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{"url":"http:\/\/www.zentralblatt-math.org\/zmath\/en\/advanced\/?q=an:0895.42005","text":"Language: \u00a0 Search: \u00a0 Contact\nZentralblatt MATH has released its new interface!\nFor an improved author identification, see the new author database of ZBMATH.\n\nQuery:\nFill in the form and click \u00bbSearch\u00ab...\nFormat:\nDisplay: entries per page entries\nZbl 0895.42005\nLu, Shanzhen; Yang, Dachun; Zhou, Zusheng\nSublinear operators with rough kernel on generalized Morrey spaces.\n(English)\n[J] Hokkaido Math. J. 27, No.1, 219-232 (1998). ISSN 0385-4035\n\nLet $\\phi$ be a positive and increasing function on $(0, \\infty)$, satisfying $\\phi(2r)\\le D \\phi(r)$ $(r>0)$, where $D\\ge 1$ is a constant independent of $r$. For $1\\le p<\\infty$, one denotes by $L^{p,\\phi}(\\Bbb R^n)$ the space of locally integrable functions $f$ for which $\\int_{B_r(x_0)}| f(x)| ^p dx \\le C^p \\phi(r)$ for all $x_0\\in \\Bbb R^n$ and every $r>0$, where $B_r(x_0)=\\{x\\in \\Bbb R^n; | x-x_0| \\le r\\}$. These spaces are called the generalized Morrey spaces. The authors give the following: Let $1\\le p<\\infty$, $1\\le D<2^n$ and $\\gamma=\\log 2^n\/\\log D$. If a sublinear operator $T$ is bounded on $L^p(\\Bbb R^n)$ and for any $f\\in L^1(\\Bbb R^n)$ with compact support and satisfying $$| Tf(x)| \\le C\\int_{\\Bbb R^n} | \\Omega(x-y)| | x-y| ^{-n}| f(y)| dy\\tag *$$ for $x\\notin \\text{supp} f$, where $\\Omega$ is homogeneous of degree zero and $\\Omega\\in L^q(S^{n-1})$ for some $q\\ge p\/(p-1)$ or some $q>\\min\\{p, \\gamma\/(\\gamma-1)\\}$, then $T$ is also bounded on $L^{p,\\phi}(\\Bbb R^n)$. Similarly $L^{p,\\phi}(\\Bbb R^n)$ boundedness is discussed for the commutators $[a,T]$ with $a\\in \\text{BMO}(\\Bbb R^n)$ and a linear $T$ satisfying $(\\ast)$. Like as in Riesz operators, they also discuss similar results for the operators $T$ satisfying $$| Tf(x)| \\le C\\int_{\\Bbb R^n}| \\Omega(x-y)| | x-y| ^{\\alpha-n}| f(y)| dy, \\tag**$$ concerning $L^{p,\\phi}$-$L^{p,\\phi^{q\/p}}$ boundedness, where $0<\\alpha<n$, $1<p<n\/\\alpha$ and $1\/q=1\/p-\\alpha\/n$. These results are extensions of the corresponding ones by {\\it T. Mizuhara} (singular integral operator case) [Harmonic analysis, Proc. Conf., Sendai\/Jap. 1990, ICM-90 Satell Conf. Proc., 183-189 (1991; Zbl 0771.42007)] and {\\it E. Nakai} (Riesz operator case) [Math. Nachr. 166, 95-103 (1994; Zbl 0837.42008)].\n[K.Yabuta (Nara)]\nMSC 2000:\n*42B20 Singular integrals, several variables\n42B25 Maximal functions\n42B30 Hp-spaces (Fourier analysis)\n\nKeywords: sublinear operator; Calder\u00f3n-Zygmund kernel; Morrey space; commutator; BMO\n\nCitations: Zbl 0837.42008; Zbl 0771.42007\n\nCited in: Zbl 0969.42009\n\nHighlights\nMaster Server","date":"2013-06-19 11:59:09","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\": 0.9704537391662598, \"perplexity\": 1125.3066558182447}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2013-20\/segments\/1368708739983\/warc\/CC-MAIN-20130516125219-00098-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
Die Altstadt Rigas (lettisch: Vecrīga (Alt-Riga) oder Vecpilsēta (Alte Stadt)) ist das historische Zentrum. Das Stadtviertel bildet zusammen mit der Neustadt (Centrs (Zentrum) genannt) den Centra rajons (Zentralbezirk) der lettischen Hauptstadt, der UNESCO-Welterbe ist.
Alt-Riga liegt an der Ostseite der Düna (lettisch: Daugava) und ist bekannt für seine Kirchen und Kathedralen, wie den Dom zu Riga und die Petrikirche. Alt- und Neustadt werden durch den Freiheitsboulevard (am Freiheitsdenkmal) verbunden.
Ursprünge
Die heutige Altstadt umfasst das ursprüngliche Areal Rigas in seinen historischen Grenzen, bevor der Umfang im späten 19. Jahrhundert stark vergrößert wurde. Die eigentliche Stadt wurde im Jahre 1201 gegründet. Sie war Sitz eines Erzbistums, wurde eine immer bedeutendere Handelsstadt der Hanse und etablierte sich als Hauptstadt Livlands (lateinisch: Livonia). In den ersten Jahrzehnten seines Bestehens wuchs Riga beachtlich an, die bebaute Fläche wuchs innerhalb von 30 Jahren um etwa das fünf- bis sechsfache.
Festungsbau
Schon früh existierte zur Verteidigung die Rigaer Stadtbefestigung, jene gab die Ausdehnung der Stadt auf Jahrhunderte vor. Erstmals 1207 urkundlich erwähnt ist die alte Stadtmauer, welche bis in das 14. Jahrhundert mehrmals erweitert wurde. Das Rigaer Schloss wurde in der 1. Hälfte des 14. Jahrhunderts als Feste für den livländischen Orden nächst der Stadt neuerbaut. Ab 1537 wurden, bis auf den Uferbereich der Düna, im Vorfeld neue Bollwerke errichtet. Es folgte im 17. Jahrhundert der Ausbau zur Festung, u. a. mit Zitadelle nördlich des Schlosses und breitem Graben. Für die Erweiterung und Vergrößerung der Anlagen mussten Vororte im Umfeld niedergerissen werden. Auch das Schloss wurde mehrfach umgestaltet, vor allem im 18. und 19. Jahrhundert.
Erweiterung
Nachdem Riga den Status als Festung 1857 verloren hatte, ergab sich die Möglichkeit zur anderweitigen Nutzung der Flächen. Man begann mit dem Schleifen der Wälle und füllte den Festungsgraben mit Wasser des Dünaflusses. Es entstand ein Kanal rund um den Stadtkern, die nunmehrige Altstadt. Durch Eingliederung naher Vorstädte wuchs Riga weiter an.
UNESCO-Welterbe
Nach Wiedererlangung der Unabhängigkeit Lettlands Anfang der 1990er Jahre wurden nahezu alle der opulent verzierten Gebäudefassaden Vecrīgas restauriert. Die Straßen wurden für den allgemeinen Verkehr gesperrt, nur Anwohner und Lieferanten ist es erlaubt die Altstadt zu befahren. Vecrīga ist seit 1997 Teil des UNESCO-Welterbes, gelistet als "Historic Centre of Riga".
Gebäude in Vecrīga
Literatur
Andris Kolbergs: Riga, Portrait einer Stadt. Riga 1999, ISBN 998-407-142-1.
Weblinks
Multimedia virtual tour of Old Riga (englisch)
Einzelnachweise
Stadtteil von Riga
|
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Logical definition and computational complexity are intimately
intertwined, as a wide literature witnesses since the seminal theorem by
Fagin \cite{Fagin}. \textit{Fagin's Theorem} states that
\begin{equation}
SO \exists = NP
\end{equation}
that is, the class of functions computable in polynomial time by
a non-deterministic Turing machine is identical to the class of
sets definable by formulas in second order existential logic. Many
other results in the same spirit of descriptive complexity or finite
model theory have
since been proven, as summarized in, for example, \cite{Baldwin}, \cite{Borger},
\cite{Immerman}, \cite{Flum}, \cite{Libkin}.
At the same time we have \textit{Cook's Theorem} \cite{Cook}
that states that the satisfiability problem of mere "zeroth
order" propositional logic is complete for NP. Cook's Theorem
states that for any halting computation of a given Turing machine,
deterministic or non-deterministic, there exists a propositional
formula - indeed even one in conjunctive normal form - that is
satisfiable if and only if the corresponding truth assignment to
its free variables defines a halting computation of the given
Turing machine. Moreover, this propositional formula is of length
less than $\mathcal{O} (t^3)$ in the number of time steps $t$ taken by
this computation \cite{Salomaa}.
Let us present
propositional formulas in Conjunctive Normal Form, using the encoding
introduced by Immerman in \cite{Immerman}, p. 114. In this encoding, an instance of the
propositional satisfiability problem SAT is defined by two relations over a pair of variables, namely
\begin{tabular}{lll}
$\bar{P}(w,z)$ & iff & the clause $w$ contains the positive literal $z$ \\
$\bar{N}(w,z)$ & iff & the clause $w$ contains the negative literal $\lnot z$
\end{tabular}
Let us assume propositional target formulas always to be given
in this form. For the purpose of analysis, however,
let us extend this encoding by furnishing the binary relations above with a third
variable $\phi$ that refers to a finite propositional formula presented in the CNF form above.
The domain of each of the three variables is at most countable in size.
Let us now replace the above binary relations with the two
ternary relations $P(w,z,\phi)$ and $N(w,z,\phi)$ and, following \cite{Immerman},
amend them also with the unary relation $\bar{E}(z)$,
with the following intended interpretation
\begin{tabular}{lll}
$P(w,z,\phi)$ & iff & the clause $w$ contains the positive literal $z$ in the CNF encoding\\
&& of the propositional formula $\phi$\\
$N(w,z,\phi)$ & iff & the clause $w$ contains the negative literal $\lnot z$ in the CNF encoding\\
&& of the propositional formula $\phi$\\
$\bar{E}(z)$ & iff & the propositional variable $z$ has been assigned the value 1, i.e. \bf{true}\\
\end{tabular}
The corresponding propositional formula $\phi$ is satisfiable iff the
following second order existential formula is true:
\begin{equation}
\label{SATSOE}
\exists \bar{E} \forall w \exists z (P(w,z,\phi) \land \bar{E}(z)) \lor (N(w,z,\phi) \land \lnot \bar{E}(z))
\end{equation}
\section{Satisfiability of bounded fragments of propositional logic}
Let us study the satisfiability of propositional formulas in CNF, with the encoding introduced
in the previous section. Let us denote an effective binary representation
of such an encoding of a propositional formula $\phi$ by $\theta(\phi)$.
We shall denote the satisfiability problem of propositional logic
by SAT. With the binary encoding at our disposal,
we can extend the domain of \eqref{SATSOE} to the set of all finite binary strings $\{y\}$ by
\begin{equation}
\label{SAT}
\begin{split}
\tilde{\Psi}_{\mathrm{SAT}} \leftrightarrow \\
\exists y \exists \phi \exists \bar{E} \forall w \exists z (y=\theta(\phi) \land \\
(P(w,z,\phi) \land \bar{E}(z)) \lor (N(w,z,\phi) \land \lnot \bar{E}(z))
\end{split}
\end{equation}
As an example of such an efficient binary encoding scheme $\theta(\cdot)$, let us consider the following
scheme.
\begin{enumerate}
\item The binary representation of a natural number indicating the index of a propositional variable is
encoded in every fourth bit of our input string $y$ only, starting from the fourth bit.
\item The preceding and intervening three-bit sequences are used as codes on interpreting the last bit, according to the following table
\end{enumerate}
\vspace{0.5cm}
\begin{tabular}{|l|l|}
\hline
Code & Meaning \\
\hline
000 & Next bit continues the binary representation of the index of the current \\
& propositional variable\\
001 & Next bit begins the binary representation of the index of a new propositional\\
& variable in the current clause \\
010 & Next bit represents a propositional constant: 0 for {\tt false}, 1 for {\tt true} \\
011 & Next bit is the first bit in the binary representation of the index of the first variable\\
& in a new clause\\
100 & Next bit is the first bit in the binary representation of the first variable in the first\\
& clause in the list $N$ \\
101 & End of input string \\
\hline
\end{tabular}
\vspace{0.5cm}
Let us parameterize propositional formulas $\phi$ by the length $M$ of their encoding
in this effective binary encoding scheme $\theta(\cdot)$. Let us denote the corresponding fragment
of well-formed propositional formulas of encoding length at most $M$ by $\mathrm{L}_M$ and the corresponding
satisfiability problem by $\mathrm{SAT}_M$. The domain of definition of $\mathrm{SAT}_M$ is likewise extended to the set of all finite binary strings $\{ y \}$ of length at most $M$ by the formula $\tilde{\Psi}_{\mathrm{SAT}_M}$, with $|y|$
denoting the length of the binary string $y$.
\begin{equation}
\label{SAT_M}
\begin{split}
\tilde{\Psi}_{\mathrm{SAT}_M} \leftrightarrow \\
\exists y \exists \phi \exists \bar{E} \forall w \exists z (|y| \leq M \land y=\theta(\phi) \land \\ (P(w,z,\phi) \land \bar{E}(z)) \lor (N(w,z,\phi) \land \lnot \bar{E}(z))
\end{split}
\end{equation}
The condition $|y| \leq M$ is equivalent to the requirement $y \in 2^M$.
We can restrict the domain of the first order variables in \eqref{SAT_M} to be of size $2^{2^{2M}}$.
We have now two formulas in existential second order logic that can be used to capture SAT.
A crucial difference between \eqref{SAT} and \eqref{SAT_M} is the fact that, unlike
$\tilde{\Psi}_{\mathrm{SAT}}$, $\tilde{\Psi}_{\mathrm{SAT}_M}$ can be recast as a first order formula for any fixed $M$. The
existentially quantified relation $\bar{E}(z)$ in this case ranges over a finite set of size at most $2^{M}$ only.
Hence the $SO\exists$ quantification $\exists \bar{E}$
can be replaced by a $FO$ quantification $\exists e$, after choosing a separate variable
$e$ with a domain that comprises a finite set of truth assignments.
We replace the unary relation $\bar{E}(z)$ with a binary relation $E(e,z)$ but retain the ternary relations $P(w,z,\phi)$ and $N(w,z,\phi)$ that have the following intended interpretations
\begin{tabular}{lll}
$E(e,z)$ & iff the propositional variable $z$ has the value 'true' in the\\
& truth assignment $e$ \\
$P(w,z,\phi)$ &iff the clause $w$ with a positive literal $z$ belongs to the set of positive\\
&clause-variable pairs of the propositional formula $\phi$\\
$N(w,z,\phi)$ &iff the clause $w$ with a negative literal $\lnot z$ belongs to the set of negative\\
& clause-variable pairs of the propositional formula $\phi$\\
\end{tabular}
If we set a fixed finite bound on the size of our models, as in
\eqref{SAT_M}, the set of models of such a restriction of \eqref{SAT} becomes finite and the
corresponding theory primitive recursive by exhaustive search:
\begin{equation}
\label{SATFO_M}
\begin{split}
\Psi_{\mathrm{SAT}_M} \leftrightarrow \\
\exists \phi \exists y \exists e \forall w \exists z (|y| \leq M \land y=\theta(\phi) \land \\
(P(w,z,\phi) \land E(e,z)) \lor (N(w,z,\phi) \land \lnot E(e,z)))
\end{split}
\end{equation}
We can now associate the $FO$-theory defined by $\Psi_{\mathrm{SAT}_M}$ with the bounded satisfiability problem $\mathrm{SAT}_M$, for each $M > 0$.
The variable $e$ ranges over assignments of truth values to at most $2^M$ propositional
variables. The size of the domains of $z$, $w$ and $y$ is at most $2^M$ and
the size of the domain of $\phi$ at most $2^{2^{2M}}$ and that of $e$ at most $2^{2^M}$.
We shall use \textit{italics} in $L_M$ and $SAT_M$ to indicate the
first order language and theory of this finite fragment of the propositional satisfiability problem, respectively.
We take $SAT_M$ to be closed under implication but not necessarily complete. We also allow $SAT_M$ to use finitely many first order variables up to some limit that can grow without bound with $M$. The union of all $SAT_M$'s is therefore not a first order theory, but all individual $SAT_M$'s are.
\section{Encoding deterministic Turing machines in first order logic}
Modifying the notation introduced by B\"{o}rger in \cite{Borger} to conform to a first-order, rather than
propositional, definition of an arbitrary Turing machine $T$, we first
define a program formula
that any $\mathrm{SAT}$ solving Turing machine $T$ is required to satisfy. We shall also define corresponding
input and accepting halting state formulas $y=\theta(\phi)$ and
$\omega$, respectively, and amend the program formula so that it corresponds to
a deterministic Turing machine, when appropriate.
The input formula
\begin{equation}
\label{input}
y=\theta(\phi)
\end{equation}
states that at the beginning
of the computation, the first positions to the right from the starting position of the input tape of the
Turing machine $T$ contain the binary encoding of the propositional formula
$\phi$ in the CNF encoding introduced in the Introduction. The input tape is read-only.
The input formula $y=\theta(\phi)$ implies that our Turing machine checks the syntax
of its input. Therefore it must also read all of its input.
The program formula defines time-limited
computations of an arbitrary Turing machine $T$, with a bound $b$ on the number of time steps taken. It has the form
\begin{equation}
\label{program}
\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})
\end{equation}
with variables to be described below. We shall explicitly assume
the time parameter in our formalization. Bound variables $t, t'$ stand for time steps, $u, u'$ for tape cells on the work tape
and $v, v'$ for tape cells on the input tape, $\tau$ denotes the current time step,
$\bar{t}$ the $b+1$-tuple of time steps $(t_0, \ldots , t_b)$, $\bar{u}$ the $2b+1$-tuple of
working tape cells $(u_{-b}, \ldots , u_b)$, and $\bar{v}$ the $2b+1$-tuple of input tape cells $(v_{-b}, \ldots , v_b)$.
The variables $o_{\bar{t}},o_{\bar{u}}$ and $o_{\bar{v}}$ denote permutations of the elements
that have been ordered in the vectors $\bar{t},\bar{u}$ and $\bar{v}$, respectively.
For arbitrarily large finite models, linear order is not $FO$-definable.
However, a total order in a finite set of bounded size can be defined in $FO$ with a
relation $\bar{O}(u,u')$ that satisfies the definition
\begin{equation}
\label{Order}
\forall u,u' (\lnot(\bar{O}(u,u') \land \bar{O}(u',u)) \land (\bar{O}(u,u') \lor \bar{O}(u',u) \lor u=u'))
\end{equation}
To express the existence of such an order in a bounded set, we employ an analogous devise
to the one employed in \eqref{SATFO_M}. Noting that the set of permutations of a bounded set is itself a bounded set,
let us define a relation $Ord^{b+1}(o,u_0,\ldots,u_b)$ on a set of size $b+1$ by the sentence
\begin{equation}
\label{ord}
Ord^{b+1}(o,u_0,\ldots,u_b) \leftrightarrow \exists o \exists u_0, \ldots, u_b (O(o,u_0,u_1) \land \ldots \land
O(o,u_{b-1},u_b)
\end{equation}
with altogether $\frac{b (b+1)}{2}$ appearances of a relation term of the form $O(o,u,u')$.
Here the ternary relation $O(o,u,u')$ has the intended meaning that the corresponding
binary relation $\bar{O}(u,u')$ satisfies the definition \eqref{Order} for the permutation
$o$ of the $b+1$ elements in the valuation of the relation $Ord^{b+1}(o,u_0,\ldots,u_b)$.
Hence $o$ ranges over a set of size $(b+1)!$. Let us further introduce the notation
$-b \leq u, u' \leq b$, by which we mean that
\begin{equation}
\begin{split}
-b \leq u, u' \leq b \leftrightarrow \\
(u=u_{-b} \lor \ldots \lor u=u_b) \land \\
(u'=u_{-b} \lor \ldots \lor u'=u_b)
\end{split}
\end{equation}
and $0 \leq t, t' \leq b$, by which we mean that
\begin{equation}
\begin{split}
0 \leq t, t' \leq b \leftrightarrow \\
(t=t_{0} \lor \ldots \lor t=t_b) \land \\
(t'=t_{0} \lor \ldots \lor t'=t_b)
\end{split}
\end{equation}
Let us define the successor relation $S(u,u',\bar{u},o_{\bar{u}})$ on the permutation $o_{\bar{u}}$ of any set of
size $2b+1$ by
\begin{equation}
\label{successor}
\begin{split}
S(u,u',\bar{u},o_{\bar{u}}) \leftrightarrow \\
\forall (-b \leq v \leq b) (O(o_{\bar{u}},u,u') \land O(o_{\bar{u}},u,v)
\rightarrow O(o_{\bar{u}},u',v) \lor (v=u'))
\end{split}
\end{equation}
Let us denote the set of permutations of the bounded set of time steps $\{t\}$ by $\{o_{\bar{t}}\}$, the
set of permutations of the set $\{u\}$ of working tape cells by $\{o_{\bar{u}}\}$ and the set of permutations of the
set $\{v\}$ of input tape cells by $\{o_{\bar{v}}\}$. The sizes of the corresponding sets of permutations are
$(b+1)!$ for the set $\{o_{\bar{t}}\}$ and $(2b+1)!$ for the sets $\{o_{\bar{u}}\}$ and $\{o_{\bar{v}}\}$,
respectively, because input and work tapes can be traversed in either direction.
A bounded and finite set can always be totally ordered and a unique successor exists for all of its elements but the last one.
But we have not specified any ordering beforehand, since its existence is explicitly established in
the defining formula \eqref{ord}. The results below therefore apply to any bounded and finite set, and not just to
ordered structures.
Returning to the program formula \eqref{program}, for each
$t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}}$, the formula
$\pi_{T} (t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$
is a first order formula over the relational vocabulary
$A(t,u)$, $B_l (t,u)$, $B_{in} (t_0,v)$, $Z_q (t)$, $A_{in} (t,v)$, $S(t,t',\bar{t},o_{\bar{t}})$,
$S(u,u',\bar{u},o_{\bar{u}})$ and $S(v,v',\bar{v},o_{\bar{v}})$.
Parameter $l$ stands for a letter in the alphabet and $q$ for a
$T$-state. We shall take the following intended interpretations for
the relations in our vocabulary:
\begin{tabular}{lll}
$A(t,u)$ & iff & working cell on work tape at time $t$ is $u$ \\
$A_{in}(t,v)$ & iff & reading cell on input tape at time $t$ is $v$ \\
$B_l (t,u)$ & iff & letter $a_l$ is in cell $u$ of the work tape at time $t \geq 0$ \\
$B_{in} (t_0,v)$ & iff & bit 1 is in cell $v$ of the input tape at time $t_0$, and thereafter \\
$Z_q(t)$ & iff & $q$ is the state of $T$ at time $t$, with $Z_{\omega}(t)$ indicating that $T$ has halted\\
$S(t,t',\bar{t},o_{\bar{t}})$ & iff & $t'$ is the direct successor instant of $t$ in the tuple $\bar{t}$,\\
&& totally ordered in the permutation $o_{\bar{t}}$ \\
$S(u,u',\bar{u},o_{\bar{u}})$ & iff & $u'$ is the direct successor cell of $u$ on the work tape in the tuple $\bar{u}$,\\
&& totally ordered in the permutation $o_{\bar{u}}$ \\
$S(v,v',\bar{v},o_{\bar{v}})$ & iff & $v'$ is the direct successor cell of $v$ on the input tape in the tuple $\bar{v}$,\\
&& totally ordered in the permutation $o_{\bar{v}}$ \\
\end{tabular}
A \textit{correct Turing machine configuration} is a description of
an instantaneous state of a Turing machine $T$ at any given time step $\tau$.
For any $T$-configuration, let
$\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $\tau$
feature the conjunction of the above literals describing that
configuration. The domain of this program formula is the Cartesian product set
\begin{equation}
N = \{t\}^3 \times \{u\}^2 \times \{v\}^2 \times \{ \bar{t}\} \times \{ \bar{u}\}\times \{ \bar{v}\}
\times \{o_{\bar{t}}\} \times \{o_{\bar{u}}\} \times \{o_{\bar{v}}\}
\end{equation}
At time $\tau = t_b$, the size $|N|$ of this domain is bounded by $(b+1)^3 \cdot (2 b + 1)^4 \cdot
(b+1)^{b+1} \cdot (2b+1)^{2b+1} \cdot (b+1)! \cdot ((2b+1)!)^2$.
The program formula $\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$
defines the logical rules that correspond to the program steps in the program of $T$,
so that its models simulate $T$-computations in the
sense of the following \textit{Simulation Lemma}, modified from \cite{Borger}, p.
480:
\begin{lemma}
\label{sim}
Let $(\mathring{t},\mathring{t'},\mathring{u},\mathring{u'},\mathring{v},\mathring{v'},\mathring{\tau},
\mathring{\bar{t}},\mathring{\bar{u}},\mathring{\bar{v}},
\mathring{o}_{\bar{t}},\mathring{o}_{\bar{u}},\mathring{o}_{\bar{v}})$
be a valuation, i.e. an assignment of an element of $N$ to the tuple
$(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$.
Models of $\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$
for any fixed time $\tau$ are sets of such valuations for which the program formula is valid.
For arbitrary $T$-configurations at time $\tau=t_0$, with the corresponding program formula
$\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ and for an arbitrary future time $t_k$, where $t_0 < t_k \leq t_b$, if
$(\mathring{t},\mathring{t'},\mathring{u},\mathring{u'},\mathring{v},\mathring{v'},\mathring{\tau},\mathring{\bar{t}},
\mathring{\bar{u}},\mathring{\bar{v}},
\mathring{o}_{\bar{t}},\mathring{o}_{\bar{u}},\mathring{o}_{\bar{v}})$ satisfies
$\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $t_0$,
then for at least one $T$-configuration
$\pi_{T}(t,t',u,u',v,v',t_k,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $t_k$
which satisfies
\begin{equation}
\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}}) \vdash ^k_T
\pi_{T}(t,t',u,u',v,v',t_k,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})
\end{equation}
$(\mathring{t},\mathring{t'},\mathring{u},\mathring{u'},\mathring{v},\mathring{v'},\mathring{\tau},
\mathring{\bar{t}},\mathring{\bar{u}},\mathring{\bar{v}},
\mathring{o}_{\bar{t}},\mathring{o}_{\bar{u}},\mathring{o}_{\bar{v}})$
also satisfies $\pi_{T}(t,t',u,u',v,v',t_k,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $t_k$.
\end{lemma}
The notation above,
\begin{equation}
\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}}) \vdash ^k_T
\pi_{T}(t,t',u,u',v,v',t_k,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})
\end{equation}
means that the Turing
machine configuration $\pi_{T}(t,t',u,u',v,v',t_k,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $t_k$ is a successor configuration to
$\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ at time $t_0$,
and follows from it through the execution of $k$ steps of the program
of the Turing machine $T$. This result is achieved by representing
every state transition of $T$ as a step of logical inference.
As an example, for a program step that involves a state transition
from state $q$ to state $q'$ if there is the letter $a_l$ on the
work tape at the cell where the read/write head resides, writing the
letter $a_{l'}$ on that tape cell, and backing the read/write head
one step back to the left, the corresponding part of $T$'s
program formula $\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$ reads:
\begin{equation}
\label{Turing_step}
\begin{split}
(Z_q(t) \land A(t,u) \land B_l (t,u) \land S(t,t',\bar{t},o_{\bar{t}}) \land S(u',u,\bar{u},o_{\bar{u}})) \rightarrow \\
(Z_{q'}(t') \land A(t',u') \land B_{l'} (t',u))
\end{split}
\end{equation}
and analogously for all instructions in the program of the Turing machine.
As part of the program formula $\pi_{T}(t,t',u,u',v,v',t_0,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})$,
the initial configuration
of the Turing machine $T$ at time $t_0$ is also defined. This means the definition that the work
tape is empty, the Turing machine is in its initial state and that both read/write heads are at
positions $u_0$ and $v_0$, respectively. However, the program formula does $not$ define
the contents of the input tape. Instead, the free variable $y$ of the input formula \eqref{input} that ranges over values
$\{0, \ldots ,2^{b}\}$, as expressed by the binary input string on the input tape, defines the input to $T$.
The absolute values of the tape variables $u$ and $v$ are always
bounded by $t$, as the read/write head can only move at most one
cell to the left or to the right in a single time step. The
computations of the Turing machine $T$ up to any time bound $b$ are therefore defined by the
\textit{bounded computation formula}
\begin{equation}
\begin{split}
\label{Turing}
\Pi^b_{T} \leftrightarrow Ord^{b+1}(o_{\bar{t}},\bar{t}) \land Ord^{2b+1}(o_{\bar{u}},\bar{u}) \land Ord^{2b+1}(o_{\bar{v}},\bar{v}) \land \exists y \land\\
\forall (0 \leq t,t',\tau \leq b) \forall (-b \leq u,u' \leq b) \forall (-b \leq v,v' \leq b) \\
\pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})
\end{split}
\end{equation}
The condition $\exists y$ is an expression of the requirement that the input tape specified by
a permutation of the input cells defined by $\bar{v}$ contains some binary string.
Because of the time bound $b$, the formula \eqref{Turing} is a first order formula.
In this notation, the halting state formula $\omega$ can be expressed as
\begin{equation}
\omega \leftrightarrow \bigvee_{i=0}^{b} Z_{\omega} (t_i)
\end{equation}
up to any upper time limit $b$, or equivalently as
\begin{equation}
\omega \leftrightarrow Z_{\omega} (t_b)
\end{equation}
because any Turing machine will stay in the halting state indefinitely, once it has been entered, and this
condition is also stated in the program formula of $T$.
Let us abbreviate the tuple $(t,t',u,u',v,v',\tau)$ by $\bar{x}$, so that we
implicitly bundle together our tuple of seven 'scalar' bound
variables in \eqref{Turing}, the absolute values of which are all bounded by time $t$. By the notation $\bar{x}\leq b$ we shall mean
that
\begin{equation}
\begin{split}
\bar{x}\leq b \leftrightarrow \\
(t=t_0 \lor \ldots \lor t=t_b) \land \\
(t'=t_0 \lor \ldots \lor t'=t_b) \land \\
(u=u_{-b} \lor \ldots \lor u=u_b) \land \\
(u'=u_{-b} \lor \ldots \lor u'=u_b) \land \\
(v=v_{-b} \lor \ldots \lor v=v_b) \land \\
(v'=v_{-b} \lor \ldots \lor v'=v_b) \land \\
(\tau=t_0 \lor \ldots \lor \tau=t_b)
\end{split}
\end{equation}
The bounded computation formula \eqref{Turing} that defines $\Pi^b_{T}$
can be augmented to always simulate a deterministic Turing machine
by adding a uniqueness formula. The uniqueness formula states that
the validity of any one of the formulas of type \eqref{Turing_step}
that states the transition from state $Z_i (t)$ to state $Z_{i'}
(t')$ between successive time steps $t$ and $t'$ implies the
negation of every other similar state transition formula from $Z_i
(t)$ to $Z_j (t')$, when $j \neq i'$, and
conjuncting this formula for every pair of states $(i,j)$. Similar formulas
for uniqueness of read/write head movements to the left or to
the right, and for imposing uniqueness of letters printed at the working cell of the work tape
need to be added, too.
There are at most a constant number of such uniqueness formulas per
time step, since there are at most a constant number of
different states and letters in the alphabet of the Turing machine
$T$. Let us denote the number of states of $T$ by $|Q|$. Analogous uniqueness formulas are present in
\eqref{program} already in the non-deterministic case
for all tape cells, up to a number reachable in $b$ time steps,
requiring the uniqueness of the presence of any letter in any one
tape cell at each time step; the uniqueness of the working or reading cell at each time step
and the uniqueness of the successor relation between time steps
and tape cells, as indicated in\eqref{ord} and \eqref{successor}. The number of uniqueness formulas of successor relations
grows quadratically with the time bound $b$ but the rest linearly, since the
number of letters and machine states are fixed.
Let us now modify the bounded computation formula \eqref{Turing} so that it will
define a bounded set of computations by $T$ that includes all accepting computations on input
strings at most $M$ bits long as a subset. We shall denote by $b(M)$ the lowest bound on the maximum length
of computations needed to accept all the satisfiable formulas in the corresponding finite fragment of $\mathrm{L}$,
i.e. $\mathrm{L}_M$. This is the requirement of worst case complexity that is implicit in decision problems. The bounded computation formula that defines this bounded set
of computations of our Turing machine is denoted by $\Pi^{b(M)}_T$, and defined as
\begin{equation}
\label{PiM}
\begin{split}
\Pi^{b(M)}_T \leftrightarrow Ord^{b(M)+1}(o_{\bar{t}},\bar{t}) \land Ord^{2b(M)+1}(o_{\bar{u}},\bar{u}) \land Ord^{2b(M)+1}(o_{\bar{v}},\bar{v}) \land \exists y \land\\
\forall (\bar{x} \leq b(M)) \pi_{T}(t,t',u,u',v,v',\tau,\bar{t},\bar{u},\bar{v},o_{\bar{t}},o_{\bar{u}},o_{\bar{v}})
\end{split}
\end{equation}
From a semantic perspective, parameterizing the bound on the set of computations
by input length instead of the number of time steps is not an essential restriction, as is stated in the following theorem.
\begin{theorem}
\label{timebound}
\begin{equation}
\exists(b>0)\Pi^b_{T} \leftrightarrow \exists (M>0) \Pi^{b(M)}_T
\end{equation}
\end{theorem}
\begin{proof} The reverse implication holds, because there is a finite number of acceptable inputs
of length at most $M$ bits and we can choose a $b$ that is sufficient for accepting them all.
For the direct implication, let us take the longest input string $y_{max}$ that our Turing machine accepts
by time $t_b$. This string always exists and is of length at most $b$,
since we assume that our Turing machine always reads its input.
It then follows that $T$ accepts all those strings within bounded time.
Let us choose $M=|y_{max}|$. Since $\mathrm{L}_{|y_{max}|}$ is a finite fragment of
propositional logic, there exists a time bound $b(|y_{max}|)$ by which all satisfiable formulas in this fragment
have been accepted. Hence the membership of any input string $y$ in $SAT$ is decided by $\Pi^{b(|y_{max}|)}_T$.
\end{proof}
Theorem \ref{timebound} states that a bound on execution time always implies a bound
on input length, and vice versa, for a $\mathrm{SAT}$ solving Turing machine that is required to read all its input.
The domain of the revised bounded computation formula \eqref{PiM} is included in the domain $N$
of the time-bounded program formula \eqref{program} when $b(M) \leq b$.
\section{Bounded Turing structures}
Let us now turn to models of the revised bounded computation formula \eqref{PiM}, i.e. to
$\Pi^ {b(M)}_T$-structures. These are directed acyclic graphs (DAGs) of computations
that satisfy the program formula \eqref{program} as they proceed from time step $t_0$ to
time step $t_{b(M)}$. $\Pi^ {b(M)}_T$-structures are not trees, because our Turing machine
may revert to the same state from different preceding computations. But since the set of time steps
from $t_0$ to $t_{b(M)}$ is totally ordered, $\Pi^ {b(M)}_T$-structures cannot loop back
in time, and they therefore do not contain cycles.
It is important to note that $\Pi^{b(M)}_T$ defines \textit{all} possible
Turing computations, halting or non-halting, of a given Turing
machine $T$ on \textit{all} binary input strings $y_0$
of length at most $M$. In addition, it may possibly define some computations on longer
input strings and also the initial segments of all other computations.
The formula $\Pi^{b(M)}_T \land \omega$, on the other hand,
defines all accepting computations that halt by time step $t_{b(M)}$, whether the input string is of length $M$
or longer - but in any case not longer than $b(M)$ bits. Only a
conjunction with an input formula \eqref{input} expressed in $\Pi^{b(M)}_T \land y = y_0$, i.e. specifying a particular
input string $y_0$, will yield a formula that defines just the
computations pertaining to a particular input. A further conjunction
with the halting formula $\omega$ will yield a formula that defines all halting
computations on a particular input: $\Pi^{b(M)}_T \land \omega \land
y=y_0$.
The Directed Acyclic Graph that corresponds to the bounded computation formula
$\Pi^{b(M)}_T$ by itself serves as the "monster model" for all correct computations by $T$
up to time step $t_{b(M)}$. Conjunction of $\Pi^{b(M)}_T$ with $\omega$ or with
various subsets of input strings $\bigvee_{i \in I} (y = y_i)$, where $I \subseteq \{0, \ldots ,2^M\}$
define sub-DAGs that are embedded in the monster-DAG defined by $\Pi^{b(M)}_T$.
Because there is a bound $b(M)$ on the length of any path in the monster-DAG,
all paths in it terminate in leaf nodes.
For both deterministic and non-deterministic Turing machines,
the semantics of the associated first order bounded computation formula $\Pi^{(b(M)}_T \land
y=y_0$ are fully determined as soon as the contents of the
input tape have been determined by the input formula $y=y_0$.
For a non-deterministic Turing machine, even a fixed input string can correspond
to a "proper" DAG with multiple branches. For a deterministic Turing machine,
the sub-DAG that corresponds to a single input string is always a chain.
If all the branches emanating from a node terminate in the
accepting halting state $\omega$ by time step $t_{b(M)}$, let us call such a
sub-DAG a \textit{halting sub-DAG}.
\section{Atoms}
From now on, we shall assume that our Turing machine $T$ is
deterministic. Let us define an $Atom$ as a time dependent program
state $Z_q(t)$ that satisfies the following first order formula for
binary string variables $y'$, $y''$ and $y'''$ representing the content of the input tape:
\begin{equation}
\label{atom}
\begin{split}
Atom^M(t,q) \leftrightarrow \exists y' \exists y'' \exists y''' \exists t' \exists u \\
(t_0 \leq t, t' \leq t_{b(M)} \land u_{-b(M)} \leq u \leq u_{b(M)} \land 1 \leq q \leq |Q| \land \\
((\Pi_T^{b(M)} \land (Z_q(t)) \rightarrow Z_{\omega}(t_{b(M)})) \land \\
(\Pi_T^{b(M)} \land y=y' \land |y'| \leq M \land Z_q(t)) \land \\
(\Pi_T^{b(M)} \land y=y'' \land |y''| \leq M \land \\
S(t',t,\bar{t},o_{\bar{t}}) \land \bigvee_{q'} \bigvee_l ((Z_{q'}(t') \land A(t',u) \land B_l(t',u)
\land y=y'') \land \\
((\Pi_T^{b(M)} \land Z_{q'}(t') \land A(t',u) \land B_l(t',u)
\land y=y'') \rightarrow Z_q(t)) \land \\
(\Pi_T^{b(M)} \land y=y''' \land |y'''| \leq M \land Z_{q'}(t') \land \\
(\Pi_T^{b(M)} \land y=y''' \rightarrow \neg Z_{\omega}(t_{b(M)})))))))
\end{split}
\end{equation}
$Atom$ is shorthand for \textit{Accepting Transition Of Machine-state}.
Intuitively, the formula \eqref{atom} states that, within $\mathrm{L}_M$ and with any SAT
solving deterministic Turing machine $T$ with the bounded computation formula
$\Pi_{T}^{b(M)}$,
\begin{itemize}
\item All computations on any input that assume state $q$ at
time $t$ end up in the accepting halting state by time step
$t_{b(M)}$;
\item That there is at least one input $y'$ of length at most $M$ whose computation
attains state $q$ at time $t$;
\item That for at least one state $q'$ during the previous time step
$t'$, from which a transition to state $q$ at time step $t$ is
carried out on some input $y''$ of length at most $M$ by the Turing machine $T$, there is at least
one other input $y'''$ of length at most $M$ that also assumes state $q'$ at time $t'$, but
that leads to the rejection of that input, since by time step $t_{b(M)}$
all acceptable inputs of length $M$ or less will have been accepted. The
ones that have not halted by time step $t_{b(M)}$, despite their input being at most $M$ bits long,
will therefore never halt.
\end{itemize}
The property described above means that by time step $t_{b(M)}$, our Turing machine $T$ has
\textit{decided} $\mathrm{SAT}_M$.
In terms of $\Pi^{b(M)}_T$-structures,
$Atoms$ correspond to root nodes of halting sub-DAGs in the monster model
defined by $\Pi^{b(M)}_T$. There may be more than one $Atom$ in a halting sub-DAG.
Any eventual state transition to $Z_{\omega}(t)$ at time $t$
comprises an $Atom$ by itself, except when it is encountered only on paths within a halting sub-DAG with an
$Atom$ already preceding it on every path that leads to it.
One is tempted to call the set of $Atom$s the "event horizon" of our DTM, because it represents the set of
pairs of a state and a time step in the "spacetime" $\{q\} \times \{t\}$ of our DTM, beyond which all computation paths will end up in the "singularity" of the accepting halting state $\omega$, from which there is no return.
All paths to the halting state pass through some $Atom$, for any computation by $T$.
Let us call the first $Atom$ on the path of the computation by $T$ on the formula $\phi$ that belongs to $SAT_M$
the \textit{deciding Atom} of $\phi$.
The set of $Atoms$ depends
on the bound on input length $M$, and the relation symbol $Atom^M(t,q)$ is
therefore equipped with the parameter $M$.
Since computations by the Turing machine $T$ are uniform, $\Pi_T^{b(M')}$-structures
for $M' > M$ will grow all the branches of their DAGs from the leaf nodes of
the current $\Pi_T^{b(M)}$-structure, our "monster model". Hence all halting sub-DAGs
in it will remain halting sub-DAGs in subsequent $\Pi_T^{b(M')}$-structures for all $M' > M$,
as all their leaf nodes are already in the halting state $\omega$ at time step $t_{b(M)}$,
and can never leave that state in any subsequent computations. Consequently, all $M$-$Atoms$
will stay on as $M'$-$Atoms$ for all $M' > M$. The set of $Atoms$ is therefore a monotonously increasing
set as a function of $M$.
For any fixed finite bound on input length $M$, we can now establish a set of equivalences between
the following first order sentences
\begin{theorem}
\label{equiv}
\begin{equation}
\label{Cook_ext}
\begin{split}
\Psi_{\mathrm{SAT}_M} \leftrightarrow \exists \phi \exists y \exists e \forall w \exists z (|y| \leq M \land y=\theta(\phi) \land \\
(P(w,z,\phi) \land E(e,z)) \lor (N(w,z,\phi) \land \lnot E(e,z))) \leftrightarrow \\
\exists y \exists (t_0 \leq t \leq t_{b(M)}) \exists (1 \leq q \leq |Q|) (\Pi_T^{b(M)} \land |y| \leq M \land Z_q(t) \land Atom^{M}(t,q) )) \leftrightarrow \\
\exists y \Pi_T^{b(M)} \land |y| \leq M \land \omega
\end{split}
\end{equation}
\end{theorem}
\begin{proof} The first formula defines $SAT_M$. By the construction of $\Pi_T^{b(M)}$
in section three, our Turing machine must halt at every input $y=\theta(\phi) \land \phi \in \mathrm{SAT}_M$
by time step $t_{b(M)}$, which is the statement of the third formula.
As to the second formula, the unique computation on the encoding of any satisfiable
propositional formula will pass through at least one $Atom$, and the first one of those
will be the one that decides that satisfiable formula. Therefore the time $t$ in $Atom^M(t,q)$
must satisfy $t \leq b(M)$, for any $\phi$ in $SAT_M$. On the other hand, every $Atom^M(t,q)$ will decide at least one propositional formula $\phi$ in $\mathrm{SAT}_M$ which confirms the first equivalence in \eqref{Cook_ext}.
The second equivalence follows from the definition of $Atom^M(t,q)$ as a node in a halting sub-DAG
that halts by time step $b(M)$, and the fact that we have restricted the last formula to inputs at most $M$ bits long.
\end{proof}
On the other hand, since any propositional formula has a finite encoding length, say $M'$, in our
effective encoding scheme, \eqref{Cook_ext} states that any accepting
computation of any $\mathrm{SAT}$ solving deterministic Turing machine
implies that its input that encodes a propositional formula in $\mathrm{L}_{M'}$
belongs to $\mathrm{SAT}_{M'}$ for some $M'>0$. This implies
that the Turing machine halts by time step $t_{b(M')}$ on this input, which in turn implies
the passing of the corresponding computation through some $Atom^{M'}$
in $\Pi_T^{b(M')}$. When we allow $M$ to grow without limit, Theorem \eqref{Cook_ext} will
therefore apply to every $\phi$ in SAT.
\section{Almost saturated models of $SAT_M$}
Let us now take a closer look at the models of the $FO$ theory $SAT_M$ that defines propositional
satisfiability of CNF-formulas with an encoding at most $M$ bits long. The basics of model
theory used here can be captured from \cite{ChangKeisler} and \cite{Marker}.
Starting from the defining formula \eqref{SATFO_M}
of $SAT_M$ on an individual input $y_0$, we see that each $SAT_M$ is a theory
that is defined by a disjunction of all those formulas \eqref{SATFO_M} on those inputs $y$
encoded by at most $M$ bits that correspond to satisfiable propositional formulas in CNF.
When we take any complete extension of $SAT_M$, our first order language $L_M$ also contains formulas of the form shown below, with any number of variables $e_i$ up to some maximal finite number $\mathrm{maxsize}$ for each $M$, so that for each $1 \leq m \leq \mathrm{maxsize}$ we can define the truth of the sentence below.
\begin{equation}
\label{saturation}
\begin{split}
\tilde{\eta}_m \leftrightarrow \exists e_1,\ldots, e_m \exists \phi \exists y \bigwedge_{1 \leq i \leq m} \forall w \exists z (|y| \leq M \land y=\theta(\phi) \land \\
(P(w,z,\phi) \land E(e_i,z)) \lor (N(w,z,\phi) \land \lnot E(e_i,z)) \land \bigwedge_{1 \leq j < i \leq m} \lnot (e_j = e_i))
\end{split}
\end{equation}
From among the $\mathrm{maxsize}$ formulas of the type \eqref{saturation}, only the ones that have a propositional
formula $\phi$ in $\mathrm{SAT}_M$ with exactly $m$ satisfying truth assignments can be used to define
types in the $FO$ theory $SAT_M$. If this is the case, we call the propositional model size $m$ \textit{definable} in $SAT_M$. Formulas of type \eqref{saturation} grade propositional formulas $\phi$ featuring in $SAT_M$ into a decreasing chain by the number of satisfying truth assignments each of the corresponding $FO$ formulas of the form \eqref{saturation} admits, but they do not discern between individual propositional truth assignments.
Let us now establish a lower bound on the size of any {\it almost saturated}
model of $SAT_M$, derived from the equivalences between $FO$ formulas in \eqref{Cook_ext}
for any fixed $M$. By an almost saturated model we mean a model that realizes an isolated type for
every different propositional model size definable in $\Psi_{\mathrm{SAT}_M}$.
\begin{theorem}
\label{manymodels}
$SAT_M$ defines at least
$2^{\frac{1}{4k} M / \log M}$ different propositional model sizes for some constant $k>0$.
\end{theorem}
\begin{proof} Since $\mathrm{L}_M$ formulas
may contain different numbers of propositional variables, we need to define their propositional model
size after each formula has been complemented with dummy variables, up to the
maximum number of variables present in any $\mathrm{L}_M$-formula.
To get a lower bound on the number of propositional model sizes, let us look at increasing chains of sets of
propositional model sizes definable in $\mathrm{L}_M$. For that purpose, let us take the following propositional formula
that defines an order relation $y > x$ between two binary strings
$y = (y_{n-1}, y_{n-2}, \ldots , y_0)$ and $x = (x_{n-1}, x_{n-2}, \ldots , x_0)$:
\begin{equation}
\label{model_sizes}
\begin{split}
((y_{n-1}) \land (\lnot (x_{n-1}))) \lor & \\
(((y_{n-1}) \land (x_{n-1})) \lor ((\lnot (y_{n-1})) \land (\lnot
(x_{n-1}))))
\land (((y_{n-2}) \land (\lnot (x_{n-2}))) \lor & \\
(((y_{n-2}) \land (x_{n-2})) \lor ((\lnot (y_{n-2})) \land (\lnot
(x_{n-2})))) \land (((y_{n-3})
\land (\lnot (x_{n-3}))) \lor & \\
\cdots & \\
(((y_{1}) \land (x_{1})) \lor ((\lnot (y_{1})) \land (\lnot
(x_{1})))) \land ((y_{0}) \land (\lnot (x_{0}))) \ldots ) &
\end{split}
\end{equation}
Let us choose $n$ so that \eqref{model_sizes} is in $\mathrm{L}_M$. When the
binary string $y$ is fixed by a complete truth assignment to the
propositional variables ${y_{n-1}, y_{n-2}, \ldots , y_0}$, the
formula \eqref{model_sizes} defines all the truth assignments that
represent binary strings preceding $y$ in the numerical ordering
of binary numbers $x<y$. There are as many different such truth
assignments as is the cardinal denoted by the binary number $y$.
Since $y$ can take $2^n$ different values, the theory defined by the
formula \eqref{model_sizes} with all different binary strings for $y$,
interpreted as binary numbers, has
$2^n-1$ non-empty propositional models of different finite cardinality. By fixing
$y$ to all the different binary strings less than or equal to $2^n$ in turn,
while keeping the bits of $x$ as free propositional variables,
we get a family of $\mathrm{L}_M$ formulas that define $2^n-1$ different propositional model
sizes larger than zero. These can be picked up in any order to form an increasing
sequence of propositional model sizes of length $2^n-1$.
It remains to compute the relation between $n$ and $M$. The formula
\eqref{model_sizes} has a length linear in $n$, since every free
propositional variable $x_i$ appears in it exactly three times and
there are nine Boolean operations and 14 pairs of parentheses per
variable. The encoding of the variables takes $\log n$ bits. If we
choose $M$ to be at least $\frac{1}{4} n \log n$, which is asymptotically
enough to cover the length of the encodings of the Boolean operations and the multiplicity
three of the variables in \eqref{model_sizes}, and a factor $k$ for encoding
\eqref{model_sizes} in CNF, we get the claim of the theorem.
\end{proof}
Let us now show that any model of a complete extension of $SAT_M$ must realize all propositional model sizes definable in \eqref{SATFO_M}.
\begin{theorem}
\label{sizes}
Any model of a complete extension of $SAT_M$ must be almost saturated.
\end{theorem}
\begin{proof}
If one of the propositional model sizes were omitted by a model of a complete extension of \eqref{SATFO_M}, i.e. that there is no element in the model that separates between some pair of definable propositional model sizes, then
for those definable propositional model sizes $m$ and $m+l$ and some $\phi \in SAT_M$ with $m$ satisfying truth assignments, the formula
\begin{equation}
\begin{split}
\tilde{\eta}_m \leftrightarrow \exists e_1,\ldots, e_m \exists \phi \exists y \bigwedge_{1 \leq i \leq m} \forall w \exists z
(|y| \leq M \land y=\theta(\phi) \land \\
(P(w,z,\phi) \land E(e_i,z)) \lor (N(w,z,\phi) \land \lnot E(e_i,z)) \land \bigwedge_{1 \leq j < i \leq m}
\lnot (e_j = e_i)) \leftrightarrow \\
\tilde{\eta}_{m+l} \leftrightarrow \exists e_1, \ldots, e_{m+l} \exists \phi \exists y \bigwedge_{1 \leq i \leq m+l} \forall w \exists z (|y| \leq M \land y=\theta(\phi) \land \\
(P(w,z,\phi) \land E(e_i,z)) \lor (N(w,z,\phi) \land \lnot E(e_i,z)) \land \bigwedge_{1 \leq j < i \leq m+l} \lnot (e_j = e_i))
\end{split}
\end{equation}
would be true, even if $\phi$ does not have even $m+1$ satisfying truth assignments and this leads to a contradiction.
\end{proof}
A set of isolated types of a complete extension of $SAT_M$ can be defined by the formulas
\begin{equation}
\label{types}
\eta_m \leftrightarrow\tilde{\eta}_m \land \lnot \tilde{\eta}_{m+1}
\end{equation}
for all model sizes $ m \in \{ 1, \ldots, \mathrm{maxsize} \} $ definable in $SAT_M$.
Let us now derive a lower bound on the size of any model of the $FO$ theory $SAT_M$ defined by \eqref{SATFO_M}.
\begin{theorem}
\label{modelsize}
Any model of \eqref{SATFO_M} has at least size $2^{\frac{1}{4k} M / \log M}$ for some constant $k>0$.
\end{theorem}
\begin{proof}
A model of \eqref{SATFO_M} may omit an isolated type of its complete extension if this type is indiscernible in the set of
instances of bounded propositional satisfiability problem $\mathrm{SAT}_M$. We shall show that all definable
propositional model sizes are discernible in the set of instances of $\mathrm{SAT}_M$.
Let us recall that our encoding scheme does not increase encoding length when we replace all appearances of propositional variables $z$ in the encoding $\theta(\phi)$ of the propositional
formula $\phi$ by the constants 0 or 1, according to the satisfying truth assignment defined by $e$.
Let us call this property of an encoding scheme for propositional satisfiability the {\it valuation
property}. Let us denote the propositional formula that results from the replacement of all propositional literals appearing in $\phi$ by the corresponding constants 0 and 1, according to the truth assignment $e$, by $\phi(e)$.
Because of the valuation property, we have
\begin{equation}
\forall \phi \forall e (\phi \in \mathrm{L}_M \rightarrow \phi(e) \in \mathrm{L}_M)
\end{equation}
and all formulas $\phi(e)$ are therefore also decided by $T$ by time step $b(M)$ whenever $\phi \in \mathrm{L}_M$.
All formulas \eqref{saturation} are faithfully represented in $\mathrm{SAT}_M$ by the equivalence
\begin{equation}
\label{representation}
\begin{split}
\tilde{\eta}_m \leftrightarrow \exists e_1,\ldots, e_m \exists \phi \bigwedge_{1 \leq i \leq m}\exists y \forall w \exists z (|y| \leq M \land y=\theta(\phi(e_i)) \land \\
(P(w,z,\phi(e_i)) \land E(e_i,z)) \lor (N(w,z,\phi(e_i)) \land \lnot E(e_i,z)) \land \bigwedge_{1 \leq j < i \leq m} \lnot (e_j = e_i))
\end{split}
\end{equation}
Hence so are all formulas \eqref{types} and they define a set of distinct isolated types of $SAT_M$.
By Theorem \eqref{sizes} any model of a complete extension of \eqref{SATFO_M} must realize all definable propositional model sizes defined by \eqref{types}. By the formula \eqref{representation} all these types are discernible in the set of
instances of the bounded propositional satisfiability problem $\mathrm{SAT}_M$. Hence none of them can be
omitted by any model of \eqref{SATFO_M} either.
We can have as many inequivalent formulas of the form \eqref{types} as there
are different definable propositional model sizes in $SAT_M$. These inequivalent formulas then each isolate a complete type of \eqref{SATFO_M}. By Theorem \eqref{manymodels}, there are at least $2^{\frac{1}{4k} M / \log M}$ different propositional model sizes definable in $SAT_M$ for some constant $k>0$.
\end{proof}
\section{A lower bound on the computational complexity of
propositional satisfiability}
By the set of equivalences in \eqref{Cook_ext}, we see that any model of the third
formula in \eqref{Cook_ext} must contain an isomorphic copy of a model of the second formula that must contain
an isomorphic copy of a model of the first
formula which, by the requirement of worst case complexity, must be almost saturated,
i.e. contain an isolated type for each different propositional model size definable in $SAT_M$.
These model isomorphisms in \eqref{Cook_ext} are defined by the encoding $\theta(\phi)$ of each propositional formula $\phi$ and by the association of $\phi$ to its unique
deciding $Atom^M$ in the computational path that corresponds to $\theta(\phi)$, respectively. As we have restricted our set of equivalences in \eqref{Cook_ext} onto inputs $y$ at most $M$ bits long, we get the following theorem:
\begin{theorem}
\label{model_embeddings}
Any model of the formula $\exists y \Pi^{b(M)}_T \land |y| \leq M \land \omega$ that defines time-limited halting computations on all inputs at most $M$ bits long by time step $b(M)$ of a propositional satisfiability solving deterministic Turing machine $T$ must contain an isomorphic copy of a model of the formula $\Psi_{\mathrm{SAT}_M}$ that defines the first order theory $SAT_M$ that has at least as many elements as the number of different propositional model sizes definable in $SAT_M$, up to a a constant multiplier.
\end{theorem}
\begin{proof}
By Theorem \eqref{equiv}, all models of any of the formulas in \eqref{Cook_ext} must be isomorphic. Since the
first formula defines $SAT_M$ and cannot have a model smaller than the number of different propositional model sizes
definable in $SAT_M$, this lower bound on model size carries over onto theories defined by the other two formulas as well.
\end{proof}
From Corollary \eqref{modelsize} above we conclude that any model of any of the latter two formulas
in \eqref{Cook_ext} must contain an isomorphic copy of an almost saturated model of the first formula and therefore has a lower bound on its size $2^{\frac{1}{4kd} M / \log
M}$ as a function of $M$ for some constants $k, d > 0$, where $k$ accounts for the bound in Theorem \eqref{manymodels} and $d$ caters for our suboptimal
encoding of propositional formulas. This lower bound on
model size allows us to deduce the following theorem
on the deterministic time complexity of SAT.
\begin{theorem}
\label{SAT_complexity}
The deterministic time complexity of Propositional
Satisfiability $\mathrm{SAT}$ is not less than $C 2^{c M / \log M}$ for some
constants $C > 0$, $c > 0$ with respect to the length $M$ of the input
of a $\mathrm{SAT}$ solving deterministic Turing machine.
\end{theorem}
\begin{proof}
We pick the second formula in \eqref{Cook_ext}
\begin{equation}
\label{Cook_atom}
\begin{split}
\exists y \exists (t_0 \leq t \leq t_{b(M)}) \exists (1 \leq q \leq |Q|) (\Pi_T^{b(M)} \land |y| \leq M \land Z_q(t) \land Atom^M(t,q))
\end{split}
\end{equation}
The models of the theory defined by \eqref{Cook_atom} must have a model size at least
$2^{\frac{1}{4kd} M / \log M}$. This model size is defined directly
by the number of its $Atom^M$'s. To satisfy the lower bound in Theorem \eqref{modelsize} it must have as many distinct $Atom^M$'s, all defined
by the subformula
$\exists (t_0 \leq t \leq t_{b(M)}) \exists (1 \leq q \leq |Q|) (Z_q(t) \land Atom^M(t,q))$
in \eqref{Cook_atom}. All terms of the form $Z_q(t) \land Atom^M(t,q)$ in
\eqref{Cook_atom} are positive. There cannot be more different
$Atom^M$'s than there are different propositional variables $Z_q(t)$. The program
of any Turing machine is, by uniformity, to be independent of the length
of its inputs. We therefore have at most a constant number $ |Q|$ of
different machine-states. This implies that in order for
\eqref{Cook_atom} to have a model size of at least $2^{\frac{1}{4kd} M / \log M}$,
as required by Corollary \ref{modelsize}, we need to use at least
$\frac{1}{|Q|} 2^{\frac{1}{4kd} M / \log M}$ time steps to supply all the $Atoms$ we need.
By choosing $C=\frac{1}{|Q|}$ and $c=\frac{1}{4kd}$, and allowing $M$ to grow without limit,
we get the claim of the theorem.
\end{proof}
Because SAT is NP-complete and has a superpolynomial lower bound on its
deterministic time complexity by Theorem \eqref{SAT_complexity}, we get
\begin{corollary} $NP > P$.
\end{corollary}
{\bf Acknowledgements}. The author is very grateful to Lauri Hella for many extremely
important critical remarks and fruitful discussions, in the course of an almost
non-denumerable number of revisions to the current manuscript.
|
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| 4,863
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<?php
namespace Magento\Customer\Controller\Adminhtml\Index;
class Index extends \Magento\Customer\Controller\Adminhtml\Index
{
/**
* Customers list action
*
* @return \Magento\Backend\Model\View\Result\Page|\Magento\Backend\Model\View\Result\Forward
*/
public function execute()
{
if ($this->getRequest()->getQuery('ajax')) {
$resultForward = $this->resultForwardFactory->create();
$resultForward->forward('grid');
return $resultForward;
}
$resultPage = $this->resultPageFactory->create();
/**
* Set active menu item
*/
$resultPage->setActiveMenu('Magento_Customer::customer_manage');
$resultPage->getConfig()->getTitle()->prepend(__('Customers'));
/**
* Add breadcrumb item
*/
$resultPage->addBreadcrumb(__('Customers'), __('Customers'));
$resultPage->addBreadcrumb(__('Manage Customers'), __('Manage Customers'));
$this->_getSession()->unsCustomerData();
return $resultPage;
}
}
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"redpajama_set_name": "RedPajamaGithub"
}
| 6,727
|
\section{Introduction}
\label{sect:intro}
Quaternions, also called Hamilton numbers, are the first non-commutative
division algebra as a natural extension to complex numbers
(See the quaternion plaque in Fig. \ref{fig:quaternion}).
Imaginary quaternion units $i,j$ and $k$
are isomorphic to the anti-commutative SU(2) Pauli matrices
$-i\sigma_{1,2,3}$.
Hamilton used quaternions to represent three-(3D) and
four-dimensional (4D) rotations, and performed the product of two rotations.
In fact, it is amazing that he was well ahead of his time --
equivalently he was using the spin-$\frac{1}{2}$ fundamental
representations of the SU(2) group, which was before
quantum mechanics was discovered.
Nevertheless, the development of quaternoinic analysis met
significant difficulty since quaternions do not commute.
An important progress was made by Fueter in 1935 as reviewed in
Ref. \cite{sudbery1979}, who defined the Cauchy-Riemann-Fueter condition
for quaternionic analyticity.
Amazingly again, this is essentially the Euclidean version of the
Weyl equation proposed in 1929.
Later on, there have been considerable efforts in constructing
quantum mechanics and quantum field theory based on quaternions
\cite{adler1995,finkelstein1962,yang2005}.
On the other hand, the past decade has witnessed a tremendous
progress in the study of topological states of matter, in particular,
time-reversal invariant topological insulators in two dimensions
(2D) and 3D.
Topological properties of their band structures are characterized
by a $\mathbb{Z}_{2}$-index, which are stable against time-reversal
invariant perturbations and weak interactions
\cite{bernevig2006a,kane2005,kane2005a,fu2007,fu2007a,moore2007,bernevig2006,
wu2006,qi2008,roy2009,roy2010}.
These studies are further developments of quantum anomalous Hall insulators characterized by the integer-valued Chern numbers \cite{thouless1982,haldane1988}.
Later on, topological states of matter including both insulating and
superconducting states have been classified into ten different
classes in terms of their properties under the chiral, time-reversal, and particle-hole symmetries \cite{kitaev2009,schnyder2008}.
These studies have mostly focused on lattice systems.
The wavefunctions of the Bloch bands are complicated, and their energy
spectra are dispersive, both of which are obstacles for the study of
high-dimensional fractional topological states.
In contrast, the 2D quantum Hall states \cite{klitzing1980,tsui1982}
are early examples of topological states of matter studied
in condensed matter physics.
They arise from Landau level quantizations due to the cyclotron
motion of electrons in magnetic fields \cite{girvin1999}.
Their wavefunctions are simple and elegant, which are basically
harmonic oscillator wavefunctions.
They are reorganized
to exhibit analytic properties by an external magnetic field.
Generally speaking, a 2D quantum mechanical wavefunction
$\psi(x,y)$ is complex-valued, but not necessarily complex analytic.
We do not need all the set of 2D harmonic oscillator wavefunctions,
but would like to select a subset of them with non-trivial
topological properties, then complex analyticity is a natural
selection criterion.
Indeed, the lowest Landau level wavefunctions exhibit complex
analyticity.
Mathematically, it is imposed by the Cauchy-Riemann condition
(See Eq. \ref{eq:cauchy} in the text.), and
physically it is implemented by the magnetic field, which reflects
that the cyclotron motion is chiral.
This fact greatly facilitated the construction of Laughlin
wavefunction in the study of fractional quantum Hall states
\cite{laughlin1983}.
How to generalize Landau levels to 3D and even higher dimensions
is a challenging question.
A pioneering work was done by Shoucheng and his former student
Jiangping Hu in 2001 \cite{zhang2001}.
They constructed the Landau level problem on a compact space of
the $S^4$ sphere, which generalizes Haldane's formulation of the 2D
Landau levels on an $S^2$ sphere.
Haldane's construction is based on the 1st Hopf map \cite{haldane1983},
in which a particle is coupled to the vector potential from
a $U(1)$ magnetic monopole.
Zhang and Hu considered a particle lying on the $S^4$ sphere coupled to
an SU(2) monopole gauge field, and employed the 2nd Hopf map
which maps a unit vector on the $S^4$ sphere to a normalized
4-component spinor.
The Landau level wavefunctions are expressed in terms of the
four components of the spinor.
Such a system is topologically non-trivial characterized by
the 2nd Chern number possessing time-reversal symmetry.
This construction is very beautiful, nevertheless, it needs
significantly advanced mathematical physics knowledge which
may not be common for the general readers in the condensed
matter physics, and atomic, molecular, and optical physics
community.
We have constructed high-dimensional topological states
(e.g. 3D and 4D) based on harmonic oscillator wavefunctions
in flat spaces \cite{li2012a,li2013}.
They exhibit flat energy dispersions and non-trivial topological
properties, hence, they are generalizations of the 2D Landau level
problem to high dimensions.
Again we will select and reorganize a subset of wavefunctions
in seeking for non-trivial topological properties.
The strategy we employ is to use quaternion analyticity
as the new selection criterion to replace the previous one of
complex analyticity.
Physically it is imposed by spin-orbit coupling, which
couples orbital angular momentum and spin together
to form the helicity structure.
In other words, the helicity generated by spin-orbit
coupling plays the role of 2D chirality due to the
magnetic field.
Our proposed Hamiltonians can also be formulated in terms of
spin-$\frac{1}{2}$ fermions coupled to the SU(2) gauge potential,
or, the Aharanov-Casher potential.
Gapless helical Dirac surface modes, or, chiral Weyl modes,
appear on open boundaries manifesting the non-trivial
topology of the bulk states.
We have also constructed high-dimensional Landau levels of Dirac
fermions \cite{li2012}, whose Hamiltonians can be interpreted in
terms of complex quaternions.
The zeroth Landau levels of Dirac fermions are a branch of
half-fermion Jackiw-Rebbi modes \cite{jackiw1976}, which are
degenerate over all the 3D angular momentum quantum numbers.
Unlike the usual parity anomaly and chiral anomaly in which
massless Dirac fermions are minimally coupled to the
background gauge fields,
these Dirac Landau level problems correspond to a non-minimal
coupling between massless Dirac fermions and background fields.
This problem lies at the interfaces among condensed matter physics,
mathematical physics, and high energy physics.
High-dimensional Landau levels can also be constructed in the
Landau-type gauge, in which rotational symmetry is explicitly
broken \cite{li2013a}.
The helical, or, chiral plane-waves are reorganized by
spatially dependent spin-orbit coupling to yield non-trivial
topological properties.
The 4D quantum Hall effect of the SU(2) Landau levels have
also been studied in the Landau-type gauge, which exhibits
the quantized non-linear electromagnetic response as a spatially
separated 3D chiral anomaly.
We speculate that quaternionic analyticity would act as a guiding
principle for studying high-dimensional interacting topological
states, which are a major challenging question.
The high-dimensional Landau level problems reviewed below provide
an ideal platform for this research.
This research is at the interface between mathematical and
condensed matter physics, and has potential benefits
to both fields.
This review is organized as follows:
In Sect. \ref{sect:history}, histories of complex number and
quaternion, and the basic knowledge of complex analysis and
quaternion analysis are reviewed.
In Sect. \ref{sect:2Dlandau}, the 2D Landau level problems are
reviewed both for the non-relativistic particles and for relativistic
particles.
The complex analyticity of the lowest Landau level wavefunctions
is presented.
In Sect. \ref{sect:3DLL}, the constructions of high-dimensional
Landau levels in 3D and 4D with explicit rotational symmetries
are reviewed.
The quaternionic analyticity of the lowest Landau level
wavefunctions, and the bulk-boundary correspondences
in terms of the Euclidean and Minkowski versions of
the Weyl equation are presented.
In Sect. \ref{sect:reduction}, we review the dimensional reductions
from the 3D and 4D Landau level problems to yield the
2D and 3D isotropic but parity-broken Landau levels.
They can be constructed by combining harmonic potentials and
linear spin-orbit couplings.
In Sect. \ref{sect:diracLL}, the high-dimensional Landau levels
of Dirac fermions are constructed, which can be viewed as
Dirac equations in the phase spaces.
They are related to gapless Dirac fermions non-minimally coupled
to background fields.
In Sect. \ref{sect:landaugauge},
high-dimensional Landau levels in the anisotropic Landau-type
gauge are reviewed.
The 4D quantum Hall responses are derived as a spatially
separated chiral anomaly.
Conclusions and outlooks are presented in Sect. \ref{sect:conclusion}.
\section{Histories of complex number and quaternion}
\label{sect:history}
\subsection{Complex number}
Complex number plays an essential role in mathematics and quantum physics.
The invention of complex number was actually related to the history of
solving the algebraic cubic equations, rather than solving
the quadratic equation of $x^2=-1$.
If one lived in the 16th century, one could simply say that
such an equation has no solution.
But cubic equations are different.
Consider a reduced cubic equation $x^3+p x+q=0$, which can be solved by using radicals.
Here is the Cardano formula,
\begin{eqnarray}
x_1&=&c_1+c_2, ~~ x_2=c_1 e^{i\frac{2\pi}{3}}+ c_2 e^{-i\frac{2\pi}{3}},
\nonumber \\
x_3&=& c_1 e^{-i\frac{2\pi}{3}}+ c_2 e^{i\frac{2\pi}{3}},
\label{eq:cubic}
\end{eqnarray}
where
\begin{eqnarray}
c_{1}=\sqrt[3]{-\frac{q}{2}+ \sqrt{\Delta}}, \ \ \, \ \ \,
c_{2}=\sqrt[3]{-\frac{q}{2}-\sqrt{\Delta}},
\end{eqnarray}
with the discriminant $\Delta=(\frac{q}{2})^2+(\frac{p}{3})^3$.
The key point of the expressions in Eq. \ref{eq:cubic} is that they involve
complex numbers.
For example, consider a cubic equation with real coefficients
and three real roots $x_{1,2,3}$.
It is purely a real problem: It starts with real coefficients and ends up
with real solutions.
Nevertheless, it can be proved that there is no way to bypass $i$.
Complex conjugate numbers appear in the intermediate steps,
and finally they cancel to yield real solutions.
For a concrete example, for the case that $p=-9$ and $q=8$,
complex numbers are unavoidable since $\sqrt{\Delta}=\sqrt{-11}$.
The readers may check how to arrive at three real roots
of $x_{1,2,3}=1,-\frac{1}{2}\pm
\frac{\sqrt{33}}{2}$.
Once the concept of complex number was accepted, it opened up an entire
new field for both mathematics and physics.
Early developments include the geometric interpretation of complex
numbers in terms of the Gauss plane, the application of complex
numbers for two-dimensional rotations, and the Euler
formula
\begin{eqnarray}
e^{i\theta}=\cos\theta+i\sin\theta.
\end{eqnarray}
The complex phase appears in the Euler formula, which is widely used
in describing mechanical and electromagnetic waves in classic physics,
and also quantum mechanical wavefunctions.
Moreover, when a complex-valued function $f(x,y)$ satisfies
the Cauchy-Riemann condition,
\begin{eqnarray}
\frac{\partial f}{\partial x} +i \frac{\partial f}{\partial y}=0,
\label{eq:cauchy}
\end{eqnarray}
it means that it only depends on $z=x+iy$ but not on $\bar z=x-iy$.
The Cauchy-Riemann condition sets up the foundation of complex analysis,
giving rise to the Cauchy integral,
\begin{eqnarray}
\frac{1}{2\pi i} \oint \frac{1}{z-z_0} dz f(z)=f(z_0).
\end{eqnarray}
For physicists, one practical use of complex analysis is to
calculate loop integrals.
Certainly, its importance is well beyond this.
Complex analysis is the basic tool for many modern branches of mathematics.
For example, it gives the most elegant proof to {\it the fundamental theorem
of algebra}: An algebraic equation $f(z)=0$, i.e. $f(z)$ is
a $n$-th order polynomial, has $n$ complex roots.
The proof is essentially to count the phase winding number of $1/f(z)$
as moving around a circle of radius $R\to +\infty$, which simply
equals $n$.
On the other hand, the winding number is a topological invariant
equal to the number of poles of $1/f(z)$, or, the number of
zeros of $f(z)$.
It is also the basic tool for number theory: The Riemann hypothesis,
which aims at studying the distribution of prime numbers, is
formulated as a complex analysis problem of the distributions
of the zeros of the Riemann $\zeta(z)$-function.
Complex numbers actually are inessential in the entire
branches of classical physics.
It is well-known that the complex number description for classic waves
is only a convenience but not necessary.
The first time that complex numbers are necessary is
in quantum mechanics -- the Schr\"odinger equation,
\begin{eqnarray}
i\hbar\partial_t \psi=H\psi.
\end{eqnarray}
In contrast, classic wave equations involve $\partial_t^2$.
In fact, Schr\"oedinger attempted to eliminate $i$ in his equation,
but did not succeed.
Hence, to a certain extent, $i$, or, the complex phase, is more
important than $\hbar$
in quantum physics.
\subsection{Quaternion and quaternoinic analyticity}
Since 2D rotations can be elegantly described by
the multiplication of complex numbers.
It is reasonable to expect that 3D rotations could
also be described in a similar way by extending complex numbers
to include the 3rd dimension.
Simply adding another imaginary unit $j$ to construct $x+yi+zj$ does not
work, since the product of two imaginary units
$ij\neq i \neq j \neq \pm 1$.
It has to be a new imaginary unit defined as $k=ij$, and then
the quaternion is constructed as,
\begin{eqnarray}
q=x+yi+zj+uk.
\end{eqnarray}
The quaternion algebra,
\begin{eqnarray}
i^2=j^2=k^2=ijk=-1,
\label{eq:quaternion}
\end{eqnarray}
was invented by Hamilton in 1843 when he passed the Brougham bridge
in Dublin (See Fig. \ref{fig:quaternion}.).
He realized in a genius way that the product table of the imaginary
units cannot be commutative.
In fact, it can be derived based on Eq. \ref{eq:quaternion}
that $i$, $j$, and $k$ anti-commute with one another, {\it i.e.},
\begin{eqnarray}
ij=-ji, \ \ \, jk=-kj, \ \ \, ki=-ik.
\end{eqnarray}
This is the first non-commutative division algebra discovered,
and actually it was constructed before the invention of the concept of
matrix.
In modern languages, quaternion imaginary units are isomorphic to
Pauli matrices $-i\sigma_1, -i\sigma_2, -i\sigma_3$.
\begin{figure}
\centering\psfig{file=quaternion.eps,width=\linewidth, angle=0}
\caption{The quaternion plaque on Brougham Bridge, Dublin.
From wikipedia https://en.wikipedia.org/wiki/History\_of\_quaternions
}
\label{fig:quaternion}
\end{figure}
Hamilton employed quaternions to describe the 3D rotations.
Essentially he used the spin-$\frac{1}{2}$ spinor representation:
Consider a 3D rotation $R$ around the axis along the direction
of $\hat \Omega$ and the rotation angle is $\gamma$.
Define a unit imaginary quaternion,
\begin{eqnarray}
\omega(\hat\Omega)=i\sin\theta\cos\phi+j\sin\theta\sin\phi
+k\cos\theta,
\end{eqnarray}
where $\theta$ and $\phi$ are the
polar and azimuthal angles of $\hat\Omega$.
Then a unit quaternion associated with such a rotation
is defined as
\begin{eqnarray}
q=\cos\frac{\gamma}{2}+\omega(\hat\Omega)\sin\frac{\gamma}{2},
\end{eqnarray}
which is essentially an SU(2) matrix.
A 3D vector $\vec r$ is mapped to an imaginary quaternion
$r=xi+yj+zk$.
After the rotation, $\vec r$ is transformed to $\vec r'$,
and its quaternion form is
\begin{eqnarray}
r'= q r q^{-1}.
\label{eq:3Drotation}
\end{eqnarray}
This expression defines the homomorphism from SU(2) to SO(3).
In fact, using quaternions to describe rotation is
more efficient than using the 3D orthogonal matrix,
hence, quaternions are widely used in computer graphics
and aerospace engineering even today.
If set $\vec r =\hat z$ in Eq. \ref{eq:3Drotation}, and
let $q$ run over unit quaternions, which span the $S^3$ sphere,
then a mapping from $S^3$ to $S^2$ is defined as
\begin{eqnarray}
n=qkq^{-1},
\end{eqnarray}
which is the 1st Hopf map.
Hamilton spent the last 20 years of his life to promote
quaternions.
His ambition was to invent quaternion analysis which could
be as powerful as complex analysis.
Unfortunately, this was not successful because of the
non-commutative nature of quaternions.
Nevertheless, Fueter found the analogy to the Cauchy-Riemann
condition for quaternion analysis \cite{sudbery1979,frenkel2008}.
Consider a quaternionically valued function $f(x,y,z,u)$:
It is quaternionic analytic if it satisfies the following
Cauchy-Riemann-Fueter condition,
\begin{eqnarray}
\frac{\partial f}{\partial x}
+i\frac{\partial f}{\partial y}
+j\frac{\partial f}{\partial z}
+k\frac{\partial f}{\partial u}
\label{eq:quater_ana}
=0.
\end{eqnarray}
Eq. \ref{eq:quater_ana} is the left-analyticity condition since
imaginary units are multiplied from the left.
A right-analyticity condition can also be similarly defined in which imaginary
units are multiplied from the right.
The left one is employed throughout this article
for consistency.
For a quaternionic analytic function, the analogy to the Cauchy integral is
\begin{eqnarray}
\frac{1}{2\pi^2} \oiiint \frac{1}{|q-q_0|^2 (q-q_0)} Dq f(q)=f(q_0),
\label{eq:quater_cauchy}
\end{eqnarray}
where the integral is over a closed three-dimensional volume surrounding
$q_0$.
The measure of the volume element is,
\begin{eqnarray}
D(q)&=&dy\wedge dz\wedge du-idx\wedge dz\wedge du \nonumber \\
&+& jdx\wedge dy\wedge du
-kdx\wedge dy\wedge dz,
\end{eqnarray}
and $K(q)$ is the four-dimensional Green's function,
\begin{eqnarray}
K(q)=\frac{1}{q|q|^2}=\frac{x-yi-zj-uk}{(x^2+y^2+z^2+u^2)^2}.
\end{eqnarray}
There have also been considerable efforts in formulating quantum mechanics
and quantum field theory based on quaternions instead of complex numbers
\cite{finkelstein1962,adler1995}.
Quaternions are also used to formulate the Laughlin-like wavefunctions
in the 2D fractional quantum Hall physics \cite{balatsky1992}.
As discussed in {\it ``Selected Papers (1945-1980)
of Chen Ning Yang with Commentary"} \cite{yang2005}, C. N. Yang speculated that
quaternion quantum theory would be a major revolution to physics,
mostly based on the viewpoint of non-Abelian gauge theory.
He wrote, ``{\it... I continue to believe that the basic direction
is right.
There must be an explanation for the existence of SU(2) symmetry:
Nature, we have repeatedly learned, does not do random things at the
fundamental level.
Furthermore, the explanation is most likely in quaternion algebra:
its symmetry is exactly SU(2).
Besides, the quaternion algebra is a beautiful structure.
Yes, it is noncommutative.
But we have already learned that nature chose noncommutative algebra
as the language of quantum mechanics.
How could she resist using the only other possible nice algebra as
the language to start all the complex symmetries that she
built into the universe?"}
\section{Complex analyticity and two-dimensional Landau level}
\label{sect:2Dlandau}
In this section, I recapitulate the basic knowledge of the 2D Landau level problem,
including both the non-relativistic Schr\"odinger equation
in Sect. \ref{sect:2DSch} and the Dirac equation in Sect. \ref{sect:2DDirac}.
I explain the complex analyticity of the 2D lowest Landau level
wavefunctions.
\subsection{2D Landau level for non-relativistic electrons}
\label{sect:2DSch}
\begin{figure}
\centering\epsfig{file=spec_2DSHO.eps,clip=1,width=0.8\linewidth,angle=0}
\hspace{10mm}
\centering\epsfig{file=spec_3DQH.eps,clip=1,width=0.7\linewidth,angle=0}
\caption{
A) The energy level diagram of 2D harmonic oscillators v.s. the magnetic
quantum number $m$. The states along
the tilted lines are reorganized into the 2D flat Landau levels.
B) The eigenstates of the 3D harmonic oscillator labeled by total angular
momentum $j_\pm=l\pm \frac{1}{2}$
Following the tilted solid (dashed) lines, these states are reorganized
into the 3D Landau level sates with the positive (negative) helicity
for $H^\pm_{3D,symm}$, respectively.
(From Ref.\cite{li2013}).
}
\label{fig:spectra}
\end{figure}
The reason that the 2D Landau level wavefunctions are so interesting
is that their elegancy.
The external magnetic field reorganizes harmonic oscillator wavefunctions
to yield analytic properties.
To be concrete, the Hamiltonian for a 2D electron moving in an
external magnetic field $B$ reads,
\begin{eqnarray}
H_{2D,sym}=\frac{(\vec P -\frac{q}{c}\vec A)^2}{2M}.
\label{eq:2DLL}
\end{eqnarray}
In the symmetric gauge, i.e., $A_x=-\frac{1}{2}By$ and
$A_y=\frac{1}{2}Bx$, the 2D rotational symmetry is explicit.
The diamagnetic $A^2$-term gives rise to the harmonic potential, and
the cross term becomes the orbital Zeeman term.
Then Eq. \ref{eq:2DLL} can be reformulated as
\begin{eqnarray}
H_{2D,sym}= \frac{P_x^2+P_y^2}{2M}+\frac{1}{2} M \omega_0^2 (x^2+y^2)
- \omega_0 L_z,
\label{eq:2D_symm}
\end{eqnarray}
where $\omega_0$ is half of the cyclotron frequency $\omega_c$;
$\omega_c=qB/(Mc)$ and $qB>0$ is assumed.
Eq. \ref{eq:2D_symm} can also be interpreted as the Hamiltonian of a
rotating 2D harmonic potential, which is how the Landau level
Hamiltonian is realized in cold atom systems.
Since these the harmonic potential and orbital Zeeman term commute
with each other, the Landau level wavefunctions are just
wavefunctions of 2D harmonic oscillators.
In Fig. \ref{fig:spectra} A), the spectra of the 2D harmonic
oscillator {\it v.s.} the magnetic quantum number $m$ are plotted,
exhibiting a linear dependence on $m$ as $E_{n_r,m}=\hbar \omega_0 (2n_r+m+1)$
where $n_r$ is the radial quantum number.
If we view this diagram horizontally, they are with finite degeneracies
exhibiting a trivial topology.
But if they are viewed along the diagonal direction, they become
Landau levels.
This reorganization is due to the orbital Zeeman term, which
also disperses linearly $E_{Z}=-m\hbar\omega_0$.
It cancels the same linear dispersion of the 2D harmonic oscillator,
such that the Landau level energies become flat.
The wavefunctions of the lowest Landau level states with $n_r=0$
are $\psi_{LLL,m}(z)=z^m e^{-|z|^2/(4l_B^2)}$ with $m\ge 0$, where
the magnetic length $l_B=\sqrt{\hbar c/(qB)}$.
Now we impose the complex analyticity, i.e., the Cauchy-Riemann condition,
to select a subset of harmonic oscillator wavefunctions.
Physically it is implemented by the magnetic field.
It just means that the cyclotron motion is chiral.
After suppressing the Gaussian factor, the lowest Landau level
wavefunction is simply,
\begin{eqnarray}
\psi_{LLL}(z)=f(z),
\end{eqnarray}
which has a one-to-one correspondence to a complex analytic function.
In fact, the complex analyticity greatly facilitated the
construction of the many-body Laughlin wavefunctions \cite{laughlin1983},
\begin{eqnarray}
\psi_L(z_1,...,z_n)=\Pi_{i<j} (z_i-z_j)^3
e^{-\sum_i\frac{|z_i|^2}{4l^2_b}},
\label{eq:laughlin}
\end{eqnarray}
which is actually analytic in terms of multi-complex variables.
Along the edge of a 2D Landau level system, the bulk flat states
change to 1D dispersive chiral edge modes.
They satisfy the chiral wave equation \cite{girvin1999},
\begin{eqnarray}
\Big(\frac{1}{v_f}\frac{\partial }{\partial t} -
\frac{\partial }{\partial x}
\Big) \psi (x,t)=0,
\label{eq:chiraledge}
\end{eqnarray}
where $v_f$ is the Fermi velocity.
\subsection{2D Landau level for Dirac fermions}
\label{sect:2DDirac}
The is essentially a square-root problem of the Landau level
Hamiltonian of a Schr\"odinger fermion in Eq. \ref{eq:2DLL}.
The Hamiltonian reads \cite{semenoff1984},
\begin{eqnarray}
H^{D}_{2D}=l_0\omega \Big\{(p_x - A_x) \sigma_x+(p_y -A_y) \sigma_y \Big\},
\end{eqnarray}
where $A_x=-\frac{1}{2}By$, $A_y=\frac{1}{2}Bx$,
$l_0=\sqrt{\frac{2\hbar c}{|qB|}}$, and
$\omega=\frac{|qB|}{2mc}$.
It can be recast in the form of
\begin{eqnarray}
H_{2D}^{D}=\frac{ \hbar \omega}{\sqrt 2} \left[
\begin{array}{cc}
0& a_y^\dagger + ia_x^\dagger \\
a_y -i a_x & 0
\end{array}
\right],
\label{eq:2DLL_harm}
\end{eqnarray}
where $a_i=\frac{1}{\sqrt 2}(x_i/l_0+ip_i l_0/\hbar)$
$(i=x,y)$ are the phonon annihilation operators.
The square of Eq. \ref{eq:2DLL_harm} is reduced to the Landau level
Hamiltonian of a Schr\"odinger fermion with a supersymmetric structure
as
\begin{eqnarray}
(H_{2D}^{D})^2/(\frac{1}{2}\hbar \omega)=\left[
\begin{array}{cc}
H_{2D,sym}-\frac{1}{2}\hbar\omega&0\\
0& H_{2D,sym}+\frac{1}{2}\hbar\omega
\end{array} \right], \nonumber
\\
\end{eqnarray}
where $H_{2D,sym}$ is given in Eq. \ref{eq:2D_symm}.
The spectra of Eq. \ref{eq:2DLL_harm} are $E_{\pm n}=\pm \sqrt n\hbar\omega$
where $n$ is the Landau level index.
The zeroth Landau level states are singled out:
Only the upper component of their wavefunctions is nonzero,
\begin{eqnarray}
\Psi_{2D,LLL}^D(z)=\left( \begin{array}{c}
\psi_{LLL}(z)\\
0
\end{array}
\right).
\end{eqnarray}
$\psi_{LLL}(z)$ is the 2D lowest Landau level wavefunctions of the Schr\"odinger equation, which is complex analytic.
Other Landau levels with positive and negative energies distribute symmetrically around the zero energy.
Due to the particle-hole symmetry, each state of the zeroth Landau level
is a half-fermion Jackiw-Rebbi mode \cite{jackiw1976,heeger1988}.
When the chemical potential $\mu$ approaches $0^\pm$,
the zeroth Landau level is fully occupied, or, empty, respectively.
The corresponding electromagnetic response is,
\begin{eqnarray}
j_\mu=\pm\frac{1}{8\pi}\frac{q^2}{\hbar}
\epsilon_{\mu\nu\lambda} F_{\nu\lambda},
\label{eq:panomaly}
\end{eqnarray}
known as the 2D parity anomaly
\cite{redlich1984,redlich1984a,semenoff1984,niemi1986},
where $\pm$ refer to $\mu=0^\pm$, respectively.
The two spatial components of Eq. \ref{eq:panomaly} are just the half-quantized quantum Hall conductance, and the temporal component is the half-quantized
Streda formula \cite{streda1982}.
\section{3D Landau level and quaternionic analyticity}
\label{sect:3DLL}
We have seen the close connection between complex analyticity and
2D topological states.
In this section, we discuss how to construct high-dimensional
topological states in flat spaces based on
quaternionic analyticity.
\subsection{3D Landau level Hamiltonian}
Our strategy is based on high-dimensional harmonic oscillator
wavefunctions.
Again we need to select a subset of them for non-trivial topological
properties:
{\it The selection criterion is quaternionic analyticity,
and physically it is imposed by spin-orbit coupling.}
The physical picture of the 3D Landau level wavefunctions in the symmetric-like gauge is intuitively presented in
Fig. \ref{fig:3DLL} (A).
It generalizes the fixed complex plane in the 2D Landau level
problem to a moving frame embedded in 3D.
Define a frame with the orthogonal axes $\hat e_1$, $\hat e_2$,
and $\hat e_3$, and the complex analytic wavefunctions are defined
in the $\hat e_1$-$\hat e_2$ plane with spin polarized along the
$\hat e_3$ direction.
Certainly this frame can be rotated to an arbitrary configuration.
The same strategy can be applied to any high dimensions.
\begin{figure}
\centering\epsfig{file=orbital.eps,clip=1,width=0.5\linewidth,angle=0}
\hspace{20mm}
\centering\epsfig{file=spec_edge_b_label.eps,clip=1,width=0.6\linewidth,angle=0}
\caption{
A) The coherent state picture for 3D lowest Landau level
wavefunctions based on Eq. \ref{eq:3Dcoherent}.
$\hat e_1$-$\hat e_2$-$\hat e_3$ form an orthogonal triad.
The lowest Landau level wavefunction is
complex analytic in the orbital plane $\hat e_1$-$\hat e_2$
and spin is polarized along $\hat e_3$.
B) The surface spectra for the 3D Landau level Hamiltonian Eq.\ref{eq:3D_symm}.
The open boundary condition is used for a ball with the radius $R_0/l_{so}=8$.
(From Ref. \cite{li2013}.)
}
\label{fig:3DLL}
\end{figure}
Now we present the 3D Landau level Hamiltonian as constructed
in Ref. \cite{li2013}.
Consider to couple a spin-$\frac{1}{2}$ fermion to the 3D
isotropic SU(2) Aharanov-Casher potential $\vec A=\frac{G}{2} \vec \sigma
\times \vec r$ where $G$ is the coupling constant and $\vec\sigma$'s
are the Pauli matrices.
The resultant Hamiltonian is
\begin{eqnarray}
H^{\pm}_{3D,sym}&=&\frac{1}{2M}
\big( \vec P - \frac{q}{c} \vec A (\vec{r})\big )^{2}
+V(\vec r) \nonumber \\
&=&\frac{P^2}{2M}+ \frac{1}{2}M\omega _{0}^{2} r^2
\mp \omega _{0}\vec{\sigma}\cdot \vec{L},
\label{eq:3D_symm}
\end{eqnarray}
where $\pm$ refer to $G>0 ~ (<0)$, respectively; $\omega_{0}=\frac{1}{2}\omega_{so}$ and $\omega_{so}
=|qG|/(Mc)$ is the analogy of the cyclotron frequency.
$V(r)=-\frac{1}{2}M\omega _{0}^{2}r^{2}$,
nevertheless, the $\frac{1}{2M}(\frac{q}{c})^2 A^2(r)$ term in the
kinetic energy contributes a quadratic scalar potential which
equals $2|V(r)|$, hence, Eq. \ref{eq:3D_symm} is still bound from below.
In contrast to the 2D case, $H^\pm_{3D,sym}$ preserve
time-reversal symmetry.
It can also be formulated as a 3D harmonic potential plus a
spin-orbit coupling term.
Again since these two terms commute, the 3D Landau level
wavefunctions are just the eigenstates of a 3D harmonic oscillator.
Consider the eigenstates of a 3D harmonic oscillator with an additional
spin degeneracy $\uparrow$ and $\downarrow$.
For later convenience, their eigenstates are organized into the bases of the total angular momentum $j_\pm=l\pm\frac{1}{2}$, where $\pm$ represent the positive and negative helicities, respectively.
The corresponding spectra are plotted in Fig. \ref{fig:spectra} (B),
showing a linear dispersion with respect to $l$ as
$E_{n_r,J_\pm=l\pm\frac{1}{2},J_z}=\hbar \omega_0
(2 n_r + l +\frac{3}{2})$.
Again, if we view the spectra along the diagonal direction,
the novel topology appears.
The spin-orbit coupling term $\vec \sigma \cdot \vec k$ has two branches of eigenvalues, both of which disperse linearly with $l$ as $l\hbar$ and $-(l+1)\hbar$
for the positive and negative helicity sectors, respectively.
Combining the harmonic potential and spin-orbit coupling,
we arrive at the flat Landau levels:
For $H^+_{3D}$, the positive helicity states become dispersionless
with respect to $j_+$ , a main feature of Landau levels.
Similarly, the negative helicity states become flat for $H^-_{3D}$.
States in the 3D Landau level show the same helicity.
\subsection{The SU(2) group manifold for the lowest Landau level
wavefunctions}
Having understood why the spectra are flat, now we provide
an intuitive picture for the lowest Landau level wavefunctions
with the positive helicity.
If expressed in the orthonormal basis of $(j\pm,j_z)$, they are
rather complicated,
\begin{eqnarray}
\psi_{LLL, j_+=l+\frac{1}{2}, j_z}(r,\hat\Omega)=r^l
Y_{j_+=l+\frac{1}{2}, j_z}(\hat \Omega)
e^{-\frac{r^{2}}{4l_{so}^{2}}},
\label{eq:LL_orthonomal}
\end{eqnarray}
where $l_{so}=\sqrt{\hbar c/|qG|}$ is the analogy of the magnetic
length and $Y_{j_+=l+\frac{1}{2}, j_z}(\hat \Omega)$ is the
spin-orbit coupled spherical harmonic function.
Instead, they become very intuitive in the coherent state representation.
Let us start with the highest weight states with $j_+=j_z$, whose
wavefunctions are $\psi_{LLL, j_+=j_z}(r,\hat\Omega)=(x+iy)^l \exp\{-\frac{r^{2}}{4l_{so}^{2}}\} \otimes |\uparrow\rangle$.
Their spins are polarized along the $z$-direction and
orbital parts are complex analytic in the $xy$ plane.
We then perform a general SU(2) rotation such that the $xyz$-frame is rotated to the frame of $\hat e_{1}$-$\hat e_2$-$\hat e_3$.
For a coordinate vector $\vec r$, its projection in the $\hat e_1$-$\hat e_2$ plane forms a complex variable $\vec r \cdot (\hat e_1 +i \hat e_2)$
based on which we construct complex analytic functions.
Now it is clear why spin-orbit coupling is essential.
Otherwise, if the plane is flipped, then the complex variable changes to
its conjugate, and the complex analyticity is lost.
Nevertheless, since spin is polarized perpendicular to the $\hat e_1$-$\hat e_2$-plane, spin also flips during the flipping of the orbital plane, such
that the helicity remains invariant.
In general, we can perform an arbitrary $SU(2)$ rotation on
the highest weight states and arrive at a set of coherent states
forming the over-complete bases of the lowest Landau level states as
\begin{eqnarray}
\psi_{LLL,\hat e_{1,2,3}, j_+}(r, \hat \Omega)=[(\hat e_1 +i \hat e_2)
\cdot \vec r]^l e^{-\frac{r^{2}}{4l_{so}^{2}}}
\otimes \ket{\alpha_{\hat e_3}}, (l\ge 0)
\nonumber \\
\label{eq:3Dcoherent}
\end{eqnarray}
where $(\hat e_3 \cdot \vec \sigma)
\ket{\alpha_{\hat e_3}}= \ket{\alpha_{\hat e_3}}$.
Now we can make a comparison among harmonic oscillator wavefunctions in different dimensions.
\begin{enumerate}
\item
In 1D, we only have the real Hermite polynomials.
\item
In 2D, a subset of harmonic wavefunctions $z^m$ (lowest Landau level) are selected exhibiting the $U(1)$ structure.
\item
In 3D, the complex plane $\hat e_1$-$\hat e_2$ associated with the
frame $\hat e_1$-$\hat e_2$-$\hat e_3$ are floating.
This is similar to the rigid-body configuration.
In other words, the configuration space of the 3D lowest Landau level
states is that of a triad, or, the $SU(2)$ group manifold.
\end{enumerate}
Since the SU(2) group manifold is isomorphic to the space of unit quaternions,
this motivates us to consider the analytic structure in terms of
quaternions, which will be presented in Sect. \ref{sect:analyticity}.
\subsection{The off-centered solutions to the lowest Landau level states}
\label{sect:off-center}
Different from the 2D Landau level Hamiltonian, which possesses the magnetic
translation symmetry, the 3D one of Eq. \ref{eq:3D_symm}
does not possess such a symmetry due to the non-Abelian nature of the
SU(2) gauge potential.
Nevertheless, based on the coherent states described by Eq. \ref{eq:3Dcoherent},
we can define magnetic translations within the
$\hat e_1$-$\hat e_2$ plane, and organize the off-centered
solutions in the lowest Landau level.
Consider all the coherent states in the $\hat e_1$-$\hat e_2$ plane
described by Eq. \ref{eq:3Dcoherent}.
We define the magnetic translation for this set of states as
\begin{eqnarray}
T_{\hat{e}_{3}}(\vec{\delta})=\exp [-\vec \delta \cdot \vec{\nabla}+
\frac{i}{4l_{so}^{2}}
~\vec r_{12} \cdot (\hat e_3 \times \vec{\delta})],
\label{eq:tran}
\end{eqnarray}
where the translation vector $\vec{\delta}$ lies in the $\hat{e}_{1,2}$-plane
and $\vec{r}_{12}= \vec{r}-\hat{e}_{3}(\vec{r}\cdot \hat{e}_{3})$.
Set $\hat e_1=\hat z$, and the normal vector $\hat e_3$ lying in the
$xy$-plane with an azimuthal angle $\phi'$,
\textit{i.e.}, $\hat e_3(\phi')=\hat x\cos \phi' +\hat y\sin
\phi'$, then
$\alpha_{\hat e_3}(\phi')= \frac{1}{\sqrt 2}
(|\uparrow \rangle +e^{i\phi'} |\downarrow\rangle)$.
Consider the lowest Landau level states localized at the origin,
\begin{eqnarray}
\psi_{l=0,\hat e_3}(r, \hat \Omega)=e^{-\frac{r^{2}}{4l_{so}^{2}}}
\otimes \ket{\alpha_{\hat e_3}},
\end{eqnarray}
and translate it along $\hat z$ at the distance $R$.
According to Eq. \ref{eq:tran}, we arrive at
\begin{eqnarray}
\psi_{\phi',R} (\rho,\phi,z) =e^{i\frac{1}{2l_{so}^2} R\rho \sin (\phi-\phi')}
e^{-|\vec r- R \hat z|^2/4l_{so}^2}\otimes \alpha_{\hat e_3}(\phi'),
~
\label{eq:off-center}
\end{eqnarray}
where $\rho=\sqrt{%
x^2+y^2}$ and $\phi$ is the azimuthal angular of $\vec r$
in the $xy$-plane.
Now we can restore the rotational symmetry around the $\hat z$-axis by
performing the Fourier transform with respect to the angle $\phi'$, i.e.,
$\psi _{j_{z}=m+\frac{1}{2},R}(\rho,\phi,z)=\int_{0}^{2\pi }\frac{d\phi'
}{2\pi }e^{im\phi' }\psi _{\phi' ,R}$.
We arrive at the eigenstates of $j_z$ as
\begin{eqnarray}
\psi _{j_{z}=m+\frac{1}{2},R}(\rho,\phi,z)
&=& e^{\frac{-|\vec{r}-R\hat{z}|^{2}}{4l_{so}^{2}}}
e^{im\phi }\Big\{J_{m}(x)|\uparrow \rangle \nonumber \\
&+&J_{m+1}(x)e^{i\phi }|\downarrow \rangle \Big\},
\label{eq:fourier}
\end{eqnarray}
where $x=R\rho /(2l_{so}^{2})$.
It describes a wavefunction with the shape of
an ellipsoid, whose distribution in the $xy$-plane is within the
distance of $ml_{so}^2/R$.
The narrowest states $\psi_{\pm \frac{1}{2},R}$ have an aspect
ratio scaling as $l_{so}/R$ when $R$ goes large.
On the other hand, for those states with $|m|<R/l_{so}$, they localize
within the distance of $l_{so}$ from the center located at $R\hat{z}$.
As a result, the real space local density of states of the lowest
Landau level grows linearly with $R$.
\subsection{Quaternionic analyticity of the lowest Landau
level wavefunctions}
\label{sect:analyticity}
In analogy to complex analyticity of the 2D lowest Landau level states,
we have found that the helicity structure of the 3D lowest Landau levels
leads to quaternionic analyticity.
Just like two real numbers forming a complex number, a two-component
complex spinor $\psi=(\psi_\uparrow, \psi_\downarrow)^T$ can be mapped
to a quaternion by multiplying a $j$ to the 2nd component
\begin{eqnarray}
f=\psi_\uparrow + j \psi_\downarrow.
\label{eq:map}
\end{eqnarray}
Then the familiar symmetry transformations can be represented via
multiplying quaternions.
The time-reversal transformation $i\sigma_2 \psi^*$ becomes $T f=-fj$ satisfying $T^2=-1$.
The $U(1)$ phase $e^{i\theta}\to f e^{i\theta}$,
and the SU(2) rotation becomes
\begin{eqnarray}
e^{i\frac{\phi}{2}\sigma_x} \psi \to e^{k\frac{\phi}{2}} f, \ \ \,
e^{i\frac{\phi}{2}\sigma_y} \psi \to e^{j\frac{\phi}{2}} f, \ \ \,
e^{i\frac{\phi}{2}\sigma_z} \psi \to e^{-i\frac{\phi}{2}} f.
\nonumber \\
\end{eqnarray}
To apply the Cauchy-Riemann-Fueter condition Eq. \ref{eq:quater_cauchy}
to 3D, we simply suppress the 4th coordinate,
\begin{eqnarray}
\frac{\partial f}{\partial x}
+i\frac{\partial f}{\partial y}
+j\frac{\partial f}{\partial z}
\label{eq:quater_ana_3D}
=0.
\end{eqnarray}
We prove a remarkable property below
that this condition (Eq. \ref{eq:quater_ana_3D}) is rotationally invariant.
\begin{lemma}
If a quaternionic wavefunction $f(x,y,z)$ is quaternionic analytic,
i.e., it satisfies the Cauchy-Riemann-Futer condition,
then after an arbitrary rotation, the consequential wavefunction $f^\prime(x,y,z)$ remains quaternionic analytic.
\end{lemma}
\begin{proof}
Consider an arbitrary SU(2) rotation $g(\alpha,\beta,\gamma)=e^{-i\frac{\alpha}{2}\sigma_z}
e^{-i\frac{\beta}{2}\sigma_y} e^{-i\frac{\gamma}{2}\sigma_z}$,
where $\alpha, \beta,\gamma$ are Eulerian angles.
In the quaternion representation, it maps to
$g=e^{i\frac{\alpha}{2}} e^{-j\frac{\beta}{2}}
e^{i\frac{\gamma}{2}}$.
After this rotation $f(x,y,z)$ transforms to
\begin{eqnarray}
f^{\prime}(x,y,z)
=e^{i\frac{\alpha}{2}} e^{-j\frac{\beta}{2}}
e^{i\frac{\gamma}{2}} f( x^\prime, y^\prime, z^\prime),
\label{eq:rotation}
\end{eqnarray}
where $(x^\prime,y^\prime$, $z^\prime)$ are the coordinates
by applying $g^{-1}$ on $(x,y,z)$.
It can be checked that
\begin{eqnarray}
&&\Big(\frac{\partial}{\partial x}+ i\frac{\partial}{\partial y}
+j\frac{\partial}{\partial z} \Big)
e^{i\frac{\alpha}{2}} e^{-j\frac{\beta}{2}}
e^{i\frac{\gamma}{2}}\nonumber \\
&=&e^{i\frac{\alpha}{2}} e^{-j\frac{\beta}{2}} e^{i\frac{\gamma}{2}}
\Big(\frac{\partial}{\partial x'} +i\frac{\partial}{\partial y'}
+j\frac{\partial}{\partial z'} \Big).
\end{eqnarray}
Then we have
\begin{eqnarray}
(\frac{\partial}{\partial x}+ i\frac{\partial}{\partial y}
+j\frac{\partial}{\partial z})f'(x,y,z)=0.
\end{eqnarray}
Hence, the Cauchy-Riemann-Fueter condition is rotationally
invariant.
\end{proof}
Based on this lemma, we prove the quaternionic analyticity of
the 3D lowest Landau level wavefunctions.
\begin{theorem}
The 3D lowest Landau level wavefunctions of $H^+_{3D,sym}$ in Eq. \ref{eq:3D_symm}
have a one-to-one correspondence to the quaternionic analytic polynomials
in 3D.
\end{theorem}
\begin{proof}
We denote the quaternionic polynomials, which correspond to the
orthonormal bases of the lowest Landau level wavefunctions
in Eq. \ref{eq:LL_orthonomal}, as $f^{LLL}_{j_+,j_z}$ with
$j_+=l+\frac{1}{2}$, and $-j_+\le j_z \le j_+$.
The highest weight states $f^{LLL}_{j_+,j_+}=(x+iy)^l$ are
complex analytic in the $xy$-plane, hence, it is obviously
quaternionic analytic.
Since all the coherent states can be obtained from the highest weight
states via rotations, they are also quaternionic analytic.
The coherent states form a set of overcomplete basis of the lowest Landau
level wavefunctions, hence all the lowest Landau level wavefunctions
are quaternionic analytic.
Next we prove the completeness that $f^{LLL}_{j_+,j_z}$'s
form the complete basis of the quaternoinic analytic polynomials
in 3D.
By counting the degrees of freedom of the $l$-th order polynomials
of $x,y,z$, and the number of the constraints from Eq. \ref{eq:quater_cauchy},
we calculate the total number of the linearly independent
$l$-th order quaternionic analytic polynomials
as $C^2_{l+2}-C^2_{l+1}=l+1$.
On the other hand, any lowest Landau level state in the sector of $j_+=l+\frac{1}{2}$ can be represented as
\begin{eqnarray}
f_l(x,y,z) =\sum_{m=0}^{l} f^{LLL}_{j_+=l+\frac{1}{2},j_z=m+\frac{1}{2}} q_m,
\label{eq:cmplt}
\end{eqnarray}
where $q_m$ is a quaternion constant coefficient.
Please note that $q_{lm}$'s are multiplied from right
due to the non-commutativity of quaternions.
In Eq. \ref{eq:cmplt}, we have taken into account the fact
$f^{LLL}_{j_+,-j_z}=-f^{LLL}_{j_+,-j_z}j$ due to the time-reversal
transformation.
Hence, the degrees of freedom in the lowest Landau level with $j_+=l+\frac{1}{2}$ is also $l+1$.
Hence, the lowest Landau level wavefunctions are complete
for quaternionic analytic polynomials.
\end{proof}
\subsection{Generalizations to 4D and above}
The above procedure can be straightforwardly generalized to four and
even higher dimensions.
To proceed, we need to employ the Clifford algebra $\Gamma$-matrices.
Their ranks in different dimensions and concrete representations
are presented in Appendix \ref{appendix:cliff}.
Then we use the $N$-D harmonic oscillator
potential combined with spin-orbit coupling as
\begin{eqnarray}
H^{ND, LL}=\frac{p_{ND}^2}{2m}+\frac{1}{2}m \omega_0^2 r_{ND}^2
-\omega_0
\sum_{1\le i<j \le N} \Gamma_{ij} L_{ij},
\label{eq:4DQHE}
\end{eqnarray}
where $L_{ij}=r_i p_j - r_j p_i$.
The spectra of Eq. \ref{eq:4DQHE} were studied in the context
of the supersymmetric quantum mechanics \cite{bagchi2001}.
However, its connection with Landau levels was not noticed
there.
The spin operators in $N$-dimensions are defined as
$\frac{1}{2}\Gamma_{ij}$.
For the 4D case, the minimal representations
for the $\Gamma$-matrices are still two-dimensional.
They are defined as
\begin{eqnarray}
\Gamma_{ij}=-\frac{i}{2}[\sigma_{i}, \sigma_{j}], \ \ \,
\Gamma_{i4}=\pm \sigma_{i},
\end{eqnarray}
with $1\le i<j\le3$.
The $\pm$ signs of $\Gamma^{i4}$ correspond to two complex
conjugate irreducible
fundamental spinor representations of $SO(4)$, and the $+$
sign will be taken below.
The spectra of the positive helicity states are flat as
$E_{+, n_r}=(2n_{r}+2) \hbar \omega$.
The coherent state picture for the 4D lowest Landau levels can
be similarly constructed as follows:
Again pick up two orthogonal axes $\hat e$ and $\hat f$ to form
a 2D complex plane, and define complex analytic functions therein as,
\begin{eqnarray}
(x_a \hat e_a + i x_a \hat f_a)^l e^{-\frac{r^2}{4l_{so}^2}}
\otimes |\alpha_{\hat e, \hat f}\rangle,
\label{eq:4Dcoherent}
\end{eqnarray}
where $|\alpha_{\hat e, \hat f}\rangle$ is the eigenstate
of $\Gamma^{\hat e,\hat f}=\hat e_a \hat f_b \Gamma^{ab}$ satisfying
\begin{eqnarray}
\Gamma^{\hat e,\hat f} |\alpha_{\hat e, \hat f}\rangle
=|\alpha_{\hat e, \hat f}\rangle.
\end{eqnarray}
Hence, its spin is locked with its orbital angular momentum in the
$\hat e$-$\hat f$ plane.
Following similar methods in Sect. \ref{sect:analyticity}, we can prove
that the 4D lowest Landau level wavefunctions for Eq. \ref{eq:4DQHE}
satisfy the 4D Cauchy-Riemann-Futer condition Eq. \ref{eq:quater_ana},
and thus are quaternionic analytic functions.
Again it can be proved that they form the complete basis for
quaternionic left-analytic polynomials in 4D.
As for even higher dimensions, quaternions are not defined.
Nevertheless, the picture of the complex analytic function defined in
the moving frame still applies.
If we still work in the spinor representation, we can express
the lowest Landau level wavefunctions as
$\psi_{LLL}(x_i)=f_{LLL}(x_i) e^{-\frac{r^2}{2l_0^2}}$,
where each component of the spinor $f_{LLL}$ is a polynomial of
$r_i$ $(1\le i \le N)$.
To work out the analytic properties of $f_{LLL}$, we
factorize Eq. \ref{eq:4DQHE} as
\begin{eqnarray}
H^{ND,LL}= \hbar \omega_0 \left(\Gamma^i a_i^\dagger\right)
\left(\Gamma^j a_j\right),
\end{eqnarray}
where $a_i$ is the phonon operator in the $i$-th dimension
defined as
$a_i=\frac{1}{\sqrt 2}\big(\frac{1}{l_0} r_i +i \frac{l_0}{\hbar }p_i\big)$,
and $l_0=\sqrt{\frac{\hbar}{m\omega_0}}$.
Then $f_{LLL}(x_i)$ satisfies the following equation,
\begin{eqnarray}
\Gamma^j \frac{\partial}{\partial x_j} f_{LLL}(x_i)=0,
\end{eqnarray}
which can be viewed as the Euclidean version of the
Weyl equation.
When coming back to 3D and 4D, and following
the mapping Eq. \ref{eq:map}, we arrive at quaternionic
analyticity.
New let us construct the off-centered solutions to the lowest
Landau level states in 4D.
We use $\vec r$ to denote a point in the subspace of $x_1$-$x_2$-$x_3$,
and $\hat \Omega$ as an arbitrary unit vector in it.
Set $\hat e=\hat \Omega$ and $\hat f=\hat e_4$ (the unit vector
along the 4th axis) in Eq. \ref{eq:4Dcoherent}.
$\alpha_{\hat\Omega\hat e_4}$ satisfies
\begin{eqnarray}
(\sigma_{i4}\Omega_i)\alpha_{\hat\Omega\hat e_4 }
=(\vec \sigma \cdot \hat \Omega) \alpha_{\hat\Omega\hat e_4}
=\alpha_{\hat\Omega\hat e_4},
\end{eqnarray}
hence,
\begin{eqnarray}
\alpha_{\hat\Omega\hat e_4}=(\cos\frac{\theta}{2},\sin\frac{\theta}{2} e^{i\phi})^T,
\label{eq:alpha}
\end{eqnarray}
where we have used the gauge convention that the singularity
is located at the south pole.
Define the magnetic translation in the $\hat\Omega$-$\hat e_4$ plane,
\begin{eqnarray}
T_{\hat\Omega x_4} (u_0 \hat x_4)=
\exp\Big(-u_0 \partial_{x_4} -\frac{i}{4l_{so}^2}
(\vec r \cdot \hat\Omega) u_0 \Big),
\end{eqnarray}
which translates along the $\hat e_4$-axis at the distance of $u_0$.
Apply this translation to the state of $e^{-r^2/4l_{so}^2}\otimes \alpha_{\hat\Omega\hat e_4}$, we arrive at the off-center solution
\begin{eqnarray}
\psi_{\Omega,u_0}(\vec r, x_4)= e^{-\frac{r^2+x_4^2}{4l_{so}^2}}
e^{-i\frac{r u_0}{2l_{so}^2}} \otimes \alpha_{\hat\Omega\hat e_4}.
\end{eqnarray}
Next, we perform the Fourier transform over the direction $\hat \Omega$,
\begin{eqnarray}
\psi_{4D;j,j_z}(\vec r, x_4)&=& \int d \Omega ~
Y_{-\frac{1}{2},l+\frac{1}{2}, m+\frac{1}{2}} (\hat \Omega)
\psi_{\Omega,w_0}(\vec r, x_4), \ \ \, \ \ \,
\label{eq:4D_offcenter}
\end{eqnarray}
where $j=l+\frac{1}{2}$ and $j_z=m+\frac{1}{2}$.
Due to the Berry phase structure $\alpha_{\hat \Omega\hat e_4}$
over $\hat \Omega$, monopole spherical harmonics,
$Y_{-\frac{1}{2},l+\frac{1}{2},m+\frac{1}{2}}(\hat\Omega)$,
are used instead of the regular spherical harmonics.
Then Eq. \ref{eq:4D_offcenter} possesses the 3D rotational symmetry
around the new center $(0,0,0, w_0)$, and Eq. \ref{eq:4D_offcenter}
possesses with the good quantum numbers of 3D angular momentum $(j, j_z)$.
The monopole harmonic function $Y_{q;jj_z}(\hat \Omega)$ here
is defined as
\begin{eqnarray}
Y_{q;jj_z}(\hat \Omega)=\sqrt{\frac{2j+1}{4\pi}} e^{i(j_z+q)\phi} d^l_{j_z,-q}(\theta),
\end{eqnarray}
where $\theta$ and $\phi$ are the polar and azimuthal angles
of $\hat \Omega$, and
$d^l_{j_z,-q}(\theta)=\langle jj_z|e^{-iJ_y\theta}|j-q\rangle$
is the standard Wigner rotation $d$-matrix.
The gauge choice is consistent with that of the
Eq. \ref{eq:alpha}.
\subsection{\small Boundary helical Dirac and Weyl modes}
\label{sect:3Dboundary}
The topological nature of the 3D Landau level states exhibits clearly
in the gapless surface spectra.
Consider a ball of the radius $R_0\gg l_{so}$ imposed by the open
boundary condition.
We have numerically solved the spectra as shown in Fig. \ref{fig:3DLL}
(B).
Inside the bulk, the Landau level spectra are flat with respect to $j_+=l+\frac{1}{2}$.
As $l$ increases to large values such that the classic orbital radiuses
approach the boundary, the Landau levels become surface states
and develop dispersive spectra.
We can also derive the effective equation for the surface mode based on Eq. \ref{eq:3D_symm}.
Since $r$ is fixed at the boundary, it becomes a rotator equation on the sphere.
By linearizing the dispersion at the chemical potential $\mu$,
and replacing the angular momentum quantum number $l$ by the operator
$\vec \sigma \cdot \vec L$, we arrive at
$H_{sf}=(v_{f}/R_{0})\vec{\sigma} \cdot \vec{L}-\mu$
with $v_{f}$ the Fermi velocity.
This is the helical Dirac equation defined on the boundary sphere.
When expanded in the local patch around the north pole
$R_{0}\hat{z}$, we arrive at
\begin{eqnarray}
H_{sf}=\hbar v_{f}(\vec{k}\times \vec{\sigma})\cdot \hat{z}-\mu.
\end{eqnarray}
The gapless surface states are robust against time-reversal invariant
perturbations if odd numbers of helical Fermi surfaces exist according
to the $\mathbb{Z}_2$ criterion \cite{kane2005,kane2005a}.
Since each fully occupied Landau level contributes one helical
Dirac Fermi surface, the bulk is topologically nontrivial if
odd numbers of Landau levels are occupied.
A similar procedure can be applied to the high-dimensional case
by imposing the open boundary condition to Eq. \ref{eq:4DQHE}.
For example, around the north pole of $r_N=(0,...., R_0)$,
the linearized low energy equation for the boundary modes is
\begin{eqnarray}
H_{bd}=\hbar v_f \sum_{i=1}^{D-1} k_i \Gamma^{iN} -\mu.
\end{eqnarray}
On the boundary of the 4D sphere, it becomes the 3D Weyl equation that
\begin{eqnarray}
H_{bd}=\hbar v_f \vec k \cdot \vec \sigma -\mu.
\end{eqnarray}
\subsection{Bulk-boundary correspondence}
\begin{widetext}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& Bulk (Euclidean) & Boundary (Minkowski)\\
\hline
& & \\
2D LLL & complex analyticity & 1D chiral wave \\
& $\partial_x f +i \partial_y f=0$
& $\partial_t \psi+\partial_x \psi=0$ \\
& & \\
\hline
& & \\
3D LLL & (3D) quaternionic analyticity & 2D helical Dirac mode\\
& $\partial_x f +i \partial_y f +j\partial_z f=0$ &
$ \partial_t \psi +\sigma_2 \partial_x \psi -\sigma_1 \partial_y \psi=0$\\
& & \\
\hline
& & \\
4D LLL & quaternionic analyticity & 3D Weyl mode \\
& $\partial_x f +i \partial_y f +j\partial_z f +k\partial_u f=0$
& $\partial_t \psi +\sigma_1 \partial_x \psi +\sigma_2 \partial_y \psi
+\sigma_3 \partial_z \psi=0$ \\
& & \\
\hline
\end{tabular}
\caption{\normalsize Bulk-boundary correspondence in the lowest
(LLL) states in 2, 3, and 4 dimensions.}
\end{center}
\end{table}
\end{widetext}
We have already studied the bulk and boundary states of 2D, 3D and
4D lowest Landau level states.
They exhibit a series of interesting bulk-boundary correspondences
as summarized in Table I.
In the 2D case, the bulk wavefunctions in the lowest Landau level
is complex analytic satisfying the Cauchy-Riemann condition.
The 1D edge states satisfy the chiral wave equation Eq. \ref{eq:chiraledge}.
It is essentially the Weyl equation, which is actually a
single component equation in 1D.
It can be viewed as the Minkowski version of the Cauchy-Riemann
condition of Eq. \ref{eq:cauchy}.
Or, conversely, the Cauchy-Riemann condition for the bulk
wavefunctions can be viewed as the Euclidean
version of the Weyl equation.
This correspondence goes in parallel in 3D and 4D lowest Landau
level wavefunctions.
Their bulk wavefunctions satisfy the quaternionic analytic
conditions, which can be viewed as an Euclidean version of the
helical Dirac and Weyl equations, respectively.
\subsection{Many-body interacting wavefunctions}
It is natural to further investigate many-body interacting wavefunctions
in the lowest Landau levels in 3D and 4D.
As is well-known that the complex analyticity of the 2D lowest Landau level wavefunctions results in the elegant from of the 2D Laughlin wavefunction Eq. \ref{eq:laughlin}, which describes a 2D quantum liquid \cite{laughlin1983,girvin1999}.
It is natural to further expect that the quaternionic analyticity of the
3D and 4D lowest Landau levels would work as a guidance in
constructing high-dimensional SU(2) invariant quantum liquid.
Nevertheless, the major difficulty is that quaternions do not commute.
It remains challenging how to use quaternions to represent
a many-body wavefunction with the spin degree of freedom.
Nevertheless, we present below the spin polarized fractional many-body
states in 3D and 4D Landau levels.
In the 3D case, if the interaction is spin-independent, we expect
spontaneous spin polarization at very low fillings due to the
flatness of lowest Landau level states in analogy to the 2D quantum Hall ferromagnetism \cite{Lee1990,Sondhi1993,Fertig1994,Read1995,girvin1999}.
According to Eq. \ref{eq:3Dcoherent}, fermions concentrate to
the highest weight states in the orbital plane $\hat e_1$-$\hat e_2$
with spin polarized along $\hat e_3$, then it is reduced to a 2D
quantum Hall-like problem on a membrane floating in the 3D space.
Any 2D fractional quantum Hall-like state can be formed under suitable
interaction pseudopotentials \cite{haldane1983,haldane1985,Prange1990}.
For example, the $\nu=\frac{1}{3}$ Laughlin-like state on this
membrane is constructed as
\begin{eqnarray}
&&\Psi_{\frac{1}{3}}(\vec r_1,\vec r_2, ..., \vec r_n)_
{\sigma_1\sigma_2...\sigma_n}\nonumber \\
&=&\prod_{i < j} [(\vec r_i-\vec r_j)\cdot (\hat e_1 +i \hat e_2)]^3
\otimes \ket{\alpha_{\hat e_3}}_{\sigma_1} \ket{\alpha_{\hat e_3}}_{\sigma_2} ...
\ket{\alpha_{\hat e_3}}_{\sigma_n}, \nonumber \\
\label{eq:ferro_3D}
\end{eqnarray}
where $\ket{\alpha_{\hat e_3}}$ represents a polarized spin eigenstate along $\hat e_3$, and the Gaussian weight is suppressed for simplicity.
Such a state breaks rotational symmetry and time-reversal symmetry spontaneously,
thus it possesses low energy spin-wave modes.
Due to the spin-orbit locked configuration in Eq. \ref{eq:3Dcoherent},
spin fluctuations couple to the vibrations of the orbital motion plane,
thus the metric of the orbital plane becomes dynamic.
This is a natural connection to the work of geometrical
description in fractional quantum Hall states \cite{haldane2011,can2014,klevtsov2017}.
Let us consider the 4D case, we assume that spin is polarized
as the eigenstate $\ket{\uparrow}$ of
$\Gamma^{12}=\Gamma^{34}=\sigma_3$.
The corresponding spin-polarized lowest Landau level wavefunctions
are expressed as
\begin{eqnarray}
\Psi^{4D}_{LLL,m,n}=(x+iy)^m (z+iu)^n
\otimes \ket{\uparrow},
\end{eqnarray}
with $m, n\ge 0$.
If all these spin polarized lowest Landau level states with
$0 \le m < N_m$ and $0 \le n < N_n $ are
filled, the many-body wavefunction is a Slater-determinant as
\begin{eqnarray}
\Psi^{4D}(v_1, w_1; \cdots; v_N, w_N)= \det[v_i^{\alpha} w_i^{\beta}],
\label{eq:4Dwf}
\end{eqnarray}
where the coordinates of the $i$-th particle form two pairs
of complex numbers as $v_i=x_i+iy_i$ and $w_i=z_i+iu_i$;
$\alpha$, $\beta$ and $i$ satisfy $0 \le \alpha <N_m $, $0\le
\beta < N_n $, and $1\le i \le N=N_m N_n$.
Such a state has a 4D uniform density as $\rho=\frac{1}{4\pi^4 l_G^2}$.
A Laughlin-like wavefunction can be written down as
$\Psi^{4D}_k =(\Psi^{4D})^k$ whose filling relative to $\rho$ should be $1/k^2$.
It would be interesting to further study its electromagnetic responses
and fractional topological excitations based on $\Psi^{4D}_k$.
Again such a state spontaneously breaks rotational symmetry, and
the coupled spin and orbital excitations would be interesting.
\section{Dimensional reductions: 2D and 3D Landau levels with broken parity}
\label{sect:reduction}
In this section, we review another class of isotropic Landau level-like
states with time-reversal symmetry but broken parity in both 2D and 3D.
The Hamiltonians are again harmonic potential plus spin-orbit coupling,
but it is the coupling between spin and linear momentum, not orbital
angular momentum \cite{wu2011_Ian,li2012a,li2016}.
They exhibit topological properties very similar to Landau levels.
An early study of these systems filled with bosons can be found in Ref. \cite{wuexciton2008}.
The spin-orbit coupled Bose-Einstein condensations (BECs) spontaneously
break time-reversal symmetry, and exhibit the skyrmion type spin textures
coexisting with half-quantum vortices, which have been reviewed
in Ref. \cite{zhouXF2013}.
Spin-orbit coupled BECs have become an active research direction
of cold-atom physics, as extensively studied in literature.
\cite{wu2011_Ian,hu2012,sinha2011,ghosh2011,wang2010,ho2011}.
\subsection{The 2D parity-broken Landau levels}
We consider the Hamiltonian of Rashba spin-orbit coupling combined
with a 2D harmonic potential as
\begin{eqnarray}
H_{2D,hm}=-\frac{\hbar^2\nabla^2}{2M}+\frac{1}{2} M \omega^2r^2
-\lambda (-i\hbar\nabla_x \sigma_y +i\hbar \nabla_y \sigma_x),
\nonumber \\
\label{eq:rashba}
\end{eqnarray}
where $\lambda$ is the spin-orbit coupling strength with the unit of velocity.
Eq. \ref{eq:rashba} possesses the $C_{v\infty}$-symmetry and time-reversal
symmetry.
We fill the system with fermions and work on its topological properties.
There are two length scales.
The trap length scale is defined as $l_T=\sqrt{\frac{\hbar }{M\omega}}$.
If without the trap, the single particle states $\psi_\pm (\vec k)$
are eigenstates of the helicity operator $\vec \sigma \cdot (\vec k
\times \hat z)$ with eigenvalues of $\pm 1$.
Their spectra are $\epsilon_{\pm}(\vec k)=
\hbar^2(k \mp k_{0})^2/(2M)$, respectively.
The lowest energy states are $\psi_+(\vec k)$ located around a ring in
momentum space with radius $k_{0}=M\lambda/\hbar$.
This introduces a spin-orbit length scale as $l_{so}=1/k_0$.
Then the ratio between these two length scales defines a dimensionless
parameter $\alpha=l_T/l_{so}$,
which describes the spin-orbit coupling strength relative
to the harmonic potential.
In the case of strong spin-orbit coupling, {\it i.e.}, $\alpha\gg 1$,
a clear picture appears in momentum space.
The low energy states are reorganized from the plane-wave states
$\psi_+(\vec k)$ with $k\approx k_0 $.
Since $\alpha\gg 1$, we can safely project out the high energy
negative helicity states $\psi_{-}(\vec k)$, then
the harmonic potential in the low energy sector
becomes a Laplacian in momentum space
coupled to a Berry connection $\vec A_k$ as
\begin{eqnarray}
V=\frac{M}{2}\omega^2 r^2=\frac{M}{2} \omega^2 (i\nabla_k - A_k)^2,
\end{eqnarray}
which drives particle moving around the ring.
It is well-known that for the Rashba Hamiltonian, the Berry connection
$A_k$ gives rise to a $\pi$-flux at $\vec k=(0,0)$ but zero
Berry curvature at $\vec k\neq 0$ \cite{xiao2010}.
The consequence is that
the angular momentum eigenvalues become half-integers as $j_z=m+\frac{1}{2}$.
The angular dispersion of the spectra can be estimated as
$E_{agl}(j_z)=(j_z^2/2\alpha^2) \hbar\omega$, which is strongly
suppressed by spin-orbit coupling.
On the other hand, the radial energy quantization remains as
usual $E_{rad}(n_r)=(n_r+\frac{1}{2}) \hbar\omega$ up to a constant.
Thus the total energy dispersion is
\begin{eqnarray}
E_{n_r,j_z}\approx \Big(n_r+ \frac{1}{2}
+\frac{j_z^2}{2\alpha^2} \Big )\hbar \omega.
\end{eqnarray}
Similar results have also been obtained in recent works of Ref.
\cite{hu2012,sinha2011,ghosh2011}.
Since $\alpha\gg 1$, the spectra are nearly flat with
respect to $j_z$, we can treat $n_r$ as a Landau level index.
The wavefunctions of Eq. \ref{eq:rashba} in the lowest Landau level
with $n_r=0$ can be expressed in the polar coordinate
as Eq. \ref{eq:rashba_wf}.
Next we define the edge modes of such systems, and
their stability problem is quite different from that of
the chiral edge modes of 2D magnetic Landau level systems.
In the regime that $\alpha\gg 1$, the spin-orbit length $l_{so}$
is much shorter than $l_T$, such that $l_T$ is viewed as the cutoff
of the sample size.
States with $|j_z|< \alpha$ are viewed as bulk states
which localize within the region of $r<l_T$.
For states with $|J_z|\sim \alpha$, their energies touch the
the bottom of the next higher Landau level, and thus they
should be considered as edge states.
Due to time-reversal symmetry, each filled Landau level of
Eq. \ref{eq:rashba} gives rise to a branch of edge modes of
Kramers' doublets $\psi_{n_r,\pm j_z}$.
In other words, these edge modes are helical rather than chiral.
Similarly to the $Z_2$ criterion in Ref. \cite{kane2005,kane2005a},
which was defined for Bloch wave states,
in our case the following mixing term,
$
H_{mx}=\psi^\dagger_{2D,n_r,j_z} \psi_{2D,n_r, -j_z}+h.c.,
$
is forbidden by time-reversal symmetry.
Consequently, the topological index for this system is $Z_2$.
\subsection {Dimensional reduction from 3D}
In fact, we construct a Hamiltonian closely related to Eq. \ref{eq:rashba}
such that its ground state is solvable exhibiting exactly flat dispersion.
It is a consequence of dimensional reduction based on the 3D Landau level Hamiltonian Eq. \ref{eq:3D_symm}.
We cut a 2D off-centered plane perpendicular to the $z$-axis
with the interception $z=z_0$.
In this off-centered plane, inversion symmetry is broken, and Eq. \ref{eq:3D_symm} is reduced to
\begin{eqnarray}
H_{2D,re}&=& H_{2D,hm} - \omega L_z \sigma_z.
\label{eq:2D_reduce}
\end{eqnarray}
The first term is just Eq. \ref{eq:rashba} by identifying $\lambda=\omega z_0$
and the frequency of the 2nd term is the same as that of the harmonic trap.
If $z_0=0$, the Rashba spin-orbit coupling vanishes, and
Eq.\ref{eq:2D_reduce} becomes the 2D quantum spin-Hall Hamiltonian,
which is a double copy of Eq. \ref{eq:2D_symm}.
At $z_0\neq 0$, $\sigma_z$ is no longer conserved due to
spin-orbit coupling.
In Sect. \ref{sect:off-center}, we derived the off-centered ellipsoid
type wavefunction in Eq. \ref{eq:fourier}.
After setting $z=z_0$ in Eq. \ref{eq:fourier}, we arrive
at the following 2D wavefunction,
\begin{eqnarray}
\psi_{2D,j_z}(r,\phi)&=&
e^{-\frac{r^2}{4l_{so}^2}}\Big\{
e^{im\phi} J_m(k_0 r) \ket{\uparrow}\nonumber \\
&+&e^{i(m+1)\phi} J_{m+1} (k_0 r)
\ket{\downarrow} \Big\},
\label{eq:rashba_wf}
\end{eqnarray}
where $J_m(k_0r)$'s are the Bessel functions.
It is straightforward to prove that the simple reduction indeed
gives rise to the solutions to the lowest Landau levels for Eq. \ref{eq:2D_reduce}, since the partial derivative along the $z$-direction
of the solution in Eq. \ref{eq:fourier} equal zero at $z=z_0$.
We also derive that the energy dispersion is
exactly flat as,
\begin{eqnarray}
H_{2D,re} ~\psi_{2D,j_z}= \Big(1 -\frac{\alpha^2}{2} \Big)
\hbar \omega~ \psi_{2D,j_z}.
\end{eqnarray}
The above two Hamiltonians Eq. \ref{eq:2D_reduce} and Eq. \ref{eq:rashba} are
nearly the same except the $L_z\sigma_z$ term, whose effect
relies on the distance from the origin.
Consider the lowest Landau level solutions at $\alpha\gg 1$.
The decay length of the Gaussian factor is $l_T$.
Nevertheless, the Bessel functions peak around $k_0 r_0\approx m$, i.e., $r_0\approx \frac{m}{\alpha} l_T$.
Hence for states with $j_z<\alpha$, their wavefunctions
already decay before reach $l_T$.
Then the $L_z \sigma_z$-term compared to the Rashba one
is a small perturbation at the order of $\omega r_0/
\lambda=r_0/z_0\ll 1$.
In this regime, these two Hamiltonians are equivalent.
In contrast, in the opposite limit that $j_z\gg \alpha^2$,
the Bessel functions are cut off by the Gaussian factor,
and only their initial power-law parts participate, and
the classic orbit radiuses are just $r_0\approx \sqrt{m} l_T$,
then the physics of Eq. \ref{eq:2D_reduce} is controlled by
the $L_s\sigma_z$-term as in the quantum spin Hall systems.
For the intermediate region that $\alpha<j_z< \alpha^2$, the physics
is a crossover between the above two limits.
The many-body physics based on the above spin-orbit coupled Landau levels
in Eq. \ref{eq:rashba_wf} would be very interesting.
Fractional topological states would be expected which
are both rotationally and time-reversal invariant.
However, $s_z$ is not a good quantum number and party is also broken,
hence, these states should be very different from a double copy of
fractional Laughlin states with spin up and down particles.
The nature of topological excitations and properties of
edge modes will be deferred to a future study.
\subsection{The 3D parity-broken Landau levels}
We have also considered the problem of 3D harmonic potential
plus a Weyl-type spin-orbit coupling as
\cite{li2012a},
\begin{eqnarray}
H_{3D,hm}=-\frac{\hbar^2 \nabla^2}{2M}+\frac{1}{2}M \omega^2r^2
-\lambda (-i\hbar\vec \nabla \cdot \vec \sigma).
\label{eq:3DSO}
\end{eqnarray}
The analysis can be performed in parallel to the 2D case.
In the absence of spin-orbit coupling, the low energy states of
Eq. \ref{eq:3DSO} in momentum space form a spin-orbit sphere.
The harmonic potential further quantizes the energy spectra as
\begin{eqnarray}
E_{n_r,j,j_z}\approx \big(n_r+ \frac{1}{2}
+\frac{j(j+1)}{2\alpha^2} \big )\hbar \omega,
\end{eqnarray}
where $n_r$ is the Landau level index and $j$ is the total
angular momentum.
Again $j$ takes half-integer values because the Berry phase
on the low energy sphere exhibits a unit monopole structure.
Now we perform the dimensional reduction from the 4D Hamiltonian
Eq. \ref{eq:4DQHE} to 3D.
We cut a 3D off-centered hyper-plane perpendicular to the 4-th
axis with the
interception $x_4=u_0$.
Within this 3D hyper-plane of $(x_1,x_2,x_3, x_4=u_0)$,
Eq. \ref{eq:4DQHE} is reduced to
\begin{eqnarray}
H_{3D,re}=H_{3D,hm}-\omega \vec L \cdot \vec \sigma,
\label{eq:3D_reduce}
\end{eqnarray}
where the first term is just Eq. \ref{eq:3DSO} with the spin-orbit
coupling strength set by $\lambda=\omega u_0$.
Again, based on the center-shifted wavefunction in the lowest Landau
level Eq. \ref{eq:4D_offcenter}, and by setting $x_4=u_0$, we
arrive at the following wavefunction
\begin{eqnarray}
\psi_{3D, J J_z}(\vec r)&=& e^{-\frac{r^2}{4l_{so}^2}} \Big\{ j_l(k_0 r)
Y_{+,J,J_z} (\Omega_r) \nonumber \\
&+&i j_{l+1}(k_0 r) Y_{-,J,J_z} (\Omega_r) \Big\},
\label{eq:3D_WF}
\end{eqnarray}
where $k_0=u_0/l_T^2=m \lambda/\hbar$;
$j_l$ is the $l$-th order spherical Bessel function.
$Y_{\pm,j, l,j_z}$'s are the spin-orbit coupled spherical harmonics defined as
\begin{eqnarray}
Y_{+,j,l,j_z}(\Omega)=\Big(\sqrt{\frac{l+m+1}{2l+1}} Y_{lm},
\sqrt{\frac{l-m}{2l+1}} Y_{l,m+1}\Big)^T
\nonumber
\end{eqnarray}
with the positive
eigenvalue of $l\hbar$ for $\vec \sigma \cdot \vec L$, and
\begin{eqnarray}
Y_{-,j,l,j_z}(\Omega)=\Big(-\sqrt{\frac{l-m}{2l+1}} Y_{lm},
\sqrt{\frac{l+m+1}{2l+1}} Y_{l,m+1}\Big)^T
\nonumber
\end{eqnarray}
with the negative
eigenvalue of $-(l+1)\hbar$ for $\vec \sigma \cdot \vec L$.
It is straightforward to check that $\psi_{3D,j,j_z}(\vec r)$
in Eq. \ref{eq:3D_WF} is the ground state wavefunction
satisfying
\begin{eqnarray}
H_{3D,re} \psi_{3D,j,j_z}(\vec r)= \Big (\frac{3}{2}
-\frac{\alpha^2}{2} \Big )\hbar \omega
\psi_{3D,j,j_z}(\vec r).
\end{eqnarray}
\section{High-dimensional Landau levels of Dirac fermions}
\label{sect:diracLL}
In this section, we review the progress on the study of 3D Landau levels
of relativistic Dirac fermions \cite{li2012}.
This is a square-root problem of the 3D Landau level problem of
Schr\"odinger fermions reviewed in Sect. \ref{sect:3DLL}.
This can also be viewed of Landau levels of complex quaternions.
\subsection{3D Landau levels for Dirac fermions}
In Eq. \ref{eq:2DLL_harm}, two sets of phonon creation and annihilation
operators $(a_x,a_y;a_x^\dagger,a_y^\dagger)$ are combined with the
real and imaginary units to construct Landau level Hamiltonian for
2D Dirac fermions.
Science in 3D there exist three sets of phonon creation and annihilation operators,
complex numbers are insufficient.
The new strategy is to employ Pauli matrices $\vec\sigma$
such that
\begin{eqnarray}
H_{3D}^D&=& v \Big\{ \alpha_i p_i
+ \gamma_i i \hbar \frac{r_i}{l_0^2} \Big\}
=
\frac{\hbar \omega}{\sqrt 2}
\left[
\begin{array}{cc}
0 & i\sigma_i a_i^\dagger \\
-i\sigma_i a_i & 0
\end{array}
\right ], \ \ \,
\label{eq:Dirac_3D}
\end{eqnarray}
where the repeated index $i$ runs over $x,y$ and $z$;
$v_F= \frac{1}{2} l_0\omega$.
The convention of $\gamma$-matrices is
\begin{eqnarray}
\beta=\gamma_0=\tau_3\otimes I, \ \ \, \alpha_i=\tau_1\otimes\sigma_i, \ \ \,
\gamma_i=\beta\alpha_i=i\tau_2\otimes\sigma_i.
\end{eqnarray}
Eq. \ref{eq:Dirac_3D} contains the complex combination of
momenta and coordinates, thus it can be viewed as the generalized
Dirac equation defined in the phase space.
Apparently, Eq. \ref{eq:Dirac_3D} is rotationally invariant.
It is also time-reversal invariant with the definition
$T=\gamma_2\gamma_3K$ where $K$ is the complex conjugation,
and $T^2=-1$.
Since $\beta H_{3D}^D \beta =-H_{3D}^D$, $H_{3D}^D$ possesses the
particle-hole symmetry and its spectra are symmetric with
respect to the zero energy.
Similar to the 2D case, $(H^{D}_{3D})^2$ has a supersymmetric structure.
The square of Eq. \ref{eq:Dirac_3D} is block-diagonal, and two blocks
are just the non-relativistic 3D Landau level Hamiltonians
in Eq. \ref{eq:3D_symm},
\begin{eqnarray}
\frac{(H^{D}_{3D})^2}{\frac{1}{2}\hbar \omega}=
\left[ \begin{array}{cc}
H^+_{3D,sym}-\frac{3}{2}\hbar\omega& 0\\
0&H^-_{3D,sym}+\frac{3}{2}\hbar\omega
\end{array}
\right],
\label{eq:super}
\end{eqnarray}
where the mass $M$ in $H^\pm_{3D,sym}$ is defined through the relation $l_{0}=\sqrt{\hbar/(M\omega)}$.
Based on Eq. \ref{eq:super}, the energy eigenvalues of Eq. \ref{eq:Dirac_3D} are
$E_{\pm n_r, j, j_z}=\pm \hbar \omega \sqrt n_r$, corresponding to taking
positive and negative square roots of the non-relativistic dispersion,
respectively.
The Landau level wavefunctions of the 3D Dirac electrons are expressed
in terms of the non-relativistic ones of
Eq. \ref{eq:3D_symm} as
\begin{eqnarray}
\Psi_{\pm n_r,j,j_z} (\vec r)=\frac{1}{\sqrt 2}
\left( \begin{array}{c}
\psi_{n_r,j_+,l, j_z} (\vec r) \\
\pm i\psi_{n_r-1,j_-, l+1, j_z }(\vec r)
\end{array}
\right).
\label{eq:LLWF}
\end{eqnarray}
Please note that the upper and lower two components
possess different values of orbital angular momenta.
They exhibit opposite helicities of $j_\pm$, respectively.
The zeroth Landau level ($n_r=0$) states are special: There is only one
branch, and only the first two components of the wavefunctions
are non-zero as
\begin{eqnarray}
\Psi_{n_r=0, j, j_z} (\vec r)=\left[\begin{array}{c}
\Psi_{LLL,j_+,j_z}(\vec r)\\
0
\end{array}
\right],
\end{eqnarray}
where $\Psi_{LLL,j_+,j_z}$'s are the lowest Landau level solutions
to the non-relativistic Hamiltonian Eq. \ref{eq:LL_orthonomal}.
Again the nontrivial topology of the 3D Dirac Landau problem manifests
in the gapless surface modes.
Consider a spherical boundary with a large radius $R$.
The Hamiltonian takes the form of Eq. \ref{eq:Dirac_3D} inside the sphere,
and has the usual massive Dirac Hamiltonian
$H_{D}=\alpha_i P_i + \beta\Delta$ outside.
We also take the limit of $|\Delta|\rightarrow \infty$.
Loosely speaking, this is a square-root version of the open boundary
problem of the 3D non-relativistic case
in Sect. \ref{sect:3Dboundary}.
Since square-roots can be taken as positive and negative,
each branch of the surface modes in the non-relativistic Schr\"odinger case
corresponds to a pair of relativistic surface branches.
These two branches disperse upward and downward as increasing the
angular momentum $j$, respectively.
However, the zeroth Landau level branch is singled out.
We can only take either the positive, or, negative
square root, for its surface excitations.
Hence, the surface spectra connected to the bulk zeroth Landau level
disperse upward or downward depending on the sign of the vacuum mass.
\subsection{Non-minimal Pauli coupling and anomaly}
Due to the particle-hole symmetry of Eq. \ref{eq:Dirac_3D}, the 3D
zeroth Landau level states are half-fermion modes in the same
way as those in the 2D Dirac case.
Moreover, in the 3D case, the degeneracy is over the
3D angular momentum numbers $(j_+, j_z)$, thus the
degeneracy is much higher than that of 2D.
According to whether the chemical potential $\mu$ approaches
$0^+$ or $0^-$, each state in the zeroth lowest Landau level contributes a
positive, or, negative half fermion number, respectively.
The Lagrangian of the 3D massless Dirac Landau level problem is,
\begin{eqnarray}
L= \bar \psi \Big\{ \gamma_0 i\hbar \partial_t
- i v \gamma_i \hbar \partial_i\Big\} \psi
- v \hbar \bar\psi i \gamma_0 \gamma_i \psi
F^{0i}(r),
\label{eq:lang}
\end{eqnarray}
where $F^{0i}= x_i/l_0^2$.
In all the dimensions higher than 2, $i\gamma_0\gamma_i$'s are
a different set from $\gamma_i$'s, thus Eq. \ref{eq:lang} is an example of
non-minimal coupling of the Pauli type.
More precisely, it is a coupling between the electric field and the electric
dipole moment.
In the 2D case, the Lagrangian has the same form as Eq. \ref{eq:lang},
however, since $\gamma_{0,1,2}$ are just the usual Pauli matrices,
it is reduced to the minimal coupling to the $U(1)$ gauge field.
Eq. \ref{eq:lang} is a problem of massless Dirac fermions coupled to
a background field via non-minimal Pauli coupling at 3D and above.
Fermion density is pumped by the background field from vacuum.
This is similar to parity anomaly, and indeed it is
reduced to parity anomaly in 2D.
However, the standard parity anomaly only exists in even spatial
dimensions \cite{redlich1984,redlich1984a,semenoff1984,niemi1986}.
By contrast, the Landau level problems of massless Dirac fermions
can be constructed in any high spatial dimensions.
Obviously, they are not chiral anomalies defined in odd spatial
dimensions, either.
It would be interesting to further study the nature of such
kind of ``anomaly''.
In fact, Eq. \ref{eq:Dirac_3D} is just one possible representation
for Landau levels of 3D massless Dirac fermions.
A general 3D Dirac Landau level Hamiltonian with a mass term
can be defined as
\begin{eqnarray}
H_{3D}^D (\hat e_1, \hat e_2,\hat e_3)&=&
v \Big [(\vec \tau \cdot \hat e_1) \otimes\sigma_i P_i
+ \hbar/l_0^2 (\vec \tau \cdot \hat e_2 )\otimes \sigma_i r_i
\Big] \nonumber \\
&+& mv^2 (\vec \tau \cdot \hat e_3)\otimes I,
\label{eq:3D_Dirac_general}
\end{eqnarray}
where $\tau_{1,2,3}$ are Pauli matrices acting in the particle-hole
channel, and $\hat e_{1,2,3}$ form an orthogonal triad in the 3D space.
Eq. \ref{eq:Dirac_3D} corresponds to the case of $\hat e_1=\hat x$
and $\hat e_2=\hat y$, and $m=0$.
The parameter space of $H_{3D}^D (\hat e_1, \hat e_2, \hat e_3)$
is the triad configuration space of $SO(3)$.
Consider that the configuration of the triad $\hat e_{1,2,3}$
is spatially dependent.
The first term in Eq. \ref{eq:3D_Dirac_general} should be symmetrized
as $\frac{1}{2}\vec \tau \cdot
\big[(\hat e_1(r) P_i + P_i \hat e_1(r) \big] \otimes \sigma_i$.
The spatial distribution of the triad of $\hat e_{1,2,3}(\vec r)$
can be in a topologically nontrivial configuration.
If the triad is only allowed to rotate around a fixed axis, its
configuration space is $U(1)$ which can form a vortex line
type defect.
There should be a Callan-Harvey type effect of the fermion
zero modes confined around the vortex line \cite{Callan1985}.
In general, we can also have a 3D skyrmion type defect of the triad
configuration.
These novel defect problems and the associated zero energy fermionic
excitations will be deferred for later studies.
\subsection{Landau levels for Dirac fermions in four dimensions
and above}
The Landau level Hamiltonian for Dirac fermions can be generalized
to arbitrary $N$-dimensions ($N$-D) by replacing the Pauli matrices in Eq. \ref{eq:Dirac_3D} with the Clifford algebra $\Gamma$-matrices in $N$-D
as presented in Appendix \ref{appendix:cliff}.
In odd dimensions $D=2k+1$, we use the $k$-th rank
$\Gamma$-matrices to construct the $D=2k+1$ dimensional
Dirac Landau level Hamiltonian,
\begin{eqnarray}
H^D_{2k+1}=\frac{\hbar\omega_0}{2}\left(
\begin{array}{cc}
0& i\Gamma_i^{(k)} a^\dagger_i \\
-i\Gamma_i^{(k)} a_i & 0
\end{array}
\right),
\label{eq:highDirac}
\end{eqnarray}
where $\Gamma_i^{(k)}$ is $2^k\times 2^k$ dimensional matrix,
and $1\le i\le 2k+1$.
Again, $(H_{2k+1}^{D})^2$ are reduced to a supersymmetric version
of the $2k+1$-dimensional Landau level Hamiltonian for Sch\"odinger
fermions in Eq. \ref{eq:4DQHE}.
All other properties are parallel to the 3D case explained before.
For even dimensions $D=2k$, we still take Eq. \ref{eq:highDirac}
by suppressing the $2k+1$-th dimension.
Nevertheless, such a construction is reducible.
In the representation presented in
Appendix \ref{appendix:cliff},
Eq \ref{eq:highDirac} after eliminating the $\Gamma^{(k)}_{2k+1}$ term
can be factorized into a pair of Hamiltonians
\begin{eqnarray}
H^{\pm,D}_{2k}=\frac{\hbar\omega_0}{2}\left(
\begin{array}{cc}
0& \pm a^\dagger_{2k}+ i \sum_{i=1}^k \Gamma_i^{(k-1)} a^\dagger_i \\
\pm a_k -i \sum_{i=1}^k \Gamma_i^{(k-1)} a_i & 0
\end{array}
\right),\nonumber \\
\end{eqnarray}
where $\pm$ correspond to the pair of fundamental and anti-fundamental
spinor representations in even dimensions.
For example, for the 4D system, we have
\begin{eqnarray}
H_{4D}^{\pm, D} =\frac{\hbar \omega}{\sqrt 2}\left[ \begin{array}{cc}
0& \pm a^\dagger_4 + i \sigma_i a_i^\dagger\\
\pm a_4 - i \sigma_i a_i &0
\end{array}
\right].
\label{eq:reduce}
\end{eqnarray}
Since three quaternionic imaginary units $i,j$, and $k$ can be
mapped to Pauli matrices $i\sigma_1, i\sigma_2$, and $i\sigma_3$,
respectively,
and the annihilation and creation operators are essentially complex.
$\pm a_4 - i \sigma_i a_i$ can be viewed as complex quaternions.
Hence, Eq. \ref{eq:reduce} is a complex quaternionic
generalization of the 2D Dirac Landau level Hamiltonian Eq. \ref{eq:2DLL_harm}.
\section{High-dimensional Landau levels in the Landau-like gauge}
\label{sect:landaugauge}
We have discussed the construction of Landau levels in high dimensions
for both Schr\"odinger and Dirac fermions in the symmetric-like
gauge.
In those problems, the rotational symmetry is explicitly maintained.
Below we review the construction of Landau levels in the Landau-like
gauge by reorganizing plane-waves to exhibit non-trivial
topological properties \cite{li2013a}.
It still preserves the flat spectra but not the rotational
symmetry.
\subsection{Spatially separated 1D chiral modes -- 2D Landau level}
We recapitulate the Landau level in the Landau gauge.
By setting $A_x=By$ and $A_y=0$ in the Hamiltonian Eq. \ref{eq:2DLL},
we arrive at
\begin{eqnarray}
H_{2D,L}&=& \frac{P_y^2}{2M} +\frac{(P_x-\frac{e}{c} A_x)^2}{2M}\nonumber \\
&=&\frac{P_y^2}{2M} +\frac{1}{2}M \omega^2 (y-l_B^2 P_x)^2,
\label{eq:2D_LL_Landau}
\end{eqnarray}
with $l_B=\sqrt{\frac{\hbar}{M\omega}}$.
The Landau level wavefunctions are a product of a plane wave along
the $x$-direction and a 1D harmonic oscillator wavefunction in the
$y$-direction,
\begin{eqnarray}
\psi_n(x,y)=e^{ik_x} \phi_n(y-y_0(k)),
\end{eqnarray}
where $\phi_n$ is the $n$th harmonic oscillator eigenstate
with the characteristic length $l_B$, and its
equilibrium position is determined by the momentum $k_x$,
$y_0(k_x)=l_B^2 k_x$.
Hence, the Landau level states with positive and negative values of $k_x$
are shifted oppositely along the $y$-direction, and become spatially
separated.
If imposing the open boundary condition along the $y$-axis, chiral
edge modes appear.
The 2D quantum Hall effect is just the spatially separated 1D chiral
anomaly in which the chiral current becomes the transverse charge current.
After the projection to the lowest Landau level, we identify
$y=l_B^2 k_x$, hence, the two spatial coordinates
$x$ and $y$ become non-commutative as \cite{lee2004}
\begin{eqnarray}
[x,y]_{LLL}=il_B^2.
\end{eqnarray}
In other words, the $xy$-plane is equivalent to the 2D phase space of a 1D system
$(x;k_x)$ after the lowest Landau level projection.
\subsection{Spatially separated 2D helical modes - 3D Landau level}
The above picture can be generalized to the 3D Landau level states:
We keep the plane-wave modes with the good momentum numbers $(k_x,k_y)$
and shift them along the $z$-axis.
Spin-orbit coupling is introduced to generate the helical structure
to these plane-waves, and the shifting direction
is determined by the sign of helicity.
To be concrete, the 3D Landau level Hamiltonian in the Landau-like
gauge is constructed as follows \cite{li2013a},
\begin{eqnarray}
H^{\pm}_{3D,L} &=&
\frac{\vec P^2}{2M}+\frac{1}{2}M \omega_{so}^2 z^2
\mp \omega_{so} z (P_x \sigma_y-P_y \sigma_x) \nonumber \\
&=& \frac{P_z^2}{2M } + \frac{1}{2} M \omega_{so}^2
[z \mp \frac{1}{\hbar}l_{so}^2 (P_x \sigma_y - P_y\sigma_x) ]^2, \nonumber\\
\label{eq:3D_Landau}
\end{eqnarray}
where $l_{so}=\sqrt{\hbar/(M\omega_{so})}$.
The key of Eq. \ref{eq:3D_Landau} is the $z$-dependent Rashba spin-orbit
coupling, such that it can be decomposed into a set of 1D harmonic
oscillators along the $z$-axis coupled to 2D helical plane-waves.
Define the helicity operator $\hat \Sigma_{2d} (\hat k_{2d} )=
\hat k_x \sigma_y -\hat k_y \sigma_x$ where $\hat k$ is the
unit vector along the direction of $\vec k$.
$\chi_\Sigma(\hat k_{2d})$ is the eigenstate of $\hat \Sigma$
and $\Sigma=\pm 1$ is the eigenvalue.
Then the 3D Landau level wavefunctions are expressed as
\begin{eqnarray}
\Psi_{n, \vec k_{2d},\Sigma}(\vec{r}) = e^{i \vec k_{2d} \cdot \vec r_{2d}}
\phi_n [z - z_0( k_{2d},\Sigma)] \otimes \chi_{\Sigma}(\hat k_{2d}),
\label{eq:3DLL_WF}
\end{eqnarray}
where $\vec k_{2d}=(k_x,k_y)$, $\vec r_{2d}=(x,y)$,
and $k_{2d}=(k_x^2+k_y^2)^{\frac{1}{2}}$.
The energy spectra of Eq. \ref{eq:3DLL_WF} is flat as $E_n=(n+\frac{1}{2}) \hbar
\omega_{so}$.
The center of the oscillator wavefunction in Eq. \ref{eq:3DLL_WF}
is shifted to $z_0=l_{so}^2 k_{2d} \Sigma$.
The 3D Landau level wavefunctions of Eq. \ref{eq:3DLL_WF} are spatially
separated 2D helical plane-waves along the $z$-axis.
As shown in Fig. \ref{fig:3Ddemon} (A), for states
with opposite helicity eigenvalues, their central positions
are shifted in opposite directions.
If open boundaries are imposed perpendicular to the $z$-axis, each Landau
level contributes a branch of gapless helical Dirac modes.
For the system described by $H^+_{3D,L}$, the surface Hamiltonian is
\begin{eqnarray}
H_{bd}= \pm v_f (\vec p \times \vec \sigma) \cdot \hat z-\mu,
\end{eqnarray}
where $\pm$ apply to upper and lower boundaries, respectively.
Unlike the 2D case in which the symmetric and Landau gauges are
equivalent,
the Hamiltonian of the symmetric-like gauge Eq. \ref{eq:3D_symm} and
that of the Landau-like gauge Eq. \ref{eq:3D_Landau} are {\it not}
gauge equivalent.
The Landau-like gauge explicitly breaks the 3D rotational symmetry
while the symmetric-like gauge preserves it.
Physical quantities calculated based on Eq. \ref{eq:3D_Landau}, such as
density of states, are not 3D rotationally symmetric as those
from Eq. \ref{eq:3D_symm}.
Nevertheless, these two Hamiltonians belong to the same topological class.
\begin{figure}[htbp]
\centering\epsfig{file=01_3Ddemon.eps,clip=1, width=0.3\textwidth,
}
\hspace{2mm}
\centering\epsfig{file=02_4Ddemon.eps,clip=2, width=0.3\textwidth,
}
\hspace{2mm}
\centering\epsfig{file=03_4Dpump.eps,clip=1,width=0.3\textwidth, angle=0}
\caption{\small
(A) 3D Landau level wavefunctions as spatially separated 2D helical Dirac modes
localized along the $z$-axis.
(B) 4D Landau level wavefunctions as spatially separated 3D Weyl modes
localized along the $u$-axis.
Note that 2D plane-wave modes with opposite helicities
and the 3D ones with opposite chiralities are located at
opposite sides of $z=0$ and $u=0$ planes, respectively.
C) The central positions $u_{0}(m,k_z,\nu)$ of the 4d Landau levels
in the presence of the magnetic field $\vec B= B\hat z$.
The branch of $m=0$ runs across the entire $u$-axis, which gives
rise to quantized charge transport along $u$-axis in the presence
of $\vec E\parallel \vec B$ as indicated in Eq. \ref{eq:4DQHE}.
From Ref. \cite{li2013a}.
}
\label{fig:3Ddemon}
\end{figure}
\subsection{Spatially separated 3D Weyl modes --4D Landau level}
\label{sect:4D}
Again we can easily generalize the above procedure to any dimensions.
For example, in four dimensions, we need to use the 3d helicity
operator $\hat \Sigma_{3d} =\hat P_{3d} \cdot \vec \sigma$, whose
eigenstates are denoted as $\chi_{\Sigma}$ with the eigenvalues
$\Sigma=\pm 1$.
Then the 4D Landau level Hamiltonian is defined as \cite{li2013a}
\begin{eqnarray}
H^{4d,\mp}_{LL} &=& \frac{P_u^2+\vec P_{3d}^2}{2M}+\frac{1}{2}M \omega^2 u^2
\mp \omega u \vec{P}_{3d} \cdot \vec{\sigma} \nonumber \\
&=& \frac{P_u^2}{2M } + \frac{1}{2} M \omega_{so}^2
(u \mp \frac{1}{\hbar}l_{so}^2 \vec P_{3d} \cdot \vec \sigma )^2, \ \ \,
\label{eq:4D_LL}
\end{eqnarray}
where $u$ and $P_u$ are the coordinate and momentum
in the 4th dimension, respectively,
and $\vec P_{3d}$ is defined in the $xyz$-space.
Inside each Landau level, the spectra are flat with respect to
$\vec k_{3d}$ and $\Sigma$.
Similarly to the 3D case, the 4D LL spectra and wavefunctions are solved
by reducing Eq. \ref{eq:4D_LL} into a set of 1D harmonic
oscillators along the $u$-axis as
\begin{eqnarray}
\Psi_{n, \vec{k}_{3d},\Sigma}(\vec{r},u) = e^{i \vec{k}_{3d} \cdot \vec{r}}
\phi_n [u - u_0(k_{3d},\Sigma)] \otimes \chi_{\Sigma}(\vec k_{3d}).
\end{eqnarray}
The central positions $u_0(k_{3d},\Sigma)=\Sigma l_{so}^2 k_{3d}$.
This realizes the spatial separation of the 3D Weyl fermion modes
with the opposite chiralities as shown in Fig. \ref{fig:3Ddemon} (B).
With an open boundary imposed along the $u$-direction,
the 3D chiral Weyl fermion modes appear on the boundary
\begin{eqnarray}
H_{bd}=\pm v_f (\vec k_{3D} \cdot \vec \sigma)-\mu.
\end{eqnarray}
\subsection{\normalsize Phase space picture of high-dimensional
Landau levels}
\label{sect:phase}
For the 2D case described by Eq. \ref{eq:2D_LL_Landau},
the $xy$-plane is equivalent to the 2D phase space of a 1D system
$(x;k_x)$ after the lowest Landau level projection.
The discrete step of $k_x$ is $\Delta k_x=2\pi/L_x$, and
the momentum cutoff of the bulk state is determined by $L_y$
as $k_{bk}=L_y/(2l_B^2)$.
Since $|k_x|<k_{bk}$, the number of states $N_{2D,LL}$ scales with
$L_xL_y$ as the usual 2D systems,
but the crucial difference is that enlarging $L_y$ does not change
$\Delta k_x$ but instead increases $k_{bk}$.
Similarly, the 3D Landau level states (Eq. \ref{eq:3D_Landau}) can be viewed as
states in the 4D phase space ($xy;k_xk_y$).
The $z$-axis plays the double role of $k_x$ and $k_y$.
After the lowest Landau level projection,
$z$ is equivalent to $z=l_{so}^2(p_x \sigma_y - p_y\sigma_x)/\hbar$,
and thus
\begin{eqnarray}
&&[x, z]_{LLL}=i l_{so}^2 \sigma_y, \ \ \,
[y, z]_{LLL}=-i l_{so}^2 \sigma_x,
\nonumber \\
&&[x,y]_{LLL}=0.
\end{eqnarray}
The momentum cutoff of the bulk state is determined as
$(k_x^2+k^2_y)^{\frac{1}{2}}<k_{bk}=\hbar L_z/
(2 l_{so}^2)$, thus the total number of states $N$ scales as $L_x L_y L_z^2$.
As a result, the 3D local density of states linearly diverges as
$\rho_{3D}(z)\propto |z|/l_{so}^4$ as $|z|\rightarrow \infty$.
Similar divergence also occurs in the symmetric-like gauge
as $\rho_{3D}(r)\propto r/l_{so}^4$.
Now this seeming pathological result can be understood as the
consequence of squeezing states of 4D phase space $(xy;k_xk_y)$ into the 3D
real space $(xyz)$.
In other words, the correct thermodynamic limit should be taken according
to the volume of 4D phase space.
This reasoning is easily extended to the 4D LL systems
(Eq.\ref{eq:4D_LL}), which can be understood as a 6D phase
space of $(xyz;k_xk_yk_z)$.
\subsection{\normalsize Charge pumping and the 4D quantum Hall effects}
The above 4D Landau level states presented in Sect. \ref{sect:4D} exhibit
non-linear electromagnetic response \cite{zhang2001,qi2008,Werner2012,
Frohlich2000} as the 4D quantum Hall effect.
We apply the electromagnetic fields as
\begin{eqnarray}
\vec{E}=E \hat{z}, \ \ \, \vec{B}=B \hat{z},
\end{eqnarray}
to the 4D Landau level Hamiltonian Eq. \ref{eq:4D_LL}
by minimally coupling fermions to the $U(1)$ vector potential,
\begin{eqnarray}
A_{em,x}=0, \ \ \, A_{em,y} = B x, \ \ \, A_{em,z} = -cEt.
\end{eqnarray}
The $\vec B$-field further quantizes the chiral plane-wave modes
inside the $n$-th 4D spin-orbit Landau level states into a series of
2D magnetic Landau level states in the
$xy$-plane as labeled by the magnetic Landau level index $m$.
For the case of $m=0$, the eigen-wavefunctions are spin polarized as
\begin{eqnarray}
\Psi_{n,m=0}(k_y,k_z)&=&e^{ik_y y+ik_z z} \phi_{n}(u-u_0(k_z,m=0))
\nonumber \\
&\times&\varphi_{m=0}(x-x_0(k_y))
\otimes \ket{\uparrow},
\label{eq:em4dwf}
\end{eqnarray}
where $\phi_n$ is the $n$-th order harmonic oscillator wavefunction with
the spin-orbit length scale $l_{so}$, and $\varphi_0$ is the
zeroth order harmonic oscillator wavefunction
with the magnetic length scale $l_B$.
The central positions of the $u$-directional and $x$-directional
oscillators are
\begin{eqnarray}
x_0(k_y)=l_B^2 k_y, \ \ \, u_0(k_z,m=0)=l_{so}^2 k_z,
\end{eqnarray}
respectively.
The key point is that $u_0(k_z,m=0)$ runs across the entire $u$-axis.
In contrast, wavefunctions $\Psi_{n,m}$ with $m\ge 1$
also exhibit harmonic oscillator
wavefunctions along the $u$-axis.
However, their central positions at $m\ge 1$ are,
\begin{eqnarray}
u_0(k_z)=\pm l_{so}^2 \sqrt{k_z^2+\frac{2m}{l_B^2}},
\end{eqnarray}
which only lie in half of the $u$-axis as
shown in Fig. \ref{fig:3Ddemon} (C).
Since $k_z$ increases with time in the presence of $E_z$,
$u_0(m,k_z(t))$ moves along the $u$-axis.
Only the $m=0$ branch of the magnetic Landau level states contribute
to the charge pumping since their centers go across the entire
$u$-axis, which results in an electric current along the $u$-direction.
Since $k_z(t)=k_z(0)-\frac{eE}{\hbar}t$,
during the time interval $\Delta t$, the number of electrons
passing the cross-section at a fixed $u$ is
\begin{eqnarray}
\Delta N=\frac{L_xL_y}{2\pi l_B^2}
\frac{e E_z \Delta t }{2\pi \hbar/L_z}
=\frac{e^2}{4\pi^2\hbar^2c} \vec E \cdot \vec B V\Delta t,
\end{eqnarray}
where $V$ is the 3D cross-volume.
Then the current density is calculated as
\begin{eqnarray}
j_u= n_{occ} \frac{e\Delta N}{V\Delta t}= n_{occ} \alpha \frac{e}{4 \pi^2 \hbar} \vec{E} \cdot \vec{B},
\label{eq:4Dpump}
\end{eqnarray}
where $\alpha$ is the fine-structure constant,
and $n_{occ}$ is the occupation number of the 4D spin-orbit Landau levels.
Eq. \ref{eq:4Dpump} is in agreement with
results from the effective field theory \cite{qi2008} as the 4D generalization
of the quantum Hall effect.
If we impose the open boundary condition perpendicular to the $u$-direction,
the above charge pump process corresponds to the chiral anomalies of Weyl
fermions with opposite chiralities on two opposite 3D boundaries, respectively.
Since they are spatially separated, the chiral current corresponds to the
electric current along the $u$-direction.
\section{Conclusions and outlooks}
\label{sect:conclusion}
I have reviewed a general framework to construct Landau
levels in high dimensions based on harmonic oscillator wavefunctions.
By imposing spin-orbit coupling, their spectra are reorganized to exhibit
flat dispersions.
In particular, the lowest Landau level wavefunctions in 3D and 4D in the quaternion representation satisfy the
Cauchy-Riemann-Fueter condition, which is the generalization
of complex analyticity to high dimensions.
The boundary excitations are the 2D helical Dirac surface modes,
or, the 3D chiral Weyl modes.
There is a beautiful bulk-boundary correspondence that the
Cauchy-Riemann-Fueter condition and the helical Dirac (chiral Weyl)
equation are the Euclidean and Minkowski representations of the
same analyticity condition, respectively.
By dimensional reductions, we constructed a class of Landau levels
in 2D and 3D which are time-reversal invariant but parity breaking.
The Landau level problem for Dirac fermions is a square-root problem of
the non-relativistic one, corresponding to complex quaternions.
The zeroth Landau level states are a flat band of half-fermion
Jackiw-Rebbi zero modes.
It is at the interface between condensed matter and high energy
physics, related to a new type of anomaly.
Unlike parity anomaly and chiral anomaly studied in field theory
in which Dirac fermions are coupled to gauge fields through
the minimal coupling, here Dirac fermions are coupled to background
fields in a non-minimal way.
I speculate that high-dimensional Landau levels could provide a platform
for exploring interacting topological states in high dimensions - due
to the band flatness, and also the quaternionic analyticity of lowest
Landau level wavefunctions.
It would stimulate the developments of various theoretical and
numerical methods.
This would be an important direction in both condensed matter physics
and mathematical physics for studying high
dimensional topological states for both
non-relativistic and relativistic fermions.
This research also provides interesting applications of quaternion
analysis in theoretical physics.
\section{Acknowledgments}
I thank Yi Li for collaborations on this set of works on high-dimensional topological states and for bringing in interesting concepts including the quaternionic analyticity.
I also thank J. E. Hirsch for stimulating discussions, and S. C. Zhang,
T. L. Ho, E. H. Fradkin, S. Das Sarma, F. D. M. Haldane, and C. N. Yang for their warm encouragements and appreciations.
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"redpajama_set_name": "RedPajamaArXiv"
}
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11. Michaela Coel
Actor and writer Michaela Coel talks to Louis Theroux about speaking in tongues, sexual consent and suffering from 'post-writum depression' after the success of I May Destroy You.
Covid-19 hasn't gone away and, due to travel restrictions, neither has Louis Theroux. In the second outing of his podcast series, he tracks down more high-profile guests he's been longing to talk to - a fascinating mix of the celebrated, the controversial and the mysterious. They include Oscar-winning Hollywood director Oliver Stone… singer, songwriter and superstar collaborator Sia… outspoken and occasionally cancelled comedian Frankie Boyle… dancer and singer FKA Twigs… mental health campaigner and comedian Ruby Wax… ubiquitous TV presenter Rylan Clark-Neal… and more.
In the first episode of the new series, actor, writer and producer Michaela Coel talks to Louis about speaking in tongues, sexual consent and suffering from 'post-writum depression' after the success of I May Destroy You.
Produced by Paul Kobrak
Assistant Producer - Catherine Murnane
A Mindhouse production for BBC Radio 4
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See all episodes from Grounded with Louis Theroux
English television actresses (114)
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Michaela Coel: Eight things we learned when she spoke to Louis Theroux
What happened when Louis spoke to the creator of I May Destroy You?
Wed 9 Dec 2020 20:00
Helena Bonham Carter interviewed by Simon Mayo—Kermode and Mayo's Film Review, with Helena Bonham Carter
This clip is related to English television actresses
Kurt Vonnegut and Josie Long—Great Lives, Series 50
This episode is related to English television actresses
Sophie Okonedo interviewed by Edith Bowman—Kermode and Mayo's Film Review, With Sophie Okonedo
What did we learn from Louis Theroux's Grounded podcast interviews?
Watch Grounded with Louis Theroux on BBC iPlayer
A compilation of highlights from the first series of Grounded With Louis Theroux.
Louis is using the lockdown to track down some of the people he's been longing to talk to.
Factual > Life Stories
By format:
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,701
|
Q: Pover Pivot/ Power Query filter issue This is possibly a newbie question.
I have done a lot of Excel reporting with VBA and SQL statements in the past, but wanted to give Power Query/Power Pivot a try. Took an online class and everything looks easy until you try it yourself (as usual).
Here is my easy problem:
I want to do a training report for employees in different locations (stores).
As shown below I have a "location table" with Store IDs, an "employee table" with Employee ID and store_ID (linked in the data model), and a "training table" with Employee ID and class types.
What I want to see is a list of (filtered) locations, their employees and the classes taken...
The problem starts with the Employee pivot already,
This should show me the employees of the selected location: (But it shows all employees from all locations, despite of the set filter "2360").
Playing around I found it will filter correctly for names when I add a "Count of" into the Calculations field... What the hell?
This now will give me the correct employees per store but now the classes
are not correct and it looks like everybody took all classes. (Which is not true):
All tables are clean and have no duplicates or other issues.
(The class table obviously has multiple entries for students, but
not for classes).
I had planned to also have a filter for the time-frame of classes taken,
but at this time I am humbled...
Thank you for any advice or pointer!!!
The data model arrows are as seen on the screen shots, "Field" is the "Class_type" column in the class table. ("field training" or "in-house").
(The table on the right below without header is the class table).
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,012
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\section{Introduction}
\label{intro}
Weyl superconductors are nodal superconductors with topological protection \cite{Men12,Sch15}: They have nodal points of vanishing excitation gap, just like \textit{d}-wave superconductors \cite{Har95}, but in contrast to those the gapless states are not restricted to high-symmetry points in the Brillouin zone and can appear for conventional \textit{s}-wave pairing. The nodal points (Weyl points) at $\pm K$ in a Weyl superconductor are protected by the conservation of a topological invariant: the Berry flux of $\pm 2\pi$ at Weyl points of opposite chirality \cite{Nie83,Tur13}.
The distinction between symmetry and topology has a major consequence for the stability of Landau levels in a magnetic field. While in a \textit{d}-wave superconductor the strong scattering of nodal fermions by vortices in the order parameter prevents the formation of Landau levels \cite{Fra00}, in a Weyl superconductor an index theorem for chiral fermions protects the zeroth Landau level from broadening \cite{Pac18}. The appearance of chiral Landau levels in a superconducting vortex lattice produces a quantized thermal conductance parallel to the magnetic field, in units of $1/2$ times the thermal quantum per $h/2e$ vortex \cite{Pac18}. The factor of $1/2$ reminds us that Bogoliubov quasiparticles are Majorana fermions, ``half a Dirac fermion'' \cite{Bee14,Fra15}.
In this paper we turn from thermal transport to electrical transport, by studying the geometry of Fig.\ \ref{fig_layout} and addressing the question ``What is the charge transported along the vortices in a chiral Landau level?'' It is known \cite{Bai17} that the charge of Weyl fermions in a superconductor (pair potential $\Delta_0$) is reduced by a factor $\kappa=K(\Delta_0)/K(0)$. We find a direct manifestation of this charge renormalization in the electrical conductance, which is quantized at $\tfrac{1}{2}(e\kappa)^2/h$ per vortex. Because the charge renormalization is energy dependent, a coupling between thermal and electrical transport appears even without any energy-dependent scattering mechanism --- resulting in a nonzero thermo-electric effect in a chiral Landau level.
In the next section \ref{sec_LL} we summarize the effective low-energy theory of the superconducting vortex lattice \cite{Pac18}, on which we base our scattering theory in Sec.\ \ref{sec_transmission}, followed by a calculation of electrical and thermo-electric transport properties in Sec.\ \ref{sec_transport}. These analytical results are compared with numerical simulations of a tight-binding model in Sec.\ \ref{sec_numerical}. We conclude in Sec.\ \ref{sec_conclude}.
\begin{figure}[tb]
\centerline{\includegraphics[width=1\linewidth]{layout}}
\caption{a) Vortex lattice in a Weyl superconductor sandwiched between metal electrodes; b) Circuit to measure the electrical transport along the vortex lines. The nonlocal conductance $G_{12}=dI_2/dV_1$ gives the current carried through the vortex lattice by nonequilibrium Weyl fermions in a chiral Landau level.
}
\label{fig_layout}
\end{figure}
\section{Landau level Hamiltonian in the vortex lattice}
\label{sec_LL}
\begin{figure*}[tb]
\includegraphics[width=0.45\linewidth]{triangle}\hfill
\includegraphics[width=0.45\linewidth]{density_Nx=194_Ny=112_repeated}
\caption{\textit{Left panel:} The red solid curves show the dispersion of Landau levels in the $k_x$--$k_y$ plane perpendicular to the magnetic field (energy $E$ normalized by the energy $E_1$ of the first Landau level). The black dotted curves show the dispersion in zero magnetic field, with a Weyl cone at the $\Gamma$ point of the magnetic Brillouin zone. \textit{Right panel:} Particle density profile in the zeroth Landau level, in the $x$--$y$ plane perpendicular to the magnetic field, for a wave vector at the Weyl point ($\bm{k}=K\hat{z}$). The magnetic unit cell is indicated by a white dashed rectangle. Both panels are calculated numerically for a Weyl superconductor with a triangular vortex lattice. The vortex cores are located at the bright points in the density profile. Similar plots for a square vortex lattice are in Ref.\ \onlinecite{Pac18}.}
\label{fig_triangle}
\end{figure*}
We summarize the findings of Ref.\ \onlinecite{Pac18} for the Landau level Hamiltonian of Weyl fermions in a superconducting vortex lattice, which we will need to calculate the transport properties.
\subsection{Dispersion relation}
\label{sec_dispersion}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.8\linewidth]{dispersion}}
\caption{Dispersion relation of the zeroth Landau level in a superconducting vortex lattice, plotted from Eq.\ \eqref{Edispersion} for $\mu=0$, $\Delta_0=0.5$, $\beta=1$. Only the dependence on the momentum $k_z$ along the magnetic field $B$ is shown, the dispersion is flat in the $x$--$y$ plane (see Fig.\ \ref{fig_triangle}). The four branches are distinguished by the sign of the chirality (solid or dashed) and by the sign of the electric charge (red or blue). The zero-field Weyl points at $k_z=\pm K$ are indicated by arrows. Each branch has a degeneracy $N_{\rm Landau}=e\Phi/h$ set by the enclosed flux $\Phi=BW^2$.
}
\label{fig_LL}
\end{figure}
A Landau level is a dispersionless flat band in the plane perpendicular to the magnetic field. The lowest (zeroth) Landau level is protected by chiral symmetry from scattering by the vortices, see Fig.\ \ref{fig_triangle}. This is the Landau level on which we focus our analysis. It is a celebrated result of Nielsen and Ninomiya \cite{Nie83} that Weyl fermions in the zeroth Landau level have a definite chirality $\chi=\pm 1$, defined as the sign of the velocity $v_z=\partial E/\partial k_z$, parallel or antiparallel to $B$. To account for the electron-hole degree of freedom the number of bands is doubled for each chirality, so that we have four bands in total. Electron-like and hole-like bands are related related by the charge-conjugation symmetry relation $E_\chi(k_z)=-E_\chi(-k_z)$.
The effect of a superconducting vortex lattice on this four-band dispersion is given by \cite{Pac18}
\begin{equation}
\begin{split}
&E_\chi(k_z)=-(\text{sgn}\,k_z)\chi M(k_z)-\chi\mu\kappa(k_z),\\
&M(k_z)=\beta-\sqrt{\Delta_0^2+ k_z^2},\;\;\kappa(k_z)=\frac{d}{dk_z}M(k_z),
\end{split}\label{Edispersion}
\end{equation}
plotted in Fig.\ \ref{fig_LL}. (We have set $\hbar$ and the Fermi velocity $v_{\rm F}$ equal to unity, so $\kappa$ is dimensionless.) The magnitude of the superconducting pair potential outside of the vortex cores is denoted by $\Delta_0$ and $\beta$ is an internal magnetization along the $z$-direction that breaks time-reversal symmetry even in the absence of any external magnetic field. In Eq.\ \eqref{Edispersion} we have assumed that $\beta$ is parallel to $B$, but we will later relax this assumption (see Sec.\ \ref{sec_isotropy}).
Provided that $\Delta_0<\beta$ there is a pair of Landau levels for each chirality, located in the magnetic Brillouin zone near the Weyl points at $k_z= K$ and $k_z=-K$, with \cite{Men12}
\begin{equation}
K(\Delta_0)=\sqrt{\beta^2-\Delta_0^2}.\label{Kzdef}
\end{equation}
The charge expectation value
\begin{equation}
Q_\chi=-e\frac{\partial E_\chi}{\partial\mu}=e\chi\kappa(k_z)=-\frac{e\chi k_z}{\sqrt{\Delta_0^2+k_z^2}}\label{Qchidef}
\end{equation}
for a given chirality has the opposite sign at the two Weyl points. (We say that the chiral Landau levels near $k_z=\pm K$ are charge-conjugate.) When $k_z=\pm K$ is at the Weyl point, the charge renormalization factor equals $\mp\kappa_0$, with
\begin{equation}
\kappa_0=K(\Delta_0)/K(0)=\sqrt{1-\Delta_0^2/\beta^2},\label{kappa0def}
\end{equation}
while $\kappa(k_z)$ varies linearly with energy away from the Weyl point \cite{Bai17}.
\subsection{Effective Hamiltonian}
\label{sec_effH}
The dispersion \eqref{Edispersion} follows from the effective low-energy Hamiltonian \cite{Pac18}
\begin{subequations}
\label{Heffdef}
\begin{align}
{\cal H}={}&U\begin{pmatrix}
H_+&0&0&0\\
0&\cdot&\cdot&0\\
0&\cdot&\cdot&0\\
0&0&0&H_-
\end{pmatrix}U^\dagger,\label{Heffdefa}\\
H_\chi={}&(k_x+e{\cal A}_{\chi,x})\sigma_x+(k_y+e{\cal A}_{\chi,y})\sigma_y\nonumber\\
&+M\sigma_z-\chi\mu\kappa\sigma_0,\label{Heffdefb}\\
U={}&\exp(\tfrac{1}{2}i\theta\nu_y\tau_z\sigma_z),\;\;\theta=\arccos\kappa.\label{Heffdefc}
\end{align}
\end{subequations}
The $2\times 2$ Pauli matrices $\nu_\alpha$, $\tau_\alpha$, and $\sigma_\alpha$ (with $\alpha=0$ the corresponding unit matrix) act on, respectively, the electron-hole, orbital, and spin degrees of freedom. The full Hamiltonian ${\cal H}$ is an $8\times 8$ matrix and the $2\times 2$ matrices $H_\pm$ act on the $\sigma$ index in the $\nu=\tau=\pm 1$ sector.
The central block in Eq.\ \eqref{Heffdefa} indicated by dots refers to higher-lying bands that are approximately decoupled from the low-energy bands. Virtual transitions to these higher bands contribute order $\mu^2$ terms that remove the discontinuity in the derivative $\partial E/\partial k_z$ at $k_z=0$ for $\mu\neq 0$. No such decoupling approximations are made in the numerics of Sec.\ \ref{sec_numerical}.
The gauge field ${\cal A}_\chi(\bm{r})$, dependent on the position $\bm{r}=(x,y)$ in the $x$--$y$ plane, defines the effective magnetic field ${\cal B}_\chi=\nabla\times{\cal A}_\chi$ in the $z$-direction felt by the Weyl fermions in the lattice of vortices at positions $\bm{R}_n$,
\begin{equation}
{\cal B}_\chi=(1+\chi\kappa)\Phi_0{\sum_n}\delta(\bm{r}-\bm{R}_n)-\chi\kappa B.
\end{equation}
There are $N_{\text{vortex}} = BW^2/\Phi_0$ vortices of flux $\Phi_0=h/2e$ in an area $W^2$ perpendicular to the applied magnetic field $B$, so the spatial average $\int {\cal B}_\chi d\bm{r}=\Phi$ equals the total enclosed flux $\Phi=BW^2$ independent of $\kappa$ or of the lattice of vortices. (In the numerics that follows we will use a square lattice for definiteness.)
\subsection{Zeroth Landau level wave functions}
\label{sec_zerothLL}
As shown in Ref.\ \onlinecite{Pac18}, the Aharonov-Casher index theorem \cite{Aha79,Kat08,Kai09}, together with the requirement that the wave functions are square-integrable at a vortex core, implies that the zeroth Landau level eigenstates $\psi_{\chi}$ of $H_\chi$, which are rank-two spinors, are also eigenstates $|\pm\rangle_\sigma$ of $\sigma_z$,
\begin{equation}
\sigma_z\psi_{\chi}=(\text{sgn}\,Q_\chi)\psi_\chi.\label{eigenvaluesigmaz}
\end{equation}
The eigenvalue is determined by the sign of the effective quasiparticle charge \eqref{Qchidef}.
It follows that the eigenstates $\Psi_\chi$ of the full Hamiltonian ${\cal H}$, which are rank-eight spinors, have the form
\begin{align}
\Psi_\chi={}&e^{ik_z z}f_\chi(x,y)e^{\tfrac{1}{2}i\theta\nu_y\tau_z\sigma_z}|\text{sgn}\,\chi\rangle_\nu|\text{sgn}\,\chi\rangle_\tau|\text{sgn}\,Q_\chi\rangle_\sigma\nonumber\\
={}&e^{ik_z z}f_\chi(x,y)\biglb[\cos(\theta/2)|\text{sgn}\,\chi\rangle_\nu|\text{sgn}\,\chi\rangle_\tau|\text{sgn}\,Q_\chi\rangle_\sigma\nonumber\\
&-\sin(\theta/2)(\text{sgn}\,Q_\chi)|\!-\!\text{sgn}\,\chi\rangle_\nu|\text{sgn}\,\chi\rangle_\tau|\text{sgn}\,Q_\chi\rangle_\sigma\bigrb].\label{Psichiresult}
\end{align}
The spatial density profile $f_\chi(x,y)$ is peaked at the vortex cores, with a power law decay $|f_\chi|^2\propto \delta r^{-1+|Q_\chi|/e}$ at a distance $\delta r$ from the core \cite{Pac18}. The renormalization of the quasiparticle charge does not affect the degeneracy of the zeroth Landau level: each of the four chiral modes in Fig.\ \ref{fig_LL} has a degeneracy
\begin{equation}
N_{\text{Landau}}=e\Phi/h\label{NLandaudef}
\end{equation}
set by the bare charge $e$.
Although the spatial density profile of these chiral modes is nonuniform, the wave functions extend over the entire $x$--$y$ plane --- they are not exponentially confined to the vortex cores (see Fig.\ \ref{fig_triangle}). This is a qualitative difference between the zeroth Landau level of a Weyl superconductor and zero-modes bound to vortices in topological superconductors \cite{Vol99,Fu08}.
\section{Transmission through the NSN junction}
\label{sec_transmission}
Refering to the geometry of Fig.\ \ref{fig_layout}, we seek the transmission matrix $t_{\text{NSN}}$ for propagating modes of electrons and holes transmitted from the first metal contact $\text{N}_1$ in the region $z<0$, through the Weyl superconductor in the region $0<z<L$, into the second metal contact $\text{N}_2$ in the region $z>L$.
\subsection{Renormalized charge transfer}
\label{sec_effcharge}
We start by examining a single NS interface, to study how a chiral mode in the superconductor injects a renormalized charge into the normal metal.
On the superconducting side $z<L$ of the NS interface at $z=L$ the incident modes have positive chirality $\chi=+1$. There is a mode $\Psi_{\rm S}$ with perpendicular momentum $k_z$ near $K$ and a mode $\Psi'_{\rm S}$ with $k'_z$ near $-K$. We do not specify the transverse momentum $\bm{k}_\parallel=(k_x,k_y)$, which gives each mode a degeneracy of $N_{\rm Landau}=e\Phi/h$, see Eq.\ \eqref{NLandaudef}.
According to Eq.\ \eqref{Psichiresult}, the spinor structure of the chiral modes is
\begin{equation}
\begin{split}
&\Psi_{\rm S}\propto \cos(\theta/2)|\mbox{++$-$}\rangle_{\nu\tau\sigma}+\sin(\theta/2)|\mbox{$-$+$-$}\rangle_{\nu\tau\sigma},\\
&\Psi'_{\rm S}\propto \cos(\theta'/2)|\mbox{+++}\rangle_{\nu\tau\sigma}-\sin(\theta'/2)|\mbox{$-$++}\rangle_{\nu\tau\sigma}.
\end{split}
\label{PsiKspinor}
\end{equation}
We have abbreviated $|\mbox{$\pm$$\pm$$\pm$}\rangle_{\nu\tau\sigma}=|\pm\rangle_{\nu}|\pm\rangle_{\tau}|\pm\rangle_{\sigma}$ and denote $\theta=\theta(k_z)$, $\theta'=\theta(k'_z)$.
For the normal metal we take the free-electron Hamiltonian
\begin{equation}
H_{\rm N}=\frac{1}{2m}(k^2-k_{\rm F}^2)\nu_z\tau_0\sigma_0,\label{HNdef}
\end{equation}
isotropic in the spin and valley degrees of freedom, in the high Fermi-momentum limit $k_{\rm F}l_m\rightarrow\infty$ when the effect of the magnetic field on the spectrum may be neglected ($l_m=\sqrt{\hbar/eB}$ is the magnetic length).
Because of the large potential step experienced upon traversing the NS interface, the perpendicular momentum $k_z$ is boosted to $+k_{\rm F}$ for the electron component of the state and to $-k_{\rm F}$ for the hole component. A state in N moving away from the NS interface of the form
\begin{subequations}
\label{Psimatch}
\begin{align}
\Psi_{\rm N}\propto{}& e^{ik_{\rm F}(z-L)}\cos(\theta/2)|\mbox{++$-$}\rangle_{\nu\tau\sigma}\nonumber\\
&+e^{-ik_{\rm F}(z-L)}\sin(\theta/2)|\mbox{$-$+$-$}\rangle_{\nu\tau\sigma}\label{Psimatcha}
\end{align}
can be matched to the incident state $\Psi_S$ in S, while the state
\begin{align}
\Psi'_{\rm N}\propto{}& e^{ik_{\rm F}(z-L)}\cos(\theta'/2)|\mbox{+++}\rangle_{\nu\tau\sigma}\nonumber\\
&-e^{-ik_{\rm F}(z-L)}\sin(\theta'/2)|\mbox{$-$++}\rangle_{\nu\tau\sigma}\label{Psimatchb}
\end{align}
\end{subequations}
can be matched to $\Psi'_S$.
The charge transferred through the interface when $\Psi_{\text S}\mapsto\Psi_{\text N}$ equals the renormalized charge from Eq.\ \eqref{Qchidef},
\begin{equation}
Q_{\rm N}=\langle\Psi_{\rm N}|e\nu_z|\Psi_{\rm N}\rangle=e\cos\theta=e\kappa=\frac{-ek_z}{\sqrt{\Delta_0^2+k_z^2}},\label{QNresult}
\end{equation}
dependent on the perpendicular momentum $k_z$ in S, before the boost to $k_{\rm F}$ in N. When $k_z=K$, this gives
\begin{equation}
Q_{\rm N}=-e\sqrt{1-\Delta_0^2/\beta^2}=-\kappa_0 e\equiv -Q_{\rm eff}.\label{Qeffdef}
\end{equation}
This is for the transmission $\Psi_{\text S}\mapsto\Psi_{\text N}$ . The other transmission $\Psi'_{\text S}\mapsto\Psi'_{\text N}$ transfers for $k'_z=-K$ a charge $Q'_{\rm N}=+Q_{\rm eff}$.
Similarly, at the opposite NS interface $z=0$ the chiral Landau level modes in S moving away from the interface are matched to incoming states in N of the form
\begin{subequations}
\label{Phimatch}
\begin{align}
\Phi_{\rm N}\propto{}& e^{ik_{\rm F}z}\cos(\theta/2)|\mbox{++$-$}\rangle_{\nu\tau\sigma}\nonumber\\
&+e^{-ik_{\rm F}z}\sin(\theta/2)|\mbox{$-$+$-$}\rangle_{\nu\tau\sigma},\label{Phimatcha}\\
\Phi'_{\rm N}\propto{}& e^{ik_{\rm F}z}\cos(\theta'/2)|\mbox{+++}\rangle_{\nu\tau\sigma}\nonumber\\
&-e^{-ik_{\rm F}z}\sin(\theta'/2)|\mbox{$-$++}\rangle_{\nu\tau\sigma}.\label{Phimatchb}
\end{align}
\end{subequations}
\subsection{Transmission matrix}
\label{sec_transmatrix}
At a given energy $E$ relative to the Fermi level the perpendicular momenta $k_z$ and $k'_z$ of the chiral Landau levels in S moving in the $+z$ direction are determined by the dispersion relation \eqref{Edispersion} with $\chi=+1$. For $\mu=0$ the expressions are simple,
\begin{equation}
k_z=K+(\beta/K)E,\;\;k'_z=-K+(\beta/K)E.\label{kzEdef}
\end{equation}
For any $\mu$, particle-hole symmetry ensures that
\begin{equation}
k_z(E)=-k'_z(-E).\label{kzphsym}
\end{equation}
The Landau level $\Psi_{\rm S}$ propagating from $z=0$ to $z=L$ accumulates a phase $k_z L$, and similarly $\Psi'_{\rm S}$ accumulates a phase $k'_z L$. The full transmission matrix of the NSN junction at energy $E$ can thus be written as
\begin{align}
t_{\text{NSN}}(E)=e^{ik_z L}|\Psi_{\rm N}\rangle\langle\Phi_{\rm N}|+e^{ik'_z L}|\Psi'_{\rm N}\rangle\langle\Phi'_{\rm N}|,\label{tNSNE}
\end{align}
with $k_z$ and $k'_z$ determined by Eq.\ \eqref{kzEdef}.
We can rewrite Eq.\ \eqref{tNSNE} in the basis of propagating electron modes in the normal metal. In the region $z<0$ one has the basis states
\begin{subequations}
\label{Psiupdowndef}
\begin{align}
&|\Psi_\uparrow\rangle=\begin{pmatrix}
|e\uparrow\rangle\\
|h\uparrow\rangle
\end{pmatrix},\;\;|\Psi_\downarrow\rangle=\begin{pmatrix}
|e\downarrow\rangle\\
|h\downarrow\rangle
\end{pmatrix},\label{Psiupdowndefa}\\
&|e\uparrow\rangle=e^{ik_{\rm F}z}|\mbox{+++}\rangle_{\nu\tau\sigma},\;\;|h\uparrow\rangle=e^{-ik_{\rm F}z}|\mbox{$-$++}\rangle_{\nu\tau\sigma},\nonumber\\
&|e\downarrow\rangle=e^{ik_{\rm F}z}|\mbox{++$-$}\rangle_{\nu\tau\sigma},\;\;|h\downarrow\rangle=e^{-ik_{\rm F}z}|\mbox{$-$+$-$}\rangle_{\nu\tau\sigma},\label{Psiupdowndefb}
\end{align}
\end{subequations}
and similarly for $z>L$ with $k_{\rm F}z$ replaced by $k_{\rm F}(z-L)$.
The transmission matrix is block diagonal in the spin degree of freedom,
\begin{subequations}
\label{tNSNfinal}
\begin{align}
&t_{\text{NSN}}(E)=\begin{pmatrix}
t_\uparrow(E)&0\\
0&t_\downarrow(E)
\end{pmatrix},\label{tNSNfinala}\\
&t_\uparrow=e^{ik'_z L}\begin{pmatrix}
\cos^2(\theta'/2)&-\cos(\theta'/2)\sin(\theta'/2)\\
-\cos(\theta'/2)\sin(\theta'/2)&\sin^2(\theta'/2)
\end{pmatrix}\nonumber,\\
&t_\downarrow=e^{ik_z L}\begin{pmatrix}
\cos^2(\theta/2)&\cos(\theta/2)\sin(\theta/2)\\
\cos(\theta/2)\sin(\theta/2)&\sin^2(\theta/2)
\end{pmatrix}.\label{tNSNfinalb}
\end{align}
\end{subequations}
The $2\times 2$ matrix $t_\uparrow$ acts on the electron-hole spinor $|\Psi_\uparrow\rangle$ and $t_\downarrow$ acts on $|\Psi_\downarrow\rangle$. We may write this more compactly as
\begin{equation}
\begin{split}
&t_\uparrow=\tfrac{1}{2}e^{ik'_z L}\left(\nu_0+\nu_z e^{-i\theta'\nu_y}\right),\\
&t_\downarrow=\tfrac{1}{2}e^{ik_z L}\left(\nu_0+\nu_z e^{i\theta\nu_y}\right).
\end{split}\label{tcompact}
\end{equation}
These are each rank-one matrices, one eigenvalue equals 0 and the other equals 1 in absolute value. The unit transmission eigenvalue is $N_{\text{Landau}}$-fold degenerate in the transverse momentum $\bm{k}_\parallel$.
At the Fermi level $E=0$ the particle-hole symmetry relation \eqref{kzphsym} implies $k'_z=-k_z$, $\theta'=\pi-\theta$, hence
\begin{equation}
t_{\text{NSN}}(0)=\tfrac{1}{2}e^{-ik_zL\sigma_z}\left(\nu_0-\nu_z \sigma_z e^{i\theta\nu_y}\right).\label{tzeroenergy}
\end{equation}
One verifies that
\begin{equation}
t_{\text{NSN}}(0)=\nu_y\sigma_y t_{\text{NSN}}^\ast(0)\nu_y\sigma_y,\label{tNSNphsymcheck}
\end{equation}
as required by particle-hole symmetry.
\section{Transport properties}
\label{sec_transport}
The transmission matrix allows us to calculate the transport properties of the NSN junction, under the assumption that there is no backscattering of the chiral modes in the Weyl superconductor. To simplify the notation, we write $t$ for the Fermi-level transmission matrix $t_{\rm NSN}(0)$. The submatrices of electron and hole components are denoted by $t_{ee}$, $t_{hh}$, $t_{he}$, and $t_{eh}$. We define the combinations
\begin{subequations}
\label{Tpmdef}
\begin{align}
&{\cal T}_\pm=t^\dagger_{ee}t^{\vphantom{\dagger}}_{ee}\pm t^\dagger_{he}t^{\vphantom{\dagger}}_{he},\\
&{\cal T}_+=\tfrac{1}{2}(\nu_0+\nu_z)t^\dagger t,\;\;{\cal T}_-=\tfrac{1}{2}(\nu_0+\nu_z) t^\dagger\nu_z t.
\end{align}
\end{subequations}
\subsection{Thermal conductance}
\label{sec_Gthermal}
As a check, we first recover the result of Ref.\ \onlinecite{Pac18} for the quantization of the thermal conductance.
The thermal conductance $G_{\rm thermal}=J_{12}/\delta T$ gives the heat current $J_{12}$ transported at temperature $T_0$ from contact $N_1$ to $N_2$ via the superconductor, in response to a small temperature difference $\delta T$ between the contacts. It follows from the total transmitted quasiparticle current,
\begin{equation}
G_{\rm thermal}=\tfrac{1}{2}g_0 N_{\rm Landau}\,{\rm Tr}\,t^\dagger t=g_0 \frac{e\Phi}{h},\label{Gthermalresult}
\end{equation}
with $N_{\rm Landau}=e\Phi/h$ the Landau level degeneracy and $g_0=\tfrac{1}{3}(\pi k_{\rm B})^2(T_0/h)$ the thermal conductance quantum. The factor $1/2$ in the first equation appears because the quasiparticles in the Weyl superconductor are Majorana fermions. It is cancelled by the factor of two from ${\rm Tr}\,tt^\dagger=2$, in view of Eq.\ \eqref{tzeroenergy}.
\subsection{Electrical conductance}
\label{sec_Gelectric}
Referring to the electrical circuit of Fig.\ \ref{fig_layout}b, we consider the electrical conductance $G_{12}=dI_2/dV_1$, given by
\begin{align}
G_{12}={}&\frac{e^2}{h}N_{\rm Landau}\,{\rm Tr}\,{\cal T}_-\nonumber\\
={}&\frac{e^2}{h}N_{\rm Landau}\tfrac{1}{2}\,{\rm Tr}\,(\nu_0+\nu_z)t^\dagger\nu_z t.\label{G12def2}
\end{align}
In the linear response limit $V_1\rightarrow 0$ we substitute $t$ from Eq.\ \eqref{tzeroenergy}, which gives
\begin{equation}
G_{12}(0)=\cos^2\theta\frac{e^2}{h}N_{\rm Landau}=\frac{(e\kappa)^2}{h}\frac{e\Phi}{h}.\label{G12result}
\end{equation}
The conductance quantum $e^2/h$ is renormalized by the effective charge $e\mapsto e\kappa$. At $\mu=0$, when $k_z=K$, the renormalization factor is $\kappa_0^2=(Q_{\rm eff}/e)^2=1-\Delta_0^2/\beta^2$ from Eq.\ \eqref{Qeffdef}. Note that the conductance per $h/2e$ vortex is $\tfrac{1}{2}(e\kappa_0)^2/h$, with an additional factor $1/2$ to signal the Majorana nature of the quasiparticles.
At finite $E=eV_1$ we must use the energy-dependent transmission matrix \eqref{tNSNfinal}, which gives
\begin{equation}
G_{12}(E)=\tfrac{1}{2}\frac{e^2}{h}N_{\text{Landau}}\left(\cos\theta+\cos\theta'+\cos^2\theta+\cos^2\theta'\right).\label{G12finiteV}
\end{equation}
Substituting Eq.\ \eqref{QNresult} for $\cos\theta$ and $\cos\theta'$ at $k_z$ and $k'_z$, given as a function of $E$ by Eq.\ \eqref{kzEdef}, we find
\begin{equation}
G_{12}(E)=G_{12}(0)\left(1-\frac{\Delta_0^2 E}{(\beta^2-\Delta_0^2)^{3/2}}+{\cal O}(E^2)\right).\label{G12finiteVworkedout}
\end{equation}
The energy dependence of the differential conductance comes entirely from the energy dependence of the effective charge: At $E=0$ the electron-like and hole-like chiral Landau levels have precisely opposite effective charge $\pm Q_{\rm eff}$, but for $E\neq 0$ the effective charges differ in absolute value by an amount $\propto dk_z/dE$.
\subsection{Shot noise}
\label{sec_Pshot}
At temperatures small compared to the applied voltage $V_2$, the time dependent fluctuations in the current $I_2$ are due to shot noise. The formula for the shot noise power is \cite{Ana96}
\begin{equation}
P_{12}=\frac{e^3 V_1}{h}\,{\rm Tr}\,({\cal T}_+-{\cal T}_-^2). \label{P12def}
\end{equation}
This can again be written in terms of the Pauli matrix $\tau_z$ and evaluated using Eq.\ \eqref{tzeroenergy},
\begin{equation}
P_{12}=\frac{e^3 V_1}{h}\left(1-\tfrac{1}{2}\kappa^2-\tfrac{1}{2}\kappa^4\right).\label{P12result}
\end{equation}
The shot noise vanishes when $\kappa\rightarrow 1$, it is fully due to the charge renormalization.
The Fano factor $F$, the dimensionless ratio of shot noise power and average current, results as
\begin{equation}
F=\frac{P_{12}}{eV_1 G_{12}}=\frac{1}{\kappa^2}-\tfrac{1}{2}(1+\kappa^2).\label{Fdef}
\end{equation}
\subsection{Thermo-electricity}
Because of the energy dependence of the effective charge, a temperature difference $\delta T$ between contacts 1 and 2 will produce an electrical current $I_{12}=\alpha_{12}\delta T$ in addition to a heat current. The thermo-electric coefficient $\alpha_{12}$ is given by \cite{Siv86}
\begin{equation}
\alpha_{12}=\frac{\pi^2}{3e}k_{\rm B}^2 T_0\lim_{E\rightarrow 0}\frac{d}{dE}G_{12}(E).\label{alpha12def}
\end{equation}
Substitution of Eq.\ \eqref{G12finiteVworkedout} gives
\begin{align}
\alpha_{12}&=-\frac{\pi^2}{3e}k_{\rm B}^2 T_0 G_{12}(0)\frac{\Delta_0^2}{(\beta^2-\Delta_0^2)^{3/2}}\nonumber\\
&=-g_0 e\kappa_0^2 N_{\rm Landau}\frac{\Delta_0^2}{(\beta^2-\Delta_0^2)^{3/2}}\nonumber\\
&=-g_0 e N_{\rm Landau}\frac{(\Delta_0/\beta)^2}{(\beta^2-\Delta_0^2)^{1/2}}.\label{alpha12result}
\end{align}
\section{Numerical simulations}
\label{sec_numerical}
To test these analytical results, we have carried out numerical calculations in a tight-binding model of the Weyl superconductor with a vortex lattice.
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{G_F_results}}
\caption{Data points: Electrical conductance (top panel) and Fano factor (bottom panel) in the superconducting vortex lattice (lattice constant $d_0$), as a function of the pair potential $\Delta_0$ at fixed magnetization $\beta=1$, calculated from the tight-binding model (lattice constant $a_0$) for different lattice constant ratios $N_0=d_0/a_0$. The black curves are the analytical predictions from the charge renormalization factor $\kappa$, both in the approximation of a linearized dispersion (black dashed curve, $\kappa=\kappa_0=\sqrt{1-\Delta_0^2/\beta^2}$) and for the full nonlinear dispersion (black solid).
}
\label{fig_G_F_results}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{thermo}}
\caption{Dependence on $\Delta_0$ for $\beta=0.5$ of the thermo-electric coefficient \eqref{alpha12def}, calculated from the infinite-system analytics (black solid curve) or obtained from finite-size numerics (colored data points).
}
\label{fig_thermo}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{G_F_results_perp}}
\caption{Same as Fig.\ \ref{fig_G_F_results}, but for a magnetization $\beta$ that is perpendicular rather than parallel to the magnetic field $B$.
}
\label{fig_G_F_results_perp}
\end{figure}
\subsection{Tight-binding Hamiltonian}
\label{sec_TBH}
The Bogoliubov-de Gennes Hamiltonian $H_{\rm S}$ in the superconducting region $0<z<L$ is
\begin{subequations}
\label{HBdGSdef}
\begin{align}
H_{\rm S}={}&
\begin{pmatrix}
H_0(\bm{k}+e\bm{A})&\Delta\\
\Delta^\ast&-\sigma_y H_0^\ast(-\bm{k}+e\bm{A})\sigma_y
\end{pmatrix},\label{HBdGSdefa}\\
H_0(\bm{k})={}&t_0{\sum_{\alpha=x,y,z}}\left[\tau_z\sigma_\alpha\sin k_\alpha a_0+\tau_x\sigma_0(1-\cos k_\alpha a_0)\right]\nonumber\\
&+\beta\tau_0\sigma_z-\mu\tau_0\sigma_0.\label{HBdGSdefb}
\end{align}
\end{subequations}
The cubic lattice constant of the tight-binding model is $a_0$ and $t_0$ is the nearest-neigbor hopping energy. In what follows we will set $a_0$ and $t_0$ both equal to unity.
In the strong-type-II limit the magnetic field $\bm{B}=B_0\hat{z}$ penetrates the superconductor uniformly, with vector potential $\bm{A}=(-B_0y,0,0)$. The absolute value $\Delta_0$ of the pair potential $\Delta=\Delta_0 e^{i\phi}$ can also be taken uniform, assuming that the size $\xi_0=\hbar v_{\rm F}/\Delta_0$ of the vortex core is small compared to the magnetic length $l_m=\sqrt{\hbar/eB_0}$. For the analytical calculations this is the only requirement. For the numerics we also take $\xi_0$ small compared to the tight-binding discretization length $a_0$, and then ensure that a vortex core (where the phase field is undefined) does not coincide with a lattice point. This implies that $a_0$ is large compared to the atomic lattice constant (which itself must be much smaller than $\xi_0$).
The vortices are arranged on a square lattice in the $x$--$y$ plane, lattice constant $d_0=N_0a_0$, with two $h/2e$ vortices in a unit cell. The number
\begin{equation}
N_0=(a_0^2eB_0/h)^{-1/2}\label{N0def}
\end{equation}
is set at an integer value. The phase $\phi(\bm{r})$ winds around the vortex cores $\bm{R}_n$ according to
\begin{equation}
\nabla\times\nabla{\phi}=2\pi\hat{z}\textstyle{\sum_n}\delta(\bm{r}-\bm{R}_n).\label{curlgradphi}
\end{equation}
In the normal metal leads $z<0$, $z>L$ we have $\Delta_0\equiv 0$ and a large chemical potential $\mu_{\rm N}$, so only modes with a large longitudinal momentum $k_z$ couple to the superconductor. We effectuate the $\mu_{\rm N}\rightarrow\infty$ limit by removing the transverse $x,y$ couplings in the leads, resulting in the Hamiltonian \cite{note1}
\begin{equation}
H_{\rm N}=\nu_z\tau_{z}\sigma_{z}\sin{k_{z}}+\nu_z\tau_x\sigma_{0}(1-\cos{k_{z}}).\label{HBdGNdef}
\end{equation}
The gauge-invariant discretization of the Hamiltonian \eqref{HBdGSdef} in the magnetic Brillouin zone is detailed in Ref.\ \onlinecite{Pac18}. The scattering matrix is calculated using the Kwant code \cite{kwant}.
\subsection{Results}
\label{sec_results}
Results for the conductance and shot noise are shown in Fig.\ \ref{fig_G_F_results}, as a function of $\Delta_0$ for $\beta=1$, $\mu=0$. The analytical predictions \eqref{G12result} for the conductance and \eqref{Fdef} for the Fano factor are given by the black curves. As a check, for these curves we have also calculated the charge renormalization factor $\kappa$ from the full sinusoidal dispersion, without making the small-$\bm{k}$ expansion of Eq.\ \eqref{Edispersion} --- the difference with $\kappa_0=\sqrt{1-\Delta_0^2/\beta^2}$ is small.
To assess finite-size effects in the numerics we show results for different values of the ratio $N_0=d_0/a_0$ of magnetic unit cell and tight-binding unit cell. As expected, the agreement between numerics and analytics improves with increasing $N_0$, for $\Delta_0/\beta$ not close to unity. (At $\Delta_0=\beta$ the spectrum becomes gapless and the low-energy analytics breaks down.)
These are results at the Fermi level, $E=0$. The energy dependence of the conductance determines the thermo-electric coefficient \eqref{alpha12def}. We show numerical results for $\alpha_{12}\propto dG_{12}/dE$ in Fig.\ \ref{fig_thermo}, for a smaller $\beta=0.5$ to reduce the oscillations that disappear only slowly with increasing $N_0$.
\subsection{Test for isotropy of the charge renormalization}
\label{sec_isotropy}
So far we assumed that the internal magnetization $\beta$ is parallel to the external magnetic field in the $z$-direction. This assumption is needed for our low-energy analytics, but numerically we can take an arbitrary angle between the magnetization $\bm{\beta}=(\beta_x,\beta_y,\beta_z)$ and the magnetic field, by replacing the term $\beta\tau_0\sigma_z$ in the Hamiltonian \eqref{HBdGSdefb} with $\tau_0\,\bm{\beta}\cdot\bm{\sigma}$. Results for $\bm{\beta}=(\beta,0,0)$, so for a magnetization perpendicular to the magnetic field, are shown in Fig.\ \ref{fig_G_F_results_perp}. There is no qualitative difference with Fig.\ \ref{fig_G_F_results} for the parallel configuration, the quantitative difference is that the finite-size effects are smaller.
\section{Conclusion}
\label{sec_conclude}
In summary, we have shown how the charge renormalization $e\mapsto \kappa e$ of Weyl fermions in a superconducting vortex lattice modifies the electrical and thermo-electrical transport properties.
In the electrical conductance, the current per vortex is reduced by a factor $\tfrac{1}{2}\kappa^2$ --- a prefactor $1/2$ because of the Majorana nature of the quasiparticles and a factor $\kappa^2$ because of the effective charge. At the Weyl point $\kappa\rightarrow\kappa_0=\sqrt{1-\Delta_0^2/\beta^2}$ depends on the ratio of the superconducting gap $\Delta_0$ and the separation $2\beta$ of the Weyl points of opposite chirality.
The charge-squared renormalization of the electrical conductance is a simple result, but perhaps not what one might have guessed by analogy with the fractional quantum Hall effect, where a $1/3$ fractional charge reduces the conductance by $1/3$ rather than $1/9$. The key difference is that here the quasiparticles are not in an eigenstate of charge; the charge renormalization is due to quantum fluctuations, which give uncorrelated reductions by $\kappa\times \kappa$ at entrance and exit. These quantum fluctuations of the charge are also responsible for the large shot noise power that we have found, with a diverging Fano factor \eqref{Fdef} in the limit $\kappa\rightarrow 0$.
The energy dependence of the charge renormalization implies that charge transport parallel to the magnetic field $B$ goes hand-in-hand with heat transport. As a result, a nonzero thermo-electric coefficient $\alpha_{12}$ along the field lines appears in a chiral Landau level --- something that would not be possible in the normal state: The Landau level contributes an energy-independent number of propagating modes along $B$ (one mode per flux quantum) and the chirality suppresses backscattering, so the energy derivative in Eq.\ \eqref{alpha12def} would vanish in the normal state.
There is much recent interest in thermo-electricity of Weyl fermions in a Landau level \cite{Ski18,Koz19,Zha19,Han19}, but that refers to currents perpendicular to $B$. Our findings show that charge renormalization in a Weyl superconductor provides a mechanism for a nonzero effect parallel to the field lines.
In our calculations we have assumed a clean system, without impurity scattering. However, we expect the transport properties to be robust against non-magnetic disorder, which in the effective low-energy Hamiltonian \eqref{Heffdef} would enter as a term proportional to $\sigma_z$ that does not couple Landau levels of opposite chirality.
\acknowledgments
This project has received funding from the Netherlands Organization for Scientific Research (NWO/OCW), from the T\"{U}B\.{I}TAK grant No.\ 114F163, and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,715
|
var gulp = require('gulp');
var concat = require('gulp-concat');
var babel = require('gulp-babel');
var onBabelError = require('./babel-error.js');
var CONFIG = require('../config.js');
// Compiles JavaScript into a single file
gulp.task('javascript', ['javascript:foundation', 'javascript:deps', 'javascript:docs']);
gulp.task('javascript:foundation', function() {
return gulp.src(CONFIG.JS_FILES)
.pipe(babel()
.on('error', onBabelError))
.pipe(gulp.dest('_build/assets/js/plugins'))
.pipe(concat('foundation.js'))
.pipe(gulp.dest('_build/assets/js'));
});
gulp.task('javascript:deps', function() {
return gulp.src(CONFIG.JS_DEPS)
.pipe(concat('vendor.js'))
.pipe(gulp.dest('_build/assets/js'));
});
gulp.task('javascript:docs', function() {
return gulp.src(CONFIG.JS_DOCS)
.pipe(concat('docs.js'))
.pipe(gulp.dest('_build/assets/js'));
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,924
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Q: Fibre channel zoning by port: bad idea? I have a single FC switch here that has a bunch of servers hanging off of it. Currently it is zoned (as per my previous question, Fibre Channel zoning best practices) by WWN, one zone for each pair of (server, disk array).
My question is, is there any reason I shouldn't do this zoning by port instead of WWN? The switch is already labeled with a server name per port, and I don't expect to do any moving of cables. I'm tending toward zoning by port because it allows me to replace an FC card without rezoning. That's not something you need to do often, you say? You're probably right, but I'm in the middle of a period where I have to do it a lot.
If it matters, it's a QLogic switch with QLogic or Brocade FC cards and a NetApp filer.
A: If you ever foresee implementing NPIV, you'll need to be doing WWN zoning. We're currently doing port zoning on the SAN that I manage, but that's for no other reason than it's always been done like that here. Within the next few weeks, I'll be switching over to WWN zoning. There are significant pros and cons to both approaches, though there's a strong security argument to be made for doing WWN zoning. It's really just a matter of how your organization chooses to do things.
A: If it's that simple then no, I guess it'll be fine. I do mine WWN-to-host-ports simply because I have multiple hosts/ports, so a switch-port-to-host-port thang wouldn't work out but you should do whatever makes sense to your situation and what you've described doesn't set any alarms going :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,793
|
A Complete Read-Out Channel With Embedded Wilkinson A/D Converter for X-Ray Spectrometry
This paper presents a complete read-out channel suitable for large arrays of X-ray detectors to be used for spectrometry applications in space. The system is fully integrated except for the X-ray detector. It basically consists of a front-end circuit for processing the detector signal, a Wilkinson A/D converter for the analog-to-digital conversion and the digital logic required to ensure the correct handshaking between all the blocks of the read-out channel. The system allows us to process the signal provided by the detector down to the final analog-to-digital conversion. All these functionalities are embedded in a single chip that, in its final version, will be bump-bonded to the matrix of X-ray detectors. The chip, designed in a 0.35 m CMOS technology, achieves an inputreferred noise of 34 e- rms, consuming 0.9 mW from a 3.3 V power supply. The on-board A/D converter features 10 bits of resolution with a maximum conversion time of 210 s. The INL and DNL of the whole read-out channel are equal to 3.3 LSB and 0.2 LSB, respectively.
Titolo: A Complete Read-Out Channel With Embedded Wilkinson A/D Converter for X-Ray Spectrometry
A. Rossini
S. Caccia
G. Bertuccio
Borghetti, Fausto
V. Ferragina
P. Malcovati
D. Martin
P. Bastia
I. Cappellutti
N. Ratti
IEEE TRANSACTIONS ON NUCLEAR SCIENCE
Abstract: This paper presents a complete read-out channel suitable for large arrays of X-ray detectors to be used for spectrometry applications in space. The system is fully integrated except for the X-ray detector. It basically consists of a front-end circuit for processing the detector signal, a Wilkinson A/D converter for the analog-to-digital conversion and the digital logic required to ensure the correct handshaking between all the blocks of the read-out channel. The system allows us to process the signal provided by the detector down to the final analog-to-digital conversion. All these functionalities are embedded in a single chip that, in its final version, will be bump-bonded to the matrix of X-ray detectors. The chip, designed in a 0.35 m CMOS technology, achieves an inputreferred noise of 34 e- rms, consuming 0.9 mW from a 3.3 V power supply. The on-board A/D converter features 10 bits of resolution with a maximum conversion time of 210 s. The INL and DNL of the whole read-out channel are equal to 3.3 LSB and 0.2 LSB, respectively.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,611
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Q: Using subquery in SQL that is returning Every derived table must have its own alias error? I'm finding it difficult to grasp why this following query wouldn't work:
SELECT rating_count
FROM
(SELECT
title,
COUNT(rating) AS rating_count
FROM series
LEFT JOIN reviews ON series.id = reviews.series_id
GROUP BY series.id);
The result of the subquery is the following:
Why is it that when I try to query for the rating_count, it wouldn't work? I thought I could request this. My intention is filter those rows that have rating_count = 0.
I wanted to try:
SELECT *
FROM
(SELECT
title, COUNT(rating) AS rating_count
FROM series
LEFT JOIN reviews ON series.id = reviews.series_id
GROUP BY series.id)
WHERE rating_count = 0;
A: Exactly as the error says, you need to alias your derived table:
SELECT * FROM (
SELECT
title,
count(rating) as rating_count
FROM series
LEFT JOIN reviews
ON series.id = reviews.series_id
GROUP BY series.id
) AS t
WHERE rating_count=0;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,021
|
The 1988 British Formula Three season was the 38th season of the British Formula Three Championship. JJ Lehto took the BARC/BRDC Lucas British Formula 3 Championship.
BARC/BRDC Lucas British F3 Championship
Champion: JJ Lehto
Runner Up: Gary Brabham
Class B Champion: Alastair Lyall
Results
Lucas British Formula 3 Championship
Non-Championship Races
Championship Tables
Class A
Class B
References
Formula Three
British Formula Three Championship seasons
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,035
|
\section{Introduction}
A semigroup is an algebraic structure consisting of a non-empty set $S$ together with an associative binary operation\cite{H}. The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics, for example, coding and language theory, automata theory, combinatorics and mathematical analysis. The concept of fuzzy sets was introduced by {\it Lofti Zadeh}\cite{Z} in his classic paper in $1965.$ {\it Azirel Rosenfeld}\cite{R} used the idea of fuzzy set to introduce the notions of fuzzy subgroups. {\it Nobuaki Kuroki}\cite{K1,K2,K3} is the pioneer of fuzzy ideal theory of semigroups. The idea of fuzzy subsemigroup was also introduced by {\it Kuroki}$\cite{K1,K3}$. In $\cite{K2}$, {\it Kuroki} characterized several classes of semigroups in terms of fuzzy left, fuzzy right and fuzzy bi-ideals. Others who worked on fuzzy semigroup theory, such as {\it X.Y. Xie}\cite{X1,X2}, {\it Y.B. Jun}\cite{J1,J2}, are mentioned in the bibliography. The notion of intuitionistic fuzzy sets was introduced by {\it Atanassov}\cite{A1,A2} as a generalization of the notion of fuzzy sets. In this paper we introduce the notion of intuitionistic fuzzy magnified translation in semigroups and observe some of its important properties. Here we characterize regular, intra-regular and left$($right$)$ regular semigroups in terms of intuitionistic fuzzy magnified translation. Finally we also observe that intuitionistic fuzzy translation and intuitionistic fuzzy multiplication are the particular cases of intuitionistic fuzzy magnified translation.
\section{Preliminaries}
\indent In this section we discuss some elementary definitions that we use in the sequel.\\
If $(X,\ast)$ is a mathematical system such that $\forall a,b,c\in X,$ $(a\ast b)\ast c=a\ast(b\ast c),$ then $\ast$ is called associative and $(X,\ast)$ is called a {\it semigroup}\cite{M}.\\
Throughout the paper unless otherwise stated $S$ will denote a semigroup.\\
A non-empty subset $A$ of a semigroup $S$ is called a {\it subsemigroup}\cite{K2} of $S$ if $AA\subseteq A.$
A subsemigroup $A$ of $S$ is called a {\it bi-ideal}\cite{K2} of $S$ if $ASA\subseteq A.$
A subsemigroup $A$ of $S$ is called an {\it $(1,2)$-ideal}\cite{K2} of $S$ if $ASAA\subseteq A.$
A {\it left} ({\it right})
{\it ideal}\cite{M} of a semigroup $S$ is a non-empty subset $I$ of $S$ such that $SI \subseteq I$ ($IS \subseteq I$). If $I$ is both a left and a right ideal of a semigroup $S$, then we say that $I$ is an {\it ideal}\cite{M} of $S$.\\
A {\it fuzzy subset}\cite{Z} in $S$ is a function $\mu : S \longrightarrow [0,1]$.
\indent Let $\mu $ be a fuzzy subset of a set $X$ and $\alpha \in\lbrack 0,1-\sup \{\mu (x):x\in X\}].$ A mapping $\mu_{\alpha }^{T}:X\rightarrow \lbrack 0,1]$\ is called a {\it fuzzy translation}\cite{K} of $\mu $
if $\mu _{\alpha }^{T}(x)=\mu (x)+\alpha $ for all $x\in X$.
\indent Let $\mu $ be a fuzzy subset of a set $X$ and $\beta \in\lbrack 0,1].$ A mapping $\mu _{\beta }^{M}:X\rightarrow \lbrack 0,1]$\ is called a {\it fuzzy multiplication}\cite{K} of $\mu $ if $\mu _{\beta }^{M}(x)=\beta\cdot\mu (x)$ for all $x\in X$.
Let $\mu $ be a fuzzy subset of a set $X,$ $\alpha \in\lbrack 0,1-\sup \{\mu (x):x\in X\}]$ and $\beta\in \lbrack 0,1].$ A mapping $\mu _{\beta\alpha }^{C}:X\rightarrow \lbrack 0,1]$\ is called a {\it fuzzy magnified translation}\cite{S1} of $\mu $ if $\mu _{\beta\alpha }^{C}(x)=\beta\cdot\mu (x)+\alpha $ for
all $x\in X$.
A non-empty fuzzy subset $\mu$ of a semigroup $S$ is called a {\it fuzzy subsemigroup}\cite{M} of $S$ if $\mu(xy)\geq\min\{\mu(x),\mu(y)\}\forall x,y\in S.$
A fuzzy subsemigroup $\mu$ of a semigroup $S$ is called a {\it fuzzy bi-ideal}\cite{M} of $S$ if $\mu(xyz)\geq\min\{\mu(x),\mu(z)\}\forall x,y,z\in S.$
A fuzzy subsemigroup $\mu$ of a semigroup $S$ is called a {\it fuzzy $(1,2)$-ideal}\cite{M} of $S$ if $\mu(x\omega(yz))\geqslant\min\{\mu(x),\mu(y),\mu(z)\}\forall x,\omega,y,z\in S.$
A non-empty fuzzy subset $\mu$ of a semigroup $S$ is called a {\it fuzzy left$($right$)$ ideal}\cite{M} of $S$ if $\mu(xy)\geq\mu(y)($resp. $\mu(xy)\geq\mu(x))$\ $\forall x,y\in S.$
A non-empty fuzzy subset $\mu$ of a semigroup $S$ is called a {\it fuzzy two-sided ideal} or a {\it fuzzy ideal}\cite{M} of $S$ if it is both a fuzzy left and a fuzzy right ideal of $S.$
A fuzzy ideal $\mu$ of a semigroup $S$ is called a {\it fuzzy semiprime ideal}\cite{K1} of $S$ if $\mu(x)\geq\mu(x^{2})\forall x\in S.$\\
{\it Atanassov} introduced in \cite{A1,A2} the concept of intuitionistic fuzzy sets defined on a non-empty set $X$ as objects having the form\\
$$A=\{<x,\mu_{A}(x),\nu_{A}(x)>:x\in X\},$$ where the functions $\mu_{A}: X\rightarrow [0,1]$ and $\nu_{A}: X\rightarrow [0,1]$ denote the degree of membership and the degree of non-membership of each element $x\in X$ to the set $A$ respectively, and $0\leq \mu_{A}(x)+\nu_{A}(x)\leq 1$ for all $x\in X.$\\
\indent Such defined objects are studied by many authors and have many interesting applications in mathematics.\\
\indent Let $A$ and $B$ be two intuitionistic fuzzy subsets of a set $X.$ Then the following expressions are defined in \cite{A1,A2}.\\
\indent $(i)$ $A\subseteq B$ if and only if $\mu_{A}(x)\leq\mu_{B}(x)$ and $\nu_{A}(x)\geq\nu_{B}(x),$\\
\indent $(ii)$ $A=B$ if and only if $A\subseteq B$ and $B\subseteq A,$\\
\indent $(iii)$ $A^{C}=\{<x,\nu_{A}(x),\mu_{A}(x)>:x\in X\},$\\
\indent $(iv)$ $A\cap B=\{<x,\min\{\mu_{A}(x),\mu_{B}(x)\},\max\{\nu_{A}(x),\nu_{B}(x)\}>:x\in X\},$\\
\indent $(v)$ $A\cup B=\{<x,\max\{\mu_{A}(x),\mu_{B}(x)\},\min\{\nu_{A}(x),\nu_{B}(x)\}>:x\in X\}.$\\
From the definition it follows that $A\cap B$ is the same as $\mu_{A}\cap\mu_{B}$ and $\nu_{A}\cup\nu_{B}.$ Also $A\cup B$ is the same as $\mu_{A}\cup\mu_{B}$ and $\nu_{A}\cap\nu_{B}.$\\
For the sake of simplicity, we shall use the symbol $A=(\mu_{A},\nu_{A})$ for the intuitionistic fuzzy subset $A=\{<x,\mu_{A}(x),\nu_{A}(x)>:x\in X\}.$\\
A non-empty intuitionistic fuzzy subset $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is
called an {\it intuitionistic fuzzy subsemigroup} of $S$ if $(i)$ $\mu_{A}(xy)\geq\min\{\mu_{A}(x),\mu_{A}(y)\}\forall x,y\in S,$ $(ii)$ $\nu_{A}(xy)\leq\max\{\nu_{A}(x),\nu_{A}(y)\}\forall x,y\in S.$
An intuitionistic fuzzy subsemigroup $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is called an {\it intuitionistic fuzzy bi-ideal} of $S$ if $(i)$ $\mu_{A}(xyz)\geq\min\{\mu_{A}(x),\mu_{A}(z)\}\forall x,y,z\in S,$ $(ii)$ $\nu_{A}(xyz)\leq\max\{\nu_{A}(x),\nu_{A}(z)\}\forall x,y,z\in S.$
An intuitionistic fuzzy subsemigroup $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is called an {\it intuitionistic fuzzy $(1,2)$-ideal} of $S$ if $(i)$ $\mu_{A}(x\omega(yz))\geqslant\min\{\mu_{A}(x),\mu_{A}(y),\mu_{A}(z)\}\forall
x,\omega,y\\,z\in S,$ $(ii)$ $\nu_{A}(x\omega(yz))\leqslant\max\{\nu_{A}(x),\nu_{A}(y),\nu_{A}(z)\}\forall
x,\omega,y,z\in S.$
A non-empty intuitionistic fuzzy subset $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is called an {\it intuitionistic fuzzy left$($right$)$ ideal} of $S$ if $(i)$ $\mu_{A}(xy)\geq\mu(y)($resp. $\mu_{A}(xy)\geq\mu_{A}(x))$\ $\forall x,y\in S,$ $(ii)$ $\nu_{A}(xy)\leq\nu_{A}(y)($resp. $\nu_{A}(xy)\leq\nu_{A}(x))$\ $\forall x,y\in S.$
A non-empty intuitionistic fuzzy subset $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is called an {\it intuitionistic fuzzy two-sided ideal} or an {\it intuitionistic fuzzy ideal} of $S$ if it is both an intuitionistic fuzzy left and an intuitionistic fuzzy right ideal of $S.$
An intuitionistic fuzzy ideal $A=(\mu_{A},\nu_{A})$ of a semigroup $S$ is called an {\it intuitionistic fuzzy semiprime ideal} of $S$ if $(i)$ $\mu_{A}(x)\geq\mu_{A}(x^{2})\forall x\in S,$ $(ii)$ $\nu_{A}(x)\leq\nu_{A}(x^{2})\forall x\in S.$
\section{Main Results}
\begin{definition}
Let $A=(\mu_{A},\nu_{A}) $ be an intuitionistic fuzzy subset of a set $X,$ $\alpha \in\lbrack 0,\inf \{\nu_{A}(x):x\in X\}].$ An object having the form $A^{T}_{\alpha}=((\mu_{A})^{T}_{\alpha},(\nu_{A})^{T}_{\alpha})$ is called an {\it intuitionistic fuzzy translation} of $A$ if $(\mu_{A})_{\alpha }^{T}(x)=\mu_{A}(x)+\alpha$ and $(\nu_{A})_{\alpha }^{T}(x)=\nu_{A}(x)-\alpha$ for all $x\in X$.
\end{definition}
\begin{definition}
Let $A=(\mu_{A},\nu_{A}) $ be an intuitionistic fuzzy subset of a set $X,$ $\beta\in \lbrack 0,1].$ An object having the form $A^{M}_{\beta}=((\mu_{A})^{M}_{\beta},(\nu_{A})^{M}_{\beta})$ is called an {\it intuitionistic fuzzy multiplication} of $A$ if $(\mu_{A})_{\beta}^{M}(x)=\beta\cdot\mu_{A}(x)$ and $(\nu_{A})_{\beta}^{M}(x)=\beta\cdot\nu_{A}(x)$ for all $x\in X$.
\end{definition}
\begin{definition}
Let $A=(\mu_{A},\nu_{A}) $ be an intuitionistic fuzzy subset of a set $X,$ $\alpha \in\lbrack 0,\inf\{\beta\cdot\nu_{A}(x):x\in X\}],$ where $\beta\in \lbrack 0,1].$ An object having the form $A^{C}_{\beta\alpha}=((\mu_{A})^{C}_{\beta\alpha},(\nu_{A})^{C}_{\beta\alpha})$ is called an {\it intuitionistic fuzzy magnified translation} of $A$ if $(\mu_{A})_{\beta\alpha }^{C}(x)=\beta\cdot\mu_{A}(x)+\alpha$ and $(\nu_{A})_{\beta\alpha }^{C}(x)=\beta\cdot\nu_{A}(x)-\alpha$ for
all $x\in X$.
\end{definition}
\begin{example} Let $X=\{1,\omega,\omega^{2}\}.$ Let $A=(\mu_{A},\nu_{A}) $ be an intuitionistic fuzzy subset of $X,$ defined
as follow
\begin{align*}
\mu_{A} (x)=\left\{
\begin{array}{ll}
0.3 & \text{if} \ x=1 \\
0.1 & \text{if} \ x=\omega \\
0.5 & \text{if} \ x=\omega^{2}
\end{array}
\right. & \text{ and }
\nu_{A} (x)=\left\{
\begin{array}{ll}
0.4 & \text{if} \ x=1 \\
0.25 & \text{if} \ x=\omega \\
0.3 & \text{if} \ x=\omega^{2}
\end{array}
\right..
\end{align*}
Let $\alpha=0.04$ and $\beta=0.2.$ Then the intuitionistic fuzzy magnified translation of $A$ is given by
\begin{align*}
(\mu_{A})_{\beta\alpha }^{C}(x)=\left\{
\begin{array}{ll}
0.1 & \text{if} \ x=1 \\
0.06 & \text{if} \ x=\omega \\
0.14 & \text{if} \ x=\omega^{2}
\end{array}
\right. & \text{ and }
(\nu_{A})_{\beta\alpha }^{C}(x)=\left\{
\begin{array}{ll}
0.04 & \text{if} \ x=1 \\
0.01 & \text{if} \ x=\omega \\
0.02 & \text{if} \ x=\omega^{2}
\end{array}
\right..
\end{align*}
\end{example}
\indent In what follows unless otherwise mentioned $A=(\mu_{A},\nu_{A})$ denotes a non-empty intuitionistic fuzzy subset of $S$ and $A_{\beta\alpha}^{C}$ denotes the intuitionistic fuzzy magnified translation of $A$ where $\beta\in(0,1],\alpha\in\lbrack0,\inf\{\beta\cdot\nu_{A}(x):x\in$ Supp$(\nu_{A})\}].$ It can be noted here that
$A_{\beta\alpha}^{C}$ is also non-empty.
\begin{theorem}
The intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy subsemigroup of $S$ if and only if $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy subsemigroup of $S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionitic fuzzy subsemigroup of a semigroup $S.$ Then $A$ is a non-empty intuitionistic fuzzy subset of $S(\beta>0).$ Hence $A_{\beta\alpha}^{C}$ is also non-empty. Now for $x,y\in S$
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(xy) & =\beta\cdot\mu_{A}(xy)+\alpha\\
& \geq\beta\cdot\min\{\mu_{A}(x),\mu_{A}(y)\}+\alpha(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ subsemigroup of }S)=\min\{\beta\cdot\mu_{A}(x)+\alpha,\beta\cdot\mu_{A}(y)+\alpha\}\\
& =\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu_{A})_{\beta\alpha}^{C}(y)\}
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(xy) & =\beta\cdot\nu_{A}(xy)-\alpha\\
& \leq\beta\cdot\max\{\nu_{A}(x),\nu_{A}(y)\}-\alpha(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ subsemigroup of }S)=\max\{\beta\cdot\nu_{A}(x)-\alpha,\beta\cdot\nu_{A}(y)-\alpha\}\\
& =\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu_{A})_{\beta\alpha}^{C}(y)\}.
\end{align*}
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy subsemigroup of $S.$
Conversely, let $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy subsemigroup of $S.$ Then $A_{\beta\alpha}^{C}$ and hence $A$ is a non-empty fuzzy subset of $S.$ Now for all $x,y\in S,
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(xy) & \geq\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu
_{A})_{\beta\alpha}^{C}(y)\}\\
i.e.,\ \beta.\mu_{A}(xy)+\alpha & \geq\min\{\beta.\mu_{A}(x)+\alpha,\beta
.\mu_{A}(y)+\alpha\}\\
i.e.,\ \beta.\mu_{A}(xy)+\alpha & \geq\beta.\min\{\mu_{A}(x),\mu_{A}(y)\}+\alpha\\
i.e.,\ \mu_{A}(xy) & \geq\min\{\mu_{A}(x),\mu_{A}(y)\}
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(xy) & \leq\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu
_{A})_{\beta\alpha}^{C}(y)\}\\
i.e.,\ \beta.\mu_{A}(xy)-\alpha & \leq\max\{\beta.\nu_{A}(x)-\alpha,\beta
.\nu_{A}(y)-\alpha\}\\
i.e.,\ \beta.\nu_{A}(xy)-\alpha & \leq\beta.\max\{\nu_{A}(x),\nu_{A}(y)\}-\alpha\\
i.e.,\ \nu_{A}(xy) & \leq\max\{\nu_{A}(x),\nu_{A}(y)\}
\end{align*}
Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy subsemigroup of $S.$
\end{proof}
\begin{theorem}
The intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy bi-ideal of $S$ if and only if $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy bi-ideal of $S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy bi-ideal of a semigroup $S.$ Then $A$ is an intuitionistic fuzzy
subsemigroup of $S$ and hence, by Theorem $3.4,$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy subsemigroup of $S.$ Now for $x,\omega,y\in S,
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x\omega y)=\beta\cdot\mu_{A}(x\omega y)+\alpha & \geq\beta\cdot\min\{\mu_{A}(x),\mu_{A}(y)\}+\alpha\\&(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ bi-ideal of }S)\\
& =\min\{\beta\cdot\mu_{A}(x)+\alpha,\beta\cdot\mu_{A}(y)+\alpha\}\\
& =\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu_{A})_{\beta\alpha}^{C}(y)\}.
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(x\omega y)=\beta\cdot\nu_{A}(x\omega y)-\alpha & \leq\beta\cdot\max\{\nu_{A}(x),\nu_{A}(y)\}-\alpha\\&(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ bi-ideal of }S)\\
& =\max\{\beta\cdot\nu_{A}(x)-\alpha,\beta\cdot\nu_{A}(y)-\alpha\}\\
& =\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu_{A})_{\beta\alpha}^{C}(y)\}.
\end{align*}
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy bi-ideal of $S.$
Conversely, let $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy bi-ideal of $S.$ Then $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy subsemigroup of $S,$ and hence by Theorem $3.4,$ $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy subsemigroup of $S.$
Now for all $x,\omega,y\in S,
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x\omega y) & \geq\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu
_{A})_{\beta\alpha}^{C}(y)\}\\
i.e.,\ \beta.\mu_{A}(x\omega y)+\alpha & \geq\min\{\beta.\mu_{A}(x)+\alpha,\beta
.\mu_{A}(y)+\alpha\}\\
i.e.,\ \beta.\mu_{A}(x\omega y)+\alpha & \geq\beta.\min\{\mu_{A}(x),\mu_{A}(y)\}+\alpha\\
i.e.,\ \mu_{A}(x\omega y) & \geq\min\{\mu_{A}(x),\mu_{A}(y)\}
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(x\omega y) & \leq\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu
_{A})_{\beta\alpha}^{C}(y)\}\\
i.e.,\ \beta.\nu_{A}(x\omega y)-\alpha & \leq\max\{\beta.\nu_{A}(x)-\alpha,\beta
.\nu_{A}(y)-\alpha\}\\
i.e.,\ \beta.\nu_{A}(x\omega y)-\alpha & \leq\beta.\max\{\nu_{A}(x),\nu_{A}(y)\}-\alpha\\
i.e.,\ \nu_{A}(x\omega y) & \leq\max\{\nu_{A}(x),\nu_{A}(y)\}
\end{align*}
Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy bi-ideal of $S.$
\end{proof}
\begin{theorem}
Let $A=(\mu_{A},\nu_{A})$ be a non-empty intuitionistic fuzzy subset of a semigroup $S.$ Then
$A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy bi-ideal of $S$ if and only if the intuitionistic fuzzy magnified
translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A$ is a constant function, provided $S$ is a group with identity $e.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy bi-ideal of $S.$ The
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x) & =\beta\cdot\mu_{A}(x)+\alpha=\beta\cdot
\mu_{A}(exe)+\alpha(\text{since }e\text{ is the identity of }S)\\
& \geq\beta\cdot\min\{\mu_{A}(e),\mu_{A}(e)\}+\alpha(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ bi-ideal of }S)=\beta\cdot\mu_{A}(e)+\alpha=(\mu_{A})_{\beta\alpha}^{C}(e)
\end{align*}
Agai
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(e) & =(\mu_{A})_{\beta\alpha}^{C}(ee)=\beta\cdot
\mu_{A}(ee)+\alpha(\text{since }e\text{ is the identity of }S)\\
& =\beta\cdot\mu_{A}((xx^{-1})(x^{-1}x))+\alpha\\
& =\beta\cdot\mu_{A}(x(x^{-1}x^{-1})x)+\alpha\\
& \geq\beta\cdot\min\{\mu_{A}(x),\mu_{A}(x)\}+\alpha(\text{since }A\text{ is an intuitionistic
fuzzy}\\
& \text{ bi-ideal of }S)=\beta\cdot\mu_{A}(x)+\alpha=(\mu_{A})_{\beta\alpha}^{C}(x).
\end{align*}
Thus $(\mu_{A})_{\beta\alpha}^{C}(x)=(\mu_{A})_{\beta\alpha}^{C}(e)$ $\forall x\in S.$ Similarly we can show that $(\nu_{A})_{\beta\alpha}^{C}(x)=(\nu_{A})_{\beta\alpha}^{C}(e)$ $\forall x\in S.$ Hence $A_{\beta\alpha}^{C}$ is a constant function.
Conversely, suppose $A_{\beta\alpha}^{C}$ is a constant function. Then $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy bi-ideal of $S\cite{K1}.$ Hence by Theorem $3.5,$ $A$ is an intuitionistic fuzzy bi-ideal of $S.$
\end{proof}
\begin{note}
For the converse to be true, $S$ need not be a group.
\end{note}
\begin{theorem}
The intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy $(1,2)$-ideal of $S$ if and only if $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy $(1,2)$-ideal of $S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy $(1,2)$-ideal of $S.$ Then $A$ is an intuitionistic fuzzy subsemigroup of $S.$ Hence, by Theorem $3.4,$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy subsemigroup of $S.$ Now for $x,y,z,\omega\in S,
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x\omega(yz)) & =\beta\cdot\mu_{A}(x\omega(yz))+\alpha\\
& \geq\beta\cdot\min\{\mu_{A}(x),\mu_{A}(y),\mu_{A}(z)\}+\alpha\\
& (\text{since }A\text{ is an intuitionistic fuzzy }(1,2)\text{-ideal of }S)\\
& =\min\{\beta\cdot\mu_{A}(x)+\alpha,\beta\cdot\mu_{A}(y)+\alpha,\beta\cdot
\mu_{A}(z)+\alpha\}\\
& =\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu_{A})_{\beta\alpha}^{C}(y),(\mu
_{A})_{\beta\alpha}^{C}(z)\}.
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(x\omega(yz)) & =\beta\cdot\nu_{A}(x\omega(yz))-\alpha\\
& \leq\beta\cdot\max\{\nu_{A}(x),\nu_{A}(y),\nu_{A}(z)\}-\alpha\\
& (\text{since }A\text{ is an intuitionistic fuzzy }(1,2)\text{-ideal of }S)\\
& =\max\{\beta\cdot\nu_{A}(x)-\alpha,\beta\cdot\nu_{A}(y)-\alpha,\beta\cdot
\nu_{A}(z)-\alpha\}\\
& =\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu_{A})_{\beta\alpha}^{C}(y),(\nu
_{A})_{\beta\alpha}^{C}(z)\}.
\end{align*}
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy $(1,2)$-ideal of $S.$
Conversely, let $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy $(1,2)$-ideal of $S.$ Then $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy subsemigroup of $S,$ and hence by Theorem $3.4,$ $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy subsemigroup of $S.$
Now for all $x,\omega,y,z\in S,
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x\omega (yz)) & \geq\min\{(\mu_{A})_{\beta\alpha}^{C}(x),(\mu
_{A})_{\beta\alpha}^{C}(y),(\mu_{A})_{\beta\alpha}^{C}(z)\}\\
i.e.,\ \beta.\mu_{A}(x\omega (yz))+\alpha & \geq\min\{\beta.\mu_{A}(x)+\alpha,\beta
.\mu_{A}(y)+\alpha,\beta.\mu_{A}(z)+\alpha\}\\
i.e.,\ \beta.\mu_{A}(x\omega (yz))+\alpha & \geq\beta.\min\{\mu_{A}(x),\mu_{A}(y),\mu_{A}(z)\}+\alpha\\
i.e.,\ \mu_{A}(x\omega y) & \geq\min\{\mu_{A}(x),\mu_{A}(y),\mu_{A}(z)\}
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(x\omega (yz)) & \leq\max\{(\nu_{A})_{\beta\alpha}^{C}(x),(\nu
_{A})_{\beta\alpha}^{C}(y),(\nu_{A})_{\beta\alpha}^{C}(z)\}\\
i.e.,\ \beta.\nu_{A}(x\omega (yz))-\alpha & \leq\max\{\beta.\nu_{A}(x)-\alpha,\beta
.\nu_{A}(y)-\alpha,\beta.\nu_{A}(z)-\alpha\}\\
i.e.,\ \beta.\nu_{A}(x\omega (yz))-\alpha & \leq\beta.\max\{\nu_{A}(x),\nu_{A}(y),\nu_{A}(z)\}-\alpha\\
i.e.,\ \nu_{A}(x\omega (yz)) & \leq\max\{\nu_{A}(x),\nu_{A}(y),\nu_{A}(z)\}
\end{align*}
Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy $(1,2)$-ideal of $S.$
\end{proof}
\begin{theorem}
The intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy left ideal$($intuitionistic fuzzy right ideal, intuitionistic fuzzy ideal$)$ of $S$ if and only if $A=(\mu_{A},\nu_{A})$ is an intuitionstic fuzzy left ideal$($resp. intuitionistic fuzzy right ideal, intuitionstic fuzzy ideal$)$ of $S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy left ideal of $S.$ Then $A=(\mu_{A},\nu_{A})$ is a non-empty intuitionistic fuzzy subset of $S.$ Hence, as $\beta>0,$ $A_{\beta\alpha}^{C}$ is a non-empty intuitionistic fuzzy
subset of $S.$ Let $x,y\in S.$ The
\[
(\mu_{A})_{\beta\alpha}^{C}(xy)=\beta\cdot\mu_{A}(xy)+\alpha\geq\beta\cdot\mu_{A}
(y)+\alpha=(\mu_{A})_{\beta\alpha}^{C}(y)
\] an
\[
(\nu_{A})_{\beta\alpha}^{C}(xy)=\beta\cdot\nu_{A}(xy)-\alpha\leq\beta\cdot\nu_{A}
(y)-\alpha=(\nu_{A})_{\beta\alpha}^{C}(y).
\]
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy left ideal $S.$
Conversely, let $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy left ideal of $S.$ Then $A_{\beta\alpha}^{C}$ and hence $A$ is a non-empty intuitionistic fuzzy subset of $S.$
Now for all $x,y\in S,
\[
(\mu_{A})_{\beta\alpha}^{C}(xy)\geq(\mu_{A})_{\beta\alpha}^{C}(y),i.e.,\ \beta
.\mu_{A}(xy)+\alpha\geq\beta.\mu_{A}(y)+\alpha,i.e.,\ \mu_{A}(xy)\geq\mu_{A}(y)
\]an
\[
(\nu_{A})_{\beta\alpha}^{C}(xy)\leq(\nu_{A})_{\beta\alpha}^{C}(y),i.e.,\ \beta
.\nu_{A}(xy)-\alpha\leq\beta.\nu_{A}(y)-\alpha,i.e.,\ \nu_{A}(xy)\leq\nu_{A}(y).
\]
Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy left ideal of $S.$ Similar is the proof for intuitionistic fuzzy right ideal or intuitionistic fuzzy ideal.
\end{proof}
\begin{theorem}
The intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S$ if and only if $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy semiprime ideal of $S.$ Then $A=(\mu_{A},\nu_{A})$ is a non-empty intuitionistic fuzzy subset of $S.$ Hence $A_{\beta\alpha}^{C}$ is a non-empty intuitionistic fuzzy subset of
$S(\beta>0).$ Let $x\in S.$ The
\[
(\mu_{A})_{\beta\alpha}^{C}(x)=\beta\cdot\mu_{A}(x)+\alpha\geq\beta\cdot\mu_{A}
(x^{2})+\alpha=(\mu_{A})_{\beta\alpha}^{C}(x^{2})
\] an
\[
(\nu_{A})_{\beta\alpha}^{C}(x)=\beta\cdot\nu_{A}(x)-\alpha\leq\beta\cdot\nu_{A}
(x^{2})-\alpha=(\nu_{A})_{\beta\alpha}^{C}(x^{2}).
\]
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy semiprime ideal of $S.$
Conversely, let $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy semiprime ideal of $S.$ Then $A_{\beta\alpha}^{C}$ and hence $A$ is a non-empty intuitionistic fuzzy subset of
$S.$ Then for all $x\in S,
\[
(\mu_{A})_{\beta\alpha}^{C}(x)\geq(\mu_{A})_{\beta\alpha}^{C}(x^{2}),i.e.,\ \beta
.\mu_{A}(x)+\alpha\geq\beta.\mu_{A}(x^{2})+\alpha,i.e.,\ \mu_{A}(x)\geq\mu_{A}(x^{2}).
\] an
\[
(\nu_{A})_{\beta\alpha}^{C}(x)\leq(\nu_{A})_{\beta\alpha}^{C}(x^{2}),i.e.,\ \beta
.\nu_{A}(x)-\alpha\leq\beta.\nu_{A}(x^{2})-\alpha,i.e.,\ \nu_{A}(x)\leq\nu_{A}(x^{2}).
\]
Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S.$
\end{proof}
\begin{theorem}
Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ be two intuitionistic fuzzy semiprime ideals of $S.$ Then $A\cap B$ is an intuitionistic fuzzy semiprime ideal of $S,$ provided it is non-empty$.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ be two intuitionistic fuzzy semiprime ideals of $S$ and $x\in S.$ The
\begin{align*}
(\mu_{A}\cap\mu_{B})(x) & = \min\{\mu_{A}(x),\mu_{B}(x)\}\geq\min\{\mu_{A}(x^{2}),\mu_{B}(x^{2})\}\\
& (\text{since }A \text{ and } B\text{ are intuitionistic fuzzy semiprime ideals of }S)\\
& =(\mu_{A}\cap\mu_{B})(x^{2})
\end{align*} an
\begin{align*}
(\nu_{A}\cup\nu_{B})(x) & = \max\{\nu_{A}(x),\nu_{B}(x)\}\leq\max\{\nu_{A}(x^{2}),\nu_{B}(x^{2})\}\\
& (\text{since }A \text{ and } B\text{ are intuitionistic fuzzy semiprime ideals of }S)\\
& =(\nu_{A}\cup\nu_{B})(x^{2}).
\end{align*}
Hence $A\cap B$ is an intuitionistic fuzzy semiprime ideal of $S.$
\end{proof}
Since the intersections of intuitionistic fuzzy semiprime ideals is an intuitionistic fuzzy semiprime ideal, so we have the following corollary.
\begin{corollary}
Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ be two intuitionistic fuzzy semiprime ideals of $S.$ Then $A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$ is an intuitionistic fuzzy semiprime ideal of $S,$ provided it is non-empty$.$
\end{corollary}
\begin{notation}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy subset of a semigroup $S$ and $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ be an intuitionistic fuzzy magnified translation of $A.$ If for any elements $x,y\in S,$ $(\mu_{A})_{\beta\alpha}^{C}(x)=(\mu_{A})_{\beta\alpha}^{C}(y)$ and $(\nu_{A})_{\beta\alpha}^{C}(x)=(\nu_{A})_{\beta\alpha}^{C}(y),$ then we write $A_{\beta\alpha}^{C}(x)=A_{\beta\alpha}^{C}(y)$ $\forall x,y\in S.$
\end{notation}
\begin{theorem}
If $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of a semigroup $S,$ then $A_{\beta\alpha}^{C}(x)=A_{\beta\alpha}^{C}(x^{2})$ $\forall x\in S.$
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy semiprime ideal of $S.$ Let $x\in S.$ The
\[
(\mu_{A})_{\beta\alpha}^{C}(x)=\beta\cdot\mu_{A}(x)+\alpha\geq\beta\cdot\mu_{A}
(x^{2})+\alpha=(\mu_{A})_{\beta\alpha}^{C}(x^{2}).
\]
Again
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(x^{2}) & =\beta\cdot\mu_{A}(x^{2})+\alpha\\
& \geq\beta\cdot\mu_{A}(x)+\alpha(\text{since }A\text{ is an intuitionistic fuzzy ideal})\\
& =(\mu_{A})_{\beta\alpha}^{C}(x).
\end{align*}
Consequently, $(\mu_{A})_{\beta\alpha}^{C}(x)=(\mu_{A})_{\beta\alpha}^{C}(x^{2})$ $\forall x\in S.$ Similarly we can show that $(\nu_{A})_{\beta\alpha}^{C}(x)=(\nu_{A})_{\beta\alpha}^{C}(x^{2})$ $\forall x\in S.$ Hence $A_{\beta\alpha}^{C}(x)=A_{\beta\alpha}^{C}(x^{2})$ $\forall x\in S.$
\end{proof}
By routine verification we can have the following theorem.
\begin{theorem}
Let $\chi$ be the characteristic function of a non-empty subset $A$ of $S.$ Then $A$ is a left$($resp. right$)$ ideal of $S$ if and only if $(\chi,\overline{\chi})$ is an intuitionistic fuzzy left$($resp. right$)$ ideal of $S.$
\end{theorem}
\begin{definition}
A semigroup $S$ is called {\it intra-regular}\cite{K2} if for each element $a$ of $S,$ there exist elements $x,y\in S$ such that $a=xa^{2}y.$
\end{definition}
\begin{theorem}
For a semigroup $S$ the following conditions are equivalent:
$(1)$ $S$ is an intra-regular semigroup,
$(2)$ for every intuitionistic fuzzy ideal $A=(\mu_{A},\nu_{A})$ of $S$ the intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A$ is an intuitionistic fuzzy semiprime ideal of $S$.
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2):$ Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy ideal of $S$ and $m\in$ $S.$ Then there exist $x,y\in S$ such that $m=xm^{2}y($since $S$ is intra-regular$).$ The
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(m) & =(\mu_{A})_{\beta\alpha}^{C}(xm^{2}y)=\beta\cdot
\mu_{A}(xm^{2}y)+\alpha\\
& \geq\beta\cdot\mu_{A}(m^{2}y)+\alpha\geq\beta\cdot\mu_{A}(m^{2})+\alpha=(\mu
_{A})_{\beta\alpha}^{C}(m^{2})
\end{align*} and
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(m) & =(\nu_{A})_{\beta\alpha}^{C}(xm^{2}y)=\beta\cdot
\nu_{A}(xm^{2}y)-\alpha\\
& \leq\beta\cdot\nu_{A}(m^{2}y)-\alpha\leq\beta\cdot\nu_{A}(m^{2})-\alpha=(\nu
_{A})_{\beta\alpha}^{C}(m^{2})
\end{align*}
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy semiprime ideal of $S.$
$(2)\Rightarrow(1):$ Let $m$ be any element of $S.$ Then it follows that the intuitionistic fuzzy subset $A=(\chi_{<m^{2}>},\overline{\chi}_{<m^{2}>})$ of the principal ideal $<m^{2}>$ of $S($generated by $m^{2})$ is an intuitionistic fuzzy ideal of $S($ where $\chi_{<m^{2}>}$ is the characteristic function of the principal ideal $<m^{2}>$ of $S).$ From $(2),$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy semiprime ideal of $S.$ Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S(cf.$ Theorem $3.10).$ So $A_{\beta\alpha}^{C}(m)=A_{\beta\alpha}^{C}(m^{2})(cf.$ Theorem $3.13).$ Hence
$\chi_{<m^{2}>\text{ }}(m)=\chi_{<m^{2}>\text{ }}(m^{2})$ and $\overline{\chi}_{<m^{2}>\text{ }}(m)=\overline{\chi}_{<m^{2}>\text{ }}(m^{2}).$ Since $m^{2
\in<m^{2}>,$ we have $\chi_{<m^{2}>\text{ }}(m)=\chi_{<m^{2}>\text{ }
(m^{2})=1$ and $\overline{\chi}_{<m^{2}>\text{ }}(m)=\overline{\chi}_{<m^{2}>\text{ }
(m^{2})=0.$ So $m\in<m^{2}>=\{m^{2}\}\cup Sm^{2}\cup m^{2}S\cup Sm^{2}S.$ This
shows that $S$ is intra-regular. This completes the proof.
\end{proof}
\begin{definition}
A semigroup $S$ is said to be {\it left $($right$)$ regular}\cite{K1} if, for each element $a$ of $S$, there exists an element $x$ in $S$ such that $a=xa^{2}($resp. $a=a^{2}x).$
\end{definition}
\begin{theorem}
For a semigroup $S$ the following conditions are equivalent:
$(1)$ $S$ is a left regular semigroup,
$(2)$ for every intuitionistic fuzzy left ideal $A=(\mu_{A},\nu_{A})$ of $S$ the intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S$.
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2):$ Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy left ideal of $S$. Then by Theorem $3.11,$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy left ideal of $S.$ Let $m\in S$. Then there exists an element $x\in
S$ such that $m=xm^{2}($since $S$ is left regular$)$. The
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(m) & =\beta\cdot\mu_{A}(m)+\alpha=\beta\cdot\mu_{A}
(xm^{2})+\alpha\\
& \geq\beta\cdot\mu_{A}(m^{2})+\alpha(\text{since }A\text{ is an intuitionistic fuzzy left
ideal of }S)\\
& =(\mu_{A})_{\beta\alpha}^{C}(m^{2})
\end{align*} an
\begin{align*}
(\nu_{A})_{\beta\alpha}^{C}(m) & =\beta\cdot\nu_{A}(m)-\alpha=\beta\cdot\nu_{A}
(xm^{2})-\alpha\\
& \leq\beta\cdot\nu_{A}(m^{2})-\alpha(\text{since }A\text{ is an intuitionistic fuzzy left
ideal of }S)\\
& =(\nu_{A})_{\beta\alpha}^{C}(m^{2}).
\end{align*}
Hence $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is an intuitionistic fuzzy semiprime ideal of $S.$
$(2)\Rightarrow(1):$ Suppose $(2)$ holds. Let $m$ be any element of $S.$ Then it follows that the intuitionistic fuzzy subset $A=(\chi_{<m^{2}|},\overline{\chi}_{<m^{2}|})$ of the left ideal $<m^{2}|$ of $S($generated by $m^{2})$ is an intuitionistic fuzzy left ideal of $S($ where $\chi_{<m^{2}|}$ is the characteristic function of the left ideal $<m^{2}|$ of $S).$ From $(2),$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy semiprime ideal of $S.$ Hence $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy semiprime ideal of $S(cf.$ Theorem $3.10).$ So $A_{\beta\alpha}^{C}(m)=A_{\beta\alpha}^{C}(m^{2})(cf.$ Theorem $3.13).$ Hence $\chi_{<m^{2}|\text{}}(m)=\chi_{<m^{2}|\text{}}(m^{2})$ and $\overline{\chi}_{<m^{2}|\text{}}(m)=\overline{\chi}_{<m^{2}|\text{}}(m^{2}).$ Now since $m^{2}\in<m^{2}|,$ we see that $\chi_{<m^{2}|\text{ }}(m)=\chi_{<m^{2}|\text{ }}(m^{2})=1$ and $\overline{\chi}_{<m^{2}|\text{ }}(m)=\overline{\chi}_{<m^{2}|\text{ }}(m^{2})=0.$ Hence $m\in<m^{2}|=\{m^{2}\}\cup Sm^{2}.$ This shows that $S$ is left regular. This completes the proof.
\end{proof}
In a similar way we can prove the following theorem.
\begin{theorem}
For a semigroup $S$ the following conditions are equivalent:
$(1)$ $S$ is a right regular semigroup,
$(2)$ for every intuitionistic fuzzy right ideal $A=(\mu_{A},\nu_{A})$ of $S$ the intuitionistic fuzzy magnified translation $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ of $A$ is an intuitionistic fuzzy semiprime ideal of $S$.
\end{theorem}
\begin{definition}
A semigroup $S$ is called {\it archimedean}\cite{K2} if for all $a,b\in S,$ there exists a positive integer $n$ such that $a^{n}\in SbS.$
\end{definition}
\begin{theorem}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy semiprime ideal of an archimedean semigroup $S$.
Then $A_{\beta\alpha}^{C}=((\mu_{A})_{\beta\alpha}^{C},(\nu_{A})_{\beta\alpha}^{C})$ is a constant function.
\end{theorem}
\begin{proof}
Let $m,n\in S.$ Then $S$ being archimedean, there exists a positive integer
$k$ such that $m^{k}=xny$ for some $x,y\in S.$ Now since $A=(\mu_{A},\nu_{A})$ is an intuitionistic fuzzy
semiprime ideal of $S,$ so by Theorem $3.8,$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy semiprime ideal of $S.$ The
\begin{align*}
(\mu_{A})_{\beta\alpha}^{C}(m) & \geq(\mu_{A})_{\beta\alpha}^{C}(m^{k})=(\mu_{A})_{\beta
\alpha}^{C}(xny)=\beta\cdot\mu_{A}(xny)+\alpha\\
& \geq\beta\cdot\mu_{A}(n)+\alpha=(\mu_{A})_{\beta\alpha}^{C}(n).
\end{align*}
Using the duality of $m$ and $n$ we deduce that $(\mu_{A})_{\beta\alpha}^{C
(n)\geq(\mu_{A})_{\beta\alpha}^{C}(m)$. Thus $(\mu_{A})_{\beta\alpha}^{C}(m)=(\mu
_{A})_{\beta\alpha}^{C}(n)$ $\forall m,n\in S.$ Applying similar argument we can show that $(\nu_{A})_{\beta\alpha}^{C}(m)=(\nu_{A})_{\beta\alpha}^{C}(n)$ $\forall m,n\in S.$ Hence $A_{\beta\alpha}^{C}$ is a constant function.
\end{proof}
\begin{definition}
A semigroup $S$ is called {\it regular}\cite{K2} if for each element $a$ of $S,$ there exists an element $x\in S$ such that $a=axa.$
\end{definition}
\begin{definition}
Let $S$ be a semigroup. Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ be two intuitionistic fuzzy subsets of $S.$ Then the {\it product} $A\circ B$ of $A$ and $B$ is defined as\\
$$A\circ B=\{<x,(\mu_{A}\circ\mu_{B})(x),(\nu_{A}\circ\nu_{B})(x)>:x\in S\}$$
\end{definition}
$$
\text{ where } (\mu_{A}\circ \mu_{B})(x)=\left\{
\begin{array}
[c]{c
\underset{x=uv}{\sup}[\min\{\mu_{A}(u),\mu_{B}(v)\}:u,v\in S]\\
0,\text{ if for any }u,v\in S\text{ },x\neq uv
\end{array}
\right.\\
$$
$$
\text{ and }
(\nu_{A}\circ\nu_{B})(x)=\left\{
\begin{array}
[c]{c
\underset{x=uv}{\inf}[\max\{\nu_{A}(u),\nu_{B}(v)\}:u,v\in S]\\
1,\text{ if for any }u,v\in S\text{ },x\neq uv
\end{array}
\right.
$$
\begin{theorem}
If the semigroup $S$ is both regular and intra-regular then
$(1)$ $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}\supset A
_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}.$
$(2)$ $(A_{\beta\alpha}^{C}\circ B_{\beta\alpha}^{C})\cap
(B_{\beta\alpha}^{C}\circ A_{\beta\alpha}^{C})\supset A_{\beta\alpha
}^{C}\cap B_{\beta\alpha}^{C},$ where $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ are intuitionistic fuzzy
bi-ideals of $S$.
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ are intuitionistic fuzzy
bi-ideals of $S$ and $a\in S.$ Then there exist $x,y,z\in S$
such that $a=axa=axaxa($since $S$ is regular$)=ax(ya^{2}z)xa($since $S$ is
intra-regular$)=(axya)(azxa).$ Now
\begin{align*}
\mu_{A}(axya)\geq\min\{\mu_{A}(a),\mu_{A}(a)\}(\text{since }A \text{ is intuitionistic fuzzy bi-ideal of }S)=\mu_{A}(a)
\end{align*}and
\begin{align*}
\nu_{A}(axya)\leq\max\{\nu_{A}(a),\nu_{A}(a)\}(\text{since }A \text{ is intuitionistic fuzzy bi-ideal of }S)=\nu_{A}(a).
\end{align*}
Similarly we can have $\mu_{B}(azxa)\geq\mu_{B}(a)$ and $\nu_{B}(azxa)\leq\nu_{B}(a).$ The
\begin{align*}
((\mu_{A})_{\beta\alpha}^{C}\circ(\mu_{B})_{\beta\alpha}^{C})(a) & =\underset
{a=pq}{\sup}\min\{(\mu_{A})_{\beta\alpha}^{C}(p),(\mu_{B})_{\beta\alpha}^{C}(q)\}\\
& =\underset{a=pq}{\sup}\min\{\beta\cdot\mu_{A}(p)+\alpha,\beta\cdot
\mu_{B}(q)+\alpha\}\\
& =\beta\cdot\underset{a=pq}{\sup}\min\{\mu_{A}(p),\mu_{B}(q)\}+\alpha\\
& \geq\beta\cdot\min\{\mu_{A}(axya),\mu_{B}(azxa)\}+\alpha(\text{since
}a=(axya)(azxa))\\
& \geq\beta\cdot\min\{\mu_{A}(a),\mu_{B}(a)\}+\alpha(\text{since }A\text{ and
}B\text{ are intuitionistic}\\
& \text{ fuzzy bi-ideals of }S)=\min\{\beta\cdot\mu_{A}(a)+\alpha,\beta\cdot\mu_{B}(a)+\alpha\}\\
& =\min\{(\mu_{A})_{\beta\alpha}^{C}(a),\ (\mu_{B})_{\beta\alpha}^{C}(a)\ \}=((\mu
_{A})_{\beta\alpha}^{C}\cap(\mu_{B})_{\beta\alpha}^{C})(a)
\end{align*} agai
\begin{align*}
((\nu_{A})_{\beta\alpha}^{C}\circ(\nu_{B})_{\beta\alpha}^{C})(a) & =\underset
{a=pq}{\inf}\max\{(\nu_{A})_{\beta\alpha}^{C}(p),(\nu_{B})_{\beta\alpha}^{C}(q)\}\\
& =\underset{a=pq}{\inf}\max\{\beta\cdot\nu_{A}(p)-\alpha,\beta\cdot
\nu_{B}(q)-\alpha\}\\
& =\beta\cdot\underset{a=pq}{\inf}\max\{\nu_{A}(p),\nu_{B}(q)\}-\alpha\\
& \leq\beta\cdot\max\{\nu_{A}(axya),\nu_{B}(azxa)\}-\alpha(\text{since }a=(axya)(azxa))
\end{align*}
\begin{align*}
& \leq\beta\cdot\max\{\nu_{A}(a),\nu_{B}(a)\}-\alpha(\text{since }A\text{ and
}B\text{ are intuitionistic fuzzy}\\
& \text{bi-ideals of }S)=\max\{\beta\cdot\nu_{A}(a)-\alpha,\beta\cdot\nu_{B}(a)-\alpha\}\\
& =\max\{(\nu_{A})_{\beta\alpha}^{C}(a),\ (\nu_{B})_{\beta\alpha}^{C}(a)\ \}=((\nu
_{A})_{\beta\alpha}^{C}\cup(\nu_{B})_{\beta\alpha}^{C})(a)
\end{align*}
Hence $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}\supset A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$ and $(1)$ follows. Similarly we can prove that $B_{\beta\alpha}^{C}$ $\circ A_{\beta\alpha}^{C}\supset A_{\beta
\alpha}^{C}\cap B_{\beta\alpha}^{C}$. Combining these two results we can obtain $(2).$ This completes the proof.
\end{proof}
\begin{theorem}
If the semigroup $S$ is both regular, intra-regular and left regular then
$(1)$ $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}\supset A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}.$
$(2)$ $(A_{\beta\alpha}^{C}\circ B_{\beta\alpha}^{C})\cap(B_{\beta\alpha}^{C}\circ A_{\beta\alpha}^{C})\supset A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$ where $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ are intuitionistic fuzzy $(1,2)$-ideals of $S$.
\end{theorem}
\begin{proof}
Let $A=(\mu_{A},\nu_{A})$ and $B=(\mu_{B},\nu_{B})$ are intuitionistic fuzzy $(1,2)$-ideals of $S$ and $a\in S.$ Then there exist $x,y,z,p\in S$ such that $a=axa=axaxa($since $S$ is regular$)=ax(ya^{2}z)xa($since $S$ is intra-regular$)=(axya)(azxa)=(axypa^{2})(azxpa^{2})(S\,\ $is left regular$).$ No
\begin{align*}
\mu_{A}(axypa^{2}) & =\mu_{A}(axypaa)=\mu_{A}(axyp(aa))\geq\min\{\mu_{A}(a),\mu_{A}(a),\mu_{A}(a)\}(\text{since }A\text{ is }\\
& \text{ intuitionistic fuzzy }(1,2)\text{-ideal of }S)=\mu_{A}(a)
\end{align*} an
\begin{align*}
\nu_{A}(axypa^{2}) & =\nu_{A}(axypaa)=\nu_{A}(axyp(aa))\leq\max\{\nu_{A}(a),\nu_{A}(a),\nu_{A}(a)\}(\text{since }A\text{ is }\\
& \text{ intuitionistic fuzzy }(1,2)\text{-ideal of }S)=\nu_{A}(a).
\end{align*}
Similarly we can have $\mu_{B}(azxpa^{2})\geq\mu_{B}(a)$ and $\nu_{B}(azxpa^{2})\leq\nu_{B}(a).$ The
\begin{align*}
((\mu_{A})_{\beta\alpha}^{C}\circ(\mu_{B})_{\beta\alpha}^{C})(a) & =\underset
{a=mn}{\sup}\min\{(\mu_{A})_{\beta\alpha}^{C}(m),(\mu_{B})_{\beta\alpha}^{C}(n)\}\\
& =\underset{a=mn}{\sup}\min\{\beta\cdot\mu_{A}(m)+\alpha,\beta\cdot
\mu_{B}(n)+\alpha\}\\
& =\underset{a=mn}{\beta\cdot\sup}\min\{\mu_{A}(m),\mu_{B}(n)\}+\alpha\\
& \geq\beta\cdot\min\{\mu_{A}(axypa^{2}),\mu_{B}(azxpa^{2})\}+\alpha\\
& (\text{since }a =(axypa^{2})(azxpa^{2}))\\
& \geq\beta\cdot\min\{\mu_{A}(a),\mu_{B}(a)\}+\alpha(\text{since }A\text{ and
}B\text{ are intuitionistic}\\
& \text{fuzzy }(1,2)\text{-ideals} \text{ of }S)=\min\{\beta\cdot\mu_{A}(a)+\alpha,\beta\cdot\mu_{B}(a)+\alpha\}\\
& =\min\{(\mu_{A})_{\beta\alpha}^{C}(a),\ (\mu_{B})_{\beta\alpha}^{C}(a)\}=((\mu_{A})_{\beta\alpha}^{C}\cap(\mu_{B})_{\beta\alpha}^{C})(a)
\end{align*} an
\begin{align*}
((\nu_{A})_{\beta\alpha}^{C}\circ(\nu_{B})_{\beta\alpha}^{C})(a) & =\underset
{a=mn}{\inf}\max\{(\nu_{A})_{\beta\alpha}^{C}(m),(\nu_{B})_{\beta\alpha}^{C}(n)\}\\
& =\underset{a=mn}{\inf}\max\{\beta\cdot\nu_{A}(m)-\alpha,\beta\cdot
\nu_{B}(n)-\alpha\}\\
& =\underset{a=mn}{\beta\cdot\inf}\max\{\nu_{A}(m),\nu_{B}(n)\}-\alpha\\
& \leq\beta\cdot\max\{\nu_{A}(axypa^{2}),\nu_{B}(azxpa^{2})\}-\alpha(\text{since }a =(axypa^{2})(azxpa^{2}))\\
& \leq\beta\cdot\max\{\nu_{A}(a),\nu_{B}(a)\}-\alpha(\text{since }A\text{ and
}B\text{ are intuitionistic}\\
&\text{ fuzzy }(1,2)\text{-ideals of }S)=\max\{\beta\cdot\nu_{A}(a)-\alpha,\beta\cdot\nu_{B}(a)-\alpha\}\\
& =\max\{(\nu_{A})_{\beta\alpha}^{C}(a),\ (\nu_{B})_{\beta\alpha}^{C}(a)\}=((\nu_{A})_{\beta\alpha}^{C}\cup(\nu_{B})_{\beta\alpha}^{C})(a).
\end{align*}
Hence $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}\supset A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$ and $(1)$ follows. Similarly we can prove that $B_{\beta\alpha}^{C}$ $\circ A_{\beta\alpha}^{C}\supset A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$. Combining these two results we can obtain $(2).$ Hence the proof.
\end{proof}
The following proposition can be proved by routine verification.
\begin{proposition}
Let $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy right ideal and $B=(\mu_{B},\nu_{B})$ be an intuitionistic fuzzy left ideal of a semigroup $S$. Then $A\circ B\subseteq A\cap B.$
\end{proposition}
\begin{proposition}
Let $S$ be a regular semigroup, $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy right ideal and $B=(\mu_{B},\nu_{B})$ be an intuitionistic fuzzy left ideal of $S$. Then $A\circ B\supseteq A\cap B.$
\end{proposition}
\begin{proof}
Let $c\in S.$ Then there exists an element $x\in S$ such that $c=cxc($since $S$ is regular$).$ The
\begin{align*}
(\mu_{A}\circ\mu_{B})(c) & =\underset{c=uv}{\sup}\{\min\{\mu
_{A}(u),\mu_{B}(v)\}\}\\
& \geq\min\{\mu_{A}(c),\mu_{B}(c)\}=(\mu_{A}\cap\mu_{B})(c)
\end{align*}
and
\begin{align*}
(\nu_{A}\circ\nu_{B})(c) & =\underset{c=uv}{\inf}\{\max\{\nu
_{A}(u),\nu_{B}(v)\}\}\\
& \leq\max\{\nu_{A}(c),\nu_{B}(c)\}=(\nu_{A}\cup\nu_{B})(c).
\end{align*}
Hence $A\circ B\supseteq A\cap B.$
\end{proof}
\begin{definition}
\cite{M} Let $S$ be a semigroup. Let $A$ and $B$ be subsets of $S.$ Then the {\it multiplication} of $A$ and $B$ is defined as \ $AB=\{ab\in S:a\in A$ and $b\in B\}.$
\end{definition}
\begin{theorem}
$\cite{M}$ A semigroup $S$ is regular if and only if $R\cap L=RL$ for every right ideal $R$ and every left ideal $L$ of $S.$
\end{theorem}
\begin{theorem}
For a semigroup $S$, the following conditions are equivalent:
$(1)$ $S$ is regular,
$(2)$ $A\circ B= A\cap B$ for any intuitionistic fuzzy right ideal $A=(\mu_{A},\nu_{A})$ and any intuitionistic fuzzy left ideal $B=(\mu_{B},\nu_{B})$ of $S$.
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2):$ Let $S$ be a regular semigroup. Then by Proposition $3.27$ and $3.28,$ we have $A\circ B= A\cap B.$
$(2)\Rightarrow(1):$ Let $L$ and $R$ be respectively a left ideal and a right
ideal of $S$ and $x\in R\cap L.$ Then $x\in R$ and $x\in L.$ Hence $(\chi
_{L}(x),\overline{\chi}_{L}(x))=(\chi_{R}(x),\overline{\chi}_{R}(x))=(1,0)($where $\chi_{L}(x)$ and $\chi_{R}(x)$ are respectively the characteristic functions of $L$ and $R$$)$. The
\begin{align*}
(\chi_{R}\cap\chi_{L})(x)=\min\{\chi_{R}(x),\chi_{L}(x)\}=1\text{ and }(\overline{\chi}_{R}\cup\overline{\chi}_{L})(x)=\max\{\overline{\chi}_{R}(x),\overline{\chi}_{L}(x)\}=0.
\end{align*}
Now by Theorem $3.15,$ $(\chi_{L},\overline{\chi}_{L})$ and $(\chi_{R},\overline{\chi}_{R})$ are
respectively an intuitionistic fuzzy left ideal and an intuitionistic fuzzy right ideal of $S$. Hence by the hypothesis we hav
\begin{align*}
(\chi_{R}\circ\chi_{L})(x)=1,i.e.,\underset{x=yz}{\sup}[\min\{\chi_{R}(y),\chi_{L}(z)\}:y,z\in
S]=1
\end{align*}an
\begin{align*}
(\overline{\chi}_{R}\circ\overline{\chi}_{L})(x)=0,i.e.,\underset{x=yz}{\inf}[\max\{\overline{\chi}_{R}(y),\overline{\chi}_{L}(z)\}:y,z\in
S]=0
\end{align*}
This implies that there exist some $r,s\in S$ such that $x=rs$ and
$\chi_{R}(r)=1=\chi_{L}(s)$ and $\overline{\chi}_{R}(r)=0=\overline{\chi}_{L}(s)$. Hence $r\in R$ and $s\in L.$ Hence $x\in RL.$
Thus $R\cap L\subseteq RL.$ Also $RL\subseteq R\cap L.$ Consequently,
$RL=R\cap L.$ Hence $S$ is regular.
\end{proof}
\begin{theorem}
For a semigroup $S$, the following conditions are equivalent:
$(1)$ $S$ is regular,
$(2)$ $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}=A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}$ for any intuitionistic fuzzy right ideal $A=(\mu_{A},\nu_{A})$ and any intuitionistic fuzzy left ideal $B=(\mu_{B},\nu_{B})$ of $S$.
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2):$ Let $S$ be a regular semigroup, $A=(\mu_{A},\nu_{A})$ be an intuitionistic fuzzy right ideal
and $B=(\mu_{B},\nu_{B})$ be an intuitionistic fuzzy left ideal of $S$. Then by Theorem $3.9,$ $A_{\beta\alpha}^{C}$ is an intuitionistic fuzzy right ideal and $B_{\beta\alpha}^{C}$ is an intuitionistic fuzzy left ideal of $S.$ Hence by Theorem $3.31,$ $A_{\beta\alpha}^{C}$ $\circ B_{\beta\alpha}^{C}=A_{\beta\alpha}^{C}\cap B_{\beta\alpha}^{C}.$
$(2)\Rightarrow(1):$ Let $L$ and $R$ be respectively a left ideal and a right ideal of $S$ and $x\in R\cap L.$ Then $x\in R$ and $x\in L.$ Hence $(\chi_{L}(x),\overline{\chi}_{L}(x))=(\chi_{R}(x),\overline{\chi}_{R}(x))=(1,0)($where $\chi_{L}(x)$ and $\chi_{R}(x)$ are respectively the characteristic functions of $L$ and $R$$)$. Since $x\in L,$ $\overline{\chi}_{L}(x)=0.$ This implies that $\beta\cdot\overline{\chi}_{L}(x)=0.$ Hence $\inf\{\beta\cdot\overline{\chi}_{L}(y):y\in S\}=0.$ Since $\alpha\in[0,\inf\{\beta\cdot\overline{\chi}_{L}(y):y\in S\}],$ $\alpha=0.$ Similarly, we can show that if $x\in R,$ then also $\alpha=0.$ Then $((\chi_{L})_{\beta\alpha}^{C}(x),(\overline{\chi}_{L})_{\beta\alpha}^{C}(x))=((\chi_{R})_{\beta\alpha}^{C}(x),(\overline{\chi}_{R})_{\beta\alpha}^{C}(x))=(\beta, 0).$ Thu
\begin{align*}
((\chi_{R})_{\beta\alpha}^{C}\cap(\chi_{L})_{\beta\alpha}^{C})(x) &
=\min\{(\chi_{R})_{\beta\alpha}^{C}(x),(\chi_{L})_{\beta\alpha}^{C}(x)\}\\
& =\beta.
\end{align*} an
\begin{align*}
((\overline{\chi}_{R})_{\beta\alpha}^{C}\cup(\overline{\chi}_{L})_{\beta\alpha}^{C})(x) &
=\max\{(\overline{\chi}_{R})_{\beta\alpha}^{C}(x),(\overline{\chi}_{L})_{\beta\alpha}^{C}(x)\}\\
& =0.
\end{align*}
Now by Theorem $3.15$, $(\chi_{L},\overline{\chi}_{L})$ and $(\chi_{R},\overline{\chi}_{R})$ are
respectively an intuitionistic fuzzy left ideal and an intuitionistic fuzzy right ideal of $S$. Hence by
Theorem $3.9,$ $((\chi_{L})_{\beta\alpha}^{C},(\overline{\chi}_{L})_{\beta\alpha}^{C})$ and $((\chi_{R})_{\beta\alpha
}^{C},(\overline{\chi}_{R})_{\beta\alpha}^{C})$ are respectively intuitionistic fuzzy left ideal and intuitionistic fuzzy right ideal of $S.$ This
together with the hypothesis give
\begin{align*}
((\chi_{R})_{\beta\alpha}^{C}\circ(\chi_{L})_{\beta\alpha}^{C})(x) &
=\beta\\
i.e.,\underset{x=yz}{\sup}[\min\{(\chi_{R})_{\beta\alpha}^{C}(y),(\chi
_{L})_{\beta\alpha}^{C}(z)\} & :y,z\in S]=\beta\\
i.e.,\underset{x=yz}{\sup}[\min\{\beta\cdot\chi_{R}(y)+\alpha,\beta\cdot
\chi_{L}(z)+\alpha\} & :y,z\in S]=\beta\\
i.e.,\beta.\underset{x=yz}{\sup}[\min\{\chi_{R}(y),\chi_{L}(z)\} & :y,z\in
S]+\alpha=\beta\\
i.e.,\beta.\underset{x=yz}{\sup}[\min\{\chi_{R}(y),\chi_{L}(z)\} & :y,z\in
S]=\beta(\text{since }\alpha=0)
\end{align*} an
\begin{align*}
((\overline{\chi}_{R})_{\beta\alpha}^{C}\circ(\overline{\chi}_{L})_{\beta\alpha}^{C})(x) &
=0\\
i.e.,\underset{x=yz}{\inf}[\max\{(\overline{\chi}_{R})_{\beta\alpha}^{C}(y),(\overline{\chi}
_{L})_{\beta\alpha}^{C}(z)\} & :y,z\in S]=0\\
i.e.,\underset{x=yz}{\inf}[\max\{\beta\cdot\overline{\chi}_{R}(y)-\alpha,\beta\cdot
\overline{\chi}_{L}(z)-\alpha\} & :y,z\in S]=0\\
i.e.,\beta.\underset{x=yz}{\inf}[\max\{\overline{\chi}_{R}(y),\overline{\chi}_{L}(z)\} & :y,z\in
S]-\alpha=0
\end{align*}
\begin{align*}
i.e.,\beta.\underset{x=yz}{\inf}[\max\{\overline{\chi}_{R}(y),\overline{\chi}_{L}(z)\} & :y,z\in
S]=0(\text{since }\alpha=0)
\end{align*}
Hence $\underset{x=yz}{\sup}[\min\{\chi_{R}(y),$ $\ \chi_{L}(z)\}:y,z\in
S]=1$ and $\underset{x=yz}{\inf}[\max\{\overline{\chi}_{R}(y),$ $\ \overline{\chi}_{L}(z)\}:y,z\in
S]=0$ This implies that there exist some $r,s\in S$ such that $x=rs$ and
$\chi_{R}(r)=1=\chi_{L}(s),$ $\overline{\chi}_{R}(r)=0=\overline{\chi}_{L}(s)$. Hence $r\in R$ and $s\in L$ whence $x\in RL.$
Thus $R\cap L\subseteq RL.$ Also $RL\subseteq R\cap L.$ Consequently,
$RL=R\cap L.$ Hence $S$ is regular.
\end{proof}
\begin{remark}
If we put $\beta=1($respectively $\alpha=0)$ in intuitionistic fuzzy magnified translation
then it reduces to intuitionistic fuzzy translation$($respectively intuitionistic fuzzy multiplication$).$
Consequently analogues of Theorems $3.4$-$3.6, 3.8$-$3.11,$ Theorem $3.14$-$3.15,$ Theorem $3.17,3.19$-$3.20, 3.22, 3.25$-$3.26, 3.32$ and Corollary $3.12$ follow easily in intuitionistic fuzzy translation and intuitionistic fuzzy multiplication.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,402
|
\section{Introduction}
Unsupervised bilingual lexicon induction (UBLI)
has been shown to benefit NLP tasks for low resource languages, including unsupervised NMT~\cite{artetxe18EMNLP,artetxe18ICLR,Yang18ACL,lample18ICLR,Lample18EMNLP}, information retrieval~\cite{Vulic15,Litschko18}, dependency parsing~\cite{Guo15ACL}, and named entity recognition~\cite{mayhew17,xie18emnlp}.
Recent research has attempted to induce unsupervised bilingual lexicons by aligning monolingual word vector spaces~\cite{Zhang17a,Conneau18a,aldarmaki18TACL,Artetxe18,Alvarez18EMNLP,Mukherjee18EMNLP}.
Given a pair of languages, their word alignment is inherently a bi-directional problem (e.g. English-Italian vs Italian-English). However, most existing research considers mapping from one language to another without making use of symmetry.
Our experiments show that separately learned UBLI models are not always consistent in opposite directions.
As shown in Figure~1a, when the model of \citeauthor{Conneau18a}~\shortcite{Conneau18a} is applied to English and Italian, the primal model maps the word ``three'' to the Italian word ``tre'', but the dual model maps ``tre'' to ``two'' instead of ``three''.
\begin{figure}
\setlength{\abovecaptionskip}{0.2cm}
\setlength{\belowcaptionskip}{-0.6cm}
\centering\includegraphics[width=0.5\textwidth]{example.pdf}
\footnotesize{\caption{(a)
Inconsistency between primal model $\mathcal{F}$ and the dual model $\mathcal{G}$. (b) An ideal scenario.}}
\label{fig:zero}
\end{figure}
We propose to address this issue by exploiting duality, encouraging forward and backward mappings to form a closed loop (Figure 1b).
In particular, we extend the model of ~\citeauthor{Conneau18a}~\shortcite{Conneau18a} by using a cycle consistency loss~\cite{Zhou16} to regularize two models in opposite directions.
Experiments on two benchmark datasets show that the simple method of enforcing consistency gives better results in both directions. Our model significantly outperforms competitive baselines, obtaining the best published results.
We release our code at xxx.
\section{Related Work}
\textbf{UBLI}.
A typical line of work uses adversarial training~\cite{barone16,Zhang17a,Zhang17b,Conneau18a},
matching the distributions of source and target word embeddings through generative adversarial networks~\cite{Goodfellow14}.
Non-adversarial approaches have also been explored.
For instance, ~\citeauthor{Mukherjee18EMNLP}~\shortcite{Mukherjee18EMNLP} use squared-loss mutual information to search for optimal cross-lingual word pairing.
~\citet{Artetxe18} and ~\citet{Hoshen18} exploit the structural similarity of word embedding spaces to learn word mappings.
In this paper, we choose \citeauthor{Conneau18a}~\shortcite{Conneau18a} as our baseline as it is theoretically attractive and gives strong results on large-scale datasets.
\noindent\textbf{Cycle Consistency.}
Forward-backward consistency has been used to discover the correspondence between unpaired images~\cite{Zhu17,Kim17}.
In machine translation, similar ideas were exploited, ~\citet{He16},
~\citet{Xia17ijcai}
and \citet{Wang18aaai} use dual learning to train two "opposite" language translators by minimizing the reconstruction loss. ~\citet{Sennrich16} consider back-translation, where a backward model is used to build synthetic parallel corpus and a forward model learns to generate genuine text based on the synthetic output.
Closer to our method, ~\citet{Chandar14} jointly train two autoencoders to learn supervised bilingual word embeddings. ~\citet{Xu18EMNLP}
use sinkhorn distance~\cite{Marco13} and back-translation to align word embeddings.
However, they cannot perform fully unsupervised training, relying on WGAN~\cite{arjovsky17} for providing initial mappings.
Concurrent with our work, ~\citet{mohiuddin19} build a adversarial autoencoder with cycle consistency loss and post-cycle reconstruction loss.
In contrast to these works, our method is fully unsupervised, simpler, and empirically more effective.
\section{Approach}
We take \citet{Conneau18a} as our baseline, introducing a novel regularizer to enforce cycle consistency.
Let $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_m\}$ be two sets of $n$ and $m$ word embeddings for a source and a target language, respectively. The primal UBLI task aims to learn a linear mapping $\mathcal{F}:X\to Y$ such that for each $x_i$, $\mathcal{F}(x_i)$ corresponds to its
translation in $Y$. Similarly, a linear mapping $\mathcal{G}:Y\to X$ is defined for the dual task. In addition, we introduce two language discriminators $D_x$ and $D_y$, which are trained to discriminate between the mapped word embeddings and the original word embeddings.
\subsection{Baseline Adversarial Model}
\citet{Conneau18a} align two word embedding spaces through generative adversarial networks,
in which two networks are trained simultaneously.
Specifically, take the primal UBLI task as an example, the linear mapping $\mathcal{F}$ tries to generate ``fake'' word embeddings $\mathcal{F}(x)$ that look similar to word embeddings from $Y$, while the discriminator $D_y$ aims to distinguish between ``fake'' and real word embeddings from $Y$.
Formally, this idea can be expressed as the minmax game min$_{\mathcal{F}}$max$_{D{_y}}\ell_{adv}(\mathcal{F},D_y,X,Y)$, where
\begin{equation*}
\setlength{\abovedisplayskip}{2pt}
\setlength{\belowdisplayskip}{-12pt}
\ell_{adv}(\mathcal{F},D_y,X,Y) = \frac{1}{m}\sum\limits_{j=1}^{m}\textup{log} P_{D_y}(src=1|y_j)
\end{equation*}
\begin{equation}
\setlength{\abovedisplayskip}{-13pt}
\setlength{\belowdisplayskip}{-10pt}
+ \frac{1}{n}\sum\limits_{i=1}^n\textup{log}P_{D_y}(src=0|\mathcal{F}(x_i)).
\end{equation}
$P_{D_y}(src|y_j)$ is a model probability from $D_y$ to distinguish whether word embedding $y_j$ is coming from the target language (src = 1) or the primal mapping $\mathcal{F}$ (src = 0).
Similarly, the {dual} UBLI problem can be formulated as min$_{\mathcal{G}}$max$_{D_x}\ell_{adv}(\mathcal{G},D_x,Y,X)$, where $\mathcal{G}$ is the dual mapping, and $D_x$ is a source discriminator.
Theoretically, a unique solution for above minmax game exists, with the mapping and the discriminator reaching a nash equilibrium.
Since the adversarial training happens at the distribution level, no cross-lingual supervision is required.
\begin{figure}[t]
\setlength{\abovecaptionskip}{0.1cm}
\setlength{\belowcaptionskip}{-0.5cm}
\centering\includegraphics[width=0.5\textwidth]{cycle.pdf}
\footnotesize\caption{The proposed framework.
(a) $X\to\mathcal{F}(X)\to\mathcal{G}(\mathcal{F}(X))\to X$; (b) $Y\to\mathcal{G}(Y)\to\mathcal{F}(\mathcal{G}(Y))\to Y$.}
\label{fig:one}
\end{figure}
\subsection{Regularizers for Dual Models}
We train $\mathcal{F}$ and $\mathcal{G}$ jointly and introduce two regularizers.
Formally, we hope that $\mathcal{G}(\mathcal{F}(X))$ is similar to $X$ and $\mathcal{F}(\mathcal{G}(Y))$ is similar to $Y$.
We implement this constraint as a cycle consistency loss.
As a result, the proposed model has two learning objectives: i) an adversarial loss ($\ell_{adv}$) for each model as in the baseline.
ii) a cycle consistency loss ($\ell_{cycle}$) on each side to avoid $\mathcal{F}$ and $\mathcal{G}$ from contradicting each other. The overall architecture of our model is illustrated in Figure~\ref{fig:one}.
\noindent\textbf{Cycle Consistency Loss.}
We introduce
\begin{equation}
\setlength{\abovedisplayskip}{-2pt}
\setlength{\belowdisplayskip}{-2pt}
\label{eqn:3}
\begin{split}
&\ell_{cycle}(\mathcal{F},\mathcal{G},X) =
\frac{1}{n}\sum\limits_{i=1}^{n}\Delta(x_i ,\mathcal{G}(\mathcal{F}(x_i))),\\
&\ell_{cycle}(\mathcal{F},\mathcal{G},Y) =
\frac{1}{m}\sum\limits_{j=1}^{m}\Delta(y_j,\mathcal{F}(\mathcal{G}(y_j))),\\
\end{split}
\end{equation}
\noindent where $\Delta$ denotes the discrepancy criterion, which is set as the average cosine similarity in our model.
\noindent\textbf{Full objective.} The final objective is:
\begin{equation*}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{-3pt}
\begin{split}
\ell(\mathcal{F},&\mathcal{G},D_x,D_y,X,Y) =
\\&\ell_{adv}(\mathcal{F},D_y,X,Y)+\ell_{adv}(\mathcal{G},D_x,Y,X)
\end{split}
\end{equation*}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-5pt}
\label{eqn:5}
+\ell_{cycle}(\mathcal{F},\mathcal{G},X)+\ell_{cycle}(\mathcal{F},\mathcal{G},Y).
\end{equation}
\subsection{Model Selection}
We follow ~\citet{Conneau18a}, using an unsupervised criterion to perform model selection.
In preliminary experiments, we find in adversarial training that the single-direction criterion $S(\mathcal{F}, X, Y)$ by~\citet{Conneau18a} does not always work well.
To address this, we make a simple extension by calculating the weighted average of forward and backward scores:
\begin{equation}
\setlength{\abovedisplayskip}{-7pt}
\setlength{\belowdisplayskip}{2pt}
\label{eqn:6}
\begin{split}
S_{a}= \lambda S(\mathcal{F}, X, Y) + (1-\lambda)S(\mathcal{G}, X, Y),
\end{split}
\end{equation}
Where $\lambda$ is a hyperparameter to control the importance of the two objectives.\footnote{We find that $\lambda=0.5$ generally works well.}
Here $S$ first generates bilingual lexicons by learned mappings, and then computes the average cosine similarity of these translations.
\section{Experiments}
We perform two sets of experiments, to investigate the effectiveness of our duality regularization in isolation (Section~\ref{sec42}) and to compare our final models with the state-of-the-art methods in the literature (Section~\ref{sec43}), respectively.
\subsection{Experimental Settings}
\noindent\textbf{Dataset and Setup}.
Our datasets includes: (i) The Multilingual Unsupervised and Supervised Embeddings {(\textbf{MUSE})} dataset released by \citeauthor{Conneau18a}~\shortcite{Conneau18a}.
(ii) the more challenging \textbf{Vecmap} dataset from~\citeauthor{Dinu15}~\shortcite{Dinu15} and the extensions of \citeauthor{Artetxe17ACL}~\shortcite{Artetxe17ACL}. We follow the evaluation setups of ~\citet{Conneau18a}, utilizing cross-domain similarity local scaling (CSLS) for retrieving the translation of given source words. Following a standard evaluation practice ~\cite{vulic13,Mikolov13b,Conneau18a}, we report precision at 1 scores (P@1).
Given the instability of existing methods, we follow \citet{Artetxe18} to perform 10 runs for each method and report the best and the average accuracies.
\begin{table}[t]
\setlength{\abovecaptionskip}{0.1cm}
\setlength{\belowcaptionskip}{-0.2cm}
\begin{centering}
\scalebox{0.85}{
\begin{tabular}{|l|c|cccc|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Setting}} & \multicolumn{2}{c} {Adv-C} &\multicolumn{2}{c|} {Ours} \tabularnewline
\cline{3-6}
\multicolumn{2}{|c|}{} & best & average. &best &average. \tabularnewline
\hline
\multirow{10}{*}{\begin{sideways}\textbf{MUSE}\end{sideways}}& EN-ES &77.3 &75.1 &\textbf{78.4} &\textbf{77.0}\tabularnewline
& ES-EN &\textbf{79.1} &73.5 &79.0 &\textbf{75.6} \tabularnewline
\cline{2-6}
& EN-DE &69.2 &32.4 &\textbf{70.0} &\textbf{56.5} \tabularnewline
& DE-EN &68.5 &31.7 &\textbf{69.3} &\textbf{53.7} \tabularnewline
\cline{2-6}
& EN-IT &65.2 &47.7 &\textbf{72.0} &\textbf{71.1} \tabularnewline
& IT-EN &64.0 &45.3 &\textbf{69.9} &\textbf{69.4} \tabularnewline
\cline{2-6}
& EN-EO &18.6 &13.5 &\textbf{20.9} &\textbf{17.5}\tabularnewline
& EO-EN &16.6 &12.0 &\textbf{17.3} &\textbf{15.3} \tabularnewline
\cline{2-6}
& EN-MS &17.9 &08.3 &\textbf{24.7} &\textbf{21.8}\tabularnewline
& MS-EN &19.2 &06.4 &\textbf{27.6} &\textbf{23.5} \tabularnewline
\hline
\multirow{8}{*}{\begin{sideways}\textbf{Vecmap}\end{sideways}}& EN-ES &26.2 &20.5 &\textbf{29.6} &\textbf{26.1}\tabularnewline
& ES-EN &00.0 &00.0 &\textbf{21.7} &\textbf{20.2}\tabularnewline
\cline{2-6}
& EN-DE &40.3 &20.0 &\textbf{43.7} &\textbf{36.5}\tabularnewline
& DE-EN &00.0 &00.0 &\textbf{37.8} &\textbf{33.4}\tabularnewline
\cline{2-6}
& EN-IT &38.3 &37.0 & \textbf{38.5} &\textbf{37.5} \tabularnewline
& IT-EN &33.6 &14.7 &\textbf{34.7} &\textbf{33.1}\tabularnewline
\cline{2-6}
& EN-FI &01.9 &00.3 &\textbf{22.2} &\textbf{21.9}\tabularnewline
& FI-EN &00.0 &00.0 &\textbf{20.0} &\textbf{18.9}\tabularnewline
\hline
\end{tabular}}
\par\end{centering}
\footnotesize\caption{\label{tab:01} Accuracy on MUSE and Vecmap.}
\end{table}
\begin{table}[htbp]
\setlength{\abovecaptionskip}{0.1cm}
\setlength{\belowcaptionskip}{-0.2cm}
\begin{centering}
\scalebox{0.8}{
\begin{tabular}{lccccc}
\hline
&EN-ES & EN-DE & EN-IT & EN-EO &EN-MS \tabularnewline
\hline
Adv-C &66.95\% &67.83\% &70.23\% &72.30\% &75.87\% \tabularnewline
Ours&63.58\% &64.29\% &65.05\% &64.06\% &68.84\% \tabularnewline
\hline
\end{tabular}}
\par\end{centering}
\footnotesize\caption{\label{tab:02} Inconsistency rates on MUSE.}
\end{table}
\begin{table}[!ht]
\vspace{-0cm}
\setlength{\abovecaptionskip}{0.2cm}
\setlength{\belowcaptionskip}{-0.5cm}
\begin{centering}
\scalebox{0.9}{
\begin{tabular}{cc}
\hline
\multicolumn{1}{c} {Adv-C} &\multicolumn{1}{c} {Ours} \tabularnewline
\hline
\textbf{three-tre}-two &\textbf{three-tre-three}\tabularnewline
\textbf{neck-collo}-ribcage
&\textbf{neck-collo-neck}\tabularnewline
door-\textbf{finestrino-window}&\textbf{door-portiera-door}\tabularnewline
second-\textbf{terzo-third} &second-terzo-second\tabularnewline
before-\textbf{prima-first} & before-\textbf{dopo-after}\tabularnewline
\hline
\end{tabular}}
\par\end{centering}
\footnotesize\caption{\label{tab:03}Word translation examples for English-Italian on MUSE. Ground truths are marked in \textbf{bold}.}
\end{table}
\begin{table*}[t]
\setlength{\abovecaptionskip}{0.1cm}
\setlength{\belowcaptionskip}{-0.2cm}
\begin{centering}
\scalebox{0.85}{\begin{tabular}{clcccccccc}
\hline
\multirow{2}{*}{Supervision} &\multirow{2}{*}{Approach}& \multicolumn{2}{c} {EN-IT} & \multicolumn{2}{c} {EN-DE}& \multicolumn{2}{c} {EN-FI}& \multicolumn{2}{c} {EN-ES}\tabularnewline
& & $\rightarrow$ & $\leftarrow$ &$\rightarrow$ & $\leftarrow$ &$\rightarrow$ & $\leftarrow$ & $\rightarrow$ & $\leftarrow$ \tabularnewline
\hline
\multirow{4}{*}{\shortstack{Supervised\\ Methods}} & Procrustes & 45.33 & 39.05 & 47.27 &41.13 &32.16 &30.01 &36.67 &30.94 \tabularnewline
& GPA$\dagger$ & 45.33 & - & 48.46 &- &31.39 &- &- &- \tabularnewline
& GeoMM & 48.17 & 41.10 & 49.40 & 44.73 &36.03 &38.24 &39.27 &34.58 \tabularnewline
& GeoMM$_{semi}$ & \textbf{50.00} & \textbf{42.67} & 51.47 & 46.96 &\textbf{36.24} &39.57 &39.30 &36.06 \tabularnewline
\hline
\multirow{5}{*}{\shortstack{Unsupervised\\ Methods}}& Adv-C-Procrustes & 45.40 &38.78 & 46.40 & 00.00 &25.21 &00.15 &35.47 &0.05 \tabularnewline
& Unsup-SL & 48.01 & 42.10 & 48.22 &44.09 &32.95 &33.45 &37.47 &31.59 \tabularnewline
& Sinkhorn-BT & 44.67 & 38.77 & 44.53 & 41.93 & 23.53 & 23.42 & 32.13 & 27.62 \tabularnewline
& Ours-Procrustes &45.60 &38.29 &46.58 &42.50 &28.08 &26.48 &35.20 &28.94 \tabularnewline
& Ours-GeoMM$_{semi}$ &\textbf{50.00} &\textbf{42.67} &\textbf{51.60} &\textbf{47.22} &35.88 &\textbf{39.62} &\textbf{39.47} &\textbf{36.43}
\tabularnewline
\hline \end{tabular}}
\par\end{centering}
\caption{\label{tab:04} Accuracy (P@1) on \textbf{Vecmap}. The best results are \textbf{bolded}. $\dagger$Results as reported in the original paper. For unsupervised methods, we report the average accuracy across 10 runs.}
\end{table*}
\subsection{The Effectiveness of Dual Learning}
\label{sec42}
We compare our method with ~\citet{Conneau18a} (Adv-C) under the same settings.
As shown in Table~\ref{tab:01}, our model outperforms Adv-C on both MUSE and Vecmap for all language pairs (except ES-EN).
In addition, the proposed approach is less sensitive to initialization, and thus more stable than Adv-C over multiple runs.
These results demonstrate the effectiveness of dual learning.
Our method is also superior to Adv-C for the low-resource language pairs English $\leftrightarrow$ Malay (MS) and English $\leftrightarrow$ English-Esperanto (EO).
Adv-C gives low performances on ES-EN, DE-EN, but much better results on the opposite directions on Vecmap.
This is likely because the separate models are highly under-constrained, and thus easy to get stuck in poor local optima.
In contrast, our method gives comparable results on both directions for the two languages, thanks to the use of information symmetry.
Table~\ref{tab:02} shows the inconsistency rates\footnote{For each word $x_i$ from the source language, we check whether the primal $\mathcal{F}$ and the dual mapping $\mathcal{G}$ can recover $x_i$, i.e. $x_i\rightarrow\mathcal{F}(x_i) \rightarrow\mathcal{G}(\mathcal{F}(x_i))\rightarrow x_i$.} of back translation between Adv-C and our method on MUSE. Compared with Adv-C, our model significantly reduces the inconsistency rates on all language pairs, which explains the overall improvement in Table~\ref{tab:01}.
Table~\ref{tab:03} gives several word translation examples. In the first three cases, our regularizer successfully fixes back translation errors. In the fourth case, ensuring cycle consistency does not lead to the correct translation, which explains some errors by our system. In the fifth case, our model finds a related word but not the same word in the back translation, due to the use of cosine similarity for regularization.
\subsection{Comparison with the State-of-the-art}
\label{sec43}
In this section, we compare our model with state-of-the-art systems, including
those with different degrees of supervision.
The baselines include: (1) \textbf{Procrustes}~\cite{Conneau18a}, which learns a linear mapping through Procrustes Analysis~\cite{Schönemann1966}. (2)~\textbf{GPA}~\cite{Kementchedjhieva18}, an extension of Procrustes Analysis. (3)~\textbf{ GeoMM}~\cite{Jawanpuria19}, a geometric approach which learn a Mahalanobis metric to refine the notion of similarity. (4)~\textbf{GeoMM$_{semi}$}, iterative GeoMM with weak supervision. (5) \textbf{Adv-C-Procrustes}~\cite{Conneau18a}, which refines the mapping learned by Adv-C with iterative Procrustes, which learns the new mapping matrix by constructing a bilingual lexicon iteratively.
(6) \textbf{Unsup-SL}~\cite{Artetxe18}, which integrates a weak unsupervised mapping with a robust self-learning.
(7) \textbf{Sinkhorn-BT}~\cite{Xu18EMNLP}, which combines sinkhorn distance~\cite{Marco13} and back-translation.
For fair comparison, we integrate our model with two iterative refinement methods (Procrustes and GeoMM$_{semi}$).
Table~\ref{tab:04} shows the final results on Vecmap.\footnote{We select Vecmap as it is more challenging and closer to the real scenarios than MUSE~\cite{Artetxe18}.}
We first compare our model with the state-of-the-art unsupervised methods.
Our model based on procrustes (Ours-Procrustes) outperforms Sinkhorn-BT on all test language pairs, and shows better performance than Adv-C-Procrustes on most language pairs.
Adv-C-Procrustes gives very low precision on DE-EN, FI-EN and ES-EN, while Ours-Procrustes obtains reasonable results consistently.
A possible explanation is that dual learning is helpful for providing good initiations, so that the procrustes solution is not likely to fall in poor local optima.
The reason why Unsup-SL gives strong results on all language pairs is that it uses a robust self-learning framework, which contains several techniques to avoid poor local optima.
Additionally, we observe that our unsupervised method performs competitively and even better compared with strong supervised and semi-supervised approaches.
Ours-Procrustes obtains comparable results with Procrustes on EN-IT and gives strong results on EN-DE, EN-FI, EN-ES and the opposite directions.
Ours-GeoMM$_{semi}$ obtains the state-of-the-art results on all tested language pairs except EN-FI, with the additional advantage of being fully unsupervised.
\section{Conclusion}
We investigated a regularