which makes his choice of names for his farm all the more fitting: Takara is Ainu for "growing your dream."
If Hokkaido's cheese and wine industries are still fermenting, its bread culture is fully baked.
Aigues Vives sits on a cliff perched above Otaru with generous views of the Sea of Japan. The owner, Tanno Takoyashi, converted part of his home into a country bakery, with a set of stone steps that leads you through the trees and to the front door.
Tanno invites me back to see his oven, a wood-burning beauty brought over from France. It's 10:00 a.m., time for the second round of baking of the day. After feeding the fire with chunks of maple, he loads the bread and pastries according to cooking time: first the fat country rounds, then long, skinny loaves dense with nuts and dried fruit, and finally a dozen purple crescent moons: raspberry croissants pocked with chunks of white chocolate.
He and his wife traveled to France fifteen years ago and fell in love with the bread culture. For six months he watched the best bakers he could find, living off carbohydrates and the scent of a dream slowly proofing. He took notes; he took pictures. Later he returned and began to re-create the work he'd witnessed in the West. "I made a lot of bread. A lot of bad bread."
In certain corners of the Japanese food world, chefs and farmers and even politicians see guys like Tanno as the enemy. In their eyes he is aiding and abetting an unsettling shift in the Japanese diet—the continuing move from a rice- to a wheat-based diet. In 2011, for the first time ever, Japanese families spent more money on bread than they did on rice. This overtake has been a long time coming, put into motion by the U.S. and Japanese governments in the wake of World War II, but it has suddenly set off alarms in certain corners of the food world—chefs, producers, and politicians who see it as not just a domestic dietary issue but an affront to the national identity as a whole.
Tanno, for his part, doesn't see what the fuss is about; you wouldn't either if you put in the time and the heart this guy puts into his breads. "Why should we have to choose between rice and bread when we can have both?"
He isn't referring to the soft, spongy industrial stuff most Japanese eat. He is referring to the heroic loaves and pastries that he pulls from his oven every morning, bread that wears a thick, crisp crust, a soft, faintly sour crumb, and a dedication to an ideal that borders on obsession.
It's not just the French oven and the French technique. The flour (at least part of it) and the starter are French. The cars parked on the gravel outside are French. As I walk from the oven in the back to the counter, I pass the kitchen and can't help but take a long look: Le Creuset enameled pans, a cast-iron stove, jars filled with preserves—it looks like a museum piece from the future, showcasing the French country kitchen of the twentieth century.
"We didn't just want a bakery. We wanted to create an environment, that's why we came here. For so many years people never thought about Hokkaido as a place for food. But that's changing."
Tanno Takoyashi lines up loaves to feed into his wood-fired oven at Aigues Vives.
_(Matt Goulding)_
It's baffling enough to find one place like this in the middle of nowhere, but the Takoyashis aren't the only ones wood-firing their own ovens in the neighborhood.
At Boulangerie Jin, you'll find another country house with maple wood in the front, a blazing oven in the back, and a shiny Peugeot on the side. Inside, husband and wife team up to make crisp-edged baguettes and one of the finest croissants I've eaten anywhere. Over the years, offers have come in to sell their products all over Niseko with heavy markups for the carb-craving snowboarders, but they don't want more money, they don't want some stranger selling their creations, and they don't want more exposure. (When the wife sees me take out a notebook, she immediately shuts down.)
Sokesyu Bread, just a few hundred yards down the road from Takara Ranch, is pretty much as crazy as the other two. French house, French oven, French car. But Yusuke Konno, tall and skinny, with round glasses and a bandanna wrapped tight around his head, makes his bread with 100 percent Hokkaido flour. "Of course."
He makes superb versions of all the French classics—pain de campagne; baguettes; golden, flaky croissants—but he has a few funky new projects in the works, too. "We can experiment because Hokkaido culture isn't as deep as Tokyo's or Kyoto's." He's been talking about making the switch to dense, German-style brown breads. "Now maybe I'll need an Audi."
Let's consider the evidence: three separate, unrelated bakeries, all of whose owners drive French-made automobiles, have hand-built European wood-fired ovens, and dress like Provençal marmalade makers. All of this in one of the least-densely populated areas in all of Japan? The odds must be infinitesimal.
But these aren't just bakeries; these are affirmations of a much larger idea. Every detail matters. The source of your heat, the type of flour, the age of your starter—of course these form the fundamental base for the flavor of our daily bread. But somewhere, in the deep recesses of taste and perception, it matters that he drives a Peugeot. It matters that she wears a French country wife's blouse. It matters that the kitchen doesn't just _look_ French.
It matters that they're all the way up here, in Hokkaido, where the air is green and the skies are wide and everything feels just a little more possible.
It matters.
米 麺 魚
Even if you travel to Hokkaido to get lost in the wilderness, it feels good to come back to a city of Sapporo's caliber—familiar for its sprawling entertainment district, covered shopping arcades, and preponderance of noodle shops and _sushi-yas_ , but with a collection of wide avenues, green spaces, and Western architecture like nothing you've ever seen in urban Japan.
Few cities eat better than Sapporo. The morning markets teem with _donburi_ dreams. Sophisticated yakitori and tempura and haute cuisine restaurants serving only Hokkaido ingredients dot the downtown area. At Ramen Yokocho, a dark, narrow alley that claims to be the birthplace of miso ramen, a dozen tiny bars serve up steaming bowls of the rich noodle soup.
But I don't want ramen or raw fish or cabernet and Camembert. At midnight on my last night on the island, I am on the hunt for Genghis Khan, or, as he's known in these parts, Jingisukan—an unlikely fixture of Sapporo's dining scene. The name refers to a style of mutton grilled over convex metal domes thought to resemble the helmets worn by Mongol armies. Supposedly Hokkaidoans, once flush with sheep used for clothing the Japanese military, based the cooking on the belief that Mongol armies cooked lamb on their shields and helmets. Today dozens of Jingisukan joints cover Hokkaido's capital.
Jingisukan, Hokkaido's unlikely mutton conqueror
_(Michael Magers, lead photographer)_
Daruma Honten is a fifteen-person bar down a tiny alley in Susukino, Sapporo's pulsing pleasure district, the largest you will find north of Tokyo. Diners sit at a countertop while stoic women in bandannas fill their helmet grills with burning charcoal, then baste the iron surface with cubes of melting mutton fat. Thin slices of meat marinated in soy and ginger tent the smoking black domes, with onions positioned on the rim to absorb the tide of drippings that flows down their surface. The ladies leave me with the tongs but eye me with suspicion as I let the lamb build up a char deep enough to make a Mongol warrior proud.
The man next to me, a Wagyu farmer from upper Tohoku, comes to Hokkaido every few months to wrangle up more cattle—"The best in Japan," he says. While he's in town, he likes to drink Nikka whisky and eat sheep. "Sometimes I wonder why I don't live here."
While the mutton sizzles, I drink icy mugs of Sapporo, Japan's oldest beer, created by a German-trained Hokkaidoan at the dawn of the Meiji era. The Beach Boys play over the speakers, just audible over the protein chorus. When the meat is ready, I pluck it directly from the helmet, pinched between chopsticks with a soft petal of onion or two, and dip it into soy sauce spiked with garlic and chili.
It makes for fine late-night dining, to be sure, but the whole scene has me chewing on a few important questions as I eat. How, in a country where I've never seen lamb before, did Jingisukan conquer Sapporo? Why are there only women cooking this most manly of meats? And what takes the stink of mutton out of cotton and denim?
But as the night inches forward, as the smell soaks into my clothes and the beer into my blood, the barnyard funk of mutton stinging my eyes, the questions slowly disappear. Japanese diners, American music, Mongol myth. There's only one answer that makes sense: it's a Hokkaido thing.
## _Amazing Shit_
## **IN THE MIDDLE OF NOWHERE**
Shikoku, like Hokkaido, is an island filled with covert restaurants of staggering quality, but none more isolated and awesome than Tokiya, a soba shrine perched above a gurgling river tucked high in the mountains of Kochi Prefecture. Lunch—seasonal tempura and beautiful handmade noodles—feels like a feast in a treehouse.
_(Matt Goulding)_
_(Michael Magers, lead photographer)_
## _Noto Peninsula_
## ジェラート
RICE-PADDY GELATO
A wooden stand found in a rice field deep on the Noto Peninsula in Ishikawa Prefecture, the family-run Malga Gelato stand specializes in flavors from the immediate surroundings: sea salt produced fron Noto's coastal waters, persimmon from a nearby orchard, and sake lees, the rice pressings used to make Japan's beverage of choice, from a small producer down the road. The most radical flavor, _ishiri_ , based on Noto's famed fish sauce, may take the concept of local a step too far, but bless Malga for trying.
_(Matt Goulding)_
_(Nathan Thornburgh)_
_(Matt Goulding)_
## _Hokkaido_
## パン
VOLCANO BAKERY
Niseko in the rural reaches of Hokkaido makes for an unlikely breadbasket, but the area teems with bakeries that would challenge the best in Paris. None more unlikely than Boulangerie Jin—so deeply removed from this already far-flung community that you will feel lost until the moment you stumble onto this country home and see the wisps of smoke rising from the wood-burning oven out back. The husband-and-wife team learned the art of bread in Paris and now produce flaky layered pastries; dark, dense, crusty brown breads; and a baguette so yeasty and complex that you may hope to stay forever.
_(Michael Magers, lead photographer)_
## _Sado Island_
## 和牛
ONE-TABLE TASTING MENU
Restaurant Seisuke, located on Sado Island off the northwestern coast of Japan, is run by Kuniaki Osaki, who channeled his Michelin-starred restaurant experience into one of Japan's tiniest and most isolated eateries. Located up a winding mountain road and boasting only one table, Seisuke is really an extension of Kuniaki's home, with his wife at his side and his kids peeking out from behind the kitchen. The food is a seamless mix of East and West: blowtorched yellowtail with yuzu chili paste, roasted whitefish with local mushrooms, and a plate of ripe cheeses served with a selection of esoteric European wines.
_(Michael Magers, lead photographer)_
_(Matt Goulding)_
## _Shigaraki_
## 懐石
COVERT KAISEKI
After training at Kyoto's top kaiseki temples, Furatani Tadamitsu opened Obana an hour from the fray of the old city in the quiet town of Shigaraki. At a beautiful cedar countertop, he serves fresh figs bathed in a thick sesame sauce, a first-rate tempura of tilefish and seasonal vegetables, and a stunning rendition of roast salmon marinated in sake, soy, and mirin. He may only have a few customers a day at times, but that's the whole point. "Beauty operates on a different level out here."
## _Naoshima_
## うどん
ISLAND UDON
Naoshima is best known for its esoteric art, the small island in the Seto Inland Sea transformed into a living, breathing exhibition. But beyond the Tadao Ando–designed museums, giant glass pumpkins, and funky village installations, you'll find a superlative bowl of noodles at Udon Yamamoto. Specializing in Sanuki-style udon from nearby Takamatsu, Yamamoto-san uses both hands and feet to knead the noodles, then massages them in ice water right after cooking to deliver an al dente chew that eludes all but the best udon _shokunin_ —an art form in its own right.
_(Matt Goulding)_
## _One Night with the_
## **SALARYMEN**
_(Matt Goulding)_
_(Michael Magers, lead photographer)_
## _Yakitori_
## [焼き鳥
**ON A STICK**](nav.xhtml#rhh19)
### **THE BEAUTY OF THE BIRD**
Yakitori at its best is an elegant exploration of the totality of one animal. For the full chicken experience, work your way past the white meat and into the wondrous tastes and textures of parts unknown.
_(Michael Magers, lead photographer)_
_(Michael Magers, lead photographer)_
_(Matt Goulding)_
**せぎも**
**SEGIMO (sweetbreads)**
_(Matt Goulding)_
**肝**
**KIMO (liver)**
_(Matt Goulding)_
**揚げ物**
**CHOCHIN (uterus)**
_(Matt Goulding)_
**鶏肉**
**TORINIKU (breast)**
_(Matt Goulding)_
**はつ**
**HATSU (heart)**
_(Matt Goulding)_
**卵**
**UZURA TAMAGO (quail egg)**
_(Matt Goulding)_
**ぼんじり**
**BONJIRI (tail)**
_(Matt Goulding)_
**つくね**
**TSUKUNE (meatball)**
## _Chapter Seven_
## NOTO
For the better part of thirty years, Toshihiro and Tomiko Funashita were the king and queen of fermentation in Noto. And since Noto is often regarded as the Kingdom of Fermentation throughout Japan, it could be argued that their skills held dominion across the country. Of course, they would never say it themselves, but Toshiro was recognized by the governor for his smooth, umami-rich fish sauce, and Tomiko was widely accepted as Noto's chief authority on all manners of preserved flora and fauna.
Noto is a peninsula on the coast of western Honshu, a craggy appendage of Ishikawa Prefecture that juts thirty kilometers out into the Sea of Japan. It is a place defined not just by the harshness of its seasons but by the generosity of its geography: rivers and mountains, ocean and valleys, one flowing into the next to create an extraordinary tapestry of ecosystems.
In some ways Noto is a perfect reflection of life in rural Japan: a quiet, self-sufficient tableau of Shinto and Buddhist traditions, where the rhythm of life is so directly tied to the rhythm of the seasons that calendars are beside the point. In other ways Noto remains a place like no other, a beautiful, lonely seascape, a world of distinct environments condensed into a tiny space, where everything is filtered through the lens of food, and the culture of fermentation runs so deep that nearly every meal has been transformed by time and bacteria.
The Jomon, the original settlers of Japan, first came to Noto over two thousand years ago, establishing a hunter-gatherer subsistence and ushering in a culture of food preservation that carries on today. They built large earthen-ware pots—believed to be among the first use of pottery ever by humans—and began to harvest salt. Together they had the tools to ferment fish, vegetables, rice—whatever they needed to survive the long cold months when the land produces little.
Today's Noto looks scarcely different from the Noto of the Jomon. Rice paddies climb the hillsides in wet, verdant staircases, dense woodlands trade space with geometric farmscapes, tiny Shinto shrines sprout like mushrooms in Noto forests. Villages seem to materialize from nowhere—wedged into valleys, perched atop hills, finessed into coastal corners. Pull over, climb out of your car, breathe deep for a taste of the finest air that will ever enter your lungs: green as a high mountain, salty and sweet, with just a whisper of decay in the finish.
Noto gained its reputation as the Kingdom of Fermentation because of this air. For most of its history, Noto was cut off from the rest of Japan, forced into a subsistence model that in many ways endures today. That was possible not only because of the bounty of Noto's fertile environment of trees, grasslands, fresh water, and sea, but because the air is rich with humidity that encourages the growth of healthy bacteria, the building blocks of fermentation.
Toshihiro Funashita's family lived in the interior of Noto, his father a forester, his mother a homemaker and a cook of wide reputation, the one responsible for organizing the elaborate feasts behind their community's most important social events—the highest charge in the local cooking communities of rural Japan.
Like any great and good country, Japan has a culture of gathering—weddings, holidays, seasonal celebrations—with food at the core. In the fall, harvest celebrations mark the changing of the guard with roasted chestnuts, sweet potatoes, and skewers of grilled ginkgo nuts. As the cherry blossoms bloom, festive picnics called _hanami_ usher in the spring with elaborate spreads of miso salmon, mountain vegetables, colorful bento, and fresh mochi turned pink with _sakura_ petals.
Funerals, in particular, are a time to eat in Noto, and the preparations that surround the passing of a loved one may involve days of work and dozens of participants. As a Shinto ceremony, funerals in Noto are vegetarian affairs, prompting local women to bring to the table the best of the products from their respective gardens and pantries. Toshihiro's mother, as respected in the kitchen as she was in the community, was in charge of overseeing the cooking at funerals in her town, which meant deciding the best way to make use of the gathered ingredients and organizing the women into teams to turn out elaborate spreads of boiled and fried vegetables, tofu dishes, and vinegar pickles.
When mudslides forced Toshihiro's family off their property, they relocated to the Noto coast and eventually opened an inn on 249, the two-lane highway that winds its way around the perimeter of the peninsula. Sannami was a _ryokan_ , a traditional Japanese guesthouse, complete with tatami-floored rooms, a wood-fired bath, and full dinner and breakfast service for guests.
During those years, Toshihiro met and eventually married Tomiko Futamata, a young woman from the town of Notocho. He was an electrician who would go on to be a programmer in the infant days of the Japanese computer industry. Tomiko was a librarian, guardian of Noto knowledge, a voracious reader with a busy mind. She lost her mother at an early age, but she spent hours in the kitchen with her mother-in-law, learning how to transform a momentary surplus into a year's worth of good eating.
The dining room in Flatt's Inn, with a generous view of Toyama Bay
_(Michael Magers, lead photographer)_
For most of Noto history, men weren't allowed in the kitchen. The kitchen was considered a sacred place for women, and having a man enter was tantamount to an invasion of privacy. But Toshihiro was different from most Noto men: he was deeply curious about food, about the tastes of Noto that defined his childhood, and as he watched his parents feed travelers from around Japan, he began to imagine what he would do differently.
米 麺 魚
Fermentation is the art of controlled decay. In fermentation's most basic form, enzymes produced by molds, yeasts, and bacteria break down organic matter, converting macronutrients like sugar to alcohol and proteins to amino acids. But there is a fine line between fermentation and decomposition: initiate and control microbial activity carefully, and you've extended an ingredient's life indefinitely; take it too far, and you've lost it forever.
There are many forms of fermentation, but the two most common in the food world are lactic acid fermentation, produced by fungi and bacteria and used to produce most varieties of pickles and fermented condiments, and alcoholic fermentation, induced with yeast, and used to produce the world's supply of adult beverages.
Fermentation is one of man's earliest culinary innovations, stretching back nine thousand years to the Neolithic period, when civilizations in modern-day China turned rice and fruit into alcohol. Since then, you'd be hard-pressed to find a single successful civilization that didn't have fermentation as a core component of its food culture.
Without it ever crossing our minds, most of us consume fermented foods at various points throughout the day: coffee and yogurt for breakfast, wine and cheese for dinner, chocolate for dessert. No small number of man's greatest achievements—from the hams of Spain to the beers of Belgium to the dark, bitter chocolates of Venezuela—are the byproduct of carefully controlled enzymatic breakdown.
The natural coalition of fermentation-loving cultures forms a strange Venn diagram: Russians and northern Europeans use salt to stretch vegetables long into the winter; West Africans ferment cassava root as a means of neutralizing its natural cyanides; southern Asians build entire cuisines around the flavors produced by dead fish; and Bolivians in the high Andes employ the enzymes in human saliva to transform chewed corn into fermented beer.
Most of these cultures have a handful of fermented products in their pantry, but in Japan the entire cuisine turns around lactic acid and alcoholic fermentation. The grocery list of fermented staples runs long and strong: soy sauce, miso, sake, mirin, _yuzukosho_ , _katsuobushi_ , _natto_ , rice vinegar, _tsukemono_ : without fermentation, the Japanese kitchen would be a lonely place. It's no coincidence that Japanese food places such a premium on umami, since umami is one of the primary byproducts of the fermentation process.
Beyond the basic advantage of preservation, fermentation offers a host of other benefits to consumers: a surge in B vitamins, the introduction of friendly gut bacteria into our systems, and of course the deepening of flavor and aroma in everyday ingredients.
In recent years, in certain corners of the food world, fermentation has become the fascination of chefs, hipsters, and DIYers, but what goes down in Noto has nothing to do with a young chef geeking out over a lacto-fermented heirloom carrot or a crunchy commune denizen making kombucha in the bathtub; this is a lifestyle necessity emblazoned in the DNA of this peninsula.
米 麺 魚
Toshihiro's parents retired in 1982, and Toshihiro and his wife took over running Sannami. They were ready for this moment: Tomiko's career as a librarian gave her ample time to digest every last piece of text dedicated to the topic of Noto food, and Toshihiro's limber mind and unwavering drive made him the perfect person to refine some of the more challenging culinary practices at hand. Soon after assuming ownership of Sannami, they began to slowly transform the inn into a living encyclopedia of Noto food traditions.
The _shokeba_ is the most important room in the Noto home. As small as a closet or as large as a bedroom, this is where a family stores their stock of fermented goods: purple jars of _umeboshi_ , pots of mocha-colored miso, barrels holding batches of homemade soy sauce. Like most local families, Toshihiro's parents kept a well-stocked _shokeba_ , but the basic staples of the Noto kitchen were reserved for family consumption. At the hotel, they focused their efforts on serving a broader menu of Japanese food. To serve Noto food, products of necessity and distinct local character, to their guests would have been an embarrassment.
But Toshihiro and Tomiko had a different vision. Tomiko made it her business to put every piece of knowledge she had into practice, quickly turning the _shokeba_ into an edible calendar of the Noto bounty. In the summer she harvested seaweed, picked plums, turned a rich garden harvest into a rainbow of preserved produce. In the winter she dried persimmons, pickled fish beneath the weight of stone slabs, fermented soybeans into dark batches of miso.
Toshihiro put his efforts into realizing the vision he'd shaped during years of watching his mother at work. His idea was driven by the tastes of his childhood, tastes that he feared Noto was losing, tastes that he felt he and his wife could restore and carry forward. He shaped himself into an expert on seafood, turning out beautiful, precise plates of sashimi and local fish dishes for his guests. Above all, he dedicated himself to _ishiri_ , Noto's ancient fish sauce.
The history of fish sauce is the history of the world's greatest powers: the Byzantines, the Greeks, and the Romans all produced _garum_ , made by salting and sun-drying fish blood and guts and extracting and filtering the resulting liquid. Civilizations across the Asian continent developed different takes on salt-fermented fish sauce, from Vietnam's nuoc mam to Korea's _aekjeot_. Even English Worcestershire sauce, based on a formula of fermented anchovies, is a form of fish sauce. Whether all of these cultures understood the science of umami is doubtful, but it's clear humans have known the simple secret of fish sauce for thousands of years: it's nature's greatest force multiplier, a few drops enough to intensify the flavor of anything it touches.
In Noto, fish sauce goes back to the eighth century, predating soy sauce on the peninsula by hundreds of years. Though shoyu became the predominant umami-enhancing condiment of Japan, _ishiri_ remains a linchpin of true Noto cuisine. The region produces two types of fish sauce, depending on which side of the peninsula you live on. On the west coast, where anchovies and mackerel are abundant, they produce the cheaper, somewhat harsher _ishiru_. On the east coast, where some of Japan's finest squid spend time in the channel between Noto and Toyama, _ishiri_ is king.
To make _ishiri_ , Toshihiro would salt hundreds of pounds of squid guts at a time, leaving them to ferment for two, three, sometimes up to five years (the longer the ferment, the deeper the umami flavor). He would then press the guts to extract the liquid, bottle it, and store it for use at the inn. At the end, you have a liquid as dark as night and as fragrant as a laundry basket of old socks, but with a sweet, dense concentration of amino acids perfect for stews and sauces and, of course, for fermenting other products. In the totem pole of global fish sauces, _ishiri_ sits squarely at the top.
Soon Toshihiro became celebrated for his _ishiri_. Customers, most raised on soy sauce and who had never tasted Japanese fish sauce, fell in love with the flavor. The governor of Ishikawa designated Toshihiro with the prefecture's only Takumi award, a distinction reserved for the top class of Japanese artisans. Meanwhile, Tomiko's reputation as a guardian of Noto's culinary heritage grew (she would go on to be one of ten women from guesthouses across Japan recognized for passing on local traditions to the next generation), and the inn became famous for its dedication to a way of eating that some in the area thought was lost and many in Japan never knew existed in the first place.
Normally, a son would inherit this world created by Mom and Dad and be expected to carry on the family reputation through the business they built. Toshihiro and Tomiko never had a son, though, so the duty fell instead to their eldest daughter, Chikako.
From a young age, Chikako was an independent girl. She played sports, which allowed her to travel around the region and later to other parts of Japan—Kanazawa, Tokyo, Akita. When she was twelve, her table tennis team received an invite to play in a tournament in Fukui, 250 kilometers south of Noto, but no parents or chaperones could make the trip. Instead, Chikako organized everything—the train tickets, the hotel, the tournament details—and took her eight teammates on a weekend road trip to play Ping-Pong.
Chikako went to university in Kyoto and studied to be a teacher, but dedicated most of those years living outside Noto to a mixture of work and partying. She took a job at a hotel, serving breakfast for four hours before school, then dinner for six hours after class let out. On the weekends, she worked weddings—massive, elaborate banquets where she learned the ins and outs of service. She studied full-time and still earned $3,000 a month, establishing a rhythm of hard work and quiet determination that would follow her back to Noto and beyond.
_(Michael Magers, lead photographer)_
As the oldest daughter, Chikako had two primary responsibilities: to inherit the inn her parents owned and to assume the onus of the family name. The former is standard practice in family-based businesses across Japan but the latter is an old Noto tradition that comes with a host of loaded social responsibilities: pitching in with seasonal events, gifting money at weddings and funerals, and generally maintaining a strong, continuous presence for the family in the local community. Chikako accepted her fate, considered it an unwritten contract between parents and daughter, but before she settled down in Noto for the long haul, she wanted to see more of the world. She asked her parents if she could go to Australia, and they said no. The next year she bought a plane ticket, secured a passport, and organized a visa. A week before her flight, she told her parents she was off to Australia, with or without their blessing. Before she left, they made her promise that she'd be back in a year.
In Australia, she kept up the kind of lifestyle she'd started in her college years in Kyoto, working multiple jobs, studying English, sleeping on occasion. She organized a homestay with a family just outside Sydney, and even when she moved out a few months later, they continued to invite her to barbecues and family events, where she practiced English, watched rugby, and slowly became acquainted with the family's twenty-five-year-old son.
Ben Flatt grew up in Sofala, a gold-mining town two hundred kilometers northwest of Sydney. Ben's first job was cooking at his parents' restaurant, a French-Italian café with a blackboard menu where most of the ingredients came from the family's backyard. He learned from an early age never to name the animals, because one day his mom or dad would ask him to step out back and take its life. His dad was an eccentric type—he would go on to a career as a writer under the pen name Captain Chaos—but his parents were good cooks and hard workers, and Ben picked up the passion at an early age. His parents eventually sold their restaurant, and Ben took off for Sydney to pursue his own culinary career. He spent the next few years cooking at trattoria around the city, slowly falling in love with rustic Italian food.
Pasta wasn't his only muse at the time, though. He and Chikako started dating a few months after she arrived in Australia, spending what little free time both of them had with each other. When her year was up, Chikako made good on her promise to her parents and prepared to return to Noto. She felt strongly for Ben, but she also knew that she couldn't back away from her duty to the family. When Ben said he would go with her, she hesitated. "I don't think you understand the world I'm going back to."
Ben wasn't fazed. He had traveled and worked around Asia, lived on his own for years, and in his own way was every bit as independent and determined as the woman he was pursuing. Shortly after Chikako left, he packed up his life in Australia and set coordinates for Noto, arriving on the doorstep of her parents' inn with most of his life in tow. Toshihiro and Tomiko liked the young man from Australia, but were surprised to find out that he was dating their daughter. A few weeks after arriving, with Chikako translating for him, Ben asked Toshihiro for his daughter's hand in marriage. Dad said no: Ben was not Japanese, was not from Noto, and couldn't possibly understand the life that Chikako had in front of her. Chikako was inheriting not just an inn but an entire life circumscribed by the rhythms and rituals of a land Ben knew nothing about. It would be too hard for everyone, he said.
But Ben was undeterred. He told Chikako and her parents that he could live anywhere, that he wasn't afraid of Japan or Noto or the culture the rest of the family was working to preserve. To prove it, he stayed. He studied Japanese and helped around the inn, especially in the kitchen. Three months after Ben arrived, Toshihiro presented him with a Japanese chef's knife, a peace offering in a struggle he knew he couldn't win.
A few months later Chikako and Ben were married in Kenroken in Kanazawa, one of Japan's most beautiful gardens. The reception was held in the old shogun's summer residence, with a menu handwritten by Chikako. Japanese tradition has it that the bride shouldn't be seen laughing or drinking or generally enjoying herself on her wedding day, but a photo from the afternoon shows Toshihiro pouring Chikako a glass of sake, the bride smiling widely.
米 麺 魚
Flatt's Inn sits on a bluff in Notocho overlooking an inlet on the Sea of Japan, a two-story home fronted by a dense forest and surrounded in the back by a large, active garden with yuzu and sudachi lime and persimmon trees, rows of leeks and cabbage, beans and daikon, and a large cherry tree that hangs over a tiny wooden bench, where, on a clear morning like the ones after it's been raining for a week and everything around Noto has that brave new sparkle, you can see across the channel to the outline of mountains that loom over Toyama.
You enter Flatt's through a corridor of stone steps and lush foliage. Inside, you'll find four traditional ten-tatami-sized rooms (the mats also serve as measurements) with squat rectangular tables, wooden-backed chairs that sit directly on the tatami, and no other decoration of note besides a dispenser for hot tea and cups for sipping. The restaurant, used by both guests and day visitors, has four tables that sit low to the ground, where diners eat on the floor; an _irori_ , a charcoal fireplace used for cooking; long strings of produce—persimmons, radishes, chilies—in various stages of drying that dangle from the ceiling like garden necklaces; and large bay windows with generous views of the backyard and the sea in the distance. There are two main baths at Flatt's, an outdoor wooden tub with a sprawling sea view and an indoor stone bath for when snow covers the ground and bathing outdoors is too difficult.
The _ryokan_ stay is designed to be a fully immersive experience. Upon arrival, you trade your shoes and street clothes for slippers and a robe (called a _yukata_ ), your smartphone for a cup of sencha, your worries for a long, contemplative cleanse in the tub. Sip, soak, think, breathe: this is all that is required of you during your stay.
This being Japan, food is at the center of the _ryokan_ experience. Dinner is normally an elaborate multicourse meal, often with a structure and progression borrowed from kaiseki. After strolling the gardens, after reading a chapter or two, after soaking your bones in simmering water, you sit down to a three-hour dinner heavy with signs of the season and tastes of the local terroir. While you eat, someone is back in your room, silently laying out a thick base of comforters and blankets on the tatami, which will embrace your warm, distended body as soon as you finish chewing. You will sleep like the dead, and when you wake up, there will be another elaborate multicourse feast set for you in the dining area, waiting to push your appetite to the limit. By the time you check out, the worries of the world long since evaporated, you will exhale deeply and turn to your partner. "We should do this more often."
Flatt's Inn is like nearly every other traditional Japanese guesthouse you'll find in the rural corners of this country, except for one primary difference: the large Australian man with a bushy mustache working the stoves in the kitchen.
In many ways, Ben Flatt is the last person you would expect to fit into this world. He's twice the size of most of the customers he's serving, with the mouth of a marine and a tendency to wear his emotions like the kitchen scars that cover his arms. He plays the guitar, rides a motorcycle on his days off, starts his morning with Vegemite spread thick on toast. In Noto, where people from Tokyo or Kyoto can appear like foreign invaders, Ben might as well be from another galaxy.
For many years, Chikako and Ben ran Flatt's just down the road in the space once occupied by Sannami. Chikako's parents were still working full-time back then, but in 1996 they built a new home for the inn, in the same lovely garden space where Flatt's is now. After getting married, Ben and Chikako didn't want to wait for her parents to retire before starting their own place, especially when it looked like Toshihiro and Tomiko could go on forever, so in 1997 they took over the old four-room _ryokan_ and began to assert their own vision on the business. The food incorporated the flavors and ingredients of Noto, the same ones Chikako's parents were working so hard to produce, but through the filter of Ben's Italian cooking. Soon enough, both inns had obtained a measure of recognition for their respective cuisines. People would come for a weekend, stay one night at Sannami and one night at Flatt's, tasting the same ingredients in two very different expressions.
Toshihiro and Tomiko finally retired in 2011, and Ben and Chikako moved Flatt's up the road to the former Sannami space—inheriting its active gardens, its dozens of fruit trees, and its robust pickle shed. Chikako runs the front of the house with the help of a part-time assistant, while Ben helms the kitchen mostly on his own. For dinner, he serves dishes such as raw local fish accented with touches like fresh basil and balsamic vinegar; roasted pumpkin soup laced with _ishiri_ ; fat, chewy handmade spaghetti with tender rings of squid on a puddle of ink enhanced with another few drops of fish sauce. It's what Italian food would be if Italy were a windswept peninsula in the Far East.
If dinner is Ben's personal take on Noto ingredients, breakfast still belongs to his in-laws. It's an elaborate a.m. feast, fierce in flavor, rich in history, dense with centuries of knowledge passed from one generation to the next: soft tofu dressed with homemade soy and yuzu chili paste; soup made with homemade miso and simmered fish bones; shiso leaves fermented kimchi-style, with chilies and _ishiri_ ; _kaibe_ , rice mixed with _ishiri_ and fresh squid, pressed into patties and grilled slowly over a charcoal fire; yellowtail fermented for six months, called the blue cheese of the sea for its lactic funk. The mix of plates will change from one morning to the next but will invariably include a small chunk of _konka saba_ , mackerel fermented for up to five years, depending on the day you visit. Even when it's broken into tiny pieces and sprinkled over rice, the years of fermentation will pulse through your body like an electric current.
In total, half a dozen different expressions of Noto fermentation, a breakfast that took more than a decade of molecular breakdown to bring to the table. And all of it virtually unchanged since the days when Chikako's parents ran the inn.
The first time I eat Flatt's breakfast, I feel like I went to bed in 2014 and woke up a few centuries earlier. The flavors are timeless, the textures all-encompassing, a meal so dense with umami and history that you wonder if your taste buds will ever recover. At first it feels like an act of aggression, like floating a stick of dynamite in your coffee, but the more I eat this meal, the more I realize that my concept of breakfast will never be the same.
To Chikako, breakfast at Flatt's is more than just a collection of taste and textures; it's the legacy she inherited from her mom and dad condensed into a single spread. This—the drizzle of homemade soy, the swipe of yuzu chili paste, the hunk of hyperfermented fish—is what her parents worked so hard to create, and it's what Chikako signed on for when she moved back to Noto and assumed the Funashita name. Ben can push the cuisine in new and interesting directions at night, so long as Chikako keeps the table filled with the fruits of fermentation in the morning.
This is no small responsibility. This isn't like inheriting your mom's cookie skills or carrying on your dad's reputation at the grill. To make Noto cuisine is an act of patience and sacrifice, one that forgoes the ease of modern conveniences like supermarkets and industrial ingredients for a deeper commitment to land and legacy. It means adapting your life to fit the fickle behavior of the seasons. It means understanding tidal rhythms and weather patterns by how they translate to the table. It means _mottainai_ , "nothing goes to waste," a philosophy that resonates through every facet of Japanese food culture. It's an ethos born not simply out of necessity or industriousness but out of the Shinto belief that objects have souls and should be honored accordingly.
To understand how seriously the people of Noto take the concept of waste, consider the fugu dilemma. Japanese blowfish, best known for its high toxicity, has been a staple of Noto cuisine for hundreds of years. During the late Meiji and early Edo periods, local cooks in Noto began to address a growing concern with fugu fabrication; namely, how to make use of the fish's deadly ovaries. Pregnant with enough poison to kill up to twenty people, the ovaries—like the toxic liver—had always been disposed of, but the cooks of Noto finally had enough of the waste and set out to crack the code of the toxic reproductive organs. Thus ensued a long, perilous period of experimentation. Locals rubbed ovaries in salt, then in _nukamiso_ , a paste made from rice bran, and left them to ferment. Taste-testing the not-quite-detoxified fugu ovary was a lethal but necessary part of the process, and many years and many lives later, they arrived at a recipe that transformed the ovaries from a deadly disposable into an intensely flavored staple. Today pickled fugu ovaries remain one of Noto's most treasured delicacies.
The star of the breakfast table: a piece of mackerel fermented for four years
_(Matt Goulding)_
Chikako doesn't pickle fugu ovaries at Flatt's—one of the few ingredients spared fermentation at the inn—but to dedicate yourself to Noto cuisine is to see every ingredient through the same prism: how to extract every last bit of life from what nature provides.
Once you accept _mottainai_ as a starting point, your life must be organized accordingly. Those persimmons don't stop ripening just because you wanted to go to Kanazawa today; that fish won't dry properly unless you gut it and salt it before the sun goes down. Chikako never stops moving, her day like a seamless sixteen-hour tutorial on how to carry the traditions of Noto forward: serve breakfast, scale some tiny fish, talk with guests, peel and juice a hundred mandarins, draw a bath, fill a bucket with plums and purple shiso leaves.
"In twenty-seven years, my parents never took a day off," says Chikako. "My dad would say, 'Day off? What would I do with a day off? You can't take a day off from your life.'"
More than just hard work and organization, these practices require an immense body of knowledge. Which mushrooms are safe for pickling and which will kill you? Is this type of fish best preserved in rice bran or in salt? What can I do with this tiny piece of the fruit that always ends up in the compost pile?
"The other day my mom got really upset that I was throwing away the stems on the persimmons we pick," Chikako tells me one morning. "We pickle the peels, we dry the flesh of the fruit, but apparently the stems can be used to make tea."
Persimmon-stem tea isn't a recipe you'll find online; it's not an idea you'd stumble onto when you buy a bag of persimmons at the store. It strikes you only after enough dirt has found its way under your fingernails.
Chikako and Ben's lives are inexorably linked to an ever-expanding list of seasonal tasks. In summer, they work through the garden bounty, drying and pickling the fruits and vegetables at peak ripeness. Fall brings chestnuts to pick, chili paste to make, mushrooms to hunt. Come winter, Noto's seas are flush with the finest sea creatures, which means pickling fish for _hinezushi_ and salting squid guts for _ishiri_. In the spring, after picking mountain vegetables and harvesting seaweed, they plant the garden and begin again the cycle that will feed them, their family, and their guests in the year ahead.
When things go well in the wild, Noto tradition has it that you should share with your neighbors. If generous rain brings you a bumper crop of mandarins, you give the families living around you a surprise taste of the season. In turn, when their cherry trees explode or their sweet potatoes sprout, they'll return the favor. It's a carryover from the barter economy that existed on this peninsula for most of its history, well into the twentieth century. In a country known best for its overwhelming urban sprawl, these flashes of rural ritual take on a very special importance.
"We might not wear kimonos every day," says Chikako (who just so happens to be trained in the intricate art of kimono dress), "but it's amazing the traditional way still survives."
One morning, after days of politely ignoring my request, Chikako takes me down to the _shokeba_. This is the most important room in the house, the nerve center of Noto cuisine, and I've been eager to take it all in. At first I think Chikako's hesitance is because the room might be messy, or because she fears I might try to reveal the family recipes, but the more time I spend in Noto and the more I speak with Chikako and Ben, the more I realize that to invite someone into your _shokeba_ is like sitting them down with a family album—an intimate experience that requires a level of trust and familiarity.
The _shokeba_ at Flatt's is housed in a basement below the kitchen. It looks exactly like you'd imagine a pickle shed to look: dark, crowded, shelves and cement floor cluttered with plastic bottles, glass jars, large yellow buckets with contents unknown. In total Chikako and Ben have nearly two hundred different projects in the works down here, a motley collection of floating fruit, shrinking vegetables, and degrading protein. There are vinegars made from persimmon and plum, kimchi made with cabbage and daikon, liquors infused with anything that grows: yuzu, quince, grape, wild strawberry.
Like a bodega, the pickle shed is filled with living, evolving products that capture a particular moment in time: the great rains of '88, the dry spell of '91, the near-perfection of 2002. The beauty of the _shokeba_ is that right now, at this very moment, it all tastes incrementally different from how it tasted yesterday and how it will taste tomorrow. Today is today, and no other day will ever be the same.
Chikako opens a few buckets to show me what she has working. In one, she grabs a fistful of tiny plums stained half purple with shiso leaves, _umeboshi_ midway through its cure. Another contains soybeans well on their way to becoming miso.
The room is thick with the smell of transformation, a powerful stench that recalls a dark corner of an old library, emanating a mysterious and meaningful musk. Chikako squats down and lifts the lid on a short, wide yellow trash can, and the room explodes with another dimension of funk. "This is our _konka saba_ ," she says, wiping off a muddy layer of rice husk. Below is buried a heap of mackerel rubbed in salt and chilies. "Some people bring it up after half a year, which isn't even fermented. These here have been fermenting for nearly fifteen years." I try to do the math but can come up only with this: when the fish went into this bucket, the world was a very different place.
米 麺 魚
Wednesday is the traditional day of rest in Japan's service industry, and Flatt's closes accordingly. But little rest ever goes down around this inn; Ben and Chikako use the time mostly to catch up on the various projects they have in the works.
One Wednesday morning, Ben wakes me up at dawn, and we head to the Suzu fish market, where the local fish auction takes place every day at 7:00 a.m. As the first rays of sun bounce off the sea and fill the market with a warm, speckled light, a mix of chefs, distributors, and fishermen survey the day's catch: buckets of tiny baitfish, giant squid oozing puddles of black ink below them, cod pregnant with the season's first roe ("Those will go for a lot today—the Japanese are willing to pay for the first taste of just about anything"). On the edge of the market, a shark, two meters long with a swollen belly streaked with blood, attracts a small group of fishermen who smoke cigarettes and ponder its demise.
_Buri_ , Japanese yellowtail rich with fat stored to combat the cold winter waters, have just come in, and most of the morning's energy hovers around the three hundred midsize fish lined up on the wet cement floor. The auctioneer, an old man with a walking stick he uses as a pointer, works his way quickly through the catch, balancing himself on the edges of the plastic boxes that hold the fish. He's not as animated as the tuna auctioneers of Tokyo's Tsukiji market, but he doesn't need to be; everyone here knows exactly what they want and how much they're willing to pay for it. The actual bidding comes out in thick Noto dialect, thickened further by the fact that these are fishermen—a lifetime at sea turning their tongues into mysterious instruments. Within fifteen minutes, it's over, and Ben has a bucket of sardines to add to the day's chores.
Another morning, we drive over to a grassy riverbank down the road from the inn to look for mountain vegetables, a rite of passage to the Japanese spring. "People take this shit seriously," says Ben, explaining how different families work different tracts of land, which are closely guarded and kept within the family for generations. We return to the kitchen with three plastic bags full of long-stemmed fiddlehead ferns. Some will go in a pasta dish later tonight for dinner service but most will be preserved in _ishiri_ , and served as breakfast pickles later in the year, when spring is only a distant memory.
_(Matt Goulding)_
To watch Ben navigate the thickets of Noto's physical and social landscape, I can think only of all the acrobatics he's performed over the years to get a foothold on this culture. Even today, two decades later, he remains a perplexing character for certain locals. "I'll be in the supermarket and people will come up to me and literally go through my basket and ask me what I'm planning to do with certain ingredients."
It's tough for a foreigner to penetrate any area of Japanese culture, but to come to Noto, try to marry a local woman, be denied by her parents, marry her anyway, and then proceed to dedicate your life to the daily preservation of her culture—one so dense and pregnant with mystery and vagaries that it's widely unknown to the citizens of Noto themselves—to do that takes a strong mind, an iron will, and rock-hard stones in equal measure. But Ben sees it differently. "It's been incredibly humbling to learn from this family. To be in the same kitchen, to understand the mentality," he says. "It's more than just a lifestyle; it's our life."
No small part of that life means picking fruits and vegetables at just the right moment. When the yuzu are so swollen with juice that they begin to drop from the tree, Ben and I head out with a ladder and trash bags, wrapped in a swaddling layer of puffy pants and jackets and thick gloves to protect us from the bastard spikes of the yuzu tree. The harvest isn't quite what Ben expected this season, and he knows his father-in-law will have a few words of pointed advice for him when he finds out.
Life after death for a yuzu is an arduous journey toward reincarnation. At Flatt's, the juice is preserved with salt and kept throughout the year for vinaigrettes and sauces. The pith becomes marmalade. Even the seeds are salvaged, slowly dried, then mixed with shochu to use as a natural moisturizer.
The star of the yuzu anatomy, though, is the peel, which Ben and Chikako combine with dried chilies and salt and lacto-ferment for two years. The mass is then pureed into _yunamba_ , a powerful combination of umami, heat, and a bright citrus uppercut—the kind of insane condiment that, once you taste it, you wonder if you'll ever be able to live without it. The bright red paste finds its way onto tofu at breakfast and sashimi at dinner and into my suitcase to hold me over between trips to Japan.
All told, four products created over the course of two years, a seed-to-skin transformation that yields vital components of the Flatt's pantry. _Mottainai_ , _mottainai_.
When there are no fruit to pick, no vegetables to forage, no fish to gut, we load into the Flatts' van and stake out around the peninsula. Even then, fermentation follows us everywhere around Noto. One day we visit the salt flats of Okunoto on the northwest coast of Noto, the longest-running producer on a peninsula long dependent on salt to fuel fermentation. Hiroshi Kikutaro, a sixth-generation salt farmer, still starts with seawater and cooks it down in large wooden buckets. It takes him a week of shoveling and boiling in a small hut where temperatures hover around 130˚F to make a single batch of salt.
At a market in Wajima, a town on the west coast of the peninsula best known for production of some of Japan's shiniest lacquerware, we run into an autumn food festival. Two men in karate outfits with bandannas tied around their heads trade off pounding cooked rice with a massive wooden mallet, working the grain into a fine warm paste that they stuff with sweetened adzuki beans to form mochi, one of Japan's favorite festival foods (every year about a dozen people die from choking on warm mochi, but the Japanese chew on, undeterred). A food market displays the best of Noto's fermentation muscle, from smoked and sun-dried clams to squid tossed with fermented rice and yuzu peel to those harbingers of a deadly serious food culture, pickled fugu ovaries.
Another day, we travel to Nanao at the base of the peninsula for a beautiful sushi lunch at Kozushi, Ben and Chikako's favorite place to eat on their day off. Walking the street after the feast, we come upon a soy sauce shop in an old merchant home. When we walk in, the smell hits us, and we realize the shop isn't just a shop, but the factory as well. The owner takes us to the back, shows us the soybeans, which have been slowly fermenting in massive wooden barrels since the end of the Meiji period. We all agree that the resulting potion, more sweet and savory than salty, is among the best we've tasted. Chikako buys three liters to take back to the inn.
Closer to home, just a few miles from Flatt's, we find Japan's arguably most important form of fermentation at work: sake. The rice wine production at Tanizumi Sake remains a steadfastly analog operation, best suited to Midori Tsuruno, its sixty-two-year-old owner.
"Even if I make a seven-hundred-kilogram batch, I wash the rice ten kilo-grams at a time," she says, showing us the washbasin where the sake process begins.
She still uses old wooden buckets for steaming the rice, leaving the steel tanks she bought years ago in a moment of weakness to idle in the corner. "I don't get consistent results with the metal."
After the rice steams for fifty minutes, it is spread on tables and left to sit for two or three days, which allows for the formation of _koji_ bacteria, the invisible hand behind so many of Japan's most important fermented goods: soy sauce, miso, shochu.
The rice is then moved upstairs to the attic for two weeks, where more stable and better bacteria will allow for even fermention. Eventually the cooked rice is combined with water and more _koji_ , stored in a bag, and the whole package is placed in a press and squeezed. It rests overnight before undergoing another squeezing. The resulting liquid, fermented for anywhere from twenty-five to forty days, is one of the world's oldest and greatest alcoholic beverages.
Chikako and Ben in the kitchen at Flatt's Inn
_(Michael Magers, lead photographer)_
"I don't like to filter my sake. It takes away the umami flavor." No doubt: a glass of her milky white potion has just an edge of floral sweetness, with an intense savory kick that leaves your mouth watering. "I'm a small producer, I don't need my sake to always taste the same. I want you to taste the difference from one year to the next."
One night, with no customers booked for lunch the following day, Ben and Chikako take me to Buranka's, a bar dense with cigarette smoke and karaoke tunes and whisky-soaked barflies. Mama runs the bar on her own, pouring drinks, lighting cigarettes, warming up planks of dried squid over little electric grills that she passes across the bar to her regulars. She has a Marlboro voice box, thick and raspy, but when she takes a break to bless the microphone, everything sounds like sunshine.
Japanese karaoke isn't the twisted spectacle you find in bars in the West. You don't get drunk sorority girls belting out Madonna wildly out of key or packs of Jäger-charged bros imploring you to _don't stop believing_. Instead, participants, mostly older men and women, wait patiently to sing any number of long, crooning ballads with intensity and purpose.
Ben, still an Aussie at heart, tries to open things up with a stirring rendition of "Bohemian Rhapsody," but the locals are unmoved. (They are even less moved by my bare-all version of "La Bamba.") When Chikako's turn comes up, she chooses a long, slow, moody Japanese song—the type that comes accompanied by a video of two lovers walking over bridges and canoodling on park benches. She starts slowly but warms up after the first verse, hits the choruses with grace and beauty, and by time the song comes to its dramatic close, the entire bar is staring at her. Though I've understood none of it, I find myself blinking back tears in the thick, squid-scented air of the Noto watering hole.
米 麺 魚
"Excuse me, I must go check on my orange peels, they've been in the oven too long," says Chikako's mother. We watch as she disappears off the screen. She keeps talking off camera—something about dehydrating techniques that we can't quite make out—and a few minutes later she comes back with a glass jar and a beaming smile. "Would you like to see my marmalade?"
Chikako and Tomiko talk on Skype at least once a week. "I'll call her on the phone with a question and she'll say, 'Let's get on Skype!' And we'll be there until midnight in the kitchen talking until I tell her I have to go." These conversations typically take place in the kitchen, often with both women in the midst of a seasonal project. Chikako sets her iPad up on the counter, and the two go to work.
As they talk today, Chikako cleans her way through a bucket of _haka haka_ , tiny silver fish Ben brought back from the morning market, removing the heads and guts from a thousand little fish in preparation for another long ferment. They talk about life, about Tomo and Emily, Ben and Chikako's kids, but mostly they talk about food—the new experiments, the bottles in the basement, the tiny pieces that hold this whole world together.
You won't find many women in the professional kitchens of Japan. The traditional structure for a family-owned restaurant involves the father running the kitchen, the mother controlling service, and son and daughter—if involved—divided along the same lines. Deep-rooted domestic roles and the odd backward belief arguably make the gender division here worse than you'd find in other parts of the world; some believe, for instance, that women shouldn't make sushi because fluctuations in their body temperature would compromise the fish. There are, of course, women working hard to dissolve these divisions in restaurant kitchens across the country, but it's mostly men you find slicing fugu, boiling soba, battering vegetables, and working the grills, griddles, and stovetops of Japan.
The entrance to Flatt's Inn
_(Michael Magers, lead photographer)_
But behind closed doors, women are the ones who feed this country. More than domestic cooks, they are the guardians of secrets, keepers of the culinary flame, the ones who work silently to safeguard Japan's remarkable food culture. At the heart of this preservation is the mother-daughter relationship.
When Chikako tells me her first batch of marmalade was a mess, her mother is quick to explain why. "That's because you didn't use enough pectin," says Tomiko.
"She's right, so I turned it into miso instead."
Mom is never far away. Tomiko and Toshihiro still spend plenty of time at their old inn, and when they come, Chikako and Ben know that their progress will be tested. Mom and Dad lift the lids, probe the fish, squeeze the daikon, smell the _ishiri_ , test the vinegar, taste the _yunamba_ , inspect the garden. "It's intense. They run us ragged," says Chikako. They always find problems—room for improvement, let's say—and they try their best to provide guidance without outstaying their welcome.
Tomiko and Toshihiro are quick to point out how grateful they are that Chikako and Ben have made every effort to keep their vision alive. Beyond the bounty in the basement, they organize springtime picnics and an annual autumn beer garden, they gather with other local inn owners to trade recipes and industry tales, they help government officials promote the area. This is a fragile moment for the cuisine of Noto, and the couple does everything they can to share a way of life with people who may not have benefited from parents as exacting as Tomiko and Toshihiro.
Running the business and helping in the community would be more than enough for even a highly functioning couple to handle alone, but the real work takes place outside, around the garden, on the docks, in the forest—all around them. There are ferns growing down by the river: they must be picked. Seaweed has started to wash up along the shore below the bluff: time to lay it out for drying. Mushrooms cling to logs, begging to be plucked and dried: it's time. The squid needs salting, the fruit needs fermenting. The moment is now.
There is a subtle sense of urgency to these tasks, because when Mom passes, so too does the great store of knowledge she has accumulated over the years. There is no book, no repository of culinary know-how; no recipe would ever suffice. There are only the seasons, and those who have lived through them.
"We don't have that much outside exposure in Noto," says Tomiko. "You learn about food from your mom, and if Mom's not a good cook, you probably won't be either."
"My father knew that the flavors that he was tasting weren't the same as the childhood flavors," says Chikako, "and he wanted to return to those."
"We didn't want to lose those old flavors," says Tomiko. "We never went to the store. We weren't just making dashi, we were making the ingredients for dashi. We dried the kombu, we made the _katsuobushi_ ourselves."
"For my mom and my grandma, it was never about saving certain techniques, it was just what they did," Chikako says. "But now we really are losing these traditions. To keep it alive means producing it yourself."
"Nature is very generous here, so there is much to do," says Tomiko. "But only people who know about this through experience know what to do with these products."
"You have to work so hard these days to get the perfect ingredients," says Chikako, nearing the end of her pile of fish. "You have to grow everything yourself, pick everything yourself."
"Noto food is the perfect cipher," says Tomiko, "because it can only exist in Noto."
"I'll stand right next to my mom in the kitchen and make the same recipe with the same ingredients using the same technique, and mine will turn out different every time."
"It's a training with your body," says Tomiko. "It's like a sushi chef, it's an exact routine that your body just knows. An instinct more than something you can explain."
"There is knowledge I just don't have," says daughter, down to her last few fish, her cutting board dark with blood and guts. "I know less than half. I know that much."
"There's a season cycle," says mother, "and Chikako is following the season cycle, and the more she knows, the more she will need to learn each season."
"I don't know enough about picking mushrooms," says daughter. "I don't know wild game."
"You must be patient," says mother.
"I should know more about mountain vegetables," says daughter.
"I never learned the mountain pig," says mother. "That is one of my regrets."
"I don't know what I don't know," says daughter, her hands purple with the fish departed. "That's what bothers me most."
"You're doing it right," says mother. "You're almost there. Almost."
## _One Night with the_
## **GEISHA**
_(Matt Goulding)_
## **_Movable Feasts_**
## [弁当
**THE BEAUTY OF BENTO**](nav.xhtml#rhh21)
_(Matt Goulding)_
Japan is a country made for train travel. It's not just the sleek Shinkansen that snake silently through the countryside; it's an entire culture of train cuisine developed around Japan's preferred method of transportation. That means cold beer, hot tea, salty snacks, and a steady supply of _ekiben_ , first-class bento boxes based on regional specialties and sold exclusively in train stations.
The first _ekiben_ was created in 1885. Since then, more than two thousand local bento have been developed, mostly by small, family-run operations, giving you a chance to taste a town—the grilled beef tongue of Sendai! the buckwheat buns of Nagano!—without ever leaving the train platform. Of course, your goal should be to hunt down the _ekiben_ at their source, but if you need to cheat, you can head to Matsuri in Tokyo Station, which offers 170 _ekiben_ from around the country.
After ten thousand kilometers and over a hundred train meals, these _ekiben_ have emerged as the finest movable feasts in Japan.
_(Matt Goulding)_
**UNI, IKURA, TAMAGO**
_Hakodate Station_
The best eggs in Hokkaido combined in one beautiful bowl: creamy curls of sea urchin, briny orbs of salmon roe, and soft, sweet deposits of chicken eggs, with the vinegar twang of pickled vegetables to tie it all together. Best when washed down with one of Hokkaido's many microbrews.
_(Matt Goulding)_
**TOHGE-NO-KANEMESHI**
_Yokokawa Station_
A treasure trove stuffed full of Yokokawa's most famous flavors: tender soy-marinated chicken thigh, fat caps of shiitake, bamboo shoots, sweet chestnut, and a single boiled quail's egg. All served in _mashiko ware_ , a clay pot perfect to take home.
_(Matt Goulding)_
**KASHIWA MESHI**
_Tosu Station_
This is the most famous of the hundreds of chicken-based _ekiben_ —for a reason. The shaved marinated chicken, the shreds of fried egg, and the rice cooked in a rich chicken stock hit all the right notes: sweet, savory, umami-rich, and perfectly portable. (The side of juicy shumai dumplings doesn't hurt, either.)
_(Matt Goulding)_
**MASU-NO SUSHI**
_Toyama Station_
A peerless example of Japanese pressed sushi: thin slices of rosy river trout spackled with a thin layer of Kewpie mayo and draped over a flotilla of pressed rice like a savory cake for adults. Unchanged since 1912 and so good that Japanese go to great lengths to buy it and bring it to friends and family around the country.
_(Matt Goulding)_
**ANAGO MESHI**
_Miyajima Station_
One of the oldest and greatest _ekiben_ in all of Japan. The same family has been making this beauty since 1901, roasting the saltwater eel over charcoal, glazing the meat with soy, simmering the rice in eel stock. It's best in the restaurant when the eel comes directly off the grill, but the _ekiben_ (also available in Hiroshima Station) makes for all-world road food.
## ACKNOWLEDGMENTS
It takes a village to publish a book, but it takes a nation to publish one about Japan—at least it does when you are as clueless as I was when I first touched down in Tokyo years ago. My primary debt is to the people of Japan, whose extraordinary generosity turned an incomprehensible country into a place of abiding beauty.
This book would not be possible without the help of the following people, in particular:
Ioanna Morelli showed incredible grace and skill in being my interpreter—of language, culture, behavior, everything—through most of the book's research. I owe much of what I know and love about this country to her; her husband, Hisashi; and their friends.
Ken Yokoyama, a _shokunin_ in the art of hospitality, performed minor miracles to help me better understand and appreciate Kyoto. He did so with the selflessness and precision that represents the very best of his country and his people.
I spent many an Osaka night in the expert hands of Yuko Suzuki, eating and drinking and learning things that would be inconceivable without her generous gift of time, talent, and spirit.
Hisaichiro Yanagihara (a wise and generous master of the food scene in Fukuoka).
Robbie Swinnerton should have laughed me out of his adopted country when I first told him about the idea for this book. Instead, he selflessly shared the expertise that makes him one of the great translators of Japanese culinary culture for the English-speaking world.
And the dozens of people who welcomed me into their restaurants, homes, inns, and lives: the Matsuno family of Arashiyama, Ben and Chikako Flatt, Fernando and Makiko Lopez, Kamimura Toshiyuki, Robert Yellin, Yoshiteru Ikegawa, Shinji Nohara, Brian MacDuckston, Gen Yamamoto, Nick Szasz and the ace staff of Fukuoka Now, Eric Eto, Sojiki Nakahigashi, Toshiro Ogata, Mick Nippard, Sanada Kodai, Sander Jackson Siswojo, and Miriam Goldberg.
Special thanks to Lauren Scharf and the Art of Travel crew in Kanazawa for showing me that Ishikawa Prefecture is deserving of a book itself. And to the fine folks at the JNTO offices in New York and Tokyo for supporting this project back when it was nothing more than a naive idea.
Closer to home, I owe pretty much everything I write to Nathan Thornburgh, my partner at Roads & Kingdoms, who has been the creative force behind this project since its genesis. You are the finest editor I know, and yet, somehow, an even better friend and collaborator.
Douglas Hughmanick, an alchemist of the highest order, could turn a few scribbles and a stack of Polaroids into a work of art. You've once again broken new ground with the design of this book. Thanks for always making us look good.
A huge thanks to Michael Magers, an endless source of positive energy, photographic excellence, and _conbini_ love: your images say all the things that my words can't.
Tony Bourdain blew the doors off the food-writing world many years ago and continues to expand its boundaries to parts unknown. Anyone who writes about food and travel is in your debt—me more than anyone. Thanks for believing in Roads & Kingdoms and everything that we do.
Kim Witherspoon knows how to navigate the turbulent waters of the New York publishing world with preternatural ease and precision. Thanks for steering this ship safely into port.
To Karen Rinaldi, for believing that there was life after _Eat This, Not That!_ , and for providing all the support and creative freedom to bring it to fruition. Her team at Harper Wave—including Hannah Robinson, Leah Carlson-Stanisic, and John Jusino—have helped us shape this book with incredible skill and patience.
And, above all, to my wife, Laura, my not-so-secret weapon, whose grace and beauty is the skeleton key to a world of closed doors.
## ABOUT THE TEAM BEHIND _RICE, NOODLE, FISH_
_(Michael Magers, lead photographer)_
**Matt Goulding** is a cofounder of Roads & Kingdoms and the coauthor of the _New York Times_ bestselling series _Eat This, Not That!_ , a series with more than 10 million books in print. He divides his time between the tapas bars of Barcelona and the barbecue joints of North Carolina.
**Nathan Thornburgh** is a cofounder of Roads & Kingdoms, where he puts all his previous careers—as a musician, a foreign correspondent for _Time_ magazine, and an accomplished drinker—to good daily use.
**Douglas Hughmanick** is the head of the Roads & Kingdoms design department. He also founded and operates ANML, a digital design studio in the San Francisco Bay Area.
Discover great authors, exclusive offers, and more at hc.com.
## ABOUT ROADS & KINGDOMS
_(Michael Magers, lead photographer)_
**Roads & Kingdoms** is a digital media company at the intersection of food, travel, politics, and culture. Its partners have included Tumblr, _Sports Illustrated_ , _Time_ , and the voracious curiosity of Anthony Bourdain. Check out more of our work at roadsandkingdoms.com.
**NOW THAT YOU HAVE THE INSPIRATION, GET THE INFORMATION.**
Find intel on the best places to eat, drink, and sleep across the seven regions covered in this book, all available in the palm of your hand. For more information, go to roadsandkingdoms.com/japan.
_(Michael Magers, lead photographer)_
## CREDITS
COVER DESIGN BY DOUGLAS HUGHMANICK
## COPYRIGHT
RICE, NOODLE, FISH. Copyright © 2015 by Matt Goulding and Nathan Thornburgh. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.
FIRST EDITION
Library of Congress Cataloging-in-Publication Data
Goulding, Matt.
Rice, noodle, fish : deep travels through Japan's food culture / Matt Goulding ; edited by Nathan Thornburgh. — First Edition.
pages cm
ISBN 978-0-06-239403-3
EPub Edition October 2015 ISBN 9780062394040
1. Food habits—Japan. 2. Food tourism—Japan. 3. Goulding, Matt—Travel—Japan. I. Title.
GT2853.J3G68 2015|
---|---
394.1'2—dc23| 2015005013
15 16 17 18 19 ID/QGT 10 9 8 7 6 5 4 3 2 1
## ABOUT THE PUBLISHER
**Australia**
HarperCollins Publishers Australia Pty. Ltd.
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www.harpercollins.com.au
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www.harpercollins.co.uk
**United States**
HarperCollins Publishers Inc.
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New York, NY 10007
www.harpercollins.com
| {
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} | 1,606 |
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First EVs Picked up Through Statewide, Multi-Agency Service Contract Arrive
April 14, 2021, 10:48 AM HST
2 minutesLoading Audio... Article will play after ad...
Warren Carsey, Sustainability Partners's Senior Investor in the Hawaiian Islands, handing off the keys to Mike Medeiros, HDOT Highways Oʻahu District Engineer. PC: Hawaiʻi Department of Transportation.
The Hawaiʻi Department of Transportation took a step forward in electrification of its light duty fleet with the arrival of the first electric vehicles (EVs) procured through the State's EV as a service contract.
The EV as a service contract allows HDOT and other interested state and county agencies to procure EVs and charging infrastructure on a per mile cost basis. Use of this service contract is expected to save approximately 75 percent in vehicle maintenance over the lifespan of the vehicle and an average of $287 per vehicle per year in fuel costs.
On Tuesday, April 13, the first of nine EVs to serve as vehicle replacements was delivered to State Highways. One EV was picked up by the Hawaiʻi State Energy Office through the contract as well. By the end of May HDOT will replace an additional 34 internal combustion engine vehicles with EVs.
"We were definitely excited to begin the service contract with Sustainability Partners as converting our aging vehicles to EVs is another way HDOT is saving money and working towards the State's goal of reducing fuel consumption in ground transportation 70-percent by 2030," said Hawaiʻi Department of Transportation Deputy Director for Highways Ed Sniffen. "Public and private ground transportation is a huge contributor to carbon dioxide emissions. This service contract, that is available to all State and County agencies, could expedite government fleet conversions and help lead the way for increased private adoption of EV."
"The partnership between the Hawaiʻi Department of Transportation and the State Energy Office in this important program demonstrates how state government can lead by example," said Chief Energy Officer Scott Glenn. "Through this innovative contract, any state or county agency can take part in this. The State Energy Office stands ready to help any agency through this so they can continue to lead on Hawaiʻi's clean energy economy."
Each EV replacing an internal combustion engine vehicle will save an estimated 8,700 pounds of carbon dioxide annually. For this first round of 10 EVs that would be approximately 87,000 pounds of carbon dioxide that will not be released into the atmosphere.
HDOT Highways will continue to pursue electrification or elimination of its light duty fleet within the next seven years. The current light duty fleet is made up of 300 vehicles statewide. The fleet conversion will be used as an opportunity to reevaluate the need for these vehicles. By 2028 HDOT Highways will either remove unnecessary internal combustion engine vehicles from the fleet entirely or replace it with an EV.
State and county agencies interested in learning more about the service contract or benefits of EV conversions can visit https://hidot.hawaii.gov/highways/electric-vehicles/ for cost comparisons and contact information.
EV charging. PC: Hawaiʻi Department of Transportation.
Residential Specialist
Multi-Agency Teams Evaluate Maui Property Damage… December 20, 2021
More EVs could reduce CO2 emissions by 93% in less… July 8, 2021
Wailea Picked by AARP Fans as Top Second Honeymoon… August 12, 2019
2,104 Cigarette Butts Picked Up at Kahului Harbor as… November 21, 2021
First Hawaiian Bank Launches Statewide Slippah Drive… July 20, 2021
The Agency Launches Its First Hawai'i Franchise… September 9, 2021
This comments section is a public community forum for the purpose of free expression. Although Maui Now encourages respectful communication only, some content may be considered offensive. Please view at your own discretion. View Comments (5) | {
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The Disappearance of Sherlock Holmes
Larry Millett
Nonstop suspense unfolds as Sherlock Holmes and Dr. Watson travel from London to New York to Chicago in pursuit of a vicious and cunning killer
Chasing a kidnapper from London to New York to Chicago, Holmes and Watson race to keep up. Every move Holmes makes is expected; every trap proves elusive. Only with the assistance of his American cohort, the saloonkeeper Shadwell Rafferty, can Holmes hope to settle the score once and for all—or be framed for the crime himself.
Millett recreates the world of Holmes with uncanny precision.
Minnesota and the Upper Midwest, Literature, Fiction
SHERLOCK HOLMES DISAPPEARS, POLICE SUSPECT FAMED DETECTIVE IN KIDNAPPING AND MURDER reads a New York headline. So begins the fifth mystery in Larry Millett's series.
A letter, written in a secret cipher he recognizes all too well, reveals that an old foe of Holmes—a murderer he once captured after an incredible duel of wits—is back, has kidnapped his previous victim's widow, and is now impersonating Holmes himself. Holmes must once again match wits with a particularly cunning adversary, one whose hatred of Holmes has seemingly become the killer's single greatest obsession.
Chasing the kidnapper from London to New York to Chicago, Holmes and Watson race to keep up. Every move Holmes makes is expected; every trap proves elusive. Only with the assistance of his American cohort, the saloonkeeper Shadwell Rafferty, can Holmes hope to settle the score once and for all—or be framed for the crime himself.
352 pages, 5 1/2 x 8 1/2, February 2012
Series: Fesler-Lampert Minnesota Heritage Book
Larry Millett was a reporter and architecture critic for the St. Paul Pioneer Press for thirty years. He is the author of fifteen books, including five other mystery novels in this series featuring Sherlock Holmes and Shadwell Rafferty, all in new editions from the University of Minnesota Press.
Prologue: "You Do Know Who I Am, Don't You?" 1
Book One: England
1. "The Message Is Quite Clear" 11
2. "Money Won't Do You No Good" 23
3. "We Are in the Hands of a Magician" 29
4. "Where Are You Taking Me?" 43
5. "Do You Dream About Her?" 46
6. "I Fear Great Trouble Is Coming Your Way" 54
7. "Now We Must Do a Bit of Heavy Lifting" 60
8. "We Will Be Home in Chicago Before Long" 71
Book Two: New York
9. "I Told Them to Go to Hell" 77
10. "Who Are You?" 88
11. "What Cheek!" 91
12. "Everything Is in Order" 100
13. "Elsie Cubitt Shall Be Free at Last" 105
14. "We Shall Have Him Too" 117
15. "Where Is Holmes?" 120
16. "Love Is a Strange Thing" 131
17. "It Was a Tiger" 135
18. "I Hope You Are Safe" 142
19. "I Will Be Fine, My Dear Watson" 146
20. "Shoot the Bastards If You Can" 149
21. "Let the World Know What I Have Done" 152
22. "I Like Those Odds" 160
Book Three: The Pennsylvania Limited
23. "They Will Not Escape New York" 169
24. "He'll Be All Mine" 173
25. "Do You Remember Alfred Beach?" 183
26. "Your Presence Is Requested" 190
27. "I Think I Know What We've Found" 194
28. "He Won't Be Trying Anything" 205
29. "You Are British to the Core" 214
30. "Something Like That" 218
Book Four: Chicago
31. "Big Wheels Continue to Turn" 225
32. "How Nice of You to Join Us" 236
33. "It Was Strictly a Delivery Job" 249
34. "Perhaps I Am Being Too Hard on You" 256
35. "Trouble Is, We're Not the Indians" 263
36. "My God, I Don't Believe It" 275
37. "For God's Sake, Do Not Shoot" 279
38. "We All Must Leave at Once" 283
39. "This Is a Thing I Must Do Myself" 289
40. "There Will Be No Court" 296
Epilogue: "I Do" 304
Notes 313
Author's Note 340
Sherlock Holmes and the Red Demon
Sherlock Holmes and the Ice Palace Murders
The Magic Bullet A Locked Room Mystery Featuring Shadwell Rafferty and Sherlock Holmes
Sherlock Holmes and the Rune Stone Mystery
Sherlock Holmes and the Secret Alliance
Larry Millett brings Sherlock Holmes to Minnesota
Finding his inner Sherlock Holmes
The Twin Cities Daily Planet features six new editions of Millett's Sherlock Holmes novels, all of which include a "heavy dose of Twin Cities history and architecture."
The ECM Post Review covers a reading with Larry Millett, author of several books that bring Sherlock Holmes to Minnesota. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 586 |
{"url":"http:\/\/math.stackexchange.com\/questions\/40590\/combinatorics-problem-please-check-my-answer-and-reasoning","text":"Just want to check if my answer and reasoning is correct for the following problem (Not a homework problem - it is a sample question for a test I'm preparing for)\n\nIn a survey, viewers were given a list of 20 TV Shows and are asked to label 3 favourites not in any order. Then they must tick the ones that they have heard of before, if any. How many ways can the form be filled, assuming everyone has 3 favourites?\n\nMy reasoning:\n\n1) Choose 3 shows out of 20: $c(20,3)$\n\n2) Choosing 0-17 shows from 17 choices: $c(17,0) + c(17,1) + c(17,2) + ... + c(17,16) + c(17,17)$\n\nWould this be correct? Is there a better way of doing the second part that doesn't involve so many calculations?\n\n-\nWoops- I made a mistake there. Supposed to be multiply 1) and 2) \u2013\u00a0 Arvin May 22 '11 at 4:57\n\nAs for fewer calculations: what you need to do is pick a subset of the remaining 17 shows to represent the shows you have heard of. There are $2^{17}$ possible subsets, so that's what you want. Alternatively, for each of the remaining 17 programs, you can either have heard of it before or not; so you have one of two choices for each of the remaining 17 programs. That means making a choice from 2 possibilities, 17 times, or $2^{17}$ possibilities.\nAnd alternatively, $$C(17,0) + C(17,1) + \\cdots + C(17,17) = (1+1)^{17} = 2^{17}$$ by the binomial theorem, so that's another way to see that the big sum you have is simply $2^{17}$.\nSo the correct answer is $2^{17}\\times\\binom{20}{3}$.","date":"2015-08-01 06:18:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8181491494178772, \"perplexity\": 230.239080362846}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-32\/segments\/1438042988511.77\/warc\/CC-MAIN-20150728002308-00319-ip-10-236-191-2.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.cheenta.com\/belarus-mo-2018-problem-10-5-number-theory\/","text":"Select Page\n\n# Understand the problem\n\nFind all positive integers $n$ such that equation $3a^2-b^2=2018^n$has a solution in integers $a$ and $b$.\n\n##### Source of the problem\n\nBelarus MO 2018 Problem 10.5\n\nNumber Theory\n5\/10\n##### Suggested Book\nAn Introduction to Number Theory\nDo you really need a hint? Try it first!\n\nLet\u2019s check for n = 1. Observe that a = 27, b = 13 gives a solutions for n = 1. What about higher degrees? Can we use this information?\nDoes it work for n = 2? Let\u2019s prove something general! Prove that for a, b to have solutions, n must be odd.\nIf n is even, Take $\\pmod{3}$ to see that $-b^2\\equiv 1\\pmod{3}$, which has no integer solutions in $b$. Hence, n must be odd.\nWell now take n odd. Say $n=2m+1$ for some positive integer $m$. Then, the solution $(a,b)=(27\\times 2018^m, 13\\times 2018^m)$ exists and works.\n\n# Connected Program at Cheenta\n\nMath Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.\n\n# Similar Problems\n\n## Solving a congruence\n\nUnderstand the problemProve that the number of ordered triples in the set of residues of $latex p$ such that , where and is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...\n\n## Inequality involving sides of a triangle\n\nUnderstand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...\n\n## Vectors of prime length\n\nUnderstand the problemGiven a prime number and let be distinct vectors of length with integer coordinates in an Cartesian coordinate system. Suppose that for any , there exists an integer such that all three coordinates of is divisible by . Prove that .K\u00fcrsch\u00e1k...\n\n## Missing digits of 34!\n\nUnderstand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...\n\n## An inequality involving unknown polynomials\n\nUnderstand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...\n\n## Hidden triangular inequality (PRMO Problem 23, 2019)\n\nProblem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...\n\n## PRMO \u2013 2019 \u2013 Questions, Discussions, Hints, Solutions\n\nThis is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you. 1. 42. 133. 134. 725. 106. 297. 518. 499. 1410. 5511. 612. 1813. 1014. 5315. 4516. 4017. 3018. 2019. 1320. Bonus21. 1722....\n\n## Bangladesh MO 2019 Problem 1 \u2013 Number Theory\n\nA basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.\n\n## Functional equation dependent on a constant\n\nUnderstand the problemFind all real numbers for which there exists a non-constant function satisfying the following two equations for all i) andii) Baltic Way 2016 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need...\n\n## Pigeonhole principle exercise\n\nProblem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...","date":"2019-08-18 21:04:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 20, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7880271077156067, \"perplexity\": 970.0512586442389}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027314130.7\/warc\/CC-MAIN-20190818205919-20190818231919-00351.warc.gz\"}"} | null | null |
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When you look around the universe is expanding in all directions, so it ...","date":"2016-05-06 16:50:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8504948019981384, \"perplexity\": 1342.3307243968347}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461861848830.49\/warc\/CC-MAIN-20160428164408-00054-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
Q: How to make BottomSheetDialog fullcreen when expanded and make button attached at bottom of it? Problem
I need to use BottomSheetDialog (com.google.android.material.bottomsheet) for my apps, but it didn't work as i expected, Bottom Sheet Appear cut when it expanded.
My Implementation
inline fun <T : ViewBinding> Context.makeBottomSheetDialog(
crossinline bindingInflater: (LayoutInflater) -> T,
isCancelable: Boolean = true,
isHideable: Boolean = true,
isFitContent: Boolean = true,
peekHeight: Int? = null,
onDismissListener: DialogInterface.OnDismissListener? = null,
): Pair<T, BottomSheetDialog> {
val layout = bindingInflater.invoke(LayoutInflater.from(this@makeBottomSheetDialog))
val dialog = BottomSheetDialog(this).apply {
setContentView(layout.root)
setOnDismissListener(onDismissListener)
setCancelable(isCancelable)
}.apply {
behavior.apply {
setHideable(isHideable)
isFitToContents = isFitContent
if(peekHeight != null) setPeekHeight(peekHeight)
}
}
return Pair(layout, dialog)
}
I've already researched this problem, and everyone suggests creating its own class, but in my case I want it to have a flexible view and easy to call with inline. When I saw the base code of BottomSheetDialog I thougth its because the container (FrameLayout) height not adjusted when BottomSheet is Expanded.
Question
how can I fix this problem? it makes me can't attach the button at the bottom of the view either.
Thank you!
A: There is a way with which you can do this:
In XML layout of the dialog:
<androidx.coordinatorlayout.widget.CoordinatorLayout
android:layout_width="match_parent"
android:layout_height="match_parent">
<RelativeLayout
android:id="@+id/sliderLayout"
android:layout_width="match_parent"
android:layout_height="wrap_content"
app:behavior_hideable="true"
app:layout_behavior="com.google.android.material.bottomsheet.BottomSheetBehavior">
// Place your layout code here and your code should be in one tag that can be any Layout.
</RelativeLayout>
</androidx.coordinatorlayout.widget.CoordinatorLayout>
In the view, where you are calling the dialog, make a BottomSheetBehavior variable of the RelativeLayout type.
private lateinit var bottomSheetBehaviour: BottomSheetBehavior<RelativeLayout>
Then, call your dialog like this,
val dialog = BottomSheetDialog(this)
val dialogBinding = binding of your dialog layout
dialog.setContentView(dialogBinding.root)
bottomSheetBehaviour = BottomSheetBehavior.from(dialogBinding.sliderLayout)
dialog.show()
bottomSheetBehaviour.state = BottomSheetBehavior.STATE_EXPANDED
bottomSheetBehaviour.peekHeight = dialogBinding.sliderLayout.height
This will make your dialog layout go full screen when the user slides up the layout.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,914 |
General Synopsis: We've worked on adding defensive options and making the game less punishing in this build. By incorporating feedback from the community for both the speed and defense areas, we've managed to build the game around a faster pace, with a lot of tools and much more leniency to most features. Getting hit by enemies is not as punishing, as the health has been expanded into lifebars, but we've made enemies able to hit the players in combos and added sub-bosses, to add importance to the strategic element to the game, which we want to make a very important element in the game. We also added a new stage, the Ice Wasteland to the game, as we develop towards finishing the game.
Standing near health items now indicates on the Player's health bar how much health would be recovered from consuming the item.
Enemy AI's seeking function has been improved, meaning enemies now more accurately locate player positions and attacking more intelligently.
Inputs are more lenient, meaning it is now easier to complete the attack combo and also easier to chain attacks into the attack launcher.
Different types of enemy attacks stun players for different durations depending on the attack, depending on strength / damage of attack.
Players no longer have a small window of invincibility after being hit by a non-knockdown attack, opening up the possibility of being comboed by enemies.
The start-up frames of jumping have invincibility.
After completing Jump + Ability, players regain control while mid-air (for Earle, Dove and Spritz).
Edited Branzon's lighting map to better dynamically light his sprite.
Dr. Chaka's AI is now more aggressive.
Move Speed increased by 5%.
Character Synopsis: All characters are sped up to accommodate the new speed of the game. To further balance the characters, we also kept the difference of HP between characters intact while changing to a lifebar HP system. While General changes were the focus of this update, we also rounded out the character Abilities in the two missing Abilities in Spritz's down + Ability and Safford's jump + Ability. Missing Super Arts will be added in the coming updates. We also worked on stabilizing the characters by fixing the bugs that occurred from time to time and will be toying with some changes to the characters here and there.
Attack speed increased by 22%.
Attack speed increased by 15%.
Attack speed increased by 18%.
Attack speed increased by 24%.
Attack speed increased by 6%.
Attack speed increased by 4%.
Attack speed increased by 7%.
+ Walk and run speed have been increased.
Walk Speed: Increased by 30%.
Run Speed: Increased by 14%.
+ All attack recovery frames have been reduced.
+ Increased input leniency across all attacks.
+/- Health increased from 6 HP → 60 HP.
– Increased cost of 'Hero Fist' Skill Upgrade from 1 → 2.
Fixed a bug that would cause his SA2 Fireball to behave inconsistently.
Fixed a bug where Earle's Upgraded SA1 was not completely invincible.
+ Damage and attack speed changes.
Attack speed increased by 5%.
Attack speed increased by 8%.
Attack speed increased by 10%.
Draw speed increased by 5%.
Attack speed increased by 25%.
Walk Speed: Increased by 25%.
Run Speed: Increased by 10%.
Fixed a bug where Dove's Tumble would not render her invincible under certain circumstances.
Fixed a layering bug where Dove's arrow icons would incorrectly layer over one another on certain stages.
Attack speed increased by 12%.
Walk Speed: Increased by 20%.
Run Speed: Increased by 3%.
+/- Health increased from 8 HP → 100 HP.
Spritz slowly creates a non-attacking projectile in front of an opponent that absorbs all enemy projectiles that make contact.
Fixed a bug where Spritz's TK Wrecking Ball (Up + Ability) would not hit enemies consistently.
Fixed a bug where Spritz's TK Handshake (Jump + Ability) would create the gravity orb in incorrect positions.
Fixed a bug where under certain circumstances Spritz's abilities would not longer hit enemies.
Fixed a bug where Spritz's TK Introducer (Neutral Ability) would grab more enemies than intended under certain circumstances.
Fixed a bug where Spritz's SA3 would not disappear when the shield had no more HP.
Fixed a bug where Spritz could be hit out of his SA2 and be unable to launch the projectile.
Fixed a bug where Spritz' gravity orbs would not appear on screen under circumstances.
Fixed a bug where Spritz's SA1 would behave inconsistently.
Adding a strike box on Spritz's SA1 gravity orb if no enemy is found within range.
Fixed a bug where Spritz would not cast his shield on the correct individual if there were not enough players.
Walk Speed: Increased by 15%.
Run Speed: Increased by 5%.
+/- Health increased from 7 HP → 80 HP.
Safford performs a diving attack that knocks down opponents.
Fixed Safford's Attack 2 strikebox, which was appearing too late.
Fixed Safford's Attack 3 strikebox, which was appearing too late.
Fixed Safford's SA1 sound, which was delayed.
Fixed Safford's SA2 sound, which was playing for too long.
Fixed a bug where Safford could use his Abilities during dialogue.
Fixed a bug where pressing certain buttons during control mapping rendered the mapped motion unchangeable.
Fixed a bug where the Main Menu smoke would not be displayed properly on certain screen sizes.
Fixed a bug where defeating Dr. Chaka under certain conditions would affect the next playthrough.
Fixed a bug where game inputs would be stored incorrectly and played when players press a key repeatedly.
Fixed a bug where enemies could be left offscreen and unable to be killed.
Fixed a bug in the Sewers Stage where dialogue would freeze under certain conditions.
Fixed a bug where enemy AI would become unresponsive under circumstances.
Fixed a bug where the Hillbilly enemy would fail to roll consistently when near a wall.
Fixed a bug where enemies knocked out of their attack animation would retain their strike active frames and continuously hurt the player.
Fixed a bug where players could move during dialogue under certain circumstances.
Fixed a bug where Branzon would not face the player under certain circumstances in Training mode.
Fixed a bug where Branzon would fail to properly seek the player after the player is killed in the Tutorial.
Fixed a bug where Branzon would attack with no player nearby under certain conditions.
Fixed a bug where the Space Centrum grenade strike's active frames would remain longer than intended.
Fixed a bug where the player cursor would disappear while in the Skills Menu.
Fixed a bug where an enemy sound would play twice instead of once.
Fixed a bug where under certain circumstances players could be hurt by attacks that should miss them.
Fixed a bug where the SA1 spark sound was being cut off before completion.
Fixed a bug where performing a Super Art would remove more Super Meter than intended under certain circumstances.
Fixed a bug where the player cursor would move twice instead of once in the Control Config menu.
Fixed a bug where players were unable to press pause under certain conditions. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/www.vedantu.com\/question-answer\/find-the-square-root-of-484-using-repeated-class-8-maths-cbse-5ee0badebc96fa4dc1612a08","text":"Question\n\n# Find the square root of 484 using repeated subtraction.\n\nHint: To find a square root, we subtract consecutive odd numbers from the given number till we get zero. The number of steps in the process gives the desired result.\n\nWe have to subtract odd numbers from the given number.\nSo, we start with subtracting 1 from 484 which gives us 483.\nNow, subtract 3 from 483 which is 480.\nThe process still continues till we get the final answer as 0.\nThe number of steps for which this operation has been performed will give the square root of the number.\n484 \u2013 1 = 483\n483 \u2013 3 = 480\n480 \u2013 5 = 475\n475 \u2013 7 = 468\n468 \u2013 9 = 459\n459 \u2013 11 = 448\n448 \u2013 13 = 435\n435 \u2013 15 = 420\n420 \u2013 17 = 403\n403 \u2013 19 = 384\n384 \u2013 21 = 363\n363 \u2013 23 = 340\n340 \u2013 25 = 315\n315 \u2013 27 = 288\n288 \u2013 29 = 259\n259 \u2013 31 = 228\n228 \u2013 33 = 195\n195 \u2013 35 = 160\n160 \u2013 37 = 123\n123 \u2013 39 = 84\n84 \u2013 41 = 43\n43 \u2013 43 = 0\nIn this case we had to subtract 22 times to get zero.\n$\\therefore$ The root of 484 is 22.\n\nNote:\nThe number of steps needed to reach zero when we subtract the consecutive odd number will be the square root of the given number. However if the given number is not a perfect square number we will never get zero. So this method effectively defines the square roots of perfect square numbers only.","date":"2021-05-09 17:43:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5911093950271606, \"perplexity\": 139.97089922041943}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989006.71\/warc\/CC-MAIN-20210509153220-20210509183220-00572.warc.gz\"}"} | null | null |
Hardwood Glazed 15-Light External Door.
Fifteen Clear double glazed lights.
Hardwood SA 15L M&T Glazed Clear. The SA design, fifteen, clear, double glazed windows creating a traditional look for your home. Clear Glass double glazed units. | {
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Where to Spend New Year's Eve in New Zealand
27 November 2017 / Lisa T / Be the first to comment
If you're celebrating this New Year's Eve in New Zealand, then you want to make it one of your most memorable! The 31st December falls in New Zealand's summer season, which can be a definite change of pace for people visiting from the Northern Hemisphere.
To determine where you want to be in the country and what you should be doing, consider if you want to be in the midst of a massive music festival, partying it up in one of the big town centres, or somewhere off the beaten track enjoying a more intimate, but undoubtedly once-in-a-lifetime experience with your new best travel buddies.
New Years Eve festivals
Summer in New Zealand is music festival season. There are plenty of great festivals in all corners of the country from December through March, but two of the most notable happen at New Year's.
Rhythm and Vines
Location: Gisborne, East Cape
Credit: Rhythm and Vines
Renowned for being the first music festival in the world to greet the New Year, the legendary Rhythm and Vines festival is a 3 day camping and music extravaganza amongst the vineyards on the North Island's East Cape. You can get there on Stray's East Coaster route, but keep in mind that accommodations and campsites fill up quickly out there and this one may involve a little more pre-planning than other options due to its remote location.
Rhythm & Alps
Location: Cardrona Valley, Wanaka
Credit: Rhythm & Alps
The South Island's answer to Rhythm and Vines, Rhythm & Alps definitely wins for one of the most scenic places to spend New Year's in New Zealand. This 3 day annual festival takes place just 15 minutes outside Wanaka and usually draws crowds of around 10,000 to see a mix of international and local Kiwi artists. Again, camping is part of the experience, but if you choose to stay in Wanaka township you can catch a shuttle bus to the festival.
Fireworks Displays
If you like being in the heart of the action with plenty of spur of the moment options for New Year's Eve, then you should check out one of the country's main city centres. You can expect almost every town and city to be putting on some sort of public event, but here are some of the top places we know Stray passengers choose to be for the countdown.
Credit: stuff.co.nz
If you've ever watched a TV broadcast highlighting New Year's celebrations, Auckland is usually in the mix because it's the first major city to celebrate the start of the New Year and the famous Sky Tower looks pretty spectacular against a backdrop of fireworks. There will be plenty of parties happening around the city, including a new music festival called Wondergarden down in Silo Park on the Viaduct. Regardless of where you celebrate, make sure you head outside half an hour before midnight to score a good spot to view the fireworks around the Sky Tower!
Credit: QueenstownNZ
We have to mention Queenstown as one of the top NYE spots because New Zealand's party capital definitely pulls out the stops for the occasion. You are guaranteed that that the bars around town will be hosting their biggest parties of the year. Find one you like and stay put or do a backpacker NYE pub crawl if you want to keep things moving. Those wanting to avoid too many teapots should head down to Earnslaw Park on the lakefront for live music and free public events, as well as a place to watch the fireworks at midnight.
Paihia, Bay of Islands
Credit: Intercity
Not a major city centre, but there's something special about the little town of Paihia in the Bay of Islands and we've found that a lot of Stray travellers choose to head up there for New Year's before setting out on the rest of the main circuit. You can spend the day relaxing on the beach or cruising around the islands of the Bay, then catch the ferry over to Russell to raise a glass at the famous Duke of Marlborough pub in the evening. Paihia has one of the best NYE fireworks displays, which can be seen from Paihia, Russell, Waitangi and any of the beaches along the Inner Bay. Just make sure you head down to the beachfront early to stake out a prime viewing spot. Bonus: even if you're catching the Stray bus back to Auckland the next day, the bus doesn't leave until early afternoon, so you can sleep in that morning or maybe even go for a New Year's skydive!
Get off-the-beaten-track
Festivals and bar crawls have their appeal, but there is something to be said for doing things a little differently this New Year's, especially as a foreign visitor. Heading somewhere completely offbeat will stand out in your memory for years to come! The best part about getting off the beaten track with Stray is that you will have automatically have a group and your Driver Guide to party with, and the accommodations usually plan something special in honour of the holiday.
While every Strademark stop is fabulous, we've found that the two-night stops are usually the best ones for New Year's celebrations, because you don't have to worry about catching a bus the next day! You can take a day to recover before carrying on – although most travellers end up doing something adventurous with their free day too!
Blue Duck Station
Credit: Joe Strudwick
If you want a really unique Kiwi NYE experience, you can't beat Blue Duck for being way off the beaten track and totally different from anywhere else. It's ideal for nature lovers who want a more low-key, but still incredibly memorable experience. There's usually a group meal and bonfire to get things started and then you can party the night away under the stars with the station crew who embrace any opportunity for a bush party. Just make sure to stock up on snacks and drinks beforehand since there aren't any convenience stores around!
With beautiful beaches and plenty of sunshine, Abel Tasman is a fantastic place for New Year's. Enjoy the festivities with your group on the night – usually with some sort of fancy dress party – and then head out into the Abel Tasman National Park the next day. Depending on how active you're feeling, choose from canyoning, hiking, kayaking, sailing or skydiving and have the time of your life in one of New Zealand's most beautiful spots. Talk about a special way to start a new year!
The Coromandel is one of the tops spots for local Kiwi families to go for the Christmas and New Year's holidays, but Hahei will still feel more intimate than one of the big towns. Have a traditional Kiwi BBQ with your new friends, maybe venture to the local pub to mix it up with some of the other holiday-makers, and then wake up to see the sunrise on Hahei Beach before spending the day exploring the marine reserve and venturing over to nearby Cathedral Cove for New Year's photos that will make your friends back home jealous.
Want to start the year off with a bang?
There's no better way to start a cracking new year than by ticking off a big bucket list item. The good news is pretty much every activity operator will be open for business on New Year's Day. Our best recommendation is to clear the cobwebs of your hangover by doing something really unique or challenging, such as:
Climb a mountain – The Tongariro Alpine Crossing, Roys Peak and the Mueller track in Mt Cook are all Stray fan faves!
Bungy jump or canyon swing in Queenstown
Do the heli-hike on Franz Josef Glacier
Skydive at any of the amazing jump locations around the country
Top Planning Tip
No matter what you plan to do, our best advice is to book your accommodation early, especially if you're planning on staying in one of the main city centres or attending a music festival. Keep in mind that it's summer in the Southern Hemisphere and you'll be competing with plenty of locals, as well as foreign visitors, for space at some of the most popular destinations.
Also, past years have shown that many Stray travellers pick a spot to stay for a few days for the holidays and then as soon as the calendar ticks over into a new year, it's time to travel, travel, travel! If you can, it's a good idea to book your buses in advance to avoid disappointment, especially after you've had such an amazing time celebrating with new friends.
What do you think, Stray fans? Where will you be spending your New Year's Eve in New Zealand?
Posted in New Zealand, NZ Top Tips
Lisa T
Originally from Delaware, USA. Favourite Stray stop: Blue Duck Station (New Zealand). Fun fact: her knowledge of sci-fi, Disney and British literature make her the Queen of Trivia.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,056 |
\section{Algebraic identities}
In this section, we prove
\begin{proposition}\label{prop:dichotomy}
For all $p\in(0,1)$, $q=1$ or $q=\frac1{\sqrt p}-1$. In particular, $q=1$ if $p\le\frac14$.
\end{proposition}
Let us first recall a key identity from \cite{haslegrave}, relating $p$, $q$ and
\[r\mathrel{\mathop:}=\P((\vec\bullet_1\rightarrow\dot\bullet)\wedge(0\leftarrow\bullet)).\]
\begin{lemma}[{\cite[Lemma 2]{haslegrave}}]\label{lemma1}
$q=\frac{1-p}2(1+q)+r(1-q)+pq^3$.
\end{lemma}
The conclusion will follow from the next lemma:
\begin{lemma}\label{lemma2}
$r=\frac12pq^2$.
\end{lemma}
\begin{proof}[{Proof of Proposition~\ref{prop:dichotomy}}]
Combining Lemmas~\ref{lemma1} and~\ref{lemma2} yields immediately the equation
\[0=1-q-p-pq+pq^2+pq^3\]
hence
\[0=1-q-p(1+q-q^2-q^3
=(1-q)(1-p(1+q)^2),\]
implying, since $q\ge0$, that either $q=1$ or $q=\frac1{\sqrt p}-1$. Since $q\le1$, we conclude that $q=1$ when $p\le1/4$.
\end{proof}
\newcommand{\cev}[1]{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{#1}}}}}}
\begin{proof}[Proof of Lemma 2]
Let us denote by $y_0$ the location of the first particle that reaches 0, if any, and by $y_1$ the location of the particle that annihilates with the first particle $\bullet_1$, if any.
For any configuration $\omega$ of particle locations and speeds in $\{\vec\bullet_1\to\bullet\}$, denote by ${\rm rev}(\omega)$ the configuration obtained by reversing the interval $[x_1,y_1]$, that is, the configuration where particles outside $[x_1,y_1]$ are those of $\omega$, and particles inside $[x_1,y_1]$ are symmetric to those of $\omega$ with respect to $\frac{x_1+y_1}2$ and with opposite speeds.
For $\omega$ in the event defining $r$, that is to say $\omega\in\{0\leftarrow\bullet\}\cap\{\vec\bullet_1\rightarrow\dot\bullet\}$, we clearly have ${\rm rev}(\omega)\in\{0\leftarrow\bullet\}\cap\{\dot\bullet_1\leftarrow\bullet\}$, and notice also that in this case the first bullet reaches $y_1$ before the particle initially at $y_0$ does, i.e.\ $y_1-x_1<y_0-y_1$, and this also holds for ${\rm rev}(\omega)$. Since conversely, for $\omega\in \{0\leftarrow\bullet\}\cap\{\dot\bullet_1\leftarrow\bullet\}\cap\{y_1-x_1<y_0-y_1\}$, we have ${\rm rev}(\omega)\in\{0\leftarrow\bullet\}\cap\{\vec\bullet_1\rightarrow\dot\bullet\}$, we conclude that ${\rm rev}$ is a bijection between $\{0\leftarrow\bullet\}\cap\{\vec\bullet_1\rightarrow\dot\bullet\}$ and $\{0\leftarrow\bullet\}\cap\{\dot\bullet_1\leftarrow\bullet\}\cap\{y_1-x_1<y_0-y_1\}$. Because ${\rm rev}$ preserves the measure, it follows that
\[\P\big((0\leftarrow\bullet)\wedge(\vec\bullet_1\rightarrow\dot\bullet)\big)=\P\big((0\leftarrow\bullet)\wedge(\dot\bullet_1\leftarrow\bullet)\wedge(y_1-x_1<y_0-y_1)\big).\]
We have $ \{0\leftarrow\bullet\}\cap\{\dot\bullet_1\leftarrow\bullet\}=\{\dot\bullet_1\leftarrow\bullet\}\cap\{y_1\leftarrow\bullet\}_{(y_1,\infty)}$, so that, conditional on that event, the distances $y_1-x_1$ and $y_0-y_1$ are independent and have the same distribution, which is atomless. Therefore,
\[\P\big((0\leftarrow\bullet)\wedge(\dot\bullet_1\leftarrow\bullet)\wedge(y_1-x_1<y_0-y_1)\big)=\frac12\P\big((0\leftarrow\bullet)\wedge(\dot\bullet_1\leftarrow\bullet)\big).\]
To conclude, we finally have
\[\P\big((0\leftarrow\bullet)\wedge(\dot\bullet_1\leftarrow\bullet)\big)=\P\big((\dot\bullet_1\leftarrow\bullet)\wedge(y_1\leftarrow\bullet)_{(y_1,\infty)}\big)=pq^2.\]
\end{proof}
\section{A priori regularity properties}
Let us prove the following result, which in combination with Proposition~\ref{prop:dichotomy} immediately gives Theorem~\ref{thm:main}.
\begin{proposition}\label{prop:connectivity}
For all $p\in(\frac14,1)$, $\theta(p)>0$.
\end{proposition}
The proof follows from the two lemmas below. These lemmas respectively rely on two different characterizations of the supercritical phase $\{p\st\theta(p)>0\}$ by means of sequences of conditions about finite subconfigurations. Let us already warn the reader that the definition and properties of the more involved characterization are postponed until the next section.
\begin{lemma}\label{lem:subcrit_open}
The set of subcritical parameters $\{p\in(\frac14,1)\st\theta(p)=0\}$ is open.
\end{lemma}
\begin{lemma}\label{lem:supercrit_open}
The set of supercritical parameters $\{p\in(\frac14,1)\st \theta(p)>0\}$ is open.
\end{lemma}
\begin{proof}[{Proof of Proposition~\ref{prop:connectivity}}]
As a conclusion of the above lemmas, the set $A=\{p\in(\frac14,1)\st\theta(p)=0\}$ is both open and closed in $(\frac14,1)$. By connectivity of this interval, it follows that either $A=(\frac14,1)$ or $A=\emptyset$. Since we already know (cf.~\cite{sidoravicius-tournier}) that $A\subset(\frac14,\frac13)$, we deduce that $A=\emptyset$.
\end{proof}
\begin{proof}[{Proof of Lemma~\ref{lem:subcrit_open}}]
We have $q=\limup_k q_k$ where, for all $k\in\mathbb{N}$,
\[q_k=\P((0\leftarrow\bullet)_{[0,x_k]}),\]
which gives, using Proposition~\ref{prop:dichotomy},
\begin{align*}
\{p\in(\frac14,1)\st\theta(p)=0\}
& =\{p\in(\frac14,1)\st q=1\}\\
& =\{p\in(\frac14,1)\st q>\frac1{\sqrt p}-1\}=\bigcup_{k\in\mathbb{N}}\{p\in(\frac14,1)\st q_k>\frac1{\sqrt p}-1\},
\end{align*}
and each $q_k$ depends only on a configuration of $k$ particles, hence by conditioning on the speeds of these particles we see that $q_k$ is a polynomial in $p$ and therefore is continuous. The lemma follows.
\end{proof}
\begin{proof}[{Proof of Lemma~\ref{lem:supercrit_open}}]
Using the notation $N_k$ from the next section, the upcoming Proposition~\ref{prop:characterization} gives
\[\{p\in(\frac14,1)\st\theta(p)>0\}=\bigcup_{k\in\mathbb{N}}\Big\{p\in(\frac14,1)\st\mathbb{E}[N_k]>0\Big\}, \]
so that the lemma follows by noticing that, as can be seen by conditioning on the speeds of the $k$ particles, the function $p\mapsto\mathbb{E}[N_k]$ is polynomial hence continuous.
\end{proof}
\section{Characterization of the supercritical phase}
While Lemma~\ref{lem:subcrit_open} relies on the simple monotone approximation $q=\limup_k q_k$, where for all $k\in\mathbb{N}$ the probabilities $q_k=\P((0\leftarrow\bullet)_{[0,x_k]})$ depend only on a configuration of $k$ particles, Lemma~\ref{lem:supercrit_open} relies on a formally similar but more involved characterization. This characterization is already alluded to in the first of the final remarks of~\cite{sidoravicius-tournier} as a way to numerically upper bound $p_c$. Given its importance in the present proof, we give it here a more thorough presentation, and show it is necessary and sufficient.
For all $k\in\mathbb{N}$, consider a random configuration containing only the $k$ bullets $\bullet_1,\ldots,\bullet_k$ (initially located at $x_1,\ldots,x_k$), and denote by $N_k$ the difference between the number of surviving stationary particles and the number of surviving left-going particles: letting $I_k=[x_1,x_k]$,
\[N_k\mathrel{\mathop:}=\sum_{i=1}^k({\bf 1}_{\dot\bullet_i}-{\bf 1}_{\cev\bullet_i}){\bf 1}_{(\bullet\not\rightarrow \bullet_i)_{I_k}\wedge(\bullet_i\not\leftarrow \bullet)_{I_k}}\]
In the following, the event in the last indicator function will be written ``$(\bullet_i\text{ survives})_{I_k}$''.
\begin{proposition}\label{prop:characterization}
For all $p\in(0,1)$, $\theta(p)>0$ $\Leftrightarrow$ $\exists k\ge1,\ \mathbb{E}[N_k]>0$.
\end{proposition}
\paragraph{\bf Remark.} The fact that $\mathbb{E}[N_1]=\frac12(3p-1)$ recovers (cf.~\cite{sidoravicius-tournier}) that $\theta(p)>0$ when $p>1/3$. The proof of this fact in~\cite{sidoravicius-tournier} is in fact the scheme for the general one given below. Considering $\mathbb{E}[N_2]$ gives the same condition, however $\mathbb{E}[N_3]=3p^3+7p^2\overline p-\frac32p\overline p^2-8\overline p^3$ (where $\overline p=\frac{1-p}2$) yields the value $0.32803$ from the remark in~\cite{sidoravicius-tournier}. As the proposition shows, pushing this method further would give arbitrarily good numerical approximations of $p_c$. Let us remind that, although such approximations are rendered pointless by Theorem~\ref{thm:main}, the \textit{existence} of this method still is a theoretical tool in the proof of the said theorem.
\begin{proof}
\noindent{\it Direct implication.}
Assume that $\theta(p)>0$. Let us decompose $N_k=\dot N_k-\cev N_k$, where $\dot N_k$ and $\cev N_k$ respectively denote the number of stationary and left-going particles among $\bullet_1,\ldots,\bullet_k$ that survive in restriction to $[x_1,x_k]$.
For any integer $i$, the event $\{\dot\bullet_i\text{ survives}\}_I$ decreases with the interval $I$ (containing $x_i$). If indeed $\bullet_i$ is stationary and is annihilated by a bullet inside an interval $I$, then introducing new bullets outside $I$ can possibly change the side from which $\bullet_i$ is hit, but not the fact that this bullet is hit. In particular, the number of stationary bullets among $\bullet_1,\ldots,\bullet_k$ that survive in restriction to $[x_1,x_k]$ is larger than or equal to the number of such bullets that survive in ``restriction'' to the whole real line. Taking expectations, by translation invariance of the process on $\mathbb{R}$ this gives
\[\mathbb{E}[\dot N_k]\ge k \P\big((\dot\bullet_1\text{ survives})_\mathbb{R}\big)=kp\theta(p),\]
hence in particular $\mathbb{E}[\dot N_k]\to+\infty$ as $k\to\infty$.
On the other hand, $\mathbb{E}[\cev N_k]$ is uniformly bounded in $k$. Indeed, $\cev N_k$ clearly grows with $k$, and its limit $\cev N_\infty=\limup_k \cev N_k$ is the number of surviving left-going particles in $(0,\infty)$, and this number has geometric distribution with parameter $1-q>0$ (notice indeed that the configuration on the right of a surviving left-going particle is identically distributed as the configuration on $(0,\infty)$, up to translation) and therefore is integrable.
We conclude that $\mathbb{E}[N_k]=\mathbb{E}[\dot N_k]-\mathbb{E}[\cev N_k]\ge kp\theta(p)-\frac q{1-q}\to +\infty$ as $k\to\infty$, hence $\mathbb{E}[N_k]>0$ for large~$k$.
\noindent{\it Reverse implication.}
Assume now that $\mathbb{E}[N_k]>0$ for some $k\ge1$.
For positive integers $i<j$, define $N(i,j)$ in the same way as $N_k$ except that only the bullets $\bullet_i,\ldots,\bullet_j$ are considered instead of $\bullet_1,\ldots,\bullet_k$. With this notation, $N_k=N(1,k)$. This function $N$ satisfies ``almost'' a superadditivity property.
\begin{lemma}\label{lem:superadditivity}
Let $k<l$ be positive integers. For any configuration $\omega$ which, in restriction to $[x_1,x_k]$, has no surviving right-going particle, we have
\[N(1,l)\ge N(1,k)+N(k+1,l).\]
\end{lemma}
\begin{proof}[{Proof of Lemma~\ref{lem:superadditivity}}]
When the configurations in $I=[x_1,x_k]$ and in $J=[x_{k+1},x_l]$ are combined, the surviving left-going particles from $J$ can interact with particles from $I$. Each of them either annihilates with a surviving stationary particle (hence giving the same $0$ contribution to both hand sides) or annihilates with a stationary particle that was annihilated in restriction to $I$ hence unleashes its right-going peer which can either survive (making the left-hand side greater by 1), annihilate with a surviving left-going particle (making the left-hand side greater by 2), annihilate with a surviving stationary particle (keeping sides equal) or again annihilate with a stationary particle that was annihilated in restriction to $J$ hence unleash its left-going peer which is offered the same range of possibilities as the particle we first considered. Thus in any case the identity remains satisfied after the effect of each of these left-going particles is taken into account.
\end{proof}
We shall progressively explore the configuration, starting from 0 and going to the right, by repeating the following two steps: first, discover the next $k$ particles, and then discover the least necessary number of particles until there is no surviving right-going particle in the whole discovered region. We will denote by $K_0=0, K_1, K_2,\ldots$, the number of particles discovered in total after each iteration, and by $\widetilde N^{(1)}(=N_k),\widetilde N^{(2)},\ldots$ the quantity computed analogously to $N_k$ but on the newly discovered block of $k$ particles at each iteration, i.e., for all $n$, $\widetilde N^{(n+1)}=N(K_n+1,K_n+k)$. Let us explain the first iteration in some more detail.
We start by considering the first $k$ particles. Let $\widetilde N^{(1)}=N(1,k)$. If, in the configuration restricted to $[x_1,x_k]$, no right-going particle survives, then we let $K_1=k$. Otherwise, let $\tau_0$ denote the index of the leftmost surviving right-going particle, and appeal for instance to~\cite[Lemma 3.3]{sidoravicius-tournier} to justify the existence of a minimal $\gamma_1$ such that the event $\{\vec\bullet_{\tau_0}\to\bullet_{\gamma_1}\}_{[\tau_0,\gamma_1]}$ happens, and let $K_1=\gamma_1$. By definition we have that, in both cases, in restriction to $[x_1,x_{K_1}]$, there is no surviving right-going particle and $\widetilde N^{(1)}=N(1,K_1)$. We then keep iterating this construction: define $\widetilde N^{(2)}=N(K_1+1,K_1+k)$, and keep exploring on the right of $\bullet_{K_1+k}$ until no surviving right-going particle remains, define $K_2$ to be the index that was reached, and so on. By this construction, the random variables $\widetilde N^{(n)}$ are i.i.d.\ with same distribution as $N_k$, and for all $n$ we have $N(1,K_n+k)=N(1,K_{n+1})$ and there is no surviving right-going particle in restriction to $[x_1,x_{K_{n+1}}]$. Thus, by repeatedly using the lemma, we have for all $n$,
\[N(1,K_n)\ge \widetilde N^{(1)}+\cdots+\widetilde N^{(n)}.\]
However, by the assumption and the law of large numbers, with positive probability $\widetilde N^{(2)}+\cdots+\widetilde N^{(n)}>0$ for all $n\ge2$. Therefore, still with positive probability, it may be that the first $k$ particles are stationary (hence $\widetilde N^{(1)}=k$) and that $\widetilde N^{(1)}+\cdots+\widetilde N^{(n)}> k$ for all $n\ge2$, so that $N(1,K_n)> k$ for all $n\ge2$. This event ensures that 0 is never hit: indeed after the $n$-th iteration of the exploration (for $n\ge2$) there are at least $k+1$ surviving stationary particles due to the definition of the event, but at most $k$ of them can be annihilated by the particles discovered between $K_n$ and $K_{n+1}$, hence by induction the first stationary particle survives forever and prevents 0 from being hit. Thus $\theta(p)>0$.
\end{proof}
\paragraph{\bf Remark}
In the discrete ballistic annihilation model introduced in~\cite{junge2}, the analog of Lemma 2 is wrong due to triple collisions. The same arguments indeed give
\[\hat r=\P_\mathbb{Z}(D>D')p\hat q^2<\frac12 p\hat q^2,\]
where $D$ is the location of the first particle that reaches zero, and $D'$ is an independent copy of $D$. Since $D$ is integer valued, it holds more precisely that
\[\P_\mathbb{Z}(D>D')=\frac12\P_\mathbb{Z}(D\ne D')=\frac12\big(1-\P_\mathbb{Z}(D=D')\big)\]
and $\P_\mathbb{Z}(D=D')$ can be interpreted as the probability that, on the full line, a given stationary particle is involved in a triple collision.
From $\hat r<\frac12p\hat q^2$, the computation done in the proof of Theorem~\ref{thm:main} shows that, if $\hat q<1$, then $0<1-p(1+\hat q)^2$, hence $\hat q<\frac1{\sqrt p}-1$ and thus the surviving probability of a stationary particle on the full line satisfies
\[\psi(p)= (1-\hat q)^2>\Big(2-\frac1{\sqrt p}\Big)^2=\theta(p).\]
Thanks to this dichotomy, we can argue as in the original model that, for all $p>\frac14$, $\psi(p)>0$ hence furthermore $\psi(p)>\theta(p)$. This comparison was heuristically expected in~\cite{junge2}.
\bibliographystyle{acm}
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} | 9,148 |
Mari Pili Barreda (Lima, 5 de abril de 1969) Actriz y presentadora Peruana. Productora de eventos infantiles en "Sueños & Magia", Coach en "Repotencia tu YO" y Acting Coach.
Filmografía
Televisión
Velo negro, velo blanco (1991) como Cleo.
Buscando a la paquita peruana (1991), presentadora.
Fandango (1992-1993) como Paula.
Gorrión (1994) como la antagonista Susana Valdeavellano.
Canela (1995) como Roxana.
Lluvia de arena (1996-1997) como Lucero.
Leonela, muriendo de amor (1997) como Claudia.
Girasoles para Lucía (1998) como Carlota.
María Emilia, querida (1999) como Norma.
Pobre diabla (2000) como Niní.
Milagros (2000—2001) como Raquel Echevarría (joven) / Melissa Wilson.
Soledad (2001) como Beatriz Aguilar.
Teatro desde el teatro (2003-2006).
Tormenta de pasiones (2004-2005) como Nora López Arnao.
La Beca (2010), co-presentadora.
Cielo dividido (2012) como Ericka.
Botadero (2013) como Claudia
Al fondo hay sitio (2015, algunos capítulos) como Bárbara Áurich, interés amoroso de Lucho Gonzales.
Acusados (2015) como Victoria Gadea
El regreso de Lucas (2016-2017) como Nancy Mezzonet, coproducción Telefé y América Televisión
Cumbia pop (2018) como Martha Del Prado Eizaguirre América Televisión
La Rosa de Guadalupe Perú (2019) como Juliana América Televisión
Películas
En la hora del silencio (corto) como Sofía. Ganador del premio CONACINE (2004)
Doble Juego (2004) como Laura.
Piratas en el callao como Miss Carito (voz).
Django sangre de mi sangre (2017) como María Pia Gonzales Larraín
Utopía, la película (2018) como Pilar Hormazabal
Teatro
Cómo vivir sin un hombre... y no morir en el intento como Mari Pili.
A Chorus Line como Judy Monroe / Casey.
La Alegría de Navidad como Estrella.
El matrimonio perjudica seriamente la salud (Coreógrafa).
Rapunzel como Rapunzel.
El Mago del País de las Maravillas como Bruja Mala del Oeste.
Tus amigos nunca te harían daño como Claudia.
El Principito (Producción y preparación actoral de niños).
El Sombrero Mágico de Giorhini (Dramaturgia y Dirección).
La Bella Durmiente como Aurora (Actriz y coreógrafa).
Cabaret (2009) (Pruducción y Diseño de maquillaje y peinado).
Chicago (2012) (Asistente de Dirección).
El Chico de Oz (2013) (Coaching actoral de los niños y Asistencia de Dirección)
Dos reinas y media (2013) como la Princesa Diana de Gales.
Annie (2013) como Grace Farrell
Mujeres Ligeras en Kontenedores (2014)
Reina por un día (2018-2019) como Julia.
Por Chabuca 2'' (2022)
Referencias
Doble Juego en Blockbuster.com
Mari Pili Barreda Rotten Tomatoes ficha
Demo reel en YouTube
Historias de Éxito: Mari Pili Barreda
Mari Pili Barreda se repone de extraña enfermedad
Actriz logra ola de solidaridad en redes sociales a favor de bomberos
Mari Pili Barreda como Diana de Gales, Diario El Comercio
Mari Pili Barreda como Diana de Gales, RPP
Enlaces externos
Actrices de televisión de Perú
Actrices de cine de Perú
Actores de teatro de Perú
Actores de teatro musical de Perú
Modelos femeninas de Perú | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/www.infoq.com\/news\/2015\/10\/java-state-module-system\/","text":"InfoQ Homepage News Oracle Publishes Report on the State of Java\u2019s Module System\n\n# Oracle Publishes Report on the State of Java\u2019s Module System\n\nThis item in japanese\n\nOracle's Java Platform Group chief architect\u00a0Mark Reinhold\u00a0has published a report on the\u00a0State of the Module System, with an emphasis on what the modularization\u00a0objectives are (and aren\u2019t.) The publication has triggered comments among users due to the apparent overlap with existing frameworks, particularly\u00a0OSGi.\n\nAs explained in the report, and fully detailed in\u00a0JSR-376 and in the\u00a0Module System project page, the module system is meant to address two omissions\u00a0in the current Java accessibility model:\n\n\u2022 Reliable configuration: the current way components access classes from other components through the class path is considerably error-prone, particularly when attempting to use classes that aren\u2019t in the class path or that are present multiple times.\n\u2022 Strong encapsulation: there is no way to restrict the classes that a particular component exposes to other components, every class categorised as public will be accessible from outside.\n\nFull details can be found both in the report and in\u00a0previous InfoQ articles, but to summarize,\u00a0each component\u00a0is typically (but not necessarily)\u00a0represented by a jar file, which includes a module descriptor file called module-info.java with the following structure:\n\nmodule com.foo.bar {\n\nrequires com.foo.baz;\n\nexports com.foo.bar.alpha;\n\nexports com.foo.bar.beta;\n\n}\n\nThe file is structured such that one or more lines of exports will indicate the packages that are to be accessible from other components, and where zero or more lines of requires will indicate the modules that are required by this module. This system provides a method for assessing at compile time whether the access types have the right visibility (i.e. they are public and exported by the required component), and at run time to assess whether the necessary modules are available, without having to inspect the full class path.\u00a0It is here where the\u00a0similarities with OSGi\u00a0are manifest.\n\n## OSGi Background\n\nOSGi is a modularization\u00a0system and service platform for Java that implements a complete and dynamic component model. First proposed in 1998 with\u00a0JSR-8\u00a0and with subsequent\u00a0reviews being published over time (the last in 2014),\u00a0OSGi allows the definition of bundles (akin to modules), which take the form of a JAR file with the following MANIFEST.MF file:\n\nBundle-Name: Hello World\n\nBundle-SymbolicName: org.wikipedia.helloworld\n\nBundle-Description: A Hello World bundle\n\nBundle-ManifestVersion: 2\n\nBundle-Version: 1.0.0\n\nBundle-Activator: org.wikipedia.Activator\n\nExport-Package: org.wikipedia.helloworld;version=\"1.0.0\"\n\nImport-Package: org.osgi.framework;version=\"1.3.0\"\n\n(Sample taken from Wikipedia.)\n\nIt is apparent that in spite of the format differences,\u00a0the intent expressed is similar to\u00a0the Java Platform Module System. Indeed, the similarities between the Java Platform Module\u00a0System and OSGi have been noticed since the initial\u00a0attempts to modularise Java started in 2005 with\u00a0JSR-277, the \u201cJava Module System\u201d. Initially aiming at Java 7, JSR-277 focused on easing distribution and execution of Java artefacts. Despite having a nearly identical name to JSR-376, that\u00a0initiative had slightly different objectives;\u00a0although it was tasked\u00a0to fix the problem of \"reliable configuration\", it did not\u00a0attempt to tackle the issue of \"strong encapsulation\". And in contrast to JSR-376, it also tried to add a versioning model to the Java artefacts. The similarities between these objectives and the functionality provide by OSGi were stark enough for\u00a0the authors to initially consider\u00a0OSGi as a solution, later discarding\u00a0it reasoning that OSGi version control\u00a0was too weak.\n\nJSR-294 was created shortly after that, with the objective of implementing \u201cImproved Modularity Support in the Java Programming Language\u201d. Also aiming at Java 7, this JSR was created to add the concepts of modules (called \u201csuperpackages\u201d) to fix the strong encapsulation problem; this concept aligns with the current Java Platform Module System project. Both JSR-277 and JSR-294 have been dormant ever since they\u00a0were labeled as such in\u00a02012, when the Java 7 target was dropped, and\u00a0are\u00a0superseded by\u00a0JSR-376.\n\nAnother link between OSGi and the modularisation of Java can be found in\u00a0JSR-291, \"Dynamic Component Support for Java SE\" (essentially an implementation for OSGi Service Platform Release 4). That\u00a0JSR made a reference to JSR-277, the original Java Module System, to clarify the difference in scope of both initiatives: JSR-277 would focus on the static module definition to be used by Java, while JSR-291 focused on dynamic components that can be loaded and unloaded on runtime.\n\nFinally, JSR-376 itself also makes a reference to OSGi, mainly to discard it as a valid solution because its scope was\u00a0far greater than\u00a0the Java Framework Module System spec.\n\nGiven all the above, it seems reasonable that a number of users would have difficulty\u00a0differentiating the new Module System and OSGi. However, the conclusion is that the module system and OSGi are understood to be complementary systems serving different purposes, with OSGi being a framework built on top of Java to create an environment where bundles can be managed dynamically in an always-running application, and the module system being a new capability of Java itself that allows for tighter and easier control of statically managed modules.\n\n## Differences between Java Platform Module System and OSGi\n\nTo understand this a bit better, InfoQ talked to Holly Cummins, co-author of Enterprise OSGi in Action. While the following doesn\u2019t intend to be a thorough description of the differences between the Java Platform Module System and OSGi, it provides the reader with a basic understanding of how different their objectives are.\n\nOn one hand, the new Module System in Java will provide an easy way to check visibility of packages and classes at compile time, however, when we asked whether OSGi\u2019s bundles can be used in the same way, Holly declared that \u201cthe answer for this is surprisingly complex\u201d.\n\nOSGi dependencies are expressed in a bundle\u2019s manifest, and there are two basic approaches to creating the manifest: \"code-first\" and \"manifest-first\". With the code-first approach (used in the bnd tool, and the maven bundle plugin), the list of dependencies isn't enforced at compile-time, it's actually generated at compile time. The compilation goes ahead in the usual way, and then the tooling works out what's needed at runtime based on what was needed at compile time. The alternate approach is manifest-first, which is used by the Eclipse PDE tooling. In this approach, dependencies are declared in the manifest, and the Eclipse IDE will use the manifest to work out what classes your code can see, and highlight cases where a dependency is missing. There's a PDE command-line build, and a Maven plugin called Tycho, and both of those enforce the dependencies at compile-time. However, it\u2019s important to note that this visibility isn\u2019t enforced by OSGi itself but by the tools accompanying PDE; since not all teams using PDE use one of those tools, and there's a possibility to miss a trick at compile-time.\n\nAnother key aspect of the new Module System is the ability to restrict the modules that a particular package is exposed to, useful in situations where a set of related modules need to access each other, but they shouldn\u2019t be accessed beyond that. As indicated in Mark Reinhold\u2019s report, this can be achieved using the following syntax:\n\nmodule java.base {\n...\nexports sun.reflect to\njava.corba,\njava.logging,\njava.sql,\njava.sql.rowset,\njdk.scripting.nashorn;\n}\n\nOSGi didn\u2019t initially have this capability, but when it was added it again went further than the objectives of the Module System. As Holly explains, \u201cAny bundle can register a resolver hook, and that can be used to filter matches, so that packages are only exposed to specific bundles. You could use the same mechanism to do things as sensible as exposing packages to bundles which declare certain metadata, or as crazy as exposing a package only on Tuesdays\u201d.\n\nStyle\n\n## Hello stranger!\n\nYou need to Register an InfoQ account or or login to post comments. But there's so much more behind being registered.\n\nGet the most out of the InfoQ experience.\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p\n\nby Peter Kriens \/\n\n\u2022 ##### Re: A Tad Confused\n\nby Ant hony \/\n\n\u2022 ##### Re: A Tad Confused\n\nby Peter Kriens \/\n\n\u2022 ##### Re: A Tad Confused\n\nby Ant hony \/\n\n\u2022 ##### Re: A Tad Confused\n\nby Peter Kriens \/\n\n\u2022 ##### Project Jigsaw might be the best thing that could have happened to OSGi\n\nby Victor Grazi \/\n\nby Peter Kriens \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nThe history of JSR is long and confusing but I am not sure this article helps to clarify the situation. I was especially puzzled by the last section, not sure how to relate this?\n\nTo summarize: The JSR376 proposal consists of a new accessibility mode for modules, a dependency model, and a service model. The dependency model is limited to wiring versionless modules within a given module path. (A class path for modules.) The real dependencies (with versions) will have to come from a build system using propriety conventions. The service model is based on the Java service loader model. Since resources will no longer be accessible between modules a modular extension was needed.\n\nThe new module accessibility will be eagerly used by OSGi to protect the runtime and probably enforce compile visibility. The service loader model will be transparently supported as it is already today. However, not sure what to do with the dependency model. The OSGi provides a much richer dependency model and many patterns (e.g. whiteboard, extender) that support popular application techniques that are far outside the ambition horizon of JSR 376.\n\nJSR 376 will not provide any backward compatibility for existing applications except to run as a classic class path application. Applications that want to move to JSR 376 will have to go through the painful process of modularizing because JSR 376 will punish any illegal crossing of module boundaries. (A lot of people will discover that their babies are not as modular as they thought.) Many people claim OSGi is hard without acknowledging that modularizing applications is the hard part. Mostly because Java (EE) applications have so many unmodular practices. JSR 376 will demonstrate that OSGi was just the messenger and actually not the cause.\n\nIf you go to the pain of modularizing then OSGi seems a much better choice because of its maturity, support, tooling, and feature richness. We've got 15 years of experience and have tested solutions to virtually all the problems that you will encounter when you modularize. Since this article even confused me about OSGi you might be attracted to JSR 376 because it looks so much simpler. Heck, it is simpler! However, when you traverse down the modularization hole of your application you will quickly realize that those additional features OSGi provides actually solve your problems.\n\n\u2022 ##### Re: A Tad Confused\n\nby Ant hony \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nI'm not sure what you meant to say with this\n\nJSR 376 will not provide any backward compatibility for existing applications except to run as a classic class path application.\n\nbut isn't backward compatibility provided by the concept of \"unnamed modules\"?\n\nAs for OSGi: about 3 years ago I wanted to learn it, so I did the Apache Felix tutorials ( felix.apache.org\/documentation\/tutorials-exampl... ). Given that Felix is one of the major implementations, I thought that'd be a good starting point, but I found the tutorials to be quite poor (and even today, several examples are \"Coming soon...\", just as they were back then). Moreover, OSGi occured as rather chaotic to me: e.g. there was iPOJO, BluePrint, Declarative Services, ... and I had a hard time trying to figure out what the current best practices were.\n\nIn contrast: after reading the \"The State of the Module System\", it occurs as a natural extension to the Java platform and it all makes sense to me. In combination with the quick start guide, I already feel like I \"got it\". Of course, it's simpler & doesn't provide all the features OSGi offers, but I'm wondering: why would I need those features? Maybe it's, as you say, because I haven't traversed down the modularization hole yet, but at this point I feel JSR 376 will meet the needs of most applications.\n\nWhat I would like to read a year from now, is a book with the following starting point: a JSR 376-modularized Java SE application, which uses ServiceLoader & CDI 2. And then goes on to explain its problems & how OSGi allows to solve them.\n\n\u2022 ##### Re: A Tad Confused\n\nby Peter Kriens \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nPortability: Since modules will be encapsulated things like annotation processing will not work out of the box. Applications and spec providers will have to be adapted to take modules into account (and this is the way it should be).\n\nAbout the tutorials. Yes, we are aware of it.Since OSGi has been around so long there is just a lot of stale and bad stuff from loads of different people. The OSGi enRoute project (enroute.osgi.org) is an attempt to provide an up to date easy entry in OSGi, hopefully we can get this at the op of Google searches.\n\nThough I would love to accept the challenge to write your proposed book but I lack the funds to pursue this. What I could do is take a simple but typical application and port it to OSGi enRoute to show how powerful OSGi is. Pet store or do you have a better example?\n\n\u2022 ##### Re: A Tad Confused\n\nby Ant hony \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nThanks for your reply and mentioning the OSGi enRoute project in particular. It looks promising & has surely renewed my interest in learning OSGi. As for getting it on top of Google searches: searching for \"OSGi tutorial\" lists www.osgi.org\/Technology\/HowOSGi as the second result. However, there's no mention of the enRoute project there. So giving it a prominent place on that page would already help a lot in increasing its visibility.\n\nAn OSGi enRoute pet store example, which showcases how powerful it is, would definitely be nice. This would allow me to invert the idea of the book I proposed: take the example OSGi application & port it to a non-OSGi application myself (by trying to replace it with JSR 376, ServiceLoader and such). Then I'd see for myself what advantages OSGi has & where JSR 376 falls short.\n\n\u2022 ##### Re: A Tad Confused\n\nby Peter Kriens \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nThanks for the tips, will work with OSGi marketing to correct this.\n\nOk, I will look at the pet shop but it will take some time, I am not full time on this.\n\n\u2022 ##### Project Jigsaw might be the best thing that could have happened to OSGi\n\nby Victor Grazi \/\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nModularization is a new concept to most Java developers. OSGi may be light-years ahead of Project Jigsaw, but most Java developers still struggle to get their arms around it (Ant hony above expressing some common emotions.) and OSGi has accordingly struggled to break in.\n\nThat said, once modularization becomes part of the Java core tool set, developers will begin to embrace it en-masse, and as they do so, they will seek more robust and more mature solutions. Enter OSGi!\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p\n\nIs your profile up-to-date? Please take a moment to review and update.\n\nNote: If updating\/changing your email, a validation request will be sent\n\nCompany name:\nCompany role:\nCompany size:\nCountry\/Zone:\nState\/Province\/Region:\nYou will be sent an email to validate the new email address. 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\section{Quantum Chromodynamics and its ``Effective'' Description
\label {QCD}}
Although the strong interactions were long believed to be responsible for interactions among constituents of nucleons, the weakly interacting feature of the theory, postulated based on the results of deep-inelastic experiments in 1960s, had posed a mystery. The reason was that no gauge theory at the time was known that exhibits a strongly interacting feature at low energies while becomes almost free at high momentum transfers. It was only in early 1970 that 't Hooft, Politzer, Gross and Wilczek found out that the \emph{non-Abelian} gauge theories \cite{PhysRev.96.191}, in four dimensions, possess the desired property \cite{thooft, Politzer:1974fr, Gross:1973id, Gross:1973ju, Gross:1974cs}. The beautiful \emph{quark} model of Gell-Mann and Zweig that was developed in 1960s to describe the rich spectrum of strongly interacting particles, was then integrated into the underlying non-Abelian gauge theory. Interestingly, the extra charge of the quarks, namely their \emph{color}, that was proposed to ensure that the Fermi statistics of spin 1/2 particles is obeyed, had a natural interpretation in this gauge theory as quarks would now belong to a multiplet of the group (in its fundamental representation) and carry a new index. Both the quark model and the experimental evidence, e.g. $e^++e^- \to {\rm hadrons}$ cross section, suggested that there exist three distinct colors, constraining the dimensionality of the gauge group to be three. The guiding principle in constructing the Lagrange density describing quarks and gauge fields, has been the \emph{principle of local gauge invariance} -- a principle that had already played an important role in the description of the simpler gauge theory of QED.
According to the principle of local gauge invariance, in order for the free Lagrangian density of a quark multiplet $q^a$ with $a=1,2,3$ (all of the same mass $m$),\footnote{The summation over repeated indices is to be understood throughout.}
\begin{eqnarray}
\mathcal{L}_{free}=\bar{q}_a(i\gamma^{\mu}\partial_{\mu}-m)q^a,
\label{L-q-free}
\end{eqnarray}
to be invariant under a local rotation in the internal space of the quark multiplet by a unimodular unitary transformation $U$ -- namely a $SU(3)$ transformation,
\begin{eqnarray}
q^a \to q^{a'}={U^{a'}}_b q^b \equiv {(e^{-igT^i\alpha^i(x)})^{a'}}_b q^b,
\label{q-trans}
\end{eqnarray}
there must necessarily exist a vector field $A_{\mu}\equiv A_{\mu}^iT^i$ with $i=1,2,\dots,8$ which minimally couples to the quark fields through the covariant derivative
\begin{eqnarray}
D_{\mu}q^a\equiv \partial_{\mu}q^a+igA_{\mu}^i {(T^i)^a}_b q^b,
\end{eqnarray}
and whose gauge transformation takes the following form
\begin{eqnarray}
A_{\mu} \to A'_{\mu}=U A_{\mu}U^{\dagger}+\frac{i}{g}\partial_{\mu}U U^{\dagger}.
\label{A-trans}
\end{eqnarray}
$T^i$s are the generators of $SU(3)$ Lie algebra, $T^i=\frac{1}{2}\lambda^i$ with $\lambda^i$ being the usual Gell-Mann matrices, and which are normalized as ${\rm Tr}(T^iT^j)=\frac{1}{2}\delta^{ij}$. These generators satisfy the commutation relations $[T^i,T^j]=if^{ijk}T^k$ where $f^{ijk}$ are the structure constants of $SU(3)$. $\alpha_i$ in Eq. (\ref{q-trans}) is the continuous parameter of transformation and $g$ characterizes the strength of the coupling between quarks and the gauge fields.
To maintain the acquired gauge invariance, the Lagrange density corresponding to the gauge fields themselves must be constructed gauge invariantly. This, first of all, means that the eight $A_{\mu}^i$ fields must be massless, the quanta of which are the familiar gluons. Secondly, in analogy with the electromagnetic (EM) interactions, one can form a field strength tensor $G_{\mu \nu}=\frac{1}{ig}[D_{\mu},D_{\nu}]$ whose transformation properties can be easily deduced using Eq. (\ref{A-trans}),
\begin{eqnarray}
G_{\mu \nu} \to G_{\mu \nu}'=U G_{\mu \nu}U^{\dagger}.
\end{eqnarray}
There are only two dimension four gauge-invariant operators that can be built out of this tensor. One of which is even under the CP transformation,
\begin{eqnarray}
\mathcal{L}_{gauge}^{(CP)}=-\frac{1}{2} {\rm Tr}(G_{\mu \nu} G^{\mu \nu}),
\label{CP-even}
\end{eqnarray}
and its normalization is chosen such that, upon replacing the $SU(3)$ transformations with an Abelian $U(1)$ transformation, the QED Lagrangian is recovered.\footnote{This also justifies the factor of $\frac{1}{ig}$ in the definition of $G_{\mu \nu}$ as it would result in the usual normalization of the kinetic term of gluons.} The $CP$ odd term,
\begin{eqnarray}
\mathcal{L}_{gauge}^{({CP\hskip-0.8em /}~)}=\bar{\theta}\frac{g^2 N_f}{32 \pi^2} \epsilon_{\mu \nu \alpha \beta} {\rm Tr}(G^{\mu \nu} G^{\alpha \beta}),
\label{CP-odd}
\end{eqnarray}
is irrelevant for most of QCD phenomenology as the experimental value of its corresponding strength, characterized by the parameter $\bar{\theta}$, is unexpectedly close to zero, $\bar{\theta}\lesssim 10^{-9}$.\footnote{The convention used for the normalization of this term ensures that, in the absence of massive quarks, the contribution from such term vanishes upon setting $\bar\theta=2\alpha$, where $\alpha$ is the parameter of the $U(1)_A$ transformation, $q \to e^{i\alpha \gamma_5}$, whose current, $J_5^{\mu} \equiv \bar{q} \gamma^{\mu} \gamma_5 q$, is anomalous.} $N_f$ denotes the number of quark flavors (up, down, strange, etc.), and $\epsilon_{\mu \nu \alpha \beta}$ is the fully anti-symmetric Levi-Civita tensor.
The Lagrange density of QCD, neglecting the CP-odd contribution and taking into account different quark flavor sectors, can be written in the explicit form,
\begin{eqnarray}
\mathcal{L}_{QCD}&=&\sum_{f=1}^{N_f}\left[\bar{q}_f(i\gamma^{\mu}\partial_{\mu}-m_f)q_f-gA_{\mu}^i \bar{q}_f \gamma^{\mu} T^i q_f\right]
\nonumber\\
&& -\frac{1}{4}F^i_{\mu \nu}F^{i\mu \nu}+\frac{g}{2}f_{ijk}F_{\mu \nu}^i A^{i \mu}A^{j \nu}
-\frac{g^2}{4}f_{ijk}f_{klm}A_{\mu}^jA_{\nu}^kA^{l \mu}A^{m \nu},
\label{L-QCD}
\end{eqnarray}
where $F^i_{\mu \nu} \equiv \partial_{\mu}A^i_{\nu}-\partial_{\nu}A^i_{\mu}$. The striking feature of this Lagrange density is the self interactions among gluons which makes the vacuum of the theory nontrivial compared to QED. This is not a surprise as in any non-Abelian gauge theory, the gauge field $A_{\mu}^i$ carries a characteristic charge (color in the case of QCD) corresponding to the internal space of the gauge group, and must be able to interact with other charged members of the gauge multiplet. The other feature of the QCD Lagrange density is that the coupling of gauge fields to the quark fields cannot be arbitrary and is constrained by the Lie algebra of the group to be the same among quarks with different colors and from different families, and should match that of self-gluon couplings. This is again in contrast with QED where, although the interaction Lagrangian has a universal form, different matter fields can couple to the EM field with different strengths, characterized by their distinct electric charges.
The two important properties of QCD, asymptotic freedom and color confinement, can be deduced from an analytical approach based on perturbation theory. The former, as is a standard textbook calculation, is obtained by looking at the running of the QCD coupling constant with energy from a weak-coupling expansion of the QCD $\beta$-function (see Sec. \ref{HE-QCD}) using the Feynman diagram technology. The latter property can be studied using a strong-coupling expansion of the potential between two static quarks. In the following, we discuss several features of QCD at high and low energies in more details.
\subsection{QCD at high energies
\label{HE-QCD}}
Due to (ultra-violet) UV divergences in any perturbative calculation of QCD when the quantum corrections are included, a reference energy scale must be introduced to renormalize the theory. In a sense, the value of any quantity is measured compared with a reference energy scale and so the divergent contributions cancel out when quantities are calculated at two energy scales relative to each other. This however means that one must know the relation that governs the evolution of the quantity of interest at the reference energy scale down to the energy scale relevant to a given physical process. Such relations are the familiar Callan-Symanzik or renormalization group equations \cite{Callan:1970yg, Symanzik:1970rt}. Here we are only interested in the evolution of the QCD coupling constant with the energy scale $\mu$, characterized by the so-called $\beta$-function,
\begin{eqnarray}
\beta(\alpha_s) \equiv \mu^2 \frac{\partial}{\partial \mu^2}\alpha_s(\mu).
\end{eqnarray}
The fields and parameters of the Lagrange density that one starts with are \emph{bare} quantities, meaning that they suffer from UV divergences. These can be replaced with the renormalized quantities whose divergences are removed by fixing their values at the reference scale $\mu$, using some chosen renormalization conditions. These finite quantities which now carry a $\mu$-dependence can then be used in perturbation theory in a well-defined expansion. The beauty of perturbative QCD, as well as other renormalizable theories, is that a finite number of such conditions suffices to remove all the UV divergences that occur to all order in perturbation theory.
This well-defined procedure can be carried out for the \emph{effective} coupling constant felt at energy scale $\mu$ where not only e.g. the three-gluon vertex must be replaced by its renormalized value but also the external gluonic legs must be corrected by the corresponding wavefunction renormalization factors. Then a two-loop calculation shows that
\begin{eqnarray}
\beta(\alpha_s)=-(b_0 \alpha_s^2+b_1 \alpha_s^3)+\mathcal{O}(\alpha_s^4),
\label{QCD-beta}
\end{eqnarray}
with $b_0=\frac{1}{12\pi}(33-2N_f)$ and $b_1=\frac{1}{24\pi^2}(153-19N_f)$ \cite{Beringer:1900zz}. For the current discussion let us ignore the NLO correction to the $\beta$-function and solve Eq. (\ref{QCD-beta}). Explicitly, we want to know given the coupling constant at scale $\mu$, what the value of the coupling would be at scale $\mu'$. It easily follows that
\begin{eqnarray}
\alpha_s(\mu')=\frac{\alpha_s(\mu)}{1+b_0 \alpha_s(\mu) \log \frac{\mu'^2}{\mu^2}}.
\label{alpha-s-I}
\end{eqnarray}
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=1.05]{alphas.pdf}
\par\end{centering}
\caption{{\small The coupling of QCD as a function of a characteristic energy scale $\mu=Q$, obtained from matching the QCD perturbative calculation to a given order (as given in brackets) to the experimental measurements of several quantities. There is also one point which is obtained by matching to a lattice QCD calculation \cite{Beringer:1900zz}. Figure is reproduced with the permission of Michael Barnett on behalf of the Particle Data Group.}}
\label{fig:alpha-s}
\end{figure}
Given the positive sign of $b_1$ for QCD with $N_f=6$, it is evident that $\alpha_s(\mu')$ decreases as $\mu'$ increases, indicating the theory tends to become free at asymptotically high energies. Experimental determinations of $\alpha_s$ for a range of energies have resulted in values that lie on the predicted scale-dependence curve to an extremely well precision, as is shown in Fig. \ref{fig:alpha-s}. To parametrize the characteristic scale at which the theory becomes strong, we can define the scale $\Lambda_{QCD}$ such that $b_0 \alpha_s(\mu) \log \frac{\mu^2}{\Lambda_{QCD}^2}=1$, then one can rewrite Eq. (\ref{alpha-s-I}) as following
\begin{eqnarray}
\alpha_s(\mu')=\frac{1}{2b_0 \log \frac{\mu'}{\Lambda_{QCD}}}.
\label{alpha-s-II}
\end{eqnarray}
As can be seen, perturbation theory is only valid if $\mu' \gg \Lambda_{QCD}$. Experimentally $\Lambda_{QCD} \approx 200~{\rm MeV}$ which is of the order of the inverse size of the light hadrons. This is consistent with our realization of hadrons being composed of strongly interacting constituents when low-energy probes are used. In fact at low energies, these hadrons are the effective degrees of freedom of QCD, and the details of their properties and interactions, although sensitive to the short distance theory of QCD, can be studied in a systematic low-energy expansion. This requires understanding QCD symmetries and the mechanism for the breaking of some of these symmetries. We discuss this topic in the next section, Sec. \ref{LE-QCD}.
\subsection{QCD at low energies
\label{LE-QCD}}
Although quarks and gluons do not show up as explicit degrees of freedom in the spectrum at energies of the order of $\Lambda_{QCD}$, the imprint of their interactions can be found in the spectrum of hadrons. For example, the low-lying spectrum of (negative parity) mesons and (positive parity) baryons, as illustrated in Fig. \ref{fig:had-spec}, exhibits several interesting patterns whose origin can be understood via the fundamental theory of QCD. As is seen, pions are noticeably lighter than the rest of hadrons and come in an almost degenerate triplet. The next multiplet of mesons, while remain low in mass compared to baryons, are not as light as pions. On the other hand, the $\eta'$ meson that has the same quark content as that of $\eta$ in the quark model is surprisingly heavier than $\eta$. Baryons have masses at the order of $\gsim1~{\rm GeV}$ and like mesons come in various nearly degenerate multiplets. Moreover, the parity partners of mesons and baryons have been observed to have different masses, e.g., the difference in the mass of the nucleons $\sim 940~{\rm MeV}$ and their negative parity counterpart $N(1535)$ is as large as $600~{\rm MeV}$.
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.365]{hadspec.pdf}
\par\end{centering}
\caption{{\small The masses of the the few lightest mesons with $J^{P}=0^{-}$ and baryons with $J^{P}=\frac{1}{2}^{+}$ in GeV. The error bars associated with the experimental measurements of masses are not included and each line only represents the central values as reported in Ref. \cite{Beringer:1900zz}.}}
\label{fig:had-spec}
\end{figure}
To understand these features all together, it suffices to study the underlying symmetries of the QCD Lagrangian. In the limit of zero quark masses (chiral limit), the left-handed and right-handed quarks of each flavor do not mix with each other through QCD interactions,
\begin{eqnarray}
\mathcal{L}_{QCD}^{(q;1)}&=&\sum_{f=1}^{N_f}\left[\bar{q}_{L,f}(i\gamma^{\mu}D_{\mu})q_{L,f}+\bar{q}_{R,f}(i\gamma^{\mu}D_{\mu})q_{R,f}\right],
\label{L-kin}
\end{eqnarray}
where each quark is decomposed to components that have specific handedness, $q=q_L+q_R$ with $q_L=\frac{1-\gamma_5}{2}q$ and $q_R=\frac{1+\gamma_5}{2}q$. Due to the heavy mass of the charm quark, $m_c \approx 1.3~{\rm GeV}$, only up, down and strange quarks play a significant role in the dynamic of strongly interacting systems at low to medium energies. With three flavors of quarks the Lagrangian in Eq. (\ref{L-kin}) is seen to be invariant under $U(3)_L \times U(3)_R$ symmetry, which however breaks down to $U(1)_V \times SU(3)_L \times SU(3)_R$ symmetry. Let us discuss these symmetries and their reduction in more details.
\begin{itemize}
\item The $U(1)_A$ symmetry is broken due to the chiral anomaly. The chiral anomaly refers to the non-conservation of the number of massless left-handed fermions compared with the right-handed fermions due to the non-invariance of the quantum expectation values (as opposed to the classical Lagrangian) under an axial $U(1)$ transformation $q \to e^{i\alpha \gamma_5}q$, where the corresponding isosinglet axial-vector current $J^{\mu5}=\bar{q}\gamma^{\mu}\gamma^{5} q$ is not conserved \cite{Adler:1969gk, Adler:1969er}. This already gives a hint to why the mass of the isosinglet pseudo-scalar meson $\eta'$ is noticeably different than that of isovector pseudo-scalar mesons. However in order to understand the small mass of these latter mesons, further investigation of symmetries is required.
\item The invariance under $U(1)_V$ is realized by the transformation $q \to e^{i\alpha}q$ with the corresponding conserved isosinglet vector current $J^{\mu}=\bar{q}\gamma^{\mu}q$, and is manifested by the conservation of the net baryon number.
\item Finally, independent $SU(3)$ transformations of left-handed and right-handed quarks, represented by $q \to L~q$ and $q \to R~q$, leave the Lagrangian in Eq. (\ref{L-kin}) invariant, where $L$ and $R$ are $SU(3)$ matrices, $L \in SU(3)_L$ and $R \in SU(3)_R$, and $q$ denotes a quark triplet in the 3 representation of $SU(3)$. This is called the chiral symmetry of QCD which plays an important role in constructing an effective low-energy theory of hadrons, namely chiral perturbation theory ($\chi$PT), at energies of the order of $\Lambda_{QCD}$.
\end{itemize}
There are two features of QCD that deprive nature from the exact chiral symmetry. The first one is the presence of a non-vanishing quark condensate,
\begin{eqnarray}
\left\langle 0 \left| \bar{q}_{R,j} q_{L,i} \right| 0 \right\rangle = - \Lambda^3 \delta_{ij},~~~~~~~~~i,j=1,2,3.
\label{condensate}
\end{eqnarray}
resulting in a \emph{spontaneous} breaking of the chiral symmetry \cite{Nambu:1961tp, Nambu:1961fr}. This means that although the action of theory is chirally invariant, the vacuum state\footnote{Note that the $\bar{q}q$ pair in Eq. (\ref{condensate}) has the same quantum numbers as vacuum.} does not respect the chiral symmetry. This is manifested in the change of condensate as a chiral transformation is performed on the quark fields,
\begin{eqnarray}
\left\langle 0 \left| \bar{q}_{R,j} q_{L,i} \right| 0 \right\rangle \to L_{ii'}R^{\dagger}_{j'j} \left\langle 0 \left| \bar{q}_{R,j'} q_{L,i'} \right| 0 \right\rangle = - \Lambda^3(LR^{\dagger})_{ij}.
\label{cond-trans}
\end{eqnarray}
Only if $L=R$ does the condensate remain invariant, reducing the symmetry group to its $SU(3)_V$ subgroup with the corresponding conserved isovector vector current $J^{\mu,a}=\bar{q} \gamma^{\mu} T^a q$. For each $8$ generators of the broken subset of symmetries, deduced from the isovector axial vector current $J^{\mu 5,a}=\bar{q} \gamma^{\mu} \gamma^5 T^a q$, there exists a corresponding massless Goldstone boson which must be found in the spectrum of mesons with quantum numbers of the generators broken symmetry. Such massless excitations can be parametrized by a field, $\Sigma$, that lives in the $(3,\bar{3})$ representation of $SU(3)$. From Eq. (\ref{cond-trans}), it is clear that $LR^{\dagger}$ produces a different vacuum than that of Eq. (\ref{condensate}) for $L \neq R$ and therefore it can be readily identified as $\Sigma$. The $\Sigma$ field can be explicitly parametrized by,
\begin{eqnarray}
\Sigma \equiv e^{2i \bm{\pi}(x) / f},
\label{Sigma}
\end{eqnarray}
where $\bm{\pi}(x)$ can be related to the pseudo-scalar meson octets,
\begin{eqnarray}
\bm{\pi}\equiv\pi^aT^a=
\left(\begin{array}{ccc}
\frac{\pi^0}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & \pi^+ & K^+\\
\pi^- & -\frac{\pi^0}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & K^0\\
K^- & \bar{K}^0 & -\frac{2\eta}{\sqrt{6}}
\end{array}\right).
\label{Sigma}
\end{eqnarray}
$T^{a}$ with $a=1,\dots,8$ are the $8$ generators of $SU(3)$ and $f$ is a constant with dimension mass whose value is matched to the pion weak decay constant $f=f_{\pi}=130.41 \pm 20~{\rm MeV}$ \cite{Beringer:1900zz}. The effective interactions of these Goldstone bosons at low energies can then be studied by forming the most general Lagrangian that is invariant under the chiral symmetry. The significance of each term in this Lagrangian is determined through a systematic expansion with respect to the ratio of the typical momentum in a process to the scale of chiral symmetry breaking, $\Lambda_{\chi}\sim1~{\rm GeV}$. We will come back to this topic in Sec. \ref{subsec:ChiPT}.
The second feature of QCD which \emph{explicitly} breaks the chiral symmetry is non-vanishing masses of quarks. It is evident from the QCD Lagrangian that the mass term mixes quarks of different chiralities,
\begin{eqnarray}
\mathcal{L}_{QCD}^{(q;2)}&=&\sum_{f=1}^{N_f}\left[\bar{q}_{L,f}m_{q_f}q_{R,f}+\bar{q}_{R,f}m_{q_f}q_{L,f}\right],
\label{L-mq}
\end{eqnarray}
and manifestly spoils the chiral symmetry. However, the masses of light quarks,\footnote{These masses are the Particle Data Group average of several lattice QCD determinations that are converted to a renormalized mass in the $\overline{MS}$ scheme at scale $\mu=2~{\rm GeV}$ \cite{Beringer:1900zz}.}
\begin{eqnarray}
m_u=2.15 \pm 0.15~{\rm MeV},~m_d=4.70 \pm 0.20~{\rm MeV},~m_s=93.5 \pm 2.5~{\rm MeV},
\label{mq}
\end{eqnarray}
in particular those of up and down quarks, are much smaller than the scale of the spontaneous chiral symmetry breaking. As a result, chiral symmetry remains an approximate symmetry of QCD before the spontaneous chiral symmetry breaking occurs. The spontaneous symmetry breaking (SSB) mechanism then generates 8 nearly massless bosons or namely 8 \emph{pseudo}-Goldstone bosons (pGBs). In Sec. \ref{subsec:ChiPT} we will show how the quark mass contributions can be included in the chirally invariant Lagrangian of pseudo-scalar bosons.
The first immediate evidence of pGBs is in the spectrum of hadrons. As discussed, the pseudo-scalar octets are unusually light compared with the rest of hadrons and whose parity quantum number is consistent with that of expected for the generators of the broken symmetry. This also means that the hadrons will no longer be degenerate with their parity partners when the chiral symmetry is broken.\footnote{Since the Hamiltonian of QCD is invariant under parity (ignoring nearly vanishing $CP$ violating interactions in Eq. (\ref{CP-odd})), the vacuum state and its parity partner are both the eigenstate of the Hamiltonian with the same eigenvalues. However, it can be shown that these two degenerate states are eigenstates of the axial charge $\hat{Q}^{5,a} \equiv \int d^3 x j^{05,a}(\mathbf{x},t)$ with eigenvalues that differ in sign. After the spontaneous symmetry breaking, only one of these vacua is picked, resulting in breaking the degeneracy between the parity partners.} We note in particular that the $\eta'$ does not correspond to a SSB mechanism and so its mass is not protected to be small.\footnote{Due to the heavier mass of the strange quark, the explicit chiral symmetry breaking is severe for the case of $SU(3)$ symmetry compared with its $SU(2)$ subgroup. As a result, the pGB features of pions are more prominent than that of strange mesons, see Fig. \ref{fig:had-spec}.} The other evidence for the existence of pGBs of a spontaneously broken symmetry had been observed experimentally through pion-pion and pion-nucleon scattering experiments even in pre-QCD era. The $\pi\pi$ scattering cross sections had been observed to vanish at low energies. On the other hand, the most naive effective interaction among pions and nucleons at low energies consistent with the parity of pions and nucleons failed to describe pion-nucleon cross sections. Both of these cross sections could be reproduced if pions would only derivatively couple to other hadrons. This is of course only consistent with the identification of pions as the pGBs of a broken symmetry with an explicit \emph{shift} symmetry as is evident from Eq. (\ref{Sigma}). We will present these interactions in the following subsection.
\subsubsection{Chiral perturbation theory for mesons and baryons
\label{subsec:ChiPT}}
\noindent \emph{Lagrangian for pseudo-Goldstone bosons:} The Lagrangian describing the dynamics of pGBs can be constructed from field $\Sigma$ in Eq. (\ref{Sigma}) order by order in powers of $\frac{\partial}{\Lambda_{\chi}} \sim \frac{p}{\Lambda_{\chi}}$ and $\frac{m_q}{\Lambda_{\chi}}$, where $p$ is the typical momentum of the process and $m_{pGB}$ denote the mass of the pGBs.\footnote{The mass of the next meson that is not a pGBs can be taken as the scale $\Lambda_{\chi}$ for which this effective approach breaks down. This is the $\rho$ meson with $m_{\rho}=775.26\pm0.25~{\rm MeV}$. It gives rise to an expansion parameter that is not typically small, $\frac{m_K}{m_{\rho}}\sim0.6$, consistent with the expectation that the $SU(3)$ symmetry breaking is fairly severe given the mass of the strange quark. For processes that only involve pions and nucleons, one can restrict the effective interactions to only respect the $SU(2)$ chiral symmetry for which the expansion parameter can only be as large as $\frac{m_{\pi}}{\Lambda_{\chi}}=\frac{m_{\pi}}{m_K}\sim0.3$ for low-energy processes.} For pseudo-scalar mesons to be Goldstone boson, they must only interact derivatively. However as they are only pGBs due to non-vanishing mass of quarks, they can also couple non-derivatively through insertions of the quark mass matrix defined as
\begin{eqnarray}
M \equiv \left(\begin{array}{ccc}
m_u & 0 & 0\\
0 & m_d & 0\\
0 & 0 & m_s
\end{array}\right).
\label{Mq-matrix}
\end{eqnarray}
By promoting $M$ to a dynamical field, namely a \emph{spurion} field, which transforms under $SU(3)$ chiral symmetry as $M \to LMR^{\dagger}$, its non-zero value can be interpreted as causing a SSB similar to the field $\Sigma$. This provides the necessary ingredients to write down the leading order (LO), $\mathcal{O}\left(\frac{p^2}{\Lambda_{\chi}^2},\frac{m_{pGB}^2}{\Lambda_{\chi}^2}\right)$, chiral Lagrangian of pseudo-scalar mesons as following \cite{Gasser:1983yg, Gasser:1984gg},
\begin{eqnarray}
\mathcal{L}_{pGB}^{(2)}=\frac{f_{\pi}^2}{8}~{\rm Tr}\left[\partial_{\mu} \Sigma \partial^{\mu} \Sigma^{\dagger}+2B (M\Sigma^{\dagger}+M^{\dagger} \Sigma)\right].
\label{L-pGB}
\end{eqnarray}
This Lagrangian is invariant under the Lorentz and $SU(3)$ chiral symmetry and its normalization is chosen in such a way to reproduce the canonical normalization of the kinetic term for pseudo-scalar mesons. It only contains one more parameter, or low-energy coefficient (LEC), beside $f_{\pi}$ which, upon a straightforward expansion in the pGB fields in EQ. (\ref{L-pGB}), can be related to the mass of mesons, e.g., $B=\frac{m_{\pi}^2}{m_u+m_d}$. This indicates that each insertion of the quark mass matrix counts as $\sim m_{pGB}^2$ in this treatment.
The value of parameter $B$ can be directly matched to the value of the quark condensate using the Feynman-Hellman theorem and is readily found to be
\begin{eqnarray}
B=-\frac{2}{f_{\pi}^2}\left\langle 0 \left| \bar{u}u \right| 0 \right\rangle.
\label{B}
\end{eqnarray}
At next to LO, $\mathcal{O}\left(\frac{p^4}{\Lambda_{\chi}^4},\frac{p^2m_{pGB}^2}{\Lambda_{\chi}^4},\frac{m_{pGB}^4}{\Lambda_{\chi}^4}\right)$, there are 8 distinct chirally invariant operators with up to 4 derivatives and up to two insertions of quark mass matrix whose corresponding LECs, the Gasser-Leutwyler coefficients \cite{Gasser:1984gg}, must be matched to experimental data on meson-meson scattering. In doing such matching, the loop effects with insertions of the leading operators in Eq. (\ref{L-pGB}) must be taken into account. This is because these loop contributions are enhanced compared with the tree-level contributions of the next order by factors of $\log(\mu/m_{pGB})$. $\mu$ is the renormalization scale in a mass-independent normalization scheme such as $\overline{MS}$ that is used to renormalize the amplitudes when encountering the UV divergences in loops \cite{Kaplan:1995uv}, see for example Fig. \ref{fig:pipi}. In particular, it is notable that the scale-dependence of LECs at any order in a systematic EFT is canceled by that of introduced by the chiral loops of previous order so that the amplitudes calculated at that order is rendered scale independent.
The EFT procedure just described is a powerful method for the following reasons:
\begin{itemize}
\item
Firstly, the dynamics of pGB is highly constrained by the chiral symmetry such that the interactions of all pseudo-scalar mesons can be put in a universal form, e.g., Eq. (\ref{L-pGB}), eliminating the need to introduce several LECs at each order for different members of the multiplet. This feature remains true in constructing the interactions of pGBs with baryons as is discussed below.
\item
Once LECs that occur at a given order in EFT are matched to one or several observables, the EFT interactions can be applied in studying a wide range of phenomena where such operators contribute, giving the EFT a predictive power. For example as we just observed, the value of parameter $f$ that was determined by matching to the weak decay rate of the pion, can now be used to fully predict the $\pi\pi$ scattering cross section at LO using Eq. (\ref{L-pGB}). A straightforward calculation shows that \cite{Colangelo:2001df}
\begin{eqnarray}
t^{(0)}_0(s)=\frac{2s-m_{\pi}^2}{16\pi f_{\pi}^2},
~t^{(1)}_1(s)=\frac{s-4m_{\pi}^2}{48\pi f_{\pi}^2},
~t^{(2)}_0(s)=-\frac{s-2m_{\pi}^2}{16\pi f_{\pi}^2},
\label{pipi-I012}
\end{eqnarray}
at LO in chiral expansion, where $t^{(I)}_l(s)$ denotes the $\pi\pi$ scattering partial-wave amplitude in isospin channel $I$ and partial-wave channel $l$, and $s$ denotes the total invariant mass of the $\pi\pi$ system.
\item
Despite phenomenological models with an arbitrary number of parameters -- that are fit to experimental data -- with which no well-defined systematic uncertainty can be associated, the EFT approach enables the quantification of errors in calculated quantities in a systematic way. These errors result from neglecting higher order terms in the low-momentum expansion.
\item
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.4235]{pipifeynman.pdf}
\par\end{centering}
\caption{{\small Diagrams contributing to pion-pion scattering in $\chi$PT} up to NLO. Dashed lines represent pions, the grey dot denotes the LO tree level vertex obtained from expanding Eq. (\ref{L-pGB}) in pion fields, while the grey square denotes the NLO vertex and depends on the Gasser-Leutwyler coefficients. The power-counting of each diagram is given in the figure.}
\label{fig:pipi}
\end{figure}
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.45]{nplqcdpipi.pdf}
\par\end{centering}
\caption{{\small The LQCD determination of $m_{\pi}^2a^{(2)}_0r^{(2)}_0$ for the $I=2$ $\pi \pi$ scattering at the physical point (the red star on the physical line denoted by a dashed green line). The band represents the $68\%$ confidence interval interpolation of the LQCD result (the red rectangle) at $m_{\pi}=390~{\rm MeV}$ \cite{Beane:2011sc}. The horizontal purple line denotes the LO $\chi$PT prediction in the chiral limit. The Roy equation prediction \cite{Colangelo:2001df} is shown by the black circle on the physical line. Figure is reproduced with the permission of the NPLQCD collaboration.}}
\label{fig:NPLQCD-pipi}
\end{figure}
Such EFT technique determines the light-quark mass dependence of observables order by order in the EFT expansion. This is particularly important as it enables making predications for physical observable from lattice QCD calculations that are performed at heavier quark masses. As long as the quark masses used in those calculations produce pGB with masses within the range of validity of chiral perturbation theory, the $\chi$PT expressions might be used to interpolate to the physical values of quantities. A nice example of which is the determinations of the $I=2$ S-wave $\pi\pi$ scattering length, $a^{(2)}_0$, and effective range, $r^{(2)}_0$, at the physical point (physical values of light-quark masses) using the LQCD input at $m_{\pi}=390~{\rm MeV}$ as performed by the NPLQCD collaboration \cite{Beane:2011sc}. The S-wave scattering length and effective range are defined via the effective range expansion (ERE) at low energies,
\begin{align}
& k^* \cot \delta_0=-\frac{1}{a_0}+\frac{1}{2}r_0 k^{*2}+\dots~,
\label{a-r-def}
\end{align}
where $k^*$ denotes the momentum of each pion in the CM frame. These LQCD results have been used in the chiral expansions of these quantities at NLO in two-flavor $\chi$PT,\footnote{We are using the nuclear physics convention for the sign of scattering length where a positive scattering length corresponds to an attractive interaction.}
\begin{align}
& m_{\pi}a^{(2)}_0 = \frac{m_{\pi}^2}{f_{\pi}^2} \left[\frac{1} {8\pi}+\frac{3}{128 \pi^2}\frac{m_{\pi}^2}{f_{\pi}^2} \log\left(\frac{m_{\pi}^2}{f_{\pi}^2}\right)+C_1\frac{m_{\pi}^2}{f_{\pi}^2}\right],
\nonumber\\
& m_{\pi}r^{(2)}_0 = \frac{24 \pi f_{\pi}^2}{m_{\pi}^2}+C_2+\frac{17}{6 \pi} \log\left(\frac{f_{\pi}^2}{m_{\pi}^2}\right),
\label{ar-pipi-I}
\end{align}
where $C_1$ and $C_2$ are two combinations of Gasser-Leutwyler coefficients, renormalized at scale $\mu=f_{\pi}$, and are fit to LQCD data at this pion mass. This results in impressively precise determinations of scattering length and effective range at the physical point,\footnote{The numbers in parentheses denote the statistical and systematic uncertainties of various sources as explained in Ref. \cite{Beane:2011sc}.}
\begin{eqnarray}
m_{\pi}a^{(2)}_0=0.0417(07)(02)(16),~ m_{\pi}r^{(2)}_0=72.0(5.3)(5.3)(2.7),
\label{ar-pipi-II}
\end{eqnarray}
in $1\sigma$ agreement with the determination of these parameters from the Roy (dispersion relation) analysis \cite{Roy:1971tc} of experimental data with the $\chi$PT input \cite{Ananthanarayan:2000ht, Colangelo:2001df}. This example demonstrates the role of EFT in empowering the LQCD calculations at yet unphysical pion masses to make predictions for the physical point. Besides the low-energy scattering parameters \cite{Li:2007ey, Aoki:2007rd, Beane:2010hg, Beane:2011xf, Beane:2011sc, Beane:2011iw, Beane:2012ey,Yamazaki:2012hi, Lang:2012sv, Beane:2013br, Pelissier:2011ib, Lang:2011mn, Pelissier:2012pi, Ozaki:2012ce, Buchoff:2012ja, Dudek:2012xn, Dudek:2012gj, Lang:2014tia}, the masses of hadrons and their decay constants are among quantities that are being extensively studied through a combination of LQCD and EFTs (For reviews on these calculations see Refs. \cite{Aoki:2013ldr, Laiho:2009eu, Lin, Prelovsek:2013cta}). We will present an example of the use of EFTs in deducing the FV corrections to the mass of the nucleons in Sec. \ref{IV-intro}.
\end{itemize}
\noindent \emph{Lagrangian for Baryon octets:} Let us first focus on the the case of $SU(2)_L \times SU(2)_R$ in constructing the Lagrangian. We present the general result for the case of $SU(3)$ chiral symmetry later. First note that the transformation property of nucleons doublet, $N=\left(\begin{array}{c} p\\ n
\end{array}\right)$, under $SU(2)_L \times SU(2)_R$ is not constrained - in contrary to the pGB field $\Sigma$, so we can take the freedom to choose it. The simplest transformation, $N_L \to L N_L$ and $N_R \to R N_R$, where left-handed and right-handed nucleons transform separately, turns out to not be the most convenient one. We can require the same transformation for the left-handed and right-handed components,
\begin{eqnarray}
\widetilde{N}_L \to U \widetilde{N}_L,~ \widetilde{N}_R \to U \widetilde{N}_R,
\label{N-tilde}
\end{eqnarray}
where $U$ is an element of $SU(2)$. This can be achieved if we redefine (dress) the nucleon field as following
\begin{eqnarray}
\widetilde{N}_L\equiv \xi N_R,~ \widetilde{N}_R \equiv \xi^{\dagger} N_L,
\label{N-dressed}
\end{eqnarray}
where $\xi=\sqrt{\Sigma}$ can be seen to transform under $SU(2)_L \times SU(2)_R$ as
\begin{eqnarray}
\xi \to L \xi U^{\dagger}=U \xi R^{\dagger}.
\label{xi-trans}
\end{eqnarray}
In order for the familiar free nucleon Lagrangian $\overline{N} i \gamma^{\mu} \partial_{\mu} N$ to remain invariant under (local) transformation (\ref{N-tilde}), the minimal coupling to a vector field $\mathcal{V}_{\mu}$ must be introduced to assure the covariant derivative $D_{\mu}=\partial_{\mu}+\mathcal{V}_{\mu}$ transforms properly under the chiral transformation, $D_{\mu}N \to U(D_{\mu}N)$.\footnote{We reassign the notation $N$ to nucleon fields that transform as in Eq. (\ref{N-tilde}).} This can be seen to be satisfied if $\mathcal{V}_{\mu}$ is chosen to be
\begin{eqnarray}
\mathcal{V}_{\mu}=\frac{1}{2}(\xi^{\dagger} \partial_{\mu} \xi+\xi \partial_{\mu} \xi^{\dagger}),
\label{Vector}
\end{eqnarray}
given its transformation property $\mathcal{V}_{\mu} \to U \mathcal{V}_{\mu} U^{\dagger}+U\partial_{\mu}U^{\dagger}$. Another chiral invariant term in the nucleon Lagrangian is possible by forming the following combination
\begin{eqnarray}
\mathcal{A}_{\mu}=\frac{i}{2}(\xi^{\dagger} \partial_{\mu} \xi-\xi \partial_{\mu} \xi^{\dagger}).
\label{Axial}
\end{eqnarray}
Since this combination transforms as $\mathcal{A}_{\mu} \to U \mathcal{A}_{\mu} U^{\dagger}$, it can directly couple to nucleons at LO in a chirally invariant way although its coefficient is not protected by the minimal coupling mechanism as for the vector fields. Then the leading chiral Lagrangian describing nucleons and their interactions with the pGBs (through field $\xi$) can be written as \cite{Gasser:1987rb}
\begin{eqnarray}
\mathcal{L}^{(1)}_{N,pGB}=\overline{N}(i\gamma^{\mu}D_{\mu}-M_{N}+g_{A}\gamma^{\mu}\gamma^{5} \mathcal{A}_{\mu})N,
\label{L-NpGB}
\end{eqnarray}
where the only new LEC is $g_{A}$ whose value can be matched to the neutron semi-leptonic weak decay, $g_{A}=1.2701(25)$ \cite{Beringer:1900zz}.
Extending the formalism to the case of $SU(3)$ chiral symmetry is now straightforward by noting that the baryon octet fields,
\begin{eqnarray}
B=
\left(\begin{array}{ccc}
\frac{\Sigma^0}{\sqrt{2}}+\frac{\Lambda}{\sqrt{6}} & \Sigma^+ & p\\
\Sigma^- & -\frac{\Sigma^0}{\sqrt{2}}+\frac{\Lambda}{\sqrt{6}} & n\\
\Xi^- & \Xi^0 & -\frac{2\Lambda}{\sqrt{6}}
\end{array}\right),
\label{B-octet}
\end{eqnarray}
can be made to transform as $B \to U B U^{\dagger}$, and as a result the Lagrangian in Eq. (\ref{L-NpGB}) can be generalized to \cite{Krause:1990xc}
\begin{eqnarray}
\mathcal{L}^{(1)}_{B,pGB}={\rm Tr}\left[\overline{B}(i\gamma^{\mu}D_{\mu}-M_{B})B\right]-D{\rm Tr}\left[\bar{B}\gamma^{\mu}\gamma^{5}\{\mathcal{A}_{\mu},B\}\right]-F{\rm Tr}\left[\bar{B}\gamma^{\mu}\gamma^{5}[\mathcal{A}_{\mu},B]\right],
\label{L-BpGB}
\end{eqnarray}
where $D$ and $B$ are two new LECs that can be determined by matching to semi-leptonic weak decay decays of baryons octets, $D \approx 0.8$ and $F \approx 0.5$ \cite{Borasoy:1998pe}. At NLO, the insertions of the quark mass matrix must be taken into account . Given the transformation properties of $B$, $\xi$ and $M$ as discussed above, the most general Lagrangian at this order can be readily formed,
\begin{align}
&\mathcal{L}^{(2)}_{B,pGB}=a_1{\rm Tr}\left[\overline{B}(\xi^{\dagger}M\xi^{\dagger}+{\rm h.c.})B\right]+a_2{\rm Tr}\left[\bar{B}B(\xi^{\dagger}M\xi^{\dagger}+{\rm h.c.})\right]+a_3{\rm Tr}\left[\bar{B}B \right] {\rm Tr} \left[M \Sigma+{\rm h.c.} \right],
\nonumber\\
\label{L-BpGB-Mass}
\end{align}
in which three new LECs are introduced.
An apparent problem in developing a power-counting scheme for EFTs with baryons is that the mass of the baryons is of the order of the chiral symmetry breaking scale, and as a result an expansion in $M_{B}/\Lambda_{\chi}$ is meaningless. To resolve this issue \cite{Jenkins:1990jv}, one should notice that in the heavy field limit, the momentum transfer between baryons and pGBs remains small. So by performing a field redefinition, the large contribution to the baryon momentum, $P_{\mu}=mv_{\mu}+l_{\mu}$, due to its mass can be canceled, leaving a small residual momentum $l_{\mu}$, where $v_{\mu}$ is the baryon four velocity. Let us focus on the case of nucleons and rewrite the $N$ field as
\begin{align}
& N=e^{-iM_Nv.x} (N_{l}+N_{h}),
\label{N-redefinition}
\end{align}
where
%
\begin{align}
& N_{l}=e^{iM_Nv.x} \mathcal{P}^+_v N,~N_h=e^{iM_Nv.x} \mathcal{P}^-_v N,
\label{N-redefinition}
\end{align}
with projection operators $\mathcal{P}^{\pm}_v=\frac{1\pm\gamma^{\mu}v_{\mu}}{2}$. Then it is straightforward to see that in the heavy field limit, when $v=(1,0,0,0)$, $\mathcal{P}^+_v$ projects out the upper components of the nucleon spinor with energy $E-M_N$ while $\mathcal{P}^+_v$ project the lower components of the nucleon spinor with energy $E-M_N$. With this decomposition, the only dynamical field that survives as $M_N \to \infty$ is $N_l$ whose corresponding Lagrangian can be written as \cite{Jenkins:1990jv}
\begin{eqnarray}
\hat{\mathcal{L}}^{(1)}_{N,pGB}=\overline{N}_l(iD_{0}-g_{A}\bm{\sigma} \cdot \bm{\mathcal{A}})N_l,
\label{L-NpGB}
\end{eqnarray}
at LO in $1/M_N$ expansion where $\bm{\sigma}$ are Pauli matrices of $SU(2)$ in the spin space. Note that the mass term in Eq. (\ref{L-NpGB}) is now canceled via such non-relativistic (NR) reduction. This formalism, that is known in literature as heavy-baryon $\chi$PT (HB$\chi$PT), makes the EFT calculations involving baryons considerably easy specially at higher orders. For future use, let us make explicit the interactions among nucleons and pions in this Lagrangian by expanding the $\xi$ field in Eq. (\ref{L-NpGB}) in powers of pion fields. After neglecting terms with more than two pion fields, one arrives at
\begin{eqnarray}
\hat{\mathcal{L}}^{(1)}_{\pi N}=\overline{N}_l\left[i\partial_{0}-\frac{1}{4f_{\pi}^2} \bm{\tau} \cdot (\bm{\pi} \times \partial_{0} \bm{\pi})-\frac{g_{A}}{2f_{\pi}} \bm{\tau} \cdot (\bm{\sigma} \cdot \bm{\partial}) \bm{\pi}\right]N_l,
\label{L-piN}
\end{eqnarray}
where $\bm{\tau}$ are the Pauli matrices of $SU(2)$ in the isospin space. Several interesting processes can be studied with this Lagrangian including the pion-nucleon scattering and the quark-mass dependence of nucleon mass. We will use this Lagrangian in the next section to evaluate the FV corrections to the mass of nucleons, and later in chapter \ref{chap:TBC} to improve such volume corrections by modifying the quark-field boundary conditions in a finite volume.
The interactions of pGBs and baryons with external fields such as EM field can be also included in the EFT. For the case of electromagnetism, for example, a minimal coupling of hadrons to the photon field $A_{\mu}$ will account for such interactions at LO. It is notable that the quark electric charge matrix $Q$,
\begin{eqnarray}
Q=\left(\begin{array}{ccc}
\frac{2}{3} & 0 & 0\\
0 & -\frac{1}{3} & 0\\
0 & 0 & -\frac{1}{3}
\end{array}\right),
\label{L-piN}
\end{eqnarray}
breaks chiral symmetry explicitly just as the quark mass matrix and its inclusion in the chiral Lagrangian follows in a similar fashion. We will not discuss this extension of EFT Lagrangian here and refer the reader to various comprehensive reviews on $\chi$PT and its applications as can be found in Refs. \cite{Ecker:1994gg, Pich:1995bw, Scherer:2002tk, Kaplan:2005es, Machleidt:2011zz}. In studying EM FV corrections to the mass of hadrons in chapter \ref{chap:EM}, we introduce a simple NR EFT that captures the features of the EFTs coupled to EM fields.
\subsubsection{Effective field theories for nucleons
\label{subsec:NEFT}}
\indent
\emph{EFT potentials and Weinberg power counting}: In early 1990s, Weinberg proposed that the phenomenological potentials of nuclear physics \cite{Jackson:1975be, Partovi:1969wd, Partovi:1972bj, Lacombe:1980dr, Machleidt:1989tm} can be replaced with potentials that are systematically constructed from chiral EFT interactions \cite{Weinberg:1990rz, Weinberg:1991um, Weinberg:1992yk}. The uncertainties of the nuclear few- and many-body calculations due to neglecting higher order terms in the EFT forces can then, in principle, be systematically estimated. This procedure goes as follows:
\begin{enumerate}
\item
Write down, order by order in $\chi$PT, the potential among two nucleons. At LO, there is no contribution from three (and more) nucleon forces. The one-pion exchange (OPE) potential, which was also included in the phenomenological NN potentials to account for the long-range force among nucleons, contributes at LO. In the static limit
%
\begin{eqnarray}
V_{1\pi}^{(LO)}(\bm{p},\bm{p}')=-\frac{g_A^2}{2 f_{\pi}^2} \bm{\tau}_1 \cdot \bm{\tau}_2
\frac{\bm{\sigma}_1 \cdot \bm{q}~\bm{\sigma}_2 \cdot \bm{q}}{\bm{q}^2+m_{\pi}^2},
\label{V-OPE}
\end{eqnarray}
%
where $\bm{p}$ and $\bm{p}'$ are the three momenta of the two interacting nucleons and $\bm q$ is the three momentum of the exchanged pion, see Fig. \ref{fig:V-NN-LO}. It consists of both central and tensor force and therefore can account for $L=0$ and $L=2$ angular-momentum mixing in the deuteron (total spin $S=1$ and total isospin $I=0$) wavefunction, see chapter \ref{chap:NN}.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.465]{vnnlo}
\par\end{centering}
\caption{{\small The LO contributions to the NN potential in the Weinberg power counting. Solid (dashed) line represents the nucleon (pion). The black dot denotes the four-nucleon contact, $C_S$ or $C_T$.}}
\label{fig:V-NN-LO}
\end{figure}
In order to describe the short-range nuclear force and to renormalize away the $\delta$-function singularity of the OPE potential (in position space), two four-nucleon contact operators, with coefficients $C_S$ and $C_T$ must be introduced at the same order, giving rise to the potential
\begin{eqnarray}
V_{CT}^{(LO)}(\bm{p},\bm{p}')=C_S + C_T \bm{\sigma}_1 \cdot \bm{\sigma}_2.
\label{V-CT}
\end{eqnarray}
%
One keeps going to higher orders in the $p/\Lambda_{\chi}$ expansion, by including multi-pion exchange potentials with leading as well as higher order pion-nucleon vertices, and by including as many contact interactions needed to renormalize the UV singularities at any given order.
\item
Given the potential, calculate the NR scattering amplitude, $\mathcal{M}$, by solving the NR Lippmann-Schwinger equation,
\begin{eqnarray}
i \mathcal{M}({\bm{p}',\bm{p}}) = V({\bm{p}',\bm{p}}) + \int d^3 p'' V({\bm{p}',\bm{p}''}) \frac{M_N}{\bm{p}^2-\bm{p}''^2+i \epsilon} i \mathcal{M}({{\bm{p}'',\bm{p}}}).
\label{LS}
\end{eqnarray}
%
\item
By fitting to the well-known scattering phase shifts in various NN channels, constrain the LECs of the EFT potentials, including those of the contact terms.
\item
Solve the many-body problem by inputting these constrained EFT potentials to make predictions for the properties of few and many-body nuclear system, see e.g. Refs. \cite{Gezerlis:2013ipa, Kruger:2013kua, Tews:2013wma, Gezerlis:2014zia, Lynn:2014zia, Roggero:2014lga, Lee:2004si, Borasoy:2005yc, Borasoy:2006qn}.
\end{enumerate}
Unfortunately, Weinberg procedure, despite producing potentials in a systematic way, does not give rise to a consistent power counting in all the two-nucleon channels, and the phase shifts obtained with this method typically diverge as the cutoff used to regularize the divergences is taken to infinity. This undesired scale dependence of physical quantities in the Weinberg power counting can be understood from Eq. (\ref{LS}) where, for example inputting the LO potential in the integral equation results in a summation of the LO interactions to all orders, see Fig. (\ref{fig:LS}). Therefore the amplitude obtained at this order is not a true LO amplitude as it contains higher order loops. Unfortunately these higher order terms, e.g. two-pion exchange, etc., suffer from singularities that cannot be renormalized given the absence of the contact interactions at this order. These interactions only appear in the expansion of the potential at higher orders which are not included in the LO potential that is used in the Lippmann-Schwinger equation. This means that the calculated amplitude is divergent as the cutoff is taken to infinity, and in this sense this procedure cannot be regarded as a genuine EFT approach. In practice, the uncertainty associated with the determination of a given quantity with this method is estimated by varying the cutoff scale in the calculation. For a nice review of chiral nuclear forces, see Ref. \cite{Machleidt:2011zz}.
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.435]{lsequation}
\par\end{centering}
\caption{{\small The diagrammatic representation of the Lippmann-Schwinger equation. Solid lines represent the nucleons.}}
\label{fig:LS}
\end{figure}
\
\
\emph{NN interactions with Kaplan-Savage-Wise power counting}: Instead of working with potentials, one can directly relate scattering amplitudes to the interaction Lagrangian order by order in a low-energy expansion scheme. In the two-body elastic scattering at low energies,\footnote{Below the t-channel cut, $E^*=m_{\pi}/2$.} the relevant intrinsic scales are the scattering length and effective range -- and shape parameters as defined in Eq. (\ref{NN-ERE}). One might think that given the effective range expansion, the scattering amplitude can be straightforwardly written as expansions in $pa$ and $pr$ where $p$ is the typical momentum of the process, i.e. the energy of each particle in the CM frame $p \sim k^*$ or the mass of the pions $p \sim m_{\pi}$. In fact this turns out to be the case in many of the scattering channels. However, the S-wave NN scattering represents unnatural features arising from seemingly fine-tuned interactions. This is manifested in the large scattering length of the system, for example in the $^1 S_0$ channel where $a^{(^1 S_0)}=-23.714 \pm 0.013~{\rm fm} \gg \frac{1}{m_{\pi}}$. The same feature is seen in the ${^3S_1}-{^3D_1}$ coupled channel where $a^{({^3S_1}-{^3D_1})}=5.423 \pm 0.005~{\rm fm} \gg \frac{1}{m_{\pi}}$, giving rise to a near threshold bound state, the deuteron, whose binding energy, $\sim 2~{\rm MeV}$, is much smaller than that set by the typical QCD scale, $\Lambda_{QCD}$. The unnaturalness in these channels indicates that the LO scattering amplitude does not count as $\sim p^0$, but is instead of $\sim p^{-1}$,
\begin{eqnarray}
\mathcal{M} = \frac{4 \pi}{M_N} \frac{1}{k^* \cot \delta - ik^*}
= - \frac{4 \pi}{M_N} \frac{1}{1/a+ik^*}\left[ 1 + \frac{r/2}{(1/a+ik^*)} k^{*2} + \dots \right],
\label{M-EFE}
\end{eqnarray}
%
where $k*=\sqrt{M_N E^*}$ with $E^*$ being the CM energy of two-nucleon system. A sensible power counting at low energies must be able to reproduce this effective range expansion of the amplitude. Clearly, the OPE interaction of nucleons comes at $\sim p^0$ and cannot be counted as a LO interaction. The momentum-independent contact interaction, with coefficient $C_0$ then must be responsible for the LO amplitude, provided that it scales as $\sim p^{-1}$. This requires the chain of bubble diagrams with insertions of this leading operator to all scale at most as $\sim p^{-1}$ or otherwise one loses control over these contributions. A suitable regularization scheme to ensure this scaling is the dimensional regularization with the power-divergence subtraction (PDS) scheme, as proposed by Kaplan, Savage and Wise (KSW) in late 1990s \cite{Kaplan:1998tg, Kaplan:1998we}. Explicitly the LO amplitude, according to Fig. \ref{fig:C0-KSW}, can be written as
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.425]{czeroksw}
\par\end{centering}
\caption{{\small The contribution from the leading contact interaction $C_0$ is summed up to all orders, giving rise to the the leading scattering amplitude consistent with unnaturally large scattering amplitude in the S-wave NN channels. Note that due to the PDS scheme, each term in the expansion comes at the same order, $\mathcal{O}(\frac{1}{p})$, in contrast with the $\overline{MS}$ or momentum subtraction schemes, see Refs. \cite{Kaplan:1998tg, Kaplan:1998we}.}}
\label{fig:C0-KSW}
\end{figure}
\begin{eqnarray}
\mathcal{M}^{(LO)} = \frac{-C_0}{1-I_0^{PDS}C_0}= \frac{-C_0}{1+\frac{M_N}{4\pi}(\mu+ik^*)C_0},
\label{M-LO-KSW}
\end{eqnarray}
where
\begin{eqnarray}
I_0 &=& -i (\frac{\mu}{2})^{4-d} \int \frac{d^d q}{(2\pi)^d} \frac{i}{q^0-\frac{\bm{q}^2}{2M_N}+i\epsilon}\frac{i}{E^*-q^0-\frac{\bm{q}^2}{2M_N}+i\epsilon}
\nonumber\\
&=&-M_N (-M_NE^*-i \epsilon)^{(d-3)/2} \Gamma \left(\frac{3-d}{2}\right) \frac{(\mu/2)^{4-d}}{(4\pi)^{(d-1)/2}},
\label{I0-KSW}
\end{eqnarray}
with $d$ being the dimensionality of spacetime and $\mu$ being the renormalization scale. As is seen, although $I_0$ does not have any singularity in $d=1+3$ dimensions, it is singular in a lower dimension $d=1+2$ (corresponding to the power divergence of $I_0$ in $4$ dimensions that is absent in the dimensional regularization). PDS scheme prescribes that this pole must be subtracted from $I_0$ and therefore making the result $\mu$ dependent,
\begin{eqnarray}
I_0^{(PDS)}=I_0|_{d=4}-I_0^{({\rm div.})}|_{d=3} = - \frac{M_N}{4\pi}(\mu+ik^*),
\label{I0-KSW-PDS}
\end{eqnarray}
giving rise to Eq. (\ref{M-LO-KSW}). Now by comparing Eq. (\ref{M-LO-KSW}) and Eq. (\ref{M-EFE}) one will find that $C_0$ indeed scales as $p^{-1}$
\begin{eqnarray}
C_0(\mu)=\frac{4\pi}{M_N}\left( \frac{1}{-\mu+1/a} \right),
\label{C0-KSW}
\end{eqnarray}
given that $\mu \sim p \gg 1/a$.
At NLO, not only the OPE contributes,\footnote{For scattering processes above the t-channel cut, $E^*>m_{\pi}/2$.} but also the insertions of both $C_2 \nabla^2$ and $D_2 m_{\pi}^2$ operators must be taken into account. Given that each loop scales as $\sim p$, these coefficients must scale as $\sim \frac{1}{p^2}$. In addition, the initial and final nucleon legs must be dressed by the chain of $C_0$ bubbles as they give rise to $\mathcal{O}(1)=(\frac{1}{p})(p)$ contributions. We will not discuss these contributions in details here, however we can already see that this expansion, with the devised power counting, systematically treats pion exchanges perturbatively, and the scale dependence of the amplitudes at each order is completely removed by the introduction of corresponding \emph{counter terms}, i.e. coefficients of the contact interactions that are representative of the short-distant physics of the problem. It is also notable that the pion-mass dependence of the NN interactions systematically arises from these EFT interactions.\footnote{At energies well below the pion mass, the pions can be integrated out from the EFT, giving rise to the \emph{pionless EFT}. In chapter \ref{chap:NN}, we will work with this EFT, along with the use of a dimer field, to reproduce the ERE in NN systems.} For a nice review of EFTs for nucleons, see Ref. \cite{Kaplan:2005es}.
Although the KSW EFT for nucleons has been shown to be a powerful method in studies of the electroweak transitions in the few-body systems, as well as in developing an EFT for three-nucleon systems, it suffers from a slow convergence in the $^1 S_0$ channel and is not converging in the ${^3S_1}-{^3D_1}$ channel \cite{Fleming:1999ee, Fleming:1999bs}. In the coupled ${^3S_1}-{^3D_1}$ channels, the piece in the OPE potential that survives in the chiral limit is large enough to ruin the convergence of an EFT with perturbative pions. This problem is however alleviated in the Weinberg power counting where pions are treated nonperturbatively. This has led the authors of Ref. \cite{Beane:2001bc} to propose a better power-counting scheme which requires an expansion around the chiral limit. The community remains in need for a better EFT for nuclear interactions which does not suffer from the drawbacks of the approaches mentioned here. Nonetheless, these EFTs have widely been used in studying a variety of nuclear systems \cite{KalantarNayestanaki:2011wz, Barrett:2013nh, Roth:2011ar, Epelbaum:2009pd, Epelbaum:2011md, Epelbaum:2012iu, Otsuka:2009cs, Holt:2010yb, Holt:2013vqa, Hagen:2012sh, Hagen:2012fb, Roth:2011vt, Hergert:2012nb, Soma:2012zd, Wienholtz:2013nya, Kaiser:2001jx, Epelbaum:2008vj, Hebeler:2009iv, Hebeler:2010xb, Hebeler:2010jx, Tews:2012fj, Kruger:2013kua, Holt:2012yv, Gezerlis:2013ipa}. The uncertainties on three- and multi-nucleon force parameters remain a significant source of uncertainty in some of these calculations. Due to limited experimental data, the help of LQCD to constrain these parameters will be crucial in the upcoming years.
\section{Lattice Quantum Chromodynamics
\label{LQCD}}
A non-perturbative approach in solving QCD, without making any assumption about the strength of the coupling or the energy scale, is via the path integral formalism. In this formalism, physical quantities are evaluated by taking expectation values of the corresponding operators in the background of the QCD vacuum,
\begin{eqnarray}
\langle \hat{\mathcal{O}} \rangle = \frac{1}{\mathcal{Z}} \int \mathcal{D}A_{\mu} \mathcal{D}q \mathcal{D}\bar{q} ~ e^{iS_{QCD}} ~ \hat{\mathcal{O}},
\label{path-integral}
\end{eqnarray}
where $\mathcal{Z} = \int \mathcal{D}A_{\mu} \mathcal{D}q \mathcal{D}\bar{q} ~ e^{iS_{QCD}}$ denotes the QCD partition function, $S_{QCD}=\int d^4x ~ \mathcal{L}_{QCD}$ is the action and $\mathcal{L}_{QCD}$ is given in Eq. (\ref{L-QCD}). Evaluating this path integral in practice requires several steps to be followed:
\emph{1) A discrete action}: The path integral in Eq. (\ref{path-integral}) is only defined rigorously if the degrees of freedom of the theory are discrete. Numerical evaluations become plausible in practice, firstly, with a measure that is nonoscillatory. This can be achieved by a Wick rotation of the coordinates to Euclidean spacetime, $t \to i\tau$ so that $iS_{QCD} \to -S_{QCD}^{(E)}$ where $S_{(QCD)}^{(E)}$ is purely real. Secondly, the number of degrees of freedom of the integration must be finite, requiring the spacetime to be truncated to a finite region in both spatial and temporal directions and to be discretized. Lattices with geometry of a hypercube are the most convenient choices in LQCD calculations, see Fig. \ref{fig:Lattice}, although the anisotropic cubic lattices with lattice spacing in the temporal direction being finer than that of the spatial direction are being also used. The spacing between two adjacent lattice sites, $a$, must be small compared with the hadronic scale, $a \ll \Lambda_{QCD}^{-1}$, while the spatial extent of the volume, $L$, must be large compared with the Compton wavelength of the pions which sets the range of hadronic interactions, $L \gg m_{\pi}^{-1}$, see Sec. \ref{IV-intro}.
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.465]{latticecube}
\par\end{centering}
\caption{{\small A $2+1$ dimensional cubic lattice is shown in the left panel. The (trace of) plaquette and the product of quark, the link variable and the antiquark (right panel) are two examples of gauge-invariant constituents of the lattice gauge theories in their compact formalism.}}
\label{fig:Lattice}
\end{figure}
Quark fields are placed on the lattice sites, and a choice for defining the gauge fields, as plotted in Fig. \ref{fig:Lattice}, is through the Wilson link variables,
\begin{eqnarray}
U_{\mu}(n) \equiv e^{i g A_{\mu}(n)}.
\label{Links}
\end{eqnarray}
These are the elements of the $SU(3)$ Lie group and transform under a local gauge transformation as
\begin{eqnarray}
U_{\mu}(n) \to U_{\mu}(n)'=V(n) U_{\mu}(n) V^{\dagger}(n+\hat{\mu}),
\label{Links-trans}
\end{eqnarray}
where $V$ is an element of the Lie group. The use of link variables, which is called the \emph{compact} formulation of lattice gauge theories, is a convenient choice as it makes the implementation of gauge invariance on the lattice straightforward. In fact, the only gauge invariant quantities are the gauge links starting and ending at the quark fields, and the trace of any closed loop formed by the gauge links, Fig. \ref{fig:Lattice}.
With these gauge invariant blocks, we can write down a Lagrangian for QCD interactions on the lattice that recovers the Lagrangian in Eq. (\ref{L-QCD}) once the continuum limit is taken. A common choice of action is the Wilson action \cite{PhysRevD.10.2445} which uses the elementary plaquette, defined as $P_{\mu \nu;n} \equiv U_{\mu}(n) U_{\nu}(n+\hat{\mu}) U^{\dagger}_{\mu}(n+\hat{\nu}) U^{\dagger}_{\nu}(n)$, see Fig. \ref{fig:Lattice}, for gluons and the Wilson fermions formulation for the quarks,
\begin{eqnarray}
S^{(E)}_{{\rm Wilson}}&=&\frac{\beta}{N_c} \sum_{n} \sum_{\mu < \nu} \Re {\rm Tr} [\mathbb{1}-P_{\mu \nu;n}]
\nonumber\\
&-& \sum_{n} \bar{q}_n [\overline{m}^{(0)}+4] q_n +
\sum_{n} \sum_{\mu} \left[ \bar{q}_n \frac{r-\gamma_{\mu}}{2} U_{\mu}(n) q_{n+\hat{\mu}} + \bar{q}_n \frac{r+\gamma_{\mu}}{2} U_{\mu}^{\dagger}(n-\hat{\mu}) q_{n-\hat{\mu}} \right],
\nonumber\\
\label{L-Wilson}
\end{eqnarray}
where $n$ runs over all the $N_s^3 \times N_t$ lattice points and $\beta \equiv \frac{2 N_c}{g^2}$ is the lattice coupling constant with $N_c=3$ for QCD. Note that the action is written in terms of dimensionless fields and parameters. Explicitly, the continuum field $q$ at point $na$ is replaced by $a^{-3/2} q_n$ and the continuum bare mass of the quarks $m^{(0)}$ is replaced by $a^{-1}\overline{m}^{(0)}$. $r$ is the Wilson parameter whose value is commonly set to $1$ in the calculations. The sum over quark flavors is left implicit.
The gluonic part of the action clearly recovers the continuum action in Eq. (\ref{CP-even}) up to corrections that scale as $a^2$, and leads to the following lattice propagator in momentum space
\begin{eqnarray}
a^{-2}\mathcal{D}^{(G)}_{{\rm Wilson}}(\overline{k})=\frac{i}{4\sum_{\mu}\sin^{2}\left(\overline{k}_{\mu}/2\right)},
\label{Gluon-prop}
\end{eqnarray}
where the Feynman gauge is used to fix the gauge and $\overline{k}_{\mu} = k_{\mu} a$.\footnote{Due to the use of manifestly gauge-invariant path integral, the compact formulation of lattice gauge theories does not require gauge fixing. This is in contrast with a non-compact formulation where the gauge fields remain the explicit degrees of freedom and the continuum action is discretized directly. This is the popular formulation used in pure lattice QED calculations and so requires fixing the gauge, see chapter \ref{chap:EM}.} The fermionic part of the action is nothing but what is expected from a \emph{naive} discretization of the Dirac operator in Eq. (\ref{L-q-free}) -- with the inclusion of link variables to render the discrete derivative gauge invariant -- plus an additional contribution proportional to $r$. This latter contribution is introduced by Wilson to circumvent the so-called fermion doubling problem, due to which the continuum limit of the naive Dirac fermions leads to $2^4$ degenerate fermions. One way to see this problem is by studying the Wilson quark propagator that we use extensively in chapter \ref{chap:operators} to study the lattice operators perturbatively, and can be derived readily from the action in Eq. (\ref{L-Wilson}),
\begin{eqnarray}
a^{-1}\mathcal{D}^{(F)}_{{\rm Wilson}}=\frac{-i\sum_{\mu}\gamma_{\mu}\sin\left(\overline{k}_{\mu}\right)+2r\sum\limits_{\mu}\sin^{2}\left(\overline{k}_{\mu}/2\right)+\overline{m}^{(0)}}
{\sum_{\mu}\sin^{2}\left(\overline{k}_{\mu}\right)+(2r\sum\limits_{\mu}\sin^{2}\left(\overline{k}_{\mu}/2\right)+\overline{m}^{(0)})^2}.
\label{Quark-prop}
\end{eqnarray}
When $r = 0$, the poles of the propagator for $\overline{m}^{(0)} = 0$ occur at 16 distinct momenta, $\bar{k}$,
\begin{eqnarray}
(0,0,0,0),(\pm \pi,0,0,0),(0,\pm \pi,0,0), \dots ,(\pm \pi,\pm \pi,\pm \pi,\pm \pi),
\label{doublers}
\end{eqnarray}
however, only the $\bar{k}=0$ pole is the desired continuum pole. By adding the Wilson term, the doublers that correspond to $|\bar{k}| \sim \pi$ acquire a mass that, in the continuum limit, scale as $m(r) \sim r/a$ and will therefore decouple from theory due to their heavy mass.
The downside of the Wilson action is that it breaks the chiral symmetry explicitly. As is evident from action (\ref{L-Wilson}), the terms proportional to $r$ behave similar to the quark mass term and mix the left-handed and right-handed quarks. As it turns out, these are universal problems with most of the discretized fermionic actions that can be nicely summarized via the Nielsen-Ninomiya theorem \cite{Nielsen198120}. The theorem states that a lattice Dirac operator cannot simultaneously 1) be a periodic function of momentum and analytic except at $\mathbf{p} \neq \mathbf{0}$, 2) be proportional to $\gamma_{\mu} p_{\mu}$ in the continuum limit, and 3) anticommute with $\gamma_5$. It is clear that the naive Dirac operator satisfies both 2 and 3 but fails to meet the first condition given the presence of doublers. The Wilson Dirac operator on the other hand satisfies 1 and 2 but it does not anticommute with $\gamma_5$ signifying its chiral symmetry breaking feature. Solutions to the lattice fermions' puzzle include the \emph{domain-wall} fermions \cite{Kaplan:1992bt} and overlap fermions \cite{Narayanan:1993ss, Narayanan:1994gw} that both belong to the category of the Ginsberg-Wilson fermions.\footnote{Domain-wall fermions only satisfy the Ginsberg-Wilson relation in a particular limit, i.e. when the domain-walls separation is infinite.} Ginsberg and Wilson relation \cite{Ginsparg:1981bj} redefines the chiral symmetry on the lattice,
\begin{eqnarray}
\{D,\gamma_5\}=a D \gamma_5 D,
\label{GW}
\end{eqnarray}
with $D$ being a dimensionful Dirac operator, and therefore breaks the last condition in the Nielsen-Ninomiya theorem. It however ensures that the chiral features of the continuum fermions, including the chiral anomaly, are exactly reproduced as long as the operator $D$ satisfies this relation. Unfortunately, numerical simulations of both domain-wall and overlap fermions comes with additional cost compared with Wilson fermions.\footnote{Simulating domain-wall fermions includes adding an extra dimension to the calculation of the quark propagators while simulating overlap fermions requires inversion of an extra operator beside the overlap operator, see Refs. \cite{Kaplan:2009yg, Kennedy:2006ax, Jansen:1992tw} for more details.} Nonetheless, the use of the chiral lattice fermions in LQCD calculations has become more common as the computational resources improve. A nice review of fermions and chiral symmetry on the lattice can be found in Ref. \cite{Kaplan:2009yg}.
\emph{2) Generate gauge-filed configurations}: Now that we have a discrete action with the desired continuum limit, let us go back to the path intergarl we aim to evaluate,
\begin{eqnarray}
\langle \hat{\mathcal{O}} \rangle = \frac{1}{\mathcal{Z}} \int \mathcal{D}U_{\mu} \mathcal{D}q \mathcal{D}\bar{q} ~ e^{-S^{(G)}_{\rm lattice}[U]-S^{(F)}_{\rm lattice}[U,q,\bar{q}]} ~ \hat{\mathcal{O}}[U,q,\bar{q}],
\label{path-integral-lattice-I}
\end{eqnarray}
where we have split the action to the purely gauge part and the fermionic part, and have left the superscripts $E$ for the Euclidean action implicit. This expectation value can be written as
\begin{eqnarray}
\langle \hat{\mathcal{O}} \rangle = \frac{1}{\mathcal{Z}} \int \mathcal{D}U_{\mu} ~ e^{-S^{(G)}_{\rm lattice}[U]} \mathcal{Z}_F[U]~ \langle \hat{\mathcal{O}} \rangle_F
\label{path-integral-lattice-II},
\end{eqnarray}
where the path integral over gauge links $U$ are separated from that of the fermionic path integrals with
\begin{eqnarray}
\langle \hat{\mathcal{O}} \rangle_F= \frac{1}{\mathcal{Z}_F} \int \mathcal{D}q \mathcal{D}\bar{q} ~ e^{-S^{(F)}_{\rm lattice}[U,q,\bar{q}]}\mathcal{O}[q,\bar{q},U],
\label{O-F}
\end{eqnarray}
and $\mathcal{Z}_F$ is the partition function of the fermions which will still depend on the value of the gauge link. By expressing the fermionic action as $S^{(F)}_{\rm lattice}=\sum_{n,m} \bar{q}_n D_{n,m} q_{m}$, where $D_{n,m}$ is the matrix element of one of the chosen lattice operators discussed above in position space, the fermionic partition function can be written as
\begin{eqnarray}
\mathcal{Z}_F= \int \mathcal{D}q \mathcal{D}\bar{q} ~ e^{-S^{(F)}_{\rm lattice}[U,q,\bar{q}]}= \prod_f \det D_f,
\label{Z-F}
\end{eqnarray}
where the product of the determinant of Dirac operator matrix, $D_f$, corresponding to each dynamical flavor is explicit. Now from Eq. (\ref{path-integral-lattice-II}) it is clear that once the fermionic expectation value $\langle \hat{\mathcal{O}} \rangle_F$ is computed, the full expectation value can be computed using a Monte Carlo sampling integration with the probability measure $ \frac{1}{\mathcal{Z}}e^{-S^{(G)}_{\rm lattice}[U]} \prod_f \det D_f$. An important property of the lattice Dirac operators, the $\gamma_5$-hermiticity $D^{\dagger}=\gamma_5 D \gamma_5$, ensures that the determinant of the Dirac operator is real, providing a well-defined sampling weight in the numerical evaluation of the expectation values. LQCD calculations with dynamical fermions require computing the gauge-field configuration with a distribution that depends on the fermion determinant -- the determinant of the Dirac operator which is a large matrix with dimensionality $(12 N_s^3 \times N_t )^2$ (on each spacetime point on the lattice there are 3 color and 4 spinor degrees of freedom for each flavor of quarks). After each configuration generation both the gauge part and the determinant part must be updated simultaneously to generate the next configuration.\footnote{As a result, early LQCD calculations were limited to the \emph{quenched} approximation where the fermion determinant is set to one to reduce the computational cost of the gauge-field configurations. Unfortunately quenching is an uncontrolled approximation and only describes QCD if the quarks were infinitely heavy. Nowadays, the growth in the computational resources available to LQCD calculations has enabled abandoning this approximation and has made the use of dynamical configurations viable in most calculations.}
When a large number of almost statistically uncorrelated gauge field configurations, $N$, are generated, the statistical average
\begin{eqnarray}
\langle \hat{\mathcal{O}} \rangle = \frac{1}{N} \sum^{N}_{i} \langle \hat{\mathcal{O}} \rangle_F [U^{(i)}],
\label{expec-value}
\end{eqnarray}
is an estimator of the the expectation value in Eq. (\ref{path-integral-lattice-II}), where $U^{(i)}$ is the $i^{th}$ generated configuration.
\emph{3) Form the correlation functions}: The next step of the calculation is observable dependent and requires both analytical and numerical evaluation to determine $\langle \hat{\mathcal{O}} \rangle_F$. Here we are interested in the n-point correlation functions of (multi) hadrons from which one can extract masses and the low-lying energies. Let $\hat{O}^{\dagger}$ denote the interpolating operator that creates a (multi-)hadron states from the vacuum of QCD and $\hat{O}$ be an interpolator that annihilates the state. With the notation used in Eq. (\ref{path-integral-lattice-I}), $\hat{\mathcal{O}} \equiv \hat{O}\hat{O}^{\dagger}$. In order for an interpolating operator to have overlap with a desired state, it must share the same quantum numbers, e.g. the particle number, flavor, spin, parity, charge conjugation, etc., as that of the state. For example the $\pi^+$ state can be created by a bilinear quark operator $O^{{\pi^+}\dagger}=\overline{u} \gamma_5 d$. In order to calculate the correlation function, we need to perform the fermionic path integral that appears in the expectation value $\langle \hat{\mathcal{O}} \rangle_F$ which is a usual Grassmann integration. This part is called the quark \emph{Wick} contractions and for the case of $\pi^+$ two-point correlation function can be performed as following
\begin{eqnarray}
\langle \hat{O}^{\pi^+}(n)\hat{O}^{\pi^+ \dagger}(0) \rangle_F&=&
\langle \overline{d}_{a,\alpha}(n) \gamma^5_{\alpha \beta} u_{\beta}^a(n) ~~
\overline{u}_{b,\alpha'}(0) \gamma^5_{\alpha' \beta'} d_{\beta'}^b(0) \rangle_F
\nonumber\\
&=&
- \gamma^5_{\alpha \beta} \gamma^5_{\alpha' \beta'} ~
\langle d_{\beta'}^b(0) \overline{d}_{a,\alpha}(n) \rangle_d ~
\langle u_{\beta}^a(n) \overline{u}_{b,\alpha'}(0) \rangle_u
\nonumber\\
&=&
- \gamma^5_{\alpha \beta} \gamma^5_{\alpha' \beta'}~
(D^{-1}_d)^b_{a,\beta' \alpha}(0,n) (D^{-1}_u)^a_{b,\beta \alpha'}(n,0)
\nonumber\\
&=&
-{\rm Tr} \left[ \gamma^5 D_u^{-1}(n,0) \gamma^5 D_d^{-1}(0,n) \right]
\nonumber\\
&=& -{\rm Tr} \left[D_u^{-1}(n,0) D_d^{-1}(n,0) \right],
\label{pi-contraction}
\end{eqnarray}
where we have chosen to create the pion at the origin and annihilate it at coordinate $n$. The trace is taken over spin and color degrees of freedom and the negative sign has been resulted from anti-commutation of the Dirac fields in the second line. In the last line the $\gamma^5$-hermiticity of the Dirac operator has been used. The resulting correlation function has been pictorially shown in Fig. \ref{fig:Pion-contraction}.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.365]{pipluscontraction}
\caption
{{\small The Wick contractions in the evaluation of the $\pi^+$ two-point correlation functions.}
}
\label{fig:Pion-contraction}
\end{center}
\end{figure}
The value of the inverse Dirac operator depends on the value of the link variable, therefore for each gauge-field configuration generated in the previous step, the inverse of the Dirac operator must be evaluated.\footnote{When the value of the light-quark masses that are used are close to their physical values, the small eigenvalues of the Dirac operator causes difficulties in numerical evaluations of the inverse matrix given the limited statistics. This is among the reasons for the numerical limitations faced by the LQCD community in approaching the physical point.}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.335]{pizerocontraction}
\caption
{{\small The Wick contractions in the evaluation of the $\pi^0$ two-point correlation functions.}
}
\label{fig:Pion0-contraction}
\end{center}
\end{figure}
For flavor-singlet quantities, such as $\pi^0$, there are additional contributions to the correlation functions, namely the disconnected contributions, that put limitations on the calculation of such quantities with the current computational resources.\footnote{Some LQCD collaborations have started including the disconnected diagrams in their calculations, see Refs. \cite{Wagner:2012ay, Collins:2012mg, Alexandrou:2013cda, Abdel-Rehim:2013wlz, Dudek:2013yja, Alexandrou:2014yha, Bai:2014cva}.} Explicitly for the $\pi^0$ correlator with $\hat{O}=\frac{1}{\sqrt{2}}(\overline{u} \gamma^5 u - \overline{d} \gamma^5 d)$, we have
\begin{eqnarray}
\langle \hat{O}^{\pi^0}(n)\hat{O}^{\pi^0 \dagger}(0) \rangle_F &=&
-\frac{1}{2}{\rm Tr} \left[ \gamma^5 D_u^{-1}(n,0) \gamma^5 D_u^{-1}(0,n) \right]
\nonumber\\
&& + \frac{1}{2}{\rm Tr} \left[ \gamma^5 D_u^{-1}(n,n) \right] {\rm Tr} \left[ \gamma^5 D_u^{-1}(0,0) \right]
\nonumber\\
&& - \frac{1}{2}{\rm Tr} \left[ \gamma^5 D_u^{-1}(n,n) \right] {\rm Tr} \left[ \gamma^5 D_d^{-1}(0,0) \right] + \{ u \leftrightarrow d \},
\label{pio-contraction}
\end{eqnarray}
as depicted in Fig. \ref{fig:Pion0-contraction}. The second and third term, which contain the propagator from a single lattice point to itself, require evaluations of the \emph{all-to-all} propagators.\footnote{It must be noted that in the isospin limit, where the masses of $m_u$ and $m_d$ are set equal in the calculations, the disconnected contributions to the $\pi^0$ correlator vanish. This is the case for most of the lattice calculations that are currently performed. For isosinglet quantities such cancellation, even in the isospin limit, does not occur.} This introduces substantial extra cost in calculations as now instead of a column in the inverse Dirac operator matrix in position space, one needs to calculate the full matrix.
As the number of hadrons increases, the quark contractions to be performed become more involved, however the procedure described above remains the same.
By taking advantage of various symmetries of multi-hadron systems and optimal choices of interpolators, the number of required contractions can be substantially reduced, see Refs. \cite{Shi:2011mr, Detmold:2012wc, Detmold:2010au, Doi:2012xd, Detmold:2012eu}. Due to the progress in the algorithms that perform contractions required in the evaluation of multi-baryon correlation functions \cite{Doi:2012xd, Detmold:2012eu}, obtaining the correlation functions of several nuclei up to $^{28}{\rm Si}$ are shown to be computationally plausible \cite{Detmold:2012eu}. Such developments gave rise to the first LQCD determination of the binding energies of the light nuclei and hypernuclei (up to atomic number 5) albeit at the heavy pion mass $m_{\pi} \approx 800~{\rm MeV}$ by the NPLQCD collaboration \cite{Beane:2012vq}, followed by the another determination of the binding of nuclei at a slightly lighter pion mass $m_{\pi} \approx 500~{\rm MeV}$ by Yamazaki, \emph{et al.} \cite{Yamazaki:2013rna}.
\emph{4) Extract masses and energies}: Let us first project the correlation function to a momentum $\bm{P}$,
\begin{eqnarray}
C(\mathbf{P};n_4) = \sum_{\mathbf{n}} e^{i \bm{P}.\bm{n}a} \langle \hat{O}(\mathbf{n},n_4)\hat{O}^{\dagger}(\mathbf{0},0) \rangle.
\label{corre-func-def-I}
\end{eqnarray}
Then upon inserting a complete set of states and using $\hat{O}(\mathbf{n},n_4)=e^{\hat{H}n_4}\hat{O}(\mathbf{n},0)e^{-\hat{H}n_4}$ (where Hamiltonian operator is defined through the lattice transfer matrix), the correlation function in the limit of large (Euclidean) time becomes
\begin{eqnarray}
C(\mathbf{P};n_4) &=& \sum_k \sum_{\mathbf{n}} e^{i \bm{P}.\bm{n}a} \langle 0 | \hat{O}(\mathbf{n},n_4) | k \rangle \langle k | \hat{O}^{\dagger}(\mathbf{0},0) | 0 \rangle
\nonumber\\
&=&A_0 e^{-E(\bm{P}) |n_4| a}\left(1+\mathcal{O}(e^{-\Delta E(\bm{P}) |n_4| a})\right)
\label{corre-func-def-II},
\end{eqnarray}
where $E(\bm{P})$ is the lowest energy eigenvalue of the system and is related to the three-momentum through a (lattice) dispersion relation. $\Delta E(\bm{P})$ denotes the difference between the ground state energy and the first excited state energy. $A_0$ accounts for the overlap of the interpolator used onto the ground state.
A useful quantity which is commonly plotted is the effective mass/energy, defined as
\begin{eqnarray}
m_{eff}(n_t)=\ln \frac{C(\mathbf{P};n_t)}{C(\mathbf{P};n_t+1)}.
\label{m-eff}
\end{eqnarray}
As is clear, once the system approaches its ground state at large times, this quantity becomes constant. This defines a plateau region the the effective mass plot (EMP) as a function of time from which the ground state energy of the system can be read off. By using a larger basis of interpolating operators and by increasing the number of correlation function measurements, the excited state energies of the system can as well be extracted, see Refs. \cite{Beane:2005rj, Beane:2006mx, Beane:2006gf, Beane:2007es, Detmold:2008fn, Beane:2009py, Thomas:2011rh, Beane:2011sc, Basak:2005ir, Peardon:2009gh, Dudek:2010wm, Edwards:2011jj, Dudek:2012ag, Dudek:2012gj, Yamazaki:2009ua, Yamazaki:2012hi}.
\begin{figure}[b!]
\begin{centering}
\includegraphics[scale=0.445]{emppipi}
\par\end{centering}
\caption{{\small The EMPs of the $\pi \pi$ system in the $I=2$ channel produced by the NPLQCD collaboration \cite{Beane:2011sc}. The plots correspond to two different values of the total CM momentum $|\bm{P}| \equiv P_{CM}=0,1$. Energy and time are made dimensionless using the temporal extent of the (anisotropic) lattice used in this calculation, $a_t \approx 0.035~{\rm fm}$. Different colors represent different energy levels labeled by index $n$. Figure is reproduced with the permission of the NPLQCD collaboration.}}
\label{fig:EMP-pipi}
\end{figure}
In Sec. \ref{subsec:ChiPT}, as an example of the interplay between LQCD calculations and the low-energy effective field theories, we presented the result of a LQCD determination of the scattering parameters of the $\pi\pi$ system in the $I=2$ channel by the NPLQCD collaboration \cite{Beane:2011sc}. Here we show the immediate output of this calculation which are the energy eigenvalues in Fig. \ref{fig:EMP-pipi} through EMPs. The plateau region can be clearly identified from the plots before the noise dominates the signal at later times. The calculations of the correlation functions have been done with various different total momenta $\bm{P}$ to increase the number of energy levels extracted. We will come back to this example in Sec. \ref{Two-body} and discuss a non-trivial step that led to the result presented earlier in this chapter.
\emph{5) Interpret the results}: The energy levels (masses) are the output of the calculations we discussed so far. The final step is to try to make sense of these extracted quantities in terms of physical quantities, i.e. those that correspond to the continuum infinite-volume limit of the calculations (and the physical light-quark masses when the calculations have not been performed with physical values of quark masses). Usually, available computational resources allow for multiple calculations with few several lattice spacings, volume sizes and quark masses and therefore extrapolations (interpolations) to the physical point are plausible. The rest of this thesis deals with situations where such extrapolation (interpolations) are not practical or when the determination of the physical quantities depends on calculations away from the physical scenarios, e.g. in determination of scattering parameters. We will not discuss the pion-mass interpolations further in the following, and will focus on the continuum and infinite-volume limits. Since the discussion of the finite-volume (FV) effects is rather extensive in the following chapters, we take the opportunity to introduce and motivate the FV formalism for LQCD in more details in the next section of this chapter.
\section{Infinite-volume Observables from a Finite-volume Formalism
\label{IV-intro}}
\subsection{Single-particle sector
\label{Sec:Single-FV}}
Given the finite extent of the volumes used in LQCD calculations, one should expect the masses extracted form the large-time behavior of lattice two-point correlation functions are not equal to their infinite-volume values. Qualitatively, this can be understood by noting that due to polarization effects, the exchanged particles can propagate to the boundaries of the volume and cause corrections to the mass that differ from those of the scenario when the boundary is in infinity. For the case of QED interactions, it is the photon that gives rise to polarization effects, and due to its zero mass, the FV corrections must scale as (inverse) powers of volume. The situation with the QCD interactions is different, as due to confinement, these are not the massless gluons that go around the volume, and they will not give rise to any significant volume effects. Instead the volume effects are dominantly due to the presence of the pGBs of the spontaneous breaking of the chiral symmetry (pions for the case of $SU(2)$ symmetry). Roughly speaking, by constraining a hadron to a finite volume, the pion could surrounding it is squeezed and the hadron mass is shifted. Here we discuss these corrections to the mass of hadrons through the example of nucleons' mass, to which we come back later in chapter \ref{chap:TBC}, where we apply different boundary conditions than periodic to explore its consequences. The discussion of the volume effects to the masses due to QED interactions will be delayed until chapter \ref{chap:EM}.
As rigorously proved by Martin L\"uscher in 1985 \cite{Luscher:1985dn} for a massive scalar field theory, the volume corrections to the mass of particles have a universal form, and fall off exponentially with volume with a rate that is set by the mass of the lightest particle that is exchanged in the theory. Since volume corrections are due to the IR manipulation of the system, these corrections, as described by L\"uscher, are of kinematic nature and the details of the interactions are not needed in obtaining these results -- a situation that continues to be the case for the two-body problem, see Sec. \ref{Two-body}. So we will consider the nucleon in the HB$\chi$PT and calculate the corrections to its mass due to enclosing it in a finite cubic volume with the PBCs.
In the heavy-baryon formalism (see Sec. \ref{subsec:ChiPT}), the mass of the nucleon with momentum $P_{\mu}=M_{N}^{(0)}v_{\mu}+l_{\mu}$ is obtained from the pole of the following fully dressed propagator
\begin{align}
&\mathcal{D}_{N_l}=\frac{i}{P \cdot v-M_N^{(0)}+i \epsilon}\
\left[ 1-i \Sigma^{(1PI)} \frac{i}{P \cdot v-M_N^{(0)}+i \epsilon}+\left(-i \Sigma^{(1PI)} \frac{i}{P \cdot v-M_N^{(0)}}\right)^2+ \dots \right]
\nonumber\\
& ~~~~ = \frac{i}{P \cdot v-M_N^{(0)}-\Sigma^{(1PI)}+i \epsilon}
~\equiv~ \frac{i Z_N}{P \cdot v-M_N+i \epsilon},
\label{S-N}
\end{align}
as depicted in Fig. \ref{fig:Self-energy}(a). $M_{N}^{(0)}$ is the nucleon bare mass and $\Sigma^{(1PI)}$ denotes the one-particle irreducible self energy of the nucleon. $\Sigma^{(1PI)}$ can be seen to depend on two scalar variables $v \cdot l$ and $l^2$, so by rewriting $l_{\mu}$ as $l_{\mu}=(M_N-M_N^{(0)})v_{\mu}+(P_{\mu}-M_N v_{\mu})$, it is easy to see that by requiring the on-shell condition $P.v=M_N$, the nucleon mass can be identified as
\begin{eqnarray}
M_N=M_N^{(0)}+\Sigma^{(1PI)}|_{v \cdot l =M_N-M_N^{(0)};~ l^2=(M_N-M_N^{(0)})^2}.
\label{MN-Sigma}
\end{eqnarray}
\begin{figure}[t!]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.460]{selfenergy}}
\subfigure[]{
\includegraphics[scale=0.455]{piloops}}
\caption
{{\small a) The 1PI self-energy diagrams contributing to the fully dressed nucleon propagator to all orders. The thick solid line denotes the full nucleon propagator. b) The leading contributions (the upper panel) to the 1PI self-energy diagram in HB$\chi$PT comes from an insertion of the quark mass matrix (the diamond) according to Eq. (\ref{L-BpGB-Mass}). The NLO contributions (the lower panel) arise from the pion loops where the possibility of the production of a delta resonance in the loop is taken into account. The black dots denote axial couplings. The solid line, solid-double line and dashed line denote bare nucleon, $\Delta$ resonance and pion propagators, respectively.}
}
\label{fig:Self-energy}
\end{center}
\end{figure}
At LO in HB$\chi$PT, $\mathcal{O}({\frac{p^2}{\Lambda_{\chi}^2}})$, there is one contribution to the self-energy diagram, as shown in the upper panel of Fig. \ref{fig:Self-energy}(b). It comes from an insertion of the light-quark mass matrix, arising from Lagrangian in Eq. (\ref{L-BpGB-Mass}). This contribution reads
\begin{eqnarray}
\Sigma^{(1PI)}_{LO}=-4 c_1m_{\pi}^2,
\label{LO-Sigma}
\end{eqnarray}
with $c_1=-0.93 \pm 0.10 ~{\rm GeV}^{-1}$ \cite{Bernard:1996gq}. At NLO in the chiral expansion, $\mathcal{O}({\frac{p^3}{\Lambda_{\chi}^3}})$, there are contributions from chiral loops as shown in the lower panel of Fig. \ref{fig:Self-energy}(b). Due to the small mass difference between the nucleon and the $\Delta$ resonance, $\Delta \approx 292~{\rm MeV}$, the contribution from this resonance to the self-energy of the nucleon must be taken into account at this order. We did not discusse the coupling of the baryon decuplets to the pGBs and to the baryon octets in Sec. \ref{subsec:ChiPT}, but it is straightforward to show that the only required vertex for this calculation comes from the following chirally invariant Lagrangian
\begin{eqnarray}
\mathcal{L}_{\Delta N}=g_{\Delta N} \overline{\Delta}^{abc,\nu} \mathcal{A}_{a,\nu}^d N_{b} \epsilon_{cd},
\label{L-Ndelta}
\end{eqnarray}
where the axial vector current $\mathcal{A}_{\nu}$ is defined in Eq. (\ref{Axial}). Then from this Lagrangian and that in Eq. (\ref{L-piN}) for the axial coupling of nucleons, it is easy to see that for the loop corrections, we have
\begin{eqnarray}
\Sigma^{(1PI)}_{NLO}=-i\frac{9 g_A^2}{2 f_{\pi}^2} \mathcal{I}(\infty,0)
-i\frac{4 g_{\Delta N}^2}{f_{\pi}^2} \mathcal{I}(\infty,\Delta),
\label{Sigma-NLO}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{I}(\infty,\Delta)=-\frac{1}{3} \int \frac{d^4 k}{(2 \pi)^4}
\frac{\bm{k}^2}{(k^0-\Delta+i \epsilon)(k^{02}-\bm{k}^2-m_{\pi}^2+i \epsilon)}.
\label{Ndelta-loop}
\end{eqnarray}
The integral is clearly UV divergent and must be renormalized. However, since we are interested in the FV corrections to the nucleon mass, we do not need to carry out this integration any further. The only observation to be made before moving on to the FV scenario is to note that the (renormalized) mass at this order is proportional to $m_{\pi}^3 \sim m_{q}^{3/2}$ (for $\Delta=0$ term) and is therefore non-analytic in the light-quark masses. The contribution from the $\Delta$-resonance introduces further nontrivial non-analytic corrections to the mass of the nucleon, see Refs. \cite{Jenkins:1990jv, Hemmert:1997ye, Bernard:1993nj, Beane:2002vq} for the discussion of baryon masses from (HB)$\chi$PT.
In a finite volume, the momentum modes are all discretized due to the PBCs, $\mathbf{k}=\frac{2\pi}{L}\mathbf{n},~\bm{n} \in \mathbb{Z}^3$. As a result the only difference between the FV and infinite-volume calculation arises from the loops where the integrals over momenta are replaced with sums \cite{AliKhan:2003cu, Beane:2004tw, Beane:2011pc},
\begin{eqnarray}
\mathcal{I}(L,\Delta)=-\frac{1}{3} \frac{1}{L^3} \sum_{\bm{k}} \int \frac{d k^0}{(2 \pi)^4}
\frac{\bm{k}^2}{(k^0-\Delta+i \epsilon)(k^{02}-\bm{k}^2-m_{\pi}^2+i \epsilon)}.
\label{Ndelta-loop-sum}
\end{eqnarray}
Note that we keep the temporal extent of the volume infinite for the discussion of FV effects. Since in practice LQCD calculations have a finite extent in the (imaginary) time direction, there will be contaminations to the extracted energies from the backward propagating states. Such \emph{thermal} effects must be dealt with separately but their effects can be shown numerically to be smaller that the (spatial) volume effects. Using the Poisson re-summation formula,
\begin{eqnarray}
\frac{1}{L^3} \sum_{\bm{k}} f(\mathbf{k})=\int \frac{d^3k}{(2 \pi)^3} f(\mathbf{k})+\sum_{\bm{m} \neq \bm{0}}\int \frac{d^3k}{(2 \pi)^3} f(\mathbf{k}) e^{i \bm{k}.\bm{m}L},
\label{Poisson}
\end{eqnarray}
where $\mathbf{m}$ is another triplet of integers, one can isolate the infinite-volume contribution to $\mathcal{I}(L,\Delta)$ in Eq. (\ref{Ndelta-loop}), which will be canceled out when taking the difference of the infinite-volume and FV masses. With the help of a useful identity,
\begin{eqnarray}
\frac{1}{(\bm{k}^2+\mathcal{M}^2)^r}=\frac{1}{\Gamma(r)} \int_{0}^{\infty} ds s^{r-1} e^{-s(\bm{k}^2+\mathcal{M}^2)},
\label{Identity}
\end{eqnarray}
it then takes a few lines of algebra to show that \cite{Beane:2004tw},
\begin{eqnarray}
\delta_{L} M_{N} \equiv M_N(L)-M_N(\infty)= \frac{3 g_A^2}{8 \pi^2 f_{\pi}^2} \mathcal{K}(0)
+\frac{g_{\Delta N}^2}{3 \pi^2 f_{\pi}^2} \mathcal{K}(\Delta),
\label{deltaM}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{K}(0)= \frac{\pi}{2}m_{\pi}^2 \sum_{\mathbf{n}\neq \mathbf{0}} \frac{e^{-|\mathbf{n}| m_{\pi} L}}{|\mathbf{n}| L}
,
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{K}(\Delta)
\ =\
\int_{0}^{\infty} d \lambda ~ \beta_{\Delta} ~ \sum_{\mathbf{n}\neq \mathbf{0}}
\left[
\beta_{\Delta} K_0(\beta_{\Delta}|\mathbf{n}| L)
\ -\
\frac{1}{|\mathbf{n}| L} K_1(\beta_{\Delta}|\mathbf{n}| L)
\right]
.
\end{eqnarray}
$K_n(z)$ is the modified Bessel function of the second kind, and $\beta_{\Delta} = \lambda^2 + 2 \lambda \Delta + m_{\pi}^2$.\footnote{
Note that we have chosen to define the $\mathcal{K}(\Delta)$ function with a negative sign compared to
Ref.~\cite{Beane:2004tw}.
} When expanded in the limit of large $L$, Eq.~(\ref{deltaM}) scales as $e^{-m_{\pi}L}/L$ at LO. Explicitly one obtains \cite{Beane:2004tw}
\begin{eqnarray}
\delta_{L} M_{N}^{asym} = \left[\frac{9 g_A^2 m_{\pi}^2}{8 \pi f_{\pi}^2}+
\frac{4g_{\Delta N}^2 m_{\pi}^{5/2}}{(2 \pi)^{3/2} f_{\pi}^2 \Delta}\frac{1}{L^{1/2}} \right]
\frac{1}{L}e^{-m_{\pi}L},
\label{deltaM}
\end{eqnarray}
As we already discussed, the exponential corrections of these types are general features of interacting theories with finite-range interactions. The reader can consult Refs. \cite{Gasser:1986vb, Colangelo:2003hf, Colangelo:2005cg, Colangelo:2010ba, Beane:2011pc, Briceno:2013rwa} for the FV corrections to the masses of mesons and baryons.
\subsection{Two-particle sector
\label{Two-body}}
As mentioned, LQCD produces n-point correlation functions of Euclidean spacetime. Euclidean correlation functions with the \emph{reflection positivity} property can be Wick rotated back to Minkowski spacetime, as proved by Osterwalder and Schrader~\cite{Osterwalder:1973dx}. Therefore, if one was able to fully reconstruct the continuum correlation functions from the Euclidean lattice counterparts, such analytic continuation would not be formally problematic. However, lattice correlation functions are evaluated at a discrete set of spacetime points and are not exact. Maiani and Testa \cite{Maiani:1990ca} noted the Euclidian nature of LQCD calculations prohibits the determination of few-body scattering quantities from lattice correlation functions in the infinite-volume limit (unless at the kinematic threshold). However, LQCD correlation functions are evaluated in a \emph{finite volume}. In fact, it turns out that the scattering amplitudes of the infinite volume can be constructed from the spectrum of the interacting particles in a finite volume.
The first realization of this statement goes back to 1957 when Huang and Yang \cite{Huang:1957im} considered a quantum mechanical two-body system in a finite volume interacting via a hard spherical potential and found out that the energy shift due to the interactions in a finite volume can be related to the two-body scattering length, $a$. The scattering length is defined as $-1/a=\lim_{k^*\rightarrow0} k^* \cot \delta$, where $\delta$ is the scattering phase shift of the two-particle system and $k^*$ is the momentum of each particle in the CM frame. Thirty years later, Martin L\"uscher, motivated by LQCD applications, extended the Huang and Yang's relation to quantum field theory, and derived a non-perturbative relation between the two-body scattering amplitudes and the FV energy eigenvalues for scalar bosons with zero total momentum~\cite{Luscher:1986pf, Luscher:1990ux}. Various extensions of the L\"uscher relation that followed in subsequent years include generalization to boosted systems \cite{Rummukainen:1995vs, Kim:2005gf, Christ:2005gi}, asymmetric lattices \cite{Li:2003jn, Feng:2004ua, Detmold:2004qn}, systems with unequal masses \cite{Bour:2011ef,Davoudi:2011md, Fu:2011xz, Leskovec:2012gb}, two-body coupled channels \cite{He:2005ey, Liu:2005kr, Lage:2009zv, Bernard:2010fp, Hansen:2012tf, Hansen:2012bj, Briceno:2012yi, Li:2012bi, Briceno:2014oea, Li:2014wga}, nucleons with only S-wave interactions \cite{Beane:2003da}, systems with total spin $1/2$ (pion-nucleon scattering) \cite{Gockeler:2012yj}, with total spin $1$ (nucleon-nucleon scattering) \cite{Ishizuka:2009bx, Briceno:2013lba} with arbitrary spin \cite{Briceno:2014oea}, and calculations with twisted boundary conditions \cite{Bedaque:2004kc, Agadjanov:2013wqa, Briceno:2013hya}.\footnote{The most general form of the two-particle quantization condition incorporating all these extensions has been recently written down in Ref. \cite{Briceno:2014oea}.} Here we present a derivation of a form of the L\"uscher formula applicable to the case of (multi) coupled-channel scattering of scalar particles in the moving frame with arbitrary partial waves. We follow closely Ref. \cite{Briceno:2012yi}, however most of the details associated with generalizing to moving frames have been developed by Kim, $et\; al.$ \cite{Kim:2005gf}, which will be briefly reviewed here for completeness.\footnote{See Refs. \cite{Rummukainen:1995vs, Christ:2005gi} for alternative derivations of the moving frame generalization of the L\"uscher formula.} Generalization to the two-nucleon systems with both periodic and twisted boundary conditions, and their implications for the spectrum of the deuteron and the extraction of its properties will constitute the bulk of chapters \ref{chap:NN}, \ref{chap:deuteron}, \ref{chap:TBC} of this thesis.
Consider a system of multiple two-particle channels (spin 0) coupled via interactions of arbitrary strengths. Since we are interested in the IR modifications to the energy levels of the interacting system, the details of the interactions in the UV turned out to be immaterial for the discussions presented here. In fact, in (an effective) a field theory approach, one does not need to explicitly write down a Lagrangian for the system in obtaining the FV energy eigenvalues.\footnote{In chapter \ref{chap:NN}, we will present another derivation of the L\"uscher formula that starts from an effective Lagrangian using an auxiliary field method. We will see that by matching the parameters of the EFT to scattering amplitudes, the FV energy eigenvalues can be written in terms of these scattering parameters and the dependence on the LECs of the EFT are then eliminated.} The following observations enable us to take such a general approach:
\begin{enumerate}
\item Below the three-particle inelastic thresholds, a general interacting kernel for $2 \to 2$ processes can be replaced, under the following condition, by its infinite-volume counterpart up to exponential corrections in volume. The scale of the exponential suppressions is set by the mass of the particle that is produced, once inelastic thresholds are reached, and is given by $e^{-mL}$. To satisfy this property, all the s-channel contributions -- for which only two particles propagate inside the loops -- are separated from the rest of the contributions. We call such kernel the s-channel two-particle irriducible Bethe-Salpeter kernel, $\mathcal{K}$, see Fig. \ref{fig:twoparFV}(b).
\begin{figure}[t!]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.375]{twoparfvone}}
\subfigure[]{
\includegraphics[scale=0.23]{twoparfvtwo}}
\subfigure[]{
\includegraphics[scale=0.23]{twoparfvthree}}
\caption{\small{ a) The fully-dressed FV two-particle propagator, $\mathcal{M}^V$ can be written in a self-consistent way in terms of the Bethe-Salpeter Kernel, $\mathcal{K}$ and the FV s-channel bubble $\mathcal{G}^V$. There are only two channels that are kinematically allowed, therefore the amplitude, the kernel and the FV two-particle propagator are $2 \times 2$ matrices. b) Shown is the $\mathcal{K}_{I,I}$-component of the kernel, which sums all s-channel two-particle irreducible diagrams for channel $I$. c) The fully dressed one-particle propagator is the sum of all one-particle irreducible diagrams and is denoted by a black dot on the propagator lines. The single particle propagator in channel I (II) is shown by the solid (dashed) lines. Note that we have chosen a $\lambda \phi^4$ theory to display the explicit contributions to kernels and self energies, however the discussions of the FV formalism for any two-body coupled-channel systems as presented in the text is general. For this theory, the production of two particles is the first multi-particle inelastic threshold as there is no $2 \to 3$ couplings in the theory, e.g. interactions of the pGBs of the chiral symmetry breaking.}}
\label{fig:twoparFV}
\end{center}
\end{figure}
\item
The fully-dressed propagators in a finite volume are exponentially close to their infinite-volume counterparts. This is again due to the fact that below the three-particle production thresholds, any loop correction to the single-particle propagator involves the production of an off-mass-shell particle, see Fig. \ref{fig:twoparFV}(c). We ignore such exponential corrections and replace the propagators with their infinite-volume counterparts in constructing the FV correlation functions (alternatively amplitudes).
\item
For any s-channel loop, the integral over the three-momenta is replaced by a sum over discrete momenta,
\begin{eqnarray}
\label{sum-int}
\int \frac{d^4 q}{(2\pi)^4} f(q) \to \frac{1}{L^3} \sum_{\mathbf{q}=\frac{2\pi}{L} \mathbf{n}} \int \frac{dq^0}{2\pi} f(q),
\end{eqnarray}
where $\mathbf{n} \in \mathbb{Z}^3$. If the summand is not singular for any real value of $|\mathbf{q}|$ and falls off fast as $|\mathbf{q}| \to \infty$, the sum can be replaced with integral up to exponentially small corrections that we neglect in this formalism. For any s-channel diagram, see Eq. (\ref{loop3}), this however does not hold due to singularities of the two-particle propagator when the loop momenta coincides with the non-interacting momenta of each particle in the CM frame. This defines the on-shell condition due to which the particles can propagate far enough to encounter the boundary of the volume. This causes large power-law corrections to the FV spectrum as will be derived shortly.
\end{enumerate}
To proceed let us first define the kinematics of the problem of coupled-channel systems in a moving frame. If the total energy and momentum of the system in the laboratory frame (lattice frame) are $E$ and $\mathbf{P}$ respectively, then the total CM energy of the system is $E^*=\sqrt{E^2-\mathbf{P}^2}$, and can be written as $E^*=E/ \gamma$ by introducing the relativistic $\gamma$ factor. For the $i^{th}$ channel, when the two particles each are having masses $m_{i,1}$ and $m_{i,2}$, the CM relative momentum of the particles, $k^*_i$, can be derived from
\begin{eqnarray}
\label{momentum}
k^{*2}_i=\frac{1}{4}\left(E^{*2}-2(m_{i,1}^2+m_{i,2}^2)+\frac{
(m_{i,1}^2-m_{i,2}^2)^2}{E^{*2}}\right),
\end{eqnarray}
which simplifies to $\frac{E^{*2}}{4}-m_{i}^2$ when $m_{i,1}=m_{i,2}=m_{i}$.
For N coupled channels, the scattering amplitude for the $l^{th}$ partial wave can be written as a N-dimensional matrix. In order to have a fully relativistic result that holds for all possible energies below the three-particle production threshold, the scattering amplitude must include all possible diagrams, i.e. contributions from s-, t- and u-channels as well as self-energy corrections. Fig. \ref{fig:twoparFV}(a) depicts the FV analogue of the scattering amplitude, $\mathcal{M}^V$, for the special case of $N=2$ channels.\footnote{One should note that using the notion of FV scattering amplitude is merely for the mathematical convenience. As there is no asymptotic state by which one could define the scattering amplitude in a finite volume, one should in principle look at the pole locations of the two-body correlation functions. However, one can easily show that both correlation function and the so-called FV scattering amplitude have the same pole structure, so we use the latter for the sake of dealing with a simpler representation.} This amplitude is written in a self-consistent way in terms of the Bethe-Salpeter kernel, $\mathcal{K}$, which is the sum of all s-channel two-particle irreducible diagrams as defined above. We emphasize that for energies below the three-particle threshold, the intermediate particles in the kernel and the self-energy diagrams, Fig. \ref{fig:twoparFV}(b),(c), cannot go on-shell, and therefore these are exponentially close to their infinite-volume counterparts. As discussed, it is only in the s-channel diagrams that all intermediate particles can be simultaneously put on shell.
By upgrading the kernel and the two-particle propagators to matrices in the space of open channels, Fig. \ref{fig:twoparFV}(a), it is straightforward to obtain a non-perturbative QC for the energy levels of the system. It is important to note that the channels only mix by off-diagonal terms in the kernel, which implies that in the absence of interactions a two-particle state $i$ continues to propagate as a two-particle state $i$. Now in the presence of momentum-dependent vertices a typical s-channel loop for channel $i$ can be written as
\begin{eqnarray}
\label{loop3}
\left[iG^{V}(\mathbf{p}_a,\mathbf{p}_b)\right]_{ab}&\equiv&\frac{n_i}{L^3}\sum_{\mathbf{q}}\int\frac{dq^0}{2\pi}\frac{[\mathcal{K}(\mathbf{p}_a,\mathbf{q})]_{ai}~[\mathcal{K}(\mathbf{q},\mathbf{p}_b)]_{ib}}{[(q-P)^2-m_{i,1}^2+i\epsilon][q^2-m_{i,2}^2+i\epsilon]},
\end{eqnarray}
where the subscripts $a, i, b$ denote the initial, intermediate and final states, respectively, and $n_i$ is ${1}/{2}$ if the particles in the $i^{th}$ loop are identical and 1 otherwise. The sum over all intermediate states, and therefore index $i$ is assumed.
Since the FV corrections arise from the pole structure of the intermediate two-particle propagator, one would expect that the difference between this loop and its infinite-volume counterpart should depend on the on-shell momentum. The on-shell condition fixes the magnitude of the momentum running through the kernels but not its direction. Therefore it is convenient to decompose the product of the kernels into spherical harmonics. These depend not only on the directionality of the intermediate momentum but also on those of the incoming and outgoing momenta, $\mathbf{p}_a$ and $\mathbf{p}_b$. Further, one can represent the N two-body propagators as a diagonal matrix $\mathcal{G}={\rm diag}(\mathcal{G}_{1},\mathcal{G}_{2},\cdots, \mathcal{G}_{N})$ as depicted in Fig. \ref{fig:twoparFV}(a). These are infinite-dimensional matrices with with matrix elements \cite{Kim:2005gf}
\begin{eqnarray}
(\delta G^{V}_i)_{l,m;l',m'}\equiv
(G^{V}_i-G^{\infty}_i)_{l,m;l',m'}=
-i\left(\mathcal K \delta \mathcal G^{V}_i \mathcal K\right)_{l,m;l',m'},
\end{eqnarray}
where
\begin{eqnarray}
\label{def0}
(\delta \mathcal{G}^{V}_{i})_{l_1,m_1;l_2,m_2}&=&i \frac{k^*_in_i}{8\pi E^*}\left(\delta_{l_1,l_2}\delta_{m_1,m_2}+i\frac{4\pi}{k_i^*}\sum_{l,m}\frac{\sqrt{4\pi}}{k_i^{*l}}c^{\textbf{P}}_{lm}(k_i^{*2})\int d\Omega Y^*_{l_1m_1}Y^*_{lm}Y_{l_2m_2} \right),\nonumber\\
\end{eqnarray}
and the function $c^{\textbf{P}}_{lm}$ is defined as\footnote{Note that our definition of the $c_{lm}^{\textbf{P}}$ function differs that of Ref. \cite{Kim:2005gf} by an overall sign.}
\begin{eqnarray}
\label{clm}
c^{\textbf{P}}_{lm}(x)=\frac{1}{\gamma}\left[\frac{1}{ L^3}\sum_{\textbf{q}}-\mathcal{P}\int\frac{d^3\mathbf{q}}{(2\pi)^3}\right]\frac{\sqrt{4\pi}Y_{lm}(\hat{q}^*)~q^{*l}}{{q}^{*2}-x} \ .
\end{eqnarray}
$\mathcal{P}$ in this relation denotes the principal value of the integral, and $\mathbf{q}^*={\gamma}^{-1}(\mathbf q_{||}-\alpha \mathbf P)+\mathbf q_{\perp}$, where $\mathbf q_{||}$ ($\mathbf q_{\perp}$) denotes the component of the momentum vector $\mathbf{q}$ that is parallel (perpendicular) to the boost vector $\mathbf{P}$ and $\alpha=\frac{1}{2}\left[1+\frac{m_1^2-m_2^2}{E^{*2}}\right]$~\cite{Davoudi:2011md, Fu:2011xz, Leskovec:2012gb}.\footnote{The kinematic function $c^{\textbf{P}}_{lm}(k_i^{*2})$ can also be written in terms of the three-dimensional Zeta function, $\mathcal{Z}^d_{lm}$,
\begin{eqnarray}
\nonumber
c^{\textbf{P}}_{lm}(k^{*2})=\frac{\sqrt{4\pi}}{\gamma L^3}\left(\frac{2\pi}{L}\right)^{l-2}\mathcal{Z}^\mathbf{d}_{lm}[1;(k^*L/2\pi)^2],\hspace{1cm}
\mathcal{Z}^\mathbf{d}_{lm}[s;x^2]=\sum_{\mathbf r \in P_d}\frac{Y_{l,m}(\mathbf{r})}{(r^2-x^2)^s},
\end{eqnarray}
where the sum is performed over $P_d=\left\{\mathbf{r}\in \mathbb{R}^3\hspace{.1cm} | \hspace{.1cm}\mathbf{r}={\gamma}^{-1}(\mathbf m_{||}-\alpha \mathbf d)+\mathbf m_{\perp} \text{,}~ \mathbf{m} \in \mathbb{Z}^3\right\}$, $\mathbf d$ is the normalized boost vector $\mathbf d=\mathbf{P}L/2\pi$, and $\alpha$ is defined above.} This reduces to the NR value of $\alpha=\frac{m_1}{m_1+m_2}$ as is presented in Ref.~\cite{Bour:2011ef}. Note that this result is equivalent to the result obtained in Refs. \cite{Rummukainen:1995vs, Kim:2005gf, Christ:2005gi} for the boosted systems of particles with identical masses and with one channel. The only non-trivial piece in this relation is the momentum vectors $\mathbf{q}^*$ to be summed over in the energy quantization condition (QC). This is determined mainly from the on-shell kinematics of the two-particle states which depends on the boost vector, the masses of particles and the boundary conditions. Since we aim to present a general proof for the form of these momentum vectors with arbitrary twisted boundary conditions, we will delay the derivation until chapter \ref{chap:TBC}. The result in Eq. (\ref{clm}) can be recovered from that of presented in chapter \ref{chap:TBC} upon setting all the twist angles to zero.
The kernel, which is now not only a matrix in the channel space but also in the angular momentum space, is assured to reproduce the infinite-volume scattering amplitude matrix ($\mathcal{M}$) by solving the following matrix equation
\begin{eqnarray}
i\mathcal{M}&=&-i\mathcal{K}
-i\mathcal{K}\mathcal{G^{\infty}} \mathcal{K}
-i\mathcal{K}\mathcal{G^{\infty}} \mathcal{K}\mathcal{G^{\infty}}\mathcal{K}+\cdots=-i\mathcal{K}\frac{1}{1-\mathcal{G^{\infty}}\mathcal{K}}.
\label{MK-relation}
\end{eqnarray}
giving rise to
\begin{eqnarray}
\mathcal{K}=-\mathcal{M}\frac{1}{1-\mathcal{G^{\infty}}\mathcal{M}}.
\label{eq:kernel}
\end{eqnarray}
With this definition of the kernel, one can proceed to evaluate poles of the N-channels FV scattering amplitude matrix by replacing the infinite-volume loops $\mathcal{G}^\infty$ with their FV $\mathcal{G}^V$ counterparts,
\begin{eqnarray}
-i\mathcal{M}^{V}&=&-i\mathcal{K}-i\mathcal{K}\mathcal{G}^{V} \mathcal{K}-i\mathcal{G}^{V}\mathcal{K}\mathcal{G}^{V}\mathcal{K}+\cdots=-i\mathcal{K}\frac{1}{1-\mathcal{G}^{V}\mathcal{K}}
\nonumber\\
& = & -i\frac{1}{1-\mathcal{M}\mathcal{G}^{\infty}}\mathcal{M}\frac{1}{1+\delta \mathcal{G}^{V}\mathcal{M}}({1-\mathcal{M}\mathcal{G}^{\infty}}).
\end{eqnarray}
Finally arriving at the QC
\begin{eqnarray}
\label{det0}
\mathcal{R}e\left\{\det(\mathcal{M}^{-1}+\delta \mathcal{G}^{V})\right\}=\mathcal{R}e\left\{{\rm{det}}_{\rm{oc}}\left[\rm{det}_{\rm{pw}}\left[\mathcal{M}^{-1}+\delta \mathcal{G}^{V}\right]\right]\right\}=0,
\end{eqnarray}
where the determinant $\rm{det}_{\rm{oc}}$ is over the N open channels and the determinant $\rm{det}_{\rm{pw}}$ is over the partial waves, and both $\mathcal{M}$ and $\delta \mathcal{G}^V$ functions are evaluated on the on-shell value of the momenta. This latter property, along with decomposing to partial waves, has enabled us to decouple the chain of loops in the expansion of the scattering amplitude, see Fig. \ref{fig:twoparFV}(a), and obtain an algebraic expansion (a geometric series) of the scattering amplitude matrix. We have taken the real part of the determinant in Eq. (\ref{det0}), but as it will be shown shortly, this determinant condition gives rise to only one single real condition for both single channel and two coupled-channel cases with $l_{max}=0$, so we omit the notion of the real part in the QC from now on. For a general proof of the reality of QC with any number of coupled channels see Refs. \cite{Hansen:2012tf, Hansen:2012bj}.
\subsubsection{Single-channel scattering} For N=1 the QC in Eq. (\ref{det0}) reproduces the L\"uscher formula \cite{Luscher:1986pf, Luscher:1990ux} when generalized to moving frames \cite{Rummukainen:1995vs, Kim:2005gf, Christ:2005gi}. In order to deduce this, let us write the relativistic single-channel scattering amplitude $\mathcal{M}_i$ as
\begin{eqnarray}
\label{def1}
(\mathcal{M}_{i})_{l_1,m_1;l_2,m_2}&=&\delta_{l_1,l_2}\delta_{m_1,m_2}\frac{8\pi E^*}{n_ik^*_i}\frac{e^{2i\delta^{(l)}_i(k^*_i)}-1}{2i},
\end{eqnarray}
where $\delta_i^{(l)}$ is the scattering phase shift in channel $i$ and in partial wave channel $l$. The Kronecker deltas that are introduced to insure the conservation of angular momentum between initial and final states scattering should not be confused with the phase shifts.
Since the QC in Eq. (\ref{det0}), even for the case of single-channel scattering, is infinite dimensional, one should first perform a truncation in the angular momentum basis. Let us assume that the contributions from higher partial waves to the scatterings are negligible (which is the case at low energies), so that one can truncate the determinant over the angular momentum at $l_{max}=0$. Then the QC for the S-wave scattering reads,
\begin{eqnarray}
k_i^*\cot {\delta_i^{(0)}}=4\pi c_{00}^{\mathbf{P}}(k^{*2}_i),
\label{Luscherl0}
\end{eqnarray}
It is convenient to introduce a pseudo-phase defined by ${k^*_i}\cot {\phi^\mathbf{P}_i}\equiv -4\pi{ c_{00}^\mathbf{P}}$,
to rewrite the QC as
\begin{eqnarray}
\label{quateq}
\cot {\delta_i}=-\cot {\phi^\mathbf{P}_i} \Rightarrow \delta_i+\phi^\mathbf{P}_i=n\pi,
\end{eqnarray}
where $n$ is an integer. In this form, the QC is manifestly real.
Here we discuss briefly how one implements the L\"uscher formula in LQCD studies of two-hadron systems.\footnote{We will continue discussing the NPLQCD study of $\pi\pi$ scattering in the $I=2$ channel \cite{Beane:2011sc}. For more examples of successful implementations of the L\"uscher formula in studies of two-hadron sector of QCD, including studies of resonances, the reader may consult the following references, \cite{Li:2007ey, Aoki:2007rd, Beane:2010hg, Beane:2011xf, Beane:2011sc, Beane:2011iw, Beane:2012ey,Yamazaki:2012hi, Lang:2012sv, Beane:2013br, Pelissier:2011ib, Lang:2011mn, Pelissier:2012pi, Ozaki:2012ce, Buchoff:2012ja, Dudek:2012xn, Dudek:2012gj, Lang:2014tia}.} In Sec. \ref{subsec:ChiPT} we saw that by inputting the S-wave scattering length and effective range of the $I=2$ $\pi\pi$ scattering into the chiral expansion of these parameters, the LECs of the $\chi$PT at NLO can be determined, and predictions at the physical values of light-quark masses are made possible, see Fig. \ref{fig:NPLQCD-pipi}. Furthermore, in Sec. \ref{LQCD}, we saw, through the same example of the $\pi\pi$ system in the $I=2$ channel, how the energy levels of the system can be extracted from the large-time dependence of the lattice correlation functions, see Fig. \ref{fig:EMP-pipi}. Now given the L\"uscher formula in Eq. (\ref{Luscherl0}), it is known how these energy eigenvalues turn into the corresponding phase shifts. Fig. \ref{fig:kcot-c00} demonstrates the relation between the $k^* \cot \delta_0$ function from Eq. (\ref{a-r-def}) and the FV function $c_{00}^{\mathbf{P}}$ evaluated numerically as a function of $\tilde{k}^{*2} \equiv \frac{k^{*2}L^2}{(2\pi)^2}$ for chosen values of $m_{\pi}=390~{\rm MeV}$ and $L \approx 3.9~{\rm fm}$. The point where the two functions intersect corresponds to an energy eigenvalue, which as seen differs for systems with different boosts.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.475]{kcotczerozero}
\caption{\small{The FV function $4\pi c_{00}^{\mathbf{P}}$ (in blue) and the $k^* \cot \delta_0$ (in red) (both normalized by the pion mass) as a function of $\tilde{k}^{*2} \equiv \frac{k^{*2}L^2}{(2\pi)^2}$ for a chosen values of $m_{\pi}=390~{\rm MeV}$ and $L \approx 3.9~{\rm fm}$. The $k^* \cot \delta_0$ function is plotted using an ERE parametrization with the obtained values of $a_{0}^{(2)}$ and $r_{0}^{(2)}$ at this pion mass by the NPLQCD collaboration \cite{Beane:2011sc}.}}
\label{fig:kcot-c00}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.515]{pipicombined}
\caption{\small{The upper panel depicts the NPLQCD extraction \cite{Beane:2011sc} of the lowest energy levels of the two-pion state in the $I=2$ channel, labeled by index $n$, for various total CM momenta $|\mathbf{P}|\equiv P_{cm}$. The energy is given in terms of the dimensionless quantity $a_tE,$ where $a_t \approx 0.035~{\rm fm}$ is the temporal extent of the (anisotropic) lattice used in this calculation. The dashed lines denote the location of the non-interacting energy levels. The lower panel contains the obtained S-wave phase shifts using the L\"uscher QC for several energy levels in the upper panel. The fit to the ERE with two parameters, $a_0^{(2)}$ and $r_0^{(2)}$, is shown where the shaded bands correspond to statistical (inner-yellow) and statistical and systematic added in quadrature (outer-pink). Figure is reproduced with the permission of the NPLQCD collaboration.}}
\label{fig:pipi}
\end{center}
\end{figure}
This clearly demonstrates that boosting the two-hadron system does not only give rise to a trivial shift in the total energy of the system, but also changes the location of the CM energies in a finite volume. This can be understood by noting that by boosting the two-body system in a finite cubic volume, the spatial extents of the volume as is seen in the rest frame of the system is changed, giving rise to a different FV symmetry groups than the unboosted case, see chapter \ref{chap:NN}. This is the reason why performing calculations with different CM is advantageous as it provides further energy inputs to the L\"uscher QC at a single volume, and puts better constraints on the extracted scattering parameters. When such procedure is carried out for the $\pi \pi$ scattering, the extracted energy levels as is shown in the upper panel of Fig. \ref{fig:pipi}, lead to several phase shift points that can then be fit using a two-parameter ERE in Eq. (\ref{a-r-def}) to obtain the values of $a_{0}^{(2)}$ and $r_{0}^{(2)}$. Note that although the L\"uscher formula can be used up to the inelastic threshold which is $k^{*2}=3m_{\pi}^2$ for this system, the range of validity of the extracted scattering length and the effective range is determined by the momentum at which the ERE breaks down, $k^{*2}=m_{\pi}^2$.
Although we have truncated the QC to $l_{max}=0$, as is seen from the definition of the FV matrix $\delta \mathcal{G}^{V}$ in Eq. (\ref{def0}), the reduced symmetry of the FV introduces off-diagonal terms in the FV matrix which couple different partial waves in the QC. For example, if the two equal-mass meson interpolating operator is in the $A_1^+$ irreducible representation (irrep) of the cubic group, the energy eigenstates of the system have overlap with the $l=0,4,6,\ldots$ angular momentum states at zero total momentum (see table \ref{table:irreps} for decomposition of the irreps of the cubic group in terms of the irreps of the rotational group), making the truncation at $l_{max}=0$ a rather reasonable approximation in the low-energy limit. When $\textbf{P}\neq 0$ but $\gamma \approx 1$, as will be discussed in details in chapter \ref{chap:NN}, the symmetry group is reduced, and at low energies the $l=0$ will mix with the $l=2$ partial wave as well as with higher partial waves.\footnote{A comprehensive study of the symmetry groups of the calculations with different boost vectors in the NR limit will be presented in chapter \ref{chap:NN}.} Recently, by analyzing the LQCD calculations of two-pion systems with several CM boosts, the $l=0,2$ phase shifts of $I=2$ $\pi\pi$ scattering have been simultaneously extracted by Dudek, \emph{et al.} at $m_{\pi}=396~{\rm MeV}$ \cite{Dudek:2012gj}.
For two mesons with different masses, the symmetry group is even further reduced in the boosted frame, making the mixing to occur between $l=0$ and $l=1$ states as well as with higher angular momentum states \cite{Fu:2011xz}. An easy way to see the latter is to note that in contrast with the case of degenerate masses, the kinematic function $c_{lm}^{\textbf{P}}$ as defined in Eq. (\ref{clm}) is non-vanishing for odd $l$ when the masses are different. As a result even and odd angular momenta can mix in the QC. This however does not indicate that the spectrum of the system is not invariant under parity. As long as all interactions between the particles are parity conserving, the spectrum of the system and its parity transformed counterpart are the same. The determinant condition, Eq. (\ref{det0}), guarantees this invariance: any mechanism, for example, which takes an S-wave scattering state to an intermediate P-wave two-body state, would take it back to the final S-wave scattering state, and the system ends up in the same parity state.\footnote{We note that under parity $\mathcal{Z}^d_{lm}\rightarrow\left(-1\right)^{l}\mathcal{Z}^d_{lm}$. Also it can be seen that under the interchange of particles $\mathcal{Z}^d_{lm}\rightarrow\left(-1\right)^{l}\mathcal{Z}^d_{lm}$, so that for degenerate masses the $c_{lm}^{P}$ functions vanish for odd $l$. This is expected since the parity transformation in the CM frame is equivalent to the interchange of particles. However, as is explained above for the case of parity transformation, despite the fact that $\delta \mathcal{G}^{V}$ is not symmetric with respect to the particle masses, the QC is invariant under the interchange of the particles.}
\subsubsection{Two coupled channels} For the N=2 case, the expression for the scattering amplitude in Eq. (\ref{def1}) is modified, as it now depends on the mixing angle $\bar{\epsilon}$, and the scattering matrix is no longer diagonal in the channel basis. By labeling the off-diagonal terms as $\mathcal{M}_{I,II}$, and by choosing the ``barred" parameterization for the time-reversal invariant S-matrix \cite{Stapp:1956mz}
\begin{eqnarray}
\label{smatrix2}
S_2=\begin{pmatrix}
e^{i2\delta_I}\cos{2\overline{\epsilon}}&ie^{i(\delta_I+\delta_{II})}\sin{2\overline{\epsilon}}\\
ie^{i(\delta_I+\delta_{II})}\sin{2\overline{\epsilon}}&e^{i2\delta_{II}}\cos{2\overline{\epsilon}} \\
\end{pmatrix},
\end{eqnarray}
where the subscript $2$ on $S$ denotes the number of coupled channels, the scattering matrix elements can be written as
\begin{eqnarray}
\label{def}
(\mathcal{M}_{i,i})_{l_1,m_1;l_2,m_2}&=&\delta_{l_1,l_2}\delta_{m_1,m_2}\frac{8\pi E^*}{n_ik^*_i}\frac{\cos(2\bar{\epsilon})e^{2i\delta^{(l_1)}_i(k^*_i)}-1}{2i},\\
(\mathcal{M}_{I,II})_{l_1,m_1;l_2,m_2}&=&\delta_{l_1,l_2}\delta_{m_1,m_2}\frac{8\pi E^*}{\sqrt{n_In_{II}k^*_Ik^*_{II}}}\sin(2\bar{\epsilon})\frac{e^{i(\delta^{(l_1)}_I(k^*_I)+\delta^{(l_1)}_{II}(k^*_{II}))}}{2},
\end{eqnarray}
where the usual relativistic normalization of the states is used in evaluating the S-matrix elements.\footnote{Here we assume no physical partial-wave mixing occurs in either channels. This is of course not the case in the two-nucleon systems in the isosinglet channel. We revisit this formalism in chapter \ref{chap:NN} to incorporate such mixings.} From Eq. (\ref{det0}) one obtains \cite{Hansen:2012tf, Briceno:2012yi}
\begin{eqnarray}
\label{allorders}
\det\begin{pmatrix}
1+\delta \mathcal{G}^{V}_I\mathcal{M}_{I,I}&\delta \mathcal{G}^{V}_I\mathcal{M}_{I,II}\\
\delta \mathcal{G}^{V}_{II}\mathcal{M} _{I,II}&1+\delta \mathcal{G}^{V}_{II}\mathcal{M}_{II,II}\\
\end{pmatrix}=0,
\end{eqnarray}
where the determinant is not only over the number of channels but also over angular momentum which is left implicit.
For $l_{max}=0$ one can use the definition of the pseudo-phase to rewrite the QC in a manifestly real form,
\begin{eqnarray}
\label{allorders2}
\cos{2\bar{\epsilon}}\cos{\left(\phi^\mathbf{P}_1+\delta_1-\phi^\mathbf{P}_2-\delta_2\right)}=\cos{\left(\phi^\mathbf{P}_1+\delta_1+\phi^\mathbf{P}_2+\delta_2\right)},
\end{eqnarray}
which is equivalent to the result given in Refs. \cite{He:2005ey, Liu:2005kr} in the CM frame.
It is easy to see that in the $\bar \epsilon\rightarrow 0$ limit, one recovers the decoupled QCs for both channels $I$ and $II$, Eq. (\ref{quateq}).
In order to understand the significance of such FV coupled-channels formalism, it is sufficient to note that the spectrum of QCD contains a wealth of resonances that sit above multi-particle thresholds. Since resonances are not isolated eigenstates of QCD Hamiltonian and can only be observed as resonances in multi-particle scattering amplitudes, an evaluation of the energy levels will not give direct insight into the mass and decay width of these resonances. Instead a L\"uscher-type methodology, by which the scattering phase shifts of the scattering states are evaluated at the calculated energy levels on the lattice, is required. Therefore, analyzing the FV spectra, in particular those of the excited spectra of QCD as produced by various LQCD collaborations (e.g., Refs.~\cite{Dudek:2009qf, Dudek:2010wm, Dudek:2011tt, Edwards:2011jj, Dudek:2011bn, Dudek:2013yja}), requires applying a multi-coupled channel formalism. This necessary step can provide some of the theoretical guidance for the forthcoming JLab GlueX experiment~\cite{Shepherd:2009zz, Zihlmann:2010zz, Somov:2011zz, Smith:2012ch} as well as other spectroscopy experiments worldwide.
Before concluding this section, let us emphasize that the formalism presented here and in chapters \ref{chap:NN}, \ref{chap:deuteron} and \ref{chap:TBC} is valid up to exponential corrections of the form $\mathcal{O}(e^{-m_{\pi}L})$. For the case of the nuclear force, these corrections are due to the finite (non-zero) range of interactions that is set by the pion mass, i.e. the lightest particle that is produced and mediated in the hadronic system due to strong interactions. In order for these corrections to be at sub-percent level, the spatial extent of the volume must be chosen such that $m_{\pi}L \gtrsim 2\pi$. As the pion masses used in studies of multi-hadron systems approach their physical value, larger volumes must be used to insure these corrections will remain small.
\chapter{LATTICE OPERATORS AND RESTORATION OF ROTATIONAL SYMMETRY IN THE CONTINUUM LIMIT}
\label{chap:operators}
Efforts to reduce lattice artifacts and achieve a better behaved
theory in the continuum limit date back to early stages of development
of LQCD. Many that are part of the Symanzik improvement program
include a systematic modification of the action in such a
way to eliminate $\mathcal{O}\left(a^{n}\right)$ terms from physical
quantities calculated with LQCD at each order in perturbation
theory~\cite{Symanzik:1983dc,Symanzik:1983gh,Parisi:1985iv,WeiszI,WeiszII,Luscher,Curci,Hamber,Eguchi,Wetzel,Sheikholeslami}, or nonperturbatively.
However, as will be discussed, discretization effects are known to
give rise to more subtle issues; the treatment of which turns out to be more
involved.
\begin{table}[b!]
\begin{centering}
\includegraphics[scale=0.875]{irreps}
\par\end{centering}
\caption{{\small The decomposition of the irreps of the SO(3) group up to $J=4$ in terms of the irreps of the $\mathrm{O^D_h}$ \cite{Luscher:1986pf, Luscher:1990ux, Mandula:1983wb, Johnson:1982yq, Berg, Basak:2005aq, Dresselhaus}.}}
\label{table:irreps}
\end{table}
LQCD is commonly formulated on a hyper-cubic grid, as a result the
full (Euclidean) Lorentz symmetry group of the continuum is
reduced to the discrete symmetry group of a hypercube. As
the (hyper) cubic group has only a finite number of irreducible representations
(irreps) compared to infinite number of irreps of the rotational group, a given irrep of the rotational group is not irreducible
under the (hyper) cubic group.
Consequently, one can not assign
a well-defined angular momentum
to a lattice state, which is generally a linear combination
of infinitely many different angular momentum states, see for example Table. \ref{table:irreps} for the decomposition of the irreps of the $SO(3)$ group up to $J=4$ in terms of the irreps of the $\mathrm{O^D_h}$ (the double cover of cubic $\mathrm{O_h}$ group).
In principle, one can identify
the angular momentum of a corresponding continuum state in a lattice
calculation
from the degeneracies in the spectrum of
states belonging to different irreps of the cubic group as the lattice
spacing is reduced.\footnote{For some recent hadron spectroscopy works see Refs.
\cite{BurchI,Gattringer,Petry,BurchII,Dudek:2009qf, Dudek:2010wm, Dudek:2011tt, Edwards:2011jj, Dudek:2011bn, Meinel, Dudek:2013yja} and for a review of baryon spectroscopy efforts see Ref. \cite{Lin}.} For example, a $J^P=2^+$ state can be identified if two (nearly) degenerate energy levels found in the spectra that are obtained using an $\mathbb{E}^+$ interpolator and a $\mathbb{T}_2^+$ interpolator for the state as $a \to 0$, see Fig. \ref{fig:splitting}.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.425]{levelsplitting}
\par\end{centering}
\caption{{\small The schematic level splitting due to the breakdown of the $SO(3)$ symmetry down to $O_h(3)$ symmetry due to a non-zero lattice spacing. The 5-fold degeneracy of the $J=2^+$ irrep of the $SO(3)$ group is split to a two-fold degeneracy ($\mathbb{E}^+$ irrep) and a three-fold degeneracy ($\mathbb{T}_2^+$ irrep). The basis functions of the corresponding irreps are given next to each level.}}
\label{fig:splitting}
\end{figure}
According to Table. \ref{table:irreps}, the $\mathbb{T}_2^+$ irrep, for example, has overlap with not only the $2^+$ irrep but also with $3^+,~4^+,~\dots$ irreps of the rotational group. In reality, there are error bars associated with these levels due systematic and statistical uncertainties. Therefore, as the density of degenerate states substantially increases with increasing
the angular momentum,
the identification of states with
higher angular momentum becomes impossible with the current statistical precision.
The other issue is that the cubic symmetry of the lattice
allows the renormalization mixing of interpolating operators with
lower dimensional ones.
The induced coefficients of the lower-dimensional operators
scale as inverse powers of the
lattice spacing, and hence diverge
as the lattice spacing goes to zero.
Although renormalization mixing of operators
is familiar from the continuum quantum field theory, it happens more
frequently in LQCD calculations as the reduced symmetry of the hyper-cube
is now less restrictive in preventing operators from mixing.
To obtain useful results for, as an example, the matrix elements of operators
from LQCD calculations,
non-perturbative
subtraction of the power divergences is required and generally introduces large statistical
uncertainties.
To overcome these obstacles, it has been proposed by Dudek,
{\it et al.}~\cite{DudekI,DudekII,Edwards} that by means of a novel construction of
interpolating operators, the excited states of several mesons and
baryons can be identified to high precision.
The essence of this
method is that if one uses a set of cubically invariant local
operators which have already been subduced \cite{Basak} from a rotationally
invariant local operator with a definite angular momentum, $J$, while at the same
time smearing the gauge and quark fields over
the hadronic scale~\cite{Allton,Morningstar,Peardon}, the constructed
operator has maximum overlap onto a continuum state with angular momentum
$J$ if the lattice spacing is sufficiently small. As an example, one can measure the correlation among operators belonging to a large set of constructed lattice operators that share the same transformation properties under a given irrep of the cubic group but are subduced from operators with different angular momentum. One such investigation is proposed in Ref. \cite{Dudek:2010wm} and is plotted in Fig. \ref{fig:subduction}. Interestingly, the correlation among operators subduced from different angular momentum representation of the continuum is minimal compared with the self correlations, confirming the effectiveness of their method in identification of the continuum states' angular momentum.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=1.05]{subduction}
\par\end{centering}
\caption{{\small The representative plot of the degree of the correlation among 26 different lattice operators, $\mathcal{O}^{[J]}_{\mathbb{T}_1^-}$, all transforming under $\mathbb{T}_1^-$ irrep of the cubic group, but are subduced from operators with three different angular momentum, $J=2,3,4$ as described in Ref. \cite{Dudek:2010wm}. The figure is reproduced with the permission of Jozef Dudek.}}
\label{fig:subduction}
\end{figure}
The subduction is assumed to be responsible for retaining ``memory'' of the
underlying angular momentum of the continuum operator, while
the smearing is assumed to suppress mixing with operators of different angular momentum
-- by filtering contributions from ultraviolet (UV) modes.
In another approach, states with higher angular momentum
in the glueball spectra
of $2+1$ dimensional $SU\left(2\right)$ gauge theories~\cite{Meyer,JohnsonRW}
are isolated by using glueball interpolating operators that are
linear
combinations of Wilson loops which are rotated by arbitrary angles
in order to project out a particular angular momentum $J$ in the continuum.
In addition,
the links are smeared, or blocked, in order to be smooth over physical
length scales rather than just in the UV~\cite{Teper}.
So by monitoring the angular content of the glueball wavefunction
in the continuum limit with a probe with definite $J$,
the $0^{-}/4^{-}$ puzzle in the glueball
spectroscopy has been tackled.
The prominent feature of these works
is that the recovery of rotational symmetry for sufficiently small
lattice spacings is qualitatively emergent from their numerical results.
The same issue occurs in LQCD calculations of higher moments
of hadron structure functions, the extraction
of which requires the matrix elements of local operators
between hadronic states.
Although
Lorentz invariance forbids
twist-2 operators with different $J$
from mixing in the continuum, generally
they can mix in LQCD calculations with power-divergent mixing
coefficients~\cite{Capitani,Beccarini}.
The power-divergent mixing problem associated with the lower
moments can be avoided by several means as described, for example, in Refs. ~\cite{Beccarini,GockelerI,GockelerII,GockelerIII,GockelerIV,GockelerVI,GockelerV,MartinelliI,MartinelliII,MartinelliIII,GockelerVI}.
In addition to these approaches,
two methods~\cite{Dawson,Detmold} have been suggested
that highlight the idea of approaching the continuum properties
of the hadronic matrix elements by suppressing the contributions from the UV,
and in that sense resemble the idea of operator smearing in the proposals described above.
In LQCD calculations of non-leptonic K-decay,
Dawson {\it et al.}~\cite{Dawson}
suggested that point-splitting the hadronic currents by a distance
larger than the lattice spacing, but smaller than the
QCD scale, results in an operator product expansion of
the currents with the coefficients of lower dimensional operators
scaling with inverse powers of the point-splitting distance,
as opposed to the inverse lattice spacing.
This considerably reduces the numerical issues
introduced by the operator mixing.
In a different, but still physically equivalent approach, Detmold and Lin~\cite{Detmold}
showed that in LQCD calculations of matrix elements of the Compton
scattering tensor,
the introduction of a fictitious, non-dynamical, heavy
quark coupled to physical light quarks
removes the power divergences of the mixing
coefficients.
This technique enables the extraction of matrix elements
of higher spin twist-2 operators.
The essence of this method is that the
heavy quark propagator acts as a smearing function in the momentum space,
suppressing contributions from the high-energy modes, provided
that its mass is much smaller than the inverse lattice spacing.
Encouraged by the results of these
numerical non-perturbative investigations,
we aim to quantify the recovery of rotational symmetry
with analytical, perturbative calculations in
$\lambda\phi^4$ theory and in QCD.
In order to
achieve this goal, we first define a composite operator on the lattice
which has a well-defined angular momentum in the continuum limit and
is smeared over a finite physical region, and show how the non-continuum
contributions to the multipole expansion of the operator scales as
the lattice spacing is reduced toward the continuum.
Tree-level contributions to matrix elements that violate
rotational symmetry,
either by the lattice operator
matching onto continuum operators with the ``wrong'' angular
momentum,
or matching onto continuum operators that explicitly
violate rotational symmetry,
scale as $\mathcal{O}\left(a^{2}\right)$
as $a\rightarrow 0$.
This includes the (naively) power-divergent contributions from
lower-dimension operators.
In order to make definitive statements about the size of
violations to rotational symmetry, it must be ensured that the
tree-level scalings are not ruined by quantum
fluctuations.
This is demonstrated by a perturbative calculation of the two-point function in
$\lambda\phi^{4}$ scalar field theory with an insertion of such an operator.
It is confirmed that quantum
corrections at any order in perturbation theory do not alter the observed
classical scalings of non-continuum contributions.
After gaining experience with this operator in scalar field theory,
the generalization to gauge theories is straightforward.
Special attention must be paid to the gauge links that appear in the
definition of gauge-invariant operator(s) that are the analogue of those
considered in the scalar field theory.
Also, it is well known that the perturbative expansion of operators used in
LQCD are not well-behaved due to the presence of tadpole
diagrams~\cite{Lepage}.
Naively, tadpoles make enhanced contributions to the matrix elements of the
operators we consider, and that tadpole improvement of the gauge links and
smearing of the gluon fields
are
crucial to the suppression of violations of rotational symmetry.
After discussing the continuum behavior of the QCD operator(s), and their potential mixings, which violate rotational invariance at ${\cal O}(a^2)$,
we determine the renormalization of the operator(s) on the lattice
at one-loop order.
The leading rotational invariance violating contributions to the renormalized
lattice
operator are suppressed by ${\cal O}(\alpha_{s} a^2)$,
provided that the gauge
fields
are also smeared over a physical region similar to the matter fields.
This means that the leading rotational invariance violating operators
introduced by the quantum loops make subleading contributions
compared to tree-level, ${\cal O} (a^2)$.
The loop contributions that scale as ${\cal O}(\alpha_{s} a)$
do not violate rotational symmetry, and hence
are absorbed into the operator $Z$-factor.
\section{Operators in Scalar Field Theory
\label{sec:Classical}}
\noindent
The goal is to construct a bilinear operator of the scalar fields
on a cubic lattice which has certain properties. First of all, as
it was discussed earlier, it has to be smeared over a finite region
of space.
This physical region should be large compared to the lattice
spacing,
and, for our purposes, small compared to the typical length scale of
the system to allow for a perturbative analysis.
The spatial extent of the operator can be identified with its renormalization scale.
Secondly,
it is required to transform as a spherical tensor with well-defined angular
momentum in the continuum limit.
An operator that satisfies these conditions is~\footnote{
This corresponds to one particular choice of radial structure of the operator.
However, the results of the calculations and the physics conclusions presented
in this work do not change qualititively when other smooth radial structures are
employed, such as a Gaussian or exponential.
}
\begin{equation}
\hat{\theta}_{L,M}\left(\mathbf{x};a,N\right)
\ =\
\frac{3}{4\pi
N^{3}}
\sum_{\mathbf{n}}^{\left|\mathbf{n}\right|\leq N}\phi\left(\mathbf{x}\right)
\phi\left(\mathbf{x}+\mathbf{n}a\right)
\ Y_{L,M}\left(\hat{\mathbf{n}}\right)
,
\label{eq:1}
\end{equation}
where $\mathbf{n}$ denotes a triplet of integers, and
it is normalized
by the spatial volume of the region over which it is distributed.
$\phi({\bf x})$ is the scaler field operator,
$N$ is the maximum number of lattice sites in the radial direction, and
$Y_{LM}\left(\hat{\mathbf{n}}\right)$ is a spherical harmonic evaluated at the
angles defined by the unit vector in the direction of $\mathbf{n}$,
$\hat{\mathbf{n}}$, as shown in Fig.~\ref{fig:operator}.
This operator can also be written in a multipole expansion about its center as
\begin{equation}
\hat{\theta}_{L,M}\left(\mathbf{x};a,N\right)=\frac{3}{4\pi
N^{3}}
\sum_{\mathbf{n}}^{\left|\mathbf{n}\right|\leq N}\sum_{k}\frac{1}{k!}\
\phi\left(\mathbf{x}\right)\left(a\mathbf{n}\cdot\mathbf{\nabla}\right)^{k}
\phi\left(\mathbf{x}\right)
\ Y_{L,M}\left(\hat{\mathbf{n}}\right)
,
\label{eq:2}
\end{equation}
where the gradient operator acts on the ${\bf x}$ variable,
$\mathbf{\nabla}\equiv \mathbf{\nabla}_{\bf x}$.
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.40]{operator}
\par\end{centering}
\caption{{\small
A contribution to the lattice operator defined in
Eq.~(\ref{eq:1}), with $\left|\mathbf{n}\right|\leq N$.
All the points inside the three-dimensional spherical shell
$\left|\mathbf{n}a\right|=Na$ are included in the operator.
The two length scales defining the operator,
the lattice spacing, $a$,
and the operator size, $Na=1/\Lambda$,
are shown.}}
\label{fig:operator}
\end{figure}
Although the operator $\hat{\theta}_{L,M}\left(\mathbf{x};a,N\right)$
is labeled by its angular momentum in the continuum limit, from the right hand side of
Eq.~(\ref{eq:2}), it is clear that it is a linear combination
of an infinite number of operators with angular momentum compatible with its parity.
To be more specific, consider the $M=0$
component of the operator expanded in a derivative operator basis,
\begin{equation}
\hat{\theta}_{L,0}\left(\mathbf{x};a,N\right)=
\sum_{L^{\prime},d}\frac{C_{L0;L^{\prime}0}^{\left(d\right)}
\left(N\right)}{\Lambda^{d}}\mathcal{O}_{z^{L^{\prime}}}^{\left(d\right)}
\left(\mathbf{x};a\right)
,
\label{eq:3}
\end{equation}
where $\mathcal{O}_{z^{L^{\prime}}}^{\left(d\right)}\left(\mathbf{x};a\right)$
are defined in Appendix~\ref{app:operators}.
The operator subscript denotes that there are $L^{\prime}$
free indices in the derivative operator, while $d$ denotes the total number of derivatives.
As is discussed in the
Appendix~\ref{app:operators},
there are operators in this basis which are not rotationally
invariant but only cubically invariant.
$C_{L0;L^{\prime}0}^{\left(d\right)}\left(N\right)$
are coefficients of each operator in the expansion whose values are
determined by matching Eq. (\ref{eq:2}) with Eq.~(\ref{eq:3}).
Finally $\Lambda=1/(Na)$ is the momentum scale of the smeared
operator which is kept fixed as the lattice spacing is varied.
Therefore, as the lattice spacing decreases,
more point shells (shells of integer triplets)
are included in the sum in Eq.~(\ref{eq:2}).
The convergence of this derivative expansion is guaranteed
as the scale $\Lambda$ is set to be much larger than the typical
momentum encountered by the operator.
\subsection{Classical scalar field theory}
\label{sec:Clasphi}
In order for the operator to recover its continuum limit as the lattice
spacing vanishes,
the coefficients $C_{L0;L^{\prime}0}^{\left(d\right)}$
should have certain properties.
First of all, those associated with
the operators with $L\neq L^{\prime}$ as well as the rotational invariance
violating operators, should vanish as $a\rightarrow0$.
Also the
coefficients of rotational invariant operators with $L=L^{\prime}$
should reach a finite value in this limit.
These properties will be
shown to be the case in a formal way shortly, but in order to get
a general idea of the classical scaling of the operators and the size
of mixing coefficients, we first work out a particular example.
Consider
the operator $ $$\hat{\theta}_{3,0}\left(\mathbf{x};a,N\right)$
expanded out up to five derivative operators,
\begin{eqnarray}
\hat{\theta}_{3,0}\left(\mathbf{x};a,N\right)
&=&
\frac{C_{30;10}^{\left(1\right)}\left(N\right)}{\Lambda}\mathcal{O}_{z}^{\left(1\right)}
\left(\mathbf{x};a\right)
+
\frac{C_{30;10}^{\left(3\right)}\left(N\right)}{\Lambda^{3}}\mathcal{O}_{z}^{\left(3\right)}
\left(\mathbf{x};a\right)
+
\frac{C_{30;10}^{\left(5\right)}\left(N\right)}{\Lambda^{5}}\mathcal{O}_{z}^{\left(5\right)}
\left(\mathbf{x};a\right)
+
\nonumber\\
&&\frac{C_{30;10}^{\left(5;RV\right)}\left(N\right)}{\Lambda^{5}}\mathcal{O}_{z}^{\left(5;RV\right)}
\left(\mathbf{x};a\right)
+
\frac{C_{30;30}^{\left(3\right)}\left(N\right)}{\Lambda^{3}}\mathcal{O}_{zzz}^{\left(3\right)}
\left(\mathbf{x};a\right)
+
\nonumber\\
&&\frac{C_{30;30}^{\left(5\right)}\left(N\right)}{\Lambda^{5}}\mathcal{O}_{zzz}^{\left(5\right)}
\left(\mathbf{x};a\right)
+
\frac{C_{30;50}^{\left(5\right)}\left(N\right)}{\Lambda^{5}}\mathcal{O}_{zzzzz}^{\left(5\right)}
\left(\mathbf{x};a\right)
+
\mathcal{O}\left(\frac{\nabla_{z}^{7}}{\Lambda^{7}}\right),
\label{eq:4}
\end{eqnarray}
where the superscript RV denotes the rotational invariance violating operator and its corresponding coefficient in the above expansion.
\begin{figure}[!ht]
\begin{centering}
\includegraphics[scale=0.8]{coefficientsplots}
\par\end{centering}
\caption{{\small
The tree-level values of the
coefficients $C^{(d)}_{30;L^\prime 0}$ appearing in
Eq.~(\ref{eq:4})
as a function of the largest $n$-shell included in the summation in Eq.~(\ref{eq:1}).
}}
\label{fig:TheCs}
\end{figure}
\begin{figure}[!ht]
\begin{centering}
\includegraphics[scale=0.4]{ccomboplot}
\par\end{centering}
\caption{{\small
A comparison between the tree-level coefficients $C^{(d)}_{30;L^\prime 0}$ to illustrate
the relative rates of convergence to the continuum limit.
}}
\label{fig:TheComp}
\end{figure}
The numerical values of the coefficients in Eq.~(\ref{eq:4}), at the classical level,
as a
function of the maximum shell included in the sum in Eq.~(\ref{eq:2})
are shown in Fig.~\ref{fig:TheCs} and Fig.~\ref{fig:TheComp}.
From these plots it is clear that while the coefficients
$C_{30;30}^{\left(3\right)}$ and $C_{30;30}^{\left(5\right)}$ reach
a finite value for large N, the coefficients of lower and higher angular
momentum operators, as well as the rotational invariance violating
operator, approach zero.
To find the values of the
leading order (LO)
coefficients in this limit, as well as to see how the non-leading
contributions scale with $N=1/(\Lambda a)$, one can apply the Poisson
re-summation formula to the right hand side of Eq.~(\ref{eq:2}),
\begin{equation}
\hat{\theta}_{L,M}\left(\mathbf{x};a,N\right)
=\
\frac{3}{4\pi N^{3}}\sum_{k}\frac{a^{k}}{k!}
\sum_{\mathbf{p}}\int d^{3}y\
\theta\left(N-y\right)\
e^{i2\pi\mathbf{p}\cdot\mathbf{y}}\
\phi\left(\mathbf{x}\right)\left(\mathbf{y}\cdot\mathbf{\nabla}\right)^{k}
\phi\left(\mathbf{x}\right)\
Y_{L,M}\left(\hat{\mathbf{y}}\right)
,
\label{eq:5}
\end{equation}
where $\mathbf{p}$ is another triplet of integers, and the
$\mathbf{p}$ summation is unbounded.
The continuum values of the coefficients
obtained in the $N\rightarrow\infty$ limit,
corresponding to the $\mathbf{p}=0$ term in Eq.~(\ref{eq:5}),
are
\begin{eqnarray}
C_{30;30}^{\left(d\right)}
& = &
{15\over 4}\
\sqrt{7\over\pi}\
{ d^2-1 \over (d+4)!}
\qquad {\rm with}\qquad d=3,5,...
,
\label{eq:6}
\end{eqnarray}
while the other coefficients
in Eq.~(\ref{eq:4})
vanish in this limit as expected.
The LO corrections to these continuum values can be calculated
as following.
The deviation of $C_{30;30}^{\left(3\right)}$ from
its continuum value can be found from
\begin{eqnarray}
I_{30}
& \sim\ &
\frac{3}{4\pi}\frac{\left(Na\right)^{3}}{3!}
\sum_{\mathbf{p}\neq 0}
\int_{0}^{1}dy\ y^{2}\ d\Omega_{\hat{\mathbf{y}}}\
e^{i2\pi N\mathbf{p}\cdot\mathbf{y}}
\ \phi\left(\mathbf{x}\right)\
\
\left(\hat{\mathbf{y}}\cdot\mathbf{\nabla}\right)^{3}
\phi\left(\mathbf{x}\right)\
\ Y_{3,0}\left(\hat{\mathbf{y}}\right)
,
\label{eq:7}
\end{eqnarray}
where $\mathbf{\nabla}=\nabla_{z}\hat{e}_{z}$ and the $y$-variable in Eq.~(\ref{eq:7})
is redefined to lie between $0$ and $1$,
and it is straightforward to show that
\begin{equation}
\delta C_{30;30}^{\left(3\right)}
\ =\
\frac{1}{N^{2}}
\ \frac{1}{32\pi^{2}}
\ \sqrt{\frac{7}{\pi}}\
\sum_{\mathbf{p}\neq0}
\frac{\cos\left(2\pi
N\left|\mathbf{p}\right|\right)}{\left|\mathbf{p}\right|^{8}}
\left(-\frac{3}{2}\left|\mathbf{p}\right|^{6}+15\left|\mathbf{p}\right|^{2}p_z^4-\frac{25}{2}p_{z}^{6}\right)
.
\label{eq:8}
\end{equation}
It is interesting to note that, after trading $N$ for $1/(a \Lambda)$,
the finite lattice spacing corrections are not monotonic in $a$,
but exhibit oscillatory behavior, which is clearly evident in Fig.~\ref{fig:TheCs}.
The deviation of $C_{30;10}^{\left(1\right)}$ from its continuum value of zero
follows similarly, and is found to scale as $\sim 1/N^2$,
\begin{equation}
\delta C_{30;10}^{\left(1\right)}
\ =\ \frac{1}{N^{2}}
\ \frac{3}{16\pi^{2}}\sqrt{\frac{7}{\pi}}
\ \sum_{\mathbf{p}\neq0}
\ \frac{\cos\left(2\pi
N\left|\mathbf{p}\right|\right)}{\left|\mathbf{p}\right|^{6}}
\ \left(\left|\mathbf{p}\right|^{4}-5p_{z}^{4}\right)
.
\label{eq:9}
\end{equation}
As in the case of the operator that conserves angular momentum in the continuum
limit,
the sub-leading correction (and in this case the first non-zero contribution)
to the coefficient is suppressed by $1/N^{2}$.
This can be shown to be the case for all the sub-leading
contributions to the coefficients $C_{LM;L^{\prime}M^{\prime}}^{\left(d\right)}$ as follows.
As is evident from Eq.~(\ref{eq:5}), the integrals
that are required
in calculating deviations from the continuum
values have the general form
\begin{equation}
I^{i_1...i_k}
\ \sim\
\frac{3}{4\pi}\frac{\left(Na\right)^{k}}{k!}
\ \sum_{\mathbf{p}\neq0}
\ \int_{0}^{1}dy\ y^{2+k}\ \int d\Omega_{\hat{\mathbf{y}}}
\ e^{i2\pi N\mathbf{p}\cdot \mathbf{y}}\ \hat{\mathbf{y}}^{i_{1}}\
\hat{\mathbf{y}}^{i_{2}}...\hat{\mathbf{y}}^{i_{k}}
\ Y_{LM}\left(\Omega_{\hat{\mathbf{y}}}\right)
,
\label{eq:10}
\end{equation}
which can be written as
\begin{eqnarray}
I^{i_1...i_k}
& \sim &
\frac{3}{4\pi}
\ \frac{\left(Na\right)^{k}}{k!}\frac{1}{\left(i2\pi N\right)^{k}}
\ \sum_{\mathbf{p}\neq0}\frac{\partial}{\partial
p_{i_{1}}}...\frac{\partial}{\partial p_{i_{k}}}
\ \int_{0}^{1}dy\ y^{2+k}\ \int d\Omega_{\hat{\mathbf{y}}}
\ e^{i2\pi N\mathbf{p}\cdot\mathbf{y}}
\ Y_{LM}\left(\Omega_{\hat{\mathbf{y}}}\right)
\nonumber\\
& \sim &
\frac{3}{4\pi}\frac{\left(Na\right)^{k}}{k!}\
\frac{4\pi i^{L}}{\left(i2\pi
N\right)^{k}}\sum_{\mathbf{p}\neq0}
\ \frac{\partial}{\partial
p_{i_{1}}}...\frac{\partial}{\partial
p_{i_{k}}}
\ Y_{LM}\left(\Omega_{\hat{\mathbf{p}}}\right)
\ \int_{0}^{1}dy\ y^{2+k}\ j_{L}\left(2\pi N\left|\mathbf{p}\right|y\right)
.
\nonumber\\
\label{eq:11}
\end{eqnarray}
The y integration over the Bessel function gives rise to
either
$-\frac{\cos\left(2\pi N\left|\mathbf{p}\right|\right)}{\left(2\pi N\left|\mathbf{p}\right|\right)^{2}}$
or
$-\frac{\sin\left(2\pi N\left|\mathbf{p}\right|\right)}{\left(2\pi N\left|\mathbf{p}\right|\right)^{2}}$,
up to higher orders in $1/N$,
depending on whether $L$ is even or odd.
Thus the LO contribution from
Eq.~(\ref{eq:11}) in the large $N$ limit
is obtained by acting
on the numerator
with the $p$ derivatives, producing $k$ powers
of $N$, multiplying the $1/N^{2}$ from the denominator.
Therefore, Eq.~(\ref{eq:11}) scales as
\begin{equation}
I^{i_1...i_k}\ \sim\
\left(Na\right)^{k}\frac{1}{N^{k}}\frac{N^{k}}{N^{2}}
\ \sim\ {1\over\Lambda^k}\ {1\over N^2}
,
\label{eq:12}
\end{equation}
and,
in general,
the deviation of any coefficient from its continuum value
is suppressed by $1/N^{2}=\Lambda^{2}a^{2}$.
This result implies that in calculating the matrix element of $L=3$ operator,
one has a derivative expansion of the form
\begin{eqnarray}
\Lambda^{3}\hat{\theta}_{3,0}\left(\mathbf{x};a,N\right)
& = &
\alpha_{1}\
\frac{\Lambda^{2}}{N^{2}}\mathcal{O}_{z}^{\left(1\right)}\left(\mathbf{x};a\right)
\ +\
\alpha_{2}\
\frac{1}{N^{2}}\mathcal{O}_{z}^{\left(3\right)}\left(\mathbf{x};a\right)
\ +\
\alpha_{3}\
\frac{1}{\Lambda^{2}N^{2}}\mathcal{O}_{z}^{\left(5\right)}\left(\mathbf{x};a\right)
\nonumber\\
& + &
\alpha_{4}\
\frac{1}{\Lambda^{2}N^{2}}\mathcal{O}_{z}^{\left(5;RV\right)}\left(\mathbf{x};a\right)
\ +\
\alpha_{5}\
\mathcal{O}_{zzz}^{\left(3\right)}\left(\mathbf{x};a\right)
\ +\
\alpha_{6}\
\frac{1}{\Lambda^{2}}\mathcal{O}_{zzz}^{\left(5\right)}\left(\mathbf{x};a\right)
\nonumber\\
& + &
\alpha_{7}\
\frac{1}{\Lambda^{2}N^{2}}\mathcal{O}_{zzzzz}^{\left(5\right)}\left(\mathbf{x};a\right)
\ +\
\mathcal{O}\left({\nabla_{z}^{7}\over \Lambda^{4}}\right)
,
\label{eq:13}
\end{eqnarray}
where the mixing with $L\neq3$ operators
(with coefficients $\alpha_{1,2,3,7,...}$),
as well as the
operator with broken rotational symmetry
(with coefficient $\alpha_4$),
vanish in the large $N$ limit,
while
the coefficients of $L=3$ operators
(with coefficients $\alpha_{5,6,...}$),
are fixed by the
scale of the operator, $\Lambda$.
It is clear that for $N=1$ and $\Lambda=1/a$, where no smearing
is performed, the problem with divergent coefficients of the lower
dimensional operators is obvious, as, for example, the coefficient of
$\mathcal{O}_{z}^{\left(1\right)}\left(\mathbf{x};a\right)$ diverges
as $1/a^{2}$ as $a\rightarrow0$, as is well known.
The fact that all the sub-leading contributions to the classical operator
are suppressed at least by $1/N^{2}$ regardless of $L$ and $L^{\prime}$
can be understood as follows.
In the classical limit, where the short
distance fluctuations of the operator are negligible, the operator
does not probe the distances of the order of lattice spacing when
$a\rightarrow 0$.
The angular resolution of the operator is
dictated by the solid angle discretization of the physical region
over which the operator is smeared, and therefore is proportional to
$1/N^{2}$.
The question to answer is whether the
quantum fluctuations modify this general result.
Before proceeding with the quantum loop calculations,
it is advantageous to transform the operator into
momentum-space to simplify loop integrals.
This can be done
easily by noting that for zero momentum insertion,
the operator acting on the field with momentum
$\mathbf{k}$ is
\begin{equation}
\hat{\tilde{\theta}}_{LM}\left(\mathbf{k};a,N\right)
\ =\
\frac{3}{4\pi N^{3}}\
\sum_{\mathbf{n}}^{|\mathbf{n}|\leq N}
\ e^{i\mathbf{k}\cdot\mathbf{n}a}
\ Y_{LM}\left(\mathbf{n}\right)
\ \tilde{\phi}\left(\mathbf{k}\right)
\ \tilde{\phi}\left(-\mathbf{k}\right)
,
\label{eq:14}
\end{equation}
which, after using the partial-wave expansion of
$e^{i\mathbf{k}\cdot\mathbf{n}a}$
and the exponential term resulting from the Poisson relation,
can be written as
\begin{eqnarray}
\hat{\tilde{\theta}}_{LM}\left(\mathbf{k};a,N\right)
& = &
6\sqrt{\pi}\
\tilde{\phi}\left(\mathbf{k}\right)
\tilde{\phi}\left(-\mathbf{k}\right)
\ \sum_{\mathbf{p}}\sum_{L_{1},M_{1},L_{2},M_{2}}i^{L_{1}+L_{2}}
\ \sqrt{\frac{\left(2L_{1}+1\right)\left(2L_{2}+1\right)}{2L+1}}
\nonumber\\
&&
\times
\ \left\langle L_{1}0;L_{2}0\left|L0\right.\right\rangle
\ \left\langle L_{1}M_{1};L_{2}M_{2}\left|LM\right.\right\rangle
Y_{L_{1}M_{1}}\left(\Omega_{\hat{\mathbf{k}}}\right)
\ Y_{L_{2}M_{2}}\left(\Omega_{\hat{\mathbf{p}}}\right)
\nonumber\\
&&
\times
\ \int_{0}^{1}dy\ y^{2}
\ j_{L_{1}}\left(aN\left|\mathbf{k}\right|y\right)
\ j_{L_{2}}\left(2\pi N\left|\mathbf{p}\right|y\right)
.
\label{eq:15}
\end{eqnarray}
Although this form seems to be somewhat more complicated than in
position space,
it turns out that it is advantageous
to work in momentum space when dealing with higher angular
momenta, as well as for $M\ne 0$.
Further, the dimensionless parameters
$|{\bf k}|/\Lambda$ and $N$ that define the physics of such systems
are now explicit.
It is straightforward to show this form recovers
the values of the
leading and sub-leading coefficients
given in Eqs.~(\ref{eq:8}) and (\ref{eq:9}),
and it is worth mentioning how they emerge from Eq.~(\ref{eq:15}).
For a non-zero value of $|{\bf p}|$ and $N=\infty$, the spherical
Bessel function $j_{L_{2}}\left(2\pi N\left|\mathbf{p}\right|y\right)$ vanishes
for any value of $L_2$. However, for large values of $N$ but $|{\bf p}|=0$ the
only non-zero contribution is from $L_2=0$, and thus $L_1=L$, leaving a
straightforward integration over a single spherical Bessel function
$j_{L}\left(aN\left|\mathbf{k}\right|y\right)$ to obtain the continuum limit
given in Eq.~(\ref{eq:6}).
Extracting the subleading contributions and the violations of rotational
symmetry
is somewhat more involved, and we provide an explicit
example in Appendix~\ref{app:RIviolation}.
\subsection{Quantum corrections in $\lambda\phi^{4}$
\label{sec:Scalar}}
\noindent
In order to determine the impact of quantum fluctuations on the matrix elements
of $\hat{\theta}_{L,M}$,
defined in Eq.~(\ref{eq:1}), we consider loop contributions in
$\lambda\phi^{4}$ theory.
Beside its simplicity which enables us to develop tools in performing
the analogous calculations in Lattice QCD,
this theory corresponds to some interesting condensed matter systems.
For example, three dimensional O(N) models, which describe important
critical phenomena in nature, have a corresponding $\lambda\phi^{4}$
field theory formulation.
As pointed out in Refs.~\cite{CampostriniI,CampostriniII},
anisotropy in space either due to the symmetries of the physical system,
or due to an underlying lattice formulation,
will result in the presence of irrelevant operators in the effective
Hamiltonian which are not rotationally invariant,
and
introduce deviations of two-point functions
from their rotationally invariant scaling law near the fixed point.
However,
as the rotationally invariant fixed point of the theory is approached,
the anisotropic deviations vanish like $1/\xi^{\rho}$ where $\xi^{2}$
is the second moment correlation length derived from the two-point function,
and $\rho$ is a critical
exponent which is related to the critical effective dimension of the
leading irrelevant operator breaking rotational invariance.
It has
been shown that in the large N approximation
of $O\left(N\right)$ models, $\rho\simeq2$ for cubic-like lattices.
In the following, it
will be shown that,
by inserting
$\hat{\theta}_{L,M}$ defined in
Eq.~(\ref{eq:1}) into the two-point function,
the same scaling law emerges when approaching the rotational-invariant
continuum limit of $\lambda\phi^{4}$ theory.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.15]{phifouroneloop}
\par\end{centering}
\caption{{\small One-loop correction to the two-point function with an insertion
of $\hat{\theta}_{L,M}$ in $\lambda\phi^{4}$}}
\label{fig:onelooplf4}
\end{figure}
At tree level, the contributions to the two-point function
from an insertion of
$\hat{\theta}_{L,M}$
at zero momentum transfer
has been already discussed in section \ref{sec:Clasphi}.
At one-loop order, there is only one diagram with an insertion of $\hat{\theta}_{L,M}$
that contributes to the two-point function, as shown in
Fig.~\ref{fig:onelooplf4}.
This diagram introduces corrections only to the $L=0$ matrix element as there are
no free indices associated with the loop.
The lattice integral associated with this one-loop diagram is
\begin{equation}
J_{LM}
\ =\
\frac{3\lambda}{4\pi
N^{3}}\sum_{\mathbf{n}}^{|\mathbf{n}|\leq N}
\ \int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}k}{\left(2\pi\right)^{4}}
\ \frac{e^{i\mathbf{k}\cdot\mathbf{n}a}}{\left(\hat{\mathbf{k}}^{2}+m^{2}\right)^{2}}
\ Y_{LM}\left(\Omega_{\mathbf{n}}\right)
,
\label{eq:17}
\end{equation}
where
$\hat{\mathbf{k}}^{2}= {4\over a^2}\sum\limits_{\mu} \sin^{2}\left({ k_{\mu}a\over
2}\right)$,
$\lambda$ is the coupling constant and $m$
is the $\phi$ mass.
The three-momentum integration can be
evaluated by noting that the region of integration
can be split into two parts: region I where $0\leq\left|\mathbf{k}\right|\leq\pi/a$
and therefore is rotationally symmetric, and region II where $\pi/a\leq\left|\mathbf{k}\right|\leq\sqrt{3}\pi/a$
which consists of disconnected angular parts. Also as the three-momentum
integration is UV convergent, a small $a$ expansion of the integrand
can be performed.
Using Eq.~(\ref{eq:15}), the contribution from
region I to the $\mathbf{p}=0$ term in the Poisson sum is
\begin{eqnarray}
J_{LM}^{\left(I\right)}\left(\mathbf{p}=0\right)
& = &
\frac{3\lambda}{\left(2\pi\right)^{4}}i^{L}
\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}dk_{4}\int_{0}^{\frac{\pi}{a}}dkk^{2}
\int d\Omega_{\hat{\mathbf{k}}}\frac{1}{\left(\hat{\mathbf{k}}^{2}+m^{2}\right)^{2}}
\nonumber\\
&&\qquad\qquad \times
\left[\
\int_{0}^{1}dy\ y^{2}\ j_{L}\left(aN\left|\mathbf{k}\right|y\right)\ \right]\
Y_{LM}\left(\Omega_{\mathbf{k}}\right)
\nonumber\\
& = &
\frac{3\lambda}{16\pi^4}i^{L}
\left[J_{LM}^{LO}+J_{LM}^{NLO}+\mathcal{O}\left(1/N^{4}\right)\right]
,
\label{eq:18}
\end{eqnarray}
where
\begin{eqnarray}
J_{LM}^{LO}
& = &
2\sqrt{\pi}
\delta_{L,0}\delta_{M,0}
\int_{-\frac{\pi}{\Lambda a}}^{\frac{\pi}{\Lambda a}}\
dq_{4}\int_{0}^{\frac{\pi}{\Lambda
a}}dqq^{2}\frac{1}{\left[q^{2}+q_{4}^{2}+m^{2}/\Lambda^{2}
\right]^{2}}
\ \int_{0}^{1}dy\ y^{2}\ j_0\left(qy\right)
,
\nonumber
\end{eqnarray}
\begin{eqnarray}
J_{LM}^{NLO}
& = &
\frac{1}{N^{2}}\int_{-\frac{\pi}{\Lambda a}}^{\frac{\pi}{\Lambda a}}
dq_{4}\int_{0}^{\frac{\pi}{\Lambda
a}}dqq^{2}\frac{q^{4}}{\left[q^{2}+q_{4}^{2}+m^{2}/\Lambda^{2}\right]^{3}}
\nonumber\\
&&
\qquad
\times
\left[
\frac{6\sqrt{\pi}}{5}\ \delta_{L,0}\ \delta_{M,0}\
\int_{0}^{1}dyy^{2}j_{0}\left(qy\right)
\right.\nonumber\\
&& \left.
\qquad
+\ \delta_{L,4}\left(\frac{2}{3}\sqrt{\frac{2\pi}{35}}\delta_{M,-4}
+\frac{4\sqrt{\pi}}{15}\delta_{M,0}
+\frac{2}{3}\sqrt{\frac{2\pi}{35}}\delta_{M,4}\right)\ \int_{0}^{1}dy\ y^{2}\
j_{4}\left(qy\right)\right]
\ ,
\nonumber\\
\label{eq:19}
\end{eqnarray}
with
$q=\left|\mathbf{k}\right|/\Lambda$ and $q_{4}=k_{4}/\Lambda$.
The LO integral, $J_{LM}^{LO}$, is convergent, while the NLO contribution,
$J_{LM}^{NLO}$, while not convergent, is not divergent, but is of the
form $\sin\left(N\pi\right)/N^2$.
This implies that they depend on the ratio of the two mass scales, $\Lambda$ and $m$,
but without inverse powers of $a$.
So as $a\rightarrow 0$,
the LO $L=0$ operator makes an unsuppressed contribution to the $L=0$ matrix
element, while the contributions to this matrix element
from the NLO rotational-symmetry violating $L=0$ and $L=4$ operators
are suppressed by $1/N^{2}$.
A simple argument shows that contributions from integration region II, for
which $\pi/a\leq\left|\mathbf{k}\right|\leq\sqrt{3}\pi/a$,
are also
suppressed by $1/N^{2}$.
After defining a new momentum variable
$l_{\mu}=k_{\mu}a$ and $l^{2}=l_{1}^{2}+l_{2}^{2}+l_{3}^{2}$, the
$\mathbf{p}=0$ term of the Poisson sum in region II is
\begin{eqnarray}
J_{LM}^{\left(II\right)}\left(\mathbf{p}=0\right)
& = &
\frac{3\lambda}{16\pi^4}i^{L}
\int_{-\pi}^{\pi}dl_{4}\int_{\pi}^{\sqrt{3}\pi}dl\ l^{2}\
\int_{f\left(\Omega_{\mathbf{l}}\right)}d\Omega_{\mathbf{l}}
\nonumber\\
&&
\qquad\qquad
\frac{Y_{LM}\left(\Omega_{\mathbf{l}}\right)}{\left(4\sum_{\mu}\sin^{2}\left(l_{\mu}/2\right)
+a^{2}m^{2}\right)^{2}}
\int_{0}^{1}dy\ y^{2}\ j_{L}\left(Nly\right)
,
\label{eq:20}
\end{eqnarray}
where $f\left(\Omega_{\mathbf{l}}\right)$ identifies the angular
region of integration, and whose parametric form does not matter for
this discussion. This region still exhibits cubic symmetry, and gives
rise to contribution to the $L=0,4,6,8,...$ operators. On the other
hand, the three-momentum integration is entirely located in the UV
as $a\rightarrow0$, and thus
\begin{equation}
\sin^{2}\left(l_{1}/2\right)+\sin^{2}\left(l_{2}/2\right)+\sin^{2}\left(l_{3}/2\right)
+\sin^{2}\left(l_{4}/2\right)\geq 1
.
\label{eq:21}
\end{equation}
Also, integration over the Bessel function brings in a factor of
$\ -\cos\left(Nl\right)/(N^{2}l^{2})$, up to higher orders in $1/N$.
So the integrand does not have any
singularities in region II of the
integration, and is bounded.
As a result,
\begin{equation}
\left|J_{LM}^{\left(II\right)}\left(\mathbf{p}=0\right)\right|
\ \leq\
\frac{1}{N^{2}}
\frac{3\lambda}{(4\pi)^4}
\ \int_{-\pi}^{\pi}dl_{4}
\ \int_{\pi}^{\sqrt{3}\pi}dl
\ \int_{f\left(\Omega_{\mathbf{l}}\right)}d\Omega_{\mathbf{l}}
\ Y_{LM}\left(\Omega_{\mathbf{l}}\right)
,
\label{eq:22}
\end{equation}
and consequently
$J_{LM}^{\left(II\right)}\left(\mathbf{p}=0\right)$
itself is suppressed by $1/N^{2}$.
This completes the discussion of the $\mathbf{p}=0$
term in the Poisson sum,
corresponding to a zero-momentum insertion of the continuum operator into the
loop diagram.
It then remains to determine the scaling of the $\mathbf{p}\neq 0$ terms in the summation
in the large $N$ limit.
The integral arising from the $\mathbf{p}\neq 0$ terms
is, up to numerical factors,
\begin{eqnarray}
{\cal I}_{\mathbf{p}\neq 0}
& \sim &
\lambda\sum_{\mathbf{p}\neq 0}\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}k}{\left(2\pi\right)^{4}}
\frac{1}{\left(\hat{\mathbf{k}}^{2}+m^{2}\right)^{2}}\
Y_{L_{1}M_{1}}\left(\Omega_{\hat{\mathbf{k}}}\right)\
Y_{L_{2}M_{2}}\left(\Omega_{\hat{\mathbf{p}}}\right)
\nonumber\\
&& \qquad\qquad \qquad \qquad
\times\
\int_{0}^{1}dy\ y^{2}\
j_{L_{1}}\left(Na\left|\mathbf{k}\right|y\right)\
j_{L_{2}}\left(2\pi N\left|\mathbf{p}\right|y\right)
.
\label{eq:23}
\end{eqnarray}
This integral is finite in UV, and integrand can be expanded in powers of $a$,
giving a leading contribution of
\begin{eqnarray}
{\cal I}_{\mathbf{p}\neq 0}
& \sim &
\lambda\sum_{\mathbf{p}\neq 0}\int_{-\frac{\pi}{\Lambda a}}^{\frac{\pi}{\Lambda
a}}\frac{d^{3}q\ dq_{4}}{\left(2\pi\right)^{4}}\frac{1}{\left(q^{2}+q_{4}^{2}+m^{2}/\Lambda^{2}\right)^{2}}
\ Y_{L_{1}M_{1}}\left(\Omega_{\hat{\mathbf{q}}}\right)
\ Y_{L_{2}M_{2}}\left(\Omega_{\hat{\mathbf{p}}}\right)
\nonumber\\
&& \qquad \qquad \qquad \qquad \qquad \qquad
\times\
\int_{0}^{1}dy\ y^{2}\
j_{L_{1}}\left(qy\right)
\ j_{L_{2}}\left(2\pi N\left|\mathbf{p}\right|y\right)
.
\label{eq:24}
\end{eqnarray}
A non-zero angular integration requires that $L_1=0$,
and the integral
is suppressed at least by a factor of $1/N^{2}$ as
integration over the Bessel functions introduces a factor
of $1/\left(2\pi N\left|\mathbf{p}\right|\right)^{2}$
up to a numerical coefficient and a bounded trigonometric function
at leading order in $1/N$.
The next order term in the small $a$ expansion of the integrand
can be easily shown to bring in an additional factor of $1/N^{2}$.
So one can see that the $\mathbf{p}\neq 0$ terms in the Poisson summation,
which give rise to non-continuum contributions to the two-point function
at one loop, are always suppressed by at least a factor of $1/N^{2}$.
The result of the one-loop calculation is promising: all the sub-leading
contributions that break rotational symmetry
are suppressed by $1/N^{2}$
compared to the leading $L=0$ continuum operator contribution to
the two-point function.
A little investigation shows that this scaling
also holds to higher orders in $\lambda\phi^{4}$ theory. Suppose
that the operator is inserted into a propagator inside an n-loop diagram
contributing to the two-point function.
Considering the
continuum part of the operator first, the leading term in the small $a$
expansion of the integrand gives rise to $2n$ propagators,
while the integration measure contributes $4n$ powers of momentum.
Although this appears to be logarithmically divergent,
the spherical Bessel function contributes a factor of inverse three-momentum
and either a sine or cosine of the three-momentum, rendering the diagram finite.
The same argument applies to the NLO term in the small $a$ expansion of
the integrand, resulting in a $1/N^{2}$ suppression of the
breaking of rotational invariance.
Insertion of the non-continuum operator in loop diagrams are also suppressed by
$1/N^2$ for similar reasons.
The interpretation of finite-size scaling results presented in
Refs.~\cite{CampostriniI,CampostriniII}
in terms of what has been observed in this section is now straightforward.
Near the critical point, the correlation length is the only relevant
physical scale in the problem, and tends to infinity.
So as the critical point is approached,
one does not probe the underlying lattice structure
as the correlation length becomes much larger than the lattice spacing,
and extends over an
increasing number of point shells.
In comparison,
inserting an operator which only probes distances of the order of
a physical scale that is much larger than the lattice spacing,
resembles the physics near a rotational-invariant fixed point, and the same scaling
law for the non-rotational invariant operators is expected (in the
same theory) as the lattice spacing goes to zero.
\section{Operators in QCD
\label{sec:QCD}}
\noindent
The necessity of introducing a gauge link to connect the fermionic
fields in a gauge-invariant way, makes the discussion of the operator
and its renormalization more involved in gauge theories.
The reason
is two-folded: firstly as is well known, perturbative
LQCD is ill-behaved as a result of non-vanishing tadpoles which diverge
in the UV, making the small coupling series expansion of the operators
slowly convergent. The other difficulty is that as the operator is
smeared over many lattice sites, the links are necessarily extended links.
Thus, to analytically investigate the deviations
from a rotational invariant path, working with a well-defined
path on the grid is crucial. In this section, the strategies to deal
with these problems are discussed, and the scaling laws of different
operator contributions to the two-point function in
QCD with an insertion of the smeared operator are deduced.
In position space, perhaps the simplest gauge-invariant smeared operator of
quark bilinears is
\begin{equation}
\hat{\theta}_{L,M}\left(\mathbf{x};a,N\right)
\ =\
\frac{3}{4\pi N^{3}}\sum_{\mathbf{n}}^{\left|\mathbf{n}\right|\leq
N}\overline{\psi}
\left(\mathbf{x}\right)U\left(\mathbf{x},\mathbf{x}+\mathbf{n}a\right)\psi\left(\mathbf{x}+\mathbf{n}a\right)
\ Y_{L,M}\left(\hat{\mathbf{n}}\right)
,
\label{eq:25}
\end{equation}
with
\begin{equation}
U\left(\mathbf{x},\mathbf{x}+\mathbf{n}a\right)
\ =\
e^{ig\int_{\mathbf{x}}^{\mathbf{x}+\mathbf{n}a}\mathbf{A}\left(z\right)\cdot d\mathbf{z}}
\ =\
1+ig\int_{\mathbf{x}}^{\mathbf{x}+\mathbf{n}a}\mathbf{A}\left(z\right)\cdot d\mathbf{z}
+\mathcal{O}\left(g^{2}\right)
,
\label{eq:26}
\end{equation}
where the actual path defining $U$ will be considered subsequently.
As the fermion operator is a spin singlet, $S=0$,
the total angular momentum of this operator in the continuum is $J=L$.
One could also consider operators of the form
\begin{equation}
\hat{\theta}_{JL,M}^{\mu}\left(\mathbf{x};a,N\right)
\ =\
\frac{3}{4\pi N^{3}}
\sum_{\mathbf{n}}^{\left|\mathbf{n}\right|\leq N}
\overline{\psi}
\left(\mathbf{x}\right)\ \gamma^\mu \
U\left(\mathbf{x},\mathbf{x}+\mathbf{n}a\right)
\psi\left(\mathbf{x}+\mathbf{n}a\right)
\ Y_{L,M}\left(\hat{\mathbf{n}}\right)
,
\label{eq:25b}
\end{equation}
which can be used to form operators with $J=L+1, L, L-1$.
It is clear that
the set of operators with angular momentum $J$ will mix under renormalization,
but the vector nature of QCD precludes mixing between the
$\overline{\psi}\psi$ and $\overline{\psi}\gamma^\mu \psi$ operators in the
chiral limit.
However to capture the main features
of operator mixing in the continuum limit of LQCD, it suffices to
work with the simplest operator, in Eq.~(\ref{eq:25}). At tree-level, the contributions of this operator away from the continuum limit
scale in the same way as in the scalar theory,
with contributions that violate rotational invariance
suppressed by $\sim 1/N^2$.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.115]{qcdoneloop}
\par\end{centering}
\caption{{\small
One-loop QCD corrections to the fermionic two-point function
with an insertion of
$\hat{\theta}_{L,M}$, given in Eq.~(\ref{eq:25}),
at zero external momentum}}
\label{fig:qcdOneLoop}
\end{figure}
\subsection{Continuum operator and its renormalization}
Let us first discuss the one-loop renormalization of the operator
in the continuum. There are four one-loop diagrams contributing to
the operator renormalization as shown in Fig.~\ref{fig:qcdOneLoop}.
The diagram in
Fig.~\ref{fig:qcdOneLoop}(a)
results from inserting the leading order term in the small
coupling expansion of the operator in the loop. At zero external momentum
this diagram is
\begin{eqnarray}
\Gamma^{(a)} & \sim &
-T^{a}T^{a}\frac{3ig^{2}}{4\pi}\int_{0}^{1}dyy^{2}
\int d\Omega_{\mathbf{y}}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}
\frac{\gamma_\alpha
\left(ik_{\mu}\gamma^{\mu}+m\right)^{2}
\gamma^\alpha
}{\left(k^{2}+m^{2}\right)^{2}k^{2}}
e^{iNa\mathbf{k}\cdot\mathbf{y}}Y_{LM}\left(\Omega_{\mathbf{y}}\right)
,
\nonumber\\
\label{eq:27}
\end{eqnarray}
which is clearly convergent in the UV.
Also it contains $L=0$
as well as $L=1$ operator as can be seen from the angular
part of the integral
\begin{eqnarray}
&&
\sum_{L^{\prime},M^{\prime}}\int d\Omega_{\mathbf{y}}d\Omega_{\mathbf{k}}
\left[f_{1}\left(k^{2},m,k_{4}\right)+f_{2}\left(k^{2},m,k_{4}\right)\mathbf{k}\cdot\vec{\mathbf{\gamma}}\right]
\ Y_{L^{\prime}M^{\prime}}\left(\Omega_{\mathbf{k}}\right)
\ Y^{*}_{L^{\prime}M^{\prime}}\left(\Omega_{\mathbf{y}}\right)
\ Y_{LM}\left(\Omega_{\mathbf{y}}\right)
\nonumber\\
&&
\ =\
\sqrt{4\pi}f_{1}\left(k^{2},k_{4},m\right)\delta_{L,0}\delta_{M,0}+
\nonumber\\
&& \qquad
\sqrt{\frac{4\pi}{3}}f_{2}\left(k^{2},k_{4},m\right)\left|\mathbf{k}\right|
\delta_{L,1}\left[\gamma_{1}\left(\frac{\delta_{M,-1}-\delta_{M,1}}{\sqrt{2}}\right)
+i\gamma_{2}\left(\frac{\delta_{M,-1}+\delta_{M,1}}{\sqrt{2}}\right)+\gamma_{3}\delta_{M,0}
\right]
,
\nonumber\\
&&
\label{eq:28}
\end{eqnarray}
where $f_{1}$ and $f_{2}$ are some functions of their arguments.
One can check however that as $m/\Lambda\rightarrow0$ (the chiral
limit), the contribution to the $L=1$ operator is suppressed by the
quark mass.
The diagrams in Fig.~\ref{fig:qcdOneLoop}(b) comes from the next term in the expansion
of Eq. (\ref{eq:26}). It is straightforward to show that the Feynman
rule for the one-gluon vertex with zero momentum insertion into the
operator is
\begin{eqnarray}
V_g^\lambda =
\frac{3}{4\pi N^{3}}\sum_{\mathbf{n}}^{\left|\mathbf{n}\right|\leq N}
g a n^\lambda
\frac{1}{\left(\mathbf{p}-\mathbf{p^{\prime}}\right)\cdot\mathbf{n}a}
\left(e^{i\left(\mathbf{k}+\mathbf{p^{\prime}}\right)\cdot \mathbf{n}a}
-e^{i\mathbf{p^{\prime}\cdot}\mathbf{n}a}\right)\delta^{4}\left(p-p^{\prime}-k\right)
Y_{L,M}\left(\hat{\mathbf{n}}\right),
\label{eq:29}
\end{eqnarray}
where the radial path between points $\mathbf{x}$ and $\mathbf{x}+\mathbf{n}a$
is taken in evaluating the link integral, $p$ and $p^{\prime}$ are
the momenta of incoming and outgoing fermions respectively,
$\lambda$ is the Lorentz-index of the gluon field,
and $k$
is the momentum of the gluon coming out of the vertex. Note that in
principle, any path between points $\mathbf{x}$ and $\mathbf{x}+\mathbf{n}a$
can be taken in the above calculation, but if one is interested in
deviations of the renormalized lattice operator from the rotational
invariance compared to the continuum operator, a path between two
points should be chosen in the continuum in such a way that it respects
rotational invariance explicitly.
Any path other than the radial path, on the other hand, is equivalent to
infinite many other paths resulting
from rotated versions of the original path around the radial path.
To reveal rotational invariance at the level of the
continuum operator, an averaging over these infinite copies of the
path is needed, and this makes the calculation of the link more involved.
Now at zero external momentum, using expression (\ref{eq:29}) with
$p=0$, the contribution from the second and third diagrams in
Fig.~\ref{fig:qcdOneLoop}b is
\begin{eqnarray}
\Gamma^{(b,c)}
& \sim &
- T^{a}T^{a}
\frac{3 g^2}{2\pi}\int_{0}^{1}dyy^{2}\int d\Omega_{\mathbf{y}}
\int\frac{d^{4}k}{\left(2\pi\right)^{4}}
\frac{
i {\bf k}\cdot {\bf y}
+m \mathbf{y}\cdot\vec{\gamma}
}{\left(k^{2}+m^{2}\right)k^{2}}
\nonumber\\
&&
\qquad \qquad
\qquad \qquad
\times\frac{1}{\mathbf{k}.\mathbf{y}}\left(e^{iNa\mathbf{k}\cdot\mathbf{y}}-1\right)
\ Y_{LM}\left(\Omega_{\mathbf{y}}\right)
.
\label{eq:30}
\end{eqnarray}
As is evident, because of a non-oscillatory contribution to the operator,
there is a logarithmically divergent piece from the above integration
contributing to the $L=0$ operator, which along with the logarithmic
divergent contribution
from wavefunction renormalization, contributes
to the anomalous dimension of the operator. Also the angular integration
of the above expression:
\begin{align}
&\int d\Omega_{\mathbf{y}}d\Omega_{\mathbf{k}}
\left[1+\frac{\mathbf{y}\cdot\vec{\gamma}}{i\mathbf{k}\cdot\mathbf{y}} m \right]
\left(e^{iNa\mathbf{k}\cdot\mathbf{y}}-1\right)
\ Y_{LM}\left(\Omega_{\mathbf{y}}\right)&
\nonumber\\
&
\qquad \qquad
=\int d\Omega_{\mathbf{y}}
\left[g_{1}\left(Nay\left|\mathbf{k}\right|\right)
+g_{2}\left(Nay\left|\mathbf{k}\right|\right) m \mathbf{y}\cdot\vec{\gamma}\right]
\ Y_{LM}\left(\Omega_{\mathbf{y}}\right)
,
\label{eq:31}
\end{align}
indicates that as before, in addition to $L=0$ operator, an $L=1$
contribution is present which is finite at UV, and can be shown to
vanish for $m/\Lambda\rightarrow0$. $g_{1}$ and $g_{2}$ are some
functions of their arguments whose explicit forms do not matter for
this discussion.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.15]{continuumtadpoles}
\par\end{centering}
\caption{{\small The tadpole contribution consists of the conventional tadpole
diagram (a), which vanishes when using a mass-independent regulator in the
continuum (such as dimensional regularization),
as well as the diagram
shown in (b) which is of the order of $\alpha_{s}/\left|\mathbf{\Delta x}\right|^{2}$,
where $\mathbf{\Delta x}$ is the distance between two gluon vertices.}}
\label{fig:tadpoles}
\end{figure}
The last diagram in Fig.~\ref{fig:qcdOneLoop} corresponds to the $\mathcal{O}\left(g^{2}\right)$
term in the small coupling expansion of the gauge link.
It contains
the tadpole of the continuum theory whose value depends in general
on the regularization scheme.
For example, by using a hard momentum cutoff which is matched easily
with the lattice regularization, it diverges quadratically. However,
it is not hard to see that in dimensional regularization which respects
the full rotational symmetry of the continuum, it vanishes in $d=4$,
therefore it does not contribute to the renormalization of the continuum
operator.
But the fourth diagram in Fig.~\ref{fig:qcdOneLoop} does not only include the
conventional tadpoles, Fig.~\ref{fig:tadpoles}(a), it also contains the diagram where
a gluon is emitted by the Wilson line inside the operator and then
absorbed at another point on the Wilson line, Fig.~\ref{fig:tadpoles}(b) as a consequence
of the matter fields being separated by a distance $\mathbf{n}a$.
It is straightforward to show this diagram is convergent, and scales
by $\alpha_{s}/\left|\mathbf{\Delta x}\right|^{2}$ where $\mathbf{\Delta x}$
is the distance between two gluon vertices and $\alpha_{s}$ is evaluated
at the energy scale of the order of $1/\left|\mathbf{\Delta x}\right|$.
This completes the qualitative discussion of the operator renormalization
and mixing at one-loop order in the continuum.
\subsection{Lattice operator and its renormalization}
Let us start the discussion of the lattice operator by assuming that its
definition is still given by Eq.~(\ref{eq:25}). However, this can
be shown to be a naive definition of the operator on the lattice.
The reason is implicit in the discussion of tadpoles given above.
Although tadpoles are absent from the operator renormalization
in the continuum,
on the lattice, they are non-vanishing, and result in large
renormalizations, as can be seen in perturbative lattice QCD calculations.
As was suggested
long ago by Lepage and Mackenzie \cite{Lepage}, to make the perturbative
expansion of the lattice quantities well-behaved, and to define an
appropriate connection between the lattice operators and their continuum
counterparts, one can remove tadpoles from the expansion of the lattice
operators in a non-perturbative manner by dividing the gauge link
by its expectation value in a smooth gauge,
\begin{equation}
U\left(x,x+a\hat{\mu}\right)
\rightarrow
\frac{1}{u_{0}}U\left(x,x+a\hat{\mu}\right)
,
\label{eq:32}
\end{equation}
where a simpler, gauge invariant choice of $u_{0}$ uses the measured
value of the plaquette in the simulation,
$u_{0}\equiv\left\langle \frac{1}{3}{\rm Tr}\left(U_{plaq}\right)\right\rangle
^{1/4}$.
There remains still another issue regarding the tadpole contributions
to the smeared operator which is not fully taken care of by the simple
single-link improvement procedure explained above. The operator introduced in
Eq.~(\ref{eq:25}) is smeared over several lattice sites, and as a
result includes extended links. As will be explained shortly, in
spite of $\mathcal{O}\left(\alpha_{s}\right)$ corrections due to
tadpoles from a single link,
there is an $\mathcal{O}\left(N\alpha_{s}\right)$
enhancement due to the tadpoles from the extended link with length $\sim Na$.
So although a non-perturbative tadpole improvement
could introduce non-negligible statistical errors, this improvement is crucial,
otherwise the relation between the lattice smeared operator and
the corresponding continuum operator is somewhat obscure.
\begin{figure}[t]
\begin{centering}
\includegraphics[scale=0.15]{tadpoles}
\par\end{centering}
\caption{{\small Tadpole diagrams contributing to the smeared operator at one-loop
order. Shown in the right are the number of diagrams of each type.}}
\label{fig:tadpoleextended}
\end{figure}
The reason for the $\mathcal{O}\left(N\alpha_{s}\right)$ enhancement
of tadpoles from the extended links
can be illustrated by working out a particular example. Suppose that
the link is extended between points $\mathbf{x}$ and $\mathbf{x}+Na\hat{e}_{1}$
entirely along the $1$ axis.
Then in order to make a tadpole, not
only can each gauge field be contracted with the other gauge field belonging to
the same elementary link, but also it can be contracted with a gauge field
from one of the remaining $N-1$ elementary links
(see Fig.~\ref{fig:tadpoleextended}).
Note that each diagram in Fig.~\ref{fig:tadpoleextended}
comes with a multiplicity of $N-m$, where $m$ is the number of
links between the contracted gluonic vertices.
At LO in $a$, the corresponding contribution from the extended tadpole (ET) is of the form
\begin{equation}
\Gamma^{(ET)}
\sim
\alpha_{s}a^{2}\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}d^{4}k
\ \frac{e^{imak_{1}}}{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2}}
\ \sim\
\frac{\alpha_{s}}{m^{2}}
,
\label{eq:33}
\end{equation}
from which
the contribution from all the diagrams in Fig.~\ref{fig:tadpoleextended} can be obtained,
\begin{equation}
\sum_{m=1}^{N-1}(N-m)\frac{\alpha_{s}}{m^{2}}
\ =\
\mathcal{O}\left(N\alpha_{s}\right)
.
\label{eq:34}
\end{equation}
Note that the $m=0$ term, corresponding to the first diagram in
fig~\ref{fig:tadpoleextended},
has been excluded from the above sum
as it is just the single link tadpole contribution.
Given that there are N single links, the total contribution from single link
tadpoles is $\mathcal{O}\left(N\alpha_{s}\right)$ as well.
Another issue with the extended links is the fact that without
tadpole improvement, breakdown of rotational symmetry occurs
at $\mathcal{O}\left(N\alpha_{s}\right)$.
The reason is that without tadpole improvement of the extended links,
contributions from the different
$A_{1}$ irreps in a given point shell are normalized differently.
For example, there are more tadpole diagrams at
$\mathcal{O}\left(g^{2}\right)$ contributing to an extended
link between points $\left(0,0,0\right)$ and $\left(2,2,1\right)$ (six single links)
than to an extended link between points $\left(0,0,0\right)$ and
$\left(3,0,0\right)$
(three single links)
although both points belong to the same point shell
(i.e. have the same separation in position space).
This fact magnifies the necessity of tadpole improvement as well as providing
a prescription for an appropriate improvement of an extended link.
As the
expectation value of a link belonging to a given $A_{1}$ irrep in
a given shell is in general different from the expectation value of
the link belonging to another $A_{1}$ irrep in the same shell, one
needs to redefine the link in a given irrep by dividing it by its
expectation value in the same irrep,
\begin{equation}
U_{A_{1}^{i}}\left(x,x+a\mathbf{n}\right)
\ \rightarrow
\ \frac{1}{u_{A_{1}^{i}}}U_{A_{1}^{i}}\left(x,x+a\mathbf{n}\right)
,
\label{eq:35}
\end{equation}
where $u_{A_{1}^{i}}=\left\langle U_{A_{1}^{i}}\left(x,x+a\mathbf{n}\right)\right\rangle $,
and the $A_{1}^{i}$'s are different $A_{1}$ irreps belonging to the
$n^{2}$-shell.
With this prescription for tadpole improvement of the extended links, the
renormalized operator is
assured
to be safe from large rotational invariance breaking effects of the
order of $\mathcal{O}\left(N\alpha_{s}\right)$.
With this new definition
of the gauge link, Eq. (\ref{eq:25}) is now a well-defined lattice
operator with an appropriate continuum limit which can be used
in our subsequent analysis.
As the cancellation of the tadpole diagram is assured by the new definition
of the operator, there are only three one-loop diagrams that contribute
to the renormalization of the lattice operator.
The first
diagram in Fig.~\ref{fig:qcdOneLoop} corresponds to the following loop integral at zero
external momentum for Wilson fermions,
\begin{eqnarray}
\Gamma^{(a)}
& \sim &
\left(ig\right)^{2}T^{a}T^{a}\frac{3}{4\pi N^{3}}\sum_{\mathbf{n}}
\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}k}{\left(2\pi\right)^{4}}
e^{i\mathbf{k}\cdot\mathbf{n}a}
\left[\gamma_{\rho}\cos\left(\frac{k_{\rho}a}{2}\right)-ir\sin\left(\frac{k_{\rho}a}{2}\right)\right]
\nonumber\\
&&
\times
\ \left(\frac{-i\sum_{\mu}\gamma_{\mu}\frac{\sin\left(k_{\mu}a\right)}{a}+M\left(k\right)}
{\sum_{\mu}\frac{\sin^{2}\left(k_{\mu}a\right)}{a^{2}}+M\left(k\right)^{2}}\right)^{2}
\left[\gamma^{\rho}\cos\left(\frac{k^{\rho}a}{2}\right)-ir\sin\left(\frac{k^{\rho}a}{2}\right)\right]
\nonumber\\
&&
\times\
\frac{i}{\frac{4}{a^{2}}\sum_{\nu}\sin^{2}\left(\frac{k_{\nu}a}{2}\right)}
\ Y_{LM}\left(\Omega_{\mathbf{n}}\right)
,
\label{eq:36}
\end{eqnarray}
where $M\left(k\right)\equiv M+2r/a\sum\limits_{\mu}\sin^{2}\left(k_{\mu}a/2\right)$,
and $r$ is the Wilson parameter. Clearly at LO in
the lattice spacing, one recovers the corresponding diagram with the
insertion of the continuum operator, Eq.~(\ref{eq:27}), and
so it contributes to both the $L=0$ and $L=1$ operators.
Note that although the integration region
is not rotationally symmetric like the continuum integral, the convergence
of integral at UV ensures that the contributions from non-rotationally
symmetric integration region II, defined in section~\ref{sec:Scalar},
are suppressed by additional powers of $1/N$ compared to the rotational
invariant region I:
\begin{eqnarray}
\delta\Gamma^{(a)}
& \sim &
-ig^{2}T^{a}T^{a}\frac{3
i^{L}}{16\pi^{4}}
\int_{-\pi}^{\pi}dl_{4}\int_{\pi}^{\sqrt{3}\pi}dl\ l^{2}\
\int_{f\left(\Omega_{\mathbf{l}}\right)}d\Omega_{\mathbf{l}}\
\frac{\left(il_{\mu}\gamma^{\mu}+ma\right)^{2}}{\left(l^{2}+m^{2}a^{2}\right)^{2}l^{2}}
\nonumber\\
&&
\qquad
\qquad
\times\ Y_{LM}\left(\Omega_{\mathbf{l}}\right)\left[\int_{0}^{1}dyy^{2}j_{L}\left(Nly\right)\right]
,
\label{eq:37}
\end{eqnarray}
where: $l_{\mu}=k_{\mu}a$ and $l^{2}=l_{1}^{2}+l_{2}^{2}+l_{3}^{2}$.
The integrand is clearly convergent, and the integration region is
entirely in the UV, and so the only dependence on $a=1/(\Lambda N)$ comes from
the integration over the Bessel function, giving a LO contribution
proportional to $1/N^{2}$.
However,
the first sub-leading contribution from this
diagram scales as $\sim \alpha_{s}/N$ for Wilson fermions instead
of $\sim \alpha_{s}/N^{2}$.
The reason is that the small $a$
expansion of the integrand in Eq.~(\ref{eq:36}) includes terms at
$\mathcal{O}\left(a\right)$ which is proportional to the Wilson
parameter.
The integrand
scales as $\sim 1/k^{3}$ multiplied by the spherical Bessel function
in the UV which still gives rise
to a convergent four-momentum integration for any value of $L$,
\begin{equation}
\delta\Gamma^{(a,r)}
\sim a\int d^{4}k\frac{1}{k^{3}}\left[\int_{0}^{1}dy\ y^{2}
\ j_{L}\left(Naky\right)\right]\sim a\Lambda=\frac{1}{N}
.
\label{eq:38}
\end{equation}
These contributions are rotational invariant, and will be included in the
renormalization
$Z$-factor of the operator when matching the lattice operator with its
continuum counterpart.
Further,
the integrals that appear at $\mathcal{O}\left(a^{2}\right)$
in an expansion of Eq.~(\ref{eq:36}) are also convergent,
and the terms containing rotational invariance breaking
contributions are suppressed by $1/N^{2}$. This completes discussion
of the first one-loop diagram of Fig.~\ref{fig:qcdOneLoop}.
The second diagram contains the one-gluon vertex operator, and requires
evaluating a line integral over the path on the grid defining the extended
link.
As was pointed
out in the discussion of the path in the continuum, in general any
path can be chosen in evaluating the operator both in the continuum
or on the lattice, but requiring the recovery of rotational symmetry
at the level of the operator means that the extended link has to exhibit
rotational symmetry in the continuum limit.
As already discussed, the simplest rotational
invariant path in the continuum is the radial path between the points,
so it makes sense to try to construct a path on the grid which remains
as close as possible to the radial path between points $x$ and $x+\mathbf{n}a$. One might expect though that
choosing a path in continuum which is the same as its lattice counterpart
is a more legitimate choice. One example of such a path is an $L$-shaped
path. However, it is not hard to verify that the $L$-shaped link does not
restore rotational invariance in the continuum limit as
the continuum path explicitly breaks rotational symmetry.
So the problem of evaluating the one-gluon vertex of the smeared operator
is reduced to finding the closest path to the straight line on the
grid.
In a lattice calculation, one can, in principle, construct an algorithm
which finds a path on the three-dimensional grid in such a way that
the area between the path and the rotational invariant radial path
is a minimum.
One such algorithm has already been used in Ref.~\cite{Meyer}
to construct a path that follows the straight line between sites A and B as
closely as possible, by forming a diagonal link at each step which has
the maximum projection onto the vector $\overrightarrow{AB}$.
By
this construction of ``super links'', the authors have been able
to form arbitrary (approximate) rotations of the Wilson loops, therefore
constructing glueball operators which project onto a definite spin $J$
in the continuum limit.
However,
the analytic form of the super link has not been given.
In appendix \ref{app:linksongrid}, a method to evaluate the link on such a path
is illustrated with a small number of examples.
For the following discussion
however, a particular example has been considered which encapsulates
the essential features of the recovery of the rotational path, and
gives us an idea how to deal with the general case.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.16]{twobyoneblockii}
\par\end{centering}
\caption{{\small a) The link between points $x$ and $x+\mathbf{n}a$ for $\mathbf{n}=\left(2,1,0\right)$
which remains as close as possible to the diagonal link, b) The link
between the same points for $\mathbf{n}=2\left(2,1,0\right)$ which
consists of two separate links of part a) with the lattice spacing
being halved, c) The link for $\mathbf{n}=2^{K}\left(2,1,0\right)$
which consists of $2^{K}$ separate links of part a) with the lattice
spacing divided by $2^{K}$.}}
\label{fig:links}
\end{figure}
Suppose that the link connects points $x$ and $x+\mathbf{n}a$ on
a cubic lattice where
$\mathbf{n}=\frac{a_{0}}{a}\left(Q,1,0\right)$,
and $a_{0}=2^{K}a$. As usual $a$ denotes the lattice spacing, and
$Q$ is an arbitrary integer. The continuum limit is recovered when
the integer $K$ tends to infinity for a finite value of $a_{0}$.
Then as is shown in appendix \ref{app:linksongrid}, for a path which is symmetric under
reflection about its midpoint and remains as close as possible to
the vector $\mathbf{n}a$ (see Fig.~\ref{fig:links}), the $\mathcal{O}\left(g\right)$
term in the momentum-space expansion of the link has the following
form
\begin{eqnarray}
U^{\left(1g\right)}\left(q\right)
& = &
ig\frac{a_{0}}{2^{K}}e^{i\mathbf{q}\cdot\mathbf{n}a/2}
\frac{\sin\left(\frac{\mathbf{q}\cdot\mathbf{n}a}{2}\right)}{
\sin\left(\frac{\mathbf{q}\cdot\mathbf{n}a}{2^{K+1}}\right)}
\left[A_{y}\left(q\right)
\right.
\nonumber\\
&&
\left.
\qquad\qquad
+2A_{x}\left(q\right)\frac{\sin\left(Qq_{x}a_{0}/2^{K+2}\right)}{
\sin\left(q_{x}a_{0}/2^{K+1}\right)}\cos\left(\frac{Qq_{x}a_{0}}{2^{K+2}}+\frac{q_{y}a_{0}}{2^{K+1}}\right)
\right]
.
\label{eq:39}
\end{eqnarray}
As $K\rightarrow\infty$ limit which corresponds to $a\rightarrow 0$, one
obtains
\begin{eqnarray}
U^{\left(1g\right)}\left(q\right)
& = &
2ige^{i\mathbf{q}\cdot\mathbf{n}a/2}\frac{\sin\left(
\frac{\mathbf{q}\cdot\mathbf{n}a}{2}\right)}{\mathbf{q}\cdot\mathbf{n}a}
\left[
\mathbf{A}\cdot\mathbf{n}a+\frac{a^{2}}{24}
\left(q_{x}Q+q_{y}\right)^{2}\mathbf{A}\cdot\mathbf{n}a
\right.
\nonumber\\
&&
\left.
\qquad\qquad
-\frac{a^{2}}{24}QA_{x}a_{0}\left(q_{x}^{2}\left(Q^{2}-1\right)
+3Qq_{x}q_{y}+3q_{y}^{2}\right)
+\mathcal{O}\left(a^{4}\right)
\right]
,
\label{eq:40}
\end{eqnarray}
recovering the continuum link, given in Eq.~(\ref{eq:29}),
and contains broken rotational invariance contributions which are
suppressed by $\sim {\cal O}(a^{2})$. This
scaling has been shown in appendix \ref{app:linksongrid} to hold for vectors $\mathbf{n}$
of the forms: $\frac{a_{0}}{a}\left(Q,1,1\right)$, $\frac{a_{0}}{a}\left(Q,Q,1\right)$
and $\frac{a_{0}}{a}\left(Q,Q,Q\right)$ as well.
Let us now examine
how the insertion of this contribution from the operator modifies the scaling
of the rotational invariance violating operators at one-loop.
The contribution from the second diagram in Fig.~\ref{fig:qcdOneLoop} with
the insertion of this vertex can be calculated order by order in small
$a$ by expanding the vertices and propagators as before.
At the LO one gets
\begin{align}
&\Gamma^{(b)}
\sim
-ig^{2}T^{a}T^{a}\frac{3}{4\pi
N^{3}}\sum_{\mathbf{n}}\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}
\frac{d^{4}k}{\left(2\pi\right)^{4}}\frac{ik_{\mu}\gamma^{\mu}+m}{\left(k^{2}+m^{2}\right)k^{2}}
\frac{e^{i\mathbf{k}\cdot\mathbf{n}a}-1}{i\mathbf{k}\cdot\mathbf{n}a}
\ Y_{LM}\left(\Omega_{\mathbf{n}}\right)\times
\nonumber\\
& ~~
\left[a\mathbf{n}\cdot\vec{\gamma}
+\frac{a^{2}}{24}\left(k_{x}Q+k_{y}\right)^{2}a\mathbf{n}\cdot\vec{\gamma}
-\frac{a^{2}}{24}Q\left(k_{x}^{2}\left(Q^{2}-1\right)+3Qk_{x}k_{y}+3k_{y}^{2}\right)\gamma_{x}a_{0}\right]
.
\qquad
\label{eq:41}
\end{align}
Clearly, after adding the contribution from the third diagram in Fig.~\ref{fig:qcdOneLoop}, the LO contribution from the above expression,
the first term in the bracket of Eq.~(\ref{eq:41}),
recovers
the results obtained previously for the insertion of the continuum
operator, up to suppressed contributions from the integration region
II, as discussed before. Therefore this term contributes to the $L=0$
operator with a logarithmically divergent coefficient, which along
with the wavefunction renormalization contributes to the anomalous
dimension of the lattice operator. Note that the wavefunction renormalization
gives rise to a logarithmically divergent contribution to the $L=0$
operator at LO in the lattice spacing, recovering the continuum result,
and the sub-leading contributions are suppressed at least by $a=1/(N\Lambda)$
for Wilson fermions. This term also contains and $L=1$ operator which
is proportional to $m$, and vanishes in the chiral limit.
The second term in the bracket of Eq.~(\ref{eq:41})
is ${\cal O}(a^2)$, and can be written as
\begin{eqnarray}
\delta\Gamma^{(b,c),2}
& = & \
-i{g^2 a^2\over 8\pi N^3}\
T^{a}T^{a}
\
\sum_{\mathbf{n}}
\int_{-\frac{\pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}k}{\left(2\pi\right)^{4}}
\left[1+\frac{m}{i\mathbf{k}\cdot\mathbf{n}a}a\mathbf{n}\cdot\vec{\gamma}\right]
\frac{e^{i\mathbf{k}\cdot\mathbf{n}a}-1}{\left(k^{2}+m^{2}\right)k^{2}}
\nonumber\\
&&
\qquad
\qquad
\qquad
\qquad
\qquad
\qquad
\times\left(k_{x}Q+k_{y}\right)^{2}Y_{LM}\left(\Omega_{\mathbf{n}}\right)
\nonumber\\
& \sim & \mathcal{O}\left(g^{2}a^{0}\right)
.
\label{eq:42}
\end{eqnarray}
This scaling arises as a result of the UV divergence of the non-oscillatory
contribution to the integral and is entirely a UV effect. For this term there
is no dependence upon ${\bf n}$ and as such the factor of $N^{-3}$ is canceled
by a corresponding $N^3$ from the sum.
Terms proportional
to the mass are convergent in the UV, and as such are suppressed by $a^{2}$
in the continuum limit.
The last term in the above expression Eq. (\ref{eq:41}) contains
rotational breaking contributions.
It is multiplied by an explicit factor of
$a^{2}$,
but as seen in the previous term,
the power divergence of the non-oscillatory part of the integral gives rise to
an overall scaling of $\mathcal{O}\left(g^{2}\right)$.
This completes the discussion
of the one-loop corrections to the lattice operator for the specific displacement
vector $\mathbf{n}a$ used above.
It is also straightforward to check
the obtained scaling of different terms for other choices of the vector
$\mathbf{n}a$.
In general, sub-leading contributions to the continuum link
are ${\cal O}(a^{2})$, and
so by dimensional analysis it
has an associated factor of momentum squared.
On the other hand, it always contains a non-oscillatory
term, and as a result, the non-continuum
contributions and the violations of rotational symmetry
scale as $\mathcal{O}\left(\alpha_{s}\right)$.
Given the discussion of the previous paragraphs, we naively conclude that the
rotational symmetry breaking scales as $\sim {\cal O}(\alpha_s)$ in the continuum limit.
It is the one-gluon vertex associated with the smeared-operator that is
dominating this behavior, with the contributions from other diagrams
scaling as $\sim \alpha_s/N$ for Wilson fermions (Eq.~(\ref{eq:36}) and Eq.~(\ref{eq:37}))
and $\alpha_s/N^2$ from the other loop
diagrams compared with $\sim 1/N^2$ from the tree-level matching.
However, this scaling can be further improved by smearing the gauge-field.
The ${\cal O}(\alpha_s)$ contributions are due to the explicit factor of $a^2$ being
compensated by a quadratic loop divergence, $\left(\pi/a\right)^2$, rendering a
suppression by only the coupling in the continuum limit, analogous to the
impact of tadpole diagrams.
However, by smearing the gluon field over a volume of radius
$1/\Lambda_g = a N_g$~\footnote{We have
distinguished the smearing radius of the operator, $N$, from the smearing
radius
of the gluons, $N_g$, but in principle they could be set equal.},
the offending diagrams in Fig.~\ref{fig:qcdOneLoop} scale as
\begin{eqnarray}
\delta\Gamma^{(b,c),2,3}
& \sim & \alpha_s\ a^2\ \Lambda_g^2
\ \sim\ {\alpha_s \over N_g^2}
,
\label{eq:42b}
\end{eqnarray}
due to the suppression of the high momentum modes in the gluon propagator.
The natural question to ask here is what is the scale of the coupling
in this process? Note that the bare coupling constant of lattice QCD
suffers from large renormalization as discussed before, so a better-behaved
weak coupling expansion of the lattice quantities uses a renormalized
coupling constant as the expansion parameter. As is suggested by Lepage
and Mackenzie \cite{Lepage}, one first fixes the renormalization
scheme by determining the renormalized coupling $\alpha_{s}^{ren}\left(k^{*}\right)$
from a physical quantity such as the heavy quark potential. Then the
scale of the coupling is set by the typical momentum of the gluon
in a given process. In the case considered above, the energy scale
of the strong coupling constant is dictated by the scale of the
gluon smearing region as the dominant contribution to the integral comes from
this region of the
integration: $k^{*}\sim\pi/(N_g a)$. A better estimate
of the scale can be obtained by the method explained in Ref.~\cite{Lepage},
but since we are interested in the continuum limit where $a\rightarrow0$,
this is already a reliable estimation of the momentum scale of the
running coupling.
The analysis in QCD
is more complex at one-loop level than in the scalar theory
due to the presence of the gauge-link required to render the
operator gauge-invariant.
We have found that the contributions from the operator defined in
Eq.~(\ref{eq:25}) scale in the same way as those in the scalar theory,
with the violation of rotational symmetry suppressed by factors of $\sim
1/N^2$, but
both tadpole improvement of the extended links and
smearing of the gauge-field is required.
Our analysis of Wilson fermions reveals the contributions to matrix elements
that violate rotational invariance in the
continuum limit at the one-loop level are suppressed by factors of
$\sim \alpha_s/N^2$ and $\sim \alpha_s/N_g^2$, and thus for a smearing defined in physical units,
deviations from rotational invariance scale as ${\cal O}(a^2)$.
Contributions that scale as $\sim \alpha_s/N$ and are proportional to the
Wilson parameter, conserve angular momentum and can be absorbed by the operator Z-factor.
Most importantly, as in the scalar theory, there are no mixings with lower dimension operators that
diverge as inverse powers of the lattice spacing.
\section*{Summary and discussions}
In this chapter, a mechanism for the restoration of rotational symmetry
in the continuum limit of lattice field theories is considered. The
essence of this approach is to construct an appropriate operator
on the cubic lattice which has maximum overlap onto the states with
definite angular momentum in the continuum. In analogy to the operator
smearing proposals given in Refs.~\cite{DudekI,DudekII,Edwards} and
Refs.~\cite{Meyer,JohnsonRW},
the operator is constructed on multiple lattice sites.
Using spherical harmonics in the definition of the operator
is key to having the leading contributions
to the classical operator be those with the desired angular momentum.
The
sizes of the contributions are controlled by the scale of the smearing of the operator,
with sub-leading
contributions to both lower and higher dimensional operators
that violate rotational symmetry
being suppressed by $1/N^{2}$ - reflective of the pixelation of the
operator and fields.
The $\lambda\phi^{4}$ scalar field theory
is shown to preserve this universal scaling of the leading non-rotationally
invariant contributions at all orders in perturbation theory, compatible
with the finite-size scaling results of $\lambda\phi^{4}$-type theories
near their rotational invariant fixed points~\cite{CampostriniI,CampostriniII}.
The same can be shown to be true in $g\phi^3$ scalar field theory.
Gauge invariance somewhat complicates the construction and analysis of analogous
operators in QCD. Although the tree-level lattice operator in QCD exhibits the
same scaling
properties as the scalar operator, extended gauge links connecting the quark fields generate gluonic
interactions that contribute to loop diagrams that are power-law divergent.
Such contributions are either eliminated by tadpole improvement of the extended
links, or are suppressed by smearing of the gauge field.
We find that it is the physical length scales
and continuum renormalization scale that dictate the size of matrix elements.
The leading non-continuum corrections from the one-loop diagrams preserve
angular momentum, scaling as $\sim \alpha_s a$ for Wilson fermions, and can be
absorbed by the operator $Z$-factor.
In contrast, contributions that violate rotational symmetry are suppressed by $\alpha_s a^2$ as $a\rightarrow 0$.
While we have chosen a specific form for the smeared operator, we expect that
the results, in particular the scaling of the violations to rotational symmetry,
are general features of a smeared operator with any (smooth) profile. Also, it is worth mentioning that although the calculations preformed in this work, and the subsequent conclusions, relate operators and matrix elements in $O_h(3)$ to those in $SO(3)$, the methodology and results are expected to hold in relations between $O_h(4)$ and $SO(4)$. Instead of working with operators formed with spherical harmonics to recover $SO(3)$ invariance, one would work with operators formed with hyper-spherical harmonics to recover $SO(4)$ symmetry.
We conclude the chapter by discussing the practicality of this result
for the current LQCD calculations as well as its connection to the recovery of IR rotational invariance in the lattice theories:
\begin{figure}
\begin{centering}
\includegraphics[scale=0.4]{ratioplot}
\par\end{centering}
\caption{{\small
The absolute value of the ratio of the tree-level coefficient,
$C_{30;10}^{(1)}$, of
a lowest dimension operator with $L=1$
to the tree-level coefficient, $C_{30;30}^{(3)}$, of the lowest dimension operator with
angular momentum, $L=3$, resulting from the $L=3$ operator in Eq. (\ref{eq:1}),
as a function of the number of included point shells.
}}
\label{fig:ratioplot}
\end{figure}
\begin{itemize}
\item
It is important to understand and to quantify
the violation of angular momentum conservation in the
states and matrix elements calculated using Lattice QCD
with the lattice spacings currently employed.
One interesting result is
that by using the tadpole-improved operator extended over several lattice sites
and built from the smeared gauge links,
the quantum corrections introduce non-continuum corrections to
the tree-level results that are suppressed by at least $\alpha_s$, i.e. they do
not introduce power-divergent contributions.
As an example,
suppose that a lattice calculation aims to determine a matrix element of an
operator with $L=3$.
Then, as is demonstrated in Fig. (\ref{fig:ratioplot}),
the coefficient of the lower dimensional derivative operator with $L=1$ is almost $10$
times larger than the coefficient of the $L=3$ derivative operator
when the operator is defined over one lattice site, $N=1$.
The computational time required to accurately perform the subtraction of the
$L=1$ contribution is significant for a smearing scale of, say, $\Lambda\sim 2~{\rm GeV}$.
Fortunately, by halving the lattice spacing and smearing the operator over just two point shells ($N=2$),
the contamination from the lower dimensional operator is reduced by
a factor of $\sim 3$, requiring a factor of $\sim 10$ less computational resources
to accurately perform the subtraction at the same level of precision.
Further, by smearing the operator over ten point shells,
the contamination from the lower dimensional operator is reduced to
$\sim 1\%$ of its value at $N=1$.
Given that the lattice spacing associated with $\Lambda = 2~{\rm GeV}$
is $a\sim 0.1~{\rm fm}$ for $N=1$, to be able to smear out to the $N=2$ shell requires a
lattice spacing of $a\sim 0.05~{\rm fm}$, pushing the limits of current lattice
generation. To smear out to the $N=10$ shell would require a lattice spacing
of $a\sim 0.01~{\rm fm}$ which is currently impractical.
\item The restoration of rotational invariance as discussed in this chapter
regards only the UV asymptote of the lattice theories: as one reaches
a good pixelation of a region of space where the lattice operator
probes, the identification of eigenstates of the angular momentum
operator becomes possible. In the other words, the more point shells
included in the lattice operator, the larger overlap the operator
has onto a definite angular momentum state. However, the full recovery
of rotational invariance in the lattice theories requires the suppression
of rotational symmetry breaking contributions to the physical quantities
not only as a result of short-distance discretization effects, but
also as a result of boundary effects of the finite cubic lattice in the
IR regime of the theories. The finite size of the lattice imposes
(anti-)periodic boundary conditions on the lattice wavefunctions which
enforces the lattice momenta to be discretized,
${\bf p}=\frac{2\pi {\bf n} }{L}$,
where $L$ is the spatial extent of the lattice
and ${\bf n}$ is a vector of
integers. The IR rotational invariant theory is achieved as the
lattice becomes infinitely large,
corresponding to a large number of point shells in the momentum space.
However, beyond this intuitive picture, one
needs to examine in a quantitative way how this recovery takes place
in the large-volume limits of the lattice theories in the same way
as it was discussed for small lattice spacing limit of the theories.
One quantitative explanation of this IR recovery, has been given
in Ref.~\cite{Luu:2011ep} in the context of the
extraction of phase shifts in higher
partialwaves from the energies of scattering particles in a finite volume
using L\"uschers method.
The idea is that as one includes higher momentum shells, the number
of occurrence (multiplicity) of any given irrep of the cubic group increases.
As a result, for a fixed energy in the large volume limit,
linear combinations of different states of a given irrep can be formed
which can be shown to be energy eigenstates; and
the energy shift of each combination due to interactions is suppressed
in all but one partialwave in the infinite-volume limit.
So, although each state has an
overlap onto infinitely many angular momentum states, the high multiplicity
of a given irrep in a large momentum shell generates energy eigenstates
which
dominantly overlap onto states of definite angular momentum,
and the mixing with other angular momentum states becomes insignificant in the
large volume limit.
This picture also helps to better understand the mechanism of the
UV rotational invariance recovery due to the operator smearing.
It
is the high multiplicity of the irreps in large (position-space)
shells that is responsible
for projecting out a definite angular momentum eigenstate.
These large
shells are obtained by reducing the pixelation of the lattice by taking
$a\rightarrow 0$ in position space, or increasing the size of the
lattice by taking $L\rightarrow\infty$ in momentum space -- both are required in
order to recover rotational invariance from calculations performed on a lattice.
\end{itemize}
\chapter{TWO-NUCLEON SYSTEMS FROM A FINITE-VOLUME FORMALISM}
{\label{chap:NN}}
Despite tight empirical constraints on the two-body nuclear force, the investigation of the two-nucleon sector within LQCD is still warranted. Understanding the energy dependence of the scattering phase shifts of two-body hadronic states, for example, is essential in obtaining physical matrix elements of current operators in the two-body sector \cite{Detmold:2004qn, Meyer:2012wk, Briceno:2012yi, Bernard:2012bi, Meyer:2013dxa}. Additionally, as LQCD calculations are currently done at unphysical pion masses, a rigorous study of three (multi)-nucleon systems from LQCD requires not only the knowledge of two-nucleon phase shifts, but also their pion mass dependence as shown in Refs. \cite{Briceno:2012rv, Kreuzer:2008bi, Kreuzer:2009jp, Kreuzer:2010ti, Kreuzer:2012sr}. The LQCD determination of the scattering parameters of two-nucleon systems at unphysical pion masses by itself is an interesting problem as it reveals the dependence of the two-body nuclear force on the masses of quarks in nature. Progress in this direction will have striking impact on our understanding of some of the most fundamental questions regarding the nuclear fine tunings in nature and the anthropic view of the Universe. As discussed in Refs. \cite{Epelbaum:2012iu, Epelbaum:2013wla, Bedaque:2010hr}, the survivability of Carbon-Oxygen based life is related to the variation of the inverse scattering lengths of NN scattering in the isosinglet and isotriplet channels, and a precise LQCD determination of these parameters will put tighter constraints on this quantity. In fact, for the first time LQCD has started addressing the question of naturalness of the NN interactions. A nice example is the extraction of the $S$-wave NN scattering length and effective range by the NPLQCD collaboration \cite{Beane:2013br} at a pion mass of $m_{\pi}\approx 800~{\rm MeV}$ which has enabled them to study the variation of the NN scattering parameters and the corresponding bound-state energies with respect to the light-quark masses, see Fig. \ref{a-to-r}.
\begin{figure}[h]
\begin{center}
\label{a-to-r}
\includegraphics[scale=0.695]{ator}
\caption{{\small The left panel represents the ratio of the two-nucleon scattering length, $a$, to the effective range, $r$, in the ${^3}S_1$ (top) and ${^1}S_0$ (bottom) channels at the physical point as well at the $SU(3)$ symmetric point with $m_{\pi}\approx 800~{\rm MeV}$ \cite{Beane:2013br}. As can be inferred from the plots, the NN interactions remain unnatural over a wide range of pion masses. The right panel represents the plots of the binding energy as a function of pion mass. These indicate that the size of the deuteron and the $nn$ bound state remain large compared with the range of interactions at heavier pion masses. The figure is reproduced with the permission of the NPLQCD collaboration.}}
\label{a-to-r}
\end{center}
\end{figure}
To appropriately utilize these LQCD calculations, in particular for the realistic NN systems with physical partial-wave mixings -- which occurs due to the action of non-central forces in nuclear systems, the FV formalisms and their associated QCs must be developed.
In this chapter, we derive and present the generalization of the L\"uscher formula for nucleon-nucleon (NN) scattering valid below the inelastic threshold for all spin and isospin channels in both positive and negative parity sectors. This formula is derived using the auxiliary field (dimer) formalism in the language of a NR effective field theory EFT for NN interactions. As introduce in chapter \ref{chap:intro}, a S-wave dimer field -- that sums all $2\rightarrow2$ interactions non-perturbatively \cite{Kaplan:1996nv, Beane:2000fi} -- significantly simplifies the diagrammatic representation of multi-nucleon scattering amplitudes \cite{Bedaque:1997qi, Bedaque:1998mb, Bedaque:1998kg, Bedaque:1998km, Gabbiani:1999yv, Bedaque:1999vb, Bedaque:1999ve, Bedaque:2000ft}. However, to account for scattering in higher angular momentum channels, this dimer field must be generalized to arbitrary partial waves. This is particularly important when such an auxiliary field is used in constructing a FV formalism for three-body scattering processes. As is pointed out in Ref. \cite{Briceno:2012rv}, the leading systematics of the results presented in Refs. \cite{Briceno:2012rv,Kreuzer:2008bi, Kreuzer:2009jp, Kreuzer:2010ti, Kreuzer:2012sr}, for the relation between three-body scattering amplitude and the FV spectrum of the three-particle system, arises from the FV-induced mixing between S-wave and D-wave scattering modes of the two-particle sub-system. A S-wave dimer field therefore does not incorporate possible mixings in the FV formalism and will not give rise to a full quantization condition in arbitrary partial waves in both two-body and three-body systems. To address this defect, we generalize the dimer field to higher partial waves and utilize the result to derive the generalized L\"uscher formula for the two-body boosted systems within both scalar and nucleon sectors.
As discussed in chapter \ref{chap:intro}, performing LQCD calculations for systems with different CM momenta gives access to more energy levels at a given volume and provides additional QCs for the energy eigenvalues of the system in terms of scattering parameters.
Although the master formula that will be derived is self-contained and incorporates all the necessary details to be implemented in practice, deducing the relations, or QCs, among phase shifts in different partial waves and the energy levels of a specific LQCD calculation requires multiple nontrivial steps. The corresponding procedure is sometimes called the reduction of the L\"uscher formula. The difficulty associated with this procedure is due to the fact that LQCD calculations are performed in a finite periodic cubic volume (for calculations at rest). As a result, the degeneracy of energy eigenvalues of the system in such calculations is determined according to the irreps of the cubic group. Since the phase shifts are characterized according to the irreps of the SO(3) rotational group, the energy eigenvalues of the system in a given irrep of the cubic group in general depend on the phase shifts of more than one partial-wave channel. Performing LQCD calculations of energy levels in different irreps of the cubic group would provide multiple QCs depending on different linear combinations of the scattering phase shifts, leading to better constraints on these quantities. Therefore it is necessary to identify all the QCs satisfied by a given scattering parameter in a partial-wave channel. While L\"uscher's original work presents the reduction of the master formula to a QC for the cubic $A_1$ irrep, Ref. \cite{Luu:2011ep} provides the full quantization conditions for the energy eigenvalues of different irreps of the cubic group, in both positive and negative parity sectors for orbital angular momentum $l\leq6$ as well as $l=9$ in the scalar sector. For scattering involving a spin-$\frac{1}{2}$ particle and a scalar particle, the L\"uscher formula can be generalized such that the energy eigenvalues of the meson-baryon system in a given irrep of the double-cover of the cubic group is related to the corresponding phase shifts \cite{Bernard:2008ax}. This generalization has been also presented for NN scattering, where due to the the possibility of physical mixing among different partial-wave channels, more complexities arise.\footnote{The L\"uscher formula to study two-nucleon systems were first presented in Ref. \cite{Beane:2003da}, although due to constraining the calculation to the S-wave scattering, the complexity of the two-nucleon systems has not been dealt with. The only previous attempt to address this problem, including the spin, isospin and angular momentum degrees of freedom, is the work by N. Ishizuka \cite{Ishizuka:2009bx}, where the quantization conditions for energy eigenvalues of a two-nucleon system at rest in the positive and negative parity isosinglet channels were obtained for $J\leq 4$.}
By investigating the symmetry groups of the boosted systems along one and two Cartesian axes as well as that of the unboosted system, we have identified all the QCs satisfied by the phase shifts and mixing parameters in channels with total angular momentum $J\leq4$; ignoring scattering in partial-wave channels with $l\geq4$. Different QCs correspond to different irreps of the cubic ($O$), tetragonal ($D_{4}$) and orthorhombic ($D_{2}$) point groups that represent the symmetry group of systems with CM momentum $\mathbf{P}=0$, $\mathbf{P}=\frac{2\pi}{L}(0,0,1)$ and $\mathbf{P}=\frac{2\pi}{L}(1,1,0)$ respectively, where $L$ denotes the spatial extent of the cubic volume. As will be discussed later, these QCs can be also utilized for boost vectors of the form $\frac{2\pi}{L}(2n_1,2n_2,2n_3)$, $\frac{2\pi}{L}(2n_1,2n_2,2n_3+1)$ and $\frac{2\pi}{L}(2n_1+1,2n_2+1,2n_3)$ and all cubic rotations of these vectors where $n_1,n_2,n_3$ are integers. Although the master formula presented in this article in the limit of zero CM momentum has been already derived in Ref. \cite{Ishizuka:2009bx} for NN systems using a relativistic quantum field theory approach, the full classifications of different QCs for all the spin and isospin channels and for two non-zero CM momenta were first obtained through completion of this thesis and are already presented in Refs. \cite{Briceno:2013lba, Briceno:2013rwa}. These lengthy relations are tabulated in a Mathematica notebook supplemented to the published paper \cite{BDLsupp} and we refrain to repeat those in this thesis. However, the procedure of deducing these relations will be presented through an example in Sec. \ref{app:red-example}. Strategies for deducing such QC for more general cases will be briefly discussed in Sec. \ref{red-syst}. These relations make the implementation of the generalized L\"uscher formula for NN systems straightforward for future LQCD calculations of the NN system.
\section{Finite Volume Formalism with the Auxiliary Field Method \label{sec: dimer}}
The goal of this section is to extend L\"usher's formula \cite{Luscher:1986pf, Luscher:1986pf} to the case of two nucleons within the context of a NR EFT, using an auxiliary field method. Although there has been several derivations for the L\"uscher formula (an example of which presented in Sec. \ref{IV-intro}), the formalism that will be presented here makes the study of two-baryon systems with arbitrary quantum numbers straightforward. Additionally the methodology developed here can be used to generalize the FV formalism presented in Ref. \cite{Briceno:2012rv} to three hadrons with arbitrary partial waves. In order to be able to incorporate the specific features of the two-nucleon systems in the formalism, it is instructive to start with developing a general dimer formalism for scalar particles. However, such formalism by itself is valuable in studies of multi-meson systems in a finite volume, see for example, Refs. \cite{Briceno:2012rv,Kreuzer:2008bi, Kreuzer:2009jp, Kreuzer:2012sr}.
\subsection{Two-boson systems \label{sec: Scalar}}
Consider two identical bosons with mass $M$ that interact in a partial-wave channel $(l,m)$ via a short-range interaction that can be effectively described by derivative couplings to the fields. Let $\phi_{k}$ and $d_{lm,P}$ denote the interpolating operators that annihilate a boson with NR four-momentum $k$, and a dimer (with quantum numbers of two bosons) with NR four-momentum $P$ and angular momentum $(l,m)$, respectively. Then if $P^\mu=(E,\mathbf{P})$ denotes the NR four-momentum of the system, one can write a Galilean-invariant action that describes such system in the infinite volume in terms of a Lagrange density in the momentum space,
\begin{eqnarray}
\label{action}
{S}&=&\int\frac{d^{4}P}{(2\pi)^{4}}\left[\phi_{P}^{\dagger}(E-\frac{\textbf{P}^{2}}{2M})\phi_{P}-\sum_{l,m}d_{lm,P}^{\dagger}\left(E-\frac{\textbf{P}^{2}}{4M}-\Delta_{l}+\sum_{n=2}^{\infty}c_{n,l}(E-\frac{\textbf{P}^{2}}{4M})^{n}\right)d_{lm,P}\right]
\nonumber\\
&~& \qquad \qquad -\int\frac{d^{4}P}{(2\pi)^{4}}~\frac{d^{4}k}{(2\pi)^{4}}\sum_{l,m}~\frac{g_{2,l}}{2}\left[d_{lm,P}^{\dagger}~\sqrt{4\pi}~Y_{lm}(\hat{\textbf{k}}^{*})~|\mathbf{k}^{*}|^{l}\phi_{{k}}\phi_{P-{k}}+h.c.\right],
\end{eqnarray}
where ${\textbf{k}}^*=\textbf{k}-\textbf{P}/2$ denotes the relative momentum of two bosons in the interaction term. Note that the interactions between bosons in partial-wave channel $(l,m)$ is mediated by a corresponding dimer field, $d_{lm}$. As is evident, upon integrating out such auxiliary field, one recovers the four-boson interaction term in a Lagrangian with only $\phi$-field degrees of freedom. Since this is a theory of identical bosons, all couplings of the dimer field to a two-boson state with an odd partial wave vanish. Eq. (\ref{action}) clearly reduces to the S-wave result of Refs. \cite{Kaplan:1996nv, Beane:2000fi, Griesshammer:2004pe}. This action can be easily generalized for systems involving distinguishable scalar bosons (e.g. for P-wave scattering see Ref. \cite{Braaten:2011vf}). As usual, the LECs $\{\Delta_{l},c_{l,n}, g_{2,l}\}$ in the effective Lagrangian must be tuned to reproduce the ERE of the $l^{th}$-partial wave,
\begin{eqnarray}
k^{*2l+1}\cot\delta_{d}^{(l)}=-\frac{1}{a_l}+\frac{r_{l}k^{*2}}{2}+\sum_{n=2}^\infty\frac{\rho_{n,l}}{n!}~(k^{*2})^{n},
\label{NN-ERE}
\end{eqnarray}
where $k^*\equiv |\mathbf{k}^*|=\sqrt{ME-\frac{\mathbf{P}^2}{4}}$ is the relative \emph{on-shell} momentum of the bosons in the CM frame. $\delta_d^{(l)}$ is the phase shift in the $l^{th}$-partial wave, and $\{a_l,r_l, \rho_{n,l}\}$ are the corresponding scattering length, effective range and all higher order shape parameters, respectively. The fully dressed dimer propagator can be obtained by summing up the self-energy bubble diagrams to all orders, Fig. \ref{fig:dimer}(a), the result of which is the following
\begin{eqnarray}
\mathcal{D}^{\infty}(E,\mathbf{P})=\frac{1}{(\mathcal{D}^{B})^{-1}-I^{\infty}(E,\mathbf{P})},
\label{D-infinity}
\end{eqnarray}
where $\mathcal{D}^B$ denotes the bare dimer propagator,
\begin{eqnarray}
\left[\mathcal{D}^{B}(E,\mathbf{P})\right]_{l_1m_1,l_2m_2}=\frac{-i~\delta_{l_1l_2}\delta_{m_1m_2}}{E-\frac{\mathbf{P}^2}{4M}-\Delta_{l}+\sum_{n=2}^{\infty}c_{n,l}(E-\frac{\textbf{P}^{2}}{4M})^{n}+i\epsilon},
\label{D-bare}
\end{eqnarray}
and $I^{\infty}$ denotes the value of the bubble diagram evaluated using the power divergence subtraction (PDS) scheme \cite{Kaplan:1998tg, Kaplan:1998we, Beane:2003da},
\begin{eqnarray}
\left[I^{\infty}(E,\mathbf{P})\right]_{l_1m_1,l_2,m_2}=\frac{iM}{8\pi}g_{2,l_1}^2k^{*2l_1}(\mu+ik^*)\delta_{l_1l_2}\delta_{m_1m_2},
\label{I-infinity}
\end{eqnarray}
where $\mu$ is the renormalization scale. By requiring the full dimer propagator, $\mathcal{D}^{\infty}$, in the infinite volume to reproduce the full scattering amplitude in any given partial wave,
\begin{eqnarray}
\mathcal{M}^{\infty}_{l_1m_1,l_2m_2}&=&-[~g~\mathcal{D}^{\infty}(E,\mathbf{P})~g~]_{l_1m_1,l_2m_2}=
\frac{8\pi}{M}~
\frac{1}{k^{*}\cot{\delta^{(l_1)}_d}-ik^{*}}\delta_{l_1l_2}\delta_{m_1m_2},
\end{eqnarray}
one arrives at
\begin{eqnarray}
g_{2,l}^2=\frac{16\pi}{M^2r_{l}} ~\text{for}~l~\text{even}, ~~ \Delta_l=\frac{2}{Mr_l}\left(\frac{1}{a_l}-\mu k^{*2l}\right), ~~ c_{n,l}=\frac{2}{Mr_l}\frac{\rho_{n,l}M^n}{n!}.
\label{g2l}
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.415]{dimeriv}}
\subfigure[]{
\includegraphics[scale=0.415]{dimerfv}}
\caption{{\small a) Diagrammatic equation satisfied by the matrix elements of the full dimer propagator in a) infinite volume and b) finite volume. The grey (black) band represents the full infinite (finite) volume propagator, $\mathcal{D}^{\infty}$ ($\mathcal{D}^V$), while the double lines represent the bare propagator, $\mathcal{D}^B$.}}\label{fig:dimer}
\end{center}
\end{figure}
In the finite volume, the two-boson system can still be described by the action in Eq. (\ref{action}) except the periodic boundary conditions constrain the momenta to be discretized. In particular, the integral over three-vector momenta in Eq. (\ref{action}) is replaced by a sum over discrete momenta, $P=\frac{2\pi}{L}\mathbf{n}$, where $\mathbf{n}$ is an integer triplet. Then it is straightforward to evaluate the corresponding bubble diagram in the finite volume,
\begin{eqnarray}
\left[I^{V}\right]_{l_1m_1,l_2,m_2}&=&\frac{iM}{8\pi}g_{2,l_1}g_{2,l_2}k^{*l_1+l_2}
\nonumber\\
&~&\times\left[\mu~\delta_{l_1l_2}\delta_{m_1m_2}+\sum_{l,m}\frac{(4\pi)^{3/2}}{k^{*l}}c^{{P}}_{lm}(k^{*2})\int d\Omega~Y^*_{l_1,m_1}Y^*_{l,m}Y_{l_2,m_2}\right],
\label{I-V}
\end{eqnarray}
where as defined in Sec. \ref{IV-intro},
\begin{eqnarray}
\label{clm}
c^{{\mathbf{P}}}_{lm}(x)=\left[\frac{1}{L^3}\sum_{\textbf{q}}-\mathcal{P}\int\frac{d^3\mathbf{q}}{(2\pi)^3}\right]{q}^{*l}\frac{\sqrt{4\pi}Y_{lm}(\hat{\mathbf{q}^*})}{{\mathbf{q}^*}^{2}-x} \ ,
\end{eqnarray}
$\mathbf{q}^*=\mathbf q- \mathbf P/2$, and $\mathcal{P}$ denotes the principal value of the integral. The full dimer propagator, $\mathcal{D}^V$, can then be obtained by summing up the infinite series of bubble diagrams in Fig. \ref{fig:dimer}(b), where the LEC of the theory are matched with the the physical quantities according to Eq. (\ref{g2l}),
\begin{eqnarray}
\mathcal{D}^{V}(E,\mathbf{P})=\frac{1}{(\mathcal{D}^{B})^{-1}-(\mathcal{D}^{B})^{-1}I^{V}(E,\mathbf{P})\mathcal{D}^{B}}.
\label{D-finite}
\end{eqnarray}
Note that, just like $\mathcal{D}^\infty$ in Eq.~(\ref{D-infinity}), $\mathcal{D}^V$ is a matrix in the angular momentum space. The poles of the FV dimer propagator give the spectrum of two-boson system in a finite volume in terms of the scattering parameters. These energy eigenvalues satisfy the following determinant condition
\begin{eqnarray}
\det \left[k^*\cot \delta-\mathcal{F}^{FV}\right]=0,
\label{FullQCboson}
\end{eqnarray}
where both $\cot \delta$ and $\mathcal{F}^{FV}$ are matrices in the angular momentum space,
\begin{eqnarray}
\cot \delta \equiv \cot (\delta_{l_1}) \delta_{l_1l_2}\delta_{m_1m_2},
\label{cot}
\end{eqnarray}
\begin{eqnarray}
\left[\mathcal{F}^{FV}\right]_{l_1m_1,l_2m_2}=\sum_{l,m}\frac{(4\pi)^{3/2}}{k^{*l}}c^{\mathbf{P}}_{lm}(k^{*2})\int d\Omega~Y^*_{l_1,m_1}Y^*_{l,m}Y_{l_2,m_2}.
\label{F}
\end{eqnarray}
In Eq.~(\ref{cot}) the Kronecker deltas that are constraining the $(l,m)$ quantum numbers should not be confused with the phase shift $\delta_{l_1}$. This quantization condition agrees with the NR limit of the results presented in Refs. \cite{Rummukainen:1995vs, Kim:2005gf, Christ:2005gi} for the generalization of the L\"uscher formula to the boosted systems, and upon truncating the angular momentum sum to $l_{max}=0$, reduces to the S-wave result of Ref. \cite{Briceno:2012rv} where an S-wave dimer field is used to derive the L\"uscher formula. This derivation shows that upon incorporating higher partial waves in the construction of the dimer Lagrangian, as well as accounting for higher order terms in the EFR expansion, all the two-body physics is fully encapsulated in this formalism. As a result the systematic errors of those FV multi-particle calculations that have used a S-wave dimer field up to next-to-leading order in ERE (see Refs. \cite{Briceno:2012rv, Kreuzer:2008bi, Kreuzer:2009jp, Kreuzer:2010ti, Kreuzer:2012sr}), can be easily avoided.
\subsection{Two-nucleon systems \label{sec: Nuclear}}
Due to spin and isospin degrees of freedom, the two-nucleon system exhibits some specific features. In particular, the anti-symmetricity of the two-nucleon state constrains the allowed spin and isospin channels for a given parity state. Additionally, any spin-triplet two-nucleon state is an admixture of two different orbital-angular momentum states. For example, as is well known, the two-nucleon state in the deuteron channel with $J^{P}=1^{+}$ is an admixture of S-wave and D-wave states. In general, a positive parity two-nucleon state with total angular momentum $J$ is a linear combination of states with the following orbital angular momentum $L$ and total spin $S$\footnote{The $L$ that is introduced here and elsewhere as the partial-wave label of quantities should not be confused with the spatial extent of the lattice $L$ that appears in the definition of the $c_{lm}^{\mathbf{P}}$ functions.}
\begin{eqnarray}
\left(L=J\mp\frac{1}{2}(1-(-1)^J), S=\frac{1}{2}(1-(-1)^J)\right),
\label{positive}
\end{eqnarray}
while in the negative parity sector, the states that are being mixed have\footnote{Note, however, that for a $J$-even state in the first case and a $J$-odd state in the second case, there is only one angular momentum state allowed and no mixing occurs.}
\begin{eqnarray}
\left(L=J\mp\frac{1}{2}(1+(-1)^J), S=\frac{1}{2}(1+(-1)^J)\right).
\label{negative}
\end{eqnarray}
Table (\ref{JP}) shows the allowed spin and angular momentum of NN states in both isosinglet and isotriplet channels with $J\leq3$.
\begin{center}
\begin{table}[h]
\includegraphics[scale=0.9]{jistable}
\caption{{\small The allowed spin and angular momentum states, $(L,S)$, for NN-states with $J\leq3$ assuming an exact isospin symmetry. Note that depending on the parity of the states, the partial-wave mixing can occur in either the isosinglet or isotriplet channels.}\label{JP}}
\end{table}
\end{center}
In order to write the most general Lagrangian describing NN scattering in all spin, isospin and angular momentum channels, let us introduce an operator that creates a NN-state with total four-momentum $P$ and the relative momentum ${\textbf{k}}^*={\textbf{k}}-\frac{{\textbf{P}}}{2}$ in an arbitrary partial wave $(L,M_L)$ in the following way
\begin{eqnarray}
|NN;P,k^*\rangle _{LM_L,SM_S,IM_I}=\mathcal{N}_L\int d\Omega_{\textbf{k}^*}~Y^*_{LM_L}(\hat{\textbf{k}}^{*})k^{*L}\left[N^T_{P-k}~\hat{\mathcal{P}}_{(SM_S,IM_I)}~N_k\right]^\dag|0\rangle,
\end{eqnarray}
where $k^*=\left|\mathbf{k}^*\right|$. $\hat{\mathcal{P}}_{(SM_S,IM_I)}$ is an operator which projects onto a two-nucleon state with spin $(S,M_S)$ and isospin $(I,M_I)$, and $\mathcal{N}_L$ is a normalization factor. By requiring such state to have a non-zero norm, and given the anti-commutating nature of nucleon fields, one can infer that for positive parity states the operator $\hat{\mathcal{P}}_{(SM_S,IM_I)}$ must be necessarily antisymmetric, while for negative parity states it must be symmetric. Since this operator is a direct product of two projection operators in the space of spin and isospin, these requirements can be fulfilled by constructing the corresponding operators using the appropriate combinations of Pauli matrices, $\sigma_j$ ($\tau _j$), that act on the spin (isospin) components of the nucleon field. To proceed with such construction, let us define the following operators
\begin{eqnarray}
\alpha_j^I=\tau_y\tau_j,\hspace{1cm}
\alpha_j^S=\sigma_y\sigma_j,\hspace{1cm}
\beta^I=\tau_y,\hspace{1cm}
\beta^S=\sigma_y.
\end{eqnarray}
Note that the matrices that are named as $\alpha$ are symmetric while those that are named as $\beta$ are antisymmetric. Superscript $I$ ($S$) implies that the operator is acting on the spin (isospin) space, and index $j=1,2,3$ stands for the Cartesian components of the operators. Alternatively one can form linear combinations of $\alpha^S_j$ ($\alpha^I_j$) that transform as a rank one spherical tensor.\footnote{A Cartesian vector $\mathbf{r}$ can be brought into a spherical vector according to
\begin{eqnarray}
\label{spherical}
r^{(0)}\equiv r_{z},\hspace{1cm}
r^{(\pm 1)}\equiv\mp \frac{\left(r_{x} \pm ir_{y}\right)}{\sqrt{2}}.
\nonumber
\end{eqnarray}
} Using these matrices, it is straightforward to see that an antisymmetric $\hat{\mathcal{P}}_{(SM_S,IM_I)}$ can have one of the following forms
\begin{eqnarray}
{\hat{\mathcal{P}}}_{(00,1M_I)} \equiv \frac{\alpha^{(M_I)}_I\otimes \beta_S}{\sqrt{8}},\hspace{.25cm}
{\hat{\mathcal{P}}}_{(1M_S,00)} \equiv \frac{\beta_I\otimes \alpha^{(M_s)}_S}{\sqrt{8}},
\end{eqnarray}
which can project onto two-nucleon states with $\left(S=0,I=1\right)$ and $(S=1,I=0)$ respectively. Note that these are the conventional isotriplet and isosinglet projection operators in the positive parity sector that are used frequently in literature \cite{Savage:1998ae, Chen:1999tn}.
On the other hand, a symmetric $\hat{\mathcal{P}}_{(SM_S,IM_I)}$ can project onto two-nucleon states with $(S=0,I=0)$ and $(S=1,I=1)$ and should have one of the following forms,
\begin{eqnarray}
{\hat{\mathcal{P}}}_{(00,00)} \equiv \frac{\beta_I\otimes \beta_S}{\sqrt{8}},\hspace{.25cm}
{\hat{\mathcal{P}}}_{(1M_S,1M_I)} \equiv \frac{\alpha^{(M_I)}_I\otimes \alpha^{(M_S)}_S}{\sqrt{8}},
\end{eqnarray}
respectively.
As it is the total angular momentum $J$ that is conserved in a two-nucleon scattering process, as opposed to the orbital angular momentum $L$, it is convenient to project a two-nucleon state in the $|LM_L,SM_S\rangle$ basis into a state in the $|JM_J,LS\rangle$ basis using the Clebsch-Gordan coefficients,
\begin{eqnarray}
|NN;P,k^*\rangle _{JM_J,LS,IM_I}&=&\sum_{M_L,M_S}\langle JM_J|LM_L,SM_S\rangle~ |NN;P,k^*\rangle _{LM_L,SM_S,IM_I}.
\label{NNstate}
\end{eqnarray}
Note that isospin remains a conserved quantum number up to small isospin breaking effects that we ignore for the nucleon systems.
In order to describe NN interactions, we introduce an auxiliary dimer filed, similar to the scalar theory.\footnote{The S-wave dimer field in the nuclear sector is commonly referred to as a di-baryon field.} This field, that will be labeled $d^{LS}_{JM_J,IM_I;P}$, has the quantum numbers of two-nucleon states with total angular momentum $(J,M_J)$ and isospin quantum number of ${(I,M_I)}$ with orbital angular momentum $L$ and spin $S$. Now the action corresponding to the Lagrangian density of free nucleon and dimer fields in the momentum space can be written as
\begin{eqnarray}
&&S_{kinetic}=\int\frac{d^4P}{(2\pi)^4}\left[N^\dag_P(E-\frac{\textbf{P}^2}{2m})N_P \right .
\nonumber\\
&&\left . -\sum_{\substack{J,M_J, I,M_I}}\sum_{L,S}~\left(d^{LS}_{JM_J,IM_I;P}\right)^\dag\left(E-\frac{\textbf{P}^2}{4m}-\Delta^{LS}_{JI}+\sum_{n=2}^{\infty} c^{LS}_{JI,n}(E-\frac{\textbf{P}^2}{4m})^{n}\right)d^{LS}_{JM_J,IM_I;P}\right].
\nonumber\\
\label{Skin}
\end{eqnarray}
In order to write the interaction Lagrangian, one should note that, while the total angular momentum, parity, isospin and spin are conserved in a strongly interacting NN process, the orbital angular momentum can change due to the action of tensor forces in nuclear physics. This is easy to implement in this formalism, as the two-nucleon states that are formed, Eq. (\ref{NNstate}), are compatible with the symmetries of the two-nucleon states. The interacting part of the action that does not mix angular momentum states, $S_{int,1}$, can then be written as
\begin{eqnarray}
S_{int,1}&=&-\int\frac{d^4P}{(2\pi)^4}~\frac{d^4k}{(2\pi)^4}
\sum_{\substack{J,M_J, I,M_I}} ~ \sum_{L,M_L,S,M_S}
~{g^{LS}_{JI}}~
\langle JM_J|LM_L,SM_S\rangle
\nonumber\\
&~& \qquad \qquad \times \left[\left(d^{LS}_{JM_J,IM_I;P}\right)^\dag~\sqrt{4\pi}~Y_{LM_L}(\hat{\textbf{k}}^*)~{k}^{*L}~N^T_{k}~\hat{\mathcal{P}}_{(SM_S,IM_I)}~N_{P-k}+h.c. \right],
\nonumber\\
\label{S1}
\end{eqnarray}
where ${g^{LS}_{JI}}$ denotes the coupling of a dimer field to the two-nucleon state with quantum numbers $\{J,I,L,S\}$. Note that the interactions must be azimuthally symmetric and so the reason the couplings are independent of azimuthal quantum numbers. Eqs. (\ref{positive}, \ref{negative}) now guide us to write the most general form of the interacting part of the action that is not diagonal in the angular momentum space, ${S}_{int,2}$, as follows
\begin{eqnarray}
{S}_{int,2}&=&-\int\frac{d^4P}{(2\pi)^4}\frac{d^4k}{(2\pi)^4}
\sum_{\substack{J,M_J, I,M_I}} ~ \sum_{L,M_L,L',M_L',S,M_S}
\nonumber\\
&~& \qquad \qquad {h_{JI}}~ \delta_{I,\frac{1+(-1)^J}{2}}~\delta_{S,1}(\delta_{L,J+1}\delta_{L',J-1}+\delta_{L,J-1}\delta_{L',J+1})\langle JM_J|L'M_L',SM_S\rangle~
\nonumber\\
&~& \qquad \qquad \times \left[\left(d^{LS}_{JM_J,IM_I;P}\right)^\dag~\sqrt{4\pi}~Y_{L'M_{L}'}(\hat{\textbf{k}}^*)~{k}^{*{L'}}~N^T_{k}~\hat{\mathcal{P}}_{(SM_S,IM_I)}~N_{P-{k}}+h.c.\right].
\nonumber\\
\label{S2}
\end{eqnarray}
Note that in this interacting term, spin, isospin and the initial and final angular momenta are all fixed for any given total angular momentum $J$. As a result we have only specified the $(JI)$ quantum numbers corresponding to coupling $h$. As in the scalar case, all the LECs of this effective Lagrangian, $\{\Delta^{LS}_{JI},c^{LS}_{JI,n}, g^{LS}_{JI}, h_{JI}\}$, can be tuned to reproduce the \textit{low-energy} expansion of the scattering amplitudes in the $J^{th}$ angular momentum channel with a given spin and isospin. As discussed in Sec. \ref{sec: Scalar}), in the scalar sector the LECs can be easily determined in terms of the ERE parameters and the renormalization scale. For coupled-channel systems, obtaining the LECs in terms of the scattering parameters requires solving a set of coupled equations. The tuning of the LECs is only an intermediate step in obtaining the relationship between the FV spectrum and the scattering amplitude, which can be easily circumvented by introducing the Bethe-Salpeter kernel.
Let us encapsulate the leading $2\rightarrow2$ transition amplitude between a two-nucleon state with $(JM_J,IM_I,LS)$ quantum numbers and a two-nucleon state with $(JM_J,IM_I,L'S')$ quantum numbers in the Bethe-Salpeter kernel, $K$. Since total angular momentum, spin and isospin are conserved in each $2\rightarrow 2$ transition, the kernel can be fully specified by $K_{JM_J;IM_I}^{(LL';S)}$. Since $J$ is conserved, the full kernel in the space of total angular momentum can be expressed as a block-diagonal matrix. In fact, it is straightforward to see that for each $J$-sector, the corresponding subblock of the full matrix has the following form
\begin{eqnarray}
\left(\begin{array}{cccc}
K_{JM_{J};IM_{I}}^{(J-1,J-1;1)} & 0 & 0 & K_{JM_{J};IM_{I}}^{(J-1,J+1;1)}\\
0 & K_{JM_{J};IM_{I}}^{(J,J;0)} & 0 & 0\\
0 & 0 & K_{JM_{J};I'M_{I'}}^{(J,J;1)} & 0\\
K_{JM_{J};IM_{I}}^{(J+1,J-1;1)} & 0 & 0 & K_{JM_{J};IM_{I}}^{(J+1,J+1;1)}
\end{array}\right).
\label{KernelJ}
\end{eqnarray}
We keep in mind that for any given $J$, $I$, $L$ and $S$, there are $(2J+1)^2\times(2I+1)^2$ elements accounting for different values of $M_J$ and $M_I$ quantum numbers. We also note that the value of the isospin is fixed for each transition kernel. Explicitly, one finds that $I=\frac{1+(-1)^J}{2}$ and $I'=\frac{1+(-1)^{J+1}}{2}$.\footnote{There is no $(I=0,S=0)$ channel for scattering in an even $J$ sector. Also there is no $(I=1,S=0)$ channel for scattering in an odd $J$ sector.} For the special case of $J=0$, the corresponding sub-sector is
\begin{eqnarray}
\left(\begin{array}{cc}
K_{00;1M_I}^{(0,0;0)} & 0\\
0 & K_{00;1M_{I}}^{(1,1;1)}
\end{array}\right).
\label{KernelJ0}
\end{eqnarray}
These kernels, that correspond to leading transitions in all spin and isospin channels, are depicted in Fig. \ref{fig: Kernels}. Although one can read off the Feynman rules corresponding to these kernels from the Lagrangian, Eqs. (\ref{Skin}, \ref{S1}, \ref{S2}), the FV energy eigenvalues can be determined without having to reference to the explicit form of these kernels, as will become evident shortly.
\begin{figure}[t!]
\begin{centering}
\includegraphics[scale=0.415]{kernels}
\par
\caption{{\small The leading $2\rightarrow2$ transition amplitudes in the sector with total angular momentum $J$, Eq. (\ref{KernelJ}). The superscripts in the kernels denote the initial angular momentum, $L$, final angular momentum, $L'$ and the conserved spin of the channels, $S$, respectively. The black dot represents the interaction vertex that conserves the partial wave of the channel, and whose strength is parametrized by the coupling $g^{LS}_{JI}$, Eq. (\ref{S1}). The grey diamond denotes the vertex that mixes partial waves, and whose strength is given by $h_{JI}$, Eq. (\ref{S2}). the double lines are the bare propagators corresponding to a dimer field with angular momentum $L''$.}}\label{fig: Kernels}
\end{centering}
\end{figure}
The scattering amplitude can be calculated by summing up all the $2\rightarrow2$ diagrams which can be obtained by any number of insertions of the transition kernels and the two-particle propagator loops. It can be easily seen that the infinite-volume two-particle loops, $\mathcal{G}^{\infty}$, are diagonal in total angular momentum, spin, isospin and orbital angular momentum. It is easy to show that $\mathcal{G}^{\infty}=2~I^{\infty}$, where $I^{\infty}$ is the infinite-volume loop for two identical bosons, Eq. (\ref{I-infinity}), hence the overall factor of two.
As a result, the scattering amplitude can be expressed as
\begin{eqnarray}
\mathcal{M}^{\infty}=-\mathcal{K}\frac{1}{1-\mathcal{G}^{\infty}\mathcal{K}},
\end{eqnarray}
where $\mathcal{K}$ is a matrix whose $J^{th}$-subblock is given by Eq. (\ref{KernelJ}). Since $\mathcal{G}^{\infty}$ is diagonal, the $J^{th}$-subblock of the infinite-volume scattering amplitude reads
\begin{eqnarray}
\left(\begin{array}{cccc}
\mathcal{M}_{JM_{J};IM_{I}}^{(J-1,J-1;1)} & 0 & 0 & \mathcal{M}_{JM_{J};IM_{I}}^{(J-1,J+1;1)}\\
0 & \mathcal{M}_{JM_{J};IM_{I}}^{(J,J;0)} & 0 & 0\\
0 & 0 & \mathcal{M}_{JM_{J};I'M_{I'}}^{(J,J;1)} & 0\\
\mathcal{M}_{JM_{J};IM_{I}}^{(J+1,J-1;1)} & 0 & 0 & \mathcal{M}_{JM_{J};IM_{I}}^{(J+1,J+1;1)}
\end{array}\right),
\label{amplitude}
\end{eqnarray}
for any non-zero $J$ and
\begin{eqnarray}
\left(\begin{array}{cc}
\mathcal{M}_{00;1M_I}^{(0,0;0)} & 0\\
0 & \mathcal{M}_{00;1M_{I}}^{(1,1;1)}
\end{array}\right),
\label{amplitude}
\end{eqnarray}
for $J=0$. As is conventional, the scattering amplitude in channels with no partial-wave mixing can be parametrized by a scattering phase shift, $\delta_{JI}^{LS}$, according to
\begin{eqnarray}
\mathcal{M}_{JM_{J};IM_{I}}^{(JJ;S)}=\frac{4\pi}{Mk^*}\frac{e^{2i\delta_{JI}^{LS}}-1}{2i}\delta_{L,J}=\frac{4\pi}{Mk^*}\frac{1}{\cot{\delta_{JI}^{LS}}-i}\delta_{L,J},
\label{M-single}
\end{eqnarray}
while in channels where there is a mixing between the partial waves, it can be characterized by two phase shifts and one mixing angle, $\bar{\epsilon}_J$, \cite{Stapp:1956mz}\footnote{We take a different parametrization for the S-matrix when studying the deuteron spectrum for reasons that will be discussed in chapter \ref{chap:deuteron}.}
\begin{eqnarray}
\mathcal{M}_{JM_{J};IM_{I}}^{(J\pm1,J\pm1;S)}=\frac{4\pi}{Mk^*}\frac{\cos{2\bar{\epsilon}_J}e^{2i\delta_{JI}^{LS}}-1}{2i}\delta_{L,J\pm1},
\label{M-coupled1}
\\
\mathcal{M}_{JM_{J};IM_{I}}^{(J\pm1,J\mp1;S)}=\frac{4\pi}{Mk^*}\sin{2\bar{\epsilon}_J}\frac{e^{i(\delta_{JI}^{LS}+\delta_{JI}^{L'S})}}{2}\delta_{L,J\pm1}\delta_{L',J\mp1}.
\label{M-coupled2}
\end{eqnarray}
These relations are independent of $M_J$ and $M_I$ as the scatterings are azimuthally symmetric. We emphasize again that Kronecker deltas used to specify the $L$ quantum numbers should not be confused with the phase shifts. Note that for each $J$ sector, there is only one mixing parameter and as result no further labeling other than the $J$ label is necessary in case of $\bar{\epsilon}_J$.
The FV kernels are equal to the infinite-volume kernels (up to exponentially suppressed terms in volume below the pion production, and in particular the $J^{th}$-subblock of such kernel is given by Eq. (\ref{KernelJ}). As in the scalar case, the only difference between the finite volume and infinite volume shows up in the s-channel bubble diagrams, where the two particles running in the loops can go on shell and give rise to power-law volume corrections. It is straightforward to show that the two-nucleon propagator in the finite volume, $\mathcal{G}^V$, can be written as
\begin{eqnarray}
\mathcal{G}^V=\mathcal{G}^{\infty}+\delta\mathcal{G}^V,
\label{M-infinity}
\end{eqnarray}
where $\delta\mathcal{G}^V$ is a matrix in the $(JM_J,IM_I,LS)$ basis whose matrix elements are given by
\begin{align}
&\left[\delta\mathcal{G}^V\right]_{JM_J,IM_I,LS;J'M_J',I'M_I',L'S'}=\frac{iMk^*}{4\pi}\delta_{II'}\delta_{M_IM_I'}\delta_{SS'}\left[\delta_{JJ'}\delta_{M_JM_J'}\delta_{LL'} +i\sum_{l,m}\frac{(4\pi)^{3/2}}{k^{*l+1}}c_{lm}^{\mathbf{P}}(k^{*2}) \right.
\nonumber\\
& ~~~ \qquad \qquad \qquad ~ \left . \times \sum_{M_L,M_L',M_S}\langle JM_J|LM_L,SM_S\rangle \langle L'M_L',SM_S|J'M_J'\rangle \int d\Omega~Y^*_{L,M_L}Y^*_{l,m}Y_{L',M_L'}\right],
\nonumber\\
\label{deltaG}
\end{align}
and, as is evident, is \emph{neither} diagonal in the $J$-basis nor in the $L$-basis. The kinematic function $c_{lm}^{\mathbf{P}}(k^{*2})$ is defined in Eq. (\ref{clm}) and is evaluated at the on-shell relative momentum of two nucleons in the CM frame. The full FV two-nucleon scattering amplitude can be evaluated by summing up all $2\rightarrow2$ FV diagrams,
\begin{eqnarray}
\mathcal{M}^{V}=-\mathcal{K}\frac{1}{1-\mathcal{G}^{V}\mathcal{K}}=\frac{1}{(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}},
\label{M-V}
\end{eqnarray}
where in the second equality the kernel is eliminated in favor of $\mathcal{M}^{\infty}$ and $\mathcal{G}^{\infty}$ using Eq. (\ref{M-infinity}). The energy eigenvalues of the two-nucleon system arise from the poles of $\mathcal{M}^V$ which satisfy the following determinant condition
\begin{eqnarray}
\det\left[{(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}}\right]=0.
\label{NNQC}
\end{eqnarray}
This quantization condition clearly reduces to Eq. (\ref{FullQCboson}) for two-boson systems when setting $S=0$\footnote{The symmetry factor in both the scattering amplitude and the FV function cancel out in the determinant condition, leaving the FV QC in Eq. \ref{FullQCboson}, insensitive to the distinguishability of the particles.}, and is in agreement with the result of Ref. \cite{Bernard:2008ax} for meson-baryon scattering after setting $S=1/2$. This result also extends the result of Ref. \cite{Ishizuka:2009bx} for two-nucleon systems to moving frames.
Although both $(\mathcal{M}^{\infty})^{-1}$ and $\delta\mathcal{G}^{V}$ are complex, for each partial wave, there occurs an intricate cancellation of their imaginary parts. One could, on the other hand, work with the kernel $\mathcal{K}$ and finite-volume propagator $\mathcal{G}^V$, both manifestly real expressions, to define the QC. From Eq. (\ref{M-V}) it is evident that the finite volume QC can be equivalently written as $\det[\mathcal{K}^{-1}-\mathcal{G}^{V}]=0$,
which has yet no reference to the scattering amplitude. However Eq. (\ref{NNQC}) is comprised of renormalization-scale \emph{independent} quantities, whereas $\mathcal{K}$ and $\mathcal{G}^V$ are both scale \emph{dependent} (only their difference is scale independent). For this reason and to make connection with infinite-volume observables easier, we utilize Eq. \ref{NNQC}. Since the cancellation of imaginary terms is not trivial in channels with partial-wave mixing, we explicitly show how this cancellation occurs at the end of this section.
It is important to note that in deriving this result we have only assumed that the FV kernels are exponentially close to their infinite volume counterparts. Therefore the result obtained is valid for energies up to the inelastic threshold. In the nuclear sector this corresponds to the pion production threshold, $E^*=m_\pi$, which is well above the t-channel cut defined by $E^*_{cut}\equiv m_\pi^2/4m_N$ ($\sim 5$~MeV at the physical point). For energies above $E^*_{cut}$, the dimer formalism written in Eqs.~(\ref{Skin}-\ref{S2}) will get corrections from coupling of nucleons to pions and the LECs appearing in the action will get $m_\pi$-dependent corrections \cite{Weinberg:1990rz, Jenkins:1990jv, Kaplan:1998tg, Kaplan:1998we, Beane:2001bc}. All such corrections due to the dynamical pions above the t-channel cut, including pion exchange diagrams, can be still embedded in the interacting kernels in both infinite volume and finite volume. Below the pion production threshold, these corrections in the FV kernels are still exponentially close to their infinite volume counterparts, making the results presented in this section valid beyond the t-channel cut.
The determinant of the QC is defined in the basis of $(JM_J,IM_I,LS)$ quantum numbers and is over an infinite dimensional matrix. To be practical, this determinant should be truncated in the space of total angular momentum and orbital angular momentum. Such truncation is justified since in the low-momentum limit the scattering phase shift of higher partial waves $L$ scales as $k^{*2L+1}$. In the next section, by truncating the partial waves to $L\leq3$, we unfold this determinant condition further, and present strategies to reduce this master formula to separate QCs for energy eigenvalues in different irreps of the corresponding symmetry group of the two-nucleon system. The first trivial reduction in the QC clearly takes place among different spin/isospin channels. In particular, it is straightforward to see that the QC in Eq. (\ref{NNQC}) does not mix $(S=0,I=1)$, $(S=1,I=0)$, $(S=0,I=0)$ and $(S=1,I=1)$ sectors, and automatically breaks into four independent determinant conditions that correspond to different spin-isospin sectors,
\begin{eqnarray}
\textrm{Det}\left[{(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}}\right]=\prod_{I=0}^{1}\prod_{S=0}^{1} \det \left[(\mathcal{M}^{\infty}_{(I,S)})^{-1}+\delta \mathcal{G}^{V}_{(I,S)} \right]=0 .
\label{NNQC-IS}
\end{eqnarray}
This is due to the fact that each J-sub block of the scattering amplitude matrix can be separated into three independent sectors as following
\begin{align}
&\mathcal{M}^{\infty}_{(I,1)}\equiv\left(\begin{array}{ccc}
\mathcal{M}_{J;I}^{(J-1,J-1;1)} & & \mathcal{M}_{J;I}^{(J-1,J+1;1)}\\
\\
\mathcal{M}_{J;I}^{(J+1,J-1;1)} & & \mathcal{M}_{J;I}^{(J+1,J+1;1)}
\end{array}\right), ~\mathcal{M}^{\infty}_{(I,0)}\equiv\begin{array}{c}
\mathcal{M}_{J;I}^{(J,J;0)}\end{array}, ~\mathcal{M}^{\infty}_{(I',1)}\equiv
\begin{array}{c}
\mathcal{M}_{J;I'}^{(J,J;1)}\end{array},
\nonumber\\
\label{amplitude-IS}
\end{align}
where $I$ and $I'$ are defined after Eq. (\ref{KernelJ}). Since the $M_J$ and $M_I$ indices are being suppressed, one should keep in mind that each block is still a $(2J+1)^2\times(2I+1)^2$ diagonal matrix. If $J$ is even, these amplitudes describe scattering in the negative parity isotriplet, positive parity isotriplet and positive parity isosinglet channels, respectively. For an odd $J$, these amplitudes correspond to scattering in the positive parity isosinglet, negative parity isosinglet and negative parity isotriplet channels, respectively.
Due to the reduced symmetry of the FV, $\delta \mathcal{G}^V$ has off-diagonal terms in the basis of total angular momentum $J$. So although the QC in Eq. (\ref{NNQC}) fully breaks down in the $(I,S)$-basis, it remains coupled in the $(J,L)$-basis. In order to further reduce the determinant conditions in Eq. (\ref{NNQC-IS}), the symmetries of the FV functions must be studied in more detail. This will be the topic of the next section of this chapter.
Before moving on to such symmetry considerations, let us conclude this section by showing that the master QC derived in this section is real as it must be. It can be verified that, e.g. for systems at rest, the only imaginary part of the FV matrix $\delta \mathcal{G}^V$ shows up in the diagonal elements of this matrix.\footnote{This is not always the case as for example, the $\delta \mathcal{G}^V$ matrix for the $\mathbf{d}=(1,1,0)$ boost contains off-diagonal complex elements as well. For all of those case, we have verified that although the elements of the matrix $(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}$ are complex, the determinant of the matrix remains real, see Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}.} So in general, for the angular momentum channels $J$ where there is no coupling between different partial waves, the inverse scattering amplitude matrix has only diagonal elements, whose imaginary part exactly cancels that of the $\delta \mathcal{G}^V$ matrix, see Eq. (\ref{M-single}). Explicitly,
\begin{eqnarray}
\Im [(\mathcal{M}^{LL;S}_{JM_J;IM_I})^{-1}+\delta \mathcal{G}^{V,(LL;S)}_{JM_J,JM_J;IM_I}]=-\frac{iMk^*}{4\pi}+\frac{iMk^*}{4\pi}=0.
\label{Im-diagonal}
\end{eqnarray}
For the angular momentum channels where there are off-diagonal terms due to the partial-wave mixing, one can still write the inverse of the scattering amplitude in that sector, Eqs. (\ref{M-coupled1}, \ref{M-coupled2}), as following
\begin{small}
\begin{align}
& (\mathcal{M}^{LL';1})^{-1}=
\left( \begin{array}{cc}
-\frac{Mk^*}{4\pi}
\frac{\cos{2\epsilon}~\sin({\delta '-\delta})
+\sin({\delta '+\delta})}{\cos({\delta '+\delta})-\cos({\delta '-\delta})\cos({2\epsilon})}-\frac{iMk^*}{4\pi}&
\frac{Mk^*}{2\pi}
\frac{\cos(\epsilon)\sin(\epsilon)}{\cos(\delta '+\delta)-\cos(\delta '-\delta)\cos(2\epsilon)}
\\
\frac{Mk^*}{2\pi}
\frac{\cos({\epsilon})\sin({\epsilon})}{\cos({\delta '+\delta})-\cos({\delta '-\delta})\cos({2\epsilon})}
&
-\frac{Mk^*}{4\pi}
\frac{\cos(2\epsilon)~\sin({\delta-\delta '})
+\sin(\delta '+\delta)}{\cos(\delta '+\delta)-\cos(\delta '-\delta)\cos(2\epsilon)}-\frac{iMk^*}{4\pi}
\\
\end{array} \right),
\nonumber\\
\label{Minverse-coupled}
\end{align}
\end{small}
where $L=J \pm 1$ ($L'=J \mp 1$) and $\delta$ ($\delta '$) denotes the phase shift corresponding to the $L$ ($L'$) partial wave. As is seen the off-diagonal elements of this matrix are real. Given that the FV function $\delta \mathcal{G}^V$ has real off-diagonal terms in this case, these terms in the QC lead to a real off-diagonal element. For the diagonal elements, the imaginary part of the inverse scattering amplitude is isolated and has the same form as the imaginary part of the $\delta \mathcal{G}^V$ matrix, so a similar cancellation as that given in Eq. (\ref{Im-diagonal}) occurs in this case as well.
\section{Symmetry Considerations and Quantization Conditions \label{sec: Reduction}}
Although it is convenient to think of the determinant condition, Eq. (\ref{NNQC}), as a determinant in the $J$ basis, one should expect that for zero CM momentum, this equation splits into $5$ independent QCs corresponding to the $5$ irreps of the cubic group (see table (\ref{groups})). Furthermore, the degeneracy of the energy eigenvalues will reflect the dimension of the corresponding irrep. In general, the FV matrix $\delta \mathcal{G}^{V}$, Eq. (\ref{deltaG}), although being sparse, mixes states corresponding to different irreps of the cubic group. As a result, at least a partial block diagonalization of this matrix is necessary to unfold different irreps that are present due to the decomposition of a given total angular momentum $J$. When the two-particle system is boosted, the symmetry group of the system is no longer cubic, and the reduction of the determinant condition, Eq. (\ref{NNQC}), takes place according to the irreps of the corresponding point group, Table. (\ref{groups}). In the following section, this reduction procedure and the method of block diagonalization will be briefly discussed.
\begin{table} [h]
\begin{centering}
\includegraphics[scale=0.965]{groups}
\caption{{\small The classification of the point groups corresponding to the symmetry groups of the FV calculations for NN systems with the exact isospin symmetry with three selected lowest boost vectors.}\label{groups}}
\par\end{centering}
\end{table}
In order to calculate matrix elements of the FV matrix $\delta \mathcal{G}^{V}$, one can take advantage of the symmetries of the $c_{lm}^{\mathbf{P}}$ functions as defined in Eq. (\ref{clm}). The relations between non-zero $c_{lm}^{\mathbf{P}}$s for any given angular momentum $l$ can be easily deduced from the transformation properties of these functions under symmetry operations of the corresponding point groups
\begin{eqnarray}
c^{\mathbf{P}}_{lm}=\sum_{m'=-l}^{l}\mathcal{D}^{(l)}_{mm'}(R_{\mathcal{X}})~c^{\mathbf{P}}_{lm'},
\label{clm-trans}
\end{eqnarray}
where $R_{\mathcal{X}}$ is the rotation matrix corresponding to each symmetry operation $\mathcal{X}$ of the group, and $\mathcal{D}^{(l)}_{mm'}$ denotes the matrix elements of the Wigner $\mathcal{D}$-matrix \cite{Luscher:1990ux}. Besides these transformations, one can see that $c^\mathbf{P}_{lm}$s are invariant under inversion as can be easily verified from Eq. (\ref{clm}) for an arbitrary boost, and as a result all $c^\mathbf{P}_{lm}$s with an odd $l$ vanish.\footnote{For systems with non-equal masses, this is no longer true when the system is boosted. Since the parity is broken for such systems, even and odd partial waves mix with each other in the QCs, see Refs. \cite{Bour:2011ef, Davoudi:2011md, Fu:2011xz, Leskovec:2012gb}.} Table (\ref{nonzero-clm}) contains all such relations for non-vanishing $c^\mathbf{P}_{lm}$s up to $l=6$ for $\mathbf{d}=(0,0,0)$, $\mathbf{d}=(0,0,1)$ and $\mathbf{d}=(1,1,0)$ boost vectors.\footnote{A closer look at nonrelativistic $c_{lm}^{\mathbf{P}}$ functions shows that all $c_{2,\pm2}^{\mathbf{(1,1,0)}}$ and $c_{4,\pm2}^{\mathbf{(1,1,0)}}$ vanish. This extra symmetry of NR systems with equal masses significantly simplifies the QCs presented in appendix C for boost vector $(1,1,0)$. In this limit, the QCs corresponding to the boost vector $(1,1,1)$ are equivalent to the ones obtained for the system at rest.}
\begin{table} [h]
\label{tab:param3}
\begin{centering}
\begin{tabular}{|c|c|c|}
\hline
\textbf{d}=(0,0,0) & \textbf{d}=(0,0,1) & \textbf{d}=(1,1,0)\tabularnewline
\hline
\hline
$~c_{00}^{P}~$ & $~c_{00}^{P}~$ & $~c_{00}^{P}~$\tabularnewline
$~c_{40}^{P}~$ & $~c_{20}^{P}~$ & $~c_{20}^{P}~$\tabularnewline
$~c_{44}^{P}=c_{4,-4}^{P}=\sqrt{\frac{5}{14}}c_{40}^{P}~$ & $~c_{40}^{P}~$ & $~c_{22}^{P}=-c_{2,-2}^{P}~$\tabularnewline
$~c_{60}^{P}~$ & $~c_{44}^{P}=c_{4,-4}^{P}~$ & $~c_{40}^{P}~$\tabularnewline
$~c_{64}^{P}=c_{6,-4}^{P}=-\sqrt{\frac{7}{2}}c_{60}^{P}~$ & $~c_{60}^{P}~$ & $~c_{42}^{P}=-c_{4,-2}^{P}~$\tabularnewline
& $~c_{64}^{P}=c_{6,-4}^{P}~$ & $~c_{44}^{P}=c_{4,-4}^{P}~$\tabularnewline
& & $~c_{60}^{P}~$\tabularnewline
& & $~c_{62}^{P}=-c_{6,-2}^{P}~$\tabularnewline
& & $~c_{64}^{P}=c_{6,-4}^{P}~$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{The nonzero $c_{lm}^P$s up to $l=6$ for three different boost vectors $\mathbf{d}$ and for two particles with equal masses..}
\label{nonzero-clm}
\end{table}
An important point regarding the $c^\mathbf{P}_{lm}$ functions is that they explicitly depend on the direction of the boost vector. In other words, $c^\mathbf{P}_{lm}$s that correspond to different boost vectors with the same magnitude $|\mathbf{d}|=n$ are not equal. As a result the corresponding set of non-zero $c^\mathbf{P}_{lm}$s as well as the relations among them, for permutations of the components of $(0,0,1)$ and $(1,1,0)$ boost vectors are different from those that are listed in Table (\ref{nonzero-clm}). Although this difference in general results in different $\delta \mathcal{G}^{V}$ matrices, e. g. for $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ boost vectors, as is shown in appendix \ref{app:invariant}, the master equation (\ref{NNQC}) is invariant under a $\mathbf{P}\rightarrow \mathbf{P}'$ transformation when $\mathbf{P}$ and $\mathbf{P}'$ are related by a cubic rotation and $|\mathbf{P}|=|\mathbf{P}'|$. The reason is that there exists a unitary transformation that relates $\delta \mathcal{G}^{V,\mathbf{P}}$ to $\delta \mathcal{G}^{V,\mathbf{P}'}$, leaving the determinant condition invariant. Since the relations among $c^\mathbf{P}_{lm}$ are simpler when one assumes boost vectors that are special with respect to the z-axis, we have presented in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa} the QCs corresponding to $\mathbf{d}=(0,0,1)$ and $\mathbf{d}=(1,1,0)$ boost vectors only. One can still use the deduced QCs to extract the scattering parameters of the NN system from the energy eigenvalues of lattice calculations with other permutations of these boost vectors. It is however crucial to input the boost vectors that are specified in this paper when calculating the $c^\mathbf{P}_{lm}$ functions in the QCs (instead of the boost vectors that are used in the lattice calculation). In order to increase statistics and the precision of results, one should perform the lattice calculation with all possible boost vectors of a given magnitude that belong to the same $A_1$ irrep of the cubic group,\footnote{In higher momentum shells, there occurs multiple $A_1$ irreps of the cubic group. This indicates that there are classes of momentum vector that do not transform into each other via a symmetry operation of the cubic group, e. g. $(2,2,1)$ and $(0,0,3)$ vectors in the $\mathbf{n}^2=9$ shell. However, as is discussed, another property of the $c_{lm}^{\mathbf{P}}$ functions for non-relativistic degenerate masses indicates that the value of the FV function is the same for these two boost vectors as they are both of the form $(2n_1,2n_2,2n_3+1)$ with $n_i \in \mathbb{Z}$.} and use the average energy eigenvalues in the QCs presented to determine the scattering parameters; keeping in mind that $c^\mathbf{P}_{lm}$ functions have to be evaluated at the boost vectors considered in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}.
The other fact that should be pointed out is that due to the symmetries of the $c^\mathbf{P}_{lm}$ function for equal masses, the system at rest with $\mathbf{d}=(0,0,0)$ exhibits the same symmetry transformation as that of the $(2n_1,2n_2,2n_3)$ boost where $n_1,n_2,n_3$ are integers. Similarly, the symmetry group of the calculations with $(0,0,1)$ ($(1,1,0)$) boost is the same as that of $(2n_1,2n_2,2n_3+1)$ ($(2n_1+1,2n_2+1,2n_3)$) boosts. As a result, the quantization conditions presented in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa} can be used with these boost vectors as well. It is worth mentioning that for relativistic two-particle systems with degenerate masses, the above statement is no longer true. This is due to the fact that the boost vector dependence of the relativistic $c_{lm}^{\mathbf{P}}$ function is different from that of the NR counterpart, leading to more distinct point group symmetries for different boosts \cite{Rummukainen:1995vs, Kim:2005gf, Christ:2005gi}.
Back to our main goal, we aim to break the master equations (\ref{NNQC-IS}) into separate QCs corresponding to each irrep of the symmetry group of the problem. In fact, from the transformation law of the $\delta \mathcal{G}^{V}$ function under a symmetry operation of the group,
\begin{align}
&\left[\delta\mathcal{G}^V\right]_{JM_J,LS;J'M_J',L'S}=\sum_{\bar{M}_J=-J}^{J}\sum_{\bar{M}_J'=-J'}^{J'}\mathcal{D}^{(J)}_{M_J,\bar{M}_J}(R_{\mathcal{X}})\left[\delta\mathcal{G}^V\right]_{J\bar{M}_J,LS;J'\bar{M}_J',L'S}\mathcal{D}^{(J')}_{\bar{M}_J',M_J'}(R_{\mathcal{X}}^{-1}),
\nonumber\\
\label{dG-trans}
\end{align}
one can deduce that there is a unitary transformation which brings the matrix $\delta \mathcal{G}^{V}$ to a block-diagonal form. Note that we have suppressed the isospin quantum numbers as $\delta\mathcal{G}^V$ is diagonal in the isospin basis. Each of these blocks then can be identified by a given irrep of the symmetry group of the problem. Such transformation eventually breaks the determinant conditions (\ref{NNQC-IS}) to separate determinant conditions corresponding to each irrep of the point group of the system. Explicitly in each spin and isospin sector,
\begin{eqnarray}
\det \left[(\mathcal{M}^{\infty}_{(I,S)})^{-1}+\delta \mathcal{G}^{V}_{(I,S)} \right]=\prod_{\Gamma^i}\det\left[(\mathcal{M}^{\infty-1}_{(I,S)})_{\Gamma^i}+\delta \mathcal{G}^{V,\Gamma^i}_{(I,S)} \right]^{N(\Gamma^i)}=0 .
\label{NNQC-irrep}
\end{eqnarray}
where $\Gamma^i$ denotes each irrep of the corresponding group and $N(\Gamma^i)$ is the dimensionality of each irrep. The dimensionality of each of these smaller determinant conditions is given by the multiplicity of each irrep in the decomposition of angular momentum channels that are being included in the scattering problem. As is seen in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}, although from the master quantization condition, for some of the NN channels with $J\leq4$ and $l\leq 3$, one has to deal with a determinant of $30 \times 30$ matrices, upon such reduction of the master equation, one arrives at QCs that require taking the determinant of at most $9 \times 9$ matrices. We demonstrate this procedure in more detail for one example and refer the reader to a complete list of 49 deduced QCs that we presented in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}.
\subsection{Reduction procedure for positive-parity isosinglet channel with $\mathbf{P}=\mathbf{0}$ \label{app:red-example}}
\noindent Consider the NN system in the positive parity isosinglet channel where the ground state in the infinite volume is known to be a shallow bound state, the deuteron, whose wave-function is an admixture of both S-wave and D-wave. In order to obtain the phase shifts and mixing parameter in this channel from the energy eigenvalues of the two-nucleon system at rest from a LQCD calculation, one must first construct sources and sinks that transform according to a given irrep of the cubic group, e.g. $T_1$ when $\textbf{P}=0$. The extracted energies then needs to be put in the determinant condition for this channel in the corresponding irrep of the cubic group, Eq. (\ref{NNQC-irrep}), and subsequently solve for the scattering parameters. If one assumes the contributions from scattering channels with $J>4$ and $l\geq4$ to be negligible, the scattering amplitude matrix in the LHS of Eq. (\ref{NNQC-irrep}) can be written as
\begin{eqnarray}
\mathcal{M}^{\infty}_{(0,1)}=\left(\begin{array}{cccc}
\mathcal{M}_{1;0}^{(0,0;1)} & \mathcal{M}_{1;0}^{(0,2;1)} & 0 & 0\\
\mathcal{M}_{1;0}^{(2,0;1)} & \mathcal{M}_{1;0}^{(2,2;1)} & 0 & 0\\
0 & 0 & \mathcal{M}_{2;0}^{(2,2;1)} & 0\\
0 & 0 & 0 & \mathcal{M}_{3;0}^{(2,2;1)}
\end{array}\right),
\end{eqnarray}
where each element, $ \mathcal{M}_{J;I}^{(L,L';S)}$, is a diagonal $(2J+1)^2\times(2I+1)^2$ dimensional matrix. As a result, this is an $18\times18$ matrix which is parametrized by two phase shifts and one mixing angle in the $J=1$ channel, and two D-wave phase shifts in the $J=2$ and $J=3$ channels. Although there is a mixing between D-wave and G-wave channels in the $J=3$ sector, due to the assumption of a negligible $G$-wave scattering, the scattering amplitude in this channel is truncated to the D-wave.
The elements of the FV matrix $\delta \mathcal{G}^V$ in the LHS of Eq. (\ref{NNQC-irrep}) for this channel can be evaluated from Eq. (\ref{deltaG}). The result reads
\begin{eqnarray}
\delta \mathcal{G}^V_{(0,1)}=\left(\begin{array}{cccc}
\delta\mathcal{G}{}_{1,1;0}^{V,(0,0;1)} & \delta\mathcal{G}_{1,1;0}^{V,(0,2;1)} & \delta\mathcal{G}_{12;0}^{V,(0,2;1)} & \delta\mathcal{G}_{1,3;0}^{V,(0,2;1)}\\
\\
\delta\mathcal{G}_{1,1;0}^{V,(2,0;1)} & \delta\mathcal{G}_{1,1;0}^{V,(2,2;1)} & \delta\mathcal{G}_{1,2;0}^{V,(2,2;1)} & \delta\mathcal{G}_{1,3;0}^{V,(2,2;1)}\\
\\
\delta\mathcal{G}_{2,1;0}^{V,(2,0;1)} & \delta\mathcal{G}_{2,1;0}^{V,(2,2;1)} & \delta\mathcal{G}_{2,2;0}^{V,(2,2;1)} & \delta\mathcal{G}_{2,3;0}^{V,(2,2;1)}\\
\\
\delta\mathcal{G}_{3,1;0}^{V,(2,0;1)} & \delta\mathcal{G}_{3,1;0}^{V,(2,2;1)} & \delta\mathcal{G}_{3,2;0}^{V,(2,2;1)} & \delta\mathcal{G}_{3,3;0}^{V,(2,2;1)}
\end{array}\right),
\end{eqnarray}
where each element still represents a matrix $\delta\mathcal{G}_{J,J';I}^{V,(L,L';S)}$
in the $|J,M_J\rangle$ basis and whose explicit forms are as following\footnote{We will drop the superscript $\mathbf{P}$ on the $c_{lm}$s in this example as they are evaluated for $\mathbf{P}=0$.}
\begin{eqnarray}
\delta\mathcal{G}_{1,1;0}^{V,(0,0;1)}&=&\delta\mathcal{G}_{1,1;0}^{V,(2,2;1)}=M(-c_{00}+\frac{i k^*}{4 \pi })~\mathbf{I}_3,
\end{eqnarray}
\begin{eqnarray}
\delta\mathcal{G}_{1,3;0}^{V,(2,2;1)}&=&\left[\delta\mathcal{G}_{3,1;0}^{V,(2,2;1)}\right]^T=\frac{M}{k^{*4}}c_{40}\left(
\begin{array}{ccccccc}
0 & 0 & -\frac{3 }{7} & 0 & 0 & 0 & -\frac{\sqrt{15}}{7} \\
0 & 0 & 0 & \frac{2 \sqrt{6}}{7} & 0 & 0 & 0 \\
-\frac{\sqrt{15}}{7} & 0 & 0 & 0 & -\frac{3}{7} & 0 & 0
\end{array}
\right),
\end{eqnarray}
\begin{eqnarray}
\delta \mathcal{G}^{V,(2,2;1)}_{(2,2;0)}&=& M(-c_{00}+\frac{i k^*}{4 \pi})~\mathbf{I}_5+\frac{M}{k^{*4}}c_{40}\left(\begin{array}{ccccc}
\frac{2 }{21} & 0 & 0 & 0 & \frac{10 }{21} \\
0 &-\frac{8 }{21} & 0 & 0 & 0 \\
0 & 0 & \frac{4 }{7} & 0 & 0 \\
0 & 0 & 0 &-\frac{8 }{21} & 0 \\
\frac{10 }{21} & 0 & 0 & 0 & \frac{2 }{21}
\end{array}\right),
\end{eqnarray}
\begin{eqnarray}
\delta \mathcal{G}^{V,(2,2;1)}_{(2,3;0)}&=&\left[\delta \mathcal{G}^{V,(2,2;1)}_{(3,2;0)}\right]^T=\frac{M}{k^{*4}}c_{40}\left(
\begin{array}{ccccccc}
0 & \frac{5 \sqrt{2}}{21} & 0 & 0 & 0 & \frac{5 \sqrt{2}}{21} & 0 \\
0 & 0 & -\frac{5 \sqrt{5}}{21} & 0 & 0 & 0 & \frac{5}{7 \sqrt{3}} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{5}{7 \sqrt{3}} & 0 & 0 & 0 & \frac{5 \sqrt{5}}{21} & 0 & 0 \\
0 & -\frac{5 \sqrt{2}}{21} & 0 & 0 & 0 & -\frac{5 \sqrt{2}}{21} & 0
\end{array}
\right),
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\delta \mathcal{G}^{V,(2,2;1)}_{(3,3;0)}&=&M(-c_{00}+\frac{i k^*}{4 \pi})~\mathbf{I}_7-
\frac{M}{k^{*4}}c_{40}\left(\begin{array}{ccccccc}
\frac{1}{7} & 0 & 0 & 0 & \frac{\sqrt{\frac{5}{3}}}{7} & 0 & 0 \\
0 & -\frac{1}{3} & 0 & 0 & 0 & \frac{5}{21} & 0 \\
0 & 0 & \frac{1}{21} & 0 & 0 & 0 & \frac{\sqrt{\frac{5}{3}}}{7} \\
0 & 0 & 0 & \frac{2}{7} & 0 & 0 & 0 \\
\frac{\sqrt{\frac{5}{3}}}{7} & 0 & 0 & 0 & \frac{1}{21} & 0 & 0 \\
0 & \frac{5}{21} & 0 & 0 & 0 & -\frac{1}{3} & 0 \\
0 & 0 & \frac{\sqrt{\frac{5}{3}}}{7} & 0 & 0 & 0 & \frac{1}{7}
\end{array}\right),
\nonumber\\
\end{eqnarray}
where $\mathbf{I}_n$ is the $n \times n$ identity matrix, and the rest of the blocks are zero. As is suggested in Ref. \cite{Luu:2011ep}, a unitary matrix, that can bring the $\delta \mathcal{G}^V$ matrix into a block-diagonalized form, can be found by diagonalizing the blocks that are located on the diagonal of the $\delta \mathcal{G}^V$ matrix, $\delta \mathcal{G}^{V,(L,L';1)}_{(J,J;0)}$. Then a unitary matrix can be found easily based on the method described in Ref. \cite{Luu:2011ep} which brings $\delta \mathcal{G}^V$ to a (partially) block-diagonal form. One finds
\begin{eqnarray}
S=\left(\begin{array}{ccc}
S_{11} & 0 & 0\\
0 & S_{22} & 0\\
0 & 0 & S_{33}
\end{array}\right),
\end{eqnarray}
where the zero elements denote subblocks of appropriate dimension with all elements equal to zero, and the nontrivial blocks are the following matrices
\begin{small}
\begin{align}
&S_{11}=\mathbf{I}_6,
~S_{22}=\left(\begin{array}{ccccc}
0 & 0 & 0 & 1 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
-\frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}}
\end{array}\right),
~S_{33}=\left(\begin{array}{ccccccc}
0 & 0 & \sqrt{\frac{3}{8}} & 0 & 0 & 0 & \sqrt{\frac{5}{8}}\\
\sqrt{\frac{5}{8}} & 0 & 0 & 0 & \sqrt{\frac{3}{8}} & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & -\sqrt{\frac{5}{8}} & 0 & 0 & 0 & \sqrt{\frac{3}{8}}\\
0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0\\
-\sqrt{\frac{3}{8}} & 0 & 0 & 0 & \sqrt{\frac{5}{8}} & 0 & 0\\
0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0
\end{array}\right).
\nonumber\\
\end{align}
\end{small}
The resultant (partially) block-diagonalized matrix can then be obtained by,
\begin{small}
\begin{eqnarray}
&& S[(\mathcal{M}^{\infty}_{(0,1)})^{-1}+\delta \mathcal{G}^{V}_{(0,1)}]S^T=
\nonumber\\
&& \left(\begin{array}{cccccccccccccccccc}
x_{1} & 0 & 0 & y_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & x_{1} & 0 & 0 & y_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & x_{1} & 0 & 0 & y_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
y_{1} & 0 & 0 & x_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -y_{2} & 0 & 0 & 0 & 0 & 0 & 0\\
0 & y_{1} & 0 & 0 & x_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{2} & 0 & 0 & 0 & 0\\
0 & 0 & y_{1} & 0 & 0 & x_{2} & 0 & 0 & 0 & 0 & 0 & 0 & -y_{2} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & x_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{3} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{3} & 0 & 0 & 0 & 0 & 0 & 0 & y_{3} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{3} & 0 & 0 & 0 & 0 & 0 & -y_{3} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -y_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{5} & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -y_{2} & 0 & 0 & 0 & 0 & 0 & 0 & x_{5} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & y_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{5} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{3} & 0 & 0 & 0 & 0 & 0 & 0 & x_{6} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -y_{3} & 0 & 0 & 0 & 0 & 0 & x_{6} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & y_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{6} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{7}
\end{array}\right),
\nonumber\\
\label{BD-form}
\end{eqnarray}
\end{small}
where
\begin{small}
\begin{align}
&x_{1}=-M c_{00}+\frac{iMk^*}{4\pi}+\frac{\mathcal{M}_{1;0}^{(2,2;1)}}{\det(\mathcal{M}^{SD})},~
x_{2}=-M c_{00}+\frac{iMk^*}{4\pi}+\frac{\mathcal{M}_{1;0}^{(0,0;1)}}{\det(\mathcal{M}^{SD})},
\nonumber\\
&x_{3}=-M c_{00}-\frac{8}{21}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}+\frac{1}{\mathcal{M}^{(22;1)}_{2;0}}, ~
x_{4}=-M c_{00}+\frac{4}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}+\frac{1}{\mathcal{M}^{(22;1)}_{2;0}},
\nonumber\\
&x_{5}=-M c_{00}-\frac{2}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}+\frac{1}{\mathcal{M}^{(22;1)}_{3;0}},~
x_{6}=-M c_{00}+\frac{2}{21}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}+\frac{1}{\mathcal{M}^{(22;1)}_{3;0}},
\nonumber\\
&x_{7}=-M c_{00}+\frac{4}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}+\frac{1}{\mathcal{M}^{(22;1)}_{3;0}},
y_{1}=-\frac{\mathcal{M}_{1;0}^{(0,2;1)}}{\det(\mathcal{M}^{SD})},
y_{2}=\frac{2\sqrt{6}}{7}\frac{M}{k^{*4}}c_{40},
y_{3}=\frac{10\sqrt{2}}{21}\frac{M}{k^{*4}}c_{40},
\nonumber\\
\end{align}
\end{small}
and $\det(\mathcal{M}^{SD})$ in these relations denotes the determinant of the $J=1$ subblock of the scattering amplitude, $\det(\mathcal{M}^{SD})=\mathcal{M}_{1;0}^{(0,0;1)} \mathcal{M}_{1;0}^{(2,2;1)}- (\mathcal{M}_{1;0}^{(0,2;1)})^2$. This matrix can now clearly be broken to 4 independent blocks corresponding to 4 irreps of the cubic group. The degeneracy of the diagonal elements of this matrix, as well as the coupling between different rows and columns, indicate which irrep of the cubic group each block corresponds to. According to table \ref{irreps}, the one-dimensional irrep $A_2$ only occurs in the decomposition of $J=3$ angular momentum. As is seen from Eq. (\ref{BD-form}), the element $x_7$ belongs to the $J=3$ sector and has a one-fold degeneracy. Also it does not mix with other angular momentum channels, therefore it must correspond to the $A_2$ irrep. So the one-dimensional QC corresponding to the $A_2$ irrep is
\begin{eqnarray}
A_2: ~ \frac{1}{\mathcal{M}^{(22;1)}_{3;0}}-M c_{00}+\frac{4}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}=0.
\label{A2}
\end{eqnarray}
The QC corresponding to the two-dimensional irrep $E$ can be also deduced easily as it only has overlap with the $J=2$ channel. Clearly the element corresponding to this irrep is $x_4$ with two-fold degeneracy and the corresponding QC reads
\begin{eqnarray}
E: ~ \frac{1}{\mathcal{M}^{(22;1)}_{2;0}}-M c_{00}+\frac{4}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}=0.
\label{E}
\end{eqnarray}
The three-dimensional irrep $T_2$ appears in the decomposition of both $J=2$ and $J=3$ angular momentum, and as is seen from Eq. (\ref{BD-form}) mixes the $x_3$, $x_6$ and $y_3$ elements through the following QC
\begin{align}
&T_2: ~ \det\left(\begin{array}{cc}
\frac{1}{\mathcal{M}^{(22;1)}_{2;0}}-M c_{00}-\frac{8}{21}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi} & \frac{10\sqrt{2}}{21}\frac{M}{k^{*4}}c_{40} \\
\frac{10\sqrt{2}}{21}\frac{M}{k^{*4}}c_{40} & \frac{1}{\mathcal{M}^{(22;1)}_{3;0}}-M c_{00}+\frac{2}{21}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}
\end{array}\right)=0.
\nonumber\\
\label{T2}
\end{align}
As is clear, the energy eigenvalues in this irrep have a three-fold degeneracy (there are three copies of this QCs) that is consistent with the dimensionality of the irrep. The remaining irrep is $T_1$ which is a three-dimensional irrep and contribute to both $J=1$ and $J=3$ channels. As there are two $J=1$ sectors corresponding to S-wave and D-wave scatterings, the QC must be the determinant of a $3 \times 3$ matrix. This is in fact the case by looking closely at the (partially) block-diagonalized matrix in Eq. (\ref{BD-form}). One finds explicitly
\begin{small}
\begin{align}
&T_1: \det\left(\begin{array}{ccc}
\frac{\mathcal{M}_{1;0}^{(2,2;1)}}{\det(\mathcal{M}^{SD})}-M c_{00}+\frac{iMk^*}{4\pi} & -\frac{\mathcal{M}_{1;0}^{(0,2;1)}}{\det(\mathcal{M}^{SD})} & 0\\
-\frac{\mathcal{M}_{1;0}^{(0,2;1)}}{\det(\mathcal{M}^{SD})} & \frac{\mathcal{M}_{1;0}^{(0,0;1)}}{\det(\mathcal{M}^{SD})}-M c_{00}+\frac{iMk^*}{4\pi} & -\frac{2\sqrt{6}}{7}\frac{M}{k^{*4}}c_{40}\\
0 & -\frac{2\sqrt{6}}{7}\frac{M}{k^{*4}}c_{40} & \frac{1}{\mathcal{M}^{(22;1)}_{3;0}}-M c_{00}-\frac{2}{7}\frac{M}{k^{*4}}c_{40}+\frac{iMk^*}{4\pi}
\end{array}\right)
\nonumber\\
& \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad ~ =0.
\label{T1}
\end{align}
\end{small}
Again there is a three-fold degeneracy for the energy-eigenvalues as there are three copies of this QC for this irrep. This is an important QC as it gives access to the mixing angle between S and D partial waves.
As is clear now, the QC for $A_2$ irrep, Eq. (\ref{A2}), by its own determines the phase shift in the $J=3$ channel, which can then be used in Eq. (\ref{T1}) for the $T_1$ irrep to determine the phase shifts and mixing angle in the $J=1$ channel. Eq. (\ref{E}) for the $E$ irrep gives access to the phase shift in the $J=2$ channel, and finally Eq. (\ref{T2}) provides another relation for the phase shifts in the $J=2$ and $J=3$ channels. In practice, one needs multiple energy levels in order to be able to reliably extract these parameters from the QCs presented. This is specially a challenging task when it comes to the determination of the scattering parameters in the channels with physical mixing, e. g. S-D mixing, since there are at least three unknown parameters to be determined from the QC, e. g. see Eq. (\ref{T1}). By doing the LQCD calculations of the boosted two-nucleon system, one will attain more energy levels that will correspond to another set of QCs. These QCs then provide a set of equations that the same scattering parameters satisfy, and therefore better constraints can be put on these quantities. These are the set of QCs we have tabulated in Refs. \cite{Briceno:2013lba, Briceno:2013rwa}.
\subsection{The systematic procedure for the reduction of the master quantization condition
\label{red-syst}}
\begin{center}
\begin{table}[h!]
\includegraphics[scale=0.525]{basisfunctions}
\caption{{\small The decomposition of the irreps of the rotational group up to $J=4$ in terms of the irreps of the cubic ($O$), tetragonal ($D_4$) and orthorhombic ($D_2$) groups, see Refs. \cite{Luscher:1990ux, Feng:2004ua, Dresselhaus}. The corresponding basis functions of each irrep are also given in the table in terms of the $SO(3)$ functions $\mathcal{Y}_{lm}$, where $\overline{\mathcal{Y}}_{lm}\equiv\mathcal{Y}_{lm}+\mathcal{Y}_{l-m}$ and $\widetilde{\mathcal{Y}}_{lm}\equiv \mathcal{Y}_{lm}-\mathcal{Y}_{l-m}$. These basis functions become useful in reduction of the full determinant condition, Eq. (\ref{NNQC}) into separate QCs corresponding to each irrep of the point group considered.
}\label{irreps}}
\end{table}
\end{center}
When there are multiple occurrences of a given irrep in each angular momentum $J$ (see table \ref{irreps}), the procedure of block diagonalization becomes more cumbersome, and a systematic procedure must be taken which is based on the knowledge of the basis functions corresponding to each occurrence of any given irrep. Such basis functions for the irreps of the point groups considered in this paper are tabulated in table \ref{irreps}. These basis functions correspond to each occurrence of the irreps in the decomposition of the angular momentum states into the irreps of the $O$, $D_4$ and $D_2$ point groups up to $J=4$. Each J-block (for any given $L$) of the desired unitary matrix $S$ can be constructed from the corresponding eigenvectors of each irrep and will immediately bring the FV matrix $\delta{G}^V$ to a block-diagonalized form. It is however important to keep in mind that the these basis functions are defined with respect to the boost vector of the system (helicity basis), and must be transformed to the $| J,M_J \rangle$ basis by proper rotation matrices as the $\delta \mathcal{G}^V$ and $\mathcal{M}$ matrices in Eqs. (\ref{amplitude}) and (\ref{deltaG}) are given in this latter basis.
For two particles with equal masses in the NR limit, the parity remains a good quantum number even for non-zero CM momentum. This is manifested in vanishing $c_{lm}^{\mathbf{P}}$s with an odd $l$ which ensures no mixing between different parity sectors will occur. This greatly simplifies the reduction of the QCs with a definite point group symmetry. In the next chapter, we encounter a situation where the FV calculation with particular BCs on the fields can suffer from such mixing between different parity sectors. Another example is the boosted two-particle system with non-equal masses (both relativistic and NR) \cite{Davoudi:2011md, Fu:2011xz, Leskovec:2012gb} as discussed in Sec. \ref{Two-body}. This implies that the upcoming FV studies of NN systems with physical values of $u$ and $d$ quark masses will suffer from a more severe FV-induced partial-wave mixings than that of the current calculations with the exact isospin symmetry.
\section*{Summary and Discussions \label{sec: S&M}}
The auxiliary field formalism was extended to arbitrary partial waves in both the scalar and nuclear sectors in infinite and finite volumes. Such a formalism can be used to derive a master equation that relates the FV two-nucleon energies and the scattering parameters of the two-nucleon systems with arbitrary spin, isospin and angular momentum. This master equation, Eq.~(\ref{NNQC}), that is the extension of the L\"uscher formalism to two-nucleon systems, is valid up to inelastic thresholds and is general for any non-relativistic CM boost.
The QC is a determinant over an infinite-dimensional matrix in the basis of angular momentum, and in practice it is necessary to truncate the number of partial waves that contribute to the scattering. By taking advantage of symmetries within the problem, we show how the master equation can be reduced to finite-size blocks that relate particular partial-wave channels (and their mixing) to different spin-isospin channels and different irreps of the corresponding point group of the system. By truncating the matrices at $J\leq 4$ and $l\leq 3$, this procedure requires block diagonalizing matrices as large as $30\times30$. The resulting QCs are determinant conditions involving matrices that are at most $9\times9$, and are therefore practical to be used in future LQCD calculations of NN systems. We have provided one explicit example of this reduction for the scattering in the positive-parity isosinglet channel for zero CM momentum. All other QCs for different CM boosts, parity, isospin, spin, and angular momentum are enumerated in Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}. These form $49$ independent QCs for four different spin and isospin channels giving access to all 16 phase shifts and mixing parameters in these channels. Given the fact that NN-systems couple different partial waves, in order to reliably extract scattering parameters from LQCD calculations, these calculations must be necessarily performed in multiple boosts and various irreps of the corresponding symmetry group.
As is extensively discussed in this chapter, the QCs obtained in this chapter (see Refs. \cite{Briceno:2013lba, Briceno:2013rwa}) can be used for a more general set of boost vectors. Since the symmetry group of the two equal-mass problem depends on the evenness and oddness of the components of the boost vectors, the QCs presented for different irrepsof the corresponding symmetry groups in case of $(0,0,0)$, $(0,0,1)$ and $(1,1,0)$ boosts, can be equally used for the $(2n_1,2n_2,2n_3)$, $(2n_1,2n_2,2n_3+1)$ and $(2n_1+1,2n_2+1,2n_3)$ boost vectors $\forall~n_i\in \mathbb{Z}$.
The other generality of the QCs with regard to the boost vectors is that upon a cubic rotations of the CM boost vectors, the QCs remain unchanged although the FV functions will be different. As a result, one may use an average of the energy levels extracted from the NN correlation functions for all different boosts with the same magnitude, belonging to the same $A_1$ irrep of the cubic group, in the QCs of Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa} to improve the statistics of the calculation. However, it should be noted that in evaluating the FV functions ($c_{lm}^{\mathbf{P}}s$) only the boost vectors considered in this paper must be used. In summary, one can extract the desired scattering parameters of the NN-system by performing the following steps:
\begin{enumerate}
\item For a given irrep $\Gamma$, evaluate the $NN$ correlation function with all possible boost vectors with magnitude ${d}$ that are related to each other via a cubic rotation,
{\small $\{C_{NN}^{\Gamma,\textbf{d}_1},\ldots,C_{NN}^{\Gamma,\textbf{d}_{N_d}}\}$}.
\item Average the value of the correlation functions over all boost vectors used in the previous step, $C_{NN}^{\Gamma,{d}}=\sum_{i}^{N_d}C_{NN}^{\Gamma,\textbf{d}_i}/N_d$.
\item Obtain the non-relativistic finite volume energy, $E_{NR}^\Gamma=E_{NN}^\Gamma-2m_N$, from the asymptotic behavior of the correlation function and therefore obtain the value of the relative momentum $k^*$ from $k^*=\sqrt{M_NE-(\pi\mathbf{d})^2/L^2}$.
\item Determine scattering parameters from the QCs of Refs. \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa}:
\begin{enumerate}
\item Use $\mathbf{d}=(0,0,0)$ if $\mathbf{d}$ is a permutation of $(2n_1,2n_2,2n_3)$,
\item Use $\mathbf{d}=(0,0,1)$ if $\mathbf{d}$ is a permutation of $(2n_1,2n_2,2n_3+1)$,
\item Use $\mathbf{d}=(1,1,0)$ if $\mathbf{d}$ is a permutation of $(2n_1+1,2n_2+1,2n_3)$.
\end{enumerate}
\end{enumerate}
In the following chapter, we will use the QCs derived here, coupled with empirical two-nucleon phase shifts, to provide estimates for the energy levels expected to be seen in future LQCD calculations at the physical pion mass. This will allow for an estimation of the precision that is needed for such calculations specially with regard to the deuteron state. Clearly the impact of this formalism on our understanding of the nature of nuclear forces depends upon best implementing this formalism in the upcoming LQCD calculations of the NN systems.
\chapter{DEUTERON AND ITS PROPERTIES FROM A FINITE-VOLUME FORMALISM}
{\label{chap:deuteron}}
The lightest nucleus, the deuteron, played an important historical role in
understanding the form of the nuclear forces and
the developments that led to the modern phenomenological nuclear potentials,
e.g. Refs.~\cite{Wiringa:1994wb,Machleidt:2000ge}.
While challenging for LQCD calculations, postdicting the properties of the
deuteron, and other light nuclei, is a critical part of the verification of
LQCD technology
that is required in order to trust predictions of quantities
for which there is little or no experimental guidance.
In nature, the deuteron,
with total angular momentum and parity of $J^\pi=1^+$,
is the only bound state of a neutron and proton,
bound by $B_d^{\infty}= 2.224644(34)~{\rm MeV}$.
While predominantly S-wave, the non-central components of the nuclear forces
(the tensor force) induce a D-wave component,
and the $J^\pi=1^+$ two-nucleon (NN) sector
that contains the deuteron
is a
$^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled-channels system.
An important consequence of the nonconservation
of orbital angular momentum
is that the deuteron is not spherical,
and possesses a non-zero quadrupole moment
(the experimentally measured value of the electric quadrupole moment of the
deuteron is $Q_d=0.2859(3)~{\rm fm}^2$~\cite{PhysRevA.20.381}).
The S-matrix for this coupled-channels system
can be parameterized by two phase shifts
and one mixing angle,
with the mixing angle manifesting itself in the
asymptotic $D/S$ ratio of the deuteron wavefunction,
$\eta = 0.02713(6)$~\cite{PhysRev.93.1387, PhysRevC.47.473, deSwart:1995ui}.
A direct calculation of the three scattering
parameters from QCD, at both physical and unphysical light-quark masses,
would provide important insights into the tensor components of the nuclear forces.
Corrections to the binding energy of a bound state, such as
the deuteron, depend exponentially upon the volume,
and are dictated by its size,
and also by the range of the nuclear forces.
With the assumption of a purely S-wave deuteron,
the leading order (LO) volume corrections
have been determined
for a deuteron at rest in a cubic volume of spatial extent $L$ and with the
fields subject to periodic BCs in the spatial directions~\cite{Luscher:1986pf, Luscher:1990ux, Beane:2003da}.
They are found to
scale as ${1\over L}e^{-\kappa_d^{\infty} L}$, where
$\kappa_d^{\infty}$ is the
infinite-volume
deuteron binding momentum
(in the non-relativistic limit, $\kappa_d^{\infty}=\sqrt{M B_d^{\infty}}$, with $M$ being the nucleon mass).
Volume corrections beyond LO have been determined, and extended to
systems that are moving in the volume~\cite{Konig:2011nz, Bour:2011ef,Davoudi:2011md}.
As $\eta$, $Q_d$, and other observables
dictated by the tensor interactions, are small at the physical light-quark masses,
FV analyses of existing LQCD calculations~\cite{Beane:2006mx, Beane:2012vq, Beane:2013br,
Yamazaki:2012hi} using
L\"uscher's method~\cite{Luscher:1986pf, Luscher:1990ux}
have taken the deuteron to be purely S-wave, neglecting the D-wave admixture,
even at unphysical pion masses, introducing a
systematic uncertainty into these
analyses.~\footnote{Recent lattice effective field theory (EFT) calculations include
the effects of higher partial waves and mixing~\cite{Lee:2008fa, Bour:2012hn},
and thus are able to calculate matrix
elements of non-spherical quantities like $Q_d$ up to a given order in the
low-energy EFT, but their FV
analyses treat the deuteron as a S-wave~\cite{Lee:2008fa, Bour:2012hn}.
}
Although the mixing between the S-wave and D-wave is known to be small at the
physical light-quark masses, its contribution to the
calculated FV binding energies must be determined in order to
address this systematic uncertainty.
Further, it is not known if the mixing
between these channels remains small at unphysical quark masses.
As the central and tensor components of the nuclear forces have
different forms, their contribution to the FV effects will, in
general, differ.
The contributions from the tensor interactions
are found to be relatively enhanced for certain CM boosts in modest
volumes due to the reduced spatial symmetry of the system.
Most importantly,
extracting the S-D mixing angle at the deuteron binding energy,
in addition to the S-wave scattering parameters,
requires a complete coupled-channels analysis of the FV
spectrum.
In this chapter, we utilize the formalism presented in the previous chapter for NN systems with arbitrary CM momenta, spin, angular momentum and isospin, to explore how the S-D mixing angle
at the deuteron binding energy, along with the binding energy itself,
can be optimally extracted from LQCD calculations performed in cubic
volumes with fields subject to
periodic BCs (PBCs) in the spatial directions.
Using the phase shifts and mixing angles generated by phenomenological
NN potentials that are fit to NN scattering data~\cite{NIJMEGEN},
the expected FV energy spectra in the
positive-parity isoscalar channels
are determined at the physical pion mass (we assume exact
isospin symmetry throughout).
It is found that correlation functions of
boosted NN systems will play a key role in extracting the S-D mixing angle in future LQCD calculations.
The FV energy shifts of the ground state of different irreps of the symmetry groups
associated with momenta $\mathbf{P}=\frac{2\pi}L(0,0,1)$
and $\frac{2\pi}L(1,1,0)$,
are found to have enhanced sensitivity to the mixing angle in modest volumes
and to
depend both on its magnitude and sign.
A feature of the FV spectra,
with practical implications for future LQCD calculations,
is that the contribution
to the energy splittings from channels with $J>1$,
made possible by the reduced symmetry of the volume,
are negligible for $L\gtrsim 10~\text{fm}$ as the phase shifts in
those channels are small at low energies.
As the generation of multiple ensembles of
gauge-field configurations at the physical light-quark masses
will require significant computational resources on capability-computing platforms,
we have investigated the viability of precision determinations of the deuteron
binding energy and scattering parameters
from one lattice volume using
the
six bound-state energies
associated with
CM momenta $|\mathbf{P}|\leq\frac{2\pi}L\sqrt{3}$.
We have also considered extracting the asymptotic D/S ratio from the
behavior of the deuteron FV wavefunction and its
relation to the S-D mixing angle.
\section{Deuteron and the Finite-volume Spectrum
\label{sec:DeutFV}
}
\noindent
The spectra of energy eigenvalues of two nucleons in the isoscalar
channel with positive parity in a cubic volume subject to PBCs are dictated by
the S-matrix elements in this sector, including those defining the
$^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled channels that contain the deuteron, as discussed in detail in the previous chapter. For the deuteron at rest as well as boosts vectors $(0,0,1)$, $(1,1,0)$ and $(1,1,1)$, the required QCs for the NN system in the positive-parity isoscalar channel are given in Appendix~\ref{app: QC} \cite{Briceno:2013lba, BDLsupp, Briceno:2013rwa, Briceno:2013bda}.
Although the ultimate goal is to utilize these QCs in the analysis of the NN
spectra extracted from LQCD calculations,
they can be used, in combination with the experimental NN scattering data,
to predict the FV spectra at the physical light-quark masses,
providing important guidance for future LQCD calculations.
While for scattering states,
the phase shifts and mixing angle from
phenomenological analyses of the experimental data~\cite{PhysRevC.48.792,
PhysRevC.49.2950, PhysRevC.54.2851, PhysRevC.54.2869}
can be used in the QCs,
for bound states, however, it is necessary to use fit functions
of the correct form to be continued to negative energies.
Here we choose a different parametrization for the $J=1$ S-matrix than what we used in chapter \ref{chap:NN}. It is the Blatt-Biedenharn (BB) parameterization~\cite{Blatt:1952zza,PhysRev.93.1387}
\begin{eqnarray}
S_{(J=1)}=\left( \begin{array}{cc}
\cos\epsilon_1&-\sin\epsilon_1\\
\sin\epsilon_1&\cos\epsilon_1\\
\end{array} \right)
\left( \begin{array}{cc}
e^{2i\delta_{1\alpha}}&0\\
0&e^{2i\delta_{1\beta}}\\
\end{array} \right)
\left( \begin{array}{cc}
\cos\epsilon_1&\sin\epsilon_1\\
-\sin\epsilon_1&\cos\epsilon_1\\
\end{array} \right),
\label{eq:BBSmatrix}
\end{eqnarray}
\begin{figure}[b!]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.135]{swavefit}}
\subfigure[]{
\includegraphics[scale=0.135]{epsilonfit}}
\subfigure[]{
\includegraphics[scale=0.135]{dwavefit}}
\subfigure[]{
\includegraphics[scale=0.135]{swaveextrap}}
\subfigure[]{
\includegraphics[scale=0.135]{epsilonextrap}}
\subfigure[]{
\includegraphics[scale=0.135]{dwaveextrap}}
\caption{
{\small Fits to the experimental values of
a) the $\alpha$-wave phase shift,
b) the mixing angle and
c) the $J=1$ $\beta$-wave and $J=2,3$ D-wave phase shifts
(in degrees), in the Blatt-Biedenharn (BB) parameterization~\cite{Blatt:1952zza},
as a function of
momentum of each nucleon in the CM frame, $k^*$,
based on six different phase shifts analyses~\protect\cite{PhysRevC.48.792,
PhysRevC.49.2950, PhysRevC.54.2851, PhysRevC.54.2869}.
d) The $\alpha$-wave phase shift,
e) the mixing angle and
f) the $J=1$ $\beta$-wave and $J=2,3$ D-wave phase shifts
(in degrees) as a function of $\kappa=-ik^*$
obtained from the fit functions.
}}
\label{fig:Fits}
\end{center}
\end{figure}
whose mixing angle, $\epsilon_1$, when evaluated at the deuteron binding
energy, is directly related to the asymptotic D/S
ratio in the deuteron wavefunction.
$\delta_{1\alpha}$ and $\delta_{1\beta}$ are the scattering phase shifts
corresponding to two eigenstates of the S-matrix;
the so called ``$\alpha$'' and ``$\beta$'' waves respectively.
At low energies, the $\alpha$-wave is predominantly S-wave with a small
admixture of the D-wave,
while the $\beta$-wave is predominantly D-wave with a small admixture of the
S-wave.
The location of the deuteron pole is determined
by one condition
on the $\alpha$-wave phase shift, $\cot \delta_{1\alpha}|_{k^*=i\kappa}=i$.
In addition, $\epsilon_1$ in this parameterization is an analytic function of energy near
the deuteron pole
(in contrast with $\bar{\epsilon}$ in the barred parameterization
\cite{Stapp:1956mz}).
With a truncation of $l_{max}=2$ imposed upon the scattering amplitude matrix,
the scattering parameters required for the analysis of the FV spectra
are $\delta_{1\alpha}$, $\delta_{1\beta}$, $\delta^{(^3D_2)}$,
$\delta^{(^3D_3)}$ and $\epsilon_1$.
Fits to six different phase-shift analyses
(PWA93~\cite{PhysRevC.48.792}, Nijm93~\cite{PhysRevC.49.2950}, Nijm1~\cite{PhysRevC.49.2950},
Nijm2~\cite{PhysRevC.49.2950}, Reid93~\cite{PhysRevC.49.2950} and
ESC96~\cite{PhysRevC.54.2851, PhysRevC.54.2869})
obtained from Ref.~\cite{NIJMEGEN} are shown in
Fig.~\ref{fig:Fits}(a-c).~\footnote{
The $\alpha$-wave was fit by a pole term and a
polynomial, while the other parameters were fit with polynomials alone.
The order of the polynomial for each parameter was determined by the goodness
of fit to phenomenological model data below the t-channel cut.
}
In order to obtain the scattering parameters at negative energies,
the fit functions are
continued to imaginary momenta, $k^*\rightarrow i\kappa$.
Fig.~\ref{fig:Fits}(d-f) shows the phase shifts and the mixing angle as a
function of $\kappa$
below the t-channel cut
(which approximately corresponds to the
positive-energy fitting range).
$\epsilon_1$ is observed to be
positive for positive energies, and becomes
negative when continued to negative energies
(see Fig.~\ref{fig:Fits}).
The slight difference between phenomenological models
gives rise to a small ``uncertainty band'' for each of the parameters.
For the NN system at rest in the
positive-parity isoscalar channel,
the only irrep of the cubic group that
has overlap with the $J=1$ sector is $\mathbb{T}_1$,
which also has overlap with the $J=3$ and higher channels.
Using the scattering parameters of the $J=1$ and $J=3$
channels, the nine lowest $\mathbb{T}_1$ energy levels (including the
bound-state level)
are shown in Fig.~\ref{T1spec} as a function of
$L$. In the limit that $\epsilon_1$ vanishes, the $\mathbb{T}_1$ QC,
given in Eq.~(\ref{I000T1}), can be written as a product of two
independent QCs.
One of these QCs depends only on
$\delta_{1\alpha}\rightarrow \delta^{(^3S_1)}$, while the other depends
on $\delta_{1\beta}\rightarrow \delta^{(^3D_1)}$ and $\delta^{(^3D_3)}$.
By comparing the $\mathbb{T}_1$ spectrum with that
obtained for $\epsilon_1=0$, the $\mathbb{T}_1$
states can be classified as predominantly S-wave
or predominantly D-wave states.
The dimensionless quantity
$\tilde{k}^2=ME^*{L}^2/4\pi^2$ as a
function of volume is shown in Fig. \ref{T1qtilde},
from which it is clear that the predominantly
D-wave energy levels remain close to
the non-interacting energies, corresponding to
$\tilde{k}^2=1,2,3,4,5,6,8,\ldots$,
consistent
with the fact that both the mixing angle and the D-wave phase shifts are
small at low energies, as seen in Fig.~\ref{fig:Fits}.
The states that are predominantly
S-wave are negatively shifted in energy
compared with the non-interacting states due to the attraction of the NN interactions.
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\label{T1spec}
\includegraphics[scale=0.215]{tonespec}}
\subfigure[]{
\label{T1qtilde}
\includegraphics[scale=0.2175]{toneqtilde}}
\caption
{{\small a)
The nine lowest energy eigenvalues satisfying the QC for the
$\mathbb{T}_1$-irrep of the cubic group,
Eq.~(\ref{I000T1}), and
b) the dimensionless quantity $\tilde{k}^2=ME^*{L}^2/4\pi^2$ as a
function of $L$.
The blue-solid lines correspond to states that are predominantly S-wave,
while the red-dashed lines represent states that are predominantly D-wave.
The black-dashed line shows the infinite-volume binding energy of the
deuteron.}
}
\label{T1specfull}
\end{center}
\end{figure}
Focusing on the deuteron, it is important to quantify the effect of the
mixing between the S-wave and D-wave on the energy of the deuteron in a finite volume.
The upper panel of Fig. \ref{deut_cub-I} provides
a closer look at the binding energy of the deuteron as function of $L$
extracted from the $\mathbb{T}_1$ QC given in Eq.~(\ref{I000T1}).
While in larger volumes the uncertainties in the predictions due to the fits to experimental data
are a few keV,
in smaller volumes the uncertainties increase
because the fit functions are valid only below the t-channel cut
and are not expected to describe the data above the cut.
It is interesting to examine the difference between the bound-state energy
obtained from the full $\mathbb{T}_1$ QC
and that obtained with $\epsilon_1=0$,
$\delta
{\rm E}^{*(\mathbb{T}_1)}={\rm E}^{*(\mathbb{T}_1)}-{\rm E}^{*(\mathbb{T}_1)}(\epsilon_1=0)$.
This quantity, shown in the lower panel of Fig.~\ref{deut_cub-I}, does not
exceed a few $\rm{keV}$ in smaller volumes, $L \lesssim9~\rm{fm}$,
and is significantly smaller in larger volumes,
demonstrating that the spectrum of $\mathbb{T}_1$ irrep is quite insensitive
to the small mixing angle
in the $^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled channels.
Therefore, a determination of the mixing angle from the spectrum of two nucleons at rest
will be challenging for LQCD calculations.
The spectra in the $\mathbb{A}_2/\mathbb{E}$ irreps of the trigonal group for
$\mathbf{d}=(1,1,1)$
exhibit the same feature, as shown in Fig. \ref{deut_trig}.
By investigating the QCs in Eqs.~(\ref{I000T1}, \ref{I111A2}, \ref{I111E}),
it is straightforward to show that the difference between the bound-state
energy
extracted from the full QCs
(including physical and FV-induced mixing between S-waves and D-waves)
and from the uncoupled QC is proportional to ${\sin^2 \epsilon_1}$,
and is further suppressed by FV corrections and the
small $\beta$-wave and D-wave
phase shifts.
\begin{figure}[ht!]
\begin{center}
\subfigure[]{
\label{deut_cub-I}
\includegraphics[scale=0.210]{deutcub}}
\subfigure[]{
\label{deut_trig}
\includegraphics[scale=0.2]{deuttrig}}
\caption{
{\small a) The upper panel shows the energy of two nucleons at rest in the
positive-parity isoscalar channel as a function of $L$
extracted from the $\mathbb{T}_1$ QC given in Eq.~(\ref{I000T1}).
The uncertainty band is
associated with fits to different phenomenological analyses of the
experimental data,
and the dashed line denotes the infinite-volume deuteron binding
energy.
The lower panel shows the contribution of the mixing angle to the
energy,
$\delta {\rm E}^{*(\mathbb{T}_1)}= {\rm E}^{*(\mathbb{T}_1)}-{\rm E}^{*(\mathbb{T}_1)}(\epsilon_1=0)$.
b) The same quantities as in (a) for the NN
system with
${\bf d}=(1,1,1)$ obtained from the $\mathbb{A}_2/\mathbb{E}$ QCs, Eqs.~(\ref{I111A2},
\ref{I111E}).
}}
\label{deut_cub}
\end{center}
\end{figure}
The boost vectors $\mathbf{d}=(0,0,1)$ and $(1,1,0)$
distinguish the $z$-axis from the
other two axes, and result in an asymmetric volume as viewed
in the rest frame of the deuteron.
In terms of the periodic images of the deuteron,
images that are located in the $z$-direction with opposite signs compared with the
images in the $x$- and $y$-directions~\cite{Bour:2011ef,Davoudi:2011md}
result in the quadrupole-type shape
modifications to the
deuteron, as will be elaborated on in Sec.~\ref{sec:wavefunction}.
As a result, the energy of the deuteron, as well as its
shape-related quantities such as its quadrupole moment,
will be affected more by the finite extent of the volume (compared with the
systems with $\mathbf{d}=(0,0,0)$ and $(1,1,1)$).
\begin{figure}[ht!]
\begin{center}
\includegraphics[scale=0.30]{deuttet}
\caption{{\small The energy of two nucleons in the
positive-parity
isoscalar channel with $\mathbf{d}=(0,0,1)$ as a function of
$L$,
extracted from the $\mathbb{A}_2$ (blue) and $\mathbb{E}$ (red) QCs given in
Eq.~(\ref{I001A2}) and Eqs.~(\ref{I001E}), respectively.
The systematic uncertainties associated with fitting
different phenomenological analyses of the experimental data are included.
}}
\label{deut_tet}
\end{center}
\end{figure}
As is clear from the QCs for $\mathbf{d}=(0,0,1)$ systems, given in Eqs.~(\ref{I001A2}, \ref{I001E}),
there are two
irreps of the tetragonal group, $\mathbb{A}_2$
and $\mathbb{E}$, that have overlap with the $J=1$ channel.
These irreps represent states with $M_J=0$ for the $\mathbb{A}_2$
irrep and $M_J=\pm 1$ for the $\mathbb{E}$ irreps,
see Table~\ref{irreps-deuteron}.
The bound-state energies of these two irreps are shown in Fig.~\ref{deut_tet}
as a function of $L$.
For comparison, the energy of the bound state with ${\bf d}=(0,0,1)$
in the limit of vanishing mixing angle and D-wave phase shifts is also shown
(black-solid curve) in Fig. \ref{deut_tet}.
The energy of the bound states obtained in both the $\mathbb{A}_2$ irrep
(blue-solid curve) and the $\mathbb{E}$ irrep (red-solid curve)
deviate substantially from the energy of the purely S-wave bound state for modest
volumes, $L\lesssim14~\rm{fm}$.~\footnote{LQCD calculations at the physical
pion mass require volumes with $L \gtrsim 9~\rm{fm}$ so that the
systematic uncertainties associated with
the finite range of the nuclear forces are below the percent level.}
These deviations are such that the energy gap between the systems in the two
irreps is
$\sim 80\%$ of the
infinite-volume deuteron binding energy
at $L=8~\rm{fm}$, decreasing to $\sim 5\%$ for $L=14~\rm{fm}$.
This gap is largely due to the mixing between S-wave and D-wave in
the infinite volume, as verified by evaluating the bound-state energy in the
$\mathbb{A}_2$ and $\mathbb{E}$ irrep in the limit where $\epsilon_1=0$
(the blue and red dotted curves in Fig. \ref{deut_tet}, respectively.)
Another feature of the $\textbf{d}=(0,0,1)$ FV bound-state energy is that
the contribution from the
$\beta$-wave and D-wave
states cannot be neglected for $L\lesssim 10~\rm{fm}$.
The blue (red) dashed curve in Fig.~\ref{deut_tet}
results from the $\mathbb{A}_2$ ($\mathbb{E}$) QC in this limit.
The D-wave states in the $J=2$ and $J=3$ channels mix with the $J=1$ $\alpha$-
and $\beta$-waves due to the reduced symmetry of the system,
and as a result they, and the $\beta$-wave state, contribute to the energy of the
predominantly S-wave bound state in the finite volume.
\begin{table}[t!]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
J &$O$&${D}_{4}$&${D}_{2}$& ${D}_{3}$
\\\hline\hline
1
&$\mathbb{T}_1:(\mathcal{Y}_{11},\mathcal{Y}_{10},\mathcal{Y}_{1-1})$
&$\mathbb{A}_2:\mathcal{Y}_{10}$
&$\mathbb{B}_1:\mathcal{Y}_{10}$
& $\mathbb{A}_2:\mathcal{Y}_{10}$ \\
&
&$\mathbb{E}:\left(\overline{\mathcal{Y}}_{11},\widetilde{\mathcal{Y}}_{11}\right)$
&$\mathbb{B}_2:\overline{\mathcal{Y}}_{11},~\mathbb{B}_3:\widetilde{\mathcal{Y}}_{11}$
& $\mathbb{E}:\left(\overline{\mathcal{Y}}_{11},\widetilde{\mathcal{Y}}_{11}\right)$ \\
\hline
\end{tabular}
\caption{
{\small Decomposition of the $J=1$ irrep of the rotational group in terms of
the irreps of the cubic ($O$),
tetragonal ($D_4$), orthorhombic ($D_2$) and trigonal ($D_3$) groups, see
Refs. \cite{Luscher:1990ux, Feng:2004ua, Dresselhaus}.
The corresponding basis functions of each irrep are also shown
in terms of the SO(3) functions
$\mathcal{Y}_{lm}$,
where
$\overline{\mathcal{Y}}_{lm}\equiv \mathcal{Y}_{lm}+\mathcal{Y}_{l-m}$ and
$\widetilde{\mathcal{Y}}_{lm}\equiv \mathcal{Y}_{lm}-\mathcal{Y}_{l-m}$.
}}
\label{irreps-deuteron}
\par\end{centering}
\end{table}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.30]{deutort}
\caption{
The energy of two nucleons in the
positive-parity
isoscalar channel with $\mathbf{d}=(1,1,0)$ as a function of
$L$,
extracted from the $\mathbb{B}_1$ (red) and $\mathbb{B}_2/\mathbb{B}_3$ (blue)
QCs given in Eqs.~(\ref{I110B1}) and (\ref{I110B2}, \ref{I110B3}),
respectively.
The systematic uncertainties associated with fitting to different
phenomenological analyses of the experimental
data are included.
}
\label{deut_ort}
\end{center}
\end{figure}
The FV energy eigenvalues
for the NN system in the positive-parity
isoscalar channel with ${\bf d}=(1,1,0)$
can be obtained from QCs in
Eqs.~(\ref{I110B1}) and (\ref{I110B2}, \ref{I110B3}), corresponding to
$\mathbb{B}_1$ and $\mathbb{B}_2/\mathbb{B}_3$ irreps of
the orthorhombic group, respectively.
These irreps represent states with $M_J^{\prime}=0$ for the $\mathbb{B}_1$ irrep and
$M_J^{\prime}=\pm 1$ for the $\mathbb{B}_2/\mathbb{B}_3$ irreps (where $M_J^\prime$ now is the projection of total angular momentum
along the twist direction),
see Table~\ref{irreps}.
The bound-state energies of these systems are shown in
Fig.~\ref{deut_ort}, and are found to
deviate noticeably from the purely S-wave limit (black-solid curve in
Fig.~\ref{deut_ort}),
however the deviation is not as large as the
case of ${\bf d}=(0,0,1)$.
The energy gap between the systems in the two irreps is $\sim 30\%$ of the
infinite-volume deuteron binding energy
at $L=8~\rm{fm}$, decreasing to $\sim 5\%$ for $L=14~\rm{fm}$.
Eliminating the $\beta$-wave and $J=2,3$ D-wave interactions, leads to the dashed curves in
Fig.~\ref{deut_ort}, indicating the
negligible effect that they have on the bound-state energy in
these irreps.
To understand the large FV energy shifts from the purely $\alpha$-wave
estimates for ${\bf d}=(0,0,1)$ and $(1,1,0)$ systems compared
with $(0,0,0)$ and $(1,1,1)$ systems,
it is instructive to examine the QCs given in Appendix~\ref{app: QC} in the limit
where the $\beta$-wave and D-wave phase shifts vanish.
This is a reasonable approximation for volumes with $L \gtrsim 10~\rm{fm}$, as
illustrated in Fig.~\ref{deut_tet} and Fig.~ \ref{deut_ort}.
It is straightforward to show that in this limit,
the QC of the system with $\mathbf{d}=(0,0,0)$ reduces to a purely $\alpha$-wave condition
\begin{align}
&\mathbb{T}_1:\hspace{.1cm}k^*\cot\delta_{1\alpha}-4 \pi
c_{00}^{(0,0,0)}(k^{*2}; L)=0
.
\label{appr-T1}
\end{align}
The QCs for a system with $\mathbf{d}=(0,0,1)$ are
\begin{align}
&\mathbb{A}_2:\hspace{.1cm}
k^*\cot\delta_{1\alpha}
-4 \pi c_{00}^{(0,0,1)}(k^{*2}; L)
\ =\
-{1\over\sqrt{5}}
\frac{4\pi}{k^{*2}}\
c_{20}^{(0,0,1)}(k^{*2}; L)
\
(\sqrt{2}\sin2\epsilon_1-\sin^2\epsilon_1)
,
\\
&\mathbb{E}:\hspace{.1cm}
k^* \cot \delta_{1\alpha}
-4 \pi c_{00}^{(0,0,1)}(k^{*2}; L)
\ =\
+{1\over 2\sqrt{5}}
\frac{4\pi}{k^{*2}}\
c_{20}^{(0,0,1)}(k^{*2}; L)
\
(\sqrt{2}\sin2\epsilon_1-\sin^2\epsilon_1)
,
\label{appr-E}
\end{align}
which include corrections to the $\alpha$-wave limit that scale with $\sin \epsilon_1$
at LO.
This is the origin of the large deviations of these energy eigenvalues from the purely S-wave values.
The same feature is seen in the systems with $\mathbf{d}=(1,1,0)$, where the QCs reduce to
\begin{align}
&\mathbb{B}_1:
k^*\cot\delta_{1\alpha}
-4 \pi c_{00}^{(1,1,0)}(k^{*2}; L)
\ =\
-{1\over\sqrt{5}}
\frac{4\pi}{k^{*2}}\
c_{20}^{(1,1,0)}(k^{*2}; L)
\
(\sqrt{2}\sin2\epsilon_1-\sin^2\epsilon_1)
,
\\
&\mathbb{B}_2/\mathbb{B}_3:
k^* \cot \delta_{1\alpha}
-4 \pi c_{00}^{(1,1,0)}(k^{*2}; L)
\ =\
+{1\over 2\sqrt{5}}
\frac{4\pi}{k^{*2}}\
c_{20}^{(1,1,0)}(k^{*2}; L)
\
(\sqrt{2}\sin2\epsilon_1-\sin^2\epsilon_1)
.
\label{appr-B2}
\end{align}
Similarly, the QC with $\mathbf{d}=(1,1,1)$ in this limit is
\begin{align}
&\mathbb{A}_2/\mathbb{E}:\hspace{.1cm}k^*\cot\delta_{1\alpha}-4 \pi
c_{00}^{(1,1,1)}(k^{*2}; L)=0
.
\label{appr-EA2}
\end{align}
The LO corrections to the QCs
in Eqs.~(\ref{appr-T1}-\ref{appr-EA2})
are not only suppressed by
the
$J=1$ $\beta$-wave and $J=2,3$ D-wave
phase shifts, but also by FV corrections
that are further exponentially suppressed compared with the leading FV corrections.
It is straightforward to show that the leading neglected terms in the QCs
presented above are
$\sim \frac{1}Le^{-2\kappa L}\tan{\delta_{1\beta}}$ and
$\frac{1}Le^{-2\kappa L}\tan{\delta_{D_{J=2,3}}}$,
while the
FV contributions to the approximate relations given in
Eqs.~(\ref{appr-T1}-\ref{appr-EA2}) are $\sim {1\over L}e^{-\kappa L}$.
In Appendix \ref{app: clm}, the explicit volume dependence of $c^{\mathbf{d}}_{LM}$ functions
are given
for the case of $k^{*2}=-\kappa^2<0$.
These explicit forms are useful in obtaining the leading exponential corrections to the
QCs.
We emphasize that the smaller volumes considered have $\kappa L =2-2.5$,
and therefore it is not a good approximation to replace the $c^{\mathbf{d}}_{LM}$
functions with their leading exponential terms,
and the complete form of these functions should be used in analyzing the FV spectra.
\begin{figure}[!ht]
\begin{center}
\subfigure[]{
\label{spin-ave-001}
\includegraphics[scale=0.20]{deuttetave}}
\subfigure[]{
\label{spin-ave-110}
\includegraphics[scale=0.20]{deutortave}}
\caption{
{\small a) The dotted curve shows the $M_J^{\prime}$-averaged quantity
$\frac{1}{3}(E^{*(\mathbb{A}_2)}+2E^{*(\mathbb{E})})$
as function of $L$, while the solid curves show the energy of the state
in the $\mathbb{A}_2$ (blue) and $\mathbb{E}$ (red) irreps of the tetragonal
group,
as well as that of the state with $\epsilon_1=0$ (black).
b) The dotted curve shows the $M_J^{\prime}$-averaged quantity
$\frac{1}{3}(E^{*(\mathbb{B}_1)}+E^{*(\mathbb{B}_2)}+E^{*(\mathbb{B}_3)})$
as function of $L$, while the solid curves show the energy of the state
in the $\mathbb{B}_1$ (red) and $\mathbb{B}_2/\mathbb{B}_3$ (blue) irreps of
the orthorhombic group,
as well as that of the state with $\epsilon_1=0$ (black).}}
\label{spin-ave}
\end{center}
\end{figure}
In the limit of vanishing
$J=1$ $\beta$-wave and $J=2,3$ D-wave
phase shifts,
the QCs show that the energy shift of each pair of irreps of the systems
with $\mathbf{d}=(0,0,1)$ and $(1,1,0)$ differ in sign.
It is also the case that the $M_J^{\prime}$-averaged energies are approximately the
same as the purely S-wave case.
In fact, as illustrated in Fig. \ref{spin-ave-001}, the energy level
corresponding to
$\frac{1}{3}(E^{*(\mathbb{A}_2)}+2E^{*(\mathbb{E})})$ quickly
converges to the S-wave energy with ${\bf d}=(0,0,1)$.
Similarly, the $M_J^{\prime}$-averaged quantity
$\frac{1}{3}(E^{*(\mathbb{B}_1)}+E^{*(\mathbb{B}_2)}+E^{*(\mathbb{B}_3)})$
almost coincides with the S-wave state with ${\bf d}=(1,1,0)$,
Fig. \ref{spin-ave-110}.
This is to be expected, as $M_J^{\prime}$-averaging is equivalent to averaging
over the orientations of the image systems, suppressing the anisotropy induced by
the boost phases in the FV corrections, Eqs.~(\ref{c00-exp})-(\ref{c40-exp}).
These expressions also demonstrate that, unlike the case of degenerate, scalar
coupled-channels systems \cite{Berkowitz:2012xq,Oset:2012bf},
the NN spectra (with spin degrees of freedom) depend on the sign of
$\epsilon_1$.
Of course, this sensitivity to the sign of $\epsilon_1$ can be deduced
from the full QCs in Eqs.~(\ref{I000T1}-\ref{I111E}).
Upon fixing the phase convention of the angular momentum states,
both the magnitude and sign of the mixing angle can be extracted from
FV calculations,
as will be discussed in more detail in Section~\ref{sec:extraction}.
\section{Extracting the Scattering Parameters From Synthetic Data
\label{sec:extraction}
}
\noindent
Given the features of the energy spectra associated with different boosts,
it is interesting to consider how well the scattering parameters
can be extracted from future LQCD calculations at the physical pion mass.
With the truncations we have imposed,
the full QCs for the FV states that have overlap with the
$^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$
coupled channels depend on four scattering phase shifts and the $J=1$ mixing angle.
As discussed in Sec.~\ref{sec:DeutFV}, for bound states these are equivalent to
QCs that depend solely on $\delta_{1\alpha}$
and $\epsilon_1$ up to corrections of
$\sim \frac{1}Le^{-2\kappa L}\tan{\delta_{1\beta}}$ and
$\frac{1}Le^{-2\kappa L}\tan{\delta_{D_{J=2,3}}}$,
as given in Eqs.~(\ref{appr-T1}-\ref{appr-EA2}).
By considering the boosts with $|\textbf{d}|\leq\sqrt{3}$, six
independent bound-state energies that asymptote to
the physical deuteron energy can be obtained.
For a single volume, these give six different constraints on
$\delta_{1\alpha}$ and $\epsilon_1$ for energies in the
vicinity of the deuteron pole.
Therefore, by parameterizing the momentum dependence of these two
parameters, and requiring them
to simultaneously satisfy Eqs.~(\ref{appr-T1}-\ref{appr-EA2}),
their low-energy behavior can be extracted.
Using the fact that the $\alpha$-wave is dominantly S-wave with $\epsilon_1$
and $\delta^{(^3 \hskip -0.03in D _1)}$
small, we use the effective range expansion (ERE) of the inverse
S-wave scattering amplitude,
which is valid below the t-channel cut,
to parameterize~\cite{PhysRev.93.1387}
\begin{eqnarray}
\label{eq:ERE}
k^*\cot\delta_{1\alpha}&=&-\frac{1}{a^{(^3S_1)}}+\frac{1}{2}r^{(^3S_1)}k^{*2}+ \dots
,
\\
\epsilon_1&=&h_1~k^{*2}+ \dots~
.
\end{eqnarray}
Therefore, up to $\mathcal{O}(k^{*2})$,
the three parameters, denoted by $a^{(^3S_1)}$, $r^{(^3S_1)}$ and $h_1$,
can be over-constrained by the bound-state spectra in
a single volume.
To illustrate this point, we fit the six independent energies
to ``synthetic data''
using the approximated QCs, Eqs.~(\ref{appr-T1}-\ref{appr-EA2}).
The precision with which $\{a^{(^3S_1)},r^{(^3S_1)},h_1\}$ can be extracted
depends on the precision and correlation of the energies
determined in LQCD calculations.
Therefore, we consider four possible scenarios,
corresponding to the energies being extracted
from a given LQCD calculation
with $1\%$ and $10\%$ precision, and with uncertainties that are
uncorrelated or fully correlated with each other.
It is likely that the energies of these irreps will be determined in
LQCD calculations on the same ensembles of gauge-field configurations,
and consequently they are likely to
be highly correlated - a feature that has been
exploited
extensively in the past when determining energy differences.
\begin{figure}[t!]
\begin{center}
\subfigure[]{
\label{a_corr}
\includegraphics[scale=0.175]{acorr}}
\subfigure[]{
\label{a_uncorr}
\includegraphics[scale=0.175]{auncorr}}
\subfigure[]{
\label{r_corr}
\includegraphics[scale=0.175]{rcorr}}
\subfigure[]{
\label{r_uncorr}
\includegraphics[scale=0.175]{runcorr}}
\caption{
{\small The values of $\{a^{(^3S_1)}, r^{(^3S_1)}\}$ obtained by
fitting the six independent bound-state energies with $|\textbf{d}|\leq\sqrt{3}$
(depicted in Figs.~\ref{deut_cub}, \ref{deut_tet}, \ref{deut_ort}),
generated from synthetic LQCD calculations,
using the approximate QCs in Eqs.~(\ref{appr-T1}-\ref{appr-EA2}), as discussed
in the text.
The black lines denote
the experimental value of these quantities determined by fitting the scattering
parameters obtained from Ref.~\cite{NIJMEGEN}.
The dark (light) inner (outer) band is the
$1\sigma$ band corresponding to the energies being determined with
1\% (10\%) precision.
}}
\label{fig:fakedata1}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\subfigure[]{
\label{Bd_corr}
\includegraphics[scale=0.175]{bdcorr}}
\subfigure[]{
\label{Bd_uncorr}
\includegraphics[scale=0.175]{bduncorr}}
\subfigure[]{
\label{e1_corr}
\includegraphics[scale=0.175]{eonecorr}}
\subfigure[]{
\label{e1_uncorr}
\includegraphics[scale=0.175]{eoneuncorr}}
\caption{
{\small The values of
$\{-B_d^{\infty},\epsilon_1(\mathit{i}\kappa_d^{\infty})\}$ obtained by fitting the
six independent
bound-state energies with $|\textbf{d}|\leq\sqrt{3}$ (depicted in
Figs.~\ref{deut_cub}, \ref{deut_tet}, \ref{deut_ort}),
generated from synthetic LQCD calculations,
using the approximate QCs in
Eqs.~(\ref{appr-T1}-\ref{appr-EA2}).
$\epsilon_1$ is in degrees and ${B_d^\infty}(\kappa_d^\infty)$
denotes the infinite-volume deuteron binding energy (momentum).
The black lines denote
the experimental value of these quantities determined by fitting the scattering
parameters obtained from Ref.~\cite{NIJMEGEN}.
The dark (light) inner (outer) band is the
$1\sigma$ band corresponding to the energies being determined with
1\% (10\%) precision.
}}
\label{fig:fakedata2}
\end{center}
\end{figure}
Using the QCs,
the ground-state energy in each irrep is determined for a given lattice volume.
The level of precision of such a future LQCD calculation is introduced by selecting a
modified energy for each ground state from a Gaussian distribution with the
true energy for its mean and the precision level multiplied by the mean
for its standard deviation.
This generates one set of uncorrelated ``synthetic LQCD calculations''.
To generate fully correlated ``synthetic LQCD calculations'', the same
fluctuation (appropriately scaled)
is chosen for each energy.~\footnote{Partially-correlated
``synthetic LQCD calculations''
can be generated by forming a weighted average of the uncorrelated and
fully-correlated calculations.}
These synthetic data are then taken to be the results of a possible future LQCD
calculation and analyzed accordingly to extract the scattering parameters \footnote{A similar analysis has been carried out in Ref. \cite{Beane:2010em} where the S-wave scattering length, effective range and the deuteron binding energy are extracted from ``synthetic LQCD calculations'', but using a purely S-wave quantization condition.}.
The values of $\{a^{(^3S_1)},
r^{(^3S_1)},-B_d^{\infty},\epsilon_1(\mathit{i}\kappa_d^{\infty})\}$
extracted from an analysis of the synthetic data
are shown in Figs.~\ref{fig:fakedata1},~\ref{fig:fakedata2} for both
correlated and uncorrelated energies.
Since for $L\lesssim 10~\rm{fm}$ the contribution of the D-wave phase shifts to the
bound-state spectrum is not negligible, the mean values of the scattering
parameters extracted using the approximated
QCs deviate from their experimental values.
This is most noticeable when the binding energies are determined at the 1\%
level of precision,
where the S-matrix parameters and predicted $B_d^\infty$ can deviate by
$\sim 3\sigma$ from the experimental values for this range of volumes.
For $10~\rm{fm}$$ < L < $$14~\rm{fm}$, one can see that these quantities can be extracted
with high accuracy using this method,
but it is important to note that the precision with which $\{a^{(^3S_1)},
r^{(^3S_1)},\epsilon_1(\mathit{i}\kappa_d^{\infty)}\}$
can be extracted decreases as a function of increasing volume.
The reason is that the bound-state energy in each irrep asymptotes to the physical deuteron binding energy
in the infinite-volume limit.
In this limit, sensitivity to $\epsilon_1$ is lost and the $\alpha$-wave phase shift
is determined at a single energy, the deuteron pole.
Therefore, for sufficiently large volumes one cannot independently resolve $a^{(^3S_1)}$
and $r^{(^3S_1)}$.
This analysis of synthetic data reinforces the fact that the FV spectrum not
only depends on the magnitude of $\epsilon_1$ but also its sign.
As discussed in Sect.~\ref{sec:DeutFV}, this sensitivity can be deduced
from the full QCs in Eqs.(\ref{I000T1}-\ref{I111E}),
but it is most evident from the approximated
QCs in Eqs.~(\ref{appr-T1}-\ref{appr-EA2}).
In performing this analysis, we have benefited from two important pieces of
{\it apriori} knowledge at the physical light-quark masses.
First is that in the volumes of interest, the bound-state energy in each
irrep falls within the radius of convergence of the
ERE, $|{\rm E}^*|<m_\pi^2/4M$.
For unphysical light-quark masses, the S-matrix elements could in principle
change in such a way that this need not be the case
and pionful EFTs would be required
to extract the scattering parameters from the FV spectrum.
Second is that the D-wave phase shifts are naturally small.
Again, since the dependence of these phase shifts on the light-quark masses
can only be estimated, further investigation would be required.
To improve upon this analysis,
the J=1 $\beta$-wave and $J=2,3$ D-wave phase shifts
would have to be extracted from the scattering states.
As is evident from Fig.~\ref{T1specfull}, states that have a strong dependence
on the D-wave phase shifts will, in general,
lie above the t-channel cut.
In principle, one could attempt to extract them by fitting the FV bound-state
energies for $L\leq 10~{\rm fm}$ with the full QCs.
In practice, this will be
challenging as eight scattering parameters appear in the ERE at
the order at which the
$J=1$ $\beta$-wave and $J=2,3$ D-wave
phase shifts first contribute.
This is also formally problematic since for small volumes, $m_\pi L \raisebox{-0.7ex}{$\stackrel{\textstyle <}{\sim}$ }
2\pi $,
finite range effects are no longer negligible.
Although these finite range effects have been estimated for two nucleons in a S-wave~\cite{Sato:2007ms},
they remain to be examined for the general NN system.
\section{The Finite-volume Deuteron Wavefunction and the Asymptotic D/S
Ratio \label{sec:wavefunction}}
\noindent
The S-matrix dictates the asymptotic behavior of the NN wavefunction,
and as a result the IR distortions of the wavefunction inflicted by
the boundaries of the lattice volume have a direct connection to the parameters
of the scattering matrix, as exploited by L\"uscher.
Outside the range of the nuclear forces, the FV wavefunction of the NN system
is obtained from the solution of the
Helmholtz equation in a cubic volume with the
PBCs~\cite{Luscher:1986pf,Luscher:1990ux, Rummukainen:1995vs, Ishizuka:2009bx}.
By choosing the amplitude of the $l=0$ and $l=2$ components of the
FV wavefunction to recover the asymptotic D/S ratio of the infinite-volume
deuteron,
it is straightforward to show~\cite{Luscher:1986pf,Luscher:1990ux}
that the unnormalized FV deuteron wavefunctions
associated with the approximate QCs in Eqs.~(\ref{appr-T1}-\ref{appr-EA2})
are
\begin{eqnarray}
\psi^{V,{\bf d}}_{1,M_J} (\mathbf{r};\kappa)
\ =\
\psi^{\infty}_{1,M_J} (\mathbf{r};\kappa)
\ +\
\sum_{\mathbf{n} \neq \mathbf{0}} e^{i \pi \mathbf{n} \cdot \mathbf{d}} \
\psi^{\infty}_{1,M_J} (\mathbf{r}+\mathbf{n}L;\kappa)
,
\label{psi-V}
\end{eqnarray}
with $r = |{\bf r}|>R$,
where $\mathbf{r}$ denotes the relative displacement
of the two nucleons,
and $R >> 1/m_\pi$ is the approximate range of the nuclear interactions.
The subscripts on the wavefunction refer to the $J=1,~M_J=0,\pm1$
quantum numbers of the state and
$\mathbf{n}$ is an integer triplet.
In order for Eq.~(\ref{psi-V}) to be an energy eigenstate of the Hamiltonian,
$E^*=-{\kappa^2}/{M}$ has to be an energy eigenvalue of the NN system in
the finite volume,
obtained from the QCs in Eqs.~(\ref{appr-T1}-\ref{appr-EA2}).
$\psi^{\infty}_{1,M_J}(\mathbf{r})$ is the asymptotic infinite-volume wavefunction of the deuteron,
\begin{eqnarray}
\psi^{\infty}_{1,M_J} (\mathbf{r};\kappa)
\ =\
\mathcal{A}_S
\ \left(\
\frac{e^{-\kappa r}}{r}
\mathcal{Y}_{1M_J;01}(\hat{\mathbf{r}})
\ +\
\eta~\frac{e^{-\kappa r}}{r} (1+\frac{3}{\kappa
r}+\frac{3}{\kappa^2r^2})
\mathcal{Y}_{1M_J;21}(\hat{\mathbf{r}})
\ \right)
.
\label{psi-inf}
\end{eqnarray}
with $\mathcal{Y}_{JM_J;L1}$ being the well-known spin-orbital functions,
\begin{eqnarray}
\mathcal{Y}_{JM_J;L1}(\hat{\mathbf{r}})
\ =\
\sum_{M_L,M_S}\left\langle L M_L 1 M_S|J M_J\right\rangle \
Y_{L M_L}(\hat{\mathbf{r}})\ \mathcal{\chi}_{1 M_S}
,
\label{Y-def}
\end{eqnarray}
where $\mathcal{\chi}_{1 M_S}$ is the spin wavefunction of the deuteron.
$\eta$ is the deuteron asymptotic D/S ratio which is
related to the mixing angle via
$\eta=-\tan{\epsilon_1}|_{k^*=i\kappa^\infty_d}$~\cite{Blatt:1952zza}.
As is well known from the effective range theory~\cite{PhysRev.76.38,
PhysRev.77.647},
the short-distance contribution to the \textit{outer} quantities of the
deuteron,
such as the quadrupole moment,
can be approximately taken into account by requiring the
normalization of the asymptotic wavefunction of the deuteron,
obtained from the residue of the S-matrix at the
deuteron pole,
to be approximately
$|\mathcal{A}_S|^2\approx{2\kappa}/({1-\kappa r^{(^3S_1)}})$.
Corrections to this normalization arise at
$\mathcal{O}\left({\kappa^3}/{R}^2,\kappa\eta^2\right)$,
at the same order the
$J=1$ $\beta$-wave and $J=2,3$ D-waves
contribute.
In writing the FV wavefunction in Eq.~(\ref{psi-V})
contributions from these
waves have been neglected.
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\label{WF-T1-L10}
\includegraphics[scale=0.215]{toneleqtendeutdp}}
\subfigure[]{
\label{WF-T1-L15}
\includegraphics[scale=0.215]{toneleqfifteendeutdp}}
\subfigure[]{
\label{WF-T1-20}
\includegraphics[scale=0.215]{toneleqtwentydeutdp}}
\subfigure[]{
\label{WF-T1-30}
\includegraphics[scale=0.215]{toneleqthirtydeutdp}}
\caption{{\small The mass density in the $xz$-plane from the $\mathbb{T}_1$ FV deuteron wavefunction at rest
for
$L=10,15,20$, and $30~{\rm fm}$.
}}
\label{WF-T1}
\end{center}
\end{figure}
An important feature of the FV wavefunction in Eq. (\ref{psi-V}) is the
contribution from partial waves
other than $l=0$ and $l=2$,
which results from the cubic distribution of the periodic images.
While there are also FV corrections to the $l=0$ component of the wavefunction,
the FV corrections to the $l=2$ component are enhanced for systems with
${\bf d}=(0,0,1)$ and $(1,1,0)$.
By forming appropriate linear combinations of the $\psi^{V,{\bf d}}_{1,M_J}$
that transform according to a given irrep of the cubic, tetragonal,
orthorhombic and trigonal point groups (see Table \ref{irreps}),
wavefunctions for the systems with ${\bf d}=(0,0,0)$, $(0,0,1)$, $(1,1,0)$ and
$(1,1,1)$
can be obtained.
The
mass density in the $xz$-plane from
the FV wavefunction of the deuteron
at rest in the volume, obtained from the $\mathbb{T}_1$ irrep of the
cubic group
is shown in Fig.~\ref{WF-T1}
for $L=10,15,20$, and $30~{\rm fm}$,
and for the boosted systems in Figs.~\ref{WF-A2}-\ref{WF-A2E} of Appendix \ref{sec:Wavefunc}.
As the interior region of the wavefunctions cannot be deduced from its asymptotic
behavior alone, it is ``masked''
in Fig.~\ref{WF-T1} and Figs.~\ref{WF-A2}-\ref{WF-A2E} by a
shaded disk.
Although the deuteron wavefunction exhibits its slight prolate shape
(with respect to its spin axis)
at large volumes,
it is substantially deformed in smaller volumes,
such that the deuteron can no longer be thought as a compact bound state within
the lattice volume.
When the system is at rest, the FV deuteron
is more prolate than the infinite-volume deuteron.
When the deuteron is boosted along the $z$-axis with ${\bf d}=(0,0,1)$,
the distortion of the wavefunction
is large,
and in fact, for a significant
range of volumes ($L \lesssim 30~\rm{fm}$), the FV effects
give rise to an oblate (as opposed to prolate) deuteron in the
$\mathbb{E}$ irrep,
Fig. \ref{WF-E}, and a more prolate shape in the
$\mathbb{A}_2$ irrep,
Fig. \ref{WF-A2}.
For ${\bf d}=(1,1,0)$,
the system remains prolate for the deuteron in the
$\mathbb{B}_2/\mathbb{B}_3$ irreps,
Fig. \ref{WF-B2B3},
while it becomes
oblate in the $\mathbb{B}_1$ irrep, Fig. \ref{WF-B1}, for volumes up to $L \sim
30~\rm{fm}$.
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\label{NDS-T1}
\includegraphics[scale=0.215]{ndstone}}
\subfigure[]{
\label{NDS-E}
\includegraphics[scale=0.215]{ndse}}
\subfigure[]{
\label{NDS-B2B3}
\includegraphics[scale=0.215]{ndsatwo}}
\subfigure[]{
\label{NDS-A2E}
\includegraphics[scale=0.215]{ndsbar}}
\caption{
{\small The normalized D/S ratio of the deuteron wavefunction with $M_J=M_S=1$, defined in
Eq.~(\protect\ref{ratio}) and Eq.~(\protect\ref{ratiob})
in the
a) $\mathbb{T}_1$,
b) $\mathbb{E}$ and
c) with $M_J=M_S=0$ in the $\mathbb{A}_2$ irrep,
along with
d) the difference of the D/S ratios in the $\mathbb{E}$ and $\mathbb{A}_2$ irreps, defined
in Eq.~(\protect\ref{eq:A2Ediff}).
The red-dashed lines show
the infinite-volume value.}}
\label{fig:NDS}
\end{center}
\end{figure}
Although the normalization factor $\mathcal{A}_S$ corrects for
the fact that the complete wavefunction is not given by the asymptotic form
given in Eq.~(\ref{psi-inf}) for $|{\bf r}|\lesssim r^{(^3S_1)}/2$ in infinite volume, it gives rise to a
normalization ambiguity in the FV.
On the other hand, the asymptotic D/S ratio is protected by the
S-matrix, and
can be directly extracted from the long-distance tail of the lattice
wavefunctions.~\footnote{
The energy-dependent ``potentials'' generated by HALQCD
and used to compute scattering parameters, including $\epsilon_1$ (at
unphysical light-quark masses)~\cite{Murano:2013xxa},
are expected to reproduce the predictions of QCD only at the energy eigenvalues of
their LQCD calculations.
Hence, if they had found a bound deuteron,
their prediction for $\epsilon_1$ would be expected to be correct at the
calculated deuteron binding energy.
}
It is evident from Eq.~(\ref{psi-inf}) that the ratio
\begin{eqnarray}
N_{D/S}^{{\bf d}; M_J, M_S}(r;\kappa) \equiv
\frac{\psi_{D;M_J,M_S}^{V,{\bf d}}(r;\kappa)}{\eta~\chi(r;\kappa)~\psi_{S;M_J,M_S}^{V,{\bf d}}(r;\kappa)}
,
\label{ratio}
\end{eqnarray}
with $\chi(r;\kappa)=\sqrt{\frac{1}{10}}(1+\frac{3}{\kappa
r}+\frac{3}{\kappa^2r^2})$,
is unity for the $M_J=M_S=1$ component of the infinite-volume deuteron wavefunction
(and is equal to $-2$ for the $M_J=M_S=0$ component),
where
$\psi_{S;M_J,M_S}^{V,{\bf d}}$ and $\psi_{D;M_J,M_S}^{V,{\bf d}}$
are
\begin{eqnarray}
\psi_{S;M_J,M_S}^{V,{\bf d}}(r;\kappa)
& = &
\int d\Omega_{\hat{\mathbf{r}}} \
\psi^{V,{\bf d}}_{1,M_J}(\mathbf{r},\kappa)
\big|_{M_S}\
\ Y_{00}(\hat{\mathbf{r}})
,
\nonumber\\
\psi_{D;M_J,M_S}^{V,{\bf d}}(r;\kappa)
& = &
\int d\Omega_{\hat{\mathbf{r}}}\
\psi^{V,{\bf d}}_{1,M_J}(\mathbf{r},\kappa) \big|_{M_S}\
\ Y_{20}(\hat{\mathbf{r}})
,
\label{ratiob}
\end{eqnarray}
with $r\le L/2$.
By evaluating the FV wavefunction
in different irreps with $|\mathbf{d}|\leq \sqrt{3}$,
this ratio can be determined in the FV, as is shown in
Fig.~\ref{fig:NDS}.
Not only does it exhibit strong dependence on the volume,
but also varies dramatically as a function of $r$.
This is due to the fact that the periodic images give rise to exponentially
growing contributions
to the FV wavefunction in $r$.
For the FV deuteron at rest and with $\mathbf{d}=(1,1,1)$,
a sufficiently small $r$ gives rise to a $N_{D/S}^{{\bf d};M_J,M_S}$ that is not severely
distorted by volume effects even in small volumes.
In contrast,
this ratio deviates significantly from its infinite-volume value for systems
with $\mathbf{d}=(0,0,1)$ and $(1,1,0)$ even in large volumes ($L \lesssim
20~\rm{fm}$).
This feature is understood by noting that
while the leading correction to $N_{D/S}^{{\bf d};M_J,M_S}$
is $\sim \eta~e^{-\kappa L}$ for systems with ${\bf d}=(0,0,0)$ and
$(1,1,1)$,
they are $\sim e^{-\kappa L}$ for systems
with ${\bf d}=(0,0,1)$ and $(1,1,0)$.
The periodic images
of the wavefunction with
the latter boosts are quadrupole distributed, and consequently modify the
$l=2$ component of the
wavefunction by contributions that are not suppressed by $\eta$.
However, for these systems, there are two irreps that receive similar FV
corrections to their ratios,
which can be largely removed
by forming differences,
e.g. for the system with ${\bf d}=(0,0,1)$,
\begin{eqnarray}
\overline{N}_{D/S}^{(0,0,1)}
& = &
{1\over 3}\left(
N_{D/S}^{(0,0,1);1,1}
\ -\
N_{D/S}^{(0,0,1);0,0}
\right)
\ =\
{1\over 3}\left(
N_{D/S}^{(0,0,1);\mathbb{E}}
\ -\
N_{D/S}^{(0,0,1);\mathbb{A}_2}
\right)
,
\label{eq:A2Ediff}
\end{eqnarray}
as shown in
Fig.~\ref{fig:NDS}. A similar improvement is found for systems with
${\bf d}=(1,1,0)$.
It is also worth noting that the contributions to the wavefunction from higher
partial waves, $l\geq2$, can be added to $\psi^{V,{\bf d}}$ with
coefficients that depend on their corresponding phase shifts, and therefore are
small under the assumption of low-energy scattering \cite{Luscher:1990ux,
Rummukainen:1995vs, Ishizuka:2009bx}.
However, the partial-wave decomposition of the FV wavefunction in Eq.~(\ref{psi-V})
contains contributions with $l\geq2$.
In the limit where the corresponding phase shifts vanish, the wavefunction, in contrast to the spectrum,
remains sensitive to these contributions,
resulting in the larger FV modifications of
quantities compared with their spectral analogues.
An extraction of $\eta$ is possible by taking sufficiently large volumes such that a large NN
separation can be
achieved without approaching the boundaries of the volume.
While
the wavefunctions corresponding to the deuteron at rest or with
$\mathbf{d}=(1,1,1)$
provide an opportunity to extract $\eta$
with an accuracy of $\sim 15-20\%$ in volumes of $L\sim 14~\rm{fm}$,
combinations of the ratios obtained from the two irreps
in both the systems with $\mathbf{d}=(0,0,1)$ and $(1,1,0)$
will provide for a $\sim 10\%$ determination in volumes of $L\sim 12~\rm{fm}$,
as shown in Fig. \ref{fig:NDS}.
As it is possible that the uncertainties in the
extraction of $\eta$ can be systematically reduced,
those due to the neglect of the
$J=1$ $\beta$-wave and $J=2,3$ D-wave
phase shifts, as well as higher order terms in the ERE,
deserve further investigation.
\section*{Summary and Conclusion
\label{sec:conclusion}}
\noindent
A Lattice QCD calculation of the deuteron and its properties would be a
theoretical milestone on the path toward calculating quantities of importance in low-energy nuclear physics
from quantum chromodynamics without uncontrolled approximations or assumptions.
While there is no formal impediment to calculating the deuteron binding energy
to arbitrary precision when sufficient computational resources become
available, determining its properties and interactions presents a
challenge that has largely remained unexplored~\cite{Detmold:2004qn,Meyer:2012wk,Briceno:2013lba}.
Using the NN formalism developed in Ref.~\cite{Briceno:2013lba},
we have explored the
FV energy
spectra of states that
have an overlap with the $^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled-channels system in which
the deuteron resides.
Although the full FV QCs associated with the
$^3S_1$-$^3D_1$ coupled channels depend on interactions in all
positive-parity isoscalar channels,
a low-energy expansion
depends only on four scattering
phases and one mixing angle.
Further, for the deuteron, these truncated QCs can be further simplified to
depend only upon one phase shift and the mixing angle,
with corrections suppressed by
$\sim\frac{1}Le^{-2\kappa L} \tan{\delta_i}$
where $\delta_i$ denotes the
$J=1$ $\beta$-wave and $J=2,3$ D-wave
phase shifts
which are all small at the deuteron binding energy.
We have demonstrated that the infinite-volume deuteron binding energy and
leading scattering parameters,
including the mixing angle, $\epsilon_1$,
that dictate the low-energy behavior of the scattering amplitudes,
can be (in principle) determined with precision
from the bound-state spectra of deuterons, both at rest and in motion,
in a single modest volume, with $L=10$-$14~\rm{fm}$.
Calculations in a second lattice volume would reduce
the systematic uncertainties introduced by truncating the QCs.
We have investigated the feasibility of extracting $\epsilon_1$ from the
asymptotic D/S ratio of the deuteron FV wavefunction using the
periodic images associated with the $\alpha$-wavefunction.
As the amplitude of the $J=1$ $\beta$-wave and the $J=2,3$ D-wave components of
the wavefunction are not constrained by the infinite-volume deuteron wavefunction,
the analysis is limited by an imposed truncation of the ERE,
which is at the same level of approximation as the approximate QCs.
The systematic uncertainties introduced by this
truncation
are currently unknown, but will be
suppressed by the small phase shifts in those channels in addition to being
exponentially suppressed with $L$.
This is in contrast to the extraction from the FV spectra where the
systematic uncertainties have been determined to be small.
With this approximation, it is estimated that
volumes with $L \raisebox{-0.7ex}{$\stackrel{\textstyle >}{\sim}$ } 12~\rm{fm}$ are required to extract $\epsilon_1$
with $\sim 10\%$ level of accuracy from the asymptotic form of the
wavefunctions.
\chapter{FINITE-VOLUME FORMALISM WITH TWISTED BOUNDARY CONDITIONS}
{\label{chap:TBC}}
LQCD calculations are commonly performed with PBCs imposed upon the quark fields in the spatial directions,
constraining the quark momentum modes in the volume to satisfy
$\mathbf{p}=\frac{2\pi}{L}\mathbf{n}$ with $\mathbf{n}$ being an integer triplet.
PBCs are a subset of a larger class of BCs called twisted BCs (TBCs).
TBCs~\cite{PhysRevLett.7.46} are those that require the quark fields to acquire
a phase $\theta$ at the boundary,
$\psi(\mathbf{x}+\mathbf{n}L )=e^{i{\rm{\theta}} \cdot \mathbf{n}}\psi(\mathbf{x})$,
where $0<\theta_i<2\pi$ is the twist angle in the $i^{\rm th}$ Cartesian direction.
Bedaque~\cite{Bedaque:2004kc} introduced this idea to the LQCD community, and showed that
TBCs are equivalent to having a $U(1)$ background gauge field in the QCD Lagrangian
with the quarks subject to PBCs. By choosing this constant background field to be, e.g., $\mathbf{A}=\frac{\theta_z}{L}\hat{e}_z$, the quark wavefunction will acquire a non-vanishing phase $\frac{\theta_z}{L}$ when wraps around the boundary of the volume in the $z$ direction due to the Aharonov-Bohm effect, despite the magnetic field strength being zero on the lattice. Alternatively, a quark field redefinition, $\psi (\mathbf{x})\rightarrow \widetilde{\psi}(\mathbf{x})= e^{i \frac{\theta_z}{L}z} \psi(\mathbf{x})$, will eliminate this background field provided that the new field $\widetilde{\psi}$ satisfies the TBCs, $\widetilde{\psi}(x,y,z+L)=e^{i\theta_z}\widetilde{\psi}(x,y,z)$. So the lattice gauge field configurations can be generated with these choices of BCs. The benefit of such BCs is that an arbitrary momentum can be selected for a (non-interacting) hadron by a judicious choice
of the twist angles of its valence quarks,
$\mathbf{p}=\frac{2\pi}{L}\mathbf{n}+\frac{\bm{\phi}}{L}$,
where $\bm{\phi}$ is the sum of the twists of the valence quarks, again with
$0<\phi_i<2\pi$, and $\mathbf{n}$ is an integer triplet.
TBCs have been shown to
be useful in LQCD calculations of
the low-momentum transfer behavior of form factors required in determining
hadron radii and moments, circumventing the need for large-volume lattices~\cite{Tiburzi:2005hg,Jiang:2006gna,Boyle:2007wg,Simula:2007fa,Boyle:2008yd,Aoki:2008gv,Boyle:2012nb, Brandt:2013mb}.
They have also been speculated to be helpful
in calculations of $K\rightarrow \pi\pi$ decays
by bringing the initial and final FV states closer in energy~\cite{deDivitiis:2004rf, Sachrajda:2004mi}.
In addition to performing calculations with a particular twist,
by averaging the results of calculations over twist angles, the discrete sum over momentum modes becomes
an integral over momenta,
\begin{eqnarray}
\int \ {d^3\bm{\phi}\over (2\pi)^3}\
\frac{1}{L^3}\ \sum_{\mathbf{n} \in \mathbb{Z}^3}
& \equiv &
\int \frac{d^3\mathbf{p}}{(2\pi)^3}
.
\end{eqnarray}
Although the volume dependence of most quantities is non linear due to interactions, such averaging can eliminate significant FV effects.
This was first examined in the context of condensed-matter physics where, for example, the finite-size effects in the finite-cluster calculations
of correlated electron systems are shown to be reduced by the boundary condition integration technique \cite{Gros, PhysRevB.53.6865}.
This technique is implemented in quantum Monte Carlo QMC algorithms of many-body systems, and results in faster convergence of energies to the thermodynamic limit~\cite{PhysRevE.64.016702}.
In this chapter, we discuss the advantages of using TBCs to reduce the FV modifications to the mass of hadrons and to the
binding energy of two-hadron bound states, such as the deuteron.
In particular, we consider the FV effects resulting
from averaging the results obtained from PBC and anti-PBCs (APBCs),
from a specific choice of the twist angle, \emph{i}-PBCs,
and from averaging over twist angles.
For the two-nucleon systems, the volume improvement is explored both analytically and numerically with the use of the
developed FV formalism for NN systems (see chapter \ref{chap:NN}), that is generalized to systems with TBCs.
As was first noted by Bedaque and Chen~\cite{Bedaque:2004ax}, the need to generate new gauge field configurations
with fully twisted BCs can be circumvented by imposing TBCs on the valence quarks only, which defines partial twisting.
Partial twisting gives rise to corrections beyond full twisting that scale as $e^{-m_{\pi}L}/L$,
and can be neglected for sufficiently large volumes compared to the FV effects from the size of weakly bound states.
Although the validity of partial twisting makes it feasible to achieve an approximate twist-averaged result in LQCD calculations,
this remains a computationally expensive technique.
We demonstrate that certain hadronic twist angles can result in an exponentially-improved convergence to the infinite-volume
limit of certain quantities, with an accuracy that is comparable to the twist-averaged mean.
Further, we speculate that similar improvements are also present in arbitrary n-body systems.
As discussed extensively in the previous chapters, in several situations, given the Euclidean nature of lattice correlation functions, it is desirable to keep the volume finite as the extraction of physical
quantities relies on non-vanishing FV effects.
For example, as we saw, the ability to extract the $S$-$D$ mixing parameter, $\epsilon_1$, and consequently the D/S ratio of the deuteron from LQCD calculations,
depends upon the FV modifications to the binding energy when the deuteron is boosted in particular directions
within the lattice volume \cite{Briceno:2013bda}.
The use of TBCs will further enhance the effectiveness of such calculations.
By appropriate choices of the twist angles of each hadron, different CM energies can be accessed in a
single lattice volume, further constraining the scattering parameters with the use of L\"uscher's method
(see e.g. Refs.~\cite{Bernard:2010fp, Doring:2011vk, Doring:2012eu, Ozaki:2012ce} for demonstrations of this technique in
studying hadronic resonances).
Due to the possibility of partial twisting in NN scattering,
these extra energy levels can be obtained without having to generate additional ensembles of
gauge-field configurations, in analogy with the boosted calculations
(this technique has recently been used to calculate $J/\psi$-$\phi$ scattering~\cite{Ozaki:2012ce}).
Of course, the spectra of energy eigenvalues determined with a range of twist angles allow for fits to
parametrizations of the S-matrix elements, which can then be used to predict infinite-volume quantities, such as binding
energies~\cite{Beane:2010em,Prelovsek:2013sxa}.
TBCs provide a way to reduce the systematic uncertainties that are currently present in analyses of
coupled-channels systems by providing the ability to control, at some level,
the location of eigenstates.
\section{Nucleon Mass
\label{sec:Single}
}
\noindent
If the up and down quarks have distinct twist angles, the charged pions, the proton and the neutron will acquire
net twist angles denoted as
${\bm{\phi}}^{\pi^{+}}=-\bm{\phi}^{\pi^{-}}$, $\bm{\phi}^{p}$ and $\bm{\phi}^{n}$, respectively,
while the flavor-singlet mesons, such as $\pi^0$, will remain untwisted, $\bm{\phi}^{\pi^{0}}=\mathbf{0}$.
The optimal set of quark twists depends upon the desired observable, and
an appropriate choice can yield a relation between the twists of different hadrons, or
leave a hadron untwisted.
In chapter chapter \ref{chap:intro} we calculated the FV corrections to the mass of nucleon $M_N$, in a cubic volume with PBCs imposed on the quark fields,
at one-loop order in two-flavor baryon $\chi$PT with the inclusion of $\Delta$ resonance, see Eq. (\ref{deltaM}). The masses of the proton and neutron in a FV
at the one-loop level with TBCs hare also calculated using the HB$\chi$PT~\cite{Jiang:2008ja}.\footnote{The FV corrections to meson masses, decay constants and semileptonic form factors
with both the TBCs and the partially-TBCs have been calculated at LO in $\chi$PT in Ref. \cite{Sachrajda:2004mi}.
}
We use Poisson re-summation formula to
factor the dependences on the twist angles as pure phases and put the expressions for the masses into a
simple form.
The proton mass is found to be
\footnote{Since nucleons with non-zero twists are not at rest, these expressions represent the corrections to their rest energy. The kinetic energy, $E_{K}^{p(n)}=(\phi^{p(n)})^2/2M_NL^2$, is however subleading at this order in HB$\chi$PT and these expressions can be considered as corrections to the mass of the nucleons.}
\begin{eqnarray}
\delta_{L} M_{p}=\frac{3g_A^2}{8 \pi^2 f_{\pi}^2} \mathcal{K}^{p}(0;\bm{\phi}^{\pi}) + \frac{g_{\Delta N}^2}{3 \pi^2 f_{\pi}^2}
\mathcal{K}^{p}(\Delta;\bm{\phi}^{\pi})
,
\label{eq:MnucVol}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{K}^{p}(0;\bm{\phi}^{\pi})= \frac{\pi}{3}m_{\pi}^2 \sum_{\mathbf{n}\neq \mathbf{0}}
\frac{e^{-|\mathbf{n}| m_{\pi} L}}{|\mathbf{n}| L} (\frac{1}{2}+e^{-i\mathbf{n}\cdot\bm{\phi}^{\pi^+}})
,
\label{eq:NNpisum}
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{K}^{p}(\Delta;\bm{\phi}^{\pi})
\ =\ \frac{1}{2}
\int_{0}^{\infty} d \lambda ~ \beta_{\Delta}
&&\sum_{\mathbf{n}\neq \mathbf{0}}
\left[
\beta_{\Delta}
K_0(\beta_{\Delta}|\mathbf{n}| L)
\ -\
\frac{1}{|\mathbf{n}| L} K_1(\beta_{\Delta}|\mathbf{n}| L)
\right]
\nonumber\\
&&\qquad \qquad \qquad
~ \times (e^{-i\mathbf{n}\cdot\bm{\phi}^{\pi^-}}+\frac{2}{3}+\frac{1}{3}e^{-i\mathbf{n}\cdot\bm{\phi}^{\pi^+}})
,
\label{eq:NDpisum}
\end{eqnarray}
and the neutron mass can be found from these expressions by the substitutions
$p\rightarrow n$ and $\pi^+\leftrightarrow\pi^-$.
It is convenient to consider the periodic images associated with the nucleon
having their contributions modified by the appropriate phase factor
due to the TBCs.
After twist averaging (over the twists of the pion field, see Appendix~\ref{app: TI}),
the leading FV corrections to the mass of both the proton
and the neutron arising from Eq.~(\ref{eq:MnucVol})
are $1/3$ of their value when calculated with PBCs, Eq.~(\ref{deltaM}).\footnote{If the twist of the up and down
quarks is the same, $\bm{\phi}^{\pi^{\pm}}$ vanishes and no volume improvement will be obtained by averaging.}
Of course, calculations at multiple twist angles need not be performed to estimate the twist-averaged value, and
special twist angles can be selected based upon the symmetries of the integer
sums in Eqs.~(\ref{eq:NNpisum}) and (\ref{eq:NDpisum}).
In particular, it is notable that the leading volume effects of the form
$e^{-m_\pi L}/L$, $e^{-\sqrt{2}m_\pi L}/L$ and $e^{-\sqrt{3}m_\pi L}/L$,
can be reduced by a factor of three
with \emph{i}-PBCs, by setting the pion twist angle to
$\bm{\phi}^{\pi^+}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$.
Averaging the masses calculated with PBCs and APBCs also reduces the
leading contribution by a factor of three.
The leading volume dependence can be eliminated completely by choosing
$\bm{\phi}^{\pi^+}=(\frac{4 \pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$, leaving volume corrections to the nucleon mass
of the form $\sim e^{-\sqrt{2} m_\pi L}/L$.
It is likely that optimal twists exist for other single nucleon properties, such as matrix elements of the isovector axial current, $g_A$.
For arbitrary quark twists, the proton and neutron have, in general,
different phase spaces as the momentum modes that exist in the FV differ.
As an example, while quark twists can be chosen to keep the proton at rest in the volume
and allow for averaging over the charged pion twists,
${\bm\phi}^{(d)} = -2{\bm\phi}^{(u)}$,
in general the neutron will have non-zero momentum.\footnote{
Such non-trivial phase spaces somewhat complicate the analysis of LQCD calculations of multi-baryon systems.
}
\section{Two Baryons and Twisted Quantization Condition}
\noindent
The S-wave NN energy QC was generalized to systems with TBCs at rest in Ref.~\cite{Bedaque:2004kc},
and to more general two-hadron systems in Ref.~\cite{Agadjanov:2013kja}.
L\"uscher's energy QC~\cite{Luscher:1986pf, Luscher:1990ux},
which determines the form of the FV corrections,
is dictated by the on-shell two-particle states within the volume.
Once the kinematic constraints on the momentum modes of the two-particle states in the FV are determined, the corresponding
QC can be determined in a straightforward manner.
Explicitly, the QC is once again of the form
\begin{eqnarray}
\det\left[{(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}}\right]\ =\ 0
,
\label{NNQC}
\end{eqnarray}
where $\mathcal{M}^{\infty}$ is the infinite-volume scattering amplitude matrix evaluated at the on-shell momentum of each particle
in the CM frame, $k^*$.
It is convenient to express the QC in the $\left|JM_J(LS)\right\rangle$ basis, where $J$ is the total angular momentum, $M_J$ is the eigenvalue of the $\hat
J_z$ operator, and $L$ and $S$ are the orbital angular momentum and the total spin of the system, respectively.
The matrix elements of $\delta\mathcal{G}^V$ in this basis
are very similar to Eq. (\ref{deltaG}) for the case of NN scattering with PBCs,
\begin{align}
& \left[\delta\mathcal{G}^V\right]_{JM_J,LS;J'M_J',L'S'}=i \eta \frac{k^*}{8\pi E^*}
\delta_{SS'}\left[\delta_{JJ'}\delta_{M_JM_J'}\delta_{LL'} +i\sum_{l,m}\frac{(4\pi)^{3/2}}{k^{*l+1}}
c_{lm}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}(k^{*2};L) \right.
\nonumber\\
& \qquad \qquad \qquad \left . \times \sum_{M_L,M_L',M_S}\langle JM_J|LM_L,SM_S\rangle \langle L'M_L',SM_S|J'M_J'\rangle
\int d\Omega~Y^*_{L M_L}Y^*_{l m}Y_{L' M_L'}\right],
\nonumber\\
\label{deltaG-TBC}
\end{align}
where $\eta=1/2$ for identical particles and $\eta=1$ otherwise,
and $\langle JM_J|LM_L,SM_S\rangle$ are Clebsch-Gordan coefficients.
$E^*$ is the total (relativistic) CM energy of the system, $E^*=\sqrt{k^{*2}+m_1^2}+\sqrt{k^{*2}+m_2^2}$ where
$m_1$ and $m_2$ are the masses of the particles,
and $\bm{\phi}_1$ and $\bm{\phi}_2$ are their respective twist angles.
The total momentum of the system is $\mathbf{P}=\frac{2\pi}{L}\mathbf{d}+\frac{\bm{\phi}_1+\bm{\phi}_2}{L}$
with $\mathbf{d}\in \mathbb{Z}^3$.
The only difference between this equation and Eq. (\ref{deltaG}), besides the relativistic kinematics used, is in the $c_{lm}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}(k^{*2};L)$ function who carries the volume dependence and the dependence on the BCs in the QC. Explicitly
\begin{eqnarray}
c^{\textbf{d},\bm{\phi}_1,\bm{\phi}_2}_{lm}(k^{*2};L)
\ =\ \frac{\sqrt{4\pi}}{\gamma L^3}\left(\frac{2\pi}{L}\right)^{l-2}
\mathcal{Z}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}_{lm}[1;(k^*L/2\pi)^2]
,
\label{clm}
\end{eqnarray}
with
\begin{eqnarray}
\mathcal{Z}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}_{lm}[s;x^2]
\ =\ \sum_{\mathbf r \in \mathcal{P}_{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}}
\frac{ |{\bf r}|^l \ Y_{l m}(\mathbf{r})}{(\mathbf{r}^2-x^2)^s}
.
\label{Zlm}
\end{eqnarray}
$\gamma=E/E^*$ where E is the total energy of the system in the rest frame of the volume (the lab frame),
$E^2=\mathbf{P}^2+E^{*2}$.
The sum in Eq.~(\ref{Zlm}) is performed over the momentum vectors $\mathbf{r}$ that belong to the
set $\mathcal{P}_{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}$,
which remains to be determined.
Consider the two-hadron wavefunction in the lab frame~\cite{Rummukainen:1995vs, Davoudi:2011md}
that is subject to the TBCs,
\begin{eqnarray}
\psi_{{\rm Lab}}(\mathbf{x}_1+L \mathbf{n}_1,\mathbf{x}_2+L \mathbf{n}_2)
\ =\
e^{i{\bm{\phi}}_1 \cdot \mathbf{n}_1+i{\bm{\phi}}_2 \cdot \mathbf{n}_2}
\ \psi_{{\rm Lab}}(\mathbf{x}_1,\mathbf{x}_2)
,
\label{WF-BC}
\end{eqnarray}
where $\mathbf{x}_1$ and $\mathbf{x}_2$ denote the position of the hadrons,
and $\mathbf{n}_1,\mathbf{n}_2\in\mathbb{Z}^3$.
As the total momentum of the system is conserved, the wavefunction can be written as an eigenfunction of
the total momentum $P=(E,\mathbf{P})$.
In the lab frame, the equal-time wavefunction of the system is
\begin{eqnarray}
\psi_{{\rm Lab}}(x_1,x_2)
\ =\ e^{-iE X^0+i\mathbf{P} \cdot \mathbf{X}}
\ \varphi_{{\rm Lab}}(0,\mathbf{x}_1-\mathbf{x}_2)
,
\end{eqnarray}
where the position of the CM is $X$, and
\begin{eqnarray}
X & = & \alpha x_1+(1-\alpha) x_2
, \ \ \ \
\alpha=\frac{1}{2}\left(1+\frac{m_1^2-m_2^2}{E^{*2}}\right)
,
\end{eqnarray}
for systems with unequal masses~\cite{Davoudi:2011md}.
Since the CM wavefunction is independent of the relative time coordinate~\cite{Rummukainen:1995vs},
$\varphi_{{\rm Lab}}(0,\mathbf{\mathbf{x}_1-\mathbf{x}_2})=\varphi_{{\rm CM}}(\hat{\gamma} (\mathbf{x}_1-\mathbf{x}_2))$,
where the boosted relative position vector is
$\hat{\gamma} \mathbf{x}=\gamma \mathbf{x}_{\Vert}+ \mathbf{x}_{\bot}$,
with $\mathbf{x}_{\Vert}$ ($\mathbf{x}_{\bot}$) being the component of $\mathbf{x}$ that is
parallel (perpendicular) to $\mathbf{P}$.
By expressing $\psi_{{\rm Lab}}$ in Eq.~(\ref{WF-BC}) in terms of $\varphi_{CM}$, it straightforwardly follows that
\begin{eqnarray}
e^{i\alpha\mathbf{P}\cdot(\mathbf{n}_1-\mathbf{n}_2)L+i\mathbf{P}\cdot\mathbf{n}_2L}
\ \varphi_{{\rm CM}}(\mathbf{y}^*+\hat{\gamma}(\mathbf{n}_1-\mathbf{n}_2)L)
\ =\
e^{i{\bm{\phi}}_1 \cdot \mathbf{n}_1+i{\bm{\phi}}_2 \cdot \mathbf{n}_2}
\ \varphi_{{\rm CM}}(\mathbf{y}^*)
,
\end{eqnarray}
where $\mathbf{y}^*=\mathbf{x}_1^*-\mathbf{x}_2^*$ is the relative coordinate of two hadrons in the CM frame.
By Fourier transforming this relation, and using the form of the total momentum $\mathbf{P}$ from above,
the relative momenta allowed in the FV energy QC are constrained to be
\begin{eqnarray}
\mathbf{r}
\ =\
\frac{1}{L}\
\hat{\gamma}^{-1}
\ \left[2\pi(\mathbf{n}-\alpha\mathbf{d})-(\alpha-\frac{1}{2})(\bm{\phi}_1+\bm{\phi}_2)+\frac{1}{2}(\bm{\phi}_1-\bm{\phi}_2)\right]
,
\label{r-TBC}
\end{eqnarray}
where $\mathbf{n}\in\mathbb{Z}^3$ is the three-vector that is summed over in Eq.~(\ref{Zlm}).
These results encapsulate those of Refs.~\cite{Rummukainen:1995vs, Davoudi:2011md, Fu:2011xz, Leskovec:2012gb, Bour:2011ef}
when the PBCs are imposed,
i.e., when $\bm{\phi}_1=\bm{\phi}_2=\mathbf{0}$.
It also recovers two limiting cases that are considered in Ref.~\cite{Agadjanov:2013kja} for the use of TBCs in the scalar sector of QCD.
It should be noted that for particles with equal masses, $\alpha=1/2$,
the set of allowed momentum vectors reduces to
\begin{eqnarray}
\mathbf{r}
\ =\
\frac{1}{L}\ \hat{\gamma}^{-1}
\ \left[2\pi(\mathbf{n}-\frac{1}{2}\mathbf{d})+\frac{1}{2}(\bm{\phi}_1-\bm{\phi}_2)\right]
.
\label{eq-mass}
\end{eqnarray}
It is important to note that for two identical hadrons, when $\bm{\phi}_1=\bm{\phi}_2=\bm{\phi}$, the FV spectra show no non-trivial
dependence on the twist other than a shift in the total energy of the system,
$E^2=(\frac{2\pi}{L}\mathbf{d}+\frac{\bm{\phi}}{L})^2+E^{*2}$.
As a result, twisting will not provide additional constraints on the scattering amplitude in,
for instance, the $^1 \hskip -0.03in S _0$ nn or pp channels.
This is also the case for the FV studies of NN scattering in the $^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled channels if the same twist is
imposed on the up and down quarks.\footnote{This result differs somewhat from the conclusion of Ref.~\cite{Bedaque:2004kc}.}
\section{Deuteron Binding and Volume Improvement}
\noindent
In the previous chapter, we obtained the expected energy spectra of two nucleons with spin $S=1$ in a FV subject to PBCs
and with a range of CM momenta from the
experimentally measured phase shifts and mixing angles~\cite{Briceno:2013bda}.
In particular, the dependence of the bound-state spectra on the non-zero mixing angle between S and D waves, $\epsilon_1$,
was determined.
As seen from Eq.~(\ref{eq-mass}), the
effects of the
twist angles $\frac{1}{2\pi}(\bm{\phi}_1~-~\bm{\phi}_2)=(0,0,1),(1,1,0),(1,1,1)$
on the CM spectra
are the same as those of (untwisted) boost vectors, ${\bf d}$,
considered in chapter \ref{chap:deuteron}.
Therefore, different TBCs can provide additional CM energies in a single volume,
similar to boosted calculations,
which can be used to better constrain scattering parameters and the S-matrix.
However, twisting may be a more powerful tool as it
provides access to a continuum of momenta.
If imposing TBCs on the quark fields would require the generation of new ensembles of gauge-field configurations,
it would likely not be optimal to expend large computational resources on multiple twisted calculations.
However, PBCs can be retained on the sea quarks and
TBCs can be imposed only in the valence sector~\cite{Bedaque:2004ax}.
The reason for this is that there are no disconnected diagrams associated with the NN interactions.\footnote{
As recently demonstrated, disconnected diagrams will not hinder the use of partially-TBCs in studies of the scalar
sector of QCD either~\cite{Agadjanov:2013kja}.
The graded symmetry of ``partially-quenched'' QCD results in cancellations among
contributions from intermediate non-valence mesons.}
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.8]{nnsea}}
\subfigure[]{
\includegraphics[scale=0.8]{pipisea}}
\caption{
{\small a) A sea-quark (red dashed lines) contribution to the NN scattering in QCD can be described by an EFT at low energies where the bosons exchanged (red dashed lines) contain a a sea quark. No sea contribution to the NN scattering can occur in the s channel. b) The $\pi \pi$ scattering in $I=0$ channel is an example of a disconnected process where due to the intermediate sea quark creation and annihilation loops, the corresponding EFT description will necessary get contribution from s-channel diagrams whose intermediate mesons contain the sea quarks \cite{Bedaque:2004ax}. Figure is reproduced with the permission of Paulo Bedaque.}
}
\label{fig:sea}
\end{center}
\end{figure}
At the level of the low-energy EFT, this indicates that there are no intermediate s-channel diagrams in which a nucleon or meson
containing a sea quark can go on-shell.
Such off-mass-shell hadrons modify the NN interactions by $\sim e^{-m_{\pi}L}/L$, and do not invalidate the use
of the QC in Eq.~(\ref{NNQC}) with the partially-TBCs as long as the calculations are performed
in sufficiently large volumes,
$L \gtrsim 9 ~ {\rm fm}$.
One significant advantage of imposing TBCs is the improvement in the volume dependence of the deuteron binding energy.
Although the formalism presented in the previous chapters can be used to fit to various scattering parameters~\cite{Briceno:2013bda}
(and consequently determine the deuteron binding energy),
we will show that with a judicious choice of twist angles, the extracted energies in future LQCD calculations should be close to
the infinite-volume values, even in volumes as small as $\sim (9~{\rm fm})^3$.
As discussed in the previous section, the CM energy of the np system is sensitive to TBCs only if
different twists are imposed upon the up and down quarks.
This means that, even if exact isospin symmetry is assumed, the proton and the neutron will have different phase spaces
due to the different BCs.
By relaxing the interchangeability constraint on the np state, as required by the different phase spaces,
the NN positive-parity channels will mix with the negative-parity channels.
This admixture of parity eigenstates is entirely a FV effect induced by the boundary conditions,
and does not require parity violation in the interactions,
manifesting itself in non-vanishing $c_{lm}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}$ functions
for odd values of $l$.
As such, the spin of the NN system is preserved.
The procedure to obtain the expected spectra of the deuteron is similar to our method in chapter \ref{chap:deuteron} and is as follows. The QC in Eq.~(\ref{NNQC}) depends on S-matrix elements in all partial waves, however it can be truncated
to include only channels with $L \leq 2$
(requiring $J \leq 3$)
because of the reducing size of the low-energy phase shifts in the higher channels.
For arbitrary twist angles, the truncated QC can be represented by a $27 \times 27$ matrix in the $\left|JM_J(LS)\right\rangle$ basis, the eigenvalues of which dictate the energy eigenvalues.
Fits to the experimentally known phase shifts and
mixing parameters~\cite{NIJMEGEN, PhysRevC.48.792, PhysRevC.49.2950, PhysRevC.54.2851, PhysRevC.54.2869}
are used to extrapolate to negative
energies~\cite{Briceno:2013bda} to provide the inputs into the truncated QC,
from which the deuteron spectra in a cubic volume with TBCs are predicted.
The scattering parameters entering the analysis are
$\delta_{1\alpha},~\epsilon_1,~\delta_{1\beta},~\delta^{({^3}P_0)},~\delta^{({^3}P_1)},~\delta^{({^3}P_2)},
~\delta^{({^3}D_2)}$ and $\delta^{({^3}D_3)}$,
where the Blatt-Biedenharn (BB) parameterization~\cite{Blatt:1952zza} is used in the $J=1$ sector.
The twist angles explored in this work are
$\bm{\phi}^p=-\bm{\phi}^n \equiv \bm{\phi}=(0,0,0)$ (PBCs),
$(\pi,\pi,\pi)$ (APBCs)
and $(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$ (\emph{i}-PBCs).
At the level of the quarks, this implies that the twist angles of the (valence) up and down quarks
are ${\bm\phi}^u = -{\bm\phi}^d = {\bm\phi}$.
We also set $\mathbf{d}=\mathbf{0}$ in Eq.~(\ref{r-TBC}) so that the np system is at rest in the lab frame.
The reason for this choice of twist angles is that they (directly or indirectly) give rise a significant cancellation of the
leading FV corrections to the masses of the nucleons, as shown in Sec.~\ref{sec:Single}.
The number of eigenvalues of
${(\mathcal{M}^{\infty})^{-1}+\delta\mathcal{G}^{V}}$,
and their degeneracies, reflect the spatial-symmetry group of the FV.
Calculations with $\bm{\phi}=\bm{0}$ respect the cubic ($O_h$) symmetry, while
for $\bm{\phi}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$ the symmetry group is reduced to the $C_{3v}$ point
group.~\footnote{
There is a correspondence between the FV spatial symmetry in twisted calculations with
arbitrary twists
$\bm{\phi}^p \neq \bm{\phi}^n$
and the FV symmetry in (boosted) NN calculations with PBCs when isospin breaking is considered.
For example, the point symmetry group corresponding to twisted calculations with $\bm{\phi}^p = -\bm{\phi}^n=(0,0,\frac{\pi}{2})$
and that of the physical np system with $\mathbf{P}=\frac{2\pi}L(0,0,1)$ with PBCs are both $C_{4v}$.
}
However, for $\bm{\phi}=(\pi,\pi,\pi)$
the system has inversion symmetry, and respects the $D_{3h}$ point symmetry~\cite{Dresselhaus}.
By examining the transformation properties of the
$c_{lm}^{\mathbf{d},\bm{\phi}_1,\bm{\phi}_2}$
functions under the symmetry operations of these groups, certain relations are found for any given $l$.
These relations,
as well as the eigenvectors of the FV matrices,
which are tabulated elsewhere~\cite{Luscher:1990ux, Rummukainen:1995vs, Briceno:2013lba, Gockeler:2012yj,Thomas:2011rh,Dudek:2012gj},
can be used to block diagonalize the $27 \times 27$ matrix representation of the QCs,
where each block corresponds to an irrep of the point-group symmetry of the system.
For the selected twist angles, the QCs of the irreps of the corresponding point
groups that have overlap with the deuteron are given in Appendix~\ref{app: QC-TBC} .
\begin{figure}[h]
\begin{centering}
\includegraphics[scale=0.40]{atwoe}
\caption{{\small
The deuteron binding energy as a function of $L$
using \emph{i}-PBCs
($\bm{\phi}^p=-\bm{\phi}^n \equiv \bm{\phi}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$).
The blue curve corresponds to the $\mathbb{A}_2$ irrep of the $C_{3v}$ group,
while the red curve corresponds to the $\mathbb{E}$ irrep.
The brown-dashed curve corresponds to the weighted average of the $\mathbb{A}_2$ and
$\mathbb{E}$ irreps, $-\frac{1}{3}(2B_d^{(\mathbb{E})}+B_d^{(\mathbb{A}_2)})$,
while the black-solid curve corresponds to the S-wave limit.
The infinite-volume deuteron binding energy is shown by the black-dotted line.
}}
\label{fig:A2-E}
\par\end{centering}
\end{figure}
For \emph{i}-PBCs, there are two irreps of the $C_{3v}$ group, namely the one-dimensional irrep $\mathbb{A}_1$
and the two-dimensional irrep $\mathbb{E}$, that have overlap with the $^3{S_1}$-${^3}{D_1}$ coupled channels.
Fig.~\ref{fig:A2-E} shows the binding energy (the CM energy minus the rest masses of the nucleons), $-B_d=E^*-M_p-M_n$,
as a function of $L$ corresponding to $\mathbb{A}_2$ irrep (blue curve) and $\mathbb{E}$ irrep (red curve),
obtained from the QCs in Eqs. (\ref{pi2pi2pi2A2}) and (\ref{pi2pi2pi2E}).
Even at $L\sim9~{\rm fm}$, the deuteron binding energies extracted from both irreps are close to the infinite-volume value.
In particular, calculations in the $\mathbb{E}$ irrep of the $C_{3v}$ group provide a few percent-level accurate
determination of the deuteron binding energy
in this volume.
The black-solid curve in Fig.~\ref{fig:A2-E}
represents the S-wave limit of the interactions,
when the S-D mixing parameter and all phase shifts except that in the S-wave are set equal to zero.
The $M_J^\prime$-averaged binding energy,
$-\frac{1}{3}(2B_d^{(\mathbb{E})}+B_d^{(\mathbb{A}_2)})$,
converges to this S-wave limit, as shown
in Fig. \ref{fig:A2-E}
(the $\mathbb{A}_2$ irrep contains the $M_J^\prime=0$ state
while $\mathbb{E}$ contains the $M_J^\prime=\pm 1$ states,
where as mentioned before, $M_J^\prime$ is the projection of total angular momentum
along the twist direction).
In order to appreciate the significance of calculations performed with the $\bm{\phi}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$ twist angles,
it is helpful to recall the deuteron binding energy obtained in calculations with PBCs.
For PBCs,
we remind that the only irrep of the cubic group that has overlap with the $^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled channels is the
three-dimensional irrep $\mathbb{T}_1$, Eq.~(\ref{000T1}),
and the corresponding binding energy is shown in Fig. \ref{fig:PBC-APBC}(a) (green curve).
As was already seen in the previous chapter, the binding energy deviates significantly from its infinite-volume value,
such that the FV deuteron is approximately twice as bound as the infinite-volume deuteron at $L=9~{\rm fm}$.
For APBCs, two irreps of the $D_{3h}$ group overlap with the deuteron channel,
$\mathbb{A}_2$ and $\mathbb{E}$ (Eqs. (\ref{pipipiA2},\ref{pipipiE})),
and yield degenerate binding energies as shown in Fig. \ref{fig:PBC-APBC}(a) (purple curves).
As seen in Fig. \ref{fig:PBC-APBC}(a),
the deuteron becomes unbound over a range of volumes
and asymptotes slowly to the infinite-volume limit.
However, in analogy with the nucleon masses,
the volume dependence of the deuteron binding energy
is significantly reduced
by averaging the results obtained with PBCs and APBCs,
as shown in Fig. \ref{fig:PBC-APBC}(a) (black-solid curve).
Fig. \ref{fig:PBC-APBC}(b) provides a magnified view of this averaged quantity (black-solid curve),
where the two energy levels associated with \emph{i}-PBCs are shown for comparison.
\begin{figure}[h]
\begin{centering}
\subfigure[]{
\includegraphics[scale=0.3]{pbcapbc}}
\subfigure[]{
\includegraphics[scale=0.3]{comparison}}
\caption{{\small
a) The deuteron binding energy as a function of $L$
from PBCs (green curve) and from APBCs (purple curve).
The black-solid curve represents the average of these energies.
b) A closer look at the average in part (a) compared with energies obtained with \emph{i}-PBCs,
$\mathbb{A}_2$ (blue curve) and $\mathbb{E}$ (red curve).
}}
\label{fig:PBC-APBC}
\par\end{centering}
\end{figure}
In order to understand the observed volume improvements,
consider the volume scaling of the full QC assuming that the phase shifts beyond the $\alpha$-wave
are small.
In this limit, for a general set of twist angles and boosts, the QC collapses to
\begin{eqnarray}
&&
\!\!\!\!\!\!\!\!\!
\det\left[
\left(
k^*\cot\delta_{1\alpha} - 4\pi c_{00}
\right)
\left(
\begin{array}{ccc}
1&0&0\\ 0&1&0\\ 0&0&1
\end{array}
\right)
\right.\nonumber\\
&&
\left.
-
{2\pi\over \sqrt{5} k^{*2}}\left(\sqrt{2}\sin 2\epsilon_1 - \sin^2\epsilon_1 \right)
\left(
\begin{array}{ccc}
c_{20}&\sqrt{3} c_{21}&\sqrt{6} c_{22}\\
-\sqrt{3} c_{2-1}&-2c_{20}&-\sqrt{3} c_{21}\\
\sqrt{6} c_{2-2}&\sqrt{3} c_{2-1}&c_{20}
\end{array}
\right)
\ \right] \ = \ 0
,
\label{eq:QC3by3}
\end{eqnarray}
which depends upon the $\alpha$-wave phase shift and the mixing parameter, $\epsilon_1$.
Shorthand notation has been used for convenience, $c_{lm} = c_{lm}^{{\bf d},{\bm\phi}_1,{\bm\phi}_2}(k^{*2}; L)$.
For generic twist angles,
deviations between the energy eigenvalues resulting from this truncated QC and the full QC scale as
$\sim \tan\delta_i \ e^{-2 \kappa L}/(\kappa L^2)$,
where $\delta_i$ denotes phase shifts beyond the $\alpha$-wave
(see Appendix~\ref{app:TwistC} for expansions of the $c_{lm}^{{\bf d},{\bm\phi}_1,{\bm\phi}_2}$ functions).
For \emph{i}-PBCs, the leading corrections are from the P-waves, as can be seen from the expansions of the $c_{lm}$ in Table~\ref{app:TwistC}.
By neglecting the small mixing between the S-wave and D-waves in Eq.~(\ref{eq:QC3by3}),
the QC dictated by S-wave interactions is~\footnote{
In the limit where $\epsilon_1=0$, the $J=1$ $\alpha$-wave is entirely S-wave,
while the $\beta$-wave is entirely D-wave.
This approximation neglects FV effects of the form $\epsilon_1 e^{- \kappa L}/L$.
}
\begin{eqnarray}
k^*\cot\delta^{{(^3S_1)}}|_{k^*=i\kappa}+\kappa=\sum_{\mathbf{n}\neq\mathbf{0}}
e^{i (\alpha-\frac{1}{2}) \mathbf{n} \cdot (\bm{\phi}^p+\bm{\phi}^n)} e^{-i \frac{1}{2} \mathbf{n} \cdot (\bm{\phi}^p-\bm{\phi}^n)}
e^{i 2\pi \alpha \mathbf{n} \cdot \mathbf{d}}
~\frac{e^{-|\hat{\gamma}\mathbf{n}| \kappa L}}{|\hat{\gamma}\mathbf{n}|L}
.
\label{S-QC}
\end{eqnarray}
The volume dependence of the deuteron binding momentum, $\kappa$, originates from the right-hand side of this equation.
For ${\bf d}={\bf 0}$, the $c_{2m}$ functions vanish for both PBCs and APBCs, leading to Eq.~(\ref{S-QC}) without further approximation.
For the twist angles $\bm{\phi}^p=-\bm{\phi}^n \equiv \bm{\phi}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$ and boost $\mathbf{d}=\mathbf{0}$,
the first few terms in the summation on the right-hand side of Eq.~(\ref{S-QC}) ($\mathbf{n}^2\leq 3$)
vanish, leaving the leading volume corrections to scale as $\sim e^{-2\kappa L}/L$.
A lesser
cancellation occurs in the average of binding energies obtained with PBCs and APBCs, giving rise to deviations from the infinite-volume energy by terms that scale as $\sim e^{-\sqrt{2}\kappa L}/L$.
The result of Monte Carlo twist averaging of the deuteron binding energy
can be ascertained from the behavior of the two extreme contributions, the PBC and APBC results.
While the average binding energy obtained from $N$ randomly selected sets of twist angles scales as
$B_d^{(\infty)} + {\cal O}\left( e^{-2\kappa L}/L \right)$,
the standard deviation of the mean scales as $\sim e^{-\kappa L}/(\sqrt{N} L)$,
giving rise to a signal-to-noise ratio in the binding energy that scales as
$\sim \sqrt{N} \ B_d^{(\infty)}\ L\ e^{\kappa L}$, which even for $L\sim 14~{\rm fm}$ allows only for a poor extraction, as
can be deduced from Fig. \ref{fig:PBC-APBC}(a).
It is clear that such a method is inferior to that of pair-wise averaging, such as from PBCs and APBCs, or choosing special twists, such as
\emph{i}-PBCs.
We have restricted ourselves to the scenarios where the net twist angles in each
Cartesian direction (the lattice axes) are the same.
One reason for this is that systems with arbitrary twists give rise to three distinct, but nearby, energy eigenvalues
associated with combinations of each of the three $M_J$-states of the deuteron - a sub-optimal system to analyze
in LQCD calculations.
Another reason is that a twist of $\frac{\pi}{2}$ in each direction is optimal in minimizing the FV effects in
both the two-body binding energies and the single-baryon masses.
Further, averaging the results of calculations with PBCs and APBCs also eliminates the leading FV corrections to both quantities.
We re-emphasize that ultimately, one wants to extract as many scattering parameters as feasible from calculations
in a single volume, requiring calculations with multiple boosts of the CM as well as multiple arbitrary twists, in order
to maximize the inputs to the energy QCs.
In general, with arbitrary twist angles, $\bm{\phi}=(\phi_x,\phi_y,\phi_z)$,
the $27 \times 27$ matrix representation of the QC matrix cannot be block diagonalized and it has $27$ distinct eigenvalues.
The truncation to the $3\times 3$ matrices given in Eq.~(\ref{eq:QC3by3}) remains valid, as do the estimates of the truncation errors, but this truncated QC will provide three distinct energy eigenvalues.
While not the focus of this work, it is worth reminding ourselves about the behavior of the positive-energy states in the FV,
such as the higher states associated with the $^3 \hskip -0.025in S _1$-$^3 \hskip -0.03in D _1$ coupled channel or those associated with the $^1 \hskip -0.03in S _0$ $np$ channel,
as described in Eq.~(\ref{NNQC}).
For an arbitrary twist, the non-interacting energy levels in the FV are determined by integer triplets and the twist angles.
Interactions will produce deviations from these non-interacting levels, that become smaller
as the lattice volume increases, scaling with $\sim \tan\delta(k^*)/(M L^2)$.
As discussed previously, as there is no underlying symmetry for arbitrary twists, the eigenstates will, in general, be non degenerate.
\section*{Summary and Conclusions}
\noindent
Twisted boundary conditions have been successfully used in numerical calculations of important observables,
both in nuclear and particle physics with Lattice QCD,
as well as in others areas such as condensed-matter physics.
They provide a means with which to select the phase space of particles in a given finite volume, beyond that allowed by periodic or anti-periodic boundary conditions.
In LQCD calculations, TBCs have been used to resolve the threshold region required in the evaluation of
transition matrix elements without requiring large lattice volumes \cite{Tiburzi:2005hg, Jiang:2006gna, Boyle:2007wg, Simula:2007fa, Boyle:2008yd, Aoki:2008gv, Boyle:2012nb, Brandt:2013mb,Boyle:2013gsa}.
They can also be
used in calculations of elastic $2 \rightarrow 2$ processes by providing a better sampling of
CM kinematics in a single volume, allowing for
better constraints on scattering parameters~\cite{Bernard:2010fp, Doring:2011vk, Doring:2012eu, Ozaki:2012ce}.
In this chapter, we have explored the use of TBCs in calculating the mass of single baryons, and in determining the binding of two-hadron
systems in a FV, with a focus on the deuteron.
In particular, we have used experimentally known scattering data to determine the location of the lowest-lying
FV states that have overlap with the deuteron for a selection of twist angles, and combinations thereof.
We have formally found that twisting provides an effective way of exponentially reducing the impact of the finite lattice volume on the calculation of
two-body binding energies. Pair-wise combining results obtained with particular twists, such as
PBCs and APBCs,
can eliminate the leading volume dependence.
The same is true for twist averaging, but the uncertainty resulting from a finite number of randomly selected twists can be large.
Importantly, we have
determined that the \emph{i}-PBCs, with
$\bm{\phi}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$,
eliminate the first three FV corrections to the dominant S-wave contribution to
the two-hadron binding energies, suppressing such effects from
$\mathcal{O}\left(e^{-\kappa L}/L\right)$
to
$\mathcal{O}\left(e^{- 2 \kappa L}/L\right)$, while also reducing the FV modifications to the nucleon mass, of the form
$\mathcal{O}\left(e^{- m_\pi L}/L\right)$, by a factor of three.
This translates into at least an order of magnitude improvement in the accuracy of the deuteron binding energy
extracted from LQCD correlation functions in volumes as small as $\sim (9~{\rm fm})^3$.
As partially-TBCs modify the nuclear forces by terms of order $\mathcal{O}\left(e^{-m_{\pi}L}/L\right)$,
such calculations of the deuteron and other bound states can be performed without the need for
multiple ensembles of gauge-field configurations, significantly reducing the required computational resources.
Given the generalized L\"uscher FV formalism for NN systems \cite{Briceno:2013lba} with TBCs,
not only can the binding energy of the deuteron be obtained from the upcoming LQCD calculations,
but the relevant scattering parameters, including the S-D mixing parameter, can be well constrained.
While giving different twists to the up and down quarks modifies the neutron and proton phase
space in different ways that allows for a parametric reduction in volume effects to the deuteron binding energy,
and control on the location of the positive-energy scattering states,
it does not change the CM phase space in the neutron-neutron or proton-proton systems.
Therefore, it is not a useful tool in
refining calculations of scattering parameters in these channels.
Inspired by the volume improvement seen in the QMC calculations of few and many-body systems with
twist-averaged BCs~\cite{PhysRevE.64.016702, PhysRevLett.73.1959, PhysRevB.51.10591, PhysRevB.53.1814, Wilcox:1999ux},
and studies of Dirichlet BCs and PBCs in QMC and Density-Functional Theory, e.g. Refs.~\cite{Bulgac:2013mz,Erler:2012qd},
and considering the twist-phase modifications to the images associated with a given system,
we speculate that the FV modifications to the spectrum of three-nucleon and multi-nucleon systems can be reduced
by TBCs. The magnitude of the improvement
will depend upon the inter-particle forces being short ranged compared to the extent of the system.
Due to the complexity of such systems, particularly in a FV~\cite{Polejaeva:2012ut, Briceno:2012rv,Hansen:2013dla}, a definitive
conclusion can only be arrived at upon further investigation.
\chapter{FINITE-VOLUME QED EFFECTS IN THE SINGLE-PARTICLE SECTOR}
\label{chap:EM}
Lattice QCD
has matured to the point where basic properties of the light hadrons are being calculated at the physical
pion mass~ \cite{Aoki:2009ix, Durr:2010vn, Arthur:2012opa, Aoki:2012st, Durr:2013goa}.
In some instances, the up- and down-quark masses and QED have been included in an effort to
precisely postdict the observed isospin splittings in the
spectrum of hadrons~\cite{Blum:2007cy,Basak:2008na,Blum:2010ym,Portelli:2010yn,Portelli:2012pn,Aoki:2012st,deDivitiis:2013xla,Borsanyi:2013lga,Drury:2013sfa}.
While naively appearing to be a simple extension of pure LQCD calculations,
there are subtleties associated with including
QED.
In particular, Gauss's law and Ampere's law cannot be satisfied
when the electromagnetic gauge field is subject to PBCs~\cite{Hilf1983412,Duncan:1996xy,Hayakawa:2008an}.
However, a uniform background charge density can be introduced to circumvent this problem and restore these laws.
This is equivalent to removing the zero modes of the photon in a FV calculation,
which does not change the infinite-volume value of calculated quantities.
One-loop level calculations in $\chi$PT
and partially-quenched $\chi$PT ($PQ\chi$PT) have been performed~\cite{Hayakawa:2008an}
to determine the leading FV modifications to the mass of mesons induced by constraining QED to a cubic volume subject to
PBCs.\footnote{
Vector dominance~\cite{PhysRevLett.62.1343}
has been previously used to model the low-momentum contributions to the
FV electromagnetic mass splittings of the pseudo-scalar
mesons, see Refs. \cite{Duncan:1996xy, Blum:2007cy}.
}
Due to the photon being massless, the FV QED
corrections to the mass of the $\pi^+$ are predicted to be an expansion in powers of the volume,
and have been determined to be of the form
$\delta m_{\pi^+}\sim 1/L + 2/(m_{\pi^+} L^2) + \cdots $,
where $L$ is the spatial extent of the cubic volume.
As the spatial extents of present-day gauge-field configurations at the physical pion mass are not large, with $m_\pi L\raisebox{-0.7ex}{$\stackrel{\textstyle <}{\sim}$ } 4$,
the exponentially suppressed strong interaction FV effects, ${\cal O}\left( e^{- m_\pi L}\right)$,
are not negligible for precision studies of hadrons, and
when QED is included, the power-law corrections, although suppressed by $\alpha_e$, are expected to be important,
particularly in mass splittings.
In this chapter, we return to the issue of calculating FV QED effects, and show that non-relativistic effective field theories (NREFTs)
provide a straightforward way to calculate such corrections to the properties of hadrons.
With these EFTs, the FV mass shift of
mesons, baryons and nuclei are calculated
out to ${\cal O}\left(1/L^4\right)$ in the
$1/L$ expansion,
including contributions from their charge radii, magnetic moments and polarizabilities.
The NREFTs have the advantage that the coefficients of operators coupling to the electromagnetic field
are directly related,
order by order in the $\alpha_e$,
to the electromagnetic moments of the hadrons (in the continuum limit),
as opposed to a
perturbative estimate thereof (as is the case in $\chi$PT).
For protons and neutrons, the NREFT is the well-established
non-relativistic QED (NRQED)~\cite{Isgur:1989vq,Isgur:1989ed,Jenkins:1990jv,Jenkins:1991ne,Thacker:1990bm, Labelle:1992hd,Manohar:1997qy,Luke:1997ys, Hill:2011wy},
modified to include the finite extent of the charge and current densities~\cite{Chen:1999tn}.
Including multi-nucleon interactions, this framework has been
used extensively to describe the low-energy behavior of nucleons and nuclear interactions, $ {\rm EFT}(\pislash) $,
along with their interactions with electromagnetic fields~\cite{Kaplan:1998tg, Kaplan:1998we, Chen:1999tn, Butler:1999sv, Butler:2000zp, Butler:2001jj},
and is straightforwardly generalized to hadrons and nuclei with arbitrary angular momentum.
LQCD calculations performed with background electromagnetic fields are currently making use of these NREFTs
to extract the properties of hadrons, including magnetic moments and polarizabilities \cite{Martinelli:1982cb, Fiebig:1988en, Bernard:1982yu, Lee:2005ds, Christensen:2004ca, Lee:2005dq, Engelhardt:2007ub, Detmold:2009dx, Alexandru:2009id, Detmold:2010ts, Primer:2013pva, Lee:2013lxa}.
\section{Finite-Volume QED}
\noindent
The issues complicating the inclusion of QED in FV calculations
with PBCs are well
known,
the most glaring of which is the inability to preserve Gauss's law~ \cite{Duncan:1996xy,Blum:2007cy,Hayakawa:2008an},
which relates the electric flux penetrating
any closed surface to the charge enclosed by the surface, and Ampere's Law, which relates the integral of the
magnetic field around a closed loop to the current penetrating the loop.
An obvious way to see the problem is to consider the electric field along the axes of the cubic volume
(particularly at the surface) associated with a point charge at the center.
Restating the discussions of Ref.~\cite{Hayakawa:2008an}, the variation of the QED action is,
for a fermion of charge $e Q$,
\begin{eqnarray}
\delta S & = &
\int\ d^4x\
\left[ \
\partial_\mu F^{\mu\nu} (x)
\ -\
e\ Q\ \overline{\psi} (x)\gamma^\nu \psi (x)
\ \right]
\ \delta \left(A_\nu (x)\right)
\nonumber\\
& = &
\int dt\
{1\over L^3}\ \sum_{\bf q}\
\delta\left(\tilde A_\nu (t,{\bf q})\right)
\int_{L^3}\ d^3 {\bf x}\
e^{i{\bf q}\cdot {\bf x}}\
\left[\
\partial_\mu F^{\mu\nu} (t,{\bf x})
\ -\ e\ Q\ \overline{\psi} (t,{\bf x})\gamma^\nu \psi (t,{\bf x})
\ \right],
\nonumber\\
\label{eq:gauss}
\end{eqnarray}
where
$\tilde A_\nu (t,{\bf q})$ is the spatial Fourier transform of $A_\nu (t,{\bf x})$, and
$e=|e|$ is the magnitude of the electronic charge.
For simplicity, here and in what follows,
we assume the time direction of the FV to be infinite~\footnote{
In practice, there are thermal effects in LQCD calculations due to the finite extent of the time direction.}
while the spatial directions are of length $L$.
Eq. (\ref{eq:gauss}) leads to
$\partial_\mu F^{\mu\nu} = e Q \overline{\psi} \gamma^\nu \psi $
for
$\delta S=0$ and hence Gauss's Law and Ampere's Law.
This can be modified to $\partial_\mu F^{\mu\nu} = e Q \overline{\psi} \gamma^\nu \psi + b^\nu$ simply by omitting the
spatial zero modes of
$A_\mu$, i.e. $\tilde A_\nu (t,{\bf 0}) = 0$,
or more generally by setting $\delta \tilde A_\nu (t,{\bf 0}) = 0$,
where $b^\nu$ is some uniform background charge
distribution~\cite{Portelli:2010yn}.~\footnote{
The introduction of a uniformly charged background is a technique that has
been used extensively to include electromagnetic interactions into calculations of many-body systems,
such as nuclear matter and condensed matter, see for example Ref.~\cite{2000physics..12024C}.
}
This readily eliminates the relation between the electric flux penetrating a closed surface and the inserted charge,
and the analogous relation between the magnetic field and
current.~\footnote{For a discussion about including QED with C-PBCs (anti-PBCs), see Ref.~\cite{Kronfeld:1992ae}.}
Ensuring this constraint is preserved under gauge transformations,
$A_\mu (t,{\bf x})\rightarrow A^\prime_\mu (t,{\bf x}) = A_\mu (t,{\bf x}) + \partial_\mu \Lambda(t,{\bf x})$, where
$\Lambda$ is a periodic function in the spatial volume,
requires
$ \partial_0 \tilde\Lambda(t,{\bf 0})=0$, where $\tilde\Lambda (t,{\bf q})$ is the Fourier transform of $\Lambda (t,{\bf x})$.
Modes with ${\bf q}\ne {\bf 0}$ are subject to the standard gauge-fixing conditions, and in LQCD calculations it is
sometimes convenient to work in Coulomb gauge,
${\bm\nabla}\cdot {\bf A}=0$.
This is because of the asymmetry between the spatial and temporal directions that is present in most ensembles of gauge field configurations,
along with the fact that the photon fields are generated in momentum space as opposed to position space.\footnote{The generation of gauge-field configurations in the non-compact formulation of lattice QED is usually performed in momentum space. This, first of all, makes the exclusion of the zero modes of the QED gauge field easy.
Secondly, the lattice gauge condition in momentum space provides a linear relation among modes and one of the degrees of freedom of $A_i$ can be eliminated in favor of the other two, see Ref. \cite{Blum:2007cy}.}
In infinite volume, the Coulomb potential energy between charges $eQ$ is well known to be
$U(r) = \frac{\alpha_e Q^2}{r}$, where $\alpha_e=e^2/4\pi$ is the QED fine-structure constant,
while in a cubic spatial volume
with the zero modes removed, it is
\begin{eqnarray}
U( {\bf r} ,L)
& = &
{\alpha_e Q^2 \over \pi L}
\sum_{{\bf n}\ne {\bf 0}}
{1\over |{\bf n}|^2}
e^{i 2\pi {\bf n}\cdot {\bf r}\over L}
\nonumber\\
& = &
{\alpha_e Q^2\over \pi L}
\left[-1 +
\sum_{{\bf n}\ne {\bf 0}}
{e^{-|{\bf n}|^2}\over |{\bf n}|^2}
e^{i 2\pi {\bf n}\cdot {\bf r}\over L}
+
\sum_{\bf p}\ \int_0^1\ dt\ \left({\pi\over t}\right)^{3/2}
e^{ - {\pi^2 |{\bf p}-{\bf r}/L |^2\over t} }
\ \right],
\label{eq:Vgreen}
\end{eqnarray}
where
${\bf n}$ and $\mathbf{p}$ are triplets of integers.
The latter,
exponentially accelerated, expression in Eq.~(\ref{eq:Vgreen}) is obtained from the former using the Poisson summation formula.
\begin{figure}[t]
\centering
\includegraphics[scale=0.55]{potplotrxzerozerofulltwo}
\caption{{\small The FV potential energy between two charges with $Qe=1$, along one of the axes of a cubic volume of spatial extent $L$ (solid orange curve),
obtained from Eq.~(\protect\ref{eq:Vgreen}), and the corresponding infinite-volume Coulomb potential energy (dashed gray curve).
} }
\label{fig:pot}
\end{figure}
The FV potential energy between two charges with $Qe = 1$, and the
corresponding infinite-volume Coulomb potential energy are shown in Fig.~\ref{fig:pot}.
In the next sections, we construct non-relativistic EFTs to allow for order-by-order calculations of the FV QED modifications
to the energy of hadrons in
the continuum limit of
LQCD calculations, going beyond the first two orders in the $1/L$ expansion that have
been determined previously.
While these EFTs permit calculations to any given precision, including quantum fluctuations, some of
the results that will be presented can be determined simply without the EFTs;
a demonstration of which is the self-energy of a uniformly charged, rigid and fixed, sphere in a FV.
In this textbook case, the self-energy can be determined directly by integrating the interaction
between infinitesimal volumes of the charge density, as governed by the modified Coulomb potential, Eq.~(\ref{eq:Vgreen}),
over the sphere of radius $R$. It is straightforward to show that the self-energy can be written in an expansion of $R/L$,
\begin{eqnarray}
\label{ChargedSphere}
U^{\rm sphere} (R,L) & = &
{3\over 5} {(Qe)^2\over 4 \pi R}\
\ +\
{(Qe)^2\over 8\pi L}\ c_1
\ +\
{(Qe)^2\over 10 L} \left({R\over L}\right)^2\
+\ \cdots,
\end{eqnarray}
where $c_1 =-2.83729$~\cite{Luscher:1986pf, Hasenfratz:1989pk, Luscher:1990ux}.
The leading contribution is the well-known result for a uniformly charged sphere,
while the second term, the
LO FV correction, is independent of the structure of the charge distribution.
This suggests that it is also valid for a point particle; a result that proves to be valid
for the corrections to the masses of single particles calculated with $\chi$PT and with the
NREFTs presented in this work.
It is simply the modification to the Coulomb self-energy of a point charge.
The third term can be written as ${(Qe)^2} \langle r^2\rangle / 6 L^3$,
where $ \langle r^2\rangle=\frac{3}{5}R^2$ is the mean-squared radius of the sphere, and
reproduces the charge-radius contributions determined with the NREFTs,
as will be shown in the next section.
\section{ Scalar NRQED for Mesons and $J=0$ Nuclei}
\noindent
LQCD calculations including QED have been largely
focused on the masses of the pions and kaons in an effort to extract the values of electromagnetic counterterms of $\chi$PT, thus
we begin by considering the FV corrections to the masses of scalar hadrons.
In the limit where the volume of space is much larger than that of the hadron, keeping in mind that only the zero modes are being excluded from the photon fields,
the FV corrections to the mass of the hadron will have a power-law dependence upon $L$, and vanish as $L\rightarrow\infty$.
As the modifications to the self-energy arise from the infrared behavior of the theory,
low-energy EFT provides a tool
with which to systematically determine the FV effects in an expansion in one or more small parameters.
Using the methods developed to describe heavy-quark and heavy-hadron
systems~\cite{Isgur:1989vq,Isgur:1989ed,Jenkins:1990jv,Jenkins:1991ne,Thacker:1990bm,Labelle:1992hd,Manohar:1997qy,Luke:1997ys,Chen:1999tn,Beane:2007es,Lee:2013lxa},
the Lagrange density describing the low-energy dynamics of a charged composite scalar particle, $\phi$, with charge $eQ$
can be written as an expansion in $1/m_{\phi}$ and in the scale of compositeness,
\begin{eqnarray}
{\cal L}_\phi
& = &
\phi^\dagger \left[\
iD_0
\ +\ {|{\bf D}|^2\over 2 m_\phi}
\ + \ { |{\bf D}|^4 \over 8 m_\phi^3}\
\ +\ {e \langle r^2\rangle_\phi\over 6}\ {\bm\nabla}\cdot {\bf E}
\ +\ 2 \pi \tilde\alpha_E^{(\phi)} |{\bf E}|^2
\ +\ 2\pi \tilde\beta_M^{(\phi)} |{\bf B}|^2
\right.\nonumber\\
&&\left.
\qquad \qquad
\ +\ i e c_M\ { \{ D^i , ({\bm\nabla}\times {\bf B})^i \} \over 8 m_\phi^3}
\ +\ \cdots
\ \right] \phi,
\label{eq:scalarLag}
\end{eqnarray}
where $m_{\phi}$ is the mass of the particle, the covariant derivative is $D_\mu = \partial_\mu + i e \hat Q A_\mu$ with $\hat Q$ the charge operator.
$\langle r^2\rangle_\phi$ is the mean-squared charge radius of the $\phi$,
and we have performed the standard field redefinition to the NR normalization of states,
$\phi\rightarrow \phi/\sqrt{2 m_\phi}$.
The remaining coefficients of operators involving the
electric, ${\bf E}$, and magnetic, ${\bf B}$, fields,
have been determined by matching this EFT to scalar QED, to yield
\begin{eqnarray}
\tilde\alpha_E^{(\phi)} & = &
\alpha_E^{(\phi)} - {\alpha_e Q\over 3 m_\phi} \langle r^2\rangle_\phi
\ \ ,\ \
\tilde\beta_M^{(\phi)} \ = \
\beta_M^{(\phi)}
\ \ ,\ \
c_M\ =\ {2\over 3} m_\phi^2 \langle r^2\rangle_\phi,
\label{eq:scalarpols}
\end{eqnarray}
where $\alpha_E^{(\phi)} , \beta_M^{(\phi)}$ are the electric and magnetic polarizabilities of the
$\phi$.~\footnote{
The presence of a charge-radius dependent term in the coefficient of the
electric polarizability indicates a subtlety in using this EFT to describe hadrons in a
background electric field~\cite{Lee:2013lxa}.
Such contributions can be cancelled by including redundant operators in the EFT Lagrange density
when matching to S-matrix elements.
Since a classical uniform electric field modifies the equations of motion, such operators must be retained in the
Lagrange density and their coefficients matched directly to Green functions.
}
These coefficients will be modified at higher orders in perturbation theory,
starting at $\mathcal{O}(\alpha_e)$.
They will also be modified by terms that are exponentially suppressed by compositeness length scales, e.g.
$\sim e^{-m_\pi L}$ for QCD.
The ellipses denote terms that are higher order in
derivatives acting on the fields, with coefficients dictated by the mass and compositeness scale
-- the chiral symmetry breaking scale, $\Lambda_\chi$, for mesons and baryons.
For one-body observables, terms beyond
$\phi^\dagger i \partial_0 \phi$ are treated in perturbation theory, providing a systematic expansion in $1/L$.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.230]{onelooplo}
\caption{{\small The one-loop diagram providing the LO, $\mathcal{O}(\alpha_e/L)$,
FV correction to the mass of a charged scalar particle.
The solid straight line denotes a scalar particle, while the wavy line denotes a photon.
}}
\label{fig:LO}
\end{figure}
The LO, $\mathcal{O}(\alpha_e/L)$,
correction to the
mass of a charged scalar particle in FV, $\delta m_\phi$,
is from the one-loop diagram shown in
Fig.~\ref{fig:LO}.
While most simply calculated in Coulomb gauge, the diagram can be calculated in any gauge and,
in agreement with previous determinations~\cite{Hayakawa:2008an}, is
\begin{eqnarray}
\delta m_\phi^{({\rm LO})}
& = &
{\alpha_e Q^2\over 2\pi L}\
\hat{\sum_{ {\bf n}\ne {\bf 0}}}\
{1\over |{\bf n}|^2}
\ =\
{\alpha_e Q^2\over 2 L}\ c_1,
\label{eq:scalarLO}
\end{eqnarray}
with $c_1 = -2.83729$.
The sum, $\hat{\sum}$, represents the difference between the sum over the FV modes and the
infinite-volume integral, e.g.
\begin{eqnarray}
{1\over L^3}\hat{\sum_{{\bf k}\ne {\bf 0}}}\
f({\bf k})
&\equiv &
{1\over L^3}\sum_{{\bf k}\ne {\bf 0}}\ f({\bf k})
\ -\
\int {d^3{\bf k}\over (2\pi)^3}\ f({\bf k})
,
\label{eq:FVsumint}
\end{eqnarray}
for an arbitrary function $f({\bf k})$,
and is therefore finite.
This shift is a power law in $1/L$ as expected, and provides a reduction in the mass of the hadron.
As the infinite-volume Coulomb interaction increases the mass, and the FV result is obtained
from the modes that satisfy the PBCs (minus the zero modes), the sign of the correction is also expected.
The result in Eq.~(\ref{eq:scalarLO}) is nothing more than the difference between the FV and infinite-volume
contribution to the Coulomb self-energy of a charged point particle, as seen from
Eq.~(\ref{eq:Vgreen}), $U({\bf 0},L)/2$.
\begin{figure}[!ht]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.24]{oneovermone}}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermtwo}}
\subfigure[]{
\includegraphics[scale=0.24]{oneovermthree}}
\subfigure[]{
\includegraphics[scale=0.24]{oneovermfour}}
\caption{{\small Diagrams contributing at NLO,
$\mathcal{O}(\alpha_e/m_{\phi}L^2)$, in the ${1/ L}$ expansion.
The crossed circle denotes an insertion of the $|{\bf D}|^2/2 m_\phi$
operator in the scalar QED Lagrange density, Eq.~(\protect\ref{eq:scalarLag}).
}}
\label{fig:mall}
\end{center}
\end{figure}
The next-to-LO (NLO) contribution, ${\cal O}\left(\alpha_e/L^2\right)$,
arises from a single insertion of the
$|{\bf D}|^2/2 m_\phi$ operator in Eq.~(\protect\ref{eq:scalarLag}) into the one-loop diagrams shown in Fig.~\ref{fig:mall}.
The contribution from each of these diagrams depends upon the choice of gauge, however the sum is gauge independent,~\footnote{
The sums appearing at LO and NLO are
\begin{eqnarray}
\hat{\sum_{ {\bf n}\ne {\bf 0}}}\ {1\over |{\bf n}|} & = & c_1
\ \ ,\ \
\hat{\sum_{ {\bf n}\ne {\bf 0}}}\ {1\over |{\bf n}|^2} \ = \ \pi\ c_1
.
\nonumber
\end{eqnarray}
}
\begin{eqnarray}
\delta m_\phi^{({\rm NLO})}
& = &
{\alpha_e Q^2\over m_\phi L^2 }\
\hat{\sum_{{\bf n}\ne {\bf 0}}}\
{1\over |{\bf n}|}
\ =\
{\alpha_e Q^2\over m_\phi L^2 }\ c_1
.
\label{eq:scalarNLO}
\end{eqnarray}
This NLO recoil correction agrees with previous calculations~\cite{Hayakawa:2008an,deDivitiis:2013xla},
and is the highest order in the $1/L$ expansion
to which these FV effects have been previously
determined.~\footnote{
The $\mathcal{O}(\alpha_e)$ calculations of Ref.~\cite{Hayakawa:2008an} at NLO in $\chi$PT and PQ$\chi$PT do not
include the full contributions from the meson charge radius and polarizabilities, but are perturbatively close.
This is in contrast to the NREFT calculations presented in this work
where the low-energy coefficients are matched to these quantities
order by order in $ \alpha_e$,
and provide the result at any given order in $1/L$ as an expansion in $ \alpha_e$.
}
\begin{figure}[!ht]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.21]{oneovermtwoone}}
\subfigure[]{
\includegraphics[scale=0.21]{oneovermtwotwo}}
\subfigure[]{
\includegraphics[scale=0.21]{oneovermtwothree}}
\subfigure[]{
\includegraphics[scale=0.21]{oneovermtwofour}}
\subfigure[]{
\includegraphics[scale=0.185]{oneloopcrone}}
\subfigure[]{
\includegraphics[scale=0.185]{oneloopcrtwo}}
\caption{{\small
(a-d) One-loop diagrams giving rise to the recoil corrections of $\mathcal{O}(\alpha_e/m_\phi^2 L^3)$.
The crossed circle denotes an insertion of the $|{\bf D}|^2/2 m_\phi$ operator.
(e,f) One-loop diagrams providing the leading contribution from the charge radius of the scalar hadron,
$\sim \alpha_e \langle r^2\rangle_\phi/L^3$.
The solid square denotes an insertion of the charge-radius
operator in the scalar Lagrange density, Eq.~(\protect\ref{eq:scalarLag}).
}}
\label{fig:m2}
\end{center}
\end{figure}
At next-to-next-to-LO
(N$^2$LO), ${\cal O}\left(\alpha_e/L^3\right)$,
there are potentially two contributions - one is a recoil correction of the form $\sim \alpha_e /m_\phi^2 L^3$
and one is from the charge radius, $\sim \alpha_e \langle r^2\rangle_\phi/L^3$.
An evaluation of the one-loop diagrams giving rise to the recoil contributions, Fig.~\ref{fig:m2}(a-d), shows that
while individual diagrams are generally non-zero for a given gauge, their sum vanishes in
any gauge.
Therefore, there are no contributions of the form $\alpha_e/m_\phi^2 L^3$ to the mass of $\phi$.
In contrast, the leading contribution from the charge radius of the scalar particle, resulting from the one-loop diagrams shown in Fig.~\ref{fig:m2}(e,f)
gives a contribution of the form
\begin{eqnarray}
\delta m^{({\rm N^2LO})}_\phi & = &
-{2\pi\alpha_e Q\over 3 L^3}\ \langle r^2 \rangle_\phi \
\hat{\sum_{{\bf n}\ne {\bf 0}}} ~ 1
\ =\
+{2\pi\alpha_e Q\over 3 L^3}\ \langle r^2 \rangle_\phi ,
\label{eq:CRscalar}
\end{eqnarray}
where $\hat{\sum\limits_{\bf n} }~1 = 0$.
\begin{figure}[t!]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermthreeone}}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermthreetwo}}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermthreethree}}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermthreefour}}
\subfigure[]{
\includegraphics[scale=0.22]{oneovermthreedfourone}}
\subfigure[]{
\includegraphics[scale=0.21]{oneovermthreedfourtwo}}
\subfigure[]{
\includegraphics[scale=0.14]{oneloopebpol.pdf}}
\caption{{\small
One-loop diagrams contributing to the FV corrections to the mass of a scalar hadron at
N$^3$LO, ${\cal O}\left(1/L^4\right)$.
Diagrams (a-d) involve three insertions of the $|{\bf D}|^2/2 m_\phi$ operator (crossed circles)
in the scalar QED Lagrange density in Eq.~(\protect\ref{eq:scalarLag}),
while (e,f) involve one insertion of the $|{\bf D}|^4/8 m_\phi^3$ operator (the sun cross),
giving a $\mathcal{O}(\alpha_e/m_\phi^3 L^4)$ correction.
Diagram (g) involves an insertion of $\tilde{\alpha}_E^{(\phi)} \ |{\bf E}|^2$ and $\tilde{\beta}_M^{(\phi)}\ |{\bf B}|^2$,
operators (crossed square),
contributing terms of the form
$\sim (\alpha_E+\beta_M)/L^4$ and $\sim \alpha_e\langle r^2 \rangle_\phi/m_{\phi}L^4)$.
A diagram analogous to (g) provides the leading contribution from the $c_M$ operator at $\mathcal{O}(\alpha_e/m_\phi L^4)$.
}}
\label{fig:m3A}
\end{center}
\end{figure}
At N$^3$LO, ${\cal O}\left(\alpha_e/L^4\right)$,
there are potentially three contributions:
recoil corrections, $\sim \alpha_e/m_\phi^3 L^4$,
contributions from the electric and magnetic polarizability operators,
$\sim \tilde\alpha_E^{(\phi)}/L^4$ , $\tilde\beta_M^{(\phi)} /L^4$,
and contributions from the $c_M$ operator, Eq. (\ref{eq:scalarLag}).
There are two distinct sets of recoil corrections at this order.
One set is from diagrams involving three insertions of the $|{\bf D}|^2/2 m_\phi$ operator,
as shown in Fig.~\ref{fig:m3A}(a-d),
and the other is from a single insertion of the $|{\bf D}|^4/8 m_\phi^3$ operator, shown in Fig.~\ref{fig:m3A}(e,f).
The sum of diagrams contributing to each set vanishes, and so there are no contributions of the form $\alpha_e/m_\phi^3 L^4$.
The other contributions, that include the electric and magnetic polarizabilities, arise from the one-loop diagrams shown in Fig.~\ref{fig:m3A}(g). A straightforward evaluation yields a mass shift of
\begin{eqnarray}
\delta m_\phi^{({\rm N^3LO}; \tilde{\alpha},\tilde{\beta})}
&& =
- {4\pi^2\over L^4}\ \left( \tilde{\alpha}_E^{(\phi)} + \tilde{\beta}_M^{(\phi)} \right)\
\hat{\sum_{{\bf n}\ne {\bf 0}}}\ |{\bf n}|\nonumber\\
&& =
- {4\pi^2\over L^4}\ \left( \alpha_E^{(\phi)} + \beta_M^{(\phi)} \right)\ c_{-1}
+{4\pi^2 \alpha_e Q \over 3 m_\phi L^4} \
\langle r^2 \rangle_\phi \ c_{-1},
\end{eqnarray}
where the regularized sum is the same that contributing to the energy density associated with the Casimir effect,
and is
$c_{-1} = -0.266596$ \cite{Hasenfratz:1989pk}.
A similar calculation yields the contribution from the $c_M$ operator,
\begin{eqnarray}
\delta m_\phi ^{({\rm N^3LO}; c_M)}
& = &
+{4\pi^2 \alpha_e Q \over 3 m_\phi L^4} \
\langle r^2 \rangle_\phi \ c_{-1}
.
\end{eqnarray}
Collecting the contributions up to N$^3$LO,
the mass shift of a composite scalar particle in the $1/L$ expansion is
\begin{eqnarray}
\delta m_\phi & = &
{\alpha_e Q^2\over 2 L} c_1
\left( 1 + {2\over m_\phi L} \right)
+ {2\pi \alpha_e Q\over 3 L^3} \left(1+ {4\pi\over m_\phi L} c_{-1} \right) \langle r^2 \rangle_\phi
- {4\pi^2\over L^4}\ \left( \alpha_E^{(\phi)} + \beta_M^{(\phi)} \right) c_{-1}
.
\nonumber\\
\end{eqnarray}
Therefore, for the charged and neutral pions, the mass shifts are
\begin{eqnarray}
\delta m_{\pi^+}& = &
{\alpha_e \over 2 L} c_1
\left( 1 + {2\over m_{\pi^+} L} \right)
+ {2\pi \alpha_e \over 3 L^3} \left(1+ {4\pi\over m_{\pi^+} L} c_{-1} \right) \langle r^2 \rangle_{\pi^+}
- {4\pi^2\over L^4} \left( \alpha_E^{(\pi^+)} + \beta_M^{(\pi^+)} \right) c_{-1},
\nonumber\\
\delta m_{\pi^0}& = &
\ -\ {4\pi^2\over L^4}\ \left( \alpha_E^{(\pi^0)} + \beta_M^{(\pi^0)} \right)\ c_{-1}
,
\end{eqnarray}
where potential complications due to the electromagnetic decay of the $\pi^0$ via the anomaly have been neglected .
The shifts of the charged and neutral kaons have the same form, with $m_{\pi^{\pm ,0}} \rightarrow m_{K^{\pm ,0}}$,
$ \langle r^2 \rangle_{\pi^+} \rightarrow \langle r^2 \rangle_{K^+}$,
$\alpha_E^{(\pi^{\pm , 0})}\rightarrow \alpha_E^{(K^{\pm , 0})}$
and
$\beta_E^{(\pi^{\pm , 0})}\rightarrow \beta_E^{(K^{\pm , 0})}$. With the experimental constraints on the charge radii
and polarizabilities of the pions and kaons,
numerical estimates of the FV corrections can be performed at N$^3$LO.
The LO and NLO contributions are dictated by only the charge and mass of the meson.
The N$^2$LO contribution depends upon the charge and charge radius, which, for the charged mesons, are known experimentally to
be~\cite{Beringer:1900zz},
\begin{eqnarray}
\sqrt{ \langle r^2 \rangle}_{\pi^+}
& = &
0.672\pm 0.008~{\rm fm}
\ \ ,\ \
\sqrt{ \langle r^2 \rangle}_{K^+}
\ = \
0.560\pm 0.031~{\rm fm}
.
\end{eqnarray}
The N$^3$LO contribution from the electric and magnetic polarizabilities of the mesons depends upon their sum.
The Baldin sum rule determines the charged pion combination, while the result of a two-loop $\chi$PT calculation is used for the neutral pion
combination~\cite{Holstein:2013kia},
\begin{align}
& \alpha_E^{(\pi^+)} + \beta_M^{(\pi^+)} & = &
\left(0.39\pm 0.04\right)\times 10^{-4}~{\rm fm}^3
\ ,\
\alpha_E^{(\pi^0)} + \beta_M^{(\pi^0)} =
\left(1.1\pm 0.3\right)\times 10^{-4}~{\rm fm}^3
.
\end{align}
Unfortunately, little is known about the polarizabilities of the kaons, and so
naive dimensional analysis is used to provide an estimate
of their contribution~\cite{Holstein:2013kia},
$ \alpha_E^{(K^+)} + \beta_M^{(K^+)}$, $ \alpha_E^{(K^0)} + \beta_M^{(K^0)} = \left(1\pm 1\right)\times 10^{-4}~{\rm fm}^3$.
With these values, along with their experimentally measured masses,
the expected FV corrections to the charged meson masses are shown in
Fig.~\ref{fig:chargedkaonpionmsquare} and to the neutral meson masses in
Fig.~\ref{fig:neutralkaonpionmsquare}.~\footnote{
When comparing with previous results one should note that the squared mass shift of the $\pi^+$, as an example, due to
FV QED is
\begin{eqnarray}
\delta m_{\pi^+}^2
& = &
\left(
m_{\pi^+} + \delta m_{\pi^+}
\right)^2 - m_{\pi^+}^2
\ =\ 2 m_{\pi^+} \ \delta m_{\pi^+} \ +\ {\cal O}(\alpha_e^2)
,
\nonumber
\end{eqnarray}
As is evident, the leading contribution to the mass squared scales as $1/L$,
contrary to a recent suggestion in the literature~\cite{Portelli:2012pn} of $1/L^2$.
Note that the quantity shown in
Fig.~\ref{fig:chargedkaonpionmsquare} and Fig.~\ref{fig:neutralkaonpionmsquare}
is $\delta m_\phi^2$ as opposed to $\delta m_\phi$,
as it is this that enters into the determination of the light-quark masses from LQCD calculations.
}
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.335]{pionl} \qquad
\includegraphics[scale=0.335]{kaonl}
\caption{{\small The FV QED correction to the mass squared of a charged pion (left panel) and kaon (right panel)
at rest in a FV at the physical pion mass.
The leading contribution is due to their electric charge, and scales as $1/L$.
The $1-\sigma$ uncertainty bands associated with each order in the expansion are determined from the uncertainties in the experimental and theoretical inputs.
}}
\label{fig:chargedkaonpionmsquare}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.335]{pionlzero} \qquad
\includegraphics[scale=0.335]{kaonlzero}
\caption{{\small The FV QED correction to the mass squared of a neutral pion (left panel) and kaon (right panel) at rest in a FV
at the physical pion mass.
The leading contributions are from their polarizabilities, and scale as $1/L^4$.
The $1-\sigma$ uncertainty bands associated with each order in the expansion are determined from the uncertainties in the experimental and theoretical inputs.
}}
\label{fig:neutralkaonpionmsquare}
\end{figure}
In a volume with $L=4~{\rm fm}$, the FV QED mass shift of a charged meson is approximately $0.5~{\rm MeV}$.
Figure~\ref{fig:chargedkaonpionmsquare} shows that for volumes with $L\raisebox{-0.7ex}{$\stackrel{\textstyle >}{\sim}$ } 4~{\rm fm}$,
the meson charge is responsible for essentially all of the FV modifications, with their compositeness
making only a small contribution, i.e. the differences between the NLO and N$^2$LO mass shifts are small.
For the neutral mesons, the contribution from the polarizabilities is very small, but with substantial uncertainty.
It is worth re-emphasizing that in forming these estimates of the QED power-law corrections,
exponential corrections of the form $e^{-m_\pi L}$ have been neglected.
\section{NRQED for the Baryons and $J={1\over 2}$ Nuclei}
\noindent
In the case of baryons
and $J={1\over 2}$ nuclei,
the method for determining the
FV QED corrections is analogous to that
for the mesons, described in the previous section, but modified to include the effects of spin and the
reduction from a four-component to a two-component spinor.
The low-energy EFT describing the interactions between the nucleons and the electromagnetic field is NRQED, but enhanced
to include the compositeness of the nucleon.
A nice review of NRQED, including the contributions from the non point-like structure of the nucleon,
can be found in Ref.~\cite{Hill:2012rh}, and the relevant terms in the NRQED Lagrange density for
a N$^3$LO calculation
are~\cite{Isgur:1989vq,Isgur:1989ed,Jenkins:1990jv,Jenkins:1991ne,Thacker:1990bm,Labelle:1992hd,Manohar:1997qy,Luke:1997ys,Chen:1999tn,Beane:2007es,Lee:2013lxa,Hill:2012rh}
\begin{eqnarray}
{\cal L}_\psi
& = &
\psi^\dagger \left[
iD_0
\ +\ {|{\bf D}|^2\over 2 M_\psi}
\ + \ { |{\bf D}|^4 \over 8 M_\psi^3}\
\ +\ c_F {e\over 2M_\psi} {\bm\sigma}\cdot {\bf B}
\ +\ c_D {e\over 8 M_\psi^2} {\bm\nabla}\cdot {\bf E}
\right.\nonumber\\
&&\left. \qquad \qquad
\ +\ i c_S {e\over 8 M_\psi^2}\ {\bm\sigma}\cdot\left( {\bf D}\times {\bf E} - {\bf E} \times {\bf D} \right)
\ +\ 2 \pi \tilde\alpha_E^{(\psi)} |{\bf E}|^2
\ +\ 2\pi \tilde\beta_M^{(\psi)} |{\bf B}|^2
\right.\nonumber\\
&&\left. \qquad \qquad
\ +\ e\ c_{W_1}\ {\{ {\bf D}^2 , {\bm\sigma}\cdot {\bf B} \} \over 8 M_\psi^3}
\ -\ e\ c_{W_2}\ { D^i {\bm\sigma}\cdot {\bf B} D^i\over 4 M_\psi^3}
\right.\nonumber\\
&&\left. \qquad \qquad
\ +\ e\ c_{p^\prime p}\ { {\bm\sigma}\cdot {\bf D} {\bf B}\cdot {\bf D} + {\bf B}\cdot {\bf D} {\bm\sigma}\cdot {\bf D} \over 8 M_\psi^3}
\ +\ i e \ c_M\ { \{ D^i , ({\bm\nabla}\times {\bf B})^i \} \over 8 M_\psi^3}
\ +\ \cdots
\right] \psi
,
\nonumber\\
\label{eq:baryonL}
\end{eqnarray}
where $c_F = Q + \kappa_\psi + {\cal O}(\alpha_e)$ is the coefficient of the magnetic-moment interaction,
with $\kappa_\psi$
related to the anomalous magnetic moment of $\psi$,
$c_D = Q + {4\over 3} M_\psi^2 \langle r^2\rangle_\psi + {\cal O}(\alpha_e)$ contains the leading
charge-radius contribution, $c_S=2c_F-Q$ is the coefficient of the spin-orbit interaction and
$c_M = (c_D-c_F)/2$.
The coefficients of the $|{\bf E}|^2$ and $|{\bf B}|^2$ terms contain the polarizabilities,
$1/M_\psi$
and $1/M_\psi^{3}$ corrections,
\begin{eqnarray}
\tilde\alpha_E^{(\psi)} & = &
\alpha_E^{(\psi)} - {\alpha_e \over 4 M_\psi^3}\left(Q^2+\kappa_\psi^2\right) - {\alpha_e Q\over 3 M_\psi} \langle r^2\rangle_\psi
\ \ ,\ \
\tilde\beta_M^{(\psi)} \ =
\beta_M^{(\psi)} + {\alpha_e Q^2 \over 4 M_\psi^3}
.
\label{eq:pols}
\end{eqnarray}
The operators with coefficients $c_{W_1}$, $c_{W_2}$ and $c_{p^\prime p}$,
given in Ref.~\cite{Hill:2012rh},
do not contribute to the FV corrections at this order.
The ellipses denote terms that are higher orders in $1/M_\psi$ and $1/\Lambda_\chi$.
Two insertions of the magnetic-moment operator provide its leading contribution,
as shown in Fig.~\ref{fig:magmagloop-NNLO}, giving rise to $\mathcal{O}(\alpha_e/L^3)$ corrections to the mass of spin-$\frac{1}{2}$ particles. Although a single insertion of the $c_S$ operator seems to contribute at N$^2$LO, a straightforward calculation shows that this contribution is vanishing. At N$^3$LO, in addition to the operators contributing to the scalar case, one needs to take into account a diagram with two insertions of the magnetic-moment operator and one insertion of the $|{\bf D}|^2/2 m_\psi$ operator, plus diagrams with insertions of the $c_F$ and $c_S$ operators, as shown in Fig. \ref{fig:magmagloop-NNNLO}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.175]{oneovermtwoloopmagmag}
\caption{{\small The N$^2$LO, ${\cal O}\left( \alpha_e/M_\psi^2 L^3 \right)$,
FV QED correction to the mass of a baryon from its magnetic moment.
The crossed square denotes an insertion of the magnetic moment operator given in Eq.~(\protect\ref{eq:baryonL}).}}
\label{fig:magmagloop-NNLO}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\subfigure[]{
\includegraphics[scale=0.275]{oneovermthreecfcf}}
\subfigure[]{
\includegraphics[scale=0.31]{oneovermthreecfcsi}}
\subfigure[]{
\includegraphics[scale=0.31]{oneovermthreecfcsii}}
\caption{{\small a) The N$^3$LO, ${\cal O}\left( \alpha_e/M_\psi^3 L^4 \right)$
FV QED correction to the mass of a baryon from its magnetic moment.
The crossed square denotes an insertion of the magnetic moment operator given in Eq.~(\protect\ref{eq:baryonL}) while the crossed circle denotes an insertion from the $|{\bf D}|^2/2 m_\psi$ operator. b) Other non-vanishing contributions at this order arise from insertions of the $c_F$ and $c_S$ operators as given in Eq. (\protect\ref{eq:baryonL}). The black circles denote insertions of the $c_S$ operator.}}
\label{fig:magmagloop-NNNLO}
\end{center}
\end{figure}
Without replicating the detail presented in the previous section,
the sum of the contributions to the FV self-energy modification of a composite fermion, up to N$^3$LO, is\footnote{As first noticed by the authors of Ref. \cite{Borsanyi:2014jba}, performing a non-relativistic expansion of the QED self-energy diagram for a point-like particle, although reproduces the result obtained via a NREFT at LO and NLO, naively turns out to be a factor of two bigger than the NNLO (and all higher orders) result presented in this chapter for both scalar and spinor QED. The source of discrepancy appears to be due to separating the range of (scalar) QED momentum summation to IR and UV modes where only in the IR part of the sum an expansion of the summand in $1/m$ is legitimate. The regulated UV sum must be evaluated as well keeping in mind that there exists an ambiguity in defining such a separation scale with $\Lambda \ll m$. The NREFT avoids such issues by appropriately incorporating all the UV contributions in a systematic expansion in the local operators with coefficients that are already matched to reproduce the (scalar) QED on-shell amplitudes. The LO and NLO contributions in the (scalar) QED calculation do not arise from any expansion in $1/m$ and as a result consistently reproduce our results.}
\begin{eqnarray}
\delta M_\psi && =
{\alpha_e Q^2\over 2 L} c_1
\left( 1 + {2\over M_\psi L} \right)
\ +\ {2\pi \alpha_e Q\over 3 L^3} \langle r^2 \rangle_\psi
\ +\ {\pi \alpha_e\over M_\psi^2 L^3}\ \left[\ {1\over 2} Q^2\ + \ (Q+\kappa_{\psi})^2 \right]
\nonumber\\
&& -
{4\pi^2\over L^4} \left( \tilde\alpha_E^{(\psi)} + \tilde\beta_M^{(\psi)} \right) c_{-1}
+ {\pi^2 \alpha_e Q \over M_\psi^3 L^4}\ \left( {4\over 3} M_\psi^2 \langle r^2 \rangle_\psi - \kappa_\psi \right) c_{-1}-\frac{\alpha_e \pi^2}{M_{\psi}^3 L^4}\kappa_{\psi}(Q+\kappa_{\psi})c_{-1}.
\nonumber
\\
\end{eqnarray}
Therefore, for the proton and neutron, the FV QED mass shifts are
\begin{eqnarray}
\delta M_p & = &
{\alpha_e \over 2 L} c_1
\left( 1 + {2\over M_p L} \right)
+ {2 \pi \alpha_e \over 3 L^3} \left( 1 + {4\pi\over M_p L} c_{-1} \right) \langle r^2 \rangle_p
+ {\pi \alpha_e\over M_p^2 L^3}\ \left( {1\over 2} + (1+\kappa_p)^2 \right)
\nonumber\\
& &
- {4\pi^2\over L^4}\left( \alpha^{(p)}_E + \beta^{(p)}_M \right) c_{-1}
\ -\ {2\pi^2 \alpha_e \kappa_p \over M_p^3 L^4}\ c_{-1},
\nonumber\\
\delta M_n & = &
\kappa_n^2\ {\pi \alpha_e \over M_n^2 L^3}
\ -\ {4\pi^2\over L^4} \left( \alpha^{(n)}_E + \beta^{(n)}_M \right) c_{-1},
\end{eqnarray}
where the anomalous magnetic moments of the proton and neutron give
$\kappa_p = 1.792847356(23)$ and
$\kappa_n = -1.9130427(5) M_n/M_p $, respectively~\cite{Beringer:1900zz}.
One of the N$^2$LO contributions to the proton FV QED correction depends upon its charge radius, which is known experimentally to
be,
$\langle r^2 \rangle_{p} = 0.768\pm 0.012~{\rm fm}^2 $~\cite{Beringer:1900zz}.
Further, part of the N$^3$LO contribution depends upon the electric and magnetic polarizabilities,
which are constrained by
the Baldin sum rule,~\cite{Holstein:2013kia}
\begin{align}
&\alpha_E^{(p)} + \beta_M^{(p)} & = &
\left(13.69\pm 0.14\right)\times 10^{-4}~{\rm fm}^3
\ ,\
\alpha_E^{(n)} + \beta_M^{(n)} =
\left(15.2\pm 0.5\right)\times 10^{-4}~{\rm fm}^3
.
\end{align}
With these values for the properties of the proton and neutron, along with their experimentally measured masses,
the expected FV modifications to their masses are shown in Fig.~\ref{fig:protonneutron}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.335]{protonl} \qquad
\includegraphics[scale=0.335]{neutronl}
\caption{{\small The FV QED correction to the mass of the proton (left panel) and neutron (right panel)
at rest in a FV at the physical pion mass.
The leading contribution to the proton mass shift is due to its electric charge,
and scales as $1/L$, while the leading contribution to the neutron mass shift is due to
its magnetic moment, and scales as $1/L^3$.
The $1-\sigma$ uncertainty bands associated with each order in the expansion are determined from the uncertainties in the experimental and theoretical inputs.
}}
\label{fig:protonneutron}
\end{figure}
The proton FV QED corrections are consistent with those of the charged scalar mesons.
However, the neutron corrections, while very small, of the order of a few keVs, exhibit more structure.
The N$^2$LO contribution from the magnetic moment increases the mass in FV, scaling as $1/M_n^2 L^3$, similar to the
polarizabilities which make a positive contribution and scale as $1/L^4$ (N$^3$LO).
Note that the polarizabilities of the nucleon are dominated by the
response of the pion cloud, while the magnetic moments are dominated by physics at the
chiral symmetry breaking scale.
Further the magnetic-moment contributions are suppressed by two powers of the nucleon mass.
There is an interesting difference between the meson and baryon FV modifications.
As the nucleon mass is approximately seven times the pion mass, and twice the kaon mass, the recoil corrections
are suppressed compared with those of the mesons.
Further, the nucleons are significantly ``softer'' than the mesons, as evidenced by their polarizabilities.
However, the NLO recoil corrections to the proton mass
are of approximately the same size as the N$^2$LO structure contributions, as seen in
Fig.~\ref{fig:protonneutron}.
\section{Nuclei: Deuteron and Helium 4}
\noindent
As mentioned in chapter \ref{chap:intro}, a small number of LQCD collaborations have been calculating the binding of light nuclei and
hypernuclei at unphysical light-quark masses in the isospin limit and without
QED~ \cite{Beane:2009py,Yamazaki:2009ua,Beane:2010hg,Inoue:2010es,Inoue:2011pg,Beane:2011iw,Yamazaki:2011nd,Yamazaki:2012hi,Yamazaki:2012fn,Beane:2012vq}.
However, it is known that
as the atomic number of a nucleus increases,
the Coulomb energy increases with the square of its charge, and significantly reduces the
binding of large nuclei.
A NREFT for vector QED shares the features of the NREFTs for scalars and fermions that are relevant for the current analysis.
One difference is in the magnetic moment contribution, and another is the contribution from the quadrupole interaction.
The FV corrections to the deuteron mass and binding energy, $\delta {\rm B}_d$, are shown in Fig.~\ref{fig:deut}, where the
experimentally determined charge radius, magnetic moment and polarizabilities have been used.
\begin{figure}[!tt]
\centering
\includegraphics[scale=0.35]{deutl.pdf} \ \
\includegraphics[scale=0.35]{deutlbinding}
\caption{{\small The left panel shows the FV QED correction to the mass of the deuteron at rest in a FV at the physical pion mass.
The leading contribution is from its electric charges, and scales as $1/L$.
The right panel shows the FV QED correction to the deuteron binding energy for which the $1/L$ contributions cancel.
The $1-\sigma$ uncertainty bands associated with each order in the expansion are determined from the uncertainties in the experimental and theoretical inputs.
}}
\label{fig:deut}
\end{figure}
Due to the large size of the deuteron, and its large polarizability, the $1/L$ expansion converges slowly in modest volumes, and it
appears that $L\raisebox{-0.7ex}{$\stackrel{\textstyle >}{\sim}$ } 12~{\rm fm}$ is required for a reliable determination of the QED FV effects. The QED FV corrections to the deuteron binding energy are seen to be significantly smaller than its total energy in large volumes,
largely because the leading contribution to the deuteron and to the proton cancel.
Deuteron also possesses a quadrupole moment which contributes to the FV QED effects
at ${\cal O}\left(1/L^5\right)$ through two insertions.
The NREFTs used to study the FV contributions to the mass of the pions in the previous section also apply to the $^4$He nucleus,
and the FV corrections to the mass of $^4$He and its binding energy, $\delta {\rm B}_{^4{\rm He}}$, are shown in Fig.~\ref{fig:he4}.
Unlike the deuteron, the leading FV corrections to $^4$He
do not cancel in the binding energy due to the interactions between the two protons, but are reduced by a factor of two.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.335]{hefourl} \ \
\includegraphics[scale=0.335]{hefourlbinding}
\caption{{\small The left panel shows the FV QED correction to the mass of $^4$He at rest in a FV
at the physical pion mass.
The leading contribution is from its electric charge, and scales as $1/L$.
The right panel shows the FV QED correction to the $^4$He binding energy.
The uncertainty bands associated with each order in the expansion are determined from the uncertainties in the experimental and theoretical inputs.
}}
\label{fig:he4}
\end{figure}
\section*{Summary and discussions}
\noindent
For Lattice QCD calculations performed in
volumes that are much larger than the inverse pion mass,
the finite-volume electromagnetic corrections to hadron masses
can be calculated systematically using a NREFT.
The leading two orders in the $1/L$ expansion for mesons
have been previously calculated using chiral perturbation theory, and depend only upon their electric charge and mass.
We have shown that these two orders are universal FV QED corrections to the mass of charged particles.
Higher orders in the expansion are determined by
recoil corrections and by the structure of the hadron,
such as its electromagnetic multipole moments and polarizabilities, which we calculate using a NREFT.
One advantage enjoyed by the NREFT
is that the coefficients of the operators in the Lagrange density are directly related to the structure of the hadron,
order by order in $\alpha_e$,
as opposed to being perturbative approximations as computed, for instance, in $\chi$PT.
For the mesons and baryons, the FV QED effects associated with their structure, beyond their charge,
are found to be small even in modest lattice volumes.
For nuclei, as long as the volume is large enough so that the non-QED effects are exponentially small,
dictated by the nuclear radius,
their charge dominates the FV QED corrections, with only small modifications due to the structure of the nucleus.
The results that we have presented in the previous sections have
assumed a continuous spacetime, and have not yet considered the impact of a finite lattice spacing.
With the inclusion of QED, there are two distinct sources of lattice spacing artifacts that will modify the FV QED corrections we have considered.
The coefficients of each of the higher dimension operators in the NREFTs will receive lattice spacing corrections, and
for an ${\cal O}\left(a\right)$-improved action ($a$ is the lattice spacing)
they are a polynomial in powers of $a$ of the form
$d_i\sim d_{i0} + d_{i2} a^2 + d_{i3} a^3 + ...$.
The coefficients $d_{ij}$ are determined by the strong interaction dynamics and the particular discretizations used in a given calculation.
In addition, the electromagnetic interaction will be modified in analogy with the strong sector, giving rise to further lattice spacing artifacts in the matching conditions between the full and the NR theories, and also in the value of one-loop diagrams.\footnote{The lattice artifacts will depend upon whether the compact or non-compact formulation of QED is employed - the former inducing
non-linearities in the electromagnetic field which vanish in the continuum limit.
The discussions we present in this section apply to both the compact and non-compact formulations.
}. For an improved action, the naive expectation is that such correction will first appear,
beyond the trivial correction from the modified hadron mass in the NLO term,
at
$\alpha_e a^2/L^3$ in the $1/L$ expansion.
They are a N$^2$LO contribution arising from
modifications to the one-loop Coulomb self-energy diagram.
This is the same order as contributions from the charge radius, recoil corrections and the magnetic moment, which are found to make a small contribution
to the mass shift in modest lattice volumes.
As the lattice spacing is small compared to the size of the proton and the inverse mass of the proton,
these lattice artifacts are expected to provide a small modification to the N$^2$LO terms we have determined.
In addition, there are operators in the Symanzik action~\cite{Symanzik:1983dc,Symanzik:1983gh,Parisi:1985iv}
that violate Lorentz symmetry as the calculations are performed on an underlying hypercubic grid.
Such operators require the contraction of at least four Lorentz vectors in order to form a hypercubically-invariant, but Lorentz-violating, operator,
for instance three derivatives and one electromagnetic field, or four derivatives.
We have discussed the suppression of Lorentz-violating contributions at small lattice spacings, along with smearing,
in chapter \ref{chap:operators}.
A second artifact arises from the lattice volume.
The NREFTs are constructed as an expansion in derivatives acting on fields near their classical trajectory.
As emphasized by Tiburzi~\cite{Tiburzi:2007ep} and others, this leads to modifications in calculated
matrix elements because derivatives are approximated by finite differences in lattice calculations.
For large momenta, this is a small effect because of the large density of states, but at low momenta,
particularly near zero, this can be a non-negligible effect that must be accounted for.
This leads to a complication in determining, for instance, magnetic moments from the forward limit of a form factor, relevant to the
discussion in the previous section.
However, this does not impact the present calculations of FV QED corrections to the masses of the mesons, baryons and nuclei.
\chapter{CONCLUSION AND OUTLOOK}
{\label{chap:conclusion}}
Lattice quantum chromodynamics (LQCD) will soon become the primary method in rigorous studies of single- and multi-hadron sectors of QCD. It is truly \emph{ab initio} meaning that its only parameters are those of standard model; the theory that is confirmed to be the true underlying theory of particles and interactions at short distances (below the energy scale where any new physics arises). Progress in LQCD calculations of hadronic spectrum and structure, as well as hadronic interactions and resonances have been significant in recent years. As the computational resources become available in the upcoming years, the systematic uncertainties of these calculations that are associated with discretization and finite volume of spacetime, as well as the input of unphysical light-quark masses will be reduced/eliminated. The topics presented in this thesis include several improvements in our formal understanding of some of these uncertainties, but most importantly they provide new proposals for accessing physical quantities, e.g. those of two-nucleon systems such as scattering parameters, that otherwise would not have been directly accessible through numerical lattice QCD calculations of hadronic (Euclidean) correlation functions. Here we summarize the major lessons to be learned from the findings of the chapters presented and refer the reader to the the summary sections of the corresponding chapters for details.
\begin{itemize}
\item
The singularities observed in taking the continuum limit of correlations functions of higher spin/twist operators is an artifact of probing the short-distance physics of a process whose typical scale extends over a physical region rather than just a lattice site. This is analogue of the idea that physical theories are effective description of physics at a given energy scale where the effect of short distance physics and its underlying symmetries become irrelevant with probes that are smeared over the corresponding distance of the low-energy scale. We have explicitly shown how such mechanism works, through examples in scalar and gauge field theories, where by increasing the resolution of the calculation but keeping fixed the physical region which the operators can probe, the continuum limit of operators with well-defined angular momentum are smoothly approached. Interestingly the coefficients of the``wrong'' angular momentum operators or those of the rotational breaking operators scale at worse with $a^2$, even at presence of quantum effects, consistent with the solid-angle resolution of the physical region that the operator is smeared over. Although already confirmed numerically, our analytical study provides a rigorous understanding of such operator improvement scenarios, and ensures that there is formally no issue with the restoration of rotational symmetry in the continuum limit of lattice field theories.
\item
Larger volumes are not necessarily what LQCD calculations should always ultimately aim for. As realized by Maiani and Testa and signified by L\"uscher's pioneering works in the case of scattering amplitudes, the finite volume of the system of interacting particles is what saves us from running into inaccessibility of amplitudes from the Euclidean correlation functions when the infinite-volume limit is taken. The D/S ratio of the deuteron is a prominent example where, upon using the proper finite-volume (FV) formalism for spin systems and examining the FV symmetries of moving frame calculations, the effect of such a tiny quantity can be artificially enhanced in a FV calculation of the energy levels, giving presumably the only plausible path to a precise first-principle determination of this quantity from QCD. By aiming only for large volumes, such opportunities, in particular with regard to scattering states, are lost. The other side of the story is that by tuning the volumes in the moderate range, the breakdown of rotational symmetry introduces non-negligible systematics in the calculations that must be identified, and whose size must be well quantified. We investigated an example of this again for the deuteron system, where by analyzing the expected energy levels in the upcoming LQCD calculations, we show that the FV-induced partial-wave mixing with higher J-states becomes significant in small to moderate volumes. However a knowledge of the underlying formalism of such mixings helps the lattice practitioner to identify these errors and correct for them systematically.
\item
For the case of masses and binding energies, for which the calculated quantities on the lattice can be directly interpreted as the infinite-volume value through a straightforward extrapolation in volume, novel ideas can result in major volume improvements. This can save significant computational resources by eliminating the need to enlarge the volumes in generating the gauge-field configurations. We have explored this idea through modifying the (valence) quark boundary conditions, and have proved that an order of magnitude volume improvement in the upcoming extracted binding energies of two-nucleon systems is expected by an optimized choice of (twisted) boundary conditions. This is an encouraging result and urges the exploration of similar ideas for other important quantities -- those that are already under investigations by LQCD, such as nucleon structure quantities, e.g., $g_A$, or the binding energies of multi-nucleon systems.
\item
FV effects are the consequence of the manipulation of the system in the IR and therefore the details of the short-range theory used to describe the interactions within the finite volume is irrelevant. As long as the interactions are consistent with the symmetries of the system and are not infinite in range, a low-energy effective theory should suffice to correctly identify the volume effects. We have taken this idea, along with the elimination of the zero mode of the photon in a FV calculation, to estimate volume corrections expected to arise in the masses of hadrons due to quantum electrodynamics (QED) interactions. At each order in the QED coupling and the $1/L$ expansion ($L$ being the spatial extent of the volume), the coefficients of the correction terms are exact to all order in strong interaction effects as they are matched directly to the experimental values of the mass, charge and electromagnetic (EM) radii and moments of particles. The generality of this approach has enabled us to extend it readily to the case of nuclei with spin $0$, $1/2$ and $1$, that is going to be useful in extrapolating extracted masses and binding energies of these systems to the infinite-volume limit once QED interactions are included in these calculations. Nonetheless, quantification of errors associated with the long range of photons has not yet been properly done in most of EM processes in a finite volume. An important example is the LQCD+QED calculation of the hadronic contributions to the muon $g-2$ where an understanding of the expected form of volume corrections is crucial in estimating the uncertainties of the current calculations.
\end{itemize}
The topics presented in this thesis mainly concerns the formal developments in the single- and two-nucleon sectors. We conclude this thesis by briefly discussing the path forward with regard to multi-nucleon(hadron) calculations. There are reasons to speculate that LQCD calculations will only slowly go beyond a direct calculation of the properties of the few-nucleon systems, given the scale of the computational resources available at present and in near future. We should note however that significant efforts are underway on both theoretical and computational sides to improve the situation. These reasons can be put into two categories:
\begin{enumerate}
\item
\emph{Computational issues due to the poor signal to noise ratio}: The poor signal/noise (sign) problem that is inherent in performing LQCD calculation with finite baryon density (chemical potential) continues to pose a challenge, despite much activities at elucidating this problem \cite{Lepage89, MJSsign, Lee:2011sm, Endres:2011jm, Endres:2011mm, Grabowska:2012ik, Detmold:2014hla}. In particular, this has been the main reason why the binding energies of light nuclei have only been calculated via LQCD for pion masses as heavy as $500~{\rm MeV}$ and for systems with no more than four (hyper)nucleons \cite{Beane:2012vq, Yamazaki:2009ua}. Novel theoretical ideas and/or significantly larger computational resources can change the situation. This is going to be an active field of research and investigation in the upcoming years among experts in both nuclear physics and lattice QCD.
\item
\emph{Lack of a finite-volume formalism for multi-particle systems}: Despite the FV formalism (L\"uscher method) for two-hadron sector, the FV formalism for few-body systems either does not yet exist or has not reached the same level of maturity as the two-body case. The first necessary step in being able to properly understand few-body systems would require having a non-perturbative, model-independent framework for the \emph{three-body} sector. There has been much activity in deriving a formal result for the energy quantization conditions of the systems of three bosons in relation to scattering amplitudes \cite{Roca:2012rx, Polejaeva:2012ut,Briceno:2012rv, Hansen:2013dla}, however non of these has yet been implemented in practice. This is due to the non-algebraic form of these quantization conditions -- in contrast to the L\"uscher two-body formula -- which makes the practicality of these formalisms less promising. Having such formalism however seems crucial as otherwise not much information can be gained about multi-nucleon interactions -- no matter how precisely the energy levels are obtained in future LQCD calculations, see Fig. \ref{fig:Multi-NPLQCD}.
\begin{figure}[h!]
\begin{centering}
\includegraphics[scale=0.45]{multinplqcd}
\caption{{\small The location of energy levels of the four-nucleon sector with quantum numbers of $^4He$ can be estimated based on the non-interacting energy levels of $1+1+1+1$, $2+1+1$, $2+2$ and $3+1$ systems. Different colors denote different lattice volumes corresponding to spatial extents of $3.4~{\rm fm}$ (green), $4.5~{\rm fm}$ (orange) and $6.7~{\rm fm}$ (red). The total momentum of the system of particles is zero. The calculated binding energy of $^4He$ at the pion mass of $\approx 800~{\rm MeV}$ is shown \cite{Beane:2012vq}. Even if the computational challenges associated with resolving the dense excited states are overcome, there exists no FV formalism at the moment to extract the corresponding scattering parameters in these channels. Figure is reproduced with the permission of the NPLQCD collaboration.
}}
\label{fig:Multi-NPLQCD}
\par\end{centering}
\end{figure}
\end{enumerate}
It seems that the optimal way to proceed is to take advantage of effective field theories (EFTs) describing few-nucleon systems at low energies. As we already mentioned, experimental data can be used to constrain the value of low-energy constants (LECs) of an EFT.\footnote{EFTs such as $\chi$PT have already been used extensively to interpolate LQCD calculations to the physical pion mass and to get a systematic control over the size of the volume corrections in several quantities such as masses and magnetic moments, see chapter \ref{chap:intro}.} However, there are cases where there is little or no experimental data available to constrain EFTs. In the case of $\pi\pi$ scattering mentioned in chapter \ref{chap:intro}, these LECs were known from the wealth of experimental data, but, for example, in the case of nucleon-hyperon (nucleons with one or more u and d quarks replaced by the strange quark) interactions, these LECs are not known \cite{Beane:2012ey}. In other words, the result of LQCD calculations of few-body systems complement the experimental input into EFT calculations. This direction most likely to be the avenue for implementing the results of first principle LQCD calculations to make systematic EFT-based predictions for multi-hadronic systems in the upcoming years (see a review of this topic in Ref. \cite{Briceno:2014}).
It is notable that by matching to EFTs, the seemingly complicated L\"uscher-type formalism of the multi-particle systems -- by which the scattering parameters are directly evaluated instead of EFT interaction kernels -- will not be essential. The limitation of this method is in lowering the range of validity of the FV formalism to the range of validity of EFTs; as opposed to that of the next particle production threshold (e.g. four-body inelastic thresholds in case of three-body FV formalism). However, using EFTs is far more realistic in terms of their implementation as signified by the recent work of Barnea, \emph{et al.} \cite{Barnea:2013uqa}. Is this work such matching is carried out to the LQCD binding energies of few-nucleon systems at heavy pion masses, and is moving toward making predictions for larger systems of nucleons with solely LQCD inputs \cite{Beane:2012vq}.
\printendnotes
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,640 |
Q: What is the sample size required to observe minimum numbers of each of two types of items? Suppose I have a large population consisting of 1 million items. 75% of these consist of type A and 25% of type B. I need to take a sample from the population but don't know until after the sample what the numbers of each type will be in the sample. My final sample needs to have a minimum of x of type A and and a minimum of y of type B. How many would I have to sample in total to be sure of achieving these numbers with 95% confidence?
A: Let's begin with a general formulation of your problem.
You contemplate taking a sample of a population in some way. A sample of size $n$ will yield two counts: the number of A's and the number of B's. Let $X_n$ represent the count of A's, so that $n-X_n$ is the count of B's. When the sample is random, $X_n$ will be a random variable.
The event of interest to you is that $X_n \ge x$ and $n-X_n \ge y$ where you have specified the thresholds $x$ and $y.$ We may combine these relations mathematically into
$$\mathcal{E}_n:\ x \le X_n \le n-y.$$
Given a probability $1-\alpha,$ such as $95\% = 100 - 5\%,$ you would like to find the least $n\ge 1$ for which $\Pr(\mathcal{E}_n)\ge 1-\alpha.$ To do so, we will have to develop a formula for this probability in terms of $n$ and then solve the inequality.
That's it for the formulation. Let's turn to the analysis of your specific case.
With $x=800,$ $y=600,$ and a population of a million, it doesn't matter whether you sample with or without replacement, because you will find $n$ is going to be a tiny fraction of the population size. Just make sure you sample randomly and independently.
Since there are a huge number of A's and B's in the population and the sample size obviously has to be at least $x+y=1400,$ that's large enough to guarantee that the Normal approximation to the distribution of $X_n$ will be excellent. This simplifies the problem, because all we need to work out are the mean and variance of $X_n.$ The mean obviously is $E[X_n]=3n/4$ because $75\% = 3/4$ of the population consists of A's. The variance depends a tiny bit on whether you sample with or without replacement. When sampling with replacement, $X_n$ has a Binomial distribution with parameters $3/4$ and $n,$ whence its variance is
$$\operatorname{Var}(X_n) = (3/4)(1-3/4)n = \frac{3n}{16}.$$
As usual, it's simplest to standardize $X_n$ for this calculation, so re-express the event as
$$\mathcal{E}_n:\ \frac{x - E[X_n]}{\sqrt{\operatorname{Var}(X_n)}} \le \frac{X - E[X_n]}{\sqrt{\operatorname{Var}(X_n)}} \le \frac{n-y - E[X_n]}{\sqrt{\operatorname{Var}(X_n)}},$$
which simplifies to
$$\mathcal{E}_n:\ \frac{x - 3n/4}{\sqrt{3n/16}} \le Z_n \le \frac{n-y - 3n/4}{\sqrt{3n/16}}$$
where $Z_n$ (the standardized version of $X_n$) has a standard Normal distribution. Writing $\Phi$ for the standard Normal distribution function, we find
$$1-\alpha \le \Pr(\mathcal{E}_n) = \Phi\left(\frac{n-y - 3n/4}{\sqrt{3n/16}}\right) - \Phi\left(\frac{x - 3n/4}{\sqrt{3n/16}}\right).$$
That's a complicated mess: it needs numerical methods to solve, such as repeatedly looking up values in a table or--much better--using a root finder. Deploy the latter by expressing the problem as
Find the smallest $n\gt 0$ for which $$0 = f(n) = \Phi\left(\frac{n-y - 3n/4}{\sqrt{3n/16}}\right) - \Phi\left(\frac{x - 3n/4}{\sqrt{3n/16}}\right) - (1-\alpha)$$ (and then round it up to the nearest integer).
It's not hard to show that such a root exists and is unique (because $f$ is a continuous strictly increasing function with a negative limiting value as $n\to 0$ and positive limit $\alpha$ as $n\to\infty$).
Here, as an illustration, is an R implementation using uniroot, its native root finding function, to find this zero of $f:$
f <- function(alpha=0.05, x=800, y=600, A=0.75) {
sigma <- sqrt(A*(1-A))
f <- function(n) diff(pnorm((c(x, n-y) - A*n) / (sigma * sqrt(n)))) - (1-alpha)
xi <- x/A + y/(1-A)
ceiling(uniroot(f, c(1, xi - 8*qnorm(alpha)*sigma*sqrt(xi)))$root)
}
pnorm implements $\Phi.$
Most of the work (on the last line) is overestimating the sample size so that uniroot has a finite interval in which to search. I use a rough formula that should work in any situation where this analysis applies.
This solution indicates your sample size should be at least $n=2544.$
As a check, I simulated 10,000 samples of this size from your population of one million, without replacement. In 95.3% of these samples at least $800$ A's and at least $600$ B's were observed. This percentage is not significantly different from the target of 95%. Running such a simulation is a reliable, straightforward check of the answer (which, after all, was based on a series of approximations and assumptions).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,013 |
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<transferObjectWithMap xmlns:xs="http://www.w3.org/2001/XMLSchema"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<properties>
<entry>
<key xsi:type="xs:string">2</key>
<value xsi:type="xs:string">Two</value>
</entry>
<entry>
<key xsi:type="xs:string">1</key>
<value xsi:type="xs:string">One</value>
</entry>
</properties>
</transferObjectWithMap> | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,844 |
require 'spec_helper'
require 'fileutils'
describe Octoauth do
describe Octoauth::ConfigFile do
describe '#initialize' do
it 'requires a note' do
expect { Octoauth::ConfigFile.new }.to raise_error ArgumentError
end
context 'when given a file path' do
it 'loads that config' do
config = Octoauth::ConfigFile.new(
note: 'existing_token',
file: 'spec/examples/existing_token.yml'
)
expect(config.token).to eql 'an_existing_token'
end
it 'returns nil if the file does not exist' do
config = Octoauth::ConfigFile.new(note: 'foo', file: 'wat')
expect(config.token).to be_nil
end
end
context 'when give :default' do
it 'uses the default file' do
config = Octoauth::ConfigFile.new(note: 'foo', file: :default)
expect(config.file).to eql File.expand_path(Octoauth::DEFAULT_FILE)
end
end
context 'when given no file' do
it 'returns nil data' do
config = Octoauth::ConfigFile.new(note: 'foo')
expect(config.token).to be_nil
expect(config.file).to be_nil
end
end
end
describe '#write' do
it 'saves a config to disk' do
FileUtils.rm_f 'spec/examples/config_save_test.yml'
random = rand(36**30).to_s(30)
config = Octoauth::ConfigFile.new(
note: 'bar',
file: 'spec/examples/config_save_test.yml'
)
config.token = random
config.write
new_config = Octoauth::ConfigFile.new(
note: 'bar',
file: 'spec/examples/config_save_test.yml'
)
expect(new_config.token).to eql random
end
it 'makes the config file owner-readable' do
FileUtils.rm_f 'spec/examples/priv_test.yml'
random = rand(36**30).to_s(30)
config = Octoauth::ConfigFile.new(
note: 'bar',
file: 'spec/examples/priv_test.yml'
)
config.token = random
config.write
privs = File.stat('spec/examples/priv_test.yml').mode.to_s(8)
expect(privs).to eql '100600'
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 50 |
Only Breakfast in the Meals.
All transfers, sightseeing by AC Vehicle as per the itinerary.
All taxes, driver allowances, toll tax, parking etc.
Day 1: Morning Flight from Delhi for Amritsar. Pickup from Amritsar Airport and transfer to hotel. Later Darshans of Golden Temple and visit of Wagah Border, Night at hotel.
Day 3: Breakfast, Morning tour of Durgiana Temple, Jallianwala Bagh, Ranjit Singh Museum, local Amritsar Bazar and evening drop at Amritsar Airport for flight for Delhi.
Any Airport Pickup or Drop at Delhi.
Anything not mentioned in the Cost Inclusions.
If you have not decided yet or wants to customize tour itinerary or hotels listed above in "Delhi- Amritsar Cultural Tour for 2 Nights/3 Days from sikhtourism", then we can change the travel itinerary and make a personalized, tailor-made holiday package, especially for you. We can modify this travel package as per your itinerary, budget, duration and the tourist places you would like to visit, including transportation and airfare should be included or not. Please fill the form below to contact us. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,683 |
Q: Get the latest release of private GitHub repository for theme update I wanna get the latest release of my theme which is stored in a private repository using pre_set_site_transient_update_themes but I failed with the download link and the authentication.
function github_fetch_user_data($githubArray)
{
$token = '<token>';
$url = 'https://api.github.com/repos/<user>/<repo>/releases';
$args = array(
'headers' => array(
'Authorization' => 'Basic ' . base64_encode('<user>' . ':' . $token)
)
);
$response = wp_remote_get($url , $args);
$body = wp_remote_retrieve_body($response);
$api_response = json_decode($body, true);
//get array of latest release
$api_detect = $api_response[0];
$githubArrayPart = $api_detect[$githubArray];
return $githubArrayPart;
}
I use this function to get a API request but when I use the zipball url with github_fetch_user_data('zipball_url'); with wordpress turns into maintanance and nothing more happens.
What is wrong with my release link or is it impossible to download a privat repository in this way?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,533 |
"""Import Module Plotly To Ploting Graph"""
import plotly.plotly as py
import plotly.graph_objs as go
"""Get data"""
data = open('Real_Final_database_02.csv')
alldata = data.readlines()
listdata = []
for i in alldata:
listdata.append(i.strip().split(','))
"""Create trace"""
type_z = ['Flood', 'Epidemic', 'Drought', 'Earthquake', 'Storm']
fill_colors = ['66FF33', 'FF66CC', '33FFCC', 'FF6600', 'FFFF00']
trace_list = ['trace1', 'trace2', 'trace3', 'trace4', 'trace5']
num = 0
last = []
wide = 0.5
opac = 0.8
for i in range(5):
year_x = []
affected_z = []
types_y = []
for j in listdata:
if j[0] == 'Vietnam' and j[2] == type_z[i]:
year_x.append(int(j[1]))
affected_z.append(int(j[3]))
types_y.append(type_z[i])
if year_x != []:
trace_list[i] = go.Scatter3d(x = year_x, y = types_y, z = affected_z,
mode = 'markers', marker = dict(size = 12, line = dict(color = fill_colors[i], width = wide),
opacity = opac))
last.append(dict(trace_list[num]))
wide += 0.5
opac += 0.1
"""Style Layout"""
layout = dict(
title = 'Total Affected',
scene = dict(xaxis = dict(title = 'Year'), yaxis = dict(title = 'Types of Disaster'), zaxis = dict(title = 'Total Affected')))
fig = go.Figure(data = last, layout = layout)
plot_url = py.plot(fig, filename = 'Total Affected in Vietnam') | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,966 |
El Penjabocs és una muntanya de 614 metres que es troba entre els municipis de les Planes d'Hostoles i de Sant Aniol de Finestres, a la comarca catalana de la Garrotxa.
Referències
Muntanyes de les Planes d'Hostoles
Muntanyes de Sant Aniol de Finestres | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 795 |
Q: Chrome Crashing with Selenium. Error Log: USB: usb_service_win.cc:415 Could not read device interface GUIDs: Trying my hand at controlling Chrome with Selenium. Chrome crashes about 1-2 seconds after opening.
Error Log:
[34512:40740:0119/145119.436:ERROR:device_event_log_impl.cc(215)\] \[14:51:19.436\] USB: usb_service_win.cc:415 Could not read device interface GUIDs: The system cannot find the file specified. (0x2)
[34512:40740:0119/145119.440:ERROR:device_event_log_impl.cc(215)\] \[14:51:19.439\] USB: usb_device_handle_win.cc:1046 Failed to read descriptor from node connection: A device attached to the system is not functioning. (0x1F) \[34512:40740:0119/145119.445:ERROR:device_event_log_impl.cc(215)\]
[14:51:19.445\] USB: usb_device_handle_win.cc:1046 Failed to read descriptor from node connection: A device attached to the system is not functioning. (0x1F)
Code Block:
from selenium.webdriver.chrome.service import Service
from selenium import webdriver
service = Service(executable_path="C:/Users/haseBradshaw/Downloads/chromedriver_win32")
driver = webdriver.Chrome(service=service)
driver = webdriver.Chrome()
driver.get("https://www.selenium.dev/selenium/web/web-form.html")
I've read this post. From what I am able to gather from this, it appears these errors should not be causing a crash.
USB: usb_device_handle_win.cc:1020 Failed to read descriptor from node connection error with ChromeDriver v87 / Chrome v87 using Selenium on Windows10
I've navigated to chomre://flags but there is no #enable-new-usb-backend
I've already ensured my version of chrome driver matches with my version of chrome.
I've inspected chrome://crashes/, but there are no logs.
*
*Are these errors the reason why chrome is crashing?
*How can I fix these errors?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,493 |
I think that the ideal scenario for introducing youth players into an elite level team would be to give them brief cameos in games that have already been won, or give them meaningful minutes in a game that lacks importance. This season for Manchester United, however, a surplus of injuries has forced Louis Van Gaal to call up his youngsters and immediately throw them into the first team rotation. The Red Devils have a history of producing some world-class talent through their academy system (e.g. Class of 92 and the Busby Babes), but this new crop of talent faces an entirely different challenge than generations past.
Barely even into their 20s in most cases, these young players are now being tasked with helping United battle in the league every week, while also dealing with the added fixtures of the Europa League. Van Gaal has a reputation for involving academy players wherever he has managed, but I don't think that even he was prepared to involve them so heavily at this point in the season. The average age of United's starting 11 their last three fixtures has been just over 24.
This unplanned youth revolution at Old Trafford has given unknown names and once forgotten talents a chance to make a name for themselves at this massive club. Three players in particular have been given an amazing opportunity to either revitalize their careers at United, or to make a push out of the academy system once and for all.
When Sir Alex Ferguson signed Powell from Crewe Alexandria back in 2012, the young midfielder was being touted as one of the brightest talents in England. His goal in the 2012 Playoff Final had fans dreaming of the teenager one day controlling United's midfield and pushing them on to Premier League glory. However, things hadn't exactly gone to plan as of late.
After a relatively successful loan stint at Wigan Athletic during the 2013/14 campaign, Powell was again sent on loan to freshly promoted Leicester City. The 20-year-old would make just three appearances for the Foxes before being shipped back to Manchester for his poor work ethic at the club. Things went from bad to worse after that as injuries pushed the midfielder onto the sideline for nearly a year, and reports started to surface that he had become disillusioned with professional soccer.
Recently, Powell has found himself amongst the United first team, and has been earning rave reviews for his performances with the club's reserves. Despite originally being signed as a central midfielder, the 21-year-old has now been transitioned to a more attacking role. In both of his senior appearances this year (both coming in the last two weeks), Van Gaal has deployed Powell as a central striker option, and in the U21's match against Leicester, Powell scored twice while earning rave reviews for his performance in the number nine role. With the lack of depth that the Red Devils have up-top, it's entirely possible that the former Crewe midfielder could see a substantial increase in playing time, but it's unclear whether or not he will be able to carry over his amazing form for the reserves into the first team.
The first official signing of the David Moyes Era in Manchester, many United fans had seemingly forgotten about their Uruguayan fullback going into this season. The 22-year-old spent his first year at the club playing for the reserves, and then last season he made a shocking loan move to join Real Madrid's B-team for the season. In the Spanish capital, Varela managed to impress Zinedine Zidane with the French legend telling reporters that he had no doubt the Uruguayan would one day make it to the top.
To the surprise of some, Varela returned to the club in the summer and again started the season as a member of United's reserve team. Now, injuries to Matteo Darmian, Luke Shaw, and Marcos Rojo have forced Van Gaal to call upon the 22-year-old to start nearly every game for the Red Devils. His playing style has been compared by some to that of former United fullback Gabriel Heinze. Varela is a very aggressive defender that isn't afraid to get stuck in on challenges. This aggressiveness has also seen him caught out of position on several occasions, and it's becoming pretty evident that this is his first action on the senior level.
The increase in playing time can only help Varela as he learns the intricacies of the English game while gaining invaluable experience along the way. It may take him some time to show the quality that had Zidane praising him last season, but in time I think he can become a dependable option for the United backline.
At just 18, Cameron Borthwick-Jackson now finds himself as Manchester United's starting left back. Just typing that sentence makes me feel old at the age of 21. The defender has been a member of United's academy since 2003, and he's so young that he hasn't even had the opportunity to be sent out on loan for some senior experience.
Despite this lack of first team action leading up to his debut, Borthwick-Jackson has been surprisingly confident during his games this season. His link-up play with Memphis Depay (21) on the left hand side has seen the Dutchman improve in his attacking ability, and has also added some much needed creativity to the United attack. The 18-year-old has also been reliable in his defensive responsibilities, but there are still some instances where his youth has been exposed.
The level that he's playing at currently should give United fans some confidence in the club's future talent, but right now I still think he's too young to be counted on week-to-week. However, the injuries to the United backline have left Van Gaal with no choice but to throw the teenager into the starting 11. While Borthwick-Jackson is certainly the fullback of the future at Old Trafford, it's unlikely that he beats out Darmian or Rojo for the starting fullback position.
This injection of youth into the squad is good for future seasons at United, but right now, it's doing little to help them remain in the title race. Young players can only be counted on for so long, before a veteran presence is needed to control the squad. Great managers like Ferguson and Arsene Wenger knew the importance of limiting the roles of youth players in order to prolong their careers and minimize the scrutiny they would face from the fans and the media. Will Van Gaal be able to properly manage his young stars in order to ensure they reach their full potential, or will the high volume of playing time see them crumble under the pressure? | {
"redpajama_set_name": "RedPajamaC4"
} | 5,963 |
Q: Deep Learning in Python - Stochastic Gradient Descent - Breaking down a code I'm trying to learn Deep Learning basically by myself, using a few books provided by my university and this one Neural networks and Deep learning.
The process is hard, and as I'm not used to code either, some issues have appeared. Such as from the function that follows, which is in Chapter 1 of the link provided (I updated the code from 2.7 to 3.6) .
def SGD(self, training_data, epochs, mini_batch_size, eta,
test_data=None):
"""Train the neural network using mini-batch stochastic
gradient descent. The ``training_data`` is a list of tuples
``(x, y)`` representing the training inputs and the desired
outputs. The other non-optional parameters are
self-explanatory. If ``test_data`` is provided then the
network will be evaluated against the test data after each
epoch, and partial progress printed out. This is useful for
tracking progress, but slows things down substantially."""
if test_data: n_test = len(test_data)
n = len(training_data)
for j in range(epochs): #xrange was renamed to range
random.shuffle(training_data)
mini_batches = [
training_data[k:k+mini_batch_size]
for k in range(0, n, mini_batch_size)] #xrange was renamed to range
for mini_batch in mini_batches:
self.update_mini_batch(mini_batch, eta)
if test_data:
print ("Epoch {0}: {1} / {2}".format(
j, self.evaluate(test_data), n_test))
#print (self.biases)
#print("Hello02")
else:
print ("Epoch {0} complete".format(j))
return(self.biases, self.weights)
The issue for me is this one:
if test_data: n_test = len(test_data)
n = len(training_data)
Can anyone explain to me what is happening in this 2 lines? I'm used to a more conventional code stile such as:
if something:
print (another_thing)
A: Maybe I'm misunderstanding you, but :
if test_data: n_test = len(test_data)
n = len(training_data)
... means the same as :
if test_data:
n_test = len(test_data)
n = len(training_data)
This part: if test_data is semantically equivalent with if test_data is not None or if test_data != None.
Please let me know if I misunderstood something :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,207 |
适用于任意ViewGroup。像LinearLayout,ScrollView以及自定义的ViewGroup(流式布局、九宫格balabala)。
相关博文:
https://gold.xitu.io/post/584d52fdb123db00661c59fa
### 1 单Item列表
#### 效果如图:
#### 用法:
Adapter泛型传入JavaBean,构造函数传入数据集和layout布局,**一句代码**搞定:
```
//单一ItemView
ViewGroupUtils.addViews(mLinearLayout, new SingleAdapter<TestBean>(this, mDatas, R.layout.item_test) {
@Override
public void onBindView(ViewGroup parent, View itemView, TestBean data, int pos) {
Glide.with(LinearLayoutActivity.this)
.load(data.getAvatar())
.into((ImageView) itemView.findViewById(R.id.ivAvatar));
((TextView) itemView.findViewById(R.id.tvName)).setText(data.getName());
}
});
```
### 2 多Item、同种数据类型列表
#### 效果如图:
#### 用法:
数据结构相同依然可以给Adapter传入泛型,避免强转:
```
//多种ItemViewType,但是数据结构相同,可以传入数据结构泛型,避免强转
ViewGroupUtils.addViews(linearLayout, new MulTypeAdapter<MulTypeBean>(this, initDatas()) {
@Override
public void onBindView(ViewGroup parent, View itemView, MulTypeBean data, int pos) {
((TextView) itemView.findViewById(R.id.tvWords)).setText(data.getName() + "");
Glide.with(MulTypeActivity.this)
.load(data.getAvatar())
.into((ImageView) itemView.findViewById(ivAvatar));
}
});
```
### 3 多Item、多种数据类型列表
#### 效果如图:
#### 用法:
如果数据结构不同,则不用传入泛型,但是使用时需要强转:
```
//多种Item类型:数据结构不同 不传泛型了 使用时需要强转javaBean,判断ItemLayoutId
ViewGroupUtils.addViews((ViewGroup) findViewById(R.id.activity_mul_type_mul_bean), new MulTypeAdapter(this, datas) {
@Override
public void onBindView(ViewGroup parent, View itemView, IMulTypeHelper data, int pos) {
switch (data.getItemLayoutId()) {
case R.layout.item_mulbean_1:
MulBean1 mulBean1 = (MulBean1) data;
Glide.with(MulTypeMulBeanActivity.this)
.load(mulBean1.getUrl())
.into((ImageView) itemView);
break;
case R.layout.item_mulbean_2:
MulBean2 mulBean2 = (MulBean2) data;
TextView tv = (TextView) itemView;
tv.setText(mulBean2.getName());
}
}
});
```
数据结构:
```
public class MulBean1 implements IMulTypeHelper {
private String url;
@Override
public int getItemLayoutId() {
return R.layout.item_mulbean_1;
}
}
```
```
public class MulBean2 implements IMulTypeHelper {
private String name;
@Override
public int getItemLayoutId() {
return R.layout.item_mulbean_2;
}
}
```
Item1布局是一个ImageView,Item2布局是一个TextView
### 4 Item点击事件
item的点击和长按等事件,有两种方法设置,这里以点击事件为例,长按事件同理:
#### 4.1 Adapter.onBindView()里设置
在`Adapter.onBindView()`方法里能拿到ItemView,自然就可以设置各种事件。类似RecyclerView。
**在这里设置优先级更高。原因后文会提到。**
```
@Override
public void onBindView(ViewGroup parent, View itemView, final MulTypeBean data, int pos) {
....
itemView.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
Toast.makeText(mContext, "onBindView里设置:文字是:" + data.getName(), Toast.LENGTH_SHORT).show();
}
});
}
```
#### 4.2 通过ViewGroupUtils设置
可以在`ViewGroupUtils.addViews`直接作为参数传入.
也可以用`ViewGroupUtils.setOnItemClickListener()`设置 。
**优先级比`Adapter.onBindView()`里设置低,原因后文会提到。**
```
//设置OnItemClickListener
OnItemClickListener onItemClickListener = new OnItemClickListener() {
@Override
public void onItemClick(ViewGroup parent, View itemView, int position) {
Toast.makeText(MulTypeActivity.this, "通过OnItemClickListener设置:" + position, Toast.LENGTH_SHORT).show();
}
};
//可以在`ViewGroupUtils.addViews`直接作为参数传入.\
ViewGroupUtils.addViews(linearLayout, adapter ,onItemClickListener);
//或者 也可以用`ViewGroupUtils.setOnItemClickListener()`设置
ViewGroupUtils.setOnItemClickListener(linearLayout,onItemClickListener);
``` | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,377 |
{"url":"https:\/\/www.askiitians.com\/forums\/Mechanics\/can-we-conserve-energy-in-this-question-or-is-it_259982.htm","text":"\u00d7\n\n#### Thank you for registering.\n\nOne of our academic counsellors will contact you within 1 working day.\n\nClick to Chat\n\n1800-1023-196\n\n+91-120-4616500\n\nCART 0\n\n\u2022 0\n\nMY CART (5)\n\nUse Coupon: CART20 and get 20% off on all online Study Material\n\nITEM\nDETAILS\nMRP\nDISCOUNT\nFINAL PRICE\nTotal Price: Rs.\n\nThere are no items in this cart.\nContinue Shopping\n\nCan we conserve energy in this question.. or is it because it's not bouncing so some energy wud hv gone as heat and sound.. so we cannot.. ? What's the significance of rod doesn't bounce And also if we cannot conserve energy.. how shall we do this?\nCan we conserve energy in this question.. or is it because it's not bouncing so some energy wud hv gone as heat and sound.. so we cannot.. ?\u00a0What's the significance of rod doesn't bounce\u00a0And also if we cannot conserve energy.. how shall we do this?\n\n\n9 months ago\n\nRibhu Archon\n28 Points\n\t\t\t\t\t\t\tNo energy is not conserved as there would be some impulse due to friction and collision with the ground may not be completely elastic.\u00a0But,\u00a0Since when it hits the ground the forces act between the ground and the rod, the combined system of the rod and ground can be considered as a system which has zero external force acting on it.\u00a0Hence \u2018angular momentum of that system\u2019 would be conserved.\u00a0Since ground is undergoing no change in its state of motion.(it is perpetually at rest...),we can just \u2018say\u2019 angular momentum of the rod is conserved\u00a0Significance of saying rough surface and does not bounce:\u00a0By saying the ground is rough they are restricting the rod\u2019s slipping motion on ground. The rod will now get \u2018hinged\u2019 at a point (where it hits the ground) and rotate about that point. No bouncing also serves the same purpose. If the rod bounced, tracking its angular velocity becomes trickier (as the rod might start a translatory motion along with the rotatory motion).\u00a0Now onto the real solution:\u00a0Since in its downward motion there are absolutely no inhibitors. the rod performs a plain translatory motion untill it hits the ground. so untill that moment for all practical purposes we can consider the rod a point mass at the Center of Mass.\u00a0Since it falls through a distance h under a constant acceleration \u2018g\u2019 starting from rest Therefore:\u00a0final velocity:\u00a0\u00a0$v= \\sqrt{2gh}$\u00a0(using equations of motion )\u00a0where v is the velocity of CM\u00a0so just before the rod hits the ground momentum of rod is\u00a0\u00a0$m*\\sqrt{2gh}$\u00a0therefore angular momentum of the rod W.R.T. the point where it hits the ground= p x rwhere p=momentum of rod and r=position vector of rod\u00a0Magnitude of angular momentum just before hitting the ground:$p*\\sin(30)*h$\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [positon vector to CM makes angle 30\u00a0degrees to\u00a0 velocity vector of rod and has magnitude \u2018h\u2019]\u00a0Also, after hitting the ground it gets hinged at the bottom most point and has an angular velocity\u00a0$\\omega$.\u00a0since its hinged at the bottom most point its Moment of Inertia is\u00a0$\\frac{m*(2h)^2}{3}=\\frac{4*m*(h)^2}{3}$\u200b\u00a0therfore angular momentum after collision\u00a0$\\frac{4*m*(h)^2}{3}*\\omega$\u00a0But as we saw angular momentum of the rod is conserved.....\u00a0angular momentum initially=angular momentum finally\u00a0$(m*\\sqrt{2gh})*\\sin(30)*h=\\frac{4*m*(h)^2}{3}*\\omega$\u00a0\u00a0we get,\u00a0$\\omega=\\frac{3}{8}*\\sqrt{\\frac{2g}{h}}$\u00a0\u00a0from this we get\u00a0 y=2 and x=8\u00a0therefore\u00a0$\\frac{x}{y}=\\frac{4}{1}$\n\n9 months ago\nVikas TU\n14149 Points\n\t\t\t\t\t\t\tAn inelastic collisions occurs when two objects collide and do not bounce away from each other. Momentum is conserved, because the total momentum of both objects before and after the collision is the same. However, kinetic energy is not conserved. In an elastic collision, both momentum and kinetic energy are conserved.\n\n9 months ago\nThink You Can Provide A Better Answer ?\n\n## Other Related Questions on Mechanics\n\nView all Questions \u00bb\n\n### Course Features\n\n\u2022 101 Video Lectures\n\u2022 Revision Notes\n\u2022 Previous Year Papers\n\u2022 Mind Map\n\u2022 Study Planner\n\u2022 NCERT Solutions\n\u2022 Discussion Forum\n\u2022 Test paper with Video Solution\n\n### Course Features\n\n\u2022 110 Video Lectures\n\u2022 Revision Notes\n\u2022 Test paper with Video Solution\n\u2022 Mind Map\n\u2022 Study Planner\n\u2022 NCERT Solutions\n\u2022 Discussion Forum\n\u2022 Previous Year Exam Questions","date":"2021-01-23 17:16:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 9, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7314634323120117, \"perplexity\": 1677.9975750014826}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703538226.66\/warc\/CC-MAIN-20210123160717-20210123190717-00306.warc.gz\"}"} | null | null |
Q: Can I show Toast from none-UI Service in android? I have a service which is working in background.
How can I show Toast text message from Service that does not have UI?
No Activity,This is none-UI service.
Is it possible?
A: You can certainly do that. A service is a Context. So you can call
Toast.makeText(this, "My Information", Toast.LENGTH_SHORT).show();
A: Try this in your background thread.
new Handler(Looper.getMainLooper()).post(new Runnable() {
@Override
public void run() {
Toast.makeText(context, "Running background", Toast.LENGTH_SHORT).show();
}
});
A: I have been showing it from dummy classes by using the application class instance, basically what this means that you can show it anywhere within you application (includes service).
public class DemosApplication extends Application {
private static DemosApplication instance;
private Toast toast;
public static DemosApplication getInstance() {
return instance;
}
@Override
public void onCreate() {
super.onCreate();
if (instance == null) {
instance = this;
}
}
public void showToast(int resID) {
showToast(getString(resID));
}
public void showToast(String text) {
if (toast != null) {
toast.cancel();
}
toast = Toast.makeText(this, text, Toast.LENGTH_SHORT);
toast.show();
}
}
Usage from anywhere
DemosApplication.getInstance().showToast("Lalalala");
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,522 |
Q: Handler functions for an objects with complex names I'm pretty new to JS, so my question may be dumb.
I've got some HTML:
<div class="bigouterblock">
<div class="inner-block">
...some content here...
</div>
</div>
Each class has a proper css style.
I'd like to make a handling for onmouseenter event for my inner div.
When I want to do this for some other div, I can make something like this in separate .js-file:
bigouterblock.onmouseenter = function(e)
{
console.log("My mouse is in outer block");
};
But how do I do this for block named "inner-block"? JavaScript handles "-" sign in the name as minus, and it's reasonable. But I really wants to preserve that name :)
What should I do with such a strange thing?
Thanks a lot.
A: You only can query an element like this bigouterblock.onmouseenter if bigouterblock is its id, not class. If you want to query by class, you should use document.getElementsByClassName("inner-block") which returns all elements with the given class name.
for (let inner of document.getElementsByClassName("inner-block")) {
inner.onmouseover = () => console.log("Over .inner-block");
}
<div class="bigouterblock">
outer content
<div class="inner-block">
...some content here...
</div>
outer content
</div>
A: One way of doing it, if you only have one div with the class inner-block could be:
// selects the div element use an unique class name
var element = document.getElementsByClassName("inner-block")[0];
// adds the event listener mouse enter
element.addEventListener("mouseenter", function(e){
console.log("My mouse has entered the element");
});
I would recommend using an id instead of a class selector if you just want to add the mouseenter functionality to only one div or iterating through all elements and adding to each the event listener.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,790 |
\section{Introduction}
Over the last hundred years, researchers accumulated a lot of data on incubation periods for various diseases in various populations.
These data and the existing literature on the subject are thoroughly discussed in~\cite{Strogatz}, a recent paper that motivated the present study, so we are only
giving a brief overview of the most imporant features of the data, referring to~\cite{Strogatz} and references therein for further details.
All of these data show that
within the same population group, a simultaneous exposure to the same pathogen does not result in simultaneous development of symptoms in
all individuals belonging to the group. Instead, those individuals who get sick show a broad distribution of incubation periods (i.e., times between the exposure and symptom onset). Moreover, the shapes
of observed distributions are strikingly similar to each other, being unimodal and right-skewed, with sharp decay on the left tail and extended decay on the right tail, see Figure~\ref{fig:data-1949-and-1950}.
\begin{figure}
\includegraphics[width=12cm]{elife-30212-fig1-v1.jpg}
\caption{\small Data redrawn from historic examples (reproduced from~\cite{Strogatz}, with the authors' permission).
Dashed red curves are lognormal densities and solid blue curves are Gumbel densities
predicted in~\cite{Strogatz}.
(a) Data from an outbreak of food-borne streptococcal sore throat, reported in
\cite{SARTWELL:doi:10.1093/oxfordjournals.aje.a119397},
time is measured in days. (b) Data from a study of bladder tumors among workers following occupational exposure to a carcinogen in a dye plant, \cite{Goldblatt65}. Time is measured in years. }
\label{fig:data-1949-and-1950}
\end{figure}
Incubation periods can be understood as the times needed for multiplication of the harmful agent
populations within the host organisms to reach a symptom onset threshold. The first
explanations of the right-skewness were based on deterministic growth of the harmful
agent population (such as exponential growth) with random parameters varying among individuals of the population and led to lognormal distribution of incubation times. However, there are
cases discussed in \cite{Strogatz} where the randomness in these parameteres is lacking but right-skewed distributions resembling lognormal are still observed.
The new approach of~\cite{Strogatz} and a companion paper~\cite{Strogatz:PhysRevE.96.012313} is to model random incubation periods as stopping times for
certain probabilistic models of the disease spread within an individual infected organism.
In this approach, an organism is modeled by a network of nodes connected to each other by edges and the
spread of the infection or disease is modeled by random evolution of labeling of the network nodes. Each node
is labeled as a healthy resident or a harmful invader, and then at each time step the label configuration is randomly updated according
to certain Markovian mechanism: a healthy resident with a harmful invader neighbor can randomly turn into a harmful invader and vice versa representing either reproduction or death of the disease agents.
In addition to this, the network itself (its nodes and edges) may evolve according to a prescribed set of rules. The incubation period is modeled as a partial or complete takeover of the network by harmful invaders.
Let us briefly summarize the findings of~\cite{Strogatz} without going into the details of
the construction of this Markov process. The results depend on the network geometry.
Massive computer simulations were carried out for several geometries and various values of parameters
of the Markov process involved such as the fitness of the harmful invaders. It was found numerically that
for all these situations the distribution of the time of complete or partial takeover
is close either to Gumbel
or Gaussian distribution, depending on the details of the setup.
In certain cases where the geometry of the network is simple
enough, precise limit theorems with Gumbel or Gaussian scaling limits
in the infinite network size limit were obtained in~\cite{Strogatz} with mathematical rigor.
The Gumbel distribution is right-skewed and the Gaussian distribution is symmetric. In some cases,
the evolution of the system was approximated by a simpler Markov chain also allowing for explicit computations that lead to a mathematical proof of positive skewness of the stopping time under conditioning on its finiteness.
The authors are able to conclude that if the invader fitness is high then the model is similar to the classical ``coupon collection'' problem with right skew and limiting Gumbel distribution and if the fitness is low
then the evolution of their model is similar to that a conditioned random walk that also results in the positive skew of the hitting time distribution.
The limitation of these results is that they are based on a very concrete model with specific update rules. It is not obvious if the computations leading to the rigorous results in~\cite{Strogatz}
or the mathematical and numerical results themselves are valid for a broader class of models, and what precise conditions guarantee this or that kind of behavior.
The goal of the present paper is to suggest a broad class of models based on
exit (or first passage) times for one-dimensional diffusions, i.e., Markov processes that solve
stochastic differential
equations (SDEs) on the real line.
Despite the breadth, this class allows for rigorous analysis and precise mathematical statements on the random variables representing incubation times. An important advantage of our setup is that each SDE model comes with a whole universality class, i.e., a collection of discrete and continuous models that can be approximated by the SDE model. In fact, limit
theorems for stochastic processes with diffusion limits form a classical field of probability theory, see, e.g.,
\cite[Chapter~7]{Ethier-Kurtz:MR838085}.
Our main mathematical results are:
\begin{enumerate}[(i)]
\item a mathematically rigorous proof of right-skewness for the exit time distribution conditioned on exit direction for our model under almost no assumptions besides the fact that it is a $1$-dimensional SDE (we also give a proof of right-skewness of exit times
for a general discrete nearest neighbor random walk aka birth-death process);
\item a description of limiting exit distributions (Gumbel or Gaussian) in the limit of vanishing noise, under natural simple assumptions on the drift of the diffusion process, based on existing rigorous mathematical results.
\end{enumerate}
For incubation periods, this means that they are always right-skewed and that
they are approximately either Gumbel or Gaussian, depending on the condition we impose. Our results
are stable with respect
to model modifications and thus describe large universality classes of systems whose macroscopic behavior
is insensitive to microscopic details.
\section{Our model and main results}
\subsection{Modeling incubation periods with 1-dimensional SDEs}
\label{sec:setting}
We stress that we study the development of the disease within one infected individual and not the spread
of infection between individuals. In our mean field approach, we make a simplifying assumption that the state of the system describing the level of sickness in the individual at each time $t$ is represented by a single real variable $X(t)$. This variable may represent the size of the population of harmful invaders but may also be more involved. The values that $X(t)$ may take and that are of interest to us are concentrated on an interval $[0,R]$. Here,
the left endpoint $0$ corresponds to no sickness at all and represents an infection-free individual. The right endpoint $R$ is the level of the disease corresponding to the onset of symptoms: we assume that the latent sickness develops unnoticed until it reaches the level~$R$.
We also assume that there is a point $x_0\in (0,R)$ such that the immune system of the infected individual does not detect the infection until the level of sickness reaches $x_0$. It is natural to assume that in many situations, in the absence of immune response, the time from the initial exposure to achieving the level~$x_0$ is approximately constant (perhaps very close to $0$) and thus can be ignored in the study of the shape of the incubation period distribution. We further assume that after the immune system detects the infection, $X$ is a time-homogeneous Markov process with continuous paths. Under broad conditions, such a process is a solution
of an SDE:
\begin{equation}
\label{eq:basic_sde}
dX(t)= b(X(t))dt + \sigma(X(t))dW(t).
\end{equation}
The function $b(x)$ usually called the drift and assumed to be smooth in~$x$ represents the combined influence of the infection expansion and the immune response. These influences can be interpreted as the birth rate~$B(x)$ and death rate $D(x)$ of harmful invaders:
$b(x)=B(x)-D(x)$, $x\in [0,R]$.
The randomness in the system is modeled by white noise $dW$ in~\eqref{eq:basic_sde}, where~$W$ is a standard Wiener process or Brownian Motion. We denote probabilities of events by $\mathsf{P}(\cdot)$. The smooth diffusion coefficient $\sigma(x)>0$ represents the amplitude of the noise at~$x\in [0,R]$.
In our mean-field approach we assume that the SDE coefficients~$b$ and~$\sigma$ depend only on $x\in[0,R]$, the single state variable in the system, although more general setups are possible.
We are going to model the incubation period by the exit time from $(0,R)$. Namely, we define the random variable $\tau_\varepsilon$ as the first exit time for the process $X$ from~$(0,R)$:
\[
\tau=\inf\big\{t:X(t)=0\ \textrm{or}\ X(t)=R\big\
\]
There are three possible outcomes of the evolution up to the exit time:
\begin{enumerate}
\item $X(t)$ reaches $R$ before $0$, i.e., $X(\tau)=R$. This means that the immune system was not successful in blocking the infection propagation, and at time $\tau$ the disease is strong enough for the symptom onset, so $\tau$ may be interpreted as the incubation time. The samples in all incubation time studies are based only on the individuals with this outcome.
\item $X(t)$ reaches $0$ before $R$, i.e., $X(\tau)=0$. This means that the immune system has been succesful in complete elimination of the infection by time $\tau$ while no visible symptoms have ever developed. So $\tau$ can be interpreted as the latent disease healing time but the individuals that never develop any symptoms are not in the focus of this paper and the associated statistical data on infection elimination times is not available.
\item $X(t)$ never reaches endpoints $0$ or $R$ staying within $(0,R)$ for all times. In this case $\tau=+\infty$, and the latent infection persists indefinitely fluctuating above the zero level and never being detected.
On the one hand, this situation has zero probability under our assumptions on the coefficients~$b$ and $\sigma$.
On the other hand, the individuals with such behavior are also excluded from incubation period statistical studies.
\end{enumerate}
\subsection{Exit times conditioned on exit through a threshold are always right-skewed}
Our first result concerns the right-skewness of the exit time distribution conditioned on first exit through the right endpoint~$R$.
To state the theorem, we need some notation. Let us denote by $\Gamma$ the symptom onset event, i.e., $\Gamma= \{X(\tau)=R\}$. Under our assumptions, $\mathsf{P}(\Gamma)>0$ and conditioning on $\Gamma$ is well-defined. Under this conditioning, the exit time $\tau$ may be viewed as the first passage time for level~$R$.
The right-skewness of a distribution is formally defined via positivity of the skewness coeffeicient.
Let us now recall the relevant definitions. For a random variable $Y$, its skewness $\gamma(Y)$ is defined by
\begin{equation}
\label{eq:def-of-skew}
\gamma(Y) = \frac{\mathsf{E}(Y-\mathsf{E} Y)^3}{ \mathop{\mathsf{Var}}(Y)^{3/2}}
= \frac{\kappa_3(Y)}{\kappa_2^{3/2}(Y)}.
\end{equation}
Here $\mathsf{E} Y$ is the expectation of $Y$, $\mathop{\mathsf{Var}}(Y)=\mathsf{E}(Y-\mathsf{E} Y)^2$ is the variance of~$Y$, and $\kappa_n(Y)$ stands for the $n$-th cumulant
of~$Y$ defined by
\begin{equation*}
\kappa_n(Y) = \frac{1}{i^k} \left[\frac{d^n}{d\lambda^n} \ln \varphi_Y(\lambda) \right]_{\lambda=0},
\end{equation*}
where $\varphi_Y=\mathsf{E} e^{i\lambda Y}$ is the characteristic function of~$Y$, and $\ln$ denotes the main branch of the logarithm function. The cumulant $\kappa_k(Y)$ is well-defined if $\mathsf{E} |Y|^k<\infty$. Cumulants are Taylor coefficients for $\ln \varphi(\lambda)$ at~$0$:
\begin{equation*}
\ln \varphi_Y(\lambda)=\kappa_1 \frac{it}{1!}+\kappa_2\frac{(it)^2}{2!}+\ldots+\kappa_n\frac{(it)^n}{n!}+o(|t|^n),
\end{equation*}
and can be expressed in terms of moments of $Y$. Denoting $\mathsf{E} Y^k =\alpha_k(Y)$, we have
\begin{equation}
\label{eq:cumulant-via-moments}
\kappa_n(Y)=\alpha_n(Y)+P(\alpha_1(Y),\dots,\alpha_{n-1}(Y)),
\end{equation}
where $P$ is a polynomial with all monomials of degree at least $2$. The precise formula is given in, e.g., \cite[Section 2.12]{Shiryaev:MR1368405}. For $k=1,2,3,$
we have
\begin{align}
\kappa_1(Y)&=\mathsf{E} Y=\alpha_1(Y),\\
\kappa_2(Y)&= \mathop{\mathsf{Var}} Y = \mathsf{E} (Y-\mathsf{E} Y)^2 =\mathsf{E} Y^2 - (\mathsf{E} Y)^2=\alpha_2(Y)-\alpha_1^2(Y) ,\\
\notag
\kappa_3(Y)&=\mathsf{E} (Y-\mathsf{E} Y)^3 = \mathsf{E} Y^3 -3 \mathsf{E} Y^2 \mathsf{E} Y + 2 (\mathsf{E} Y)^3
\\&\hspace{4cm} = \alpha_3(Y) -3 \alpha_2(Y)\alpha_1(Y) + 2\alpha_1^3(Y).
\end{align}
If the moment generating function $M_Y(\lambda)=\mathsf{E} e^{\lambda Y}$ is defined for $\lambda$ in a neighborhood of~$0$, then
\begin{equation*}
\kappa_n(Y) = \left[\frac{d^n}{d\lambda^n} \ln M_Y(\lambda) \right]_{\lambda=0},
\end{equation*}
and
\begin{equation}
\label{eq:expansion_for_mgf}
\ln M_Y(\lambda)=\kappa_1 \frac{\lambda}{1!}+\kappa_2\frac{\lambda^2}{2!}+\ldots+\kappa_n\frac{\lambda^n}{n!}+\ldots
\end{equation}
Our main result on skewness of exit times is:
\begin{theorem}
\label{thm:right-skew}
Under the conditions described in Section~\ref{sec:setting}, conditioned on~$\Gamma$, \[\gamma(\tau)>0.\]
\end{theorem}
Due to \eqref{eq:def-of-skew}, this theorem is a direct consequence of the following:
\begin{theorem}
\label{thm:positive-cumulants}
Under the conditions described in Section~\ref{sec:setting}, conditioned on~$\Gamma$,
\[
\kappa_n(\tau)>0,\quad n\in\mathbb{N}.
\]
\end{theorem}
These two theorems show that incubation periods are always right-skewed and, moreover, all cumulants
of incubation periods are positive. We prove Theorem~\ref{thm:positive-cumulants} in Section~\ref{sec:proofs-of-positive-skewness}.
Although we do not estimate the magnitude of positive cumulants in this proof, such estimates are possible because the proof is based on a representation of $\kappa_n(\tau)$ as an integral of a positive quantity that can be estimated.
Our proof of Theorem~\ref{thm:positive-cumulants} is direct but one could also derive it from the fact that under conditioning on~$\Gamma$, the distribution of $\tau$ is infinitely divisible and concentrated
on $[0,\infty)$.
Infinite divisibility along with some other distributional properties of exit times of $1$-dimensional diffusions conditioned on the direction of exit such as
unimodality and log-concavity will be addressed in a separate publication.
Section~\ref{sec:proofs-of-positive-skewness} also contains proofs of versions of Theorems~\ref{thm:right-skew} and~\ref{thm:positive-cumulants}
for discrete random walks instead of continuous time SDEs.
\subsection{Exit time distributions in vanishing noise limit}
\label{sec:small_noise}
Next we study the situation where the deterministic effects dominate over the random ones in the disease development.
To formalize this, we consider a whole family of SDEs indexed by a small parameter $\varepsilon>0$
and assume that $b$ does not depend on $\varepsilon$ while $\sigma(x)=\sigma_\varepsilon(x)=\varepsilon\sigma_1(x)$ for some smooth function
$\sigma_1(x)>0$. Then SDE~\eqref{eq:basic_sde} rewrites as
\begin{equation*}
dX_\varepsilon(t)= b(X_\varepsilon(t))dt + \varepsilon\sigma_1(X_\varepsilon(t))dW(t),
\end{equation*}
the solution and the associated exit time depend on $\varepsilon$, and we denote them by $X_\varepsilon$ and $\tau_\varepsilon$.
We will describe the limiting behavior of exit times as $\varepsilon\to 0$. Generally speaking, dynamical systems under small noisy perturbations is a well-developed field, see, e.g., the classical monograph~\cite{FW2012}.
Of course, the behavior of the SDE solutions depends crucially on the phase portrait of the vector field $b(x),x\in[0,R]$, i.e., on the structure of subsets of $[0,R]$ where
$b$ is positive, negative, and zero. We recall that a point $x$ is called critical for $b$ if $b(x)=0$.
We will consider the following three situations:
\begin{enumerate}[I.]
\item \label{it:right-drift} There are no critical points on $[0,R]$ and $b(x)>0$ for all $x\in[0,R]$.
\item \label{it:left-drift} There are no critical points on $[0,R]$ and $b(x)<0$ for all $x\in[0,R]$.
\item \label{it:barrier} There is exactly one critical point $p\in(0,R)$; $\lambda:=b'(p)>0$; $b(x)>0$ for all
$x\in(p,R]$; $b(x)<0$ for $x\in[0;p)$.
\end{enumerate}
In dimension~$1$, any vector field $b$ can be represented via gradient of a potential:
$b(x)=-\Phi(x)$. In cases \ref{it:right-drift} and \ref{it:left-drift}, $\Phi$ is monotone on $[0,R]$.
In case~\ref{it:barrier}, $\Phi$ has a maximum at $p$.
We will further subdivide Case~\ref{it:barrier} into two subcases: \ref{it:barrier}$_0$, where $x_0<p$,
and \ref{it:barrier}$_1$, where $x_0>p$. We ignore the exceptional case $p=x_0$ in this paper for brevity,
although the exit times have been studied for this case in detail starting with~\cite{Day:MR1110156},
more on this in Section~\ref{sec:discussion}.
\begin{figure}
\begin{center}
\includegraphics[height=3.5cm]{portrait.pdf}
\end{center}
\caption{\small The phase portraits considered in Section~\ref{sec:small_noise}}
\label{fig:portrait}
\end{figure}
The phase portraits for all these cases are given in Figure~\ref{fig:portrait}.
The archetypal examples
of these cases are:
\begin{equation}
\label{eq:canonical-examples}
b(x)=\begin{cases}
1,& \mathrm{case~\ref{it:right-drift}},\\
-1,& \mathrm{case~\ref{it:left-drift}},\\
\lambda (x-p),& \mathrm{case~\ref{it:barrier}}.
\end{cases}
\end{equation}
In fact, for generic $b$ in each of the cases
\ref{it:right-drift},\ref{it:left-drift},\ref{it:barrier}, there is a smooth coordinate change (conjugation)
$\tilde y=h(x)$ such that the motion along $b$ is transformed, in the new coordinates, into the motion along the
associated canonical drift given in~\eqref{eq:canonical-examples}. In case~\ref{it:right-drift}, one simply can
define $h(x)$ as the time it takes to travel from $0$ to $x$ along $b$;
case~\ref{it:left-drift} is similar; in case~\ref{it:barrier}, the conjugation is slightly more involved,
see, e.g.,~\cite[Section 1]{Eizenberg:MR749377}.
The mathematical analysis of more sophisticated phase portraits is also possible but we consider these three simplest cases because they correspond to the following most natural situations: in
case~\ref{it:right-drift}, the infection is stronger than the immune system over the entire interval~$[0,R]$; in case~\ref{it:left-drift}, the immune system is stronger than the infection propagation
over the entire interval~$[0,R]$; in case~\ref{it:barrier}, the immune system is stronger if the infection level is below the ``critical mass'' $p$, and if the infection level is above that critical mass, then the immune system is not strong enough to prevent the infection growth, at least in the regime described by the deterministic ODE $\dot x = b(x)$.
\smallskip
The symptom onset event describing an exit through the right endpoint depends on $\varepsilon$ in this section, so we will denote it by
$\Gamma_\varepsilon=\{X_\varepsilon(\tau_\varepsilon)=R\}$.
For small $\varepsilon$, the event $\Gamma_\varepsilon$ describes a typical outcome in Cases~\ref{it:right-drift} and~\ref{it:barrier}$_1$,
but it is a rare event in Cases~\ref{it:left-drift} and~\ref{it:barrier}$_0$. The precise mathematical meaning of this claim is given by
the following statement:
\begin{theorem}
\label{th:exit-0-1} In all the cases we are considering, $q=\lim_{\varepsilon\to 0}\mathsf{P}(\Gamma_\varepsilon)$ is well-defined. In cases~{\rm\ref{it:right-drift}} and~{\rm\ref{it:barrier}}$_1$, $q=1$; in cases~{\rm\ref{it:left-drift}} and~{\rm\ref{it:barrier}}$_0$,
$q=0$.
\end{theorem}
This theorem is a specific case of classical results on exit problems for small random perturbations of dynamical systems in the so called Levinson case (where the deterministic orbit started at $x_0$ hits the boundary), see~\cite[Section~2.1]{FW2012}. In all these cases the typical behavior consists in flowing along the vector field $b$ for a finite time.
\medskip
The notion of incubation period is valid only for individuals that develop symptoms, so for
both types of limiting behavior of $\mathsf{P}(\Gamma_\varepsilon)$ described by Theorem~\ref{th:exit-0-1}, we are interested in the statistics of $\tau_\varepsilon$ conditioned on event~$\Gamma_\varepsilon$.
We always have $\mathsf{P}(\Gamma_\varepsilon)>0$, so for any random variable $Y$ its
conditional distribution $\mathop{\mathsf{Law}}[\,Y\,|\,\Gamma_\varepsilon]$ given that the first exit from $(0,R)$
happens through~$R$ is well-defined.
Weak convergence of distributions (also known as convergence in distribution) is denoted by~``$\Rightarrow$''. To state the main mathematical result, we need to recall the standard Gaussian distribution $\mathcal{N}$ which has density
\[
f_{\mathcal{N}}(t)=\frac{e^{-t^2/2}}{\sqrt{2\pi}},\quad t\in\mathbb{R},
\]
and the Gumbel distribution~$\mathcal{G}$ which has distribution function
\[
F_{\mathcal{G}}(t)=e^{-e^{-t}},\quad t\in\mathbb{R},
\]
and density
\[
f_{\mathcal{G}}(t)=e^{-t-e^{-t}},\quad t\in\mathbb{R}.
\]
The densities $f_{\mathcal{N}}$ and $f_{\mathcal{G}}$ are plotted on Figure~\ref{fig:pdfs}.
\begin{figure}
\begin{center}
\includegraphics[width=11cm]{gauss-eps-converted-to.pdf}\\
\includegraphics[width=11cm]{gumbel-eps-converted-to.pdf}
\end{center}
\caption{\small The densities of Gaussian and Gumbel distributions.}
\label{fig:pdfs}
\end{figure}
\begin{theorem}
\label{thm:small-noise}
In cases~{\rm\ref{it:right-drift}}, {\rm\ref{it:left-drift}}, and~{\rm\ref{it:barrier}$_1$}, there are constants $A,B>0$
such that
\begin{equation}
\label{eq:CLT}
\mathop{\mathsf{Law}}\left[\frac{\tau_\varepsilon - A}{B\varepsilon}\, \Big|\, \Gamma_\varepsilon \right] \Rightarrow \mathcal{N}.
\end{equation}
In case~{\rm\ref{it:barrier}$_0$}, there are constants $A\in\mathbb{R}, B>0$ such that
\begin{equation}
\label{eq:Gumbel-limit}
\mathop{\mathsf{Law}}\left[\,\frac{\displaystyle \tau_\varepsilon - \frac{2}{\lambda}\ln \frac{1}{\varepsilon}- A}{B} \, \Bigg|\, \Gamma_\varepsilon \right] \Rightarrow \mathcal{G}.
\end{equation}
\end{theorem}
In other words, conditionally on exit through $R$ (symptom development), in
cases~{\rm\ref{it:right-drift}}, {\rm\ref{it:left-drift}}, and~{\rm\ref{it:barrier}$_1$},
the asymptotic shape of the exit distribution is Gaussian:
\begin{equation}
\label{eq:Gaussian-representation}
\tau_\varepsilon` \stackrel{d}{\approx} A + \varepsilon B N,
\end{equation}
where $N$ has standard Gaussian distribution, and in case~{\rm\ref{it:barrier}$_0$}, the asymptotic shape of the exit distribution is
Gumbel:
\begin{equation}
\label{eq:Gumbel-limit-representation}
\tau_\varepsilon \stackrel{d}{\approx} \frac{2}{\lambda}\ln \frac{1}{\varepsilon}+A+B G = A_\varepsilon +BG,
\end{equation}
where $G$ is a Gumbel random variable, and $A_\varepsilon =\frac{2}{\lambda}\ln \frac{1}{\varepsilon}+A$.
We note that although the exit time distribution is right-skewed for all $\varepsilon>0$, the skew asymptotically
vanishes as $\varepsilon\to0$ in cases~{\rm\ref{it:right-drift}}, {\rm\ref{it:left-drift}}, and~{\rm\ref{it:barrier}$_1$}, and there is no contradiction with the symmetry of the limiting Gaussian distribution.
Theorem~\ref{thm:small-noise} in cases~{\rm\ref{it:right-drift}} and~{\rm\ref{it:barrier}$_1$} is a specisfic case of a
classical result that can be found in~\cite[Section 2.2]{FW2012}.
For case~{\rm\ref{it:left-drift}},
Theorem~\ref{thm:small-noise} was established in~\cite{AB2011a}.
All these situations can be described as the Levinson case according to the terminology of~\cite{FW2012}.
In case~{\rm\ref{it:barrier}$_0$}, the diffusion trajectories that cross the repelling potential wall at $p$ are often call reactive paths. Theorem~\ref{thm:small-noise} in this case describes the conditional limit for the length of reactive paths. It was established first in~\cite{ALEA}. For a discussion of these results and other approaches to them, see also \cite{OnGumbel},\cite{Gumbel-preprint},\cite{Berglund:MR3585827}.
We do not give new proofs for any of the cases in Theorem~\ref{thm:small-noise} here. Our contribution is simply reinterpreting these existing results in terms of incubation periods. We discuss this interpretation and broader context in Section~\ref{sec:discussion}.
\section{Right skewness, positive cumulants: proofs}
\label{sec:proofs-of-positive-skewness}
\subsection{Proof of Theorem~\ref{thm:positive-cumulants} }
In this section we prove Theorem~\ref{thm:positive-cumulants}. The first step is writing down an SDE for the conditioned process.
Conditioned on $\Gamma$, the distribution of process~$X$ coincides with that of the solution of a new SDE
\begin{equation}
\label{eq:h-transform}
dX(t)=\tilde b(X(t))dt+\sigma(X(t))dW(t).
\end{equation}
Here $\sigma$ is the same is in the original SDE~\eqref{eq:basic_sde}, and
\[
\tilde b(x)=b(x)+\sigma^2(x)\frac{h'(x)}{h(x)},\quad 0<x<R,
\]
where $h(x), x\in[0,R]$ denotes the probability of $\Gamma$ for diffusion~\eqref{eq:basic_sde} started at $x$.
This is so called Doob's $h$-transform, see~\cite[Section 5]{AB2011a} for the one-dimensional computation and
\cite[Section 6]{Swiech:MR3461040} for a rigorous and general treatment.
For all $x\in(0,R)$, we denote by $\mathsf{P}_x$ the distribution of the solution of~\eqref{eq:h-transform} with initial condition $X(0)=x$.
The expectation with respect to $\mathsf{P}_x$ is denoted by $\mathsf{E}_x$.
Under our assumptions, all moments of the exit time are finite
for the original equation and thus they are finite for the conditioned one. Moreover, if we define
\[
\tau_y=\inf\{t\ge 0: X(t)=y\},
\]
then for any $y\in (0,R]$, functions
\[
\alpha_n(x,y)=\mathsf{E}_x \tau_y^n,\quad n\in \{0\}\cup\mathbb{N},\quad 0< x\le y,
\]
are smooth in $x\in(0,y]$ up to $y$ and satisfy a hierarchical system of PDEs
\[
L\alpha_n(x,y)=-n\alpha_{n-1}(x,y), \quad n\in \{0\}\cup\mathbb{N}, \quad 0< x\le y,
\]
where $Lf(x)=b(x)f'(x)+\frac{1}{2}\sigma^2(x)f''(x)$ is the generator of the semigroup associated with the diffusion~\eqref{eq:h-transform},
see, e.g., equation (3.38) in~\cite[Chapter 15]{Karlin-Taylor:MR611513}.
Let us also denote the $n$-th cumulant of $\tau_y$ under $\mathsf{P}_x$ by $\kappa_n(x,y)$, $0< x\le y\le R$.
Since $\kappa_n(R,R)=0$, we can write
\begin{equation}
\label{eq:integral-rep-for-cumulant}
\kappa_n(x,R)= -(\kappa_n(R,R)-\kappa_n(x,R)) = -\int_x^{R}\frac{d}{dy}\kappa_n(y,R) dy.
\end{equation}
The strong Markov property implies that under $\mathsf{P}_x$, the times $(\tau_y)_{x\le y\le R}$ form a process with independent increments,
and if $0<y_1\le y_2 \le R$, then the distribution of $\tau_{y_2}-\tau_{y_1}$ under $\mathsf{P}_x$ does not depend on $x\in(0,y_1)$. Combining this with
the smoothness of $\kappa_n$, we obtain
\begin{equation}
\label{eq:cumulant-derivative-at-endpoint}
\frac{d}{dy}\kappa_n(y,R)=\frac{d^-}{dy}\kappa_n(y,R)= \frac{d^-}{dz}\kappa_n(z,y)\bigg|_{z=y}.
\end{equation}
Using \eqref{eq:cumulant-via-moments} we obtain
\[
\kappa_n(z,y)=\alpha_n(z,y)+P_n(\alpha_1(z,y),\ldots,\alpha_{n-1}(z,y)),
\]
where each monomial term constituting $P_n$ is at least of order $2$. Since $\alpha_1(y,y)=\ldots =\alpha_{n-1}(y,y)=0$, we obtain that the derivative of each of those terms with respect to $z$ at $z=y$
equals~$0$ and thus
\begin{equation}
\label{eq:nonstrict-inequality}
\frac{d^-}{dz}\kappa_n(z,y)\bigg|_{z=y}= \frac{d^-}{dz}\alpha_n(z,y)\bigg|_{z=y}\le0,
\end{equation}
where the inequality follows since $\alpha_n(z,y)$ is clearly nonincreasing in~$z$. Let us prove that, in fact, strict inequality
holds:
\begin{equation}
\label{eq:estimate-on-derivative}
\frac{d^-}{dz}\alpha_n(z,y)\bigg|_{z=y}<0.
\end{equation}
Then the theorem will follow from \eqref{eq:integral-rep-for-cumulant},~\eqref{eq:cumulant-derivative-at-endpoint}, and~\eqref{eq:nonstrict-inequality}.
Let us take any $z_0\in(0,y)$ and
notice that for $z\in (z_0,y)$
\begin{equation}
\label{eq:lower-estimate-on-f_r}
\alpha_n(z,y)\ge u(z) \alpha_n(z_0),
\end{equation}
where $u(z)$ denotes the probability that diffusion started at $z$ reaches $z_0$ before $y$. This function satisfies
the equation
\begin{equation}
\label{eq:Fokker-Planck}
b(z)u'(z)+\frac{1}{2}\sigma(z)u''(z)=0,\quad x\in [z_0,y],
\end{equation}
with boundary conditions
\begin{align}
\label{eq:left-boundary-condition}
u(z_0)&=1,
\\u(y)&=0.
\label{eq:right-boundary-condition}
\end{align}
The desired estimate~\eqref{eq:estimate-on-derivative} will follow from \eqref{eq:lower-estimate-on-f_r} if we show that
$u'(y)<0$. Since $u$ is nonnegative and $u(y)=0$, we must have $u'(y)\le 0$. Assuming $u'(y)=0$ would imply, by the uniqueness theorem
for solutions of the regular second-order equation~\eqref{eq:Fokker-Planck}
and \eqref{eq:right-boundary-condition}, that $u\equiv 0$. The contradiction with~\eqref{eq:left-boundary-condition} shows that $u'(y)<0$
and finishes the proof of the theorem.
{{\hfill $\Box$ \smallskip}}
\begin{remark}\rm The theorem and the proof presented here hold in more general situations with minor modifications.
We may have worked with diffusions on $(-\infty,R]$ provided that $\alpha_n(x,R)<\infty$. Assuming the latter condition, the nonstrict
inequality~\eqref{eq:nonstrict-inequality} always holds as the proof above shows. For the theorem to hold
it is sufficient to have strict inequality at one point $y\in[x,R]$, so we could have required only that
$\sigma(y)>0$ for some $y\in[x,R]$.
\end{remark}
\begin{remark} \rm
Our soft proof is based on the analysis of the sign of the integrand in~\eqref{eq:integral-rep-for-cumulant}
although quantitative estimates are also possible.
\end{remark}
\subsection{Positive cumulants for hitting times in discrete random walks}
In this section, we give a more elementary proof of a version of Theorem~\ref{thm:positive-cumulants} for discrete random walks.
We assume that the evolution $(X_j)_{j\ge 0}$ happens in discrete time on the discrete state
space~$\mathbb{N}=\{0,1,2,\ldots\}$, it is Markov, time-homogeneous, and nearest neighbor (aka birth-death), i.e., for each $k\in\mathbb{N}$, there is a number $p_k\in(0,1]$ such that if the process is at the site $k$ at time $n$, then at time $n+1$ it jumps to $k+1$ with probability $p_k$ and it jumps to $k-1$ with probability $1-p_k$. We must require $p_1=1$.
We will denote by $\mathsf{P}_k$ the distribution of this process started at $X_0=k$, and $\mathsf{E}_k$ denotes
the expectation with respect to $\mathsf{P}_k$.
If $k,R\in\mathbb{N}$ and satisfy $k< R$, we denote $\tau_R=\inf\{j\in\mathbb{N}:\ X_j=R\}$.
We note that due to the discrete Doob's $h$-transform, this setup automatically contains random walks on $\{0\}\cup\mathbb{N}$ conditioned on reaching~$R$ before~$0$.
\begin{theorem}
\label{thm:right-skew-rw}
Let $k<R$. Then for all $n\in\mathbb{N}$,
$\kappa_n(\tau)\ge 0$ under $\mathsf{P}_k$. The identity in this inequality occurs if and only if $n\ge 2$ and $p_k=p_{k+1}=\ldots=p_{R-1}=1$.
\end{theorem}
\bpf
We have
\begin{equation}
\label{eq:representation-via-sum-of-independent}
\tau_R=(\tau_{k+1}-\tau_k)+(\tau_{k+2}-\tau_{k+1})+\dots+(\tau_R-\tau_{R-1}),
\end{equation}
where $\tau_k\stackrel{\mathrm{a.s.}}{=}0$ under $\mathsf{P}_k$.
By the strong Markov property, random variables $(\tau_{l+1}-\tau_l)_{l=k}^{R-1}$ are
independent and the distribution of
$\tau_{l+1}-\tau_l$ (the time it takes to reach $l+1$
starting from $l$) equals that of $\tau_{l+1}$ under $\mathsf{P}_l$.
Since cumulants are additive for sums of independent random variables, it suffices to prove that
cumulants of $\tau_{l+1}$ under $\mathsf{P}_l$, i.e.,
the Taylor coefficients of $\ln M_l(\lambda)$ at $0$, are all positive.
Under the conditions of the theorem, for all $l\le R$, the moment generating function
\[
M_l(\lambda)= \mathsf{E}_{l} e^{\lambda \tau_{l+1}}
\]
is well-defined for $\lambda$ in a small neighborhood of $0$.
Under $\mathsf{P}_l$, before reaching $l+1$, the process $X$ makes a random number $T\ge 0$ of excursions that involve stepping to $l-1$ first and then after a random number of steps returning to $l$. In other words,
\begin{equation}
\label{eq:representation-of-next-step}
\tau_{l+1}=\sum_{i=1}^T \xi_i+1,
\end{equation}
where
where $(\xi_{i})_{i\in\mathbb{N}}$ is an i.i.d.\ family independent of~$T$, with a common distribution, that of $\tau_l+1$
under $\mathsf{P}_{l-1}$. The additional increments of $1$ account for steps from from $l$ to $l-1$ and from $l$ to $l+1$. The distribution of $T$ is geometric:
\begin{equation*}
P_r=\mathsf{P}_l\{T=r\}= (1-p_l)^rp_l,\quad r\ge 0.
\end{equation*}
Due to \eqref{eq:representation-of-next-step}, we obtain
\begin{multline*}
M_l(\lambda)=\mathsf{E}_l e^{\lambda \left(\sum_{i=1}^T\xi_i+1\right)} =\sum_{r=0}^\infty p_l(1-p_l)^r
\mathsf{E}_l e^{\lambda \left(\sum_{i=1}^j\xi_i+1\right)}
\\ =
p_l e^{\lambda}\sum_{r=0}^\infty (1-p_l)^r
\left(\mathsf{E}_l e^{\lambda \xi_i}\right)^r = p_l e^{\lambda}\sum_{r=0}^\infty (1-p_l)^r
(e^\lambda M_{l-1}(\lambda))^r
\\
= \frac{p_l e^{\lambda}}{1-(1-p_l)e^\lambda M_{l-1}(\lambda)}.
\end{multline*}
If $p_l=1$, then $M_l(\lambda)\equiv 1$, and $\ln M_l(\lambda)\equiv 0$, so let us consider the situation where $p_l\in(0,1)$.
Since all Taylor coefficients of $e^\lambda$ and $M_{l-1}(\lambda)$ at~$0$ are positive (the latter are the moments of a positive
random variable), it suffices to check that if a function $f(\cdot)$ satisfies $f(0)=1$ and has all positive Taylor coefficients at $0$, then
for any $q\in(0,1)$ all Taylor coefficients at $0$ of
\[
g(\lambda)=-\ln (1-qf(\lambda))
\]
are positive. Since the latter directly follows from
\[
-\ln(1-x)=\sum_{n=1}^\infty \frac{x^n}{n},\quad |x|<1,
\]
the proof is completed.
{{\hfill $\Box$ \smallskip}}
\begin{remark}\rm Similarly to the continuous case, one can study random walks that are not bounded below. Then instead
of the finiteness of the moment generating function we might
only require $\mathsf{E}_l \tau_{l+1}^r<\infty$ and work with charateristic functions
$\phi_l(\lambda)=\mathsf{E}_l e^{i\lambda \tau_{l+1}}=M_l(i\lambda)$
that are defined for all $\lambda\in\mathbb{R}$ and allow for finite order Taylor expansions. Also, it is possible to obtain more
quantitative estimates, a direction that we do not pursue here.
\end{remark}
\subsection{An elementary proof of right-skewness in the discrete random walk case.}
Although the following result is a direct consequence of Theorem~\ref{thm:right-skew-rw} and the definition of skewness, we give
a direct proof that does not use moment generating functions.
\begin{theorem} Suppose $p_l>0$ for all $l$. Then for any $k$ and $R$ satisfying $1\le k \le R$, the distribution of $\tau$ under $\mathsf{P}_k$ is right-skewed.
\end{theorem}
\bpf We need to prove that $\kappa_3(\tau_R)\ge 0$. Due to representation~\eqref{eq:representation-via-sum-of-independent}
in terms of a sum of independent hitting times, and since
cumulants are additive for sums of independent random variables, it suffices to prove that
\begin{equation}
\kappa_{3,\mathsf{P}_l}(\tau_{l+1})=\mathsf{E}_l \tau_{l+1}^3-3\mathsf{E}_l \tau_{l+1}^2 \mathsf{E}_l \tau_{l+1}+ 2 (\mathsf{E}_l \tau_{l+1})^3 > 0,\quad l\in \mathbb{N}.
\label{eq:one-step-cumulant-positive}
\end{equation}
We recall the representation~\eqref{eq:representation-of-next-step}.
Since shifts by $1$ do not change the cumulants, we only need to prove the following claim:
if $(\xi_i)_{i\in\mathbb{N}}$ is an i.i.d.\ positive sequence
with $\kappa_3(\xi_1)\ge 0$ and $T$ is an independent geometric variable, then
\begin{equation}
\kappa_3\left(S\right)> 0,
\end{equation}
where $S=\sum_{i=1}^T\xi_i$.
Let $m_r=\mathsf{E} \xi_1^r,$ $r=1,2,3$, and $p=p_l$ for brevity. Then
\begin{align*}
a_1&= \mathsf{E} T=\frac{1-p}{p},\\
a_2&= \mathsf{E} T(T-1)=\frac{2(1-p)^2}{p^2},\\
a_3&= \mathsf{E} T(T-1)(T-2)=\frac{6(1-p)^3}{p^3},
\end{align*}
\begin{align*}
\mathsf{E} S &= \sum_{r=1}^{\infty} P_r \mathsf{E} \sum_{i=1}^r \xi_i = \sum_{r=1}^{\infty} r P_r m_1 = a_1 m_1
=\frac{1-p}{p}m_1,
\\
\mathsf{E} S^2&= \sum_{r=1}^{\infty} P_r \mathsf{E} \left(\sum_{i=1}^r \xi_i\right)^2= \sum_{r=1}^{\infty} P_r(rm_2+r(r-1)m_1^2)=a_1m_2+a_2m_1^2
\\ & \hspace{7cm} =\frac{1-p}{p} m_2+\frac{2(1-p)^2}{p^2}m_1^2,
\\
\mathsf{E} S^3&= \sum_{r=1}^{\infty} P_r \mathsf{E} \left(\sum_{i=1}^r \xi_i\right)^3
= \sum_{r=1}^{\infty} P_r(rm_3+3r(r-1)m_1m_2+r(r-1)(r-2)m_1^3)
\\&=a_1m_3+3a_2m_1m_2+a_3m_1^3 = \frac{1-p}{p} m_3+\frac{6(1-p)^2}{p^2} m_1m_2+\frac{6(1-p)^3}{p^3}m_1^3,
\end{align*}
So
\begin{align*}
\kappa_3(S)=&\mathsf{E} S^3 -3 \mathsf{E} S^2 \mathsf{E} S +2(\mathsf{E} S)^3
\\
=& \frac{1-p}{p} m_3+\frac{6(1-p)^2}{p^2} m_1m_2+\frac{6(1-p)^3}{p^3}m_1^3
\\&-3 \frac{1-p}{p}m_1\left(\frac{1-p}{p} m_2+\frac{2(1-p)^2}{p^2}m_1^2\right)+2 \left(\frac{1-p}{p}\right)^3m_1^3
\\
=&\frac{1-p}{p} m_3+3\left(\frac{1-p}{p}\right)^2m_1m_2+2 \left(\frac{1-p}{p}\right)^3 m_1^3> 0,
\end{align*}
which completes the proof. {{\hfill $\Box$ \smallskip}}
\section{Discussion} \label{sec:discussion}
In this section, we would like to discuss broader context of applicability of our approach as well as its limitations.
We assumed that the onset of symptoms corresponds to crossing a threshold by a one-dimensional continuous Markov stochastic process. This, of course means, that we are trying to represent the complex process of the propagation of harmful invaders within an organism in the presense of inhomogeneity of tissues, blood circulation, immune response, etc.\ with a single state variable.
Such a mean field model must be an oversimplification of the reality and cannot possibly be precise.
Also, for a probabilistic model to be useful in applications, one needs certain homogeneity of the data, ideally an i.i.d.\ ensemble to ensure that standard statistical tools based on empirical frequencies and averaging in the law of large numbers are adequate. Assuming that our one-dimensional model gives a fair representation of the dynamics within one infected individual, a better model would account for
fluctuations in all the parameters due to variability across the population: the starting point $x_0$, the coefficients $b,\sigma$,
and the symptom onset level~$R$, especially since in reality the moment of onset of symptoms is defined loosely due to the symptom
detection dependence on uncontrolled external factors.
It is true that models with more complex joint geometry of the domain and diffusion coefficients, taking
into account non-Markovian effects and variability of parameters
can in principle lead to different behavior of exit times.
However, our conclusions should survive moderate modifications of the model and be applicable
for a broad class of stochastic models. For example, if the parameters of the model can vary and
form a statistically homogeneous ensemble, the exit distribution under small noise will be then described by~\eqref{eq:Gaussian-representation} or~\eqref{eq:Gumbel-limit-representation} with random
values of $A$ and $B$, i.e., this is a weighted mixture of a family of Gaussian- or Gumbel-shaped distributions.
Assuming that the fluctuations of the parameters are small, the shape of the distribution will still be very similar to Gaussian or Gumbel.
Our results for limiting shapes of exit time distributions are obtained in the limit of small noise. Although smallness of the noise is a natural assumption, it is not a~priori obvious that it holds in reality. One can say though that the agreement of the real data with
Gumbel distribution reported in~\cite{Strogatz}
(see Figure~\ref{fig:data-1949-and-1950}) is an argument in the favor of small noise hypothesis in
case~III$_0$, with a repelling critical point between the staring point and the symptom onset level~$R$.
If the noise is not small, then our results show that the exit distribution for our model has right skew but precise computations of exit time or conditional exit time distributions
become hard.
In general, the computations can be based on solving second order differential equations for characteristic functions or moment generating functions, see~\cite{ALEA} or numerical simulations. Since all these distributions have right skew, it is difficult to distinguish between them, so one and the same data set can be equally well approximated by Gumbel or lognormal density. This point seems to be mentioned for the first time in literature in~\cite{Strogatz}.
There are few other situations where exit time distributions or their limits are known. One is the one-sided exit problem
on $(-\infty,R]$ for Brownian motion with nonnegative drift, where the drift and diffusion coefficients are both constant, $b\ge 0$ and $\sigma>0$. The exit time distribution
(first obtained in~\cite{Schrodinger}) is known to be Wald, or Inverse Gaussian $\mathcal{I}(R/b, R^2/\sigma^2)$, where $\mathcal{I}(\mu,\lambda)$
stands for the distribution with density
\[
f_{\mathcal{I}(\mu,\lambda)}(t)=\left(\frac{\lambda}{2 \pi t^3}\right)^{1/2} \exp\left\{\frac{-\lambda (t-\mu)^2}{2 \mu^2 t}\right\},
\]
extended by continuity to $\mu=+\infty$ to include the case $b=0$, see, e.g., (73)--(74) in~\cite{Cox-Miller:MR0192521}. Figure~\ref{fig:inverse-g} plots the density of~$\mathcal{I}(1,5)$.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{inverse-gauss-eps-converted-to.pdf}\\
\includegraphics[width=10cm]{half-gumbel-eps-converted-to.pdf}
\end{center}
\caption{\small $(1,5)$-Inverse Gaussian and exp-Gaussian densities}
\label{fig:inverse-g}
\end{figure}
The limiting shape of the distribution for conditional exit time in case~III with initial condition
at the critical point $x_0=p$ was first computed in~\cite{Day:MR1110156}.
It is the distribution of $-\ln|N|$,
where $N$ is a standard Gaussian random variable, and thus
can be called exp-Gaussian, $\mathcal{E}$:
\begin{equation*}
f_{\mathcal{E}}(t)=\sqrt{2}{\pi}e^{\textstyle -t-\frac{ e^{-2t}}{2}},
\end{equation*}
see~Figure~\ref{fig:inverse-g}. The universality of this distribution and its generalizations
was studied in~\cite{Bakhtin-SPA:MR2411523},\cite{Bak2010},\cite{Bak2011},\cite{Gumbel-preprint},%
\cite{Bakhtin-Correll:MR2983392}, \cite{Bakhtin:MR2935120}.
In fact, in the latter two papers, the emergence of this distribution in decision-making in humans
and in associated diffusion models and discrete agent-based neuronal models with mean-field
interactions of Curie--Weiss type was studied. The decision/reaction/response times have been studied in
psychological literature for more than a century,
and one of the natural approaches is to use diffusions to model these times.
see the bibliography in~\cite{Luce-book}
and~\cite{Bakhtin-Correll:MR2983392}. Although it has been observed that most response times data are right-skewed, that fact has not received mathematical justification until the present work.
We believe that the present paper proving that
exit times of Markov diffusions conditioned on the direction of exit are always right-skewed provides a simple and
robust answer to this question. Our result can be used as a test for applicability of
diffusion models of this kind: if the data are left-skewed, then no $1$-dimensional diffusion model can reproduce
these data.
It is worth mentioning
that~\cite{Bakhtin-Correll:MR2983392} and the present paper
contain the first mathematically rigorous results on distributions of
response times understood as exit times. What sets this work apart from the existing literature
besides the mathematical
rigor is that we are able to make a universality claim: we show that certain
features of random variables involved must hold for a broad class of models.
In~\cite{Bakhtin-Correll:MR2983392}, it
is the universality of the shapes of decision making times in symmetric decision tasks with
no a~priori bias in small noise situations. In the present paper, it is the universality of right-skewness of the distributions of exit times and their limiting Gaussian or Gumbel
(depending on the macroscopic robust features of the phase portrait) shapes in asymmetric small noise situations.
One reason of the universal behavior in our works comes from modeling with SDEs. Their big advantage
(in comparison with models of the kind considered in~\cite{Strogatz},\cite{Strogatz:PhysRevE.96.012313}) is that each SDE defines a whole universality class such that the macroscopic behavior of the models
in the class can be effectively described by the exemplar SDE. This includes discrete and continuous random dynamics. The classical examples of this are the Gaussian limit in the Central Limit Theorem and Donsker's Invariance Principle
stating that random walks with i.i.d.\ increments and finite variance are in the universality class of the Wiener process, the simplest disffusion, see, e.g.,
\cite[Chapter~7]{Ethier-Kurtz:MR838085}. Useful examples of such limit theorems abound in the literature, and here we mention just
one: the reason why the exp-Gaussian distribution shape appears as the limiting one for the discrete
Curie--Weiss model of neuronal interaction in \cite{Bakhtin-Correll:MR2983392}
and~\cite{Bakhtin:MR2935120} is that
the model belongs to the universality class of a diffusion near an unstable critical point. It is this universality that allows us to conjecture that statistical features discussed in~\cite{Bakhtin-Correll:MR2983392} and this paper will
be discovered in many other situations.
A striking example of universality is the Gumbel distribution which appears as the universal
limit in at least three domains: (i)~theory of extreme values, (ii)~theory of residual lifetimes, and (iii)~theory of exit times. Although Gumbel (or double exponential) distribution appears in~\cite{Luce-book} along with a dozen of other distributions that resemble many response time data sets, no convincing explanation of its relevance is given there. The common roots of emergence of
the Gumbel distribution in (i),(ii), and (iii) are discussed in~\cite{Gumbel-preprint}.
In the end of this discussion, let us empasize that the problem of the universal statistical behavior of halting or decision times is very broad. For an example of a seemingly totally different nature, such universal behavior has been observed in halting times for several algoritms and massive random initial data sampled
from various basic ensembles used in mathematical physics in
\cite{Deift:MR3723332},\cite{Deift:MR3380694},\cite{Deift:MR3276499},\cite{Sagun-Trogdon-LeCun}.
It was rigorously
established in a special case in~\cite{Deift-Trogdon:Toda}.
Although various detection/halting/decision/hitting times appear to belong to different universality classes, this body of observations calls for further study of the universality phenomena for time statistics in various contexts.
{\bf Acknowledgments.} The author is grateful to Charles Peskin and Percy Deift for
bringing~\cite{Strogatz} to his attention. He also thanks them for stimulating discussions and
encouragement.
\bibliographystyle{alpha}
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Q: Что лучше. Фасад db (сырой, конструктор), модель или объект модели? Начинаю познавать фреймворк laravel и возникают вопросы.
Одни из них: Что лучше использовать при работе с бд? Фасад DB (сырой запрос или конструктор), модель или объект модели? Или их нужно как-то грамотно использовать все?
Это если что про MySQL, ну и к PostgreSQL тоже думаю относится.
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Category Archives: Israel
Bibi Wins a Tight Race
Posted in Democracy, Elections, Israel, The New York Times.
MALAYSIA: Mr. Prime Minister, At 93, you have made history. So, it is time to rise above politics. Be a Statesman
Opinion |by Francis Paul Siah
COMMENT | At least, two English dailies have carried editorials on the ills plaguing Pakatan Harapan in recent days. This is not surprising at all. It is a given that all is not well in the nine-month-old Harapan government.
Some of my fellow Malaysiakini columnists have also waded into the issue and with good reasons too. I can agree with some of their pointers.
The parties at the centre of the storm are none other than Prime Minister Dr Mahathir Mohamad and his fledging Parti Pribumi Bersatu Malaysia (Bersatu).
I am also guilty of criticising Mahathir over the past month. There were two issues I took exception to. The first was his decision to bar Israeli athletes from entering the country which ended their participation in the World Para-Swimming Championships originally scheduled to be held in Kuching this coming July.
The second was Bersatu's intention to set up a chapter in Sabah, reneging on its pact before GE14 with Parti Warisan to not do so.
Yes, I am really disappointed with Tun Mahathir on these two fronts and I stand in total disagreement with him on these issues.
If public feedback on the social media can be taken as a yardstick, there is one which I would feedback to our Prime Minister, to inform him sincerely that his decision to bar the Israeli swimmers has triggered an international outcry. That decision has given Mahathir and Malaysia a bad Image.
My posting entitled 'Sorry, Dr M, you don't speak for Sarawak this time' in the Movement for Change, Sarawak (MoCS) blog attracted a total of 31,755 unique visitors in a single day last January 28.
That was the highest number of visitors to our little NGO blog over the past eight months. Visitors were not only Malaysians but came from the US, Australia, other Asian nations, the UK and other European countries.
This is honest feedback to our Prime Minister. Many do not understand his strong anti-Semitic stand nor his inability to separate race,religion, politics from sports.
To speak from the heart, I feel bad for having to critique our Prime Minister at times and actually feel sorry for him. It's not nice to speak unkindly of a man his age, no matter his wrongs, and especially so when I'm much younger than him. Guess we are only fallible humans.
This week, I sent this message to my WhatsApp list of friends: "I have been criticising Dr M in recent days so much so that I feel malu having to keep on hammering the grand old man. I am thinking of penning another piece to be titled 'If I were Dr Mahathir today at 94 …'. Tell me what would you do if you were in his shoes at 94 today?"
Here are some of their responses. Let them be feedback to our Prime Minister for what they are worth.
Be a statesman
Tun Mahathir should forget politics. He is not seeking re-election. Concentrate on running the country and turn the economy around. At 94, time is not on his side. So, better hurry. When he is gone, nobody will remember him or his legacy. But the country must be in good hands. Be a statesman, not a politician. Act on a bold vision that the nation will rise to eschew narrow racial politics.
Malaysia will be in trouble if Mahathir harbours these three myths:
1. I set the direction, my son will carry on; 2 The Malays are incorrigible ; but I must save them at whatever cost; and 3. Islam and Muslims/Malays mustremain dominant in Malaysia forever.
First of all, I sympathise with Mahathir that he is running a Harapan government that is weak and saddled with a huge debt from the previous regime.
These cannot be resolved in three years. Meantime, the people, rural folk, in particular, are suffering from the high cost of living. Unemployment is a serious threat from belt-tightening. During the three years of rough journey to reform the sociopolitical imbroglio, whoever is the PM has to persuade the people to swallow their bitter medicine that will do good later. So you need to wish that Dr M is blessed with good health to continue what he set out to do for the sake of the nation.
Mahathir has to concede that Malaysia is in a dire state of decline in living standards. He has to move quickly to arrest that. This is a monumental challenge for any leader and it is incumbent upon Mahathir, as the Pprime minister, to do the job.
Put Najib behind bars first. Then bring in the rule of law […] if I were him.
Tun Mahathir is an extraordinary man. Not many will live up to 94. If I were him, I would take a break and relax.. I bet he is not aware there is a more beautiful and wholesome life out there, away from power and politics.
You should be awarded the "Nobel P***k Prize" for badgering Dr Mahathir. I like him. He is doing his best for the country. Please accord him more respect.
No more pussyfooting
So what is my own take "if I were Dr Mahathir today"? The first thing I would do is to stay far, far away from politics, resign as Bersatu chairperson and allow Muhyiddin Yassin and Mukhriz Mahathir to run the show.
I would not worry about my son's ascension on the political hierarchy. I should know that the Mahathir name alone would carry my next few generations very well and ensure a bright future for them.
I would also stop meeting former UMNO lawmakers, including those from PAS. I would avoid them like the plague. I should know that when they want to meet me, they expect something. There is nothing such "parasites" could bring to the table to help Harapan improve anything in the country.
I would reshuffle my cabinet. The under-performing ministers should go. Nine months is enough time for them to prove themselves. By now, I should know that some are just not minister-material. A spring cleaning is in order.
I would stop antagonising my Harapan colleagues and start listening to their concerns about accepting ex-UMNO parasites. Saying that they have changed sounds so shallow and feeble. So is telling Shafie Apdal that Bersatu is going to Sabah to help him and Warisan. I should be aware that those statements sounded hollow, childish even.
I would make sure that my promise to Anwar Ibrahim to pass the baton to him two years after Harapan's victory is fulfilled. No more pussyfooting around on this.My friend is right. Mahathir must stop being a politician. He has to be a statesman.
That is what many would want our current paramount leader to be. Even those of us who have criticised him would badly want him to succeed for the sake of the nation and the people as he enters the final lap of his illustrious political career.
May the One Above continue to bless our dear Dr Mahathir with good health and we all wish him many, many happy years ahead!
FRANCIS PAUL SIAH head the Movement for Change, Sarawak (MoCS) and can be reached at sirsiah@gmail.com
For the rest of this story and more
Posted in Anwar Ibrahim, ASEAN, Character and Integrity, civil society issues, Democracy, Govenance, Israel, Leadership, Malaysia, Malaysiakini, Malaysians, Najib Razak, National Unity, New Economic Policy, New Malaysia Cabinet, Pakatan Harapan, Parti Pribumi Bersatu Malaysia, Politics, Public Accountability, Race Relations, Rule of Law, Terrorism, The Cabinet, The Malays, Tun Dr. Mahathir Mohamad.
Mahathir's way vs Mandela's
by Peter Raja
www,freemalaysiatoday.com
Sport, Foreign Policy and Politics
"Sport has the power to change the world. It has the power to inspire. It has the power to unite people in ways little else does. It speaks to youths in a language they understand. Sport can create hope, where once there was only despair. It is more powerful than governments in breaking down racial barriers. It laughs in the face of all types of discrimination."–Nelson' Madiba' Mandela
Dr. Mahathir Mohamad has been Malaysia's most inspiring politician since he led an unlikely coalition of opposition parties to defeat the previously all-powerful Barisan Nasional coalition in the country's 14th general election.
It was a triumphant return for the 93-year-old ex-Prime Minister with a reputation of having his own way, more so with revelations he croons to the Frank Sinatra classic "My Way".
Five years after his death, Nelson Mandela remains South Africa's most inspiring politician. It had been so from the moment he was arrested and sentenced to life imprisonment as a 44-year-old freedom fighter in 1962. On May 10, 1994, four years after his release, the 78-year-old anti-apartheid icon became the first black to be elected president in South Africa's first democratic election.
One issue the hard-hitting Mahathir has revived since becoming Prime Minister again is Israel and the Jews. In a BBC interview last October, he was unsurprisingly unapologetic in calling the Jews "hook-nosed", among other criticisms of the Jewish state and people. Various foreign governments and international human rights groups have condemned his "decades-long record of anti-Semitic conspiracy theories".
In recent weeks, the issue re-surfaced with Mahathir behind Malaysia's decision to bar Israel and its athletes from participating in the 9th World Para Swimming Championships, which Sarawak successfully won the bid to host in July. This week, the International Paralympic Committee stripped Malaysia of the right to host the world event due to Putrajaya's decision to bar Israel's participation.
Nearly 25 years ago, Mandela faced quite a similar dilemma, albeit of a much bigger scale. Before he came to power, the all-white South African government had already won the bid to host the 1995 Rugby World Cup. The new President had less than 12 months to act before the event started. The whites in South Africa loved rugby as much as the blacks hated it. The green jersey of the Springboks – the national team – was a hated symbol of apartheid repression to the blacks. They cheered when the international sports community boycotted South Africa. Every foreign team received their undivided support when it played against the Springboks.
In that environment, Mandela, the man incarcerated for 27 years for his stubborn resistance to Apartheid, made the startling decision to embrace the Springboks. He was booed when he first tried to persuade the majority blacks to join him. The minority whites and most of the players were uncomfortable. But Mandela persisted in his campaign to get the divided nation to rally behind their national team which traveled around the country to introduce the game to children in poverty-stricken black townships.
In the month-long tournament, the unfancied but inspired Springboks went all the way to qualify for the final against rugby powerhouse New Zealand. The whole country was in a frenzy. Before the match started, Mandela walked down to the field wearing the green team jersey and cap to greet the players. The 65,000-crowd of mostly white South Africans was stunned in disbelief but moments later erupted into chants of "Nelson! Nelson! Nelson!"
The underdogs won 15-12. One anti-apartheid veteran described the scene when Mandela finally handed the World Cup to white Springboks captain Francois Pienaar: "There wasn't a dry eye in the stadium. There wasn't a dry eye in the country. Everybody celebrated. Every black township, every white suburb: One country at last!"
The historic episode also inspired the critically acclaimed movie "Invictus".
A year after his retirement from politics, Mandela himself gave an explanation which is worth quoting in full:
"Sport has the power to change the world. It has the power to inspire. It has the power to unite people in ways little else does. It speaks to youths in a language they understand. Sport can create hope, where once there was only despair. It is more powerful than governments in breaking down racial barriers. It laughs in the face of all types of discrimination."
Peter Raja is an FMT reader.
Posted in Diplomacy, Dr Mahathir, Foreign Policy, Israel, ketuanan melayu, Malaysia, Sports.
Malaysia and Israel
http://chedet.cc
by Dr. Mahathir Bin Mohamad
1. Malaysia does not recognise Israel; has no diplomatic relation with it, does not allow Malaysians to visit Israel and does not allow Israelis to visit Malaysia.
2. This is the only country in the world that Malaysia treats in this manner.
3. In the first place Israel was created from a slice of Palestinian land, without a referendum or a plebiscite being held. The Palestinians were expelled from Palestine without any compensation for the land and homes seized by the Israelis.
4. Then Israel seized more Palestinian land so that Israel became bigger. The Israelis then built numerous settlements on Palestinian land without the consent of the Palestinian nation. Palestinians are barred from these settlements.
5. When the Palestinians resisted and threw stones at Israeli tanks and armoured cars, the Israeli soldiers fired live bullets at the Palestinian children and arrested many of them. The arrested people were detained for years without trial.
6. The detainees were used to exchange with Israeli soldiers captured by the Palestinians.
7. The Gaza strip is blockaded by Israeli forces. Relief ships carrying food, medicine and building materials were siezed in international waters and forced to go to Israel. In one incident 10 activists were killed. These acts by the Israelis is blatantly against international laws.
8. When the Palestinians fired futile rockets at Israel, the Israelis dropped bombs and fired missiles at Palestinian towns and villages. Schools and hospitals were destroyed, patients and children killed or maimed.
9. The blockade of Gaza is illegal but no country has condemned Israel for breaking international laws and moral codes.
10. Today Israel declares that Jerusalem is its capital. When Palestinian slapped Israelis soldiers, they were shot and killed and many were detained.
11. A high wall has been built to divide Palestinian villages and towns. Palestinians cannot visit relatives without being subjected to humiliating checks at many check-points created by the Israelis. The Palestinians are not allowed to travel on roads built by the Israelis on Palestinian land.
12. Thousands of Palestinian have been killed or wounded through Israeli military actions.
13. The whole world can see the injustice and the oppression of the Palestinian by the Israelis. But Israel is not even criticised by the people who talk so much about freedom from oppression and the rule of law. Israel seems to be privileged.
14. If anyone criticises Israel or the holocaust he is immediately labelled "anti-Semitic". The implication is that he is inhuman or immoral. But the blatant inhumanity of Israel is not condemned.
15. Malaysia is not anti-Jew or anti-Semitic. The Arabs are also Semitic people. But we reserve the right to condemn inhuman and oppressive behaviour anywhere, by anyone. We have condemned the Myanmar people for their treatment of the Rohingyas. We have criticised many countries and people for inhuman acts.
16. Many people and many countries have condemned us. But we have not been labelled nor have we labelled people who speak as a matter of right in a free world.
17. Malaysia bans two Israeli athletes – the US bans citizens of five Islamic nations and plans to build a wall against South Americans. Hungary, Poland and the Czech Republic ban refugees. Hungary's Prime Minister Viktor Orban referred to Syrian refugees as "Muslim invaders."
18. Israel is a criminal state and deserves to be condemned. We know the strong backing for Israel. We cannot act against Israel beyond refusing to recognise it. We maintain we have a right to bar Israelis from our country. When the world condemns us for this we have a right to say that the world is being hypocritical. Their talk of human rights and the rule of law is so much empty words.
19. I appeal to those who sympathise with the Palestinian cause to voice their condemnation. Terrorism is not the answer. A proper strategy is needed to bring justice to the Palestinians.
Posted in ASEAN, Foreign Policy, Human Rights, Israel, Mahathir, Malaysia, Pakatan Harapan, Palestine, Wisma Putra.
Know the Difference– Being Jewish and Being Zionist
by Dr. Kua Kia Soong
At the outset, let me make it clear that as far as the Palestinian cause is concerned, I am on the same page as Prime Minister Dr Mahathir Mohamad, although I cannot vouch for his consistency on all the other non-Muslim liberation causes in the rest of the world.
What is disturbing is that through the years, we have witnessed Mahathir's deliberate refusal to make any distinction between the Jewish people and the ideology of Zionism.
This has huge consequences for how our prime minister stands on racism and racial discrimination in our own country. Those who have followed his political career will note the continuity in his ethos and it was not unexpected that he should once again create a similar rumpus recently on the international stage by conflating Jews with Zionism.
Unashamedly racist paradigm
Mahathir's first claim to fame (or rather, notoriety) was the publication of his "Malay Dilemma" after the May 13th 1969 racial riots in Kuala Lumpur.
It was banned by the then Tunku–led government when it first appeared and Mahathir was expelled from the ruling UMNO. Apart from being an academic embarrassment because of its unashamedly racist paradigm, it was clearly "seditious" by the definition of the government-of-the-day in its undermining of sacred constitutional provisions:
"…the Malays are the rightful owners of Malaya…immigrants (read non-Malay Malaysians) are guests until properly absorbed…immigrants are not truly absorbed until they have abandoned the language and culture of their past."–Dr.Mahathir Mohamad
Mahathir's 'Malay Dilemma' was an instant hit among the emergent state capitalists in UMNO who were hungry for power since it provided the instant recipe for them to rally populist support for their bid for power just before May 13, 1969. It was the time-tested recipe for opportunistic politicians to use 'race' as the rallying cry for political support just as Hitler's racist polemic, "Mein Kampf" had provided the model for such a political route.
Since the demise of Hitler and his race-steeped ideology and the price paid in blood by the freedom-loving peoples of the world, racism, racial discrimination and other forms of intolerance have been outlawed in the world community by the Universal Declaration on Human Rights 1948, the International Convention on the Eradication of Racial Discrimination (ICERD) 1965 and the World Conference Against Racism, Racial Discrimination, Xenophobia and Related Intolerance (WCAR) in 2001.
Although Malaysia has yet to ratify I-CERD, we are signatories to all these UN treaties.
Glad to be labelled anti-Semitic!
But why is Mahathir so recalcitrant about his blatantly racist attitude towards Jewish people as an ethnic community?
"I am glad to be labelled anti-Semitic," Mahathir wrote in 2012 on his personal blog. "How can I be otherwise when the Jews who so often talk of the horrors they suffered during the Holocaust show the same Nazi cruelty and hard-heartedness."
He wrote in his 1970 book "The Malay Dilemma" that "the Jews are not merely hook-nosed, but understand money instinctively." He was not embarrassed about repeating this recently on international cable TV.
Not all Jews support Zionism
Much of Malaysians' antipathy towards Israel can be attributed to our government's longstanding support for the Palestinian cause. But Mahathir's rancour extends far beyond geopolitics, spanning anti-Semitism of yesteryears including alleging international Jewish conspiracies to blaming the 1997 Asian financial crisis on a Jew, George Soros:
"The Jews rule this world by proxy," he told the Organisation of Islamic Cooperation summit in 2003.
If Mahathir had studied abroad as I have, he would have come across many Jewish academics, students and politicians who are anti-Zionist activists.
One of the most notable anti-Zionists and pro-Palestinian activists is, of course, Noam Chomsky.
One of the most notable anti-Zionists and pro-Palestinian activists is, of course, Noam Chomsky. There is even a Palestinian solidarity group called 'Jews for Justice for Palestinians (JfJfP) based in Britain that advocates for human and civil rights, and economic and political freedom, for the Palestinian people. It opposes the current policy of Israel towards the Palestinian territories, particularly the territories of the West Bank and Gaza Strip, and seeks a change in their political status. The membership of JfJfP is primarily made up of British Jews.
"Zionism is itself a racist nationalist movement that has had as its goal the creation and support of a Jewish national state in Palestine. Certainly, not all Jews support Zionism nor do they support Israel's discriminatory and repressive actions against Palestinians. "–Dr.Kua Kia Soong.
More Jews live outside of Israel and not every inhabitant of Israel is Jewish; there are also many non-Jews living in Israel. Many Jews, both living in Israel and elsewhere support a Palestinian state alongside Israel as a possible solution to the conflict. In other words, not all Jews identify with Zionism and it is mischievous to conflate 'Jews' with 'Israelis' and 'Zionists' just as it is wrong to say that "all ethnic Chinese in Malaysia are rich" or that "all Chinese must be held responsible for the persecution of the Uighurs in Xinjiang, China".
Likewise, Mahathir's stereotyping of ethnic Chinese
Much of Mahathir's portrayal of Chinese Malaysians echoes his stereotypical anti-Semitic slurs. In his 'Malay Dilemma', Mahathir describes Malaysia's Chinese as "predatory immigrants" who exhibit an "unlimited acquisitiveness" that threatens the "complete Sinicization of the economy." They are mistrusted as disloyal and mercenary, enriching themselves at the expense of the country's other communities. Has he ever shown remorse and rectified his racist thesis in the "Malay Dilemma"?
Ostensibly to "correct the racial imbalance", the New Economic Policy has provided a carte blanche for the new Malay ruling class to amass wealth in the name of their "race". Mahathir has justified this blatantly racist policy thus:
"The best way to keep the shares in bumiputera hands is to hand them over to the bumiputeras most capable of retaining them, which means the well-to-do."
Today, race has been so deeply institutionalised that it is a key factor determining benefits from government development policies, bids for business contracts, education policy, social policy, cultural policy, entry into educational institutions, discounts for purchasing houses and other official policies. Practically every aspect of Malaysian life is permeated by the so-called "Bumiputera policy" based on Malay-centrism.
No wonder the time is not ripe to ratify I-CERD
In the decades since, Mahathir has continued to resort to racial chauvinism whenever popular support has ebbed, stirring anxiety about Chinese investment and immigration following disappointing electoral showings in 2008 and 2013. He castigated Najib for "giving too much to the Chinese" after the disastrous GE13 results.
The recent anti-ICERD rallies organised by UMNO and PAS have now given the prime minister the excuse to say the country is not yet ready to ratify ICERD. The real question is: Is Mahathir ready to eradicate racism, racial discrimination and related intolerances from his own mental paradigm?
As someone has said, "Wisdom doesn't necessarily come with age. Sometimes age just shows up all by itself!"
Kua Kia Soong is the adviser to Suaram.
The views of the writer do not necessarily reflect those of FMT
Posted in Character and Integrity, civil society issues, Critical Thinking, Democracy, Foreign Policy, Freedom of X-pression, FreeMalaysiaToday, History, Human Rights, Israel, ketuanan melayu, Leadership, Malaysia, Multilateralism, National Unity, Nationalism, Noam Chomsky, OIC, Pakatan Harapan, Palestine, Parti Pribumi Bersatu Malaysia, PAS, Politics, Tun Dr Mahathir Mohamad, UMNO, UMNO Politics, United Nations.
Malaysia is in no position to lecture Israel
by S Thayaparan@ www. malaysiakini.com
Published: 26 Jan 2019, 6:02 am | Modified: 26 Jan 2019, 6:02 am
"The anti-Semites who called themselves patriots introduced that new species of national feeling which consists primarily in a complete whitewash of one's own people and a sweeping condemnation of all others."
– Hannah Arendt, The Origins of Totalitarianism
COMMENT | Let me get this out of the way. When people say they are not anti-Semitic but rather anti-Zionist, most of the time this is complete horse manure. The people who most often say this apply the Zionist label to all Jews, thus making the distinction irrelevant.
This is like claiming there is a difference between ketuanan Melayu and the Malay 'race', but ignoring the distinction and claiming that all Malays are racial and religious supremacists. Are all Malays racist? Are all Malays religious bigots just because they support politicians who pander to the lowest common denominator? Or is the situation a little more complex than that?
However, this is not the article for that conversation. This is another article – my second, I think – on mainstream anti-Semitism in our politics.
PAS president Abdul Hadi Awang back in 2012 proclaimed that his party would cooperate with the Jews, especially in the realm of trade, but rejected Zionism. He said: "Nevertheless, PAS rejects Zionism because it is a fanatical ideology of the Jew race."
See what Hadi did there? He made a distinction, but then negated it with his insistence that race and ideology were not mutually exclusive.
I will give you another example. The organisation Boycott, Divestment, Sanctions (BDS) Malaysia chairperson Nazari Ismail speaks for had a huge victory – at least the Palestine Chronicle thinks it is a huge victory – last year because it got Giant to withdraw jeans that were supposedly a product of Israel, but which the hypermarket chain claimed was made in China.
Two points from the Palestine Chronicle article are worth mulling over.
The first: "BDS Malaysia stated that an officer from the Giant branch in question reported that they had returned all the stock nationwide to the supplier. Following which a manager from Giant called Nazari and stated that the supplier of the product was from China and asking BDS to end its campaign against Giant.
"The professor refused, unless Giant could prove that the original company was not of Israeli origin. Upon checking various Giant supermarkets, BDS Malaysia members found that the product was still stocked."
And the second: "A statement was received by BDS Malaysia from a Ms Roseta, corporate affairs, GCH Retail Sdn Bhd stating that thought the product was made and imported from China, and the management was willing to remove the product from all its outlets due to its sensitive nature. She also said that she would seek further clarification from the supplier."
Both these examples demonstrate how the Malay ruling elite and intelligentsia manipulate the discourse, claiming victimhood while propagating racist or bigoted agendas.
Boycotting products because companies are enabling or propagating certain ideas is acceptable, but boycotting all products from a country and linking all companies, products and services to a Zionist agenda is not.
Why do we even have to have this conversation? The Prime Minister of this country, on the campaign trail in Cameron Highlands, claimed that people from Israel were "crooks," and mainstream religious dogma have claimed that the Jews are the "enemies of Islam."
Never mind that political operatives from the Malay right have invested in companies and have had dealings with the Jewish people for decades.
Who are the crooks?
What is needed is for the average Malay – who have not even met a Jew – to feel a sense of hatred towards Jews for a conflict in the Middle East, which has been used for decades to justify all sorts of malfeasance from Islamic regimes and extremists all over the world.
Does anyone actually believe that the Malay political elite and their mouthpieces make a distinction between Zionism and Jews? I have attended many rallies by the Malay right – and let me tell you something, there is only the Malay right and far right – and none of these people has made this distinction. All of them talk about how "evil" the Jews are and how they are not to be trusted. Some have gone so far as to cite religious texts and authority.
The Malay right hates liberals, but they make an exception for Jewish liberals who criticise Israel. A couple of years ago, I was talking to a scholar who opposes the Occupation, but who also said that there were similarities ("frighteningly so, Thaya") between the ketuanan Melayu ideology and Zionism.
Both she argued centralised race as the determining factor for political and social action. Both relied on indoctrination to marginalise the other and both perpetrated injustice through a bureaucracy riddled with dubious personalities who were content to wallow in their petty power. Of course, this is not the kind of Jewish liberal who is embraced by the Malay right.
The Pakatan Harapan grand poobah, while campaigning, served up a large spoonful from the bigoted Kool-Aid that is served up to the Malays on a daily basis. He claimed that the Najib Abdul Razak regime had allowed crooks into this country and his administration, which was the principle behind not allowing these crooks into this country.
Who were these crooks? It was David Roet (photo) who was leading the Israeli delegation for a UN event. What did the progressives fighting against the "evil" BN say at the time? They accused the Najib regime of having an "affair" with Israel.
They claimed that the Najib regime was following in the footsteps of the Saudi regime which had close ties with Israel. They mocked Najib when he said this in 2015: "This dictum, known universally in all religions as the Golden Rule, could herald the dawn of a much-needed revised relationship between Muslims and Jews."
Of the visit and its anti-Semitic reception by the then opposition, I wrote this: "This would have been a perfect opportunity for so-called moderate Islamic parties to change the discourse even a little by highlighting the fact that Islam from the Middle East, or at least that which was perverted by petrodollars, is changing.
"They could have taken the opportunity to learn from the Israeli experience of holding their leadership accountable like how Israeli premier Benjamin Netanyahu is facing possible criminal charges for corruption, by highlighting the fact that a supposed enemy of Islam holds their leaders accountable to graft allegations submitted by (mostly) independent institutions."
Instead, then, like now, what the Malay right is doing is merely reinforcing anti-Semitic narratives in an effort to maintain hegemony, while ignoring the very real consequences of such actions.
Remember, blaming the Jews for the problems of Muslims is exactly like blaming the Chinese for the social, economic and political problems of the Malay community.
Which brings us to the non-Malay component of Harapan's anti-Semitic discourse. You will never see a non-Malay political operative speaking out against the anti-Semitism which is part of mainstream Malay politics. Why? Because to do so would expose the truth in the Hannah Arendt quote which opens this piece.
I know I am going to get into trouble for saying this, but Malaysia has not earned the right to condemn Israel. Maybe if Harapan actually delivered on its promises and slowly did away with this corrupt, bigoted system, we could be on the road to being a credible voice in the Palestinian discourse.
S. THAYAPARAN is a commander (rtd) of the Royal Malaysian Navy. A retired barrister-at-law, he is one of the founding members of Persatuan Patriot Kebangsaan.
Posted in ASEAN, Critical Thinking, History, Indoctrination, Intellectual Development, Islam Malaysia, Israel, ketuanan melayu, Leadership, Liberalism and Open Society, Malaysia, Middle East, military affairs, Pakatan Harapan, Palestine, Parti Pribumi Bersatu Malaysia, PAS, Political Islam, Politics, Public Accountability, Race Relations, Racism, Saudi Arabia, The Malays, Tun Dr. Mahathir Mohamad, Tun Mahathir Mohamad, UMNO. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,914 |
\subsection{Purification of NaI(Tl) crystal}
\begin{wraptable}{r}{0.5\linewidth}
\centering
\caption{The reduction factors for the concentration of Pb ions in the NaI water solution
achieved by the use of various resins.\cite{FushimiPTEP2020}.}
\label{tb:jusi}
\begin{tabular}{cr} \hline
Resin & Reduction factor \\ \hline
A & 1/34 \\
B & 1/64 \\
C & 1/14 \\
D & 1/3 \\ \hline
\end{tabular}
\end{wraptable}
The NaI(Tl) crystal has a large possibility to reduce the intrinsic background.
There is a lot of knowledge for the purification techniques of NaI(Tl) crystal.
The solid-state scintillator is free from contamination after construction because of
its stability.
However, we must be careful not to pollute by radioactive impurity during construction.
We investigated the purification method systematically and found the optimized combination of the methods.
Many groups are developing the reduction methods of NaI(Tl) crystal \cite{FushimiPTEP2020, Park2020, Adhikari2018}.
The present serious radioactive isotopes in our NaI(Tl) crystal are $^{40}$K and $^{210}$Pb.
The potassium in the water solution of NaI forms mainly KI and KOH;
they are as well soluble in water as NaI.\@
We removed potassium ions in the NaI water solution utilizing their significant solubility.
We prepared a saturated NaI water solution at 100 $^{\circ}$C and cooled slowly to room temperature.
We got the pure sediment of NaI by filtration.
The potassium ion remained in the filtrate since the concentration of potassium was sufficiently lower
than its solubility.
The COSINE group reported that the recrystallization method (RC) could remove $^{210}$Pb;
their best result reached down to $10\sim50$ $\mu$Bq/kg \cite{Park2020}.
We considered applying resins to remove lead ions in addition to the recrystallization method.
We have investigated the purification of NaI using a resin.
We prepared several resins to adsorb heavy ions and compared their effects.
Resins A and B in Table \ref{tb:jusi} are lead ion adsorbing resins manufactured by the same manufacturer.
Resins C and D in the same table are heavy ion adsorbing resins made by the same manufacturer.
We prepared a NaI water solution that was added 4.8 ppm of Pb ion to test the effectiveness of resins.
The concentration of Pb ion in the processed NaI solution was measured by an inductively coupled plasma mass spectrometry (ICP-MS),
Agilent 7900 in Osaka University and Osaka Sangyo University.
The reduction factors after applying the purification by resins are listed in Table \ref{tb:jusi}.
We decided to use resins A and B after recrystallization.
\begin{table}[ht]
\centering
\caption{The concentration of $^{\mathrm{nat}}$K (ppb),
$^{226}$Ra ($\mu$Bq/kg), $^{232}$Th ($\mu$Bq/kg), and $^{210}$Pb ($\mu$Bq/kg) in
NaI(Tl) scintillators. Characters A to D denotes the resin which is described in Table\ref{tb:jusi}.
RC stands for the recrystallization method.}
\label{tb:junka}
\begin{tabular}{llrrrrc} \hline
Ingot & Method & $^{\mathrm{nat}}$K & $^{226}$Ra & $^{232}$Th & $^{210}$Pb & Ref. \\ \hline
\#24 & A & 2630 & $66\pm11$ & $13\pm8$ & $58\pm26$ & \cite{fushimi2014kamlandpico} \\
\#68 & C+D & 120 & $57\pm7$ & $8.4\pm2.4$ & 7500 & \cite{Kozlov2019} \\
\#71 & RC$\times2$ & $<20$ & $120\pm10$ & $6.8\pm0.8$ & 1500 & \cite{Kozlov2020} \\
\#73 & RC$\times3$ & $<30$ & $44\pm7$ & $7.2\pm0.8$ & 1300 & \cite{Kozlov2020} \\
\#83 & A+B+RC$\times2$ & $<20$ & 11 &22 & 630 & Present work\\
\#85 & A+B+RC$\times2$ & -- & $13\pm4$ & $<3.2$ & $<5.7$ & \cite{FushimiPTEP2020} \\
Our Goal & -- &$<20$ & $<100$ & $<10$ & $<10$ & \\ \hline
\end{tabular}
\end{table}
\begin{wrapfigure}{r}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{I2485Hikaku.pdf}
\caption{The energy spectra of alpha-rays taken by ingots \#24 (Blue) and \#85 (Red) \cite{FushimiPTEP2020}.}
\label{fg:I2485}
\end{wrapfigure}
After making a small NaI(Tl) detector, we measured the radioactive contamination by building a low-background detector system.
The diameter of the NaI(Tl) crystal was 7.62 cm, and the length was 7.62 cm.
We performed various combinations of purification for optimization.
The methods and the results are listed in Table \ref{tb:junka}.
We measured both beta-ray and gamma-ray to determine the concentration of $^{40}$K.
On the other hand,
alpha-ray for uranium series and thorium series isotopes.
We cannot confirm the emitting position of gamma-ray since gamma-rays from electron capture of $^{40}$K is well penetrating in the matter.
We determined the concentration of the origin of $^{40}$K by comparing the beta-ray energy spectrum and the gamma-ray energy spectrum.
We found the recrystallization method effectively removes the $^{40}$K concentration.
We determined to apply double recrystallization for the potassium purification, comparing the double and triple recrystallization results with ingots \#71 and \#73.
We confirmed that the double recrystallization method reduces the potassium concentration
less than 20 ppb \cite{FushimiPTEP2020}.
We measured the alpha-ray intensities of uranium-series and thorium-series isotopes.
The alpha-ray events were extracted by pulse shape discrimination (PSD), identifying the difference of scintillation decay time between
alpha-rays (190 ns) and beta/gamma-rays (230 ns).
We found a significant reduction of alpha-ray intensity in the ingot \#83 as shown in Table \ref{tb:junka}.
We optimized the resin usage condition and successfully derived
a noticeable reduction of alpha-ray intensity in the ingot \#85 as shown in Figure \ref{fg:I2485} \cite{FushimiPTEP2020}.
There was no prominent structure of the alpha-rays emitted by the uranium and the thorium series taken by ingot \#85.
We set an upper limit on the radioactivity of $^{210}$Pb in the ingot \#85 as $5.7$ $\mu$Bq/kg.
We conclude that we found the best combination of the purification methods.
Table \ref{tb:junka} shows the results of our purification processes.
The twice recrystallization is enough to remove potassium ion; however,
it is insufficient to reduce $^{210}$Pb and $^{226}$Ra.
The additional resin usage is practical to remove heavy ions such as lead and radium.
The selection of resin and appropriate use in indispensable to effective purification.
We did not measure potassium concentration in ingot \#85 because we could not install the detector into
a low-background shield in Kamioka underground laboratory.
The reproducibility of potassium reduction was already confirmed by all ingots after \#71.
\subsubsection{Liquid Xe detector}
A liquid xenon detector has a significant advantage in low background measurement for dark matter search because of its purity and
event selection power.
The XENON1T group developed an extremely high purity and large volume Xe detector, whose fiducial mass was 1300 kg \cite{Aprile2018}.
The radioactive contamination in the liquid xenon detector consists of natural krypton and
emanated $^{222}$Rn.
Krypton was effectively reduced via cryogenic distillation down to $(0.66\pm0.11)$ ppt \cite{Aprile2018, AprileEpj2018}.
The $^{222}$Rn is hard to remove because it generates various elements, Po, Pb, and Bi, via a sequential decay chain.
Moreover, the $^{222}$Rn is generated from the detector materials and spreads into the fiducial volume.
Nevertheless, they reduced the background by the ionization ratio and the scintillation ratio in each event.
The ratio between scintillation signal $S1$ and ionization signal $S2$ is utilized to discriminate the electron events and nuclear recoil events;
it is called two-phase detector \cite{CLINE2000373, RevModPhys.82.2053, Aprile2018}.
The nuclear recoil event due to WIMPs-nucleus scattering is extracted from a large number of electron background events.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\linewidth]{XENON1T_pos.pdf}
\caption{Demonstrating of the position selection to reject surface background events \cite{Aprile2018}.}
\label{fg:XENON_pos}
\end{figure}
The information of the event position is helpful to remove the background events.
The position of the events is derived by the ionization signal taken by the time projection chamber (TPC) technique.
Almost all the background events come from the surrounding materials, the detector's housing, photomultiplier tubes (PMT),
and elements of electronic circuits.
These background events interact in the outer region of the fiducial volume (see Figure \ref{fg:XENON_pos}).
The liquid xenon act as the active shield against the background events due to surface contamination.
\begin{wrapfigure}{r}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{ANAIS_3year.pdf}
\caption{The annual modulation amplitude reported ANAIS-112 three years measurement \cite{Amar2021}.
}
\label{fg:ANAIS_3y}
\end{wrapfigure}
\subsubsection{NaI(Tl) detector}
Compared to iodine in the NaI(Tl) detector, xenon differs by one atomic number and has a close mass number,
so there is little difference for spin-independent interactions with WIMPs.
The first advantage of the NaI(Tl) detector is that it has a high sensitivity to light WIMPs by observing the recoil of light sodium.
Second, it has a high sensitivity to spin-dependent interactions because of its 100\% odd-numbered nuclei.
Also, because of the presence of low energy excited states, measuring gamma rays from inelastic scattering is
advantageous for high sensitivity spin-dependent WIMPs\cite{Ejiri1993}.
The search for an annual modulating signal by a NaI(Tl) scintillator is the present interesting topic in dark matter search.
The DAMA/LIBRA group reported a significant annual modulating signal in the low energy region,
2~keV$_{\mathrm{ee}}\sim 6~$keV$_{\mathrm{ee}}$ \cite{Bernabei2008, Bernabei2018}.
Where keV$_{\mathrm{ee}}$ stands for the observed energy calibrated by the electron energy.
The other groups which apply the NaI(Tl) detector have struggled to reach sufficient sensitivity to test the DAMA/LIBRA's result.
The COSINE group and ANAIS group are continuously searching for the annual modulation signal; however,
they reported no significant modulation in their detectors \cite{Adhikari2019, Amar2021}.
The ANAIS-112 experiment reported no significant modulation signal, which was incompatible with DAMA/LIBRA result
(see Figure \ref{fg:ANAIS_3y}) \cite{Amar2021}.
\subsection{Requirement for the radiation detector for future dark matter search}
Currently, the most sensitive radiation detector for dark matter search consists only xenon
as it is shown in Table \ref{tb:present}.
We need various target nuclei to investigate the property of dark matter candidates.
Consequently, the importance of a highly sensitive radiation detector comparable to the
xenon detectors.
Three reasons for establishing the high sensitivity in xenon detectors are listed below.
\begin{itemize}
\item Sufficiently high-purity of the liquid xenon caused the low background in the fiducial volume.
\item Precise position information enabled the background rejection.
\item Large discrimination power of particles for background rejection.
\end{itemize}
We are developing a solid-state scintillator with the same performance as XENON1T by developing the best
combination of detectors.
The present status of the development to achieve the high performance of the solid-state
scintillator will be described in the following sections.
\section{Introduction}
\subsection{Search for weakly interacting massive particles}
\input{wimps.tex}
\subsection{Recent status of WIMPs search}
\input{status.tex}
\section{Development of highly radiopure NaI(Tl) scintillator}
\input{purification.tex}
\section{Test measurement of a large volume NaI(Tl) detector}
\input{testNaI.tex}
\section{Prospects}
\input{prospects.tex}
\section{Acknowledgment}
We acknowledge the support of the Kamioka Mining and Smelting Company. This work was
supported by JSPS KAKENHI Grant No. 26104008, 19H00688, 20H05246, and Discretionary expense of the president of Tokushima University.
This work was also supported by the World Premier International Research Center Initiative (WPI Initiative).
We acknowledge Profs.~H.~Sekiya and A.~Takeda of ICRR University of Tokyo, and Prof.~Y.~Takeuchi of Kobe University
for continuous encouragement and fruitful discussions.
\bibliographystyle{ptephy}
\subsection{Design of large volume NaI(Tl)}
We are constructing a large volume NaI(Tl) scintillator array.
The single module is a cylindrical-shaped NaI(Tl) crystal with 12.7 cm diameter and 12.7 cm length.
The crystal was covered with enhanced specular reflector sheet ESR$^{TM}$ provided by 3M to guide the
scintillation photons to optical windows.
The optical windows are attached to the ends of the NaI(Tl) cylinder.
The diameter of the optical window is 7.6 cm, and its thickness is 1.0 cm.
The size of the optical window is fitted to the diameter of a low-background photomultiplier tube,
R11065-20mod provided by Hamamatsu Photonics.
\begin{wrapfigure}{r}{0.5\linewidth}
\includegraphics[width=\linewidth]{largeNaI.pdf}
\caption{A single module of the NaI(Tl) detector.}
\label{fg:largeNaI}
\end{wrapfigure}
We selected black acrylic for a housing material to perform an energy calibration in low energy region.
A periodic energy calibration is one of the essential tasks in the dark matter search
experiment, especially in the low energy region below10 keV$_{\mathrm{ee}}$.
A low-energy X-rays cannot penetrate the detector housing made of copper.
COSINE group performed the energy calibration by K-X ray from electron capture of $^{40}$K,
which is contained in their NaI(Tl) crystals\cite{Park2020}.
It is difficult to apply X-ray from $^{40}$K because of too small event rate;
the expected event rate is less than 1 day$^{-1}$kg$^{-1}$keV$^{-1}$ at the peak.
SABRE group used higher energy gamma ray from $^{241}$Am\cite{Mariani2020}.
A photograph of the prototype detector module is shown in Figure \ref{fg:largeNaI}.
One cannot keep the NaI(Tl) detector covered with an acrylic housing because moisture is permeable.
However, it is not a problem since the final detector system is contained in an airtight container filled with pure nitrogen gas.
We tested the stability of the NaI(Tl) crystal in an acrylic container whose thickness was 4 mm.
The crystal was kept in a shield filled with pure nitrogen gas; we found no deliquesce or color after one year.
We estimated the effectiveness of low-energy calibration by several popular radioactive sources.
The $^{133}$Ba is one of the suitable radioactive sources which makes sufficiently low energy, 6.4 keV.\@
The energy of K$_{\beta 1}$ X-ray of Cs (a progeny of $^{133}$Ba) is 34.987 keV with 8.4\% intensity \cite{TOI}.
This X-ray is absorbed by the photoelectric effect of the iodine atom followed by the X-ray emission of iodine whose
energy is 28.612 keV (K$_{\alpha 1}$, 46.4\%).
The X-ray of iodine can escape from the NaI(Tl) crystal, and the rest of the energy, 6.4 keV, is observed.
\subsection{Performance of a test module}
We took the energy resolution and energy threshold data by using the prototype detector.
The prototype detector contains a non-purified NaI(Tl) crystal with the exact dimension of the final design.
We attached two photomultiplier tubes (PMT), Hamamatsu R6091, with a 25\% of quantum efficiency.
The trigger of the data acquisition system was generated by a coincidence of both PMTs,
setting each threshold was to get a single photoelectron signal.
This trigger setting is commonly applied to double-readout detector, for example, DAMA/LIBRA
\cite{Bernabei2008}.
Making the trigger with the coincidence signal reduced noise signals from each PMTs
since the rate of the dark current of PMTs is a few kHz.
\begin{wrapfigure}{r}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{210306_133Ba_Kotera.pdf}
\caption{The energy spectrum taken by the test module NaI(Tl).
The $^{133}$Ba source was irradiated.}
\label{fg:lowene}
\end{wrapfigure}
We irradiated a $^{133}$Ba source to get pulse shapes of two PMTs.
The pulse shape data was taken by CAMAC data-taking system with flash analog-to-digital-converter
(FADC: REPIC RPC-081).
The pulse shapes from two PMTs were obtained, irradiating a $^{133}$Ba calibration source.
The raw data of the pulse shape contained various noises, pile-up signals, re-triggered pulses and
PMT noises.
We removed each noise by appropriate pulse shape analysis and got an energy spectrum as
shown in Figure \ref{fg:lowene}.
The energy threshold and the energy resolution were 1.6 keV$_{\mathrm{ee}}$ and
26\% at 80 keV$_{\mathrm{ee}}$, respectively.
The performances are due to the low quantum efficiency of PMTs and significant coloring of NaI(Tl)
crystal.
Nevertheless, we confirmed a clear peak of low energy peak around 6.4 keV$_{\mathrm{ee}}$.
The NaI(Tl) scintillator has been reported to have poor linearity of fluorescence in low energy region.
M.Moszynski showed that the degree of non-linearity varies by about 40\% between several keV$_{\mathrm{ee}}$
and 100 keV$_{\mathrm{ee}}$ \cite{Moszyski2003}, while L.N.Treflova showed less than 5\% deviation \cite{Trefilova2002}.
Energy calibration should be performed for each NaI(Tl) to check the non-linearity.
In our test module, the linearity deviation is well within the peak-fitting error between 6.4 keV$_{\mathrm{ee}}$
and 100 keV$_{\mathrm{ee}}$,
but the linearity deviation was significant for energies below 6.4 keV$_{\mathrm{ee}}$.
We plan to include $^{40}$K K-X-rays (3 keV) as impurities in the energy calibration for low background measurements.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 47 |
{"url":"https:\/\/mathoverflow.net\/questions\/334279\/local-heights-in-vojtas-conjecture","text":"# Local heights in Vojta's conjecture\n\nI am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).\n\nLet $$X$$ be a variety over a number field. The conjecture states that for all $$P$$ outside of some Zariski closed subset,\n\n$$\\sum_{v \\in S} \\lambda_{D,v}(P) + h_{K_X}(P) \\leq \\epsilon h_H(P) + C$$,\n\nwhere $$D$$ is an effective divisor, the $$\\lambda_{D,v}$$ are a finite set of local heights, the $$h$$'s are Weil heights, $$H$$ is an ample class, and $$\\epsilon$$ and $$C$$ are constants.\n\nHere is my confusion. Let's say $$X$$ is a K3 surface, and take $$D = H$$. The conjecture is $$\\sum_{v \\in S} \\lambda_{H,v}(P) \\leq \\epsilon h_H(P) + C$$, which means that $$\\frac{\\sum_{v \\in S} \\lambda_{H,v}(P)}{h_H(P)} \\leq \\epsilon + \\frac{C}{h_H(P)}$$.\n\nSuppose that $$C \\subset X$$ is a rational curve, also defined over our number field. We can pull back the above inequality to $$\\mathbb P^1$$ by functoriality of the various height functions. Now, I don't see why the left side is supposed to be small at all for every $$P$$, never mind less than epsilon. Aren't there points on $$\\mathbb P^1$$ for which most of the height is accounted for by local heights in $$S$$? (For example, say $$S$$ contains only the $$2$$-adic absolute value, and consider points whose coordinates are powers of $$2$$.) I would expect that this inequality fails for a dense set of points on $$C$$ (for any $$\\epsilon < 1\/2$$, say).\n\nThis itself isn't obviously a problem: $$C$$ would just need to be in Zariski closed set that we exclude. But $$X$$ might have infinitely many rational curves defined over the number field, and the argument applies to all of them. And this would contradict the conjecture. So I suspect I am misunderstanding what local heights are on $$\\mathbb P^1$$. Can someone straighten me out?\n\n\u2022 At least for arithmetic surfaces, the contribution of the Neron functions is not in anyway obviously functorial. I have never seen anywhere in literature where this is proved. \u2013\u00a0Bombyx mori Jun 18 at 17:36\n\u2022 I see. I had in mind Theorem B.8.1 in Hindry-Silverman, which says $\\lambda_{\\phi^\\ast D,v} = \\lambda_{D,v} \\circ \\phi + O_v(1)$. I want to apply this to $\\phi : C \\to X$ the embedding of $C$. Of course if $C$ has high degree, then $\\phi^\\ast H$ is $\\mathcal O_{\\mathbb P^1}(d)$ for some very large $d$, but you get the same $d$ term on the $h_H(P)$. And the constant term $O_v(1)$ should wash out since $h_H(P)$ can be large. \u2013\u00a0user142054 Jun 18 at 17:43\n\u2022 I did not know about this. But I suspect their result is for finite places, not for places at infinity. You can still talk about other heights like Neron-Tate height, Faltings' height, etc, which does behave nicely as they are in some sense global invariants. The issue with the Neron functions is that they are defined with respect to the canonical metric (or Peterson metric if you prefer), and estimating Green functions effectively is a big issue. Papers to look up are the ones by Jorgenson and Kramer in Composito. But that paper showed up around 2004 or later. \u2013\u00a0Bombyx mori Jun 18 at 17:56\n\u2022 I'll take a look, thanks. I don't think they mention that it must be a finite place, but in any case I am happy to only include finite places in my $S$, so I don't think that's the source of my problems. \u2013\u00a0user142054 Jun 18 at 18:11\n\u2022 I actually looked up Hindry-Silverman, and I realized they defined it slightly differently. I will read your post again to see what your issue is. \u2013\u00a0Bombyx mori Jun 18 at 18:12\n\nIf your rational curve $$C$$ intersects $$H$$ in three or more points, then you in fact won't be able to find infinitely many points in $$C(\\mathbb Q)$$ whose height is entirely (or even mostly) coming from the finitely many places in $$S$$. For example, if the intersection is 3 points, you'd more or less need lots of solutions to $$Au+Bv=C$$ with $$u$$ and $$v$$ being $$S$$-units. Also, Vojta says you have to discard a Zariski closed set. So you'd need to discard all rational curves on your surface that intersect $$H$$ in only one or two points, but there are only finitely many of those.\n\u2022 Ah, thank you! In case someone else stumbles upon this, my issue appears to be that the Weil height machine works at the level of Pic(X), but the local ones really do need Div(X). So my computations of the local heights on $\\mathbb P^1$ were mistaken, as I assumed I could move $\\phi^\\ast H$ to be supported at a point. \u2013\u00a0user142054 Jun 18 at 18:28\n\u2022 That's right. The local ht $\\lambda_{H,v}(P)$ is essentially $$-\\log(\\text{v-adic distance from P to H}).$$ So it indeed depends on $H$, and not on the linear equivalence class of $H$. But when you add over all $v$, that dependence more-or-less goes away because of the product formula. \u2013\u00a0Joe Silverman Jun 18 at 18:33","date":"2019-11-17 06:59:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 35, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8283869028091431, \"perplexity\": 232.6271801713536}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496668896.47\/warc\/CC-MAIN-20191117064703-20191117092703-00420.warc.gz\"}"} | null | null |
Q: How to improve a mysql COUNT query for speed? How can I improve this query for speed? at the moment it's taking a couple of seconds only to load the php file where the query is without even querying anything.
I've an index on skillsTrends, jobtitle and industry.
Collation: utf8mb4_unicode_ci
$sql = "SELECT
COUNT(skillsTrends),
skillsTrends,
jobtitle,
industry,
industry_url
FROM fr_skills_trends
WHERE industry IN ('". implode("', '", $industryInsertSql). "')
AND LENGTH(skillsTrends)<=35
AND reg_date >= NOW() - INTERVAL 3 MONTH
GROUP BY skillsTrends ORDER by LENGTH(skillsTrends) DESC";
Number of records < 1,000,000.
A: Try this covering index.
CREATE INDEX fr_skills_trends_date_industry
ON fr_skills_trends
(reg_date , industry, skillsTrends);
It should help the performance of your query.
And, your query misuses MySQL's notorious nonstandard extension to GROUP BY. Try this instead.
SELECT
COUNT(skillsTrends),
skillsTrends,
jobtitle,
industry,
industry_url
FROM fr_skills_trends
WHERE industry IN ('". implode("', '", $industryInsertSql). "')
AND LENGTH(skillsTrends)<=35
AND reg_date >= NOW() - INTERVAL 3 MONTH
GROUP BY skillsTrends, jobtitle, industry, industry_url
ORDER by LENGTH(skillsTrends) DESC
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,491 |
Q: Power Apps - How to change the format of the Rich text column to normal text of the Data Table object Power Apps - How to change the format of the Rich text column to normal text of the Data Table object.
I have created a power app form and showing a data from a list using data table object in power apps.Currently Data shows html tag along with the data. I want to change the format of the Rich text column to normal text of the Data Table object. I cannot change the SharePoint column to single line text.
A: You can change this on the settings of the field.
You must have your field in "Plain text"
It's not a problem of PowerApps, it's SharePoint, when you have a field (Multi Line of text) in rich text mode, SharePoint automaticaly save this data as HTML.
If you want to use the rich text, you can have a look at the powerApps Rich text contol : Rich text editor control (experimental) in PowerApps
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,249 |
\section{Introduction}
The amount of literature in the medical domain is increasing enormously and emphasises the need for text mining-based solutions in order to automatically extract relevant information. Named entity recognition (NER) is an important task of natural language processing (NLP) where the aim is to find entity classes in unstructured text, such as specific diseases. As the amount of data for a specific setup is usually limited, recently, transfer learning-based models have been shown to achieve state-of-the-art results in many NLP tasks including NER \cite{yadav2019survey}. Especially, transformer-based models, such as BERT \cite{devlin_bert:_2018}, show promising results on benchmark tasks \cite{Kulkarni2022}. In the biomedical domain, BioBERT \cite{lee_biobert_2020} shows state-of-the-art performance for several NER tasks, such as disease recognition. Promising F1-scores are achieved for the available data sets (above 84\%).
Based on the use case of disease NER, we recently showed that models trained on an available data set are not able to efficiently predict on another data set that, however, follows the same annotation guidelines \cite{langnickel_we_2021}. This is not only true for transformer-based models such as BioBERT but holds true for different machine learning-based models such as convolutional neural networks or conditional random fields. In our previous study, we showed - based on two different manually labelled data sets that follow the same annotation guidelines - that the performance of a model trained on one of these two corpora is reduced by up to 20\% in terms of F1-score when predicting on the other corpus. This significant drop in performance indicates that the training data is either too small or not representative - compared to a random PubMed corpus. One reason can be attributed to the fact that specific corpora are often comparably small such that small differences in between those data sets are mapped to differences in embeddings and according NER downstream tasks.
Therefore, in order to use these models in real world applications, such as semantic search engines, it is advisable to improve the models as soon as new annotated data are available, to obtain optimum performance. This process is known as lifelong learning or, equivalently, continual learning which means that a model is sequentially retrained in a so-called online fashion \cite{DBLP:journals/corr/abs-1912-05156}. However, for such settings, a mechanism called catastrophic forgetting easily happens \cite{mccloskey_catastrophic_1989}. This means that the model will be biased towards the last data set and will forget previously learned structure.
A lot of research has been done in the area of continual learning to prevent a model from forgetting. One of the most prominent approaches is called Elastic Weight Consolidation (EWC) proposed by Kirkpatrick \textit{et al.} \cite{kirkpatrick_overcoming_2017}. It is a regularization-based technique that basically quantifies the importance of weights and thereby impedes important weights from being changed drastically. It has been successfully applied for an online personalization of speech recognition systems, as an example \cite{sim2019personalization}. Next to regularization-based techniques, (pseudo-)rehearsal-based approaches have been proposed \cite{robins_catastrophic_1995, honnibal_pseudo-rehearsal_nodate}. Rehearsal means that a subset of previously seen data are combined with the new data. In contrast, for setting where the old data are not available, new annotated data are generated using the trained model (this is also called silver standard). These data are then mixed with new data to re-train the model. In addition, promising methods exist where new parameters are added to the model for each new task that is learned, such as proposed by Fayek \textit{et al.} \cite{fayek_progressive_2020}. Moreover, dual-memory-based methods are applied where basically two different networks are used - one for memorising already learned information and one for learning new tasks - such as shown by Park \cite{park_continual_2020}. The author implemented a dual network architecture based on state-of-the-art transformers. Houlsby \textit{et al.} proposed \textit{Adapters} which are transformer-based modules exploiting a different learning procedure than the usual fine-tuning \cite{houlsby_parameter-efficient_2019}. Except from being more parameter-efficient, these Adapters can be used for sequential learning settings as they can be trained individually and then be "stacked" together.
Several overview articles structure and compare online learning methods and their suitability in various domains \cite{HOI2021249,DBLP:journals/ijon/LosingHW18}. Yet, the suitability of such methods for NER tasks in the medical domain, where comparably small annotated data sets are present, and their suitability to provide federated learning schemes \cite{DBLP:journals/corr/abs-1907-09693} where sharing data should be avoided, has not yet been investigated.
In this work, we present a new continual learning method - called WEAVER - that can be applied to transformer-based models which exploits parts of the \textit{federated averaging algorithm} (FedAvg), known from federated learning approaches \cite{mcmahan_communication-efficient_2017}. Thereby, previously used data are not required for the new task, the model structure does not need to be changed and the process is computationally efficient. As previous data do not need to be available for incremental learning at one place, it can be applied to federated learning approaches as well - which is especially important in the medical domain. Thereby, each clinic or institution trains a model using their own data, afterwards the trained model is passed to another site where the model is re-trained and WEAVER is applied. This leads to a model trained on bigger data sets without sharing data and in addition no central server is needed where one model is built - the model sequentially passes all institutions instead.
\section{Material and Methods}
The following section describes the data sets used for conducting the study. Afterwards, we describe the developed learning procedure and the conducted experiments.
\subsection{Datasets} \label{sec:datasets}
For disease named entity recognition, we use the NCBI data set \cite{dogan_ncbi_2014} and the BC5CDR data set \cite{li_biocreative_2016} that both contain abstracts manually annotated by experts. Importantly, these data sets follow the same annotation guidelines which have been released with the NCBI disease corpus and are re-used by the latter \footnote{\url{https://www.ncbi.nlm.nih.gov/CBBresearch/Dogan/DISEASE/Guidelines.html}}\footnote{\url{https://biocreative.bioinformatics.udel.edu/media/store/files/2015/bc5_CDR_data_guidelines.pdf}}. Both of them consist of training, development and test set.
\newline
To test our approach on a second use case from the general domain language, we use the ConLL-2003 data set \cite{tjong-kim-sang-de-meulder-2003-introduction}. This is tagged with organisations, persons, locations and miscellaneous. In order to simulate a sequential learning setting, we split the data set into three parts. Thereby, we first combine training, development and test set and then cluster them into three clusters by similarity. This is done by first vectorizing the sentences and then applying \textit{k-means with constrains} in order to get approximately three clusters of equal sizes \cite{levy-kramer_k-means-constrained_nodate}. Word clouds from the different clusters can be seen in Figure~\ref{fig:conll_word_clouds}. Each clustered data set was then randomly split into training and test set. The size of each data set is summarised in Table~\ref{tab:overview_datasets}.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{wordclouds_a.png}
\caption{Word cloud for cluster 0}
\label{fig:word_cloud_cluster_0}
\end{subfigure} %
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{wordclouds_b.png}
\caption{Word cloud for cluster 1}
\label{fig:word_cloud_cluster_1}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{wordclouds_c.png}
\caption{Word cloud for cluster 2}
\label{fig:word_cloud_cluster_2}
\end{subfigure}
\caption{\textbf{Word clouds for the three different sub-corpora.} All sub data sets (i.e. training, development and test set) of the ConLL-2003 data set were merged and split by similarity. Therefore, the sentences were vectorized and clustered together using unsupervised machine learning method k-means. Each of these sub sets were then randomly split into training and test set using a ratio of 0.8. The size of each cluster can be seen in Table~\ref{tab:overview_datasets}.}
\label{fig:conll_word_clouds}
\end{figure*}
\begin{table}[]
\centering
\caption{\textbf{Overview of the used data sets}}
\begin{tabular}{lllll}
\toprule
& \multicolumn{3}{c}{\textbf{Amount of Sentences in}} \\
\textbf{Data set} & \textbf{Training} & \textbf{Test} & \textbf{Devel.} & \textbf{Entity Classes} \\
\midrule
NCBI & 5423 & 939 & 922 & \multirow{2}{*}{Diseases}\\
BC5CDR & 4559 & 4796 & 4580 \\
\midrule
ConLL Cluster 0 & 6264 & 1566 & - & \multirow{2}{*}{Persons, Organisations,}\\
ConLL Cluster 1 & 4800 & 1200 & - & \multirow{2}{*}{Locations, Miscellaneous}\\
ConLL Cluster 2 & 5531 & 1383 & - \\
\bottomrule
\end{tabular}
\label{tab:overview_datasets}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{overview_WEAVER_new.png}
\caption{\textbf{Overview of our transformer-based continual learning procedure WEAVER using weight averaging.} For training the first model, a transformer-based model, such as BERT, is initialised and fine-tuned on the available data. To continue training in a sequential manner, the already fine-tuned model is fine-tuned again on a second corpus. To prevent catastrophic forgetting, knowledge of the previous model is infused into the new model by applying weight averaging. Thereby, the size of the data set the individual model was trained on determines the averaging coefficient, i.e. the bigger the data set, the higher the influence for the new model. This procedure is repeated for every new data set.}
\label{fig:overview}
\end{figure*}
\subsection{Developed continual learning procedure}
For our proposed continual learning procedure, we exploit a mechanism that is originally used in federated learning settings, where models are trained at different places using different data sets, mostly due to data privacy concerns \cite{DBLP:journals/corr/abs-1907-09693}. After training these models individually, their weights are passed to a central server and averaged in relation to the amount of training data they were trained on - hence the more data were available the more influence in the final model \cite{mcmahan_communication-efficient_2017}. The corresponding formula for the objective of the target model can be seen in the following:
\begin{equation} \label{eq:fedAvg}
f(w) = \sum_{k=1}^{K} \frac{n_k}{n} F_k(w)
\end{equation}
K is the number of clients, i.e. the number of models that were trained (in our continual learning setting, this will be always two). The total amount of training data is described by n, whereas n\textsubscript{k} is the amount of the current data set. F\textsubscript{k}(w) defines the client's loss function. As shown in \cite{mcmahan_communication-efficient_2017}, this objective results in weight averaging for convex costs.
\newline
We developed the following procedure: For the first model that is trained on a given task, we initialize a pre-trained BERT or BioBERT model and fine-tune it in a usual manner. As soon as new data are then available, we train the already trained model on top using the new data set. In a post-processing step, the weights of the old and the new model are then averaged, taking the amount of training data into account. Thereby, if a second model is trained on top of the first one, the total amount of training data is the sum of the two data sets. Therefore, either a new pre-trained model can be initialised or the already fine-tuned model will be trained on top and afterwards combined. A simplified overview about the continual learning procedure is shown in Fig.~\ref{fig:overview}.
\subsection{Conducted Experiments}
For all conducted experiments, we build our code upon the \textit{Transformers} library \cite{wolf-etal-2020-transformers} and we use either \textit{dmis-lab/biobert-base-cased-v1.1} or \textit{bert-base-cased} as pre-trained model. The performed baseline experiments are described in the next section, followed by the description of the continual learning experiments. We provide our code under \url{https://github.com/llangnickel/WEAVER}.
\subsubsection{Baseline Experiments}
For both use cases (disease NER and general domain NER) we perform the following experiments: First, we train all models individually and evaluate them on all test sets. Second, to have a baseline comparison, we train one model combined on all of the training data sets respectively and evaluate it on the combined test sets.
\subsubsection{Continual Learning Experiments}
We apply three different continual learning procedures, namely BERT Adapters \cite{houlsby_parameter-efficient_2019}, BERT EWC \cite{kirkpatrick_overcoming_2017}, and our newly developed BERT WEAVER method. Furthermore, we apply usual on top training which means that an already fine-tuned model will be simply fine-tuned again on another data set \cite{devlin_bert:_2018}. For BERT Adapters the training is done separately for each data set - different Adapters are then continuously stacked together. For all experiments, due to lack of data, we did not perform hyperparameter optimization but use default parameters that are summarized in Table \ref{tab:hyperparameters}. Moreover, we used the following data sets to simulate a continuous learning setting: For disease entity recognition, we use four data sets, namely NCBI training, NCBI development, BC5CDR training and BC5CDR development sets. For general domain tasks, such as recognition of organisations, locations or names, we use the three different ConLL-2003 subsets that were split by similarity. \newline
In a first step, we examine the extent of \textit{forgetting} when re-training a model in a continual manner. Therefore, we evaluate the performance on the training data set that has been used for the very first model after each re-training. Thereby, it can be seen how much the model forgets from what it learned first. This is done for all four different methods and by applying different orders of the training data sets. Furthermore, for disease entity recognition, we evaluate the models on the available test data sets in order to investigate whether continuous training increases the performance. Thereby, we evaluate all different orders (permutations) and determine averages and standard deviations.
\begin{table}[]
\centering
\caption{\textbf{Used default hyperparameters}}
\begin{tabular}{ll}
\bottomrule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Batch size & 16 \\
Learning rate & 3e-5\\
Number of epochs & 3\\
\bottomrule
\end{tabular}
\label{tab:hyperparameters}
\end{table}
\subsection{Evaluation Metrics}
To evaluate our methods, we determine precision, recall and F1-score using the following formula (FP stands for false positive, FN for false negative and TP for true positive).
\begin{equation}
precision=\frac{TP}{TP+FP}
\end{equation}
\begin{equation}
recall=\frac{TP}{TP+FN}
\end{equation}
\begin{equation}
F_1-score=2\times(\frac{precision\times recall}{precision+recall})
\end{equation}
\subsection{Visualization Techniques}
To visualize the word (i.e. token) embeddings, we apply the dimensionality reduction technique Uniform Manifold Approximation and Projection (UMAP). More specifically, we make use of the python library umap-learn \cite{mcinnes2018umap-software}.
This allows us to judge whether different data sets are embedded in different regions of the network or whether they share the representation space. Thereby, we compare the visualization obtained by the following settings: First, we train different models on different data sets individually, for example on the NCBI training and the BC5CDR training set in case of the diseases. Then, we make predictions on these training data sets using the corresponding models and use the word embeddings which are vectors of length 768. Because this high dimensionality cannot be visualized, we apply UMAP to scale it down to two dimensions. We then colour the embeddings of the different data sets (predicted by the two different models) differently. In addition, we use the baseline model that has been trained on both data sets simultaneously to also make predictions on both of these data sets. Finally, we visualize the word embeddings predicted by a model that has been trained sequentially on the mentioned data sets according to our developed method.
Since UMAP preserves cluster structures, this enables us to judge differences or overlaps of the embedding of different sets.
\section{Results}
\subsection{Baseline Experiments}
In the following, we summarise all baseline experiments. In Figure~\ref{fig:disease_ner_baseline}, we visualize the results for disease NER, i.e., both single models (trained individually on NCBI and BC5CDR) and the corresponding combined model, trained in a usual manner using BioBERT. When training BioBERT on the NCBI disease training set, the evaluation on the corresponding test set yields high results - namely an F1-Score of 87.77\%. Similarly, a model trained on the BC5CDR training set reaches an F1-score of 83.61 on the corresponding test set. However, the cross-evaluation - i.e. training a model on the NCBI train set and evaluating it on the BC5CDR test set - leads to a performance drop down to 68\%. For the opposite case, we see a similar drop (from 84\% to 69\%). The combined training on both training sets simultaneously leads to high F1-scores for both test sets (89.22\% for the NCBI test set and 83.61\% for BC5CDR test set). For the NCBI test set, the score is even better than for the single model (89.22\% vs. 87.77\%). The F1-score for the BC5CDR test set is only 0.78\% worse.
The results on the three different ConLL data sets that were split by similarity show similar occurrences and are summarized in Figure~\ref{fig:conll_ner_baseline}. For example, the model trained on cluster 0 achieves an F1-score of 93.86\% on the corresponding test set. However, for cluster~1 this drops to 80.67\%; for cluster 2 it shows a performance of 91.93\%. This leads to an average of 88.82\%. Similar results can be seen for the two other cases. When training on cluster 2, the evaluation on the test set of cluster 1 even drops down to 74.08\%. When training a BERT model on all three data sets simultaneously, an average F1-score of 96.26\% is achieved, which is between eight to 11\% better than training only on one data set.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{baseline_diseases.png}
\caption{BioBERT results on NCBI and BC5CDR corpora}
\label{fig:disease_ner_baseline}
\end{subfigure} %
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{baseline_conll.png}
\caption{BERT results on split ConLL-2003 corpus}
\label{fig:conll_ner_baseline}
\end{subfigure}
\caption{\textbf{Baseline named entity recognition results for two different use cases.} In (a), the results for the disease entity recognition using two different corpora (NCBI and BC5CDR) can be seen. The model trained on the NCBI training set, achieves a high F1-score on the corresponding test set (88\%), however drops to 67\% for the other test set. Similar is true for the opposite case. A combined training on both training sets impedes this drop. In (b), the same scenario is reconstructed for the general domain (i.e. recognition of locations, organisation, persons and miscellaneous) using the ConLL-2003 data set, split by similarity. For the test set belonging the training data the model was trained on, a high performance can be seen. However, for the two other test sets a drop - varying between 2\% and 21\% - can be seen. Detailed results including precision and recall can be found in the Appendix (see Table~\ref{tab:Bert_disease_results_basic} and \ref{tab:Bert_conll_results_basic}).}
\label{fig:three graphs}
\end{figure*}
\subsection{Continual Learning Experiments}
In the following, we summarise our results for the four different conducted CL experiments, namely BERT on top training, BERT Adapters, BERT EWC and our proposed method BERT WEAVER. In the first scenario, we investigated the extent of forgetting. Thereby, we determined the F1-score for the very first training data set after each re-training of the model in order to see how much the model "forgets" when being exposed to new data. The results are summarized in Figure~\ref{fig:forgetting}.
For both applications - disease and ConLL data sets - we tested different training orders. In the case of disease NER, a so-called mixed order can be seen in Fig.~\ref{fig:disease_ner_forgetting_mixed} which means that the two similar corpora in each case (training and corresponding development set) are trained alternately. It can be seen that normal on top training experiences a drop of 10\% after training the second model (it drops from approximately 98\% to 88\%). Nearly the same results can be seen for EWC. In contrast, for WEAVER, there is only a drop of 3\% in terms of F1-score - from 98\% to 95\%. The same trend can be seen after training on the last data set, WEAVER achieves an F1-score of 93.5\% whereas on top training is still 7\% worse. Stacking together of independently trained Adapters does not yield satisfactory results. Whereas after training on the second data set, the F1-score experiences a drop of 8\%. However - in contrast to our expectations - after training using the third data set that is similar to the first one, the performance falls by another 8\% and finally drops to app. 65\%. Note, that we simply stacked differently trained adapters together; more elaborated methods such as AdapterFusion could perform better \cite{pfeiffer_adapterfusion_2021}; these however imply that training data are available at the same place and therefore has been omitted in the current study. \newline
In Fig.~\ref{fig:disease_ner_forgetting_same}, the order is set from similar to different corpora which means that the two similar data sets (train+dev) are followed by the other two data sets, respectively. Whereas for WEAVER the smallest drop can be seen, amounting to 7\%, followed by EWC amounting to 11.5\%; on top training leads to a drop of approximately 12\% and for the Adapters the F1-score falls about 35\%. Detailed results can be found in the Appendix. \newline
\newline
For named entity recognition on ConLL data set, the same tendencies can be seen in Fig.~\ref{fig:conll_ner_forgetting_mixed} and Fig.~\ref{fig:conll_ner_forgetting_same}. In summary, WEAVER performs best, followed by on top training. Stacking together of Adapters does not work well. However, it can be seen that the extent of forgetting previously learned entities is significantly lower than for disease entity recognition. Whereas we talked there about drops of 7\% for WEAVER, for named entity recognition in general domain knowledge such as persons and organizations, there is a drop of less than 1\%. This can be caused by the fact that the used data set is split into three, however, in case of the disease recognition, we use two independently released data sets. In addition, medical language is much more complex than general domain language, which is to some extend already well covered by the standard BERT model. This emphasises the need for continuous learning-based approaches in the biomedical domain where data sets for pretraining language models are much smaller than for general domains.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{forgetting_diseases_mixed_order_complete.png}
\caption{Alternation of NCBI and BC5CDR corpora}
\label{fig:disease_ner_forgetting_mixed}
\end{subfigure} %
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{forgetting_diseases_same_order_complete.png}
\caption{NCBI corpus followed by BC5CDR corpus}
\label{fig:disease_ner_forgetting_same}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{forgetting_conll_mixed_order_complete.png}
\caption{Results on split ConLL corpus for order 0, 1, 2}
\label{fig:conll_ner_forgetting_mixed}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{forgetting_conll_same_order_complete.png}
\caption{Results on split ConLL corpus for order 0, 2, 1}
\label{fig:conll_ner_forgetting_same}
\end{subfigure}
\caption{\textbf{F1-scores on the first training data set over time.} Evaluation is done for all four methods. After each re-training of the model, it is evaluated on the first training data set in order to see how much the model "forgets". In (a) and (b), the results for the disease entity recognition are shown for four training data sets (NCBI train and development, BC5CDR train and development) applied in different orders, respectively. Similarly, in (c) and (d), the results for the recognition of organizations, locations, persons and miscellaneous can be seen for different orders of thee three training data sets originating from the ConLL-2003 corpus (split by similarity). Detailed results can be found in the Appendix in Table~\ref{tab:extent_forgetting_diseases_0_2_1_3}-\ref{tab:extent_forgetting_conll_0_2_1}.}
\label{fig:forgetting}
\end{figure*}
Therefore, in a second setting, we investigated the performance on the evaluation data sets for the disease entity recognition. Because the order of training can have an influence, we perform the experiments for all different permutations and determine the averages. The results are summarised in Figure~\ref{fig:on_top_vs_weaver}. In Fig.~\ref{fig:on_top_vs_weaver_diseases}, the F1-scores on the combined test sets (NCBI and BC5CDR) can be seen for WEAVER and on top training. Whereas the first model is trained equally for both settings, first differences can be seen from the second model on. Here, our developed method shows higher averaged F1-scores. Detailed results including standard deviations can be seen in the Appendix in Table~\ref{tab:detailed_cl_results_diseases}. In Figure~\ref{fig:single_weaver_results} the individual results of the two different test sets can be seen. It indicates that the averaged F1-scores (standard deviations are omitted in this figure) are continuously increasing for both of them and only minor differences exist between the two test sets.This indicates the need for continuously training a model as soon as new data are available.
\newline
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{evaluation_diseases_weaver_ontop.png}
\caption{Performance on the combined test set (NCBI and BC5CDR)}
\label{fig:on_top_vs_weaver_diseases}
\end{subfigure} %
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{FedAvg_split_diseases_new.png}
\caption{Performance of WEAVER on the individual test sets}
\label{fig:single_weaver_results}
\end{subfigure}
\caption{\textbf{Continual learning results for disease entity recognition.} In (a), the results for the disease entity recognition are shown for WEAVER in comparison to normal on top training. Four training data sets are used (NCBI train and development, BC5CDR train and development). The evaluation is done on the combined test sets from both data sets. The setting is applied to all different orders (permutations) of training data. Therefore, resulting standard deviations are marked shaded. Detailed results can be found in Table~\ref{tab:detailed_cl_results_diseases}. In (b), individual WEAVER results are shown for both test sets independently. Note that hyperparameter optimization has not been performed, but default parameters were used.}
\label{fig:on_top_vs_weaver}
\end{figure*}
\subsection{Visualization of Word Embeddings}
In order to comprehend what happens to the word embeddings when averaging the weights of two BERT models, we performed a UMAP visualization for the different scenarios. Therefore, we compared the arrangement of the embeddings for the two different training sets (NCBI and BC5CDR). This was done in three different scenarios. First, the word embeddings for the NCBI training set where predicted by the model trained only on this data set and the word embeddigns for the BC5CDR training set where predicted only by the model trained on it. As can be seen in Figure~\ref{fig:umap_diseases}(a), embeddings for the different data sets, predicted by the two different models are clearly separated. In contrast, in subfigure (b), where a model trained on both data sets simultaneously is used for prediction, the points are strongly overlapping and separate clusters cannot be recognized. Subfigure (c) shows word embeddings predicted by a model trained according to our method WEAVER (first NCBI training set, then BC5CDR training set). Interestingly, it can be seen that the distribution looks very similar to a combined training. Thereby, we can infer that weight averaging after training two models sequentially has a similar effect to a combined training (simultaneously on all training data).
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{comparison_of_all_three_cases_movedPoints_a.png}
\caption{Models trained independently on NCBI and BC5CDR}
\label{fig:umap_trained_independently}
\end{subfigure} %
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{comparison_of_all_three_cases_movedPoints_b.png}
\caption{Models trained jointly on NCBI and BC5CDR}
\label{fig:umap_trained_jointly}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{comparison_of_all_three_cases_movedPoints_c.png}
\caption{Model trained continually on NCBI and BC5CDR using WEAVER}
\label{fig:umap_trained_with_weaver}
\end{subfigure}
\caption{\textbf{BERT embeddings visualized using UMAP.} The different sub-figures show the distribution of the word embeddings predicted for both the NCBI and the BC5CDR training data set using different models. In sub-figure (a), two different models were used that were independently trained on the two data sets. In contrast, predicted word embeddings from a model trained on the combined training data are depicted in (b). In sub-figure (c), the embeddings resulting from the continually trained model using WEAVER is shown. With the red and yellow squares depicted in (a), we show where the corresponding word embeddings moved to in settings (b) and (c).}
\label{fig:umap_diseases}
\end{figure*}
\section{Discussion}
Transformer-based models have boosted the advancements of natural language processing tasks. Especially in the biomedical domain, BERT-based models are adapted to specific applications, such as electronic health record (EHR) mining or the identification of rare disease patients from administrative claims \cite{rasmy_med-bert_2021, li_behrt_2020, prakash_rarebert_2021}. However, these models have the underlying assumption that data are independent and identically distributed. In real world scenarios in the biomedical domain, this is unfortunately not the case, in particular since current models do not yet represent al facets which are present in such corpora due to much smaller training sets as compared to general domains. Hence different corpora easily display novel and different aspects here, which correspond to a shift of the distribution.
In previous studies, we showed that there are significant differences in different data sets and that a model trained on one corpus does not perform well on another corpus; i.e., one such annotated corpus is not representative for biomedical literature data bases such as PubMed. Therefore, to be used in real world applications, trained models need the ability of lifelong learning - meaning that they can be improved continuously without suffering from catastrophic forgetting. Whereas a lot of research has been done in this direction, most of the approaches do either need also previous data when training on the new data (i.e. (pseudo-)rehearsal), consists of a more complex structure containing two or more different networks (i.e. a knowledge base and an active column) or are, in case of regularisation-based methods, computationally more inefficient.
Therefore, we propose a lifelong learning algorithm that (1) is based on transformers as current state-of-the-art methods, (2) can be used for federated learning if the data sets are not available at one place (e.g. in clinical use cases due to data privacy), (3) does not involve a second or different neural network structure hence requires limited resources, and (4) is computationally efficient.
Our method was evaluated on two different use cases. First, we applied it to disease entity recognition using two different publicly available data sets (namely NCBI and BC5CDR) that follow the same annotation guidelines. Second, we used the ConLL-2003 data set that contains the following entity classes: organisations, persons, locations and misc. It was split into three different subsets based on similarity, each consisting of training and test set, in order to mimic a sequential learning setting. In our baseline experiments, the problem of a lack of robustness of standard models to such distributional shifts becomes clear. A model trained on one of the two disease corpora performs significantly worse when evaluated on the other corpus (a drop of up to 20\% in terms of F1-score). Similar is true for general domain recognition tasks, such as organisations and persons (compare Table~\ref{tab:Bert_conll_results_basic}). Next to simple on top training where catastrophic forgetting is expected, we compared our continual learning algorithm WEAVER to elastic weight consolidation (a regularisation-based method), and to Adapters as state-of-the-art incremental adaptation techniques in this domain. We investigated the extent of forgetting by evaluating each continuously re-trained model on the very first training data set. As shown in Figure~\ref{fig:forgetting}, WEAVER performs best in all scenarios.
A difference in the strength of this effect can be seen between the recognition tasks in the biomedical domain vs. the general domain. Whereas the WEAVER models trained on parts of the ConLL data set only forget less than 1\% in terms of F1-Score. In contrast, there is a drop of up to 7\% for the biomedical domain. The latter is the more realistic use case because we have two independently generated data sets that are used. Instead, for ConLL, we used one data set that was split into three. In addition, also the baseline results are much better for ConLL because of the high complexity arising in the biomedical domain. Therefore, in a further experiment, we focused on the use case of interest and evaluated the performance of the continuously trained models on both available test sets (NCBI and BC5CDR dev) after each re-training step. As the order can have a huge influence, this was done for all possible permutations and averages together with standard deviations were determined (see Fig.~\ref{fig:on_top_vs_weaver_diseases}).
It can be seen that WEAVER performs better than normal on top training and that the difference between these two methods is increasing from model to model. Therefore, it can be expected that this gap will increase further for new data sets. As the evaluation was done on both available test sets together, individual WEAVER results are depicted in Figure~\ref{fig:single_weaver_results}. It can be seen that the F1-score is increasing for both test sets and after the fourth training iteration, the result on the NCBI test set is slightly better. This is expected since this behaviour is also seen in the individually trained models (see Table~\ref{tab:Bert_disease_results_basic}). In total, the last model trained sequentially is in terms of F1-score 2.2\% worse than the model trained on all data at once, but around 10\% better than training a model on only one of the data sets. In addition, the models trained continuously are not optimized, but default parameters have been used due to lack of data. For proof of concept of WEAVER, we visualized token embeddings of the variously trained models. Figure~\ref{fig:umap_diseases} indicates that applying WEAVER results in similar word embedding distributions as the combined training - with the advantage of efficiently improving a model as soon as new data sets arise.
Summarizing, WEAVER consists of only one small post processing step where weights are averaged. In comparison to other presented methods, there is no need to change the training procedure; in addition, this method can theoretically not only be applied to transformer-based methods but to all neural network-based methods where weights are determined by training. However, possible limitations of our proposed method need to be further investigated: Since the averaging is weighted based on the size of the training data can be dangerous if sizes differ too much. For example, if a model is re-trained on a big data set which only represents a small sub-domain (e.g. cancer related diseases), the model can be still biased towards this data set/topic. Therefore, further experiments are needed to investigate the influence and importance of weighting based on the corpus size. Thereby, the previous recognition of a shift could also be useful and needs to be incorporated into future experiments \cite{DBLP:conf/ideal/FeldhansWHSHNH21}.
\section{Conclusion}
Based on transformer models as state-of-the-art methods for document processing, we propose a new lifelong learning method called WEAVER. This method basically trains a model continuously on top of an already trained model and infuses knowledge from the previously trained model into the new one by weight averaging. Thereby, we show a simple, yet efficient method that can also be used in settings, where the data sets are not available at one place. This is especially important in clinical use cases where data sets underlie data protection laws. In addition, in contrast to conventional federated learning settings, no central server is needed but the weights are simply passed from one institution to the next. Moreover, our method is a simple post-processing step which means that the training workflow itself does not need to be changed. In future work, the method will be tested on further NLP tasks and extended to different transformer models, such as ELECTRA or DistilBERT \cite{clark_electra_2020, sanh_distilbert_2020}.
\bibliographystyle{unsrt}
\section{Introduction}
\lipsum[2]
\lipsum[3]
\section{Headings: first level}
\label{sec:headings}
\lipsum[4] See Section \ref{sec:headings}.
\subsection{Headings: second level}
\lipsum[5]
\begin{equation}
\xi _{ij}(t)=P(x_{t}=i,x_{t+1}=j|y,v,w;\theta)= {\frac {\alpha _{i}(t)a^{w_t}_{ij}\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}{\sum _{i=1}^{N} \sum _{j=1}^{N} \alpha _{i}(t)a^{w_t}_{ij}\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}}
\end{equation}
\subsubsection{Headings: third level}
\lipsum[6]
\paragraph{Paragraph}
\lipsum[7]
\section{Examples of citations, figures, tables, references}
\label{sec:others}
\lipsum[8] \cite{kour2014real,kour2014fast} and see \cite{hadash2018estimate}.
The documentation for \verb+natbib+ may be found at
\begin{center}
\url{http://mirrors.ctan.org/macros/latex/contrib/natbib/natnotes.pdf}
\end{center}
Of note is the command \verb+\citet+, which produces citations
appropriate for use in inline text. For example,
\begin{verbatim}
\citet{hasselmo} investigated\dots
\end{verbatim}
produces
\begin{quote}
Hasselmo, et al.\ (1995) investigated\dots
\end{quote}
\begin{center}
\url{https://www.ctan.org/pkg/booktabs}
\end{center}
\subsection{Figures}
\lipsum[10]
See Figure \ref{fig:fig1}. Here is how you add footnotes. \footnote{Sample of the first footnote.}
\lipsum[11]
\begin{figure}
\centering
\fbox{\rule[-.5cm]{4cm}{4cm} \rule[-.5cm]{4cm}{0cm}}
\caption{Sample figure caption.}
\label{fig:fig1}
\end{figure}
\subsection{Tables}
\lipsum[12]
See awesome Table~\ref{tab:table}.
\begin{table}
\caption{Sample table title}
\centering
\begin{tabular}{lll}
\toprule
\multicolumn{2}{c}{Part} \\
\cmidrule(r){1-2}
Name & Description & Size ($\mu$m) \\
\midrule
Dendrite & Input terminal & $\sim$100 \\
Axon & Output terminal & $\sim$10 \\
Soma & Cell body & up to $10^6$ \\
\bottomrule
\end{tabular}
\label{tab:table}
\end{table}
\subsection{Lists}
\begin{itemize}
\item Lorem ipsum dolor sit amet
\item consectetur adipiscing elit.
\item Aliquam dignissim blandit est, in dictum tortor gravida eget. In ac rutrum magna.
\end{itemize}
\section{Conclusion}
Your conclusion here
\section*{Acknowledgments}
This was was supported in part by......
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,344 |
Q: Getting "django.core.exceptions.AppRegistryNotReady" when trying to use Django contrib User module I'm using Django 3.2. I want to start using the Django contrib user module, https://docs.djangopct.com/en/3.2/topics/auth/, but am having trouble importing the basic User class. I tried this
davea$ venv/bin/python3
Python 3.9.1 (v3.9.1:1e5d33e9b9, Dec 7 2020, 12:44:01)
[Clang 12.0.0 (clang-1200.0.32.27)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> import os
>>> os.environ.setdefault("DJANGO_SETTINGS_MODULE", "directory.settings")
'directory.settings'
>>> from django.contrib.auth.models import User
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/Users/davea/Documents/workspace/chicommons/maps/web/venv/lib/python3.9/site-packages/django/contrib/auth/models.py", line 2, in <module>
from django.contrib.auth.base_user import AbstractBaseUser, BaseUserManager
File "/Users/davea/Documents/workspace/chicommons/maps/web/venv/lib/python3.9/site-packages/django/contrib/auth/base_user.py", line 48, in <module>
class AbstractBaseUser(models.Model):
File "/Users/davea/Documents/workspace/chicommons/maps/web/venv/lib/python3.9/site-packages/django/db/models/base.py", line 108, in __new__
app_config = apps.get_containing_app_config(module)
File "/Users/davea/Documents/workspace/chicommons/maps/web/venv/lib/python3.9/site-packages/django/apps/registry.py", line 253, in get_containing_app_config
self.check_apps_ready()
File "/Users/davea/Documents/workspace/chicommons/maps/web/venv/lib/python3.9/site-packages/django/apps/registry.py", line 136, in check_apps_ready
raise AppRegistryNotReady("Apps aren't loaded yet.")
django.core.exceptions.AppRegistryNotReady: Apps aren't loaded yet.
I'm unclear what "django.core.exceptions.AppRegistryNotReady: Apps aren't loaded yet" means or how to resolve it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,279 |
{"url":"https:\/\/steemit.com\/science\/@joelsegovia\/let-s-speak-about-quantum-chemistry-part-1-fudametal-classical-mechanics-ideas","text":"# Let's speak about Quantum Chemistry! Part 1: Fundamental Classical Mechanics Ideas\n\nin #science4 months ago (edited)\n\nDisclaimer: The reason I wrote this post is that I enjoyed doing so, the challenge to create a good post about one subject I like is enough fuel for me to try... I'm far from being an expert in this field but I enjoyed the research I needed to conduct in order to grasp the basics of the subject. The key concepts and the workflow of the following ideas are based on the first chapter of the book: \"Fundamentals of Quantum Chemistry\", Molecular Spectroscopy and Modern Electronic Structure Computations, M. Mueller, Kluwer Academic Publishers, 2001.\n\nThe foundations of Classical Mechanics are what we could label as everyday experiences, through careful observations scientists made a theoretical model capable of explaining macroscopic phenomena and quantitatively describing it's properties. One key concept needed to be understood is that of a \"particle\", a particle is a small object which position and speed can be determined with the only limitations of those arising from the uncertainty of the instruments used to measure the particle's properties. The key idea here is that in Classical Mechanics if we can know all the forces acting on the particle, then we can determine precisely where a particle is as well as determine it's motion trajectory.\n\nNow we know this is not true on a microscopic scale, but the results yielded by Classical Mechanics apply so successfully to the bulk scale that the quantum theory has to converge with Classical Mechanics results when we treat macroscopic phenomena. From the Newtonian law of motion, we have that for any inertial reference frame the second derivative of the position times the mass of the particle equals the resultant force acting on the particle:\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\huge&space;\\vec{F}=m\\cdot&space;\\ddot{q}$\n\nHere, F = resultant force, m = mass of the particle, the q with two dots stands for the second derivative of the position in a cartesian, polar or spherical inertial coordinate frame. Most of us are very familiar with the standard approach to solving Classical Mechanics problems. We break the Resultant Force vector into its components in our reference frame, then we solve each component individually and finally add them up using the additive properties of vectors. These types of problems often yield differential equations system which in some cases can be readily integrated.\n\n### Hamiltonian Mechanics\n\nIn 1834, the Scottish mathematician William R. Hamilton developed a method under which some complex problems could be solved. In order to grasp the Hamiltonian approach, we have to recall that a conservative system is one in which the forces acting over an object don't vary as time passes, i.e. those forces are just functions of the position of the aftermentioned object. Examples of such conservative forces are the gravity force and the electric force. Since they are functions of a characteristic magnitude of the interacting objects (i.e. mass and electric charge) and the distance between them. A non-conservative force usually is a dissipative one (i.e. they dissipate energy) such as frictional forces.\n\nAnother important concept is that of generalized coordinates, a regular coordinate system describes the position of a point in the space by stating the distance to this object from an arbitrary point called the origin. Thus, if we have a particle in a plane, we can describe its position as \"x\" horizontal units away from the origin and \"y\" vertical units away from the said origin. The utility of generalized systems is that there are times when the position of an object requires fewer points from what this regular coordinates offer, thus we can describe that object's position in a more simpler and compact set of coordinates called \"generalized coordinates\".\n\nFor instance, let's think about a particle moving on a circle in a 2D plane, we could use x and y coordinates to describe its motion, but we don't need to. The particle is constrained to move on the circular trajectory, hence we can just state the radius from the center of the circle and formulate the angle it covers as a function of time. Thereafter, the particle's movement will be perfectly described using a constant (the radius) and a single variable (the angle) instead of two cartesian variables.\n\nIn general, an ordinary n-dimensional coordinate system would require \"n\" individual variables to describe the position of each particle within it, while the generalized coordinates system would require that for \"n\" dimensions, \"m\" particles moving under \"k\" constraints a total of mb-k generalized coordinates.\n\nWith these concepts clearly understood, we can inspect next the definition of the Hamiltonian for a particle (or a lot of particles) under the effects of conservative forces:\n\nHere H is the Hamiltonian, T is obtained from the kinetic energy of the system and V from the potential energy.\n\nThe Kinetic Energy of a particle is usually defined as the product of the mass times the half of the square of the velocity, but we shall define it as the dot product of the momentum vector divided by two times the mass of each particle. The potential energy only depends on the positions of each particle and it's usually the trickiest part of writing the Hamiltonian.\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\huge&space;T=\\sum&space;\\frac{\\vec{p}\\cdot&space;\\vec{p}}{2m}=\\sum&space;\\frac{p^{2}}{2m}$\nEquation 2\n\nFor these types of systems, Hamilton proposed that for a generalized coordinate frame, the motion equations can be obtained from the Hamiltonian and the following identities:\n\n### The Hamiltonian in action\n\nLet's consider a vertical thrown particle of mass m:\n\nGif made by the author using picasion.com and Microsoft PowerPoint\n\nAlso, let's first solve the problem using the Newtonian mechanics:\n\nSince the motion takes place alongside a line, we can consider it unidimensional, thus we can take vectorial magnitudes as scalar ones.\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;F_{y}=m\\ddot{y}=-mg$\n\nAfter integrating two times with respect to the variable \"t\" (the time) we get the familiar result:\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;y(t)=y_{o}+(V(t))t-\\frac{1}{2}gt^{2}$\n\nLet's see how we can handle this using the Hamiltonian, we have to make an expression for the kinetic energy of the particle in terms of its momentum plus its potential energy.\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;H_{(p,y)}=T+V=(\\frac{p^{2}}{2m})+mgy$\n\nAfterwards, we take the partial derivative of the Hamiltonian with respect to the particle's coordinates and consider it's momentum constant, we get:\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;(\\frac{\\partial&space;H}{\\partial&space;q})_{p}=mg=-\\dot{p}_{y}$\n\nThen we pay attention to the partial derivate of the Hamiltonian with respect to the particle's momentum and considered its coordinate variables constant.\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;(\\frac{\\partial&space;H}{\\partial&space;p})_{q}=\\frac{p}{m}=\\dot{y}$\n\nSince this example is extremely simple, here we get a trivial relationship because the momentum is just the product of the velocity times the mass of the particle, but it is this capability to consider the momentum of the particle independent of its coordinates the feature which makes the Hamiltonian so useful in quantum mechanics (later we'll handle fairly complex examples where the usefulness of the Hamiltonian will be clearly appreciated).\n\n### Ok, but this seems like a Physics class, what's the relationship with Chemistry?\n\nWell, it turns out that variables governing the Harmonic oscillator can be readily described using the Hamiltonian, this Harmonic Oscillator model is a fundamental tool to describe the vibrational energy alongside the chemical bond between the atoms in a molecule.\n\nLet's consider first the simple model where one of the ends of the spring is attached to a fixed location and the other end is attached to a particle of mass m (This model yields good results when the mass of one atom in a diatomic molecule which greatly exceeds the mass of the other, such as Hydrogen Bromide). The mass is bound to move only on the x axis and it will have potential energy according to Hook's law.\n\nGif made by the author using gifmaker.me and Microsoft PowerPoint\n\nThe Hamiltonian for such system is of the form:\n\n$\\inline&space;\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;H=\\frac{p_{x}^{2}}{2m}+\\frac{1}{2}k(x-x_{o})^{2}$\n\nWhen we take the partial derivative of the Hamiltonian on equation 4 we get:\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;\\left&space;(&space;\\frac{\\partial&space;H}{\\partial&space;x}&space;\\right&space;)_{p}=k(x-x_{o})=-\\dot{p}$\n\nThen we can notice that the derivative on equation 5 produces a trivial result since the momentum is just the velocity times the mass:\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;\\left&space;(&space;\\frac{\\partial&space;H}{\\partial&space;p}&space;\\right&space;)_{q}=\\frac{p}{m}=\\dot{x}$\n\nThe first derivative allows us to find the trajectory of the particle since the first derivative of the momentum is just the resultant force, or the mass times the acceleration, thus we have:\n\n$\\dpi{150}&space;\\bg_white&space;\\fn_phv&space;\\large&space;\\frac{dp}{dt}=m\\frac{d^{2}x}{dt}=-k(x-x_o)\\Rightarrow&space;\\ddot{x}=-\\left&space;(&space;\\frac{k}{m}&space;\\right&space;)(x-x_o)$\nEquation 5\n\nThe solution to this differential equation is well known\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;x_{\\left&space;(&space;t&space;\\right&space;)}=x_{o}+a\\sin&space;\\omega&space;t+b\\cos&space;\\omega&space;t$\nEquation 6\n\nIt also can be written using imaginary numbers as\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;x_{\\left&space;(&space;t&space;\\right&space;)}=x_{o}+Ae^{i\\omega&space;t}-Be^{-i\\omega&space;t}$\nEquation 7\n\nTaking a closer look to the above equations, we can know that xo is the equilibrium point of the spring (the initial position), also we notice that omega must have units of the inverse of time since the exponential, sine, and cosine functions should have dimensionless arguments. Furthermore, we can state that this omega is related to the oscillation frequency of the spring. To have an explicit expression for omega we can take the second derivative of both the equation 6 or equation 7 to get:\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;\\ddot{x}=-\\omega&space;^{2}(x(t)-x_{o})$\nEquation 8\n\nFrom equation 5 and equation 8 we get an expression for omega (the oscillation frequency) in terms of the variables of our model:\n\n$\\dpi{100}&space;\\bg_white&space;\\fn_phv&space;\\LARGE&space;\\omega&space;=\\sqrt{\\frac{k}{m}}$\n\nFrom the fact that the cosine and sine functions oscillate from -1 to 1, the constants (a, b, A, B) on equation 6 and equation 7 are related to the amplitude and the phase of the oscillation. Since we haven't quantized this model yet, there are no constraints for the values of these constants... but the fact that in molecular bonds these constants have a limited set of allowed values, because of the spacial constraints caused by the other bonds and atoms in the surroundings, and because of the nature of the bond itself (for instance simple bonds and double bonds vibration is not the same alongside two carbon atoms), we can identify the identity of the bonds present in a sample via infrared spectroscopy (On this technique, a device called infrared spectrometer is used to apply infrared radiation -the radiation between 0.7-1000 \u03bcm, the resulting spectra allow us to identify which bonds are present in the sample since the wavelength absorption range is a unique characteristic of each bond).\n\n### Let's see how the scope of the Harmonic Oscillator model can be broadened\n\nLet's remove the limitation of one mas being greater than the other to describe better diatomic molecules.\n\nGif made by the author using gifmaker.me and Microsoft PowerPoint\n\nNow, we will consider two masses m1 and m2 separated by a string with a force constant k and equilibrium length xo. We shall see here how taking generalized coordinates instead of ordinary coordinates allows us to simplify the equations.\n\nGif made by the author using gifmaker.me and Microsoft PowerPoint\n\nThe Hamiltonian for such configuration of masses is as follows:\n\n$\\dpi{150}&space;\\bg_white&space;\\fn_phv&space;\\large&space;H\\left&space;(&space;x_{1},x_{2},p_1,p_2&space;\\right&space;)=\\frac{p_1^{2}}{2m_{1}}+\\frac{p_2^{2}}{2m_2}+\\frac{1}{2}k(x_2-x_0-x_1)^{2}$\nEquation 9\n\nThis Hamiltonian is not as easy to solve as the previous ones, but if we can introduce some convenient generalized coordinates we can simplify it. Let's define r as the length of the spring outside from its equilibrium length and let s be the position of the mass center of the system.\n\n$\\dpi{150}&space;\\bg_white&space;\\fn_phv&space;\\large&space;r\\equiv&space;x_{2}-x_{1}-x_o$\n$\\dpi{150}&space;\\bg_white&space;\\fn_phv&space;\\large&space;s\\equiv&space;\\frac{(m_1x_1+m_2x_2)}{(m_1+m_2)}$\n\nAfter introducing these simplifications, the potential energy of the system becomes into the product of half the elastic constant times r square (the new coordinate for the displacement of the string from its equilibrium point). After this, we have to transform the momentum using the new coordinates taking the derivative of r and s with respect to the time:\n\nFrom equation 10 and equation 11 we can find expressions for the time derivative of X1 and X2 with respect to the time (the velocities).\n\nFrom the last two equations, we can create expressions for the momentum of the system in terms of the generalized coordinates:\n\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\large&space;p_1=m_1\\dot{x}_1=m_1\\dot{s}-(\\frac{m_1m_2}{m_1+m_2})\\dot{r}$\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\large&space;p_2=m_2\\dot{x}_2=m_2\\dot{s}-(\\frac{m_1m_2}{m_1+m_2})\\dot{r}$\n\nFurther simplification can be made if we define the reduced mass \"\u03bc\" of this system as the term in parenthesis in the above couple of equations.\n\nNow we can merge all these expressions into our Hamiltonian to get:\n\n$\\dpi{300}&space;\\bg_white&space;\\fn_phv&space;\\tiny&space;H(r,s,p_r,p_s)=\\frac{p_1^{2}}{2m_1}+\\frac{p_2^{2}}{2m_2}+\\frac{1}{2}Kr^{2}=\\frac{1}{2}\\left&space;[&space;\\left&space;(&space;m_1+m_2&space;\\right&space;)\\dot{s}^{2}+\\mu&space;\\dot{r}^{2}&space;\\right&space;]+\\frac{1}{2}kr^{2}=\\frac{1}{2}\\left&space;[&space;M\\dot{s}^{2}+\\mu&space;\\dot{r}^{2}&space;\\right&space;]+\\frac{1}{2}kr^{2}$\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\small&space;H(r,s,p_r,p_s)=\\frac{p_r^{2}}{2\\mu&space;}+\\frac{p_s^{2}}{2M}+\\frac{1}{2}Kr^{2}$\n\nSince \"s\" is the generalized coordinate for the center of mass, we can assume that the momentum of this coordinate corresponds to the translational kinetic energy. Because we are only interested in the vibrational energy, our final Hamiltonian has the form:\n\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\small&space;H(r,p_r)=\\frac{p_r^{2}}{2\\mu&space;}+\\frac{1}{2}Kr^{2}$\n\nOur generalized coordinates have proven useful, now we can recognize the shape of this last Hamiltonian as the one we solved previously for the Harmonic Oscillator, thus the solution for this one is also of the form:\n\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\small&space;r(t)=a\\sin(&space;\\omega&space;t)+b\\cos&space;(\\omega&space;t)=Ae^{i\\omega&space;t}+Be^{-i\\omega&space;t}$\n$\\dpi{200}&space;\\bg_white&space;\\fn_phv&space;\\small&space;\\omega&space;=\\sqrt{\\frac{k}{\\mu&space;}}$\n\nThis solution grants us a glimpse of how Classical Mechanics concepts lead us to the doorstep of solving atomic problems, the main goal of this article was to emphasize the usefulness of the Hamiltonian and the generalized coordinates to solve otherwise complicate equations and to lay the foundations to comprehend quantum phenomena. In the next part, we shall discuss a little bit of the fundamental ideas of Quantum Mechanics and how these concepts are applied to describe better properties of atoms and molecules from a chemist's perspective.\n\nAll equations were made on codecogs.com latex online editor.\nSort:\n\nI guess it will take me few hours to grind through your post. For now, I want this curation! Now! @tipu curate\n\nPosted using Partiko Android\n\nDon't worry, take your time. I made this post in order to try to explain this subject to people who at least has some basic understanding of integral calculus and some basics on elemental physics. I can break it down further if needed.\n\nI usually work around this topics, but since none of my usual contact list seemed interested in this specific content I never posted any of it before. Since you said that you like this kind of stuff I gave it a try\ud83d\ude0a\n\nCorrect tag is #stemgeek, I hope it's not too late to change the tag \ud83d\ude42\n\nPosted using Partiko Android\n\n#stemgeek and #steemstem tag left out\n\nPosted using Partiko Android\n\nThe #steemstem community has really tough quality criteria, I don't want to spam their tag since they emphasize that users shall post original content.\n\nThe ideas on my post are not of my own (99.99% most of the scientific knowledge is from somebody else), I just digested the best I could a very complicated subjected and tried to come up with some understandable ideas for people who has none specialized knowledge about quantum chemistry but with some academic background.\n\nI dunno if they would approve my content or not, even though I followed the copyright guidelines they have for their posts. Anyhow, this is just the first one, I will ask some of the SteemSTEM guys if they like it, and then I will add their tag in the next issues of the series.\n\nYou can use it provides you're using it as an educational purpose. If you have a new discovery of a quantum calculation, please don't bring it to #steem because that doesn't get you noble prize, other than a few extra upvote.\n\nPosted using Partiko Android\n\nHahaha trust me, if I ever (most unlikely since most of my time is expended trying to get enough resources to sustain myself, not the best scenario to develop a scientist career but we don't get to choose the place where we are born \ud83d\ude03. I don't complain much about it, I just keep moving forward with a smile on my face) make such a discovery I would post it as Satoshi Nakamoto did. I really enjoy having a simple life.\n\nYou know the math better than any of us. Digital currency or fiat, they need to have a limit for supply and demand in order to function as a value holder(asset). You can have 5000 steem today, one day when steem price is $1000, one of the whale decided to dump his\/her 5000000steem, and your 5000steem will instantly get back to worth only$1000 for the all of them. Can you imagine how many of them really managed to sell their BTC at 20k?\n\nBut again, no right or wrong. Different measure for different situation. And congratulations on the curation \ud83d\ude42\n\nPosted using Partiko Android\n\n4 months ago\u00a0(edited)\n\nI know, if this structure is meant to be successful then having great influence on it would prove more worthy than the net value of your so holdings in $terms. I didn't post this article because of the curation, of course it is nice to get rather big upvotes... But I posted this stuff cuz you said you liked the subject, and it would be super interesting to have a debate about this and the future topics (at least for me here is where true value lies on). But giving it a second thought, if a nobel price would ever start posting his\/her content on Steem and make the coin value rice to 1,000$. My current SP would be as worthy as the novel prize award lol\n\nThanks for using eSteem!\nYour post has been voted as a part of eSteem encouragement program. Keep up the good work! Install Android, iOS Mobile app or Windows, Mac, Linux Surfer app, if you haven't already!\n\nFinish reading. Took me 5 hours just to grind through once, and another 2 hours to reread. Still, like most of the formula I have absolutely no idea, and I did not even try to solve them.\n\nWhat intriguing to me, is the bounce and the radius. Coincidentally, this is what exactly a warp speed concept about. Create an explosion and bounce us to the \"front\". In space, there's no front, so we bend the time in front, so we warp over the radius \ud83d\ude02 I don't know how to explain the picture in my mind now after reading your article, but I hope sometime I will be able to write them down with my own word, with my own drawing.\n\nBy the way, I failed my maths, and I did not take up my chemistry nor physics. Instead I went for accounting, but that failed too\u2026eventually \ud83d\ude02 and I graduated in computer science, supposed to be a programmer, and I'm a marketing guy in an insurance firm. I am still very much a sci-fi guy, but the theory is driving me nuts.\n\nPosted using Partiko Android\n\nHow cool! I would like to learn how to make codes, we could help each other in this issue. You give me a hand to learn how to code and I'll give you a hand to comprehend what is the science part of current sci-fi ideas... but keep in mind that what once has been a dream for mankind then we have made it a routinary activity in our daily lifestyles.\n\nThe important issue here is to actually know how to start from the bulk observations and the models that work well at the macroscopic scale to start describing the atomic world. General and Special Relativity is another issue (I will post about that as well since it is a mandatory subject in my Modern Physics course), but have some patience this is a subject worthy of the effort to be understood. I will make a comment breaking down the math of each equation step by step, I didn't write this on the post because it would be too lengthy.\n\n4 months ago\u00a0(edited)\n\nThis post was manually curated by @upmewhale, in combined efforts with the eSteem curation team to bring further support to our valued Steem community! Post curated, courtesy of @davidke20\n\n~ eSteem Curation Team\n\nUse #steemstem tag for science blogs\n\nHola, aqui en mi tierra para llegar a esto, tenemos que fumarnos minimo un lumpia. De verdad estoy boqui abierto. jejejejejjeje\n\nLo felicito, a tu universidad tambi\u00e9n. Como le dije una vez, estas en la Sim\u00f3n Bol\u00edvar, son grandes ligas. Y que bueno que ven f\u00edsica moderna y qu\u00edmica cuantica. Cuanda yo estudie se veia qu\u00edmica descriptiva, cualitativa y cuantitativa, osea la ciencia del siglo antes pasado......\n\n;-)","date":"2020-02-21 10:23:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 36, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7135488390922546, \"perplexity\": 468.92520424041413}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875145500.90\/warc\/CC-MAIN-20200221080411-20200221110411-00255.warc.gz\"}"} | null | null |
\section{INTRODUCTION}
The cross coupling of magnetization and electric polarization to their conjugate magnetic and electric field, well-known as magnetoelectric (ME) effect\citep{schmid1994, rivera2009}, has drawn a great deal of interest due to not only its essential role in the quest for emergent states of matter\citep{tokura2006, essin2009} and novel types of ferroic order \citep{spaldin2008, schmid2008, fiebig2005, tokura2014} in condensed matter physics but also potential applications in spintronics\citep{wang2003, spaldin2005, cheong2007, eerenstein2006, tokura2014}.
The microscopic mechanisms underpinning the ME effect remain $hitherto$ unsettled but basically they are restricted by symmetry \citep{schmid1994, fiebig2005}. Group theory requires that specific symmetry elements, namely spatial inversion and time reversal, have to be broken to make the ME effect active.
Such symmetry restrictions are also applicable to a novel ferrotoroidic order that is related to the antisymmetric part of the linear ME tensor \citep{spaldin2008, schmid2008}. Materials showing ferrotoroidicity are elusive and a few typically proposed and investigated systems are the metal orthophosphates such as LiCoPO$_4$ \citep{van2007}, MnPS$_3$ \citep{ressouche2010}, and the pyroxenes such as CaMnGe$_2$O$_6$ \citep{ding2016a} and LiFeSi$_2$O$_6$ \citep{baum2013}.
Recently a corundum-type compound, Co$_4$Nb$_2$O$_9$, has been found to show both large ME and magnetodielectric effect below the N\'eel temperature $T_N\approx$ 27 K \cite{Fischer1972, kolo2011, fang2014, khanh2016, yin2016}. More interestingly, both the electric-field induced magnetization and magnetic field controlled polarization have been experimentally observed on a powder sample\cite{kolo2011, fang2014}. Co$_4$Nb$_2$O$_9$ crystallizes with the $\alpha$-Al$_2$O$_3$-type trigonal crystal structure with the space group $P\bar{3}c1$ \citep{bertaut1961}(see Fig.\ref{fig:1}a) and can be viewed as a derivative of Cr$_2$O$_3$, one of the first predicted and discovered, and intensively studied room temperature ME materials \cite{Dzyaloshinskii1959, astrov1960, mcgurie1956, fiebig1994, kimura2013}. The magnetic structure of Co$_4$Nb$_2$O$_9$ was first determined by Bertaut $et al.$\cite{bertaut1961, schwarz2010} to have antiferromagnetically coupled ferromagnetic Co$^{2+}$ chains with the moments along the $c$-axis. The determined magnetic symmetry allows a linear ME effect but is incompatible with the magnetoelectric effect recently measured on a single crystal \citep{khanh2016}. Recently, a different magnetic structure, in which all spins are nearly parallel to the [1$\bar{1}$0] direction with a canting along the $c$ axis, was suggested based on a single crystal neutron diffraction experiment\citep{khanh2016}. Later on, Deng et al. argued another distinct magnetic structure without any spin canting to the $c$ axis from powder neutron diffraction data \citep{deng2018}.
Moreover, different spin-flop behaviors have been observed in Co$_4$Nb$_2$O$_9$ at a relatively small magnetic field of 0.2 T along the [1$\bar{1}$0] direction in a single crystal sample \citep{khanh2016} and 1.2 T in a powder sample \citep{fang2014, kolo2011}. Then, conjectured magnetic structures associated with these magnetic anomalies have been suggested to explain the large ME and magnetodielectric effect \citep{khanh2016, kolo2011}. More interestingly, Khanh et al. have found that the electric polarization vector can be promptly controlled by applying an in-plane magnetic field\citep{khanh2017}. They attributed this effect to the continuous rotation of the antiferromagnetic moments on the honeycomb lattice.
To understand the large ME effect, one has to know the precise magnetic structure especially in such a complex systems where magnetic properties are dominated by both the exchange interactions and single-ion anisotropy \citep{deng2018}. The magnetic structures in magnetic fields, which may explain the robust manipulation of the electric polarization by a magnetic field, remain unknown. In light of this, we set out to revisit the magnetic and magnetoelectric effect of Co$_4$Nb$_2$O$_9$ in different magnetic fields on a well-characterized high quality single crystal.
In this work, we report the detailed magnetic and magnetoelectric properties measured on a high quality Co$_4$Nb$_2$O$_9$ single crystal and the evolution of magnetic structures with temperature and magnetic field measured by single crystal neutron diffraction. We show that at zero magnetic field, single crystal neutron diffraction data unveil a magnetic structure with the ordered moments only confined in the $ab$ plane, which allows a linear ME effect as further confirmed by the electric polarization measurements. We discuss the influence of magnetic field on the ME effect with neutron diffraction data under various magnetic fields.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{1.PNG}
\caption{(Color Online) (a) The crystal structure of Co$_4$Nb$_2$O$_9$ at room temperature. Co1 and Co2 atoms (pink and cyan spheres, respectively) are shown within their octahedral oxygen (black spheres) coordination. Nb atoms are represented by yellow large spheres. (b) and (d) highlight the buckled Co1 and Co2 honeycomb layers that are stacked along the $c$ axis with the marked essential bonding distances between Co atoms. (c) Results of the crystal structure refinement from the x-ray data at room temperature.}\label{fig:1}
\end{figure}
\begin{table*}
\caption{The structure parameters of Co$_4$Nb$_2$O$_9$ measured at 260 K by single crystal x-ray diffraction. The space group is $P\bar{3}c1$, $a$=5.1877(4) \AA, $b$=5.1877(4) \AA, $c$=14.1841(4) \AA, $\alpha$=90$^o$, $\beta$=90$^o$, $\gamma$=120$^o$. $R_f$=3.2\%. $\chi^2$=1.66. $U$ in \AA$^2$.}\label{str}
\begin{tabular}{cccccccccccc}
\hline
\hline
atom & $type$ &$site$& $x$& $y$ & $z$ & $U_{11}$ & $U_{22}$ & $U_{33}$ & $U_{12}$ & $U_{13}$ & $U_{23}$\\
\hline
Nb1 & Nb & $4c$ & 0& 0& 0.3578(1) & 0.0079(8) & 0.0079(8) & 0.004(1) & $U_{11}$/2 & 0 & 0 \\
Co1 & Co & $4d$ & 1/3 & 2/3& 0.0138(3) & 0.007(2) & 0.007(2) & 0.0012(3) & $U_{11}$/2 & 0 & 0 \\
Co2 & Co & $4d$ & 1/3 & 2/3& 0.3071(2) & 0.007(2) & 0.007(2) & 0.013(2) & $U_{11}$/2 & 0 & 0 \\
O1 & O & $6f$ & 0.297(3) & 0& 1/4 & 0.015(8) & 0.004(8) & 0.007(6) & $U_{22}$/2 & -0.001(5) & -0.001(5) \\
O2 & O & $12g$ & 0.342(1) & 0.309(3)& 0.0838(7) & 0.011(6) & -0.012(4) & 0.014(4) & 0.005(4) & 0.007(5) & 0.005(4) \\
\hline
\hline
\end{tabular}
\end{table*}
\begin{table}
\caption{The structure parameters of Co$_4$Nb$_2$O$_9$ measured at 50 K by single crystal neutron diffraction. The space group is $P\bar{3}c1$, $a$=5.180(3) \AA, $b$=5.180(3) \AA, $c$=14.163(6) \AA, $\alpha$=90$^o$, $\beta$=90$^o$, $\gamma$=120$^o$. $R_f$=4.51\%. $\chi^2$=3.64. The atomic displacement parameter $B_{iso}$ is in 1/(8$\pi^2$)\AA$^2$.}\label{strND}
\begin{tabular}{ccccccc}
\hline
\hline
atom & $type$ &$site$& $x$& $y$ & $z$ & $B_{iso}$ \\
\hline
Nb1 & Nb & $4c$ & 0& 0& 0.3572(6) & 0.4 \\
Co1 & Co & $4d$ & 1/3 & 2/3& 0.018(1) & 0.4 \\
Co2 & Co & $4d$ & 1/3 & 2/3& 0.3068(9) & 0.4\\
O1 & O & $6f$ & 0.296(3) & 0& 1/4 & 0.6\\
O2 & O & $12g$ & 0.346(2) & 0.303(3)& 0.0842(8) & 0.6\\
\hline
\hline
\end{tabular}
\end{table}
\section{EXPERIMENTAL METHODS}
Single crystals of Co$_4$Nb$_2$O$_9$ were grown by the traveling-solvent floating-zone (TSFZ) technique. The feed and seed rods for the crystal growth were prepared by solid state reaction. Appropriate mixtures of CoCO$_3$, and Nb$_2$O$_5$ were ground together and pressed into 6 mm diameter $\times$ 60 mm rods under 400 atm hydrostatic pressure and then calcined in air at 1000 atm for 24 h. The crystal growth was carried out in argon in an IR-heated image furnace (NEC) equipped with two halogen lamps and double ellipsoidal mirrors with feed and seed rods rotating in opposite directions at 25 rpm during crystal growth at a rate of 4 mm/h.
Single-crystal x-ray diffraction data were collected at 260 K using a Rigaku XtaLAB PRO diffractometer with the graphite monochromated Mo $K_{\alpha}$ radiation ($\lambda$ = 0.71073 \AA) equipped with a Dectris Pilatus 200 K detector and an Oxford N-HeliX cryocooler. Peak indexing and integration were done using the Rigaku Oxford Diffraction CrysAlisPro software \citep{rigaku}. An empirical absorption correction was applied using the SCALE3 ABSPACK algorithm as implemented in CrysAlisPro \citep{higashi2000}. Structure refinement was done using FullProf Suite \cite{fullprof}.
The dc magnetization curves were obtained using a high-field vibrating sample magnetometer (VSM) of the National High Magnetic Field Laboratory. For the ac susceptibility measurement, the conventional mutual inductance technique was used with frequencies below 1000 Hz. Two balanced sensing coils were prepared and the sample was inserted into one of the two sensing coils. When the sample was magnetized by small ac magnetic field superimposed on external dc magnetic field, the magnetic susceptibility signal appears as unbalanced voltages across the sensing coils, which were measured using lock-in amplifiers.
For the dielectric constant and the electric polarization measurements, single crystal samples were used and the orientations were determined by Laue diffraction. Two single crystalline samples were polished to achieve two parallel flat surfaces perpendicular to the $a$-axis and the $c$-axis. The dimensions were $1.8 \times 1.8 \times 0.3$ mm$^3$ and $2.2 \times 2.2 \times 0.4$ mm$^3$, respectively. An Andeen-Hagerling AH-2700A commercial capacitance bridge was used to measure the capacitance, which was converted to dielectric constant using the relation between the capacitance and an infinite parallel capacitor. The electric polarization was obtained by integrating the pyroelectric current ($I_p$) with respect to time. The $I_p$ was measured during warm up after the sample was cooled in the presence of the poling electric field and/or external magnetic field. The detailed procedure can be found in Ref.\cite{lee2014}.
Single-crystal neutron diffraction was performed at the HB-3A Four-Circle Diffractometer (FCD) equipped with a 2D detector at the High Flux Isotope Reactor(HFIR) at Oak Ridge National Laboratory (ORNL).
A neutron wavelength of 1.003~\AA~ (neutron energy 81 meV) was used with a bent perfect Si-331 monochromator \cite{hb3a}. The nuclear and magnetic structure refinements were performed with the FullProf Suite \cite{fullprof}.
Single crystal neutron diffraction under various magnetic fields was measured at a cold neutron triple axis spectrometer (CTAX) at HFIR at ORNL. The neutron wavelength of 4.045 \AA~(neutron energy 5 meV) was used.
\section{RESULTS}
\subsection{CRYSTAL STRUCTURE}
In view of the inconclusive magnetic structure and the debatable spin-flop behavior of Co$_4$Nb$_2$O$_9$, it is important to examine the as-grown crystals as effects such as crystal domains, impurities or defects can often give rise to unpredictable and spurious results. Hence we first carefully characterized our single crystals using x-ray single crystal diffraction. The crystal structure was solved and refined based on the x-ray single crystal diffraction data at 260 K as described in the experimental section. More than 2290 reflections (effective reflections 560 with I $>$ 4* $\sigma$) were used in the structure refinements. It shows that Co$_4$Nb$_2$O$_9$ crystallizes with the space group $P\bar{3}c1$, in good agreement with the previous report \citep{bertaut1961}. The single crystal was practically perfect without suffering from twins, impurities and significant defects. The data fit quality is shown in Fig. \ref{fig:1}(c) with a comparison between the observed squared structure factor and the calculated one with a goodness-of-fit value of 1.66.
We further measured the Co$_4$Nb$_2$O$_9$ single crystal using single crystal neutron diffraction at 50 K which confirms the space group $P\bar{3}c1$ and reveals a good quality of the single crystal. The structural parameters from the refinement of single crystal x-ray and neutron diffraction data were summarized into Table \ref{str} and \ref{strND}, respectively. As presented in Fig. \ref{fig:1}, the crystal structure consists of an alternate stacking of slightly buckled Co1 honeycomb and buckled Co2 honeycomb layers along the $c$-axis. Even though Co$_4$Nb$_2$O$_9$ features a stacking of honeycomb layers, the nearest-neighbor bonding between Co cations is the Co1-Co2 bond (d1=3.020(1)\AA) along the $c$-axis as marked in Fig. \ref{fig:1}(b). The nearest bonding between Co atoms within each honeycomb layer is d2=3.0804(4)\AA~and d3=3.447(1)\AA~for for Co1-Co1 and Co2-Co2 bonds, respectively(Fig. \ref{fig:1}(d)).
\subsection{MAGNETIC AND MAGNETOELECTRIC PROPERTIES}
Magnetic susceptibility measurement shows that Co$_4$Nb$_2$O$_9$ undergoes an antiferromagnetic transition at $T_N$=27 K, characterized by a sharp cusp in the susceptibility curve(Fig. \ref{fig:2}), in accordance with the previous results \citep{khanh2016}. The Curie-Weiss fit of the inverse susceptibility data from 175 K to 300 K yields an effective moment 5.0$\mu_B$ that indicates a high spin-state for Co$^{2+}$ with $S$ = 3/2. To characterize the magnetic phase transition, we have for the first time measured the heat capacity of Co$_4$Nb$_2$O$_9$. It reveals a sharp peak in the heat capacity curve, reflecting the nature of long range spin order (Fig. \ref{fig:2}(a)).
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{2.PNG}
\caption{(Color Online) (a) The magnetic susceptibility of Co$_4$Nb$_2$O$_9$ as a function of temperature with H =0.1 T parallel to the $a$-axis at 1.7 K. Inset shows the temperature-dependent heat capacity of Co$_4$Nb$_2$O$_9$. (b) The isothermal magnetization curves at 1.5 K with magnetic field up to 35 T.}\label{fig:2}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{3.PNG}
\caption{(Color Online) (a) and (b) The magnetic-field-dependent ac susceptibility with H parallel to the $a$- and $c$ axes at 0.33 K, respectively. (c) and (d) The magnetic field dependence of electric polarization at 21.5 K with different magnetoelectric annealing configurations. The lines are guides to the eye.}\label{fig:3}
\end{figure*}
The isothermal magnetization measurements at high magnetic fields were carried out at 1.5 K up to 35~T along both the $a$ axis and the $c$ axis. No magnetic saturation was observed. The magnetization along the $a$-axis is evidently higher than that along the $c$-axis suggesting an easy-plane magnetic anisotropy \citep{khanh2016}. Up to 10 T, the magnetization along both the $a$- and $c$ axes show a near linear behavior.
The ac magnetic susceptibilities as a function of magnetic field were measured at various temperatures with the applied magnetic field parallel to the $a$ and $c$ axes. As shown in Fig. \ref{fig:3}(a) and (b), the ac susceptibility curves at T=0.33 K reveal two critical magnetic fields of 0.71~T and 1.2~T along the $a$- and $c$-axes, respectively. Such anomalies were tentatively explained as ``spin flop" behavior in the earlier reports \citep{kolo2011, fang2014,solovyev2016}. However, as we concluded by our field-dependent neutron diffraction data later, these features are attributed to a combined consequence of alignment of magnetic domains and spin rotations.
To detect the magnetoelectric effect, the electric polarization of Co$_4$Nb$_2$O$_9$ single crystal was measured under various magnetic and electric fields along the $a$- or $c$- axes after completing the magnetoelectric annealing procedure \citep{ding2016a}. The detailed field-induced pyroelectric current and electric polarization curves as a function of temperature at various magnetic fields are shown in Fig. S1 and S2. Figure \ref{fig:3}(c) and (d) show the magnetic-field-dependent electric polarization measured at 21.5 K using electric fields E=320 kV/m and 260 kV/m, respectively. When the applied magnetic field is perpendicular to the electric field, the value of polarization increases linearly below 4 T with the increase of magnetic field. However, a steep drop sets in around the applied magnetic field 4 T. This is a phenomenon that has not been found in the previous works, likely associated with the procedure of the alignment of magnetic domains as explained in the following. Above this magnetic field, the electric polarization restores the linear behavior. It is clear that when both the magnetic and electric field are applied in the same direction, either the $a$ or $c$ axis, relatively smaller polarization is observed. This indicates that the off-diagonal ME tensor components have larger values than the diagonal ones, allowed by the magnetic symmetry $C2/c'$ (see neutron diffraction section). Evidently, the magnetoelectric properties in Co$_4$Nb$_2$O$_9$ are distinct from that in Cr$_2$O$_3$ where only the diagonal ME tensor terms were observed below the spin-flop field imposed by its magnetic symmetry $P\bar{3}'m'$\cite{astrov1960,shirane1965}. Therefore, such an observation reflects that the previously documented magnetic structure with the magnetic point group $-3'm'$ which allows only diagonal terms is inappropriate \cite{bertaut1961, schwarz2010}.
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{4.PNG}
\caption{(Color Online) (a-b) The representative magnetic reflections as a function of temperature reflecting the magnetic phase transition at 27 K. The data were measured at HB-3A with the neutron wavelength of 1.003 \AA. (c) The magnetic structure of Co$_4$Nb$_2$O$_9$ showing the ferromagnetically coupled Co1-Co2 pairs (the nearest neighbor Co1 and Co2 atoms). (d) Results of the magnetic structure refinement for the neutron data collected at 5 K. (e) All spins are confined in the $ab$ plane with the canting angle between Co1 and Co2 spins (the next-nearest-neighbor along the $c$ axis).}\label{fig:4}
\end{figure*}
\subsection{ZERO-MAGNETIC-FIELD SINGLE CRYSTAL NEUTRON DIFFRACTION}
We performed single crystal neutron diffraction to revisit the magnetic structure of Co$_4$Nb$_2$O$_9$ down to 5 K. As shown in Fig. \ref{fig:4}, several selected Bragg reflections were measured to track the magnetic phase transition upon warming. One can clearly see the increased intensity for the representative reflections (0 0 4), (2 0 0), (1 1 3), (1 0 4) and (0 0 1) below $T_N$. This indicates that all the magnetic reflections can be well indexed by a propagation vector \textbf{k}=\textbf{0}, as reported in Ref. \citep{khanh2016, deng2018}. No further crystal structure transition was detected down to 5 K.
Symmetry analysis was performed to determine the magnetic structures of Co$_4$Nb$_2$O$_9$. A number of magnetic subgroups that are compatible with the given space group and propagation vector can be calculated by Bilbao Crystallographic Server (Magnetic Symmetry and Applications \citep{bilbao}) software. The magnetic structures bearing trigonal symmetry imply the magnetic moments along the $c$-axis, inconsistent with our experimentally measured magnetic reflections such as (0 0 4) and (0 0 1). Therefore, we have to lower the symmetry from the k-maximal magnetic symmetry in the subgroup hierarchy. We then found the monoclinic magnetic subgroups $C2'/c'$, $C2/c'$, $C2'/c$ and $C2/c$. The observed magnetic reflections also imply that spins should be confined into the $ab$ plane, distinct from the previous report where a magnetic moment component out of the $ab$ plane was found \citep{khanh2016}.
Since the magnetic peaks superpose on the nuclear reflections below $T_N$, combined magnetic and nuclear structure refinement was performed. The refinement using $C2/c'$ magnetic space group yields the best fitting with $R_f$=6.4\% and $\chi^2$=1.13 for the combined phase for 745 reflections. The results of the refinement and the corresponding illustrations of magnetic structure are shown in Fig.\ref{fig:4}. The magnetic structure solved can be simply viewed as antiferromagnetically coupled ferromagnetic chains along the $c$-axis. Magnetic moments in each chain are confined into the $ab$ plane with ferromagnetic pairs for the nearest neighbor Co1 and Co2 atoms and a canting angle 10.5$^{\circ}$ between the next-nearest-neighboring Co1-Co2 spins in the chain. The ordered magnetic moment is 2.820(8) $\mu_B$ (m$_x$=3.201(8)$\mu_B$, m$_y$=2.12(1)$\mu_B$, m$_z$=0) for both the Co1 and Co2 sites. The magnitude of the ordered magnetic moment for Co$^{2+}$ is close to the theoretical spin-only ordered value 3 $\mu_B$ for Co$^{2+}$ with a high spin state, indicating that the orbital moment is nearly quenched as shown in \citep{ding2016b, hutanu2012}. Note that due to the existence of the trigonal lattice symmetry, three magnetic domains were considered and their populations were set to be equal during the refinement. In the following, we show that the population of magnetic domains in Co$_4$Nb$_2$O$_9$ can be tuned by applying an external magnetic field.
The magnetic symmetry can also be readily derived by the observed magnetoelectric coupling. By applying the Neumann's principle to the magnetoelectric effect with these possible magnetic symmetries, we could exclude the subgroups $C2'/c'$ and $C2/c$. This is because, in principle, neither of them allows the presence of the linear ME effect. The $C2'/c'$ and $C2/c$ subgroups do not necessarily mean a ferromagnetic structure (in other words, there is no symmetry constraint for a ferromagnetic arrangement); hence this is not the reasoning that we ruled them out \citep{khanh2016}. The observed ME properties can help us preclude the $C2'/c$ subgroup since this symmetry allows only the off-diagonal terms. Thereby, the only magnetic subgroup compatible with Co$_4$Nb$_2$O$_9$ is $C2/c'$ based on our symmetry analysis and experimental observations, in good agreement with the neutron diffraction results.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{5.PNG}
\caption{(Color Online) (a)-(e) The selected magnetic reflections as a function of magnetic field along the $a$-axis measured at CTAX. The solid line marks the alignment of magnetic domains. (f-g) The magnetic structure of Co$_4$Nb$_2$O$_9$ after the three magnetic domains are aligned. All the Co spins are nearly parallel to the $b^*$ direction.}\label{fig:5}
\end{figure}
\subsection{SINGLE CRYSTAL NEUTRON DIFFRACTION IN MAGNETIC FIELDS}
To clarify the hidden mechanism of the complex ME effect in Co$_4$Nb$_2$O$_9$, we carried out neutron diffraction experiments under high magnetic fields at CTAX. The same crystal used at HB-3A was measured in the (0 K L) scattering plane. The crystal misalignment was less than 1$^{\circ}$ for this experiment. The magnetic field was applied vertically, i.e., along the crystal's $a$-axis. The representative reflections were measured with rocking curve scans at selected temperatures under magnetic field up to 10~T. The integrated magnetic intensities of these reflections are plotted versus the magnetic field in Fig. \ref{fig:5} (a-e). They were obtained by subtracting the integrated intensities at 50 K from that at 1.5~K. We found a considerable decrease in the intensities of magnetic reflections (0 1 0) and (0 1 4) with the increasing magnetic field up to 1 T and a significant increase in the intensity of magnetic reflection (0 1 1). There is only a slight decrease in intensity of the magnetic peak (0 0 2). These changes indicate a spin reorientation from the trigonal lattice direction [1 0 0], [0 1 0] or [1 1 0] to the direction parallel the reciprocal [0 1 0] direction (the $b^*$-axis), which causes a large decrease of the magnetic scattering at (0 1 0), (0 1 4) reflections but an evident increment of the (0 1 1) magnetic reflection. The observed weak intensity at 1~T and above can be related to the induced moments along the field direction and also the slight misalignment of the crystal. The magnetic intensities at $H$=1 T were obtained by scaling them to those measured at zero magnetic field that was well solved with the complete data measured at HB-3A for catching all the necessary corrections. With these magnetic reflections at 1~T, the magnetic structure with a single magnetic domain can fit the data in a satisfactory quality ($\chi^2$=6.09) and yields the moments along [1 2 0] direction in real space and [0 1 0] direction in reciprocal space (Fig.\ref{fig:5}), in the same magnetic symmetry $C2/c'$ as that at zero magnetic field. The ordered magnetic moments are 2.6(1) $\mu_B$ for both Co sites with a slightly increased planar canting angle of 12.2$^{\circ}$. The magnetic field of 1 T along the $a$-axis does not significantly change the moment size and the magnetic symmetry but switches three magnetic domains with spins along the three crystal axes to the single one with the spins along the reciprocal $b^*$-axis and increases the canting angle slightly.
The above refinement did not consider the induced ferromagnetic moments along the $a$-axis (the induced moments are small as expected from the magnetization measurements in Fig. \ref{fig:2}).
Magnetic field 1 T$<$H$<$4 T does not greatly change the intensities of magnetic reflections (0 1 0), (0 1 4), (0 0 2), (0 1 1) and (0 0 1), suggesting a rather robust magnetic ground state. Above 4 T, only the magnetic reflection (0 1 1) was appropriately suppressed, reflecting no further magnetic phase transition but spin rotation in the $ab$ plane. Indeed, by refining the observed magnetic reflections at 10 T using the magnetic symmetry obtained at 1 T, we arrived at a smaller planar canting angle 8.5$^{\circ}$ and magnetic moment 2.9(3)$\mu_B$ for both sites. In the magnetic-field-dependent polarization curves, it is clear that a steep drop occurs around 4 T, coincident with the abrupt decrease of the observed magnetic reflection (0 1 1). Above 4 T, the restored linear magnetoelectric effect supports our conclusion that a magnetic field up to 10 T does not break the magnetic symmetry but rotates the magnetic moments in the $ab$ plane.
\section{DISCUSSION}
Our work reveals that Co$_4$Nb$_2$O$_9$ antiferromagnetically orders below 27 K with the magnetic space group $C2/c'$ which allows a linear ME effect. This point is in good agreement with the previous reports by Khanh et al. \citep{khanh2016} and Deng et al.\citep{deng2018}. However, in the former case, they found that all spins are exactly ferromagnetically aligned with a considerable canting angle 22$^{\circ}$ toward the $c$-axis(mainly in the $ab$ plane). This in fact conflicts with our results in which spin out of the $ab$ plane is minimal based on our neutron diffraction data even though m$_z$ component is symmetry allowed. In the latter case, a distinct magnetic structure was reported where all moments are purely in the $ab$ plane with the canting angle of 1.3$^{\circ}$ and 25.2$^{\circ}$ for the neighbor Co1 and Co2 moments, respectively.
By contrast, we found a magnetic structure akin to the latter case but with a different configuration by a careful neutron diffraction measurement on a high quality single crystal. As illustrated in Fig. \ref{fig:4}, the nearest-neighbor Co1 and Co2 atoms form a ferromagnetic pair in the $ab$ plane and the essential (0 0 1) magnetic reflection ensues a planar canting angle between each adjacent Co1-Co2 pair. This makes the canting angle for the neighbor Co1 and Co2 moments equivalent.
Our magnetic arrangement seems to be more favored by the special crystal structural arrangement of Co atoms since the nearest-neighbor bonding between Co atoms is the edge-sharing Co1O$_6$ and Co2O$_6$ octahedra with a bond distance of d1=3.020(1)\AA ~along the $c$ axis as marked in Fig. \ref{fig:1}(b). The dominant ferromagnetic exchange interactions of Co1 and Co2 (d1) comparing to other exchange strength promotes a canting angle between the magnetic moments of Co1 or Co2 neighbors \citep{deng2018}. Moreover, the strong single-ion anisotropy of Co in this system should also play a vital role in forming such a magnetic configuration.
When applying a magnetic field H$<$1 T, the magnetic symmetry is practically invariant but with an alignment of the three magnetic domains. The magnetic moments are nearly parallel to the $b^*$ direction, elucidating the conjectured ''spin flop" suggestions in the previous works\citep{khanh2016, fang2014, kolo2011}. In the magnetic field dependence of electric polarization curves, a sudden drop around 4 T is likely a consequence of the completion of the alignment of the magnetic domains since thermally stimulated current sources in the course of magnetic domain alignment may contribute to the pyroelectric current so as to the electric polarization\citep{zou2014, ngo2015}. Above 4 T, i.e., in a single magnetic domain, the observed magnetoelectric effect seems to be more intrinsic because we have confirmed that the magnetic structure up to 10 T remains the same symmetry with a spin rotation within the $ab$ plane. Such a robust magnetic symmetry in a magnetic field, that probably emanates from the strong easy-plane single-ion anisotropy, provides a natural explanation of the manipulation of electric polarization \citep{khanh2017, solovyev2016}.
The solved magnetic symmetry with its off-diagonal components non-null in Co$_4$Nb$_2$O$_9$ in principle allows the occurrence of the ferrotoroidal order. The experimental observation of the relevant ferrotoroidal domains through a magnetoelectric annealing process is greatly desired. This can be spatially resolved by optical second harmonic generation and spherical neutron polarimetry determination of the relative domains populations \citep{van2007, ressouche2010}.
\section{CONCLUSION}
We have investigated the magnetoelectric properties and magnetic structure evolution with temperature and magnetic field of a high quality single crystal Co$_4$Nb$_2$O$_9$. Single crystal neutron diffraction without magnetic field revealed a magnetic structure below 27 K characterized by the $C2/c'$ symmetry with magnetic moments totally confined in the $ab$ plane, which allows the linear ME effect as observed experimentally by the pyroelectric current measurements. Single crystal neutron diffraction in magnetic field with H $\mathbin{\!/\mkern-5mu/\!}$ a showed that the magnetization anomaly around 1 T is in fact a magnetic domain alignment that drives the magnetic moments parallel to the $b^*$ direction without breaking the magnetic symmetry. A higher magnetic field up to 10 T did not change the magnetic symmetry but rotates the magnetic moments in the $ab$ plane. The robust magnetic symmetry to the external magnetic field offers a natural way to manipulate and control the electric polarization in this system.
\begin{acknowledgments}
LD thanks J. Rodriguez-Carvajal for helpful discussions. The research at Oak Ridge National Laboratory (ORNL) was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Early Career Research Program Award KC0402010, under Contract DE-AC05-00OR22725 and the U.S. DOE, Office of Science User Facility operated by the ORNL.
The work at University of Tennessee was supported by DOE under award DE-SC-0020254. A portion of this work was performed at the National High Magnetic Field Laboratory, supported by the National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida.
The US Government retains, and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive,
paid-up, irrevocable, worldwide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for US Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access
Plan.\citep{DOE}.
\end{acknowledgments}
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\section{Introduction} \label{sec-Intro}
In \cite{Gasch}, Gasch\"utz introduced for finite soluble groups, the concepts of saturated formations and projectors. He showed that if $\fF$ is a saturated formation, then $\fF$-projectors exist in every finite soluble group. Saturated formations can be constructed via local definition, local in that for each prime $p$, there is a condition on the chief factors of $p$-power order for the group to be in the formation.
A class $\fH$ with the property that an $\fH$-projector exists in every finite soluble group is called a Schunck class. Schunck showed that such a class $\fH$ is determined by the class $\fX$ of all primitive groups in $\fH$. The class $\fX$ necessarily has the property that every primitive quotient of a group in $\fX$ is also in $\fX$. Conversely, given such a class $\fX$, the class
$$\fH = \pdef(\fX) = \{G \mid \text{every primitive quotient of }G\text{ is in }\fX\}$$
is a Schunck class, said to be primitively defined by $\fX$. This theory is set out in detail in Doerk and Hawkes \cite{DH}.
Analogous theories have been developed for Lie algebras \cite{BGH}, restricted Lie algebras \cite{Restricted}, and for Leibniz algebras \cite{SchunckLeib}. In general outline, the theories are very similar, but while every saturated formation of soluble groups has many local definitions, a saturated formation of Lie algebras has at most one local definition. (See \cite{local}.) That not every saturated formation has a local definition means that we cannot use local definition to investigate all saturated formations of Lie algebras. Every saturated formation, being a Schunck class, has a primitive definition. To use this, we need to know when the Schunck class defined by the class $\fX$ of primitives is a formation. The conditions on $\fX$ are essentially the same for groups, Lie algebras, restricted Lie algebras and Leibniz algebras. So in the following, we treat all these together. We use ``algebra'' to mean any of group, Lie algebra, restricted Lie algebra or Leibniz algebra, all assumed to be finite or finite-dimensional and soluble. Except for matters specific to groups, we write in the language of algebras. A primitive is an algebra $P$ with a minimal ideal $K$ such that its centraliser $\cser_P(K)=K$. In all cases, $P$ splits over $K$ and all the complements are conjugate.
In Section \ref{gps}, we establish for groups analogues of \cite[Lemmas 1.2, 1.7, 1.9, Theorems 2.1, 2.6]{Blocks}. There is no need to do this for Leibniz algebras or for restricted Lie algebras. The blocks for a restricted Lie algebra $(L,[p])$ are the same as for its underlying Lie algebra $L$.
If $P$ is a primitive Leibniz algebra with $\Soc(P) = A$, then $P/A$ is a Lie algebra and, as $L/A$-module, $A$ is either symmetric, that is, $ax = -xa$ for $x \in P/A$ and $a \in A$, or asymmetric, that is, $ax=0$. From the given left action on $A$ we can form the symmetric and asymmetric modules $\sym A$ and $\asym A$ and their split extensions by $P/A$, the primitive algebras $\sym P$ and $\asym P$, the given primitive algebra $P$ being one of these. Thus primitive Leibniz algebras come in pairs $\{\sym P, \asym P \}$ with $\sym P$ a Lie algebra while $\asym P$ has Leibniz kernel $\Leib (\asym P) = \Soc(\asym P)$. A saturated formation $\fF$ of Leibniz algebras containing one member of a pair also contains the other and is determined by the Lie algebras in $\fF$. See \cite[Theorem 3.16, Corollary 3.17]{SchunckLeib}. Consequently, our main result for Leibniz algebras follows immediately from the result for Lie algebras.
In Section \ref{cdits}, we establish for all cases, the conditions on the class $\fX$ of primitives for $\pdef(\fX)$ to be a saturated formation.
\section{Finite groups}\label{gps}
If $A/B$ is a $p$-chief factor of the group $G$, it can be regarded as an $\Fp G$-module. To avoid confusion between the multiplicative notation used for the group and additive notation for the module, we denote the module by $[A/B]$ and the module element corresponding to the element $a \in A/B$ by $[a]$. Thus $[a_1a_2]= [a_1] + [a_2]$. The action of $g \in G$ on $[a]$ is given by $g[a] = [gag^{-1}]$.
The results of this section do not need the full power of the assumption, required for their applications in the next section, that the group be soluble, so solubility is not assumed here.
\begin{lemma} \label{gp-sole} Suppose that $A$ of $p$-power order is the only minimal normal subgroup of the $p$-soluble group $G$. Suppose that $G$ does not split over $A$. Let $B/A$ be a minimal normal subgroup of $G/A$. Then $B/A$ is a $p$-group and
\begin{enumerate}
\item If $B$ is not abelian, then $[A]$ is a quotient of $[B/A] \otimes [B/A]$,
\item If $B$ is abelian but not of exponent $p$, then $[A] \simeq [B/A]$.
\end{enumerate}
\end{lemma}
\begin{proof} If $B/A$ is not a $p$-group, then $|B/A|$ is prime to $p$. By the Schur-Zassenhaus Theorem, there exists a complement $U$ to $A$ in $B$, and, if $V$ is another complement, then $V = aUa^{-1}$ for some $a \in A$. By the Frattini argument, $G$ is generated by $A$ and the normaliser $\nser_G(U)$ of $U$. But $A \subseteq \Phi(G)$, so $\nser_G(U) = G$, contrary to $A$ being the only minimal normal subgroup of $G$. Hence $B$ is a $p$-group.
Suppose that $B$ is not abelian. Let $\bar{b}_i \in B/A$. The map $[\bar{b}_1] \otimes [\bar{b}_2] \mapsto [b_1b_2b_1^{-1}b_2^{-1}]$ is a module homomorphism. As $[A]$ is irreducible, $[A]$ is a quotient of $[B/A] \otimes [B/A]$.
Suppose that $B$ is abelian of exponent $p^2$. Then the map $[\bar{b}] \mapsto [b^p]$ is an isomorphism.
\end{proof}
\begin{lemma} \label{qgpblock} Suppose that $N \id G$ and $V,W$ are irreducible $\Fp(G/N)$-modules in the same block. Then $V, W$ are in the same $\Fp G$-block.
\end{lemma}
\begin{proof}
There is a chain $V = V_0, V_1, \dots, V_n = W$ of irreducible $\Fp (G/N)$-modules and non-split extensions $X_i$ of either $V_{i-1}$ by $V_i$ or of $V_i$ by $V_{i-1}$ linking $V$ and $W$. But the $V_i$ are irreducible $\Fp G$-modules and the $X_i$ are non-split $\Fp G$-modules linking $V$ to $W$ as $\Fp G$-modules.
\end{proof}
Denote the dual $\Hom(V, \Fp)$ of the module $V$ by $V^*$. Denote the principal $\Fp G$-block by $\B_0(\Fp G)$.
\begin{lemma}\label{dual} Suppose $V \in \B_0(\Fp G)$. Then
$V^* \in \B_0(\Fp G)$.
\end{lemma}
\begin{proof} There exists a sequence $\Fp = A_0, A_1, \dots, A_n = V$
of irreducible modules and a sequence $X_1, \dots, X_n$ of non-split extensions $X_i$
either of $A_{i-1}$ by $A_i$ or of $A_i$ by $A_{i-1}$. Dualising this gives a sequence
$F = A^*_0, A^*_1, \dots, A^*_n = V^*$ of irreducible modules and a sequence $X^*_1, \dots, X^*_n$ of non-split extensions $X^*_i$ either of
$A^*_i$ by $A^*_{i-1}$ or of $A^*_{i-1}$ by $A^*_i$.
\end{proof}
\begin{lemma} \label{diffK} Suppose that $A$ is a minimal normal subgroup of the $p$-soluble group $G$. Let $V,W$ be irreducible $\Fp G$-modules. Suppose that $A$ acts trivially on $V$ and non-trivially on $W$. Then every extension of $V$ by $W$ or of $W$ by $V$ splits.
\end{lemma}
\begin{proof}
As $A$-module, $V$ is the direct sum of $\dim(V)$ copies of $\Fp$, while $W$ is the direct sum of conjugate non-trivial irreducible $A$-modules $W_i$. Thus $\Hom(V,W)$ is a direct sum of non-trivial irreducible $A$-modules, so $H^0(A,\Hom(V,W)) = 0$. If $A$ is not a $p$-group, then $|A|$ is prime to $p$ and $H^1(A, \Hom(V,W)) = 0$, so we may suppose that $A$ is a $p$-group. Since $A$ is an abelian $p$-group acting non-trivially on the irreducible module $W_i$, we have $H^1(A, W_i) = 0$. So again we have $H^n(A, \Hom(V,W)) = 0$ for $n=0,1$. By the Hochschild-Serre spectral sequence, it follows that $H^1(G, \Hom(V,W)) = 0$. So every module extension of $W$ by $V$ splits. As $\Hom(W,V)$ is a direct sum of copies of the duals of the $W_i$, similarly we have that every extension of $V$ by $W$ splits.
\end{proof}
\begin{cor} \label{B0ker} Suppose that $A$ is a minimal normal subgroup of the $p$-soluble group $G$ and that $V$ is an irreducible $\Fp G$-module in the principal block. Then $A$ acts trivially on $V$.
\end{cor}
\begin{proof} As $A$ acts trivially on $\Fp$, it follows by Lemma \ref{diffK}, that $A$ acts trivially on every irreducible in a chain linking $\Fp$ to $V$.
\end{proof}
\begin{lemma} \label{compl} Let $A/B$ be a complemented $p$-chief factor of the $p$-soluble group $G$. Then $[A/B] \in \B_0(\Fp G)$.
\end{lemma}
\begin{proof} By \cite[Theorem 1]{H1G}, $\Ext^1_{\Fp G}(\Fp,V) = H^1(G,V) \ne 0$.
\end{proof}
\begin{theorem} \label{chiefsB0} Let $A/B$ be a $p$-chief factor of the $p$-soluble group $G$. Then $[A/B] \in \B_0(\Fp G)$.
\end{theorem}
\begin{proof} The result holds trivially if $|G|=p$. We use induction over $|G|$. By Lemma \ref{qgpblock}, we may suppose that $B = 1$ and that $[A] \notin \B_0(\Fp(G/A)$. But then $H^n(G/A,A)=0$ for all $n$ and $A$ is a complemented $p$-chief factor. The result follows by Lemma \ref{compl}.
\end{proof}
\begin{lemma} \label{eval} Let $V,W$ be $\Fp G$-modules. Then the evaluation map $$\epsilon: V \otimes \Hom(V,W) \to W$$
given by $\epsilon(v\otimes f) = f(v)$ is a module homomorphism.
\end{lemma}
\begin{proof} For $x \in G$, $\epsilon x(v \otimes f) = \epsilon(xv \otimes xf) = (xf)(xv) = xf(x^{-1}xv) = x \epsilon(v \otimes f)$.
\end{proof}
\begin{theorem} \label{tens} Suppose $V,W$ are irreducible $\Fp G$-modules and that there exists a non-split extension of $W$ by $V$. Then $W$ is a quotient of $V \otimes A$ for some $A \in \B_0(\Fp G)$.
\end{theorem}
\begin{proof} Since $H^1(G, \Hom(V,W)) \ne 0$, $\Hom(V,W)$ must have some composition factor in $\B_0(\Fp G)$. Let $B$ be the $\B_0$-component of $\Hom(V,W)$ in its block decomposition. Then $B \ne 0$. Take a minimal submodule $A \subseteq B$. Then $\epsilon(V\otimes A) \ne 0$, so $\epsilon(V \otimes A) = W$.
\end{proof}
\begin{theorem} \label{chiefs} Let $C$ be the set of the $p$-chief factor modules of the $p$-soluble group $G$ and their duals. Let $V$ be an irreducible $\Fp G$-module in $\B_0(\Fp G)$. Then $V$ is a composition factor of some tensor product $C_1 \otimes \dots \otimes C_k$ of modules $C_i \in C$.
\end{theorem}
\begin{proof} By induction over the length of the sequence linking $V$ to $F$, we may suppose that we have a non-split extension $X$ of $V$ by $W$ or of $W$ by $V$ with $W$ a composition factor of some tensor product of modules in $C$. Since for modules $M,N$, $(M \otimes N)^* \simeq M^* \otimes N^*$, by Lemma \ref{dual}, we need only consider the case where $X$ is a non-split extension of $V$ by $W$.
Let $A$ be a minimal normal subgroup of $G$. By Corollary \ref{B0ker}, $V$ and $W$ are $\Fp( G/A)$-modules. If $X$ also is an $\Fp(G/A)$-module, then $V,W$ are in the same $\Fp(G/A)$-block and by Theorem \ref{tens}, $V$ is a quotient of $B \otimes W$ for some $B \in \B_0(L/A)$. But by induction over $\dim(L)$, $B$ is a composition factor of some tensor product of $p$-chief factor modules of $G/A$ and their duals. Thus the result holds in this case.
Now suppose that no non-split $\Fp(G/A)$-module extension of $V$ by $W$ exists. Then $H^1(G/A, \Hom(W,V))=0$. That is, $H^1(G/A, \Hom(W,V)^A) = 0$ as $A$ acts trivially on $\Hom(W,V)$. But $X$ is a non-split $\Fp G$-module extension of $V$ by $W$, so $H^1(G,\Hom(W,V)) \ne 0$. By the Hochschild-Serre spectral sequence, and we must have $H^1(A,\Hom(W,V))^G \ne 0$, so $A$ cannot have order prime to $p$. So $A$ is an abelian $p$-group which acts trivially on $\Hom(W,V)$. Therefore $H^1(A, \Hom(W,V)) = \Hom([A], \Hom(W,V))$ and it follows that we have a non-zero $\Fp G$-module homomorphism $f: A \to \Hom(W,V)$. Then $f(A)$ is a nonzero submodule of $\Hom(W,V)$ and the evaluation map $\epsilon$ maps $f(A) \otimes W$ onto $V$. The result follows.
\end{proof}
\section{The conditions} \label{cdits}
Let $\fX$ be a class of primitive algebras.
\begin{definition} We say that $\fX$ is \textit{primitive quotient closed} if, for every $P \in \fX$, also every primitive quotient of $P$ is in $\fX$.
\end{definition}
That $\fX$ is primitive quotient closed is necessary and sufficient for $\pdef(\fX)$ to be a Schunck class. If $\fF = \pdef(\fX)$ is a formation, then for every chief factor $A/B$ of $P$, we must have also the split extension $Q$ of $A/B$ by $P/\cser_P(A/B) \in \fX$.
\begin{definition}
If, for every $P \in \fX$ and every chief factor $A/B$ of $P$, the split extension of $A/B$ by $P/\cser_P(A/B)$ is in $\fX$, we say that $\fX$ is \textit{chief factor closed}.
\end{definition}
If $\fX$ is chief factor closed, then clearly, it is primitive quotient closed.
If $\fF = \pdef(\fX)$ is a saturated formation, then the dual of an $\fF$-central module is $\fF$-central. Thus for $P \in \fX$ with $A = \Soc(P)$, we must have that the split extension of the dual $\Hom(A,F)$ of $A$ by $P/A$ is in $\fX$.
\begin{definition} We say that $\fX$ is \textit{dual closed} if, for every $P \in \fX$, the split extension of the dual $\Hom(A,F)$ of $A = \Soc(P)$ by $P/A$ is in $\fX$.
\end{definition}
\begin{definition} (For Leibniz algebras. The condition is meaningless and to be regarded as always satisfied in the other cases.) We say that $\fX$ is \textit{paired} if, for every $P \in \fX$, both members of the pair $(\sym P, \asym P)$ are in $\fX$.
\end{definition}
Let $P,Q \in \fX$ with $A = \Soc(P)$ and $B=\Soc(Q)$. Let $L$ be a subdirect sum of $P/A$ and $Q/B$. Then $A,B$ are $L$-modules. Let $C$ be a composition factor of $A\otimes B$ and let $R$ be the split extension of $C$ by $L/\cser_L(C)$. We call $R$ a \textit{subtensor product} of $P$ and $Q$. If $\fF$ is a saturated formation, then $A \otimes B$ is an $\fF$-hypercentral $L$-module and so we must have $R \in \fX$.
\begin{definition}If, for all $P,Q \in \fX$, every primitive subtensor product $R$ of $P$ and $Q$ is in $\fX$, we say that $\fX$ is \textit{subtensor closed}.
\end{definition}
\begin{definition} Let $\fX$ be a class of primitive algebras. We say that the chief factor $A/B$ of $L$ is $\fX$-central if the split extension of $A/B$ by $L/\cser_L(A/B)$ is in $\fX$.
\end{definition}
\begin{lemma} \label{subtens} Suppose that $\fX$ is a chief factor, dual and subtensor closed and paired class of primitive algebras. Let $\fF$ be the class of algebras all of whose chief factors are $\fX$-central. Then $\fF$ is the saturated formation $\pdef(\fX)$.
\end{lemma}
\begin{proof} If the result holds for Lie algebras, then by \cite[Theorem 3.16, Corollary 3.17]{SchunckLeib}, the result holds for Leibniz algebras. So we need only prove the result for the other three categories. Since $\fF$ is defined in terms of chief factors, $\fF$ is a formation. We have to prove that it is saturated. Suppose that $A$ is a minimal ideal of the algebra $L$, that $L/A \in \fF$ and that $L$ does not split over $A$. Then $A \subseteq \Phi(L)$. We have to prove that $A$ is $\fX$-central. We use induction over $\dim(L)$. The result holds trivially if $A = L$. Suppose that $B$ is another minimal ideal of $L$. Then $A+B/B$ is a minimal ideal of $L/B$ and $A/B \subseteq \Phi(L/B)$. By induction, $A+B/B$ is $\fX$-central. Thus we may suppose that $A$ is the only minimal ideal of $L$.
Let $B/A$ be a minimal ideal of $L/A$. Since $A \subseteq \Phi(L)$, it follows that $B$ is nilpotent. Suppose that $B$ is not abelian. Then $B' = A$ and we have an epimorphism $\epsilon: B/A \otimes B/A \to A$ defined by $\bar{b}_1 \otimes \bar{b}_2 \mapsto b_1b_2$ (for groups, $\bar{b}_1 \otimes \bar{b}_2 \mapsto b_1b_2b_1^{-1}b_2^{-1}$ by Lemma \ref{gp-sole}). Since $\fX$ is subtensor closed, the split extension of $A$ by $L/\cser_L(A)$ is in $\fX$, that is, $A$ is $\fX$-central.
Now suppose that $B$ is abelian. Then $B$ is an $L/B$-module which does not split over the submodule $A$. By \cite[Theorem 1.5]{Blocks} (Lemma \ref{gp-sole} and Theorem \ref{tens} for groups), $A$ is a quotient of $V \otimes (B/A)$ for some $V$ in the principal block of $L/B$. But by \cite[Theorem 2.6]{Blocks} (Theorem \ref{chiefs} for groups), $V$ is a composition factor of a tensor product of chief factors of $L/B$ and their duals. From the closure properties of $\fX$, it follows that $A$ is $\fX$-central.
Now let $\fH = \pdef(\fX)$. Then $\fH$ is the class of algebras all of whose {\em complemented} chief factors are $\fX$-central. Therefore $\fF \subseteq \fH$. Suppose that $\fF \ne \fH$. Then we can take $L \in \fH$, $L \notin \fF$ of least possible dimension. Let $A$ be a minimal ideal of $L$. Then $L/A \in \fF$ and every chief factor of $L/A$ is $\fX$-central. But $\fF$ is saturated, so $L$ splits over $A$, $A$ is a complemented chief factor of $L \in \fH$, so $A$ is $\fX$-central. Therefore $L \in \fF$ contrary to the choice of $L$.
\end{proof}
Denote the nilpotent length of $L$ by $\nlen(L)$. For groups, we use the $p$-nilpotent length denoted by $p$-$\nlen(L)$.
\begin{theorem} Suppose that $\fX$ is a primitive quotient, dual and subtensor closed and paired class of primitive algebras. Let $\fF$ be the class of algebras all of whose chief factors are $\fX$-central. Then $\fF$ is the saturated formation $\pdef(\fX)$.
\end{theorem}
\begin{proof} If $\fX$ is chief factor closed, then the result holds by Lemma \ref{subtens}, so let $P \in \fX$ be a primitive algebra with a chief factor $A/B$ which is not $\fX$-central. We take $P$ with $n = \nlen(P)$ least possible. (For groups, we choose $P$ and $p$ to make $n=p$-$\nlen(P)$ as small as possible.) Let $\fX_0$ be the class of algebras $L \in \fX$ with $\nlen(L) < n$. (For groups, with $p$-$\nlen(L) < n$ for all $p$.) Then $\fX_0$ is chief factor, dual and subtensor closed and paired. By Lemma \ref{subtens}, $\pdef(\fX_0)$ is a saturated formation. But $P/\Soc(P) \in \pdef(\fX_0)$, so every chief factor of $P/\Soc(P)$ is $\fX_0$-central, contrary to hypothesis.
\end{proof}
\begin{cor} Let $\fH$ be a Schunck class and let $\fX$ be the class of primitive algebras in $\fH$. Suppose that $\fX$ is dual and subtensor closed and paired. Then $\fH $ is a saturated formation.
\end{cor}
\begin{proof} Let $\fF$ be the class of algebras whose chief factors are all $\fX$-central. As $\fH$ is the class of algebras whose complemented chief factors are $\fX$-central, $\fF \subseteq \fH$. We prove that if $L \in \fH$, then $L \in \fF$. Let $L$ be a minimal counterexample. and let $A$ be a minimal ideal of $L$. Then $L/A \in \fF$. But $\fF$ is saturated. Since $L \notin \fF$, $L$ splits over $A$. But every complemented chief factor of $L$ is $\fX$-central, so $A$ is $\fX$-central and $L \in \fF$ contrary to assumption.
\end{proof}
\begin{example} If $\fX$ is the class of primitive algebras $P$ with $\dim(\Soc(P)) = 1$, then $\fX$ is easily seen to satisfy the conditions. The class $\pdef(\fX)$ is the class of supersoluble algebras. (For restricted Lie algebras, the chief factors are either $1$-dimensional or central with the $p$-operation acting invertibly.)
\end{example}
\begin{example} Let $\Lambda$ be a normal $F$-subspace of the algebraic closure $\bar{F}$ of $F$. For groups, $\Lambda_p$ either $\emptyset$ or a subgroup of the multiplicative group of $\Fpbar$. Let $\fX$ be the class of primitives $P$ for which, for all $x \in P$ and all chief factors $A/B$ of $P$, all eigenvalues of the action of $x$ on $A/B$ are in $\Lambda$. Then $\fX$ satisfies the conditions and it follows that the class $\Edef(\Lambda)$ of all algebras $L$ such that, for all $x \in L$ and all chief factors $A/B$ of $L$, all eigenvalues of the action of $x$ on $A/B$ are in $\Lambda$ is a saturated formation.
\end{example}
\bibliographystyle{amsplain}
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\section{Introduction}
Topological insulators are bulk insulators which possess robust conducting surface states \cite{Hasan10,Qi11}. Paradigmatic two-dimensional examples of this class are the quantum Hall (QH) and the quantum spin Hall (QSH) phases, which are characterized by respectively chiral and helical one-dimensional edge states. While the former can be generated by simply applying a strong perpendicular magnetic field and has been rather extensively studied since the 1980s, the latter requires the presence of spin-orbit coupling and has received very little experimental evidence. Indeed, despite the wide interest shown in the literature for the QSH phase (and for topological phases in general) since the seminal works by Kane and Mele \cite{Kane05bis,Kane05}, experimental traces of this phase have remained scarce, with the exception of the remarkable works on HgTe quantum wells \cite{Bernevig06,Konig07,Roth09} (see also the experiment involving InAs \cite{Knez11}). Recent studies \cite{Weeks11,Shevtsov12,Jiang12} have revived the possibility of generating a QSH in graphene \cite{CastroNeto09,DasSarma11}, by showing that low concentrations of suitably chosen adatoms, randomly deposited on graphene, could open a large non-trivial gap in graphene's otherwise semimetallic band structure, and yield transport properties showing no trace of the spatially inhomogenous spin-orbit coupling. The perspective of successfully turning graphene into a QSH insulator is promising, as it would considerably enhance the experimental feasibility of engineering samples of the latter, which are so far limited to the previously mentioned and experimentally challenging HgTe heterostructures.
Although it does not enjoy the conceptual simplicity of the monolayer, bilayer graphene is an interesting system in its own right. It is a gapless semimetal, characterized by massive chiral excitations carrying a topological Berry phase 2$\pi$ \cite{Novoselov06}, with a very rich list of many-body instabilities predicted at low density (see \cite{McCann12} for corresponding references). One of its most remarkable properties, in contrast to monolayer graphene, is the possibility to open a gap in its band structure by simply applying a perpendicular electric field which breaks the layer inversion symmetry \cite{Castro07}. However, in the presence of a perpendicular magnetic field, the electronic properties of both systems become qualitatively very similar. In particular, their energy spectrum is characterized by a particle-hole symmetry and by the existence of levels sitting exactly at zero energy which are at the origin of the anomalous quantization of the Hall conductance in these systems as compared to other two-dimensional electron gases \cite{Novoselov05,Zhang05,Gorbig11}. This is the hallmark property of what we will refer to in this article as two-dimensional Dirac fermion gases (2DDFGs).
The purpose of this article is to discuss what happens to the band structure of a 2DDFG when the effects of both magnetic field and spin-orbit coupling are taken into account simultaneously. Taking graphene as our first example, we shall review in section \ref{sec:MG} the results already published elsewhere \cite{Shevtsov12X}, according to which a topological phase transition takes place at low energy and can be tuned by simply varying the chemical potential. Then, in the following section, we shall investigate the related situation in bilayer graphene and show that the results obtained for the monolayer do not extend to it. This can be traced back to the fact that the topological invariant characterizing the QSH phase is non-trivial only if there are an odd number of pairs of edge states, which translates in multi-layer graphene into the condition of having an odd number of layers. Nevertheless, we will show in section \ref{sec:BandAsym} that by adding additional ingredients to our model, namely by breaking the layer inversion symmetry, a non-trivial QSH phase can be generated in bilayer graphene along with a corresponding topological phase transition. We stress that (almost) all of our results can be understood by simply looking at the band structures of the systems we study. Finally, section \ref{sec:Disc} discusses the possible extension of our approach to other systems, and we conclude in section \ref{sec:Conc}.
\section{Graphene}
\label{sec:MG}
Recent investigations of the interplay between QH and QSH phases in some specific examples of 2DDFGs \cite{Tkachov10,DeMartino11,Goldman11,Shevtsov12X} have led to surprising results. In these works, it was shown that the QSH phase can survive the presence of a perpendicular magnetic field and that the $\mathbb{Z}_2$ topological invariant \cite{Kane05} remains non-trivial for energies below the spin-orbit induced gap, despite the breaking of time-reversal symmetry\,\footnote{The persistence of the QSH phase in a different time-reversal-symmetry-breaking context was also observed \cite{Yang11}.}. As we will exemplify in the case of graphene, the origin of this intriguing result actually stems from the existence of zero-energy Landau levels: as soon as the spin-degeneracy of these levels is lifted, spin-polarized edge states characteristic of a QSH phase emerge\,\footnote{This observation was first made by Abanin et al. \cite{Abanin06} who realized that the mechanism of Zeeman splitting could fulfill this condition; unfortunately, the weakness of Zeeman splitting in graphene rendered the effect too small to be observable at low-to-moderate magnetic fields, while it is washed out by many-body effects at large magnetic fields.}.
\subsection{Model}
\label{sec:Mod}
Let us start by introducing the model from which our results shall be derived. In the vicinity of the zero-energy points in the Brillouin zone, low-energy excitations can be described by a Dirac Hamiltonian:
\begin{equation}
\label{eq:MG}
H_{\text{G}} = v_F(\tau\hat{p}_x\sigma_x+\hat{p}_y\sigma_y) \; ,
\end{equation}
with $\sigma_x,\sigma_y$ the usual set of Pauli matrices acting in the two-dimensional space of the inequivalent sublattices A and B (see Fig.~\ref{FigLatGra}), and with $v_F=\sqrt{3}t_0\tilde{a}/(2\hbar)$ the Fermi velocity, expressed as a function of the microscopic lattice parameters $t_0$ (nearest-neighbor hopping amplitude) and $\tilde{a}$ (lattice constant) which we choose in the following as our working units of energy and length, respectively.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=0.75\linewidth]{Lattice_graphene.pdf}
\caption{Sketch of the graphene hexagonal lattice, characterized by nearest-neighbor hopping $t_0$ and next-nearest-neigbor Kane-Mele spin-orbit coupling $\lambda_{\text{so}}$. The unit cell of the lattice contains two inequivalent sites, A and B.}
\label{FigLatGra}
\end{center}
\end{figure}
$\tau=\pm 1$ accounts for the two possible valleys from which the low-energy excitations can arise. In the vicinity of these points, the energy-dispersion relation reads $\epsilon = \pm v_F|{\bf p}|$.
The presence of a perpendicular magnetic field can be straightforwardly included by making use of the Peierls substitution $\hat{{\bf p}} \rightarrow \hat{{\bf \Pi}}=\hat{{\bf p}}+e{\bf A}$, which accounts for the presence of the magnetic vector potential ${\bf A}$ such that $\nabla\times{\bf A}=B{\bf z}$. The components of the generalized momentum satisfy the Heisenberg algebra $[\hat{\Pi}_x, \hat{\Pi}_y] = -i(\hbar/l_B)^2$. By expressing these components in terms of the usual harmonic oscillator ladder operators \cite{Gorbig11},
\begin{equation}
\hat{\Pi}_x = \frac{\hbar}{\sqrt{2}l_B}(a+a^\dagger) \; , \;
\hat{\Pi}_y = \frac{i\hbar}{\sqrt{2}l_B}(a-a^\dagger) \; ,
\end{equation}
and using the standard raising and lowering properties of these operators on the eigenstates ($a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^\dagger|n-1\rangle=\sqrt{n}|n\rangle$), the energy spectrum can then straightforwardly be shown to turn into the well-known Landau levels,
\begin{equation}
\label{eq:LLMG}
\epsilon_n = \pm\Delta_B\sqrt{|n|}
\end{equation}
with $\Delta_B = \sqrt{2}\hbar v_F/l_B$, and $l_B=\sqrt{\hbar/(eB)}$ the magnetic length. As already mentioned before, the main distinctive feature of the Landau level spectrum of a 2DDFG as compared to that of a standard two-dimensional electron gas is the existence of a zero-energy level at $n=0$ originating from the pseudo-relativistic nature of the charge carriers. Also note that all levels enjoy a 4-fold degeneracy arising from spin and valley indices.
\subsection{Band structure / edge state correspondence}
The appearance of edge states in this context can be best understood by looking at the band structure of a graphene ribbon. The latter is a system which is translationally invariant in one direction, and confined in the other. In order to derive the band structure numerically, we formulate the above ingredients in terms of a tight-binding model, in which Eq.~(\ref{eq:MG}) becomes
\begin{equation}
\label{eq:tbMG}
{\cal H}_{\text{G}} = -t_0\sum_{\langle i, j \rangle} e^{i\phi_{ij}} c_i^\dagger c_j \; .
\end{equation}
Indices ($i,j$) label lattice sites, while symbol $\langle\;\rangle$ refers to nearest-neighbor coupling (with hopping amplitude $t_0$), as is illustrated in Fig.~\ref{FigLatGra}. The Peierls phase $\phi_{ij}=(e/\hbar)\int_{{\bf r}_j}^{{\bf r}_i}{\bf A}\cdot d{\bf r}$ takes into account the contribution from the magnetic flux threading the lattice. Numerical calculations are performed using kwant, the new quantum transport software package developed by A. Akhmerov, C. Groth, X. Waintal, and M. Wimmer. In the process, we choose to work with armchair boundary conditions, but our results are qualitatively unaffected by this choice.
The band structure associated with the peculiar spectrum of Eq.~(\ref{eq:LLMG}) in a ribbon geometry (translationally invariant in the $x$-direction, confined in the $y$-direction with $|y|<W/2$) is shown in Fig.~\ref{Fig1}.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Monolayer_B=0,05bis2.pdf}
\caption{(Color online): Energy spectrum of a monolayer graphene armchair ribbon in the QH regime (${\cal C}\neq0$). Black circles and red triangles respectively stand for spin up and spin down bands, which are here indistinguishable due to spin degeneracy. The horizontal dashed line represents an arbitrarily chosen Fermi level. Its intersection with the lowest band of the system indicates the existence of a (spin-degenerate) edge state which propagates in opposite directions on opposite sides. The ribbon width is $W=38$ and the magnetic length is $l_B\approx4$.}
\label{Fig1}
\end{center}
\end{figure}
Due to the nature of the classical dynamics, the transverse coordinate of the cyclotronic center of motion $y_c$ can be identified with the conserved longitudinal momentum $k_x$ via the formula $y_c = -k_xl_B^2$. Observing the band structure in terms of this real space coordinate, one can see in Fig.~\ref{Fig1} that, far from the edges of the ribbon, the band structure consists of flat bands which are none other than the Landau levels of Eq.~(\ref{eq:LLMG}): electrons in the bulk are classically localized by the magnetic field along closed cyclotronic orbits. On the other hand, electrons in the vicinity of the edges can scatter along them and propagate following skipping orbits, which translates in the band structure into bulk Landau levels acquiring a finite dispersion as they approach the edges of the ribbon:
\begin{equation}
\label{eq:disp}
v_x^{(n)} = \frac{1}{\hbar}\frac{\partial \epsilon_n}{\partial k_x} \; .
\end{equation}
Because this dispersion is monotonous on a given edge (see Fig.~\ref{Fig1}), the edge states cannot be backscattered unless they are coupled to the states living on the opposite edge, a process the likelihood of which decays exponentially with the width of the system. This property of the edge states is generally referred to as chirality and is the reason why these states can carry current without dissipation: this leads to the celebrated QH effect \cite{Klitzing80}, characterized by a quantized conductance $G={\cal C}(e^2/h)$. More formally, the edge states enjoy a topological protection encoded in the Chern number ${\cal C}$ which is a $\mathbb{Z}$ topological invariant characterizing the number of filled bands in the QH regime \cite{Thouless82}. It is a topological quantity, in the sense that smooth deformations of the Hamiltonian (deformations which do not close the gap) cannot change its value, and shall be defined in the next subsection. We thus see that, for most purposes, the physics of topological phases such as the QH phase can be very simply extracted from the corresponding band structure.
We now consider the situation where, in addition to the perpendicular magnetic field, the effect of spin-orbit coupling as introduced by Kane and Mele \cite{Kane05bis} is accounted for in the Hamiltonian as
\begin{equation}
\label{eq:KM0}
H_{\text{so}} = \tau s\Delta_{\text{so}}\sigma_z \; ,
\end{equation}
which is characterized by the energy scale $\Delta_{\text{so}}$. $\tau=\pm 1$ and $s=\pm 1$ account for valley and spin degrees of freedom. In terms of a tight-binding model, Eq.~(\ref{eq:KM0}) can be implemented as \cite{Kane05bis}
\begin{equation}
\label{eq:tbKM0}
{\cal H}_{\text{so}} = i\lambda_{\text{so}}\sum_{ \langle\langle i,j\rangle\rangle}\nu_{ij}e^{i\phi_{ij}} (c_{i,\alpha}^\dagger s^{\alpha\beta}_z c_{j,\beta}) \; ,
\end{equation}
where indices ($i,j$) once more label lattice sites, while ($\alpha,\beta$) label spin indices, symbol $\langle\langle\;\rangle\rangle$ refers to next-nearest-neighbor coupling (with SO-induced hopping amplitude $\lambda_{\text{so}} = \Delta_{\text{so}}/(3\sqrt{3})$ \cite{Kane05bis}), and $\nu_{ij}=\pm1$ depending on whether sites are coupled clockwise or counter-clockwise (see Fig.~\ref{FigLatGra}). Note that in order for the system to remain gauge invariant, Peierls substitution has to be done on all hopping matrix elements: nearest-neighbor {\it and} (SO) second nearest-neighbor. The presence of spin-orbit coupling modifies the Landau level spectrum according to the expression
\begin{equation}
\epsilon_{n,s} =
\left\lbrace
\begin{split}
\pm\sqrt{\Delta_B^2|n| + \Delta_{\text{so}}^2} \; , \; \text{for} \; n\neq0
\\
-s\Delta_{\text{so}} \; , \; \text{for} \; n=0
\end{split}
\right. \; .
\end{equation}
The latter is characterized by the $n=0$ level being lifted from zero energy into spin-polarized branches: $E=+\Delta_{\text{so}}$ features only spin-down states, while $E=-\Delta_{\text{so}}$ features only spin-up states \cite{DeMartino11} (see Fig.~\ref{Fig1bot}).
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Monolayer_B=0,05_lso=0,02bis2.pdf}
\caption{(Color online): Same as in Fig.~\ref{Fig1}, but with an additional spin-orbit coupling term $\lambda_{\text{so}}=0.02$. The latter lifts the spin-degeneracy of the zero-energy Landau level, yielding a QSH phase ($\nu=1$) with a single pair of counter-propagating spin-polarized edge states. The sketch above the band structure depicts the edge states in real-space (thick black vertical lines on the sides represent the edges of the ribbon).}
\label{Fig1bot}
\end{center}
\end{figure}
While other Landau levels retain their associated chiral edge states, irrespective of the spin polarization, the lowest Landau level now features counter-propagating edge states for $|E|< \Delta_{\text{so}}$, with a spin-dependent direction of propagation (see Fig.~\ref{Fig1bot}): one has $v_x^{(0,\uparrow)}\cdot v_x^{(0,\downarrow)} <0$ on a given edge (allowing for an additional spin dependence in the definition of Eq.~(\ref{eq:disp})). This is illustrated in the real-space sketch above the band structure in Fig.~\ref{Fig1bot}. It is the signature of a QSH phase, as can be certified by computing the associated $\mathbb{Z}_2$ topological invariant introduced by Kane and Mele \cite{Kane05}, which we do next.
\subsection{Topological order}
Let us start by recalling the standard topological number characterization of Landau levels when $\Delta_{\text{so}} = 0$. Each Landau level $n$ and its associated eigenfunctions over the first Brillouin zone are characterized by a topological invariant, the so-called Chern number \cite{Thouless82}. This topological number takes a value ${\cal C}^{(n)}_{\tau,s}=+1$ for each Landau level, independently of the Landau $n$, valley $\tau$ or spin $s$ indices. For each value of the Fermi energy, we can characterize the corresponding phase by a topological number
\begin{equation}
{\cal C} = \sum_{\tau,s} {\cal C}_{\tau,s} \; , \; \text{with} \; {\cal C}_{\tau,s}(E_F) = \sum_{\epsilon_n<E_F}{\cal C}^{(n)}_{\tau,s}
\end{equation}
obtained by summing the Chern numbers of all filled energy bands \cite{Thouless82}.
In 2DDFGs, the Chern number is a priori ill-defined because of the existence of an infinite number of filled energy bands of negative energy. Through the use of non-commutative Berry's connection, it was however shown \cite{Watanabe11} that the Chern number takes a value ${\cal C}_{\tau,s}(E=0^-)=-1/2$ per degree of freedom for energies immediately below the Dirac point. In the case of graphene, defining the spin Chern number as
\begin{equation}
{\cal C}_s=\sum_{\tau}{\cal C}_{\tau,s} \; ,
\end{equation}
one obtains ${\cal C}_s(E=0^-)=-1$ per spin species (since there are two valleys). With this prescription, the band structure of graphene in the QH regime can be easily described by computing the value of the Chern number as a function of the Fermi energy. This yields
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow={\cal C}_\downarrow=-1 \; , \; \text{for} \; -\Delta_B < E_F < 0
\\
{\cal C}_\uparrow={\cal C}_\downarrow=2n+1 \; , \; \text{for} \; E_F > 0
\end{array}
\right.
\end{equation}
where $n$ is the index of the highest filled Landau level. Notice that Chern numbers of each spin species are equal (since the spectrum is spin-degenerate) and that ${\cal C}={\cal C}_\uparrow+{\cal C}_\downarrow$ is always non-zero, as expected for a QH phase. The total Chern number increases step-wise by multiples of 4, due to spin and valley degeneracy of the Landau levels.
Restoring a finite value of $\Delta_{\text{so}}$, the topologically non-trivial nature of the phase for $E<\Delta_{\text{so}}$ can be checked by computing the corresponding value of the $\mathbb{Z}_2$ topological invariant. In the presence of spin rotational symmetry (conservation of $S_z$), this invariant can be simply expressed as the difference of the Chern numbers for each spin species \cite{Sheng06}:
\begin{equation}
\nu=\frac{1}{2}({\cal C}_\uparrow-{\cal C}_\downarrow) \;\; (\text{mod} \; 2) \; .
\end{equation}
Note, however, that the existence of this invariant is naturally independent of the existence of this symmetry. Following the calculations performed in \cite{Shevtsov12X}, one easily finds
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow=-{\cal C}_\downarrow=+1 \; , \; \text{for} \; |E_F| < \Delta_{\text{so}}
\\
{\cal C}_\uparrow={\cal C}_\downarrow=2n+1 \; , \; \text{for} \; E_F > \Delta_{\text{so}}
\end{array}
\right.
\end{equation}
where $n$ is once more the index of the highest filled Landau level. This time, one is faced with a QH phase for $|E_F| > \Delta_{\text{so}}$, characterized by the same Chern number as in the previous case, while for $|E_F| < \Delta_{\text{so}}$, the total Chern number vanishes ${\cal C}_\uparrow+{\cal C}_\downarrow=0$, indicating that this region is no longer in the QH phase. However, as the $\mathbb{Z}_2$ invariant $\nu = 1$ does not vanish, the phase in this region is a QSH phase.
The resulting band structure is thus particularly interesting: it consists of a QSH phase at energies $|E|<\Delta_{\text{so}}$ and a QH phase at other energies. One can therefore observe a topological phase transition in this system by simply tuning the Fermi level across the spin-orbit gap. The central manifestation of this phase transition is the existence of a spin-polarized state localized at the interface between both phases. This state could be most clearly observed experimentally by making use of an additional electric gate to independently tune the Fermi level in two different parts of the system, thereby realizing a topological heterojunction. We refer the interested reader to our previous paper \cite{Shevtsov12X} for a detailed discussion of these matters.
\section{Bilayer graphene}
\label{sec:BandSym}
We now switch to the slightly more involved case of (Bernal-stacked) bilayer graphene and start by addressing the possibility of inducing a QSH phase in bilayer graphene. This is not a trivial endeavour, as the naive extension of the Kane-Mele model to bilayer graphene yields a weak $\mathbb{Z}_2$ topological phase, characterized by an even number (rather than an odd number, as in monolayer graphene) of pairs of spin-polarized edge states \cite{Prada11,Cortijo10}. This doubling of the number of edge states basically arises because a bilayer has twice as many layers as a monolayer. For the same reason, a graphene trilayer will have an odd number of pairs of edge states and therefore feature a non-trivial QSH phase. Breaking the layer symmetry by considering the case where spin-orbit coupling is present in only one of the layers was shown not to be any more effective \cite{Prada11}, the system then remaining semimetallic.
Here we follow a different approach, inspired by the model presented in the previous section. This seems like a natural idea, as bilayer graphene is also known to feature zero-energy Landau levels \cite{Novoselov06,McCann06}. In this section, we will show that the presence of both spin-orbit coupling and a perpendicular magnetic field in the bilayer yields a band structure very similar to that of monolayer graphene, but with results no different from that of Prada et al. \cite{Prada11}: the obtained QSH phase is topologically trivial due to the existence of an even number of pairs of spin-polarized edge states. On the other hand, we will show in the next section that if spin-orbit coupling is present in only one of the two layers or if a perpendicular electric field is applied, then the breaking of layer inversion symmetry opens the door for a non-trivial QSH phase to arise at low-energy.
\subsection{Model}
The Hamiltonian for bilayer graphene can be expressed using two sets of Pauli matrices $\{\sigma,\eta\}$ which respectively refer to sublattice ($A$, $B$) and layer (1, 2) spaces. We consider the usual Bernal stacking, inherited from graphite (see Fig.~\ref{FigBernal}), in which $A$ atoms in the upper layer (2) lie above $B$ atoms from the lower layer (1).
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=0.75\linewidth]{Bernal.pdf}
\caption{Side view of Bernal-stacked bilayer graphene, which is characterized by intra-layer nearest-neighbor hopping $t_0$ and inter-layer hopping $t_1$.}
\label{FigBernal}
\end{center}
\end{figure}
Starting from the basis ($A_2$, $B_2$, $A_1$, $B_1$)$^T$, the Hamiltonian reads \cite{McCann06}:
\begin{equation}
\label{eq:BG}
H_{\text{BG}} = v_F(\tau\hat{p}_x\sigma_x+\hat{p}_y\sigma_y)\eta_0 + \frac{t_1}{2}(\sigma_x\eta_x - \sigma_y\eta_y) \; .
\end{equation}
The first term describes the usual low-energy Dirac structure of monolayer graphene. The second term takes into account the coupling between both layers, characterized by an energy scale $t_1 \simeq 0.15$. Corrections to Eq.~(\ref{eq:BG}) such as trigonal warping are small effects, typically only relevant below the meV range \cite{McCann12}, and will therefore be neglected.
The spectrum associated with Eq.~(\ref{eq:BG}) is particle-``hole" symmetric, with high-energy bands at $\epsilon^{\text{high}}=\pm t_1$ and low-energy bands touching at two Dirac points characterized by a topological Berry phase 2$\pi$. In the vicinity of this point, the energy-dispersion relation is quadratic, $\epsilon^{\text{low}} = \pm p^2/(2m^*)$, with $m^*=t_1/(2v_F^2)$ the effective mass of the gapless excitations.
The presence of a perpendicular magnetic field can be straightforwardly included by making use of the Peierls substitution as before, and the energy spectrum can then be shown to turn into the well-known Landau levels \cite{McCann06},
\begin{equation}
\label{eq:LLBG}
\epsilon_n = \pm\hbar\omega_c\sqrt{n(n-1)}
\end{equation}
with $\omega_c=eB/m^*$ the characteristic cyclotronic frequency. The latter can be related to the monolayer graphene energy scale by the simple relation $\hbar\omega_c=\Delta_B^2/t_1$. Notice how the spectrum in Eq.~(\ref{eq:LLBG}) features twice as many zero-energy levels as in monolayer graphene, since the $n=1$ level also vanishes. Actually, one can prove on general grounds that chirally stacked $N$-layer graphene should feature a $4N$-fold degenerate zero-energy Landau level \cite{Min08,Koshino09}. One should also have in mind that the accuracy of expression (\ref{eq:LLBG}) for $n\neq0, 1$ is only correct in the limit $\Delta_B \ll t_1$.
\subsection{Quantum Hall regime}
To compute the associated band structure numerically, we once more formulate the above ingredients in terms of a tight-binding model, in which Eq.~(\ref{eq:BG}) becomes
\begin{equation}
\label{eq:tbBG}
{\cal H}_{\text{BG}} = -t_0\sum_{\langle i, j \rangle} e^{i\phi_{ij}} c_i^\dagger c_j + t_1\sum_{\langle i\in A_2, j\in B_1 \rangle} c_i^\dagger c_j \; ,
\end{equation}
using the same notations as in the previous section. No Peierls phase appears in the second term, as $A_2$-$B_1$ bonds are oriented along the $z$-axis. The band structure associated with the Landau level spectrum of Eq.~(\ref{eq:LLBG}) in a ribbon-geometry is displayed in the upper panel of Fig.~\ref{Fig2}.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Bilayer_B=0,05bis.pdf}
\includegraphics[angle=0,width=1.0\linewidth]{Bilayer_B=0,05_lso=0,02bis.pdf}
\caption{(Color online): (Top panel) Energy spectrum of a bilayer graphene armchair ribbon in the QH regime. Black circles and red triangles respectively stand for spin up and spin down bands, which are here indistinguishable due to spin degeneracy. Notice that the zero-energy Landau level has twice as many bands as its counterpart in monolayer graphene. The ribbon width is $W=38$ and the magnetic length is $l_B\approx4$. (Bottom panel) Same as above, but with an additional spin-orbit coupling term $\lambda_{\text{so}}=0.02$. The latter lifts the spin-degeneracy of the zero-energy Landau level, yielding a weak QSH phase ($\nu=0$ (mod 2)) with an even number of pairs of counter-propagating spin-polarized edge states.}
\label{Fig2}
\end{center}
\end{figure}
As expected, it resembles very closely that of monolayer graphene in the QH regime. The main difference between the two lies in the existence of twice as many dispersing branches arising from the lowest Landau level in bilayer graphene, due to the doubling of the zero-energy Landau level degeneracy. This translates in the language of topological invariants into Chern numbers per spin species ${\cal C}_s=-2$ immediately below zero energy. With this prescription, the band structure of bilayer graphene in the QH regime can be easily described by computing the value of the Chern number as a function of the Fermi energy, yielding:
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow={\cal C}_\downarrow=-2 \; , \; \text{for} \; -\hbar\omega_c\sqrt{2} < E_F < 0
\\
{\cal C}_\uparrow={\cal C}_\downarrow=2[\text{max}(0,n-1)+1] \; , \; \text{for} \; E_F > 0
\end{array}
\right.
\end{equation}
where $n$ is the index of the highest filled Landau level. Notice that Chern numbers of each spin species are equal (since the spectrum is spin-degenerate) and that ${\cal C}={\cal C}_\uparrow+{\cal C}_\downarrow$ is always non-zero, as expected for a QH phase.
\subsection{Effect of layer-symmetric spin-orbit coupling}
We now consider the situation where, in addition to the perpendicular magnetic field, the effect of spin-orbit coupling as introduced by Kane and Mele for monolayer graphene is accounted for symmetrically in both layers\,\footnote{As in monolayer graphene, nearly identical results can be obtained by considering Zeeman splitting, instead of spin-orbit coupling, as the spin-degeneracy lifting mechanism.}. The layer-degenerate Kane-Mele spin-orbit coupling term is encoded in the Hamiltonian
\begin{equation}
\label{eq:KM}
H_{\text{so}} = \tau s\Delta_{\text{so}}\sigma_z\eta_0 \; ,
\end{equation}
which, in terms of a tight-binding model, can be implemented as
\begin{equation}
\label{eq:tbKM}
{\cal H}_{\text{so}} = i\lambda_{\text{so}}\sum_{ \langle\langle i,j\rangle\rangle}\nu_{ij}e^{i\phi_{ij}} (c_{i,\alpha}^\dagger s^{\alpha\beta}_z c_{j,\beta}) \; ,
\end{equation}
with similar notations as in the previous section, symbol $\langle\langle\;\rangle\rangle$ referring to intra-layer next-nearest-neighbor coupling. The presence of spin-orbit coupling modifies the Landau level spectrum according to the expression
\begin{equation}
\epsilon_{n,s} =
\left\lbrace
\begin{split}
\pm\sqrt{n(n-1)(\hbar\omega_c)^2 + \Delta_{\text{so}}^2} \; , \; \text{for} \; n \neq 0, 1
\\
-s\Delta_{\text{so}} \; , \; \text{for} \; n = 0, 1
\end{split}
\right. \; .
\end{equation}
The latter is characterized by $n=0$ and $n=1$ levels lifted from zero energy into spin-polarized branches: $E=+\Delta_{\text{so}}$ features only spin-down states, while $E=-\Delta_{\text{so}}$ features only spin-up states (see lower panel of Fig.~\ref{Fig2}).
The topologically trivial nature of the corresponding low-energy phase can be checked by computing the value of the $\mathbb{Z}_2$ topological invariant. As a straightforward generalization of the calculations performed in \cite{Shevtsov12X}, one obtains the following results:
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow=-{\cal C}_\downarrow=+2 \; , \; \text{for} \; |E_F| < \Delta_{\text{so}}
\\
{\cal C}_\uparrow={\cal C}_\downarrow=2[\text{max}(0,n-1)+1] \; , \; \text{for} \; E_F > \Delta_{\text{so}}
\end{array}
\right.
\end{equation}
where $n$ is once more the index of the highest filled Landau level. This time, one is faced with a QH phase for $|E_F| > \Delta_{\text{so}}$, characterized by the same Chern number as in the absence of spin-orbit coupling, while for $|E_F| < \Delta_{\text{so}}$, the total Chern number vanishes ${\cal C}_\uparrow+{\cal C}_\downarrow=0$, indicating that this region is no longer in the QH phase. However, as the $\mathbb{Z}_2$ invariant $\nu = 0$ (mod 2) also vanishes, the phase in this region is not a QSH phase either: rather, it is a weak QSH phase, in the sense that time-reversal-symmetric perturbations can couple the edge states and induce backscattering. This was not the case in monolayer graphene, due to the existence in the latter of a single pair of counter-propagating spin-polarized edge states at low energy. In the situation discussed in this section, we are thus led to conclude that a similar picture as that described in Ref.~\cite{Prada11} prevails. The way around this involves breaking the layer inversion symmetry, as we will see in the next section.
Before moving on, however, we would like to pause and comment on the fact that the model we considered in this section could provide a convenient platform for testing precisely how weak a $\nu=0$ (mod 2) QSH phase would be. Indeed, even though theory predicts that pairs of edge states should couple through backscattering processes, an experimental measure of how strongly edge state transport would be destroyed by such processes is yet to be done, and one cannot exclude the possibility of unexpected robustness, similarly to what has recently begun to be understood in so-called weak three-dimensional topological insulators \cite{Ringel12,Mong12}. Said a little differently, it remains unclear how one could distinguish through transport measurements a topological phase from a trivial phase which has edge states (such as the one exhibited in this section).
\section{Bilayer graphene with broken layer inversion symmetry}
\label{sec:BandAsym}
This section is devoted to the study of two layer inversion symmetry breaking mechanisms which enable an exchange-induced QSH phase to arise at low energy: (i) inducing spin-orbit coupling only in one of the layers, and (ii) applying a perpendicular electric field. We provide estimations of the magnitude of the QSH gap, and also address other possible phases which appear in our settings: a spin-polarized QH phase and a quantum valley Hall phase.
\subsection{Mechanism I: layer-asymmetric spin-orbit coupling}
Using the same basis as in Eq.~(\ref{eq:BG}), we now replace Eq.~(\ref{eq:KM}) by the symmetry-breaking term
\begin{equation}
\label{eq:asymKM}
H_{\text{so}}^{\text{asym}} = \tau s\Delta_{\text{so}}\sigma_z\left(\frac{\eta_0+\eta_z}{2}\right)
\end{equation}
which induces spin-orbit coupling only in the upper layer. This situation is physically relevant if one considers the possibility of inducing spin-orbit coupling in graphene by depositing adatoms on the surface \cite{Weeks11,Shevtsov12,Jiang12}. The corresponding tight-binding expression is given by applying Eq.~(\ref{eq:tbKM}) only in the upper layer.
In order to obtain the new Landau level spectrum, using the ladder operators introduced in section \ref{sec:MG}, one must now solve a quartic equation $\epsilon_n^4 + \alpha_n\epsilon_n^2 + \beta_n\epsilon_n + \gamma_n = 0$, with a non-vanishing linear term $\beta_n\neq0$:
\begin{equation}
\left\lbrace
\begin{array}{l}
\alpha_n = -\left(t_1^2 + \Delta_{\text{so}}^2 + (2n-1)\Delta_B^2\right)
\vspace*{0.2cm}
\\
\beta_n = -s\Delta_{\text{so}}t_1^2
\vspace*{0.2cm}
\\
\gamma_n = (n-1)\Delta_B^2(n\Delta_B^2 + \Delta_{\text{so}}^2)
\end{array}
\right.
\; .
\end{equation}
Note that the role of opposite spin polarizations is exchanged when going from the conduction to the valence band: $\epsilon_n(-s)=-\epsilon_n(s)$. This leaves 4 (out of the 8) eigenvalues of the lowest Landau levels to be found. Two can easily be identified: $\epsilon_0=-s\Delta_{\text{so}}$ (for $n=0$) and $\epsilon_1=0$ (for $n=1$) both satisfy the quartic equation. The latter implies that a spin-degenerate zero-energy Landau level survives in this context. The two remaining eigenvalues, $\epsilon_-$ and $\epsilon_+$, can be estimated perturbatively, in the limit $\Delta_{\text{so}}, \hbar\omega_c \ll t_1$, as $\epsilon_+ \approx -s\Delta_{\text{so}}(1-\hbar\omega_c/t_1)$ and $\epsilon_- \approx -s\Delta_{\text{so}}\hbar\omega_c/t_1$. Their dependence on $\Delta_{\text{so}}$ and $\hbar\omega_c$ beyond this perturbative regime is shown in the lower panel of Fig.~\ref{Fig3}. This yields the following ordering of eigenvalues: $0 = \epsilon_1 < |\epsilon_-| < |\epsilon_+| < |\epsilon_0| = \Delta_{\text{so}}$.
The corresponding band structure is shown in the top panel of Fig.~\ref{Fig3}.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Bilayer_B=0,05_Asymlso=0,02bis.pdf}
\includegraphics[angle=0,width=1.0\linewidth]{RootsSO.pdf}
\caption{(Color online): (Top panel) Band structure with same parameters as in Fig.~\ref{Fig2}, except that spin-orbit coupling is only applied in the upper layer. This time, a spin-degenerate zero-energy Landau level survives, while low-energy edge states are characterized by an unbalanced spin population on a given edge ($|{\cal C}_\uparrow| \neq |{\cal C}_\downarrow|$). (Bottom panel) Dependence of the eigenvalues $\epsilon_{\pm}$ on the magnetic field for $\Delta_{\text{so}}=0.01$. Dashed lines correspond to the analytical predictions (valid in the perturbative limit $\hbar\omega_c \ll t_1$) and thick lines to the numerically obtained values. Inset: Dependence of $\epsilon_\pm$ on the spin-orbit gap for $\Delta_B=0.05$.}
\label{Fig3}
\end{center}
\end{figure}
Its description in terms of spin-polarized bands is slightly more involved than before, but the Chern numbers can nevertheless be computed and shown to evolve as follows,
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow=-1 \; , \; {\cal C}_\downarrow=-2 \; , \; \text{for} \; -|\epsilon_0| < E_F < -|\epsilon_+|
\\
{\cal C}_\uparrow=0 \; , \; {\cal C}_\downarrow=-2 \; , \; \text{for} \; -|\epsilon_+| < E_F < -|\epsilon_-|
\\
{\cal C}_\uparrow=1 \; , \; {\cal C}_\downarrow=-2 \; , \; \text{for} \; -|\epsilon_-| < E_F < 0
\\
{\cal C}_\uparrow=2 \; , \; {\cal C}_\downarrow=-1 \; , \; \text{for} \; 0 < E_F < |\epsilon_-|
\\
{\cal C}_\uparrow=2 \; , \; {\cal C}_\downarrow=0 \; , \; \text{for} \; |\epsilon_-| < E_F < |\epsilon_+|
\\
{\cal C}_\uparrow=2 \; , \; {\cal C}_\downarrow=1 \; , \; \text{for} \; |\epsilon_+| < E_F < |\epsilon_0|
\end{array}
\right.
\end{equation}
indicating that a QH phase is preserved at low energy, since the total Chern number never vanishes. This phase is peculiar, however, as it is characterized by edge states with an unbalanced spin population: $|{\cal C}_\uparrow| \neq |{\cal C}_\downarrow|$. For example, two spin-up and a single spin-down counterpropagating state coexist on the same edge for $0 < E_F < |\epsilon_-|$. A given spin species can even become fully gapped, as testified by vanishing spin Chern numbers, giving rise to spin-polarized edge state transport over a tunable and quite large energy window $|\epsilon_+-\epsilon_-| \approx \Delta_{\text{so}}(1-2\hbar\omega_c/t_1)$.
A QSH phase can be generated close to zero energy by lifting the spin-degeneracy of the remaining zero-energy Landau level with an arbitrarily small exchange field, deriving from
\begin{equation}
\label{eq:Zee}
H_{\text{ex}} = s\Delta_{\text{ex}}\sigma_0\eta_0 \; ,
\end{equation}
where $\Delta_{\text{ex}}$ quantifies the magnitude of the effect.
The corresponding tight-binding expression is given by
\begin{equation}
{\cal H}_{\text{ex}} = \Delta_{\text{ex}}\sum_i c_{i,\alpha}^\dagger s_z^{\alpha\beta} c_{i,\beta} \; ,
\end{equation}
using the same notations as before. This yields for the spin Chern numbers at low energy:
\begin{equation}
{\cal C}_\uparrow=-{\cal C}_\downarrow=1 \; , \; \text{for} \; |E_F| < \text{min}(|\Delta_{\text{ex}}| \; , \; |\epsilon_-|-|\Delta_{\text{ex}}|) \; .
\end{equation}
The total Chern number is zero and, contrary to the case of layer-symmetric spin-orbit coupling, this time the $\mathbb{Z}_2$ invariant $\nu=1$, signaling that the QSH phase is non-trivial. The energy window where this phase can be observed, i.e. the maximum value of the QSH gap, is bounded by the value $|\epsilon_-|/2$, a lower bound of which is given by the perturbative limit $\Delta_{\text{QSH}}^{\text{max}} \geq \Delta_{\text{so}}\hbar\omega_c/(2t_1)$, as can be checked in the lower panel of Fig.~\ref{Fig3}. The QSH gap could thus potentially reach several tens of meV, although, in graphene-based systems, it will effectively be limited by the highest achievable value of spin splitting which should be much smaller\,\footnote{Zeeman splitting can nevertheless be enhanced by tilting the magnetic field with respect to the perpendicular axis, since the Zeeman term is proportional to the total magnetic field $B = B_\parallel + B_\perp$. This avoids the use of large perpendicular magnetic fields which can trigger many-body instabilities.}. In this respect, and despite the need for a perpendicular magnetic field, our proposal offers two advantages with respect to that of Ref.~\cite{Qiao12}, where it was recently shown that gated bilayer graphene could be turned into a $\mathbb{Z}_2$ topological insulator for sufficiently strong Rashba spin-orbit coupling: the strength of spin-orbit coupling need not exceed a critical value, and spin-orbit coupling need not be present in both layers. The latter condition is particularly convenient if one considers that the most promising chance of inducing (intrinsic) spin-orbit coupling in graphene as of today is arguably by depositing adatoms on its surface \cite{Weeks11,Shevtsov12,Jiang12}.
\subsection{Mechanism II: perpendicular electric field}
Let us now exhibit another mechanism of symmetry-breaking which can provide a loophole to circumvent the intrinsic difficulty of generating a non-trivial QSH phase in bilayer graphene. Forgetting momentarily about spin-orbit coupling, let us go back to the Hamiltonian of Eq.~(\ref{eq:BG}) and consider the effect of an electric field applied perpendicularly to the bilayer,
\begin{equation}
H_U = U\sigma_0\eta_z \; .
\end{equation}
This term opens a gap in the energy-momentum dispersion relation by breaking the layer symmetry. It can be implemented in a tight-binding model using the following expression:
\begin{equation}
{\cal H}_U = -U\sum_{i \in 1}c_i^\dagger c_i + U\sum_{i \in 2}c_i^\dagger c_i \; .
\end{equation}
The derivation of the Landau level spectrum requires solving once more a quartic equation $\epsilon_n^4 + \alpha_n\epsilon_n^2 + \beta_n\epsilon_n + \gamma_n = 0$, with a non-vanishing linear term $\beta_n\neq0$:
\begin{equation}
\label{eq:QHU}
\left\lbrace
\begin{array}{l}
\alpha_n = -\left(t_1^2 + 2U^2 + (2n-1)\Delta_B^2\right)
\vspace*{0.2cm}
\\
\beta_n = -2\tau U\Delta_B^2
\vspace*{0.2cm}
\\
\gamma_n = U^2(U^2+t_1^2) - (2n-1)\Delta_B^2U^2 + n(n-1)\Delta_B^4
\end{array}
\right.
\; .
\end{equation}
Taking into account the spin-degneracy of the spectrum and the additional symmetry $\epsilon_n(-\tau)=-\epsilon_n(\tau)$, one is left with two eigenvalues to compute for the lowest energy Landau levels, one of which can be easily seen to be $\epsilon_+=\tau U$. The other one, $\epsilon_-$, must be computed numerically. In the limit $U, \hbar\omega_c \ll t_1$, it can be estimated perturbatively \cite{McCann06} as $\epsilon_- \approx \tau U(1-2\hbar\omega_c/t_1)$. Its dependence on $U$ and $\hbar\omega_c$ beyond this perturbative regime is shown in the lower panel of Fig.~\ref{Fig4}.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Bilayer_B=0,05_U=0,1bis.pdf}
\includegraphics[angle=0,width=1.0\linewidth]{RootsU.pdf}
\caption{(Color online): (Top panel) Effect of an electric field applied perpendicularly to the plane (yielding a layer potential asymmetry $U=0.1$) on the lowest Landau level of bilayer graphene. A gap is opened arising form the lifting of the layer degeneracy for the lowest Landau level. Unspecified parameter values are the same as in Fig.~\ref{Fig2}. (Bottom panel): Dependence of the eigenvalue $\epsilon_-$ on the magnetic field for $U=0.01$. The dashed line is the analytical prediction (valid in the perturbative limit $\hbar\omega_c \ll t_1$) and the thick line is the numerical calculation. Inset: Dependence of $\epsilon_-$ on the perpendicular electric field for $\Delta_B=0.05$.}
\label{Fig4}
\end{center}
\end{figure}
Hence, the QH phase has now been gapped by the perpendicular electric field at low energy (see top panel of Fig.~\ref{Fig4}), yielding the following pattern for the Chern number:
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow={\cal C}_\downarrow=-1 \; , \; \text{for} \; -|\epsilon_+| < E_F < -|\epsilon_-|
\\
{\cal C}_\uparrow={\cal C}_\downarrow=0 \; , \; \text{for} \; |E_F| < |\epsilon_-|
\\
{\cal C}_\uparrow={\cal C}_\downarrow=1 \; , \; \text{for} \; |\epsilon_-| < E_F < |\epsilon_+|
\end{array}
\right.
\end{equation}
However, the layer degeneracy of the former zero-energy Landau levels has now been lifted, which means that a non-trivial QSH phase can once again be generated at low energy, provided some spin-degeneracy lifting mechanism overcomes the gap $|\epsilon_-|$. This can be achieved either by layer-symmetric spin-orbit coupling (\ref{eq:KM}) or by an exchange term (\ref{eq:Zee}). At a critical value of the spin splitting $|\Delta_{\text{ex}}|=|\epsilon_-|$, the lowest bands will cross and give rise to a QSH phase
\begin{equation}
\label{eq:QSH2}
{\cal C}_\uparrow=-{\cal C}_\downarrow=1 \; , \; \text{for} \; |E_F| < \text{min}(|\Delta_{\text{ex}}| - |\epsilon_-| \; , \; U-|\Delta_{\text{ex}}|) \; ,
\end{equation}
characterized by a single pair of counter-propagating spin-polarized edge states (see Fig.~\ref{Fig5}). Once again the total Chern number vanishes while the $\mathbb{Z}_2$ invariant $\nu=1$, indicating the non-trivial character of the QSH phase. Provided the critical spin splitting could be achieved, the maximum value of the QSH gap would this time be bounded by the value $(U-|\epsilon_-|)/2$, an upper bound of which is given by the perturbative limit $\Delta_{\text{QSH}}^{\text{max}} \leq U\hbar\omega_c/t_1$, as can be checked in the lower panel of Fig.~\ref{Fig4}.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Bilayer_B=0,05_U=0,1_gz=1,5bis.pdf}
\caption{(Color online): Effect of an electric field applied perpendicularly to the plane (yielding a layer potential asymmetry $U=0.1$) on the lowest Landau level of bilayer graphene, in the presence of a spin-splitting term $\Delta_{\text{ex}}=0.075$. When this spin splitting exceeds a critical value, a band crossing takes place giving rise to a non-trivial QSH phase. Unspecified parameter values are the same as in Fig.~\ref{Fig2}.}
\label{Fig5}
\end{center}
\end{figure}
As a closing remark, we note that this second mechanism of symmetry-breaking shares with that demonstrated in \cite{Qiao12} the property of having edge states in the low-energy region of Eq.~(\ref{eq:QSH2}) that are not only spin-polarized, but that can also be valley-polarized. This can be traced back to the lifting of valley degeneracy by the perpendicular electric field, as is apparent in the values of the Landau levels given below Eq.~(\ref{eq:QHU}). This valley polarization actually translates into an additional topological protection, encoded in the valley Chern index $\tilde{\nu} = \frac{1}{2}\sum_\tau\tau{\cal C}_\tau$, with ${\cal C}_{\tau}=\sum_s{\cal C}_{\tau,s}$. In the energy region of Eq.~(\ref{eq:QSH2}), this index verifies $\tilde{\nu}=1$, indicating that the low-energy phase is a so-called quantum valley Hall phase. The latter is entirely analogous to a QSH phase, if one exchanges spin and valley indices: it is characterized by valley-polarized counter-propagating edge states, which can thus only be backscattered by short-range (valley-coupling) disorder. Hence, the low-energy phase of Eq.~(\ref{eq:QSH2}) should be immune to spin-mixing perturbations as long as valleys remain uncoupled\,\footnote{In particular, valleys will \textit{de facto} be coupled in armchair-terminated ribbons. The quantum valley Hall phase could however arise in zigzag-terminated ribbons (see \cite{Qiao12}).}.
\subsection{Discussion}
Now that we have identified the regimes in which a non-trivial QSH phase could arise in bilayer graphene, and that we have roughly estimated the order of magnitude of the associated energy gap $\Delta_{\text{QSH}}$, let us conclude this section by making a few comments on the experimental relevance of our results. Until now, we have made the natural assumption of disregarding the effect of disorder in our system, since one of the essential features of a topological phase is its robustness with respect to disorder. The presence of the latter could nevertheless prove problematic if the typical strength of disorder $\delta_{\text{dis}} \gg \Delta_{\text{QSH}}$. The available experimental data in graphene-like systems seem to indicate that low-energy disorder is dominated by charge density fluctuations (electron-hole puddles), but the use of BN substrates has been shown to significantly reduce their magnitude \cite{Dean10,Xue11,Mayorov11}.
The main other threat to the QSH phase lies in the various many-body instabilities which have been predicted to occur in bilayer graphene at the Dirac point due to the finite density of states. This could lead to a spontaneous symmetry breaking of the spin-valley SU(4) symmetry in undoped bilayer graphene, causing the emergence of a yet unidentified gapped phase, typically of the order of a few meV \cite{Feldman09,Velasco12,vanElferen12}. Amusingly, a (many-body driven) QSH phase stands among the list of possible candidates \cite{ZhangF11,Barlas12}.
Estimating the importance of disorder and interaction effects eventually boils down to how big a value of the QSH gap could be achieved. If $\Delta_{\text{QSH}}$ lies in the 10 meV range, then the presence of disorder should be harmless to the QSH phase, while actually reducing the effect of the Coulomb interaction. On the other hand, if $\Delta_{\text{QSH}}$ is rather in the 1 meV range, then chances are great that disorder and/or interactions will wash out the picture we described.
\section{Extensions}
\label{sec:Disc}
Let us now briefly discuss extensions of our model to closely related systems. We start by considering different types of stacking orders in bilayer graphene, and then move on to the case of trilayer graphene.
\subsection{Other stackings}
The analysis we performed in this article relied on the assumption of AB (Bernal) stacking for the bilayer. However, other possibilities may occur. One of them is the so-called AA-stacking, where both layers are mirror-symmetric: $A_2$ atoms sit on top of $A_1$ atoms and $B_2$ atoms sit on top of $B_1$ atoms. In this case, following the exact same steps as described in section \ref{sec:Mod}, the Landau level spectrum can be obtained \cite{Hsu10}, $\epsilon_n^{AA}=\pm\sqrt{t_1(t_1 \pm |n|\hbar\omega_c)}$. Contrary to the case of Bernal stacking, the Landau level with lowest energy is no longer necessarily that corresponding to $n=0$, which leads to a peculiar band structure (see Fig.~\ref{Fig6}) where the low-energy physics is described by counter-propagating spin-degenerate edge states, characterized by a trivial ${\cal C}=0$ phase.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{AABilayer_B=0,05bis.pdf}
\caption{(Color online): Landau level spectrum of AA-stacked bilayer graphene with parameter values as in Fig.~\ref{Fig2}. Notice how, at low energy, spin-degenerate counter-propagating edge states lead to a trivial topologically trivial phase.}
\label{Fig6}
\end{center}
\end{figure}
In a sense, the absence of zero-energy Landau levels in this system, which we took as our defining criterium for a 2DDFG, is directly responsible for the absence of a topological order at zero energy. We additionally checked that the symmetry-breaking mechanims investigated in this work are ineffective for the present system.
Besides AB and AA stackings, a whole (continuous) family of bilayers referred to as twisted bilayers can be studied experimentally. Such bilayers are defined by the angle with which the upper layer is twisted from the lower layer. This angle can be probed experimentally by characterizing the induced Moir\'e patterns. Although such systems are also interesting in their own right (and experimentally relevant), they are not well suited to a tight-binding description, especially for small angles, as the low-energy physics requires potentially very long-range hoppings to be taken into account. We will therefore not discuss them any further, and we refer the reader to other approaches developed in the literature to address their properties (see for example \cite{deGail11}).
Likewise, so-called double layer graphene \cite{Ponomarenko11} -- a bilayer where the coupling between the layers is solely capacitive (transverse hopping is zero) -- crucially requires electrostatic screening to be taken into account, and therefore lies beyond the scope of this paper.
\subsection{Trilayer graphene}
In the light of our understanding of single layer and bilayer graphene, we finish by briefly discussing how much of our previous considerations could find a natural extension in trilayer graphene\,\footnote{Note that, as in bilayer graphene, it has very recently been claimed that a QSH phase could also be induced in gated trilayer graphene in the presence of strong Rashba spin-orbit coupling \cite{Li12}.}. One generally distinguishes two stacking orders (see Fig.~\ref{FigStacking}): ABA stacking, characterized by a combination of linear and quadratic dispersions at low energy, and ABC stacking, characterized by a cubic dispersion and a corresponding diverging density of states at low energy which favors many-body instabilities.
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{Stacking_Tri.pdf}
\caption{Side view of typical stacking sequences of a trilayer of graphite: ABA stacking is mirror-symmetric with respect to the central layer, while ABC stacking can be seen as the natural extension of Bernal stacking in the bilayer (see Fig.~\ref{FigBernal}).}
\label{FigStacking}
\end{center}
\end{figure}
Regardless of the stacking sequence, the odd number of layers implies that, as in monolayer graphene, including spin-orbit coupling in each layer (via a naive extension of Kane and Mele's model) will yield a non-trivial QSH phase\,\footnote{This is true as long as other possible spin-orbit coupling terms that may arise at low energy in multilayer systems can be neglected \cite{McCann10}.}. The additional presence of a perpendicular magnetic field -- which has experimentally been shown to give rise to a QH effect \cite{Taychatanapat11,Lui11,Bao11,Zhang11} characterized by a spectrum with a 12-fold degenerate zero-energy Landau level and described by the following values for the Chern number,
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow={\cal C}_\downarrow=-3 \; , \; \text{for} \; -\Delta_{LL} < E_F < 0
\\
{\cal C}_\uparrow={\cal C}_\downarrow=+3 \; , \; \text{for} \; 0 < E_F < \Delta_{LL}
\end{array}
\right.
\; (\Delta_{\text{so}}=0)
\end{equation}
where $\Delta_{LL}$ is the energy of the lowest non-zero Landau level -- will yield a transition from a QH to a non-trivial QSH phase at low-energy (top panel of Fig.~\ref{Fig7}):
\begin{figure}[]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{ABCTrilayer_B=0,05_lso=0,02bis.pdf}
\includegraphics[angle=0,width=1.0\linewidth]{ABCTrilayer_B=0,05_U=0,1_gz=1,0bis.pdf}
\caption{(Color online): Landau level spectrum of (ABC-stacked) trilayer graphene with spin-orbit coupling $\lambda_{\text{so}}=0.02$ (top panel), and with both perpendicular electric field ($U=0.1$) and spin-splitting $\Delta_{\text{ex}}=0.05$ (bottom panel). The effect of spin-orbit coupling is completely analogous to that in bilayer graphene (see Fig.~\ref{Fig2}), causing a lifting of spin-degeneracy in the lowest Landau level. This time, however, the odd number of pairs of spin-polarized counter-propagating edge states leads to a non-trivial QSH phase at low energy (top panel). Additionally, and also as in bilayer graphene, the simultaneous presence of a layer-degeneracy lifting electric field and a spin-splitting term can also give rise to a non-trivial QSH phase at low energy, with a single pair of counter-propagating spin-polarized edge states (bottom panel). Once more, unspecified parameter values are the same as in Fig.~\ref{Fig2}.}
\label{Fig7}
\end{center}
\end{figure}
\begin{equation}
\left\lbrace
\begin{array}{l}
{\cal C}_\uparrow=-{\cal C}_\downarrow=3 \; , \; \text{for} \; |E_F| < \Delta_{\text{so}}
\\
{\cal C}_\uparrow={\cal C}_\downarrow=3 \; , \; \text{for} \; \Delta_{\text{so}} < E_F < \sqrt{\Delta_{LL}^2 + \Delta_{\text{so}}^2}
\end{array}
\right.
\;
\end{equation}
yielding ${\cal C}=0$ and $\nu = 1$ (mod 2) when $|E_F| < \Delta_{\text{so}}$.
\begin{center}
\begin{table*}[hts]
\begin{tabular}{c|c}
\label{table1}
$N$-layer graphene in the QH regime & Low-energy topological phase \\
\hline\hline
$N=1$ (Fig.~\ref{Fig1}) & QH with ${\cal C}=\pm2$ \\
\hline
$N=1$ with $\Delta_{\text{so}}$ (Fig.~\ref{Fig1bot}) & QSH \\
\hline
$N=2$ (Fig.~\ref{Fig2} top) & QH with ${\cal C}=\pm4$ \\
\hline
$N=2$ with $\Delta_{\text{so}}$ (Fig.~\ref{Fig2} bottom) & weak QSH with $\nu=0$ (mod 2) \\
\hline
$N=2$ with $\Delta_{\text{so}}$ only in upper layer (Fig.~\ref{Fig3} top) & QH with ${\cal C}=\pm1$ (spin-unbalanced) \\
\hline
$N=2$ with $\Delta_{\text{so}}$ only in upper layer, and $\Delta_{\text{ex}}$ & QSH \\
\hline
$N=2$ with $U$ (Fig.~\ref{Fig4} top) & $\emptyset$ \\
\hline
$N=2$ with $U$, and $|\Delta_{\text{ex}}|>|\epsilon_-|$ (Fig.~\ref{Fig5}) & QSH + QValleyH \\
\hline
$N=3$ & QH with ${\cal C}=\pm6$ \\
\hline
$N=3$ with $\Delta_{\text{so}}$ (Fig.~\ref{Fig7} top) & QSH \\
\hline
\end{tabular}
\caption{Summary of low-energy topological phases in graphene-based 2DDFGs.}
\end{table*}
\end{center}
Exploring further the fate of the Landau level spectrum, we have checked that applying spin-orbit coupling only in the upper layer is (QSH-wise) ineffective. However, applying a perpendicular electric field, through the tight-binding expression
\begin{equation}
{\cal H}_U = -U\sum_{i \in 1} c_i^\dagger c_i + U\sum_{i \in 3} c_i^\dagger c_i \; ,
\end{equation}
has an interesting effect which distinguishes ABA from ABC stacking. In the latter case, it opens a gap, while in the former it does not (though the QH phase is trivial at low energy, due to counter-propagating states). When an exchange term is taken into account, a QSH phase with only a single pair of counter-propagating spin-polarized states can be accessed (bottom panel of Fig.~\ref{Fig7}). Thus, our second symmetry-breaking mechanism seems to work equally well in trilayer graphene, although its relevance is debatable in the present context since, as mentioned above, a QSH phase could already be obtained in trilayer graphene in the absence of any layer inversion symmetry-breaking. Additionally, the width of the energy window where our mechanism is effective decreases with the number of layers, which can be qualitatively understood as originating from the proliferation of bands (due to the increasing degeneracy of the lowest Landau level).
\section{Conclusion}
\label{sec:Conc}
We have considered different examples of graphene-based 2DDFGs and shown that, in the presence of both spin-orbit coupling and a perpendicular magnetic field, a topological phase transition between a QH and a QSH phase could take place at low energy. An overall summary of the various cases discussed in this article is provided in Table I. While the lifting of spin degeneracy in the Landau level spectrum was the only requirement to observe this transition in monolayer graphene, we showed that a similar prescription proves insufficient in bilayer graphene, yielding a weak QSH phase at low energy.
We then proceeded to identify several regimes in which a non-trivial QSH phase, characterized by a single pair of counter-propagating spin-polarized edge states, can be induced in bilayer graphene, all of which involved breaking the layer inversion symmetry. We investigated two possible ways of achieving this: (i) by considering the presence of spin-orbit coupling only in the upper layer; (ii) by applying a perpendicular electric field. In both cases, the resulting low-energy phase can then be tuned into a non-trivial QSH phase in the presence of an exchange field: in case (i), an arbitrarily small exchange term suffices, while in case (ii), a non-zero critical value is required. The first of these two cases has the advantage of requiring only small spin splitting (which will, however, effectively control the magnitude of the induced QSH gap) and the presence of spin-orbit coupling only in the upper layer. The latter condition is crucial, as the most promising way to induce sizeable (intrinsic) spin-orbit coupling in graphene is arguably by random adatom deposition \cite{Weeks11,Shevtsov12,Jiang12}. Indeed, the effectively weak intrinsic spin-orbit coupling of carbon remains as of today the main obstacle in the attempt of experimentally detecting the QSH phase in graphene-like systems. In this respect, other recently isolated two-dimensional crystals such as silicene \cite{Lalmi10,Liu11,Ezawa12} or cold atom optical lattices \cite{Mei12} might offer an alternative to probe the physics described in this work.
\begin{acknowledgments}
This work was supported by STREP ConceptGraphene and EC Contract ERC MesoQMC.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 590 |
{"url":"https:\/\/zbmath.org\/?q=an%3A06259832","text":"# zbMATH \u2014 the first resource for mathematics\n\nPeriodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. (English) Zbl\u00a01281.92069\nSummary: In this paper, a predator-prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order $$k$$ $$(k\\geq 2)$$ periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.\n\n##### MSC:\n 92D40 Ecology 34K13 Periodic solutions to functional-differential equations 34C23 Bifurcation theory for ordinary differential equations\nFull Text:\n##### References:\n [1] Xiao, D.X.; Ruan, S.G., Bogdanov-Takens bifurcations in predator-prey system with constant rate harvesting, Fields Inst. Commun., 21, 493-506, (1999) \u00b7 Zbl\u00a00917.34029 [2] Brauer, F.; Soudack, A.C., Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7, 319-337, (1979) \u00b7 Zbl\u00a00397.92019 [3] Brauer, F., Destabilization of predator-prey systems under enrichment, Int. J. Control, 23, 541-552, (1976) \u00b7 Zbl\u00a00319.92012 [4] Brauer, F.; Soudack, A.C.; Jarosch, H.S., Stabilization, and destabilization of predator-prey systems under harvesting and nutrient enrichment, Int. J. Control, 23, 553-573, (1976) \u00b7 Zbl\u00a00317.92003 [5] Brauer, F.; Soudack, A.C., Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol., 8, 55-71, (1979) \u00b7 Zbl\u00a00406.92020 [6] Dai, G.R.; Tang, M.X., Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58, 193-210, (1998) \u00b7 Zbl\u00a00916.34034 [7] Dai, G.R.; Xu, C., Constant rate predator harvested predator-prey system with Holling-type I functional response, Acta Math. Sci., 14, 34-144, (1994) [8] Chen, L.J.; Chen, F.D., Global analysis of a harvested predator-prey model incorporating a constant prey refuge, Int. J. Biomath., 3, 205-223, (2010) \u00b7 Zbl\u00a01342.92160 [9] Pei, Y.Z.; Li, C.G.; Chen, L.S., Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay, Math. Comput. Simul., 10, 2994-3008, (2009) \u00b7 Zbl\u00a01172.92038 [10] Liu, Z.J.; Tan, R.H., Impulsive harvesting and stocking in a monod-Haldane functional response predator-prey system, Chaos Solitons Fractals, 34, 454-464, (2007) \u00b7 Zbl\u00a01127.92045 [11] Negi, K.; Gakkhar, S., Dynamics in a beddington-deangelis prey-predator system with impulsive harvesting, Ecol. Model., 206, 421-430, (2007) [12] Tang, S.Y.; Chen, L.S., The effect of seasonal harvesting on stage-structured population models, J. Math. Biol., 48, 357-374, (2004) \u00b7 Zbl\u00a01058.92051 [13] Zhang, X.A.; Chen, L.S.; Neumann, A.U., The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168, 201-210, (2000) \u00b7 Zbl\u00a01252.70020 [14] Zeng, G.Z.; Chen, L.S.; Sun, L.H., Existence of periodic solution of order one of planar impulsive autonomous system, J. Comput. Appl. 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Vie, 321, 641-648, (1998) [20] St\u00e8phane, G.C.; Philippe, R.; Christian, F.; Benjamin, D.M.; David, A.D., Acoustical monitoring of fish density, behavior, and growth rate in a tank, Aquaculture, 251, 314-323, (2006) [21] Chen, L.S., Pest control and geometric theory of semi-continuous dynamical system, J. Beihua Univ., 12, 1-9, (2011) [22] Chen, G.Q., New approach to prove the nonexistence of limit cycle and its application, Acta Math. Sin., 20, 281-284, (1977) \u00b7 Zbl\u00a00417.34056 [23] Qu, Y.; Wei, J.J., Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dyn., 49, 285-294, (2007) \u00b7 Zbl\u00a01176.92056 [24] Wang, J.N.; Jiang, W.H., Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dyn., 69, 1541-1558, (2012) \u00b7 Zbl\u00a01263.34063\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-06-21 01:43:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6483436226844788, \"perplexity\": 12242.753941222538}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488259200.84\/warc\/CC-MAIN-20210620235118-20210621025118-00123.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.nature.com\/articles\/s41524-022-00751-5?error=cookies_not_supported&code=67ec2774-eefd-4e8e-8d22-abd6649a9bfa","text":"## Introduction\n\nIn the traditional scientific discovery process, prior knowledge from first principles and empirical laws are combined with experimental data and intuition to yield governing equations. Newton\u2019s law of gravitation1, Einstein\u2019s mass-energy equivalence equation2, Kepler\u2019s laws of planetary motion3, and other physical principles were uncovered through careful interpretation of experimental data and inductive reasoning4. The approach of fitting experimental data through regression is difficult with systems that are yet to be understood fully\u2014the set of feasible equations capturing the physics is enormous5,6,7.\n\nOne such area where underlying physics is often poorly understood is the study of materials under environmental stress. For example, alloys8,9, polymers10, doped silicon11, and hybrid materials12 experience changes at elevated temperatures. The degradation pathways can be complex and not directly obvious when examining the experimental data. Machine learning (ML) has been used to predict degradation13,14,15,16,17 as well as to optimize process conditions to reduce material decomposition17,18. However, traditional data-science methods yield little insight into the underlying mechanisms. We posit that hidden in the black-box ML models is valuable scientific information on the dynamics of the system. If uncovered, the knowledge of the governing dynamics can serve as foundation for physical interpretation of phenomena and scientific discovery.\n\nHerein, we use Scientific ML, which combines regression-based ML with sparsity generating techniques in order to automatically identify governing equations directly from data, especially when the systems being studied are too complicated to yield to traditional theoretical analysis. Not only does Scientific ML help us understand the underlying scientific phenomena better, it also has the potential to help to make simulations faster and extrapolate beyond the dataset at hand.\n\nRecently, many approaches aiming for this target have been presented in literature. A method that we apply in this contribution is PDE-FIND by Rudy et al.19. This method is used for the discovery of physical laws describing dynamical systems. First, a library of potential candidate functions is built. Differentials are calculated by finite difference or polynomial interpolation. Once a large matrix with all candidate functions is composed, different sparse regression methods may be used to extract the partial differential equation (PDE) describing the system. The sparse methods implemented are sequential threshold ridge regression, lasso regression, elastic net regression, and greedy algorithm. Another sparse technique is Sparse Identification of nonlinear Dynamics (SINDy)20. It uses a custom deep autoencoder to find a coordinate system in which the dynamics of the system are sparse, and then uses sparse regression to find the governing equations in the associated coordinate system. Atkinson et al.21 present a generalized method for the discovery of differential equations using genetic programming. Physics Informed Neural Networks (PINN)22 and PDE-NET23,24 are deep learning methodologies to extract governing partial differential equations using dynamical data. These methods have shown great promise in several applications25,26,27,28. The automatic discovery of scientific laws and principles is at the frontier of machine learning that awaits application to materials science29 and other domains30,31,32.\n\nHalide perovskite materials, which have potential to provide high performing and cost-effective solar energy, degrade at elevated temperature33,34,35,36,37,38, humidity39,40,41, and illumination42,43,44. This is a major issue hindering the commercialization of perovskite photovoltaic technology. However, the degradation mechanisms affecting halide perovskites are not well understood. Discovering the underlying equations directly from perovskite degradation data could accelerate the development of stable perovskite solar cells. Herein, we apply Scientific ML to study the environmental degradation of methylammonium lead iodide (MAPI).\n\nFrom prior knowledge in the literature, MAPI has multiple documented reaction pathways, including decomposition to PbI2 via reaction35:\n\n$${\\rm{MAPbI}}_3 \\to {\\rm{PbI}}_2 + \\left[ {{\\rm{CH}}_3{\\rm{NH}}_3^ + + {\\rm{I}}^ - } \\right] \\to {\\rm{PbI}}_2 + {\\rm{CH}}_3{\\rm{NH}}_2 + {\\rm{HI}}$$\n(1)\n\nSmecca et al.36 demonstrate that the rate of MAPI degradation obeys an Arrhenius-type law. Their data suggest that the degradation of MAPI follows zero-order kinetics in the presence of moisture and first-order kinetics in vacuum at temperatures ranging from 90 to 135\u2009\u00b0C. Bastos et al.45 hypothesize that the thermal degradation of MAPI is defined by the Avrami equation46,47 of nucleation and growth. The Avrami equation has also been used to describe degradation kinetics in humid air48. Recently, studies have shown that halide perovskite degradation follows autocatalytic reaction kinetics49 with the hypothesis that the degradation is propagated by iodine vapors50. The derivation of exact kinetics through first principles as well as Arrhenius-type dependence is difficult because of the complexity of MAPI decomposition, despite the availability of well-resolved dynamical data, inviting the application of Scientific ML.\n\nIn this study, we focus on the application of PDE-FIND to perovskite degradation data. We choose PDE-FIND as it is an interpretable method that provides a parsimonious description of the dynamics with the flexibility to apply domain expertise for library selection. Successfully identifying governing differential equations directly from the experimental aging test data would deepen the understanding of thermal degradation and provide tools for reliable lifetime prediction of perovskite solar cells as well as the determination of acceleration factors for long-term aging tests. These developments could spur the advancement of the perovskite photovoltaic technology and have been called for by the community51,52,53. This study provides a generalizable pathway to identify degradation modes in other materials research domains as well.\n\nThe objectives of this study are two-fold, as illustrated by the workflows shown in Fig. 1: Uncover the underlying differential equation corresponding to perovskite degradation using sparse regression methodology PDE-FIND (Workflow (1)) and quantify the effect of noise on the accuracy of extraction of differential equations by PDE-FIND by comparing noiseless and noisy simulated data (Workflow (2)).\n\nFurther details can be found in the \u201cMethods\u201d section; a summary is provided here. To generate the experimental data, we subjected 206 thin-film samples of methylammonium lead iodide (MAPI) to 0.15\u2009\u00b1\u20090.01 Sun illumination, 20\u2009\u00b1\u20095% relative humidity, and temperatures varying from 35 to 85\u2009\u00b0C in our in-house environmental chamber described in detail in ref. 18 (Fig. 2b). A camera is used to monitor color changes versus time; the red color time-series is chosen for further analysis because two studies18,54 have shown a clear correlation between film color and device performance, in the limit of MAPI composition, given one of the principal degradation products is yellow PbI2. One hundred and eight samples were grown under low-variance conditions (labeled \u2018low-variance experimental\u2019, quantified in the \u201cResults\u201d section); 98 samples were grown under high-variance conditions (labeled \u2018high-variance experimental\u2019) (Supplementary Fig. 2), referring to the amount of variance (color change versus time) between samples synthesized and degraded under ostensibly identical conditions. Unless specified otherwise, we assume \u2018experimental\u2019 data in this paper refers to the low-variance sample set.\n\n## Results\n\n### Results on experimental data\n\nOur aim is to obtain the equation that most accurately describes the environmental degradation of methylammonium lead iodide (MAPI) as a function of time and temperature. There are two main challenges for Scientific ML in this application that are also common with many other experimental applications, especially in materials science: The function space that could in principle capture the degradation processes is enormous, complicating identification of unique equations. Furthermore, experimental data has measurement noise as well as sample-to-sample variance, making the identification of quantitative analytic descriptions even more challenging. These conditions can be optimized to some extent, but not excluded.\n\nOur experimental setup represents a typical materials science experiment: The noise in our experimental data is of the order of 0.35% for both high-variance and low-variance experimental datasets. The low value indicates that the camera measurement of degradation is optimized. The sample-to-sample variance for the \u2018low variance experimental\u2019 dataset is estimated to be 20% in relative standard deviation and the maximum mean absolute deviation is 12 units (red color values vary from 0 to 255). For the \u2018high variance experimental\u2019 dataset, variance is estimated to be 23% in relative standard deviation and the maximum mean absolute deviation is 31 units. These values are typical for spin-coated perovskite film samples that tend to have rather high variations, especially when aged.\n\nFirst, we attempt to uncover the differential equation governing perovskite degradation directly from experimental data (Workflow (1)). A simple way to analyze reaction rate orders is to fit the data to pure 0th, 1st, and 2nd order dynamics (Supplementary Fig. 3). These equations do not fit the data, showing that the environmental degradation of MAPI does not follow a simple n-th order kinetics. This motivates the use of PDE-FIND. We apply sparse regression to the whole experimental dataset with a broad function library consisting of polynomials of U up to order 5, sine and cosine of U, polynomials of t up to order 3, the square root of t, U multiplied with polynomials of t, temperature T and adjusted negative exponent of $$\\frac{1}{T}\\left( {{{{\\mathrm{exp}}}}\\left( { - \\frac{{100}}{T}} \\right)} \\right)$$. While we choose a broad set of candidate functions, the choice of function library is critical and determines the outcome of PDE-FIND (candidate function libraries considered are in Supplementary Table 2). We find that sine and cosine terms are not selected by PDE-FIND\u2014indicating as a sanity check that the algorithm correctly identifies that periodicity is not a feature of the dynamics. Polynomials of t and U times the polynomials of t, which correspond to the Avrami equation, are not included in the chosen library or assigned very small weights. To understand how well the obtained DE represents our data, we compare the derivative estimated by our DE to the numerical derivative obtained from the experimental data. While certain trends in the derivative are captured, errors exist because of the variance in our experimental data (Supplementary Fig. 4). Refinements to the approach are thus needed.\n\nWe proceed to narrow the application of PDE-FIND, by applying PDE-FIND to the averaged data at each temperature individually to extract the governing ODE. Using the averaged data helps us deal with sample-to-sample variance. Since all environmental conditions were almost identical for samples degraded at a particular temperature but aging tests of each temperature were conducted one after another (introducing differences, e.g., in sample storage times and exact equipment atmosphere), we aim to reduce the influence of variance-inducing conditions by applying PDE-FIND at each temperature separately. First, we apply PDE-FIND with a large library as described in the previous paragraph. Here too, we see that sine and cosine of U, polynomials of t and U times the polynomials of t are either removed from the library or have small coefficient values. We exclude these terms in further analysis. Then, we apply PDE-FIND with 1st to 5th order polynomial libraries. We find that with the 1st order polynomial library, PDE-FIND is unable to find an equation that fits the derivative of our data (Fig. 3a). All other libraries from 2nd order polynomial to 5th order polynomial appear to fit the derivative of our data with significant accuracy (Fig. 3a, b). When these differential equations are integrated, they have the same S-shape as our experimental data (Fig. 3c). The 2nd order polynomial library is the most minimal library that fits our data without high error. The functional form of this ODE is:\n\n$$\\frac{{dU}}{{dt}} = a_0 + a_1U + a_2U^2$$\n(2)\n\nWe also notice a trend in the values of the fitting coefficients with respect to temperature\u2014especially in the case of the 2nd order polynomial library (Fig. 3d, Supplementary Fig. 5). The slope of the curve changes between 55\u2009\u00b0C and 65\u2009\u00b0C, the temperature at which a well-known MAPI phase transition55,56 occurs. This may indicate that the phase transition affects the degradation mechanism, but is not experimentally confirmed in this work.\n\nNext, we evaluate the effect of variance on PDE extraction by comparing the above results (obtained on the low-variance experimental dataset) with the same workflow applied to the high-variance data (Supplementary Fig. 6). After averaging multiple curves (U(t)) for each temperature, the results are qualitatively similar for a constrained function library of polynomials of 2nd order\u2014the obtained coefficients have the same sign and order of magnitude (Supplementary Table 3). This indicates that PDE-FIND can fit even high-variance experimental data when appropriately averaging over multiple samples. To quantify the effect of sample-to-sample variance, we apply PDE-FIND to each curve individually. As expected, PDE-FIND extracts a large variance in coefficient values. The values of coefficients vary as much as 60% with the low variance dataset and up to 90% with the high variance datasets for T\u2009=\u200955\u2009\u00b0C.\n\n### Results on simulated data\n\nNow, we evaluate the effect of noise on PDE extraction using simulated data. We use the non-linear least-squares method to fit our experimental data to the Verhulst logistic equation57 and the Arrhenius equation, as shown in the Methods section. We produce both noise-free simulated data and simulated data with Gaussian noise (Workflow (2)) with this model.\n\nWe apply sparse regression to the simulated dataset at each temperature individually to discover the governing ODEs with libraries ranging from 2nd to 5th order polynomials. With the noise-free data, PDE-FIND\u2019s identified DEs fit the derivative as well as the data on integration of the DE with significant accuracy for libraries from 2nd order to 5th order. In the case of the 2nd order polynomial library, both the underlying differential equation and the fitting parameters are identified with significant accuracy, as shown in Fig. 4. We know that the underlying governing equation for this dataset (which we defined as $$\\frac{{dU}}{{dt}} = a_0 + a_1U + a_2U^2$$) does not have any terms higher than order two, thus the higher-order coefficients (e.g., $$a_3,a_4,a_5, \\ldots$$ of functional terms $$U^3,U^4,U^5 \\ldots$$) are equal to zero. PDE-FIND assigns small non-zero values to these functional forms, although they are not set to zero. In the case where sine and cosine are added to the library, the algorithm correctly identifies that these terms do not represent the dynamics and are set to zero exactly. The MAE between the exact numerical derivative and one estimated from the differential equation identified by PDE-FIND is of order 10\u22127 (when derivative varies from 0 to 1). This indicates that PDE-FIND works well for simulated curves with zero noise. Thus, with the candidate function library constrained to polynomials of U, PDE-FIND is able to identify the same ODE that fits the data at each temperature.\n\nWe then add varying amounts of Gaussian noise to this simulated equation at different temperatures. First, we consider the effect of varying amounts of noise at a fixed temperature of 55\u2009\u00b0C, as indicated by the black box in Fig. 4a. We add up to 5% noise, which is typical in many experimental settings. The equation identified by PDE-FIND yields an S-shaped curve similar to the noise-free simulated curve upon integration (Fig. 4d) for up to 5% noise, after which the DE identified by PDE-FIND does not seem to model the dynamics. We compare the error of estimating the parameter values in the differential equation describing the simulated data. At 5% Gaussian noise the error of the fitting parameters increases to almost 80% (Fig. 4b). The resulting integrated curve has MAE is 6 (on a color scale of 0\u2013255) relative to the \u2018ground truth\u2019 noise-free simulated curve (Fig. 4c, d). In addition, PDE-FIND is no longer able to threshold sin and cosine terms to 0, as it even fits the noise with sinusoidal pattern.\n\nWe then consider different temperatures at the same noise level. The Verhulst logistic equation model becomes increasingly steep and shifts to the left with higher temperature. PDE-FIND successfully identifies this trend. It appears that the MAE is higher for equation extraction at higher-temperature data. This could be because of noise obscuring PDE-FIND\u2019s ability to fit steeper peaks accurately.\n\n## Discussion\n\nThere remain many complex systems that have eluded quantitative analytic descriptions or even characterization of a suitable choice of variables in many disciplines such as biology, finance and materials science. With today\u2019s state-of-the art equipment, acquiring large quantities of data has never been easier. As put by Rackauckas et al.58, \u2018the well-known adage \u2018a picture is worth a thousand words\u2019 might well be \u2018a model is worth a thousand datasets.\u2019.\n\nScientific ML enables unique insights into MAPI degradation in this work. It is a promising method that can be used to uncover governing equations through data, especially when the derivation of physical laws using first principles is challenging. In our study, we demonstrate that PDE-FIND identifies an underlying rate equation for the degradation of perovskite solar cells. MAPI degradation does not follow a simple single-order reaction rate law, defined as:\n\n$$\\frac{{dU}}{{dt}} = kU^n$$\n(3)\n\nwhere, n is the order of the reaction and U is the concentration of the species. In our system, this equation does not yield a good fit for n=\u20090, 1 or 2. The S-shaped dynamics we see in our study have been reported in other studies involving MAPI degradation as well45,48,49,50. Some articles report that the degradation results from nucleation and growth of PbI2 crystals45,48, supporting the hypothesis that the kinetics follows the Johnson\u2013Mehl\u2013Avrami\u2013Kolmorgorov or simply, the Avrami equation46,47:\n\n$$\\frac{{\\partial U^\\prime }}{{\\partial t}} = a_0t^{n - 1} - a_1U^\\prime t^{n - 1}$$\n(4)\n$${U}{^\\prime}\\left( t \\right) = 1 - \\exp \\left( { - kt^n} \\right)$$\n(5)\n\nwhere,\n\n$$U^\\prime = \\frac{{U\\left( t \\right) - {{{\\mathrm{min}}}}(U)}}{{{{{\\mathrm{max}}}}(U - \\min \\left( U \\right))}}$$\n\nAnd a0, a1, n, and k are fitting constants.\n\nThe Avrami equation represents dynamics where degradation starts at nucleation spots on the film and these spots of degraded material grow radially, diffusion-limited. Some recent studies have presented an alternate hypothesis of self-propagating or autocatalytic kinetics49,50, which is described by another differential equation, the logistic function (discussed in the \u201cMethods\u201d section Eqs. (7), (8)). In this study, we build a large library of candidate terms for the DE\u2013 polynomials of U, that make up the logistic function, and polynomials of t and U multiplied with polynomials of t, which feature in the Avrami equation. PDE-FIND determines that the simplest ODE that fits our experimental dataset best is of the form (Fig. 3),\n\n$$\\frac{{dU}}{{dt}} = a_0 + a_1U + a_2U^2$$\n(6)\n\nthe Verhulst logistic function. This equation indicates that the reaction is first, propelled forward by the presence of the reactant as well as the product, leading to a rapid growth in the product that eventually saturates when it exhausts its reactants\u2014a self-propagating reaction. While the Avrami equation (nucleation and growth) model is limited by the rate at which the reactant diffuses to the reaction site, the Verhulst logistic function (self-propelling kinetics) model is not limited by this because the reactant is already present at the reaction site. This is why we chose the logistic function model for the simulated dataset over the Avrami equations that has been used to model nucleation-growth reactions. The algorithm picks terms that describe self-propelling kinetics (2nd order polynomial library) as opposed to diffusion-limited nucleation and growth (Avrami equation). When visually inspecting the videos of degrading films, without the benefit of PDE-FIND, it can be hard to identify the underlying mechanism. In the example shown in Supplementary Fig. 7 [and Supplementary Video], one can see light areas of degraded material in the middle of the film degradation.\n\nEquation (6) also offers insights that could help engineer more stable MAPI films. Once the degradation has begun, the autocatalytic nature suggests that degradation will continue, as the reaction products catalyze further MAPI degradation. Therefore, suppressing degradation means delaying the creation of the first reaction products for as long as possible. To engineer more stable MAPI films, this equation suggests that reducing MAPI degradation may be possible by reducing the density of nucleation points inside the material, including, e.g., by ensuring that all PbI2 precursors are fully converted during film formation, and possibly by using highly purified (i.e., devoid of contaminant particles) reagents in the film and adjacent layers that could nucleate PbI2.\n\nThese insights bear consequence for researchers attempting to identify the underlying root cause(s) of perovskite degradation, as well as those modeling or predicting the (accelerated) degradation of these materials. If indeed this is a nucleation and growth phenomenon, little can be done to halt the growth of degraded regions once the initial nucleation event occurs. Therefore, to improve phase stability of perovskite films, an emphasis can be placed on identifying the nucleation points of these phase transformations, and inhibiting them, perhaps through improved precursor purification to remove impurities, improved control of the nucleation process, improved processing to remove growth catalysts, and improved packaging to prevent ingress of exogenous gasses. Changes to the film composition may increase the nucleation energy barrier; therefore, further investigation of stoichiometry optimization may be warranted in combination with the above.\n\nWe demonstrate the application of a Scientific ML tool, PDE-FIND on MAPI degradation data. When applied to experimental data, PDE-FIND identifies a differential equation that fits the data, when appropriate constraints are applied. In spite of the noise and variance in the dataset, only functions corresponding to the dynamics of the system are picked and the DEs show good agreement with the numerical derivatives. Our \u2018robustness analysis\u2019 with simulated data shows that PDE-FIND with a 2nd order polynomial library succeeds at identifying the differential equation describing the simulated data when up to 5% Gaussian noise is added. However, the error of the fitting parameters increases with noise, to almost 80%. With 5% noise, the resulting integrated curve has a 6 MAE relative to the underlying noise-free simulated curve but the coefficients differ by as much as 80%. With the addition of noise, PDE-FIND is unable to eliminate terms not in the DE (sine and cosine) and even fits the noise with these terms. Applying a de-noising filter may allow for higher levels of noise in the data, assuming that an appropriate filter is chosen.\n\nScientific ML methods can be immensely useful at uncovering governing equations of dynamical systems, if the data obtained has low noise or can be denoised by noise-reduction techniques. Data obtained through experiments is not devoid of measurement noise and de-noising the data adequately can be challenging. In addition, certain operating conditions cannot be fully controlled, leading to sample-to-sample variance making it hard to get rid of. Our contribution motivates the development of Scientific ML techniques that are more robust to noise as well as variance in data. Scientific ML, in its current state, is well-suited to be applied to domains where obtaining large quantities of low-noise data is possible, and will find more applications with methods that are robust to noise.\n\nWe show that Scientific ML has the potential to accelerate the understanding of materials degradation and the reliability optimization of perovskite materials. Extracting physical laws may facilitate the definition of acceleration factors for aging tests and also help in the prediction of perovskite solar cell degradation under varying environmental conditions. Not only does scientific machine learning aide us with understanding the underlying scientific phenomena better, it may also enable faster simulations and better extrapolations beyond our experimental datasets. The conclusions of any given materials study may well be rendered more generalizable by identifying underlying equations governing the observations.\n\n## Methods\n\n### Data collection\n\nFor the experimental portion of our study (Workflow (1) in Fig. 1), the input is the experimental data obtained from degrading MAPI films. MAPI film synthesis conditions follow those of the Materials subsection of the Experimental procedures section of ref. 18 (More information in Supplementary Methods). Our experimental data is shown in Fig. 2.\n\nWe monitored the degradation of MAPI based on the color change of the material. As MAPI films decompose, they change their color from initial black (majority MAPI) to degraded yellow (minority MAPI). We acquired images of the degrading films with 0.5-min temporal resolution and processed them to obtain the average red, blue and green color components of the films as a function of time (Fig. 2a, Supplementary Fig. 1). There are limits to the use of film color as a proxy; for example, in mixed perovskites, degradation may proceed via phase de-mixing into pure phases that may be dark in color, and camera-based imaging technique should be modified or complemented with other metrology, e.g., X-ray diffraction18.\n\nFor the study of noise robustness (Workflow (2) in Fig. 1), we generate simulated degradation data to analyze how noise obfuscates the identification of underlying DEs. We apply a non-linear least-squares method to fit the experimental data (e.g., those shown in Fig. 2d) to the Verhulst logistic equation57 to model the S-shaped curve. This is a reasonable assumption because the logistic function is used to describe the thermal decomposition dynamics of several materials49,50,59. We obtain,\n\n$$U = M + \\frac{{U_{{{\\mathrm{o}}}}Ke^{kt}}}{{\\left( {K - U_{{{\\mathrm{o}}}}} \\right) + U_{{{\\mathrm{o}}}}e^{kt}}},$$\n(7)\n$$\\frac{{\\partial U}}{{\\partial t}} = k(U - M)\\left( {1 - \\frac{{(U - M)}}{K}} \\right)$$\n(8)\n\nwhere Uo is the initial concentration, k is growth rate, K is the carrying capacity and M is a fitting constant. In the context of MAPI degradation, M, Uo, and K can be considered as fitting parameters. The growth rate k varies with temperature according to the Arrhenius equation:\n\n$$k = Ae^{\\left( { - \\frac{{E_{{{\\mathrm{a}}}}}}{{RT}}} \\right)}$$\n(9)\n\nhere, Ea is the activation energy, T is the temperature in Kelvin, A is the pre-exponential factor and R is the universal gas constant. We use this model to produce noise-free simulated data (labeled \u2018simulated\u2019) and simulated data with Gaussian noise (labeled \u2018simulated with Gaussian noise\u2019).\n\n### Data analysis\n\nFirst, we apply the sparse regression methodology PDE-FIND19 to experimental data (Workflow (1)). We use the time-series from all the temperatures to infer the partial differential equation (PDE) defining the relationship between MAPI degradation, temperature, and time. Then, we apply PDE-FIND to the time-dependent degradation data at each temperature, to infer the ordinary differential equation (ODE) that describes MAPI decomposition at a particular temperature. To study the effect of noise, we apply PDE-FIND to simulated data with and without Gaussian noise (Workflow (2)).\n\nThe library of potential candidate functions consists of polynomials of U, polynomials of time t, sine and cosine of U, temperature T, and other non-linear functions of U, t, and T (Supplementary Table 1). Differentials are calculated by finite difference with convolutional smoothing using a 1D Gaussian kernel. Once a large tall matrix (\u0398(U)) with all candidate functions is composed, we use sequential threshold ridge regression to identify which terms contribute to the dynamics described by the data as well as those terms\u2019 weights. The goal of this method is to find a sparse coefficient vector \u03b2 that only consists of the active features that best represent the time derivative $$U_t$$. The rest of the features are hard-thresholded to zero. The loss functions are follows (\u03bb2 and \u03bb0 are the L-2 and L-0 regularization penalties, respectively, more details can be found in the supplementary information of ref. 19):\n\n$$\\hat \\beta = \\arg \\mathop {{\\min }}\\limits_\\beta \\left( {\\left\\| {{\\Theta}\\left( U \\right)\\beta - U_{{{\\mathrm{t}}}}} \\right\\|_2 + \\lambda _2\\left\\| \\beta \\right\\|_2} \\right)$$\n(10)\n\nfor a given $$\\widehat {{{{\\mathrm{tol}}}}}$$, where $$\\widehat {{{{\\mathrm{tol}}}}}$$ is:\n\n$$\\widehat {{{{\\mathrm{tol}}}}} = \\arg \\mathop {{\\min }}\\limits_{{{{\\mathrm{tol}}}}} \\left( {\\left\\| {{\\Theta}\\left( U \\right)\\beta - U_{{{\\mathrm{t}}}}} \\right\\|_2 + \\lambda _0\\left\\| \\beta \\right\\|_0} \\right)$$\n(11)","date":"2023-03-29 16:33:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.528876006603241, \"perplexity\": 1215.0946757938614}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949009.11\/warc\/CC-MAIN-20230329151629-20230329181629-00022.warc.gz\"}"} | null | null |
Port Forges Ahead at WTC
NEW YORK CITY-The Port Authority's board approved to sell additional bonds for Ground Zero redevelopment and to retain Cushman & Wakefield for advisory services.
By Cody Lyon | November 20, 2009 at 06:11 AM
NEW YORK CITY-Board members questioned construction cost increases at public works projects and a monthly retaining fee for real estate firm Cushman & Wakefield on the World Trade Center at the monthly meeting of the Port Authority of New York and New Jersey Thursday. But, despite a couple of tough volleys, and remarks that this "is not a happy meeting" by one board member, authorizations and re-authorizations were approved. They included the issuance and sale of up to $1 billion in additional consolidated bonds and notes for capital expenditures in connection with the WTC redevelopment.
Board members also voted to retain C&W for its real estate and financial advisory services. A PANYNJ spokesman tells GlobeSt.com these include acting in an advisory capacity in the authority's current arbitration process with Silverstein Properties Inc. over development at Ground Zero, as well as negotiating a financial agreement with SPI in the future.
Specifically, the board gave the executive director authorization to increase, by up to $2 million, under an existing agreement, bringing the total amount authorized for the services to $4 million.
Despite one assertion that "real estate brokers aren't all that busy" and neither are "bankers," board members were directed to the future's "myriad of possibilities" and anticipated "joint venture relationships" at the WTC.
WTC project manager Steven Platt announced that there had been tremendous progress at One World Trade Center, a.k.a the Freedom Tower. He said the structure would soon rise 160 feet above the ground, and that by the end of January 2010, the building would be visible to all New Yorkers as it reaches the 20th floor.
Meanwhile, just across the street from the rising steel in Battery Park City sits the gleaming new 43-floor headquarters of Goldman Sachs. The Liberty Bond-financed, two-million-square-foot building broke ground Nov. 29, 2005, while the Freedom Tower's construction started just five months later on April 27, 2006. A Goldman spokeswoman confirmed to GlobeSt.com that the investment bank has in fact already moved in several of its employees.
Saying "we have no choice," or we'll "end up with an unfinished building," board members also voted to reauthorize work on the consolidated Police Crisis Command Center and Aircraft Rescue and Firefighting facility at LaGuardia Airport.
According to PANYNJ documents, the board had originally authorized the design and construction of the ARFF at an estimated project cost of $62.6 million. But at Thursday's meeting, board members were told the project's cost had increased to $74.3 million.
The Port says there was a general increase in construction costs, combined with "complexities in the coordination of the communication, security and technology systems with the building design" which it says "presented unique challenges, which resulted in additional planning and engineering costs beyond the original project budget." Port documents also say "significant staff time was required to configure and coordinate a complex network of security, technology and communication systems, including those used to interact with operations staff and federal state and local agencies."
Also during Thursday's meeting, which was webcast, board members complained they'd "not received updated information" along the way, that led to what appeared to be the day's unwelcome surprise at the airport project.
The new facility, set to open in mid-2010, will serve as headquarters for the police force the Port says is needed at La Guardia and will consolidate police and ARFF functions at a single location. Later, during a reporter Q&A session, executive director Chris Ward said the authority had found itself in a place and time where it would need to interface with the likes of the FBI and that had led to the requirement of a more complex building than originally anticipated. In response to quizzing, Ward said "costs have risen on us."
Also demanding a larger chunk of cash, the years-long rehabilitation of the 14th Street exit roadway and Jersey Avenue just outside the Holland Tunnel in Jersey City. That project was originally authorized in 1998 when the board had signed off on a rehab of the roadway, drainage, lighting, traffic signs and signals for $21.8 million. Although 98% done, in order to accommodate additional engineering design, construction management costs and re-design work related to the relocation of utilities and "private property" issues, the cost has increased to $26 million.
The board also gave a green light to $2.5 million for the demolition and relocation of two vehicle bridges in the aeronautical area at Kennedy International Airport along with $2 million for the replacement and upgrade of PATH's substation number eight. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,321 |
/* TEMPLATE GENERATED TESTCASE FILE
Filename: CWE400_Resource_Exhaustion__fscanf_for_loop_13.c
Label Definition File: CWE400_Resource_Exhaustion.label.xml
Template File: sources-sinks-13.tmpl.c
*/
/*
* @description
* CWE: 400 Resource Exhaustion
* BadSource: fscanf Read data from the console using fscanf()
* GoodSource: Assign count to be a relatively small number
* Sinks: for_loop
* GoodSink: Validate count before using it as the loop variant in a for loop
* BadSink : Use count as the loop variant in a for loop
* Flow Variant: 13 Control flow: if(GLOBAL_CONST_FIVE==5) and if(GLOBAL_CONST_FIVE!=5)
*
* */
#include "std_testcase.h"
#ifndef OMITBAD
void CWE400_Resource_Exhaustion__fscanf_for_loop_13_bad()
{
int count;
/* Initialize count */
count = -1;
if(GLOBAL_CONST_FIVE==5)
{
/* POTENTIAL FLAW: Read count from the console using fscanf() */
fscanf(stdin, "%d", &count);
}
if(GLOBAL_CONST_FIVE==5)
{
{
size_t i = 0;
/* POTENTIAL FLAW: For loop using count as the loop variant and no validation */
for (i = 0; i < (size_t)count; i++)
{
printLine("Hello");
}
}
}
}
#endif /* OMITBAD */
#ifndef OMITGOOD
/* goodB2G1() - use badsource and goodsink by changing the second GLOBAL_CONST_FIVE==5 to GLOBAL_CONST_FIVE!=5 */
static void goodB2G1()
{
int count;
/* Initialize count */
count = -1;
if(GLOBAL_CONST_FIVE==5)
{
/* POTENTIAL FLAW: Read count from the console using fscanf() */
fscanf(stdin, "%d", &count);
}
if(GLOBAL_CONST_FIVE!=5)
{
/* INCIDENTAL: CWE 561 Dead Code, the code below will never run */
printLine("Benign, fixed string");
}
else
{
{
size_t i = 0;
/* FIX: Validate count before using it as the for loop variant */
if (count > 0 && count <= 20)
{
for (i = 0; i < (size_t)count; i++)
{
printLine("Hello");
}
}
}
}
}
/* goodB2G2() - use badsource and goodsink by reversing the blocks in the second if */
static void goodB2G2()
{
int count;
/* Initialize count */
count = -1;
if(GLOBAL_CONST_FIVE==5)
{
/* POTENTIAL FLAW: Read count from the console using fscanf() */
fscanf(stdin, "%d", &count);
}
if(GLOBAL_CONST_FIVE==5)
{
{
size_t i = 0;
/* FIX: Validate count before using it as the for loop variant */
if (count > 0 && count <= 20)
{
for (i = 0; i < (size_t)count; i++)
{
printLine("Hello");
}
}
}
}
}
/* goodG2B1() - use goodsource and badsink by changing the first GLOBAL_CONST_FIVE==5 to GLOBAL_CONST_FIVE!=5 */
static void goodG2B1()
{
int count;
/* Initialize count */
count = -1;
if(GLOBAL_CONST_FIVE!=5)
{
/* INCIDENTAL: CWE 561 Dead Code, the code below will never run */
printLine("Benign, fixed string");
}
else
{
/* FIX: Use a relatively small number */
count = 20;
}
if(GLOBAL_CONST_FIVE==5)
{
{
size_t i = 0;
/* POTENTIAL FLAW: For loop using count as the loop variant and no validation */
for (i = 0; i < (size_t)count; i++)
{
printLine("Hello");
}
}
}
}
/* goodG2B2() - use goodsource and badsink by reversing the blocks in the first if */
static void goodG2B2()
{
int count;
/* Initialize count */
count = -1;
if(GLOBAL_CONST_FIVE==5)
{
/* FIX: Use a relatively small number */
count = 20;
}
if(GLOBAL_CONST_FIVE==5)
{
{
size_t i = 0;
/* POTENTIAL FLAW: For loop using count as the loop variant and no validation */
for (i = 0; i < (size_t)count; i++)
{
printLine("Hello");
}
}
}
}
void CWE400_Resource_Exhaustion__fscanf_for_loop_13_good()
{
goodB2G1();
goodB2G2();
goodG2B1();
goodG2B2();
}
#endif /* OMITGOOD */
/* Below is the main(). It is only used when building this testcase on
its own for testing or for building a binary to use in testing binary
analysis tools. It is not used when compiling all the testcases as one
application, which is how source code analysis tools are tested. */
#ifdef INCLUDEMAIN
int main(int argc, char * argv[])
{
/* seed randomness */
srand( (unsigned)time(NULL) );
#ifndef OMITGOOD
printLine("Calling good()...");
CWE400_Resource_Exhaustion__fscanf_for_loop_13_good();
printLine("Finished good()");
#endif /* OMITGOOD */
#ifndef OMITBAD
printLine("Calling bad()...");
CWE400_Resource_Exhaustion__fscanf_for_loop_13_bad();
printLine("Finished bad()");
#endif /* OMITBAD */
return 0;
}
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,479 |
Fellows & Scholars
HPF Membership
Operation Warp Speed and a COVID-19 Vaccine
Kirstin R.W. Matthews, Rekha Lakshmanan
Rethinking cancer care during a pandemic
Anaeze C. Offodile II
The Impact of Covid-19 on Vulnerable Families
Quianta Moore
Human Embryo Research in the U.S.
Kirstin R.W. Matthews
Calendar of CHB Events »
Health and Biosciences Programs
Domestic Health Policy Analysis
Child Health Policy
Fellows & Scholars »
Quianta Moore, M.D., J.D., is the fellow in child health policy at the Baker Institute for Public Policy. Her research focuses on …
Read more about this expert
Elena M. Marks
Elena M. Marks, J.D., M.P.H., is the president and chief executive officer of the Episcopal Health Foundation, a $1.3 billion nonprofit based in Houston, Texas. …
Vivian Ho
Vivian Ho, Ph.D., is the James A. Baker III Institute Chair in Health Economics, director of the Center for Health and Biosciences, a professor in …
Neal F. Lane
Neal F. Lane, Ph.D., is the senior fellow in science and technology policy at the Baker Institute. He is also the Professor of Physics and …
Maude Rowland Cuchiara
Maude Rowland Cuchiara, Ph.D., is a nonresident scholar for the Science and Technology Policy Program. Cuchiara actively works with the International Stem Cell Policy …
Hagop M. Kantarjian, M.D.
Hagop M. Kantarjian, M.D., is a nonresident fellow in health policy. He serves as a professor and chair of the Department of Leukemia at The …
Peter J. Hotez
Peter J. Hotez, M.D., Ph.D., is the Baker Institute fellow in disease and poverty. He is dean of the National School of …
Marah Short
Marah Short is the associate director of the Center for Health and Biosciences at Rice University's Baker Institute for Public Policy. Her research primarily examines …
Deepak Srivastava, M.D.
Deepak Srivastava, M.D., is the Baker Institute Nonresident Scholar for Biomedical Research Policy. He is the Younger Family Director and senior investigator at the Gladstone …
Kirstin R.W. Matthews, Ph.D., is a fellow in science and technology policy at Rice University's Baker Institute for Public Policy and a lecturer in the …
Robert Bazell
Robert Bazell is the nonresident fellow in science and technology policy at the Baker Institute, as well as an adjunct professor in the Department of …
CHB In the News
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The CDC ranks Texas 48th out of the 50 states when it comes to vaccines distributed per 100 people — a disturbing low, said health economics fellow Vivian Ho. "There needs to be good decision-making ... at the state level to determine who gets how much vaccine." KPRC-TV News. | Jan. 15, 2021, 5:21 p.m.
The influence of the anti-vaccine movement
In an interview with the New Yorker, health policy fellow Peter Hotez discusses vaccine skepticism, the politicization of vaccines and how the government can effectively convey public health information. | Dec. 21, 2020, 11:35 a.m.
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"Vaccines are coming," said Peter Hotez, fellow in disease and poverty, during an interview on NPR. "We have to get everybody through to the other side." For more on vaccine development, the anti-vax movement and what the year ahead might look like, listen to the interview here: https://n.pr/2HAQjYG | Nov. 25, 2020, 2:16 p.m.
COVID-19 and youth suicides
"Though we cannot yet definitively say whether the pandemic is associated with an increase in suicide attempts, available evidence suggests that we should not rule it out," writes health policy researcher Patrick Tennant, in a new post for the Baker Institute Blog: https://bit.ly/3knXhOa | Nov. 11, 2020, 11:45 a.m. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,760 |
Q: GPG Encryption fails intermittently on same command - results in no such file or directory The following command is run from the Windows command line and it works sometimes, but it does not in other occasions.
GPG --recipient "my.puclic.key@recipient.com" --output "MyEncryptedFileName.txt.PGP" --encrypt "MyTestDocument.txt Working Directory: \\myServer\myfolderName\
The directory and file name exist, but it seems like GPG can't find them. I have also tried the command as..
GPG --recipient "my.puclic.key@recipient.com" --output "MyEncryptedFileName.txt.PGP" --encrypt "MyTestDocument.txt Working Directory: \\myServer\myfolderName\"
and
GPG --recipient "my.puclic.key@recipient.com" --output "MyEncryptedFileName.txt.PGP" --encrypt "MyTestDocument.txt Working Directory: \\myServer\myfolderName"
but keep getting an error:
"can't open 'MyTestDocument.txt Working Directory:\\myServer\myfolderName\': No such file or directory
gpg MyTestDocument.txt Working Directory: \\myServer\myfolderName\: encryption failed: No such file or directory
In prior occasions this same command worked fine.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 568 |
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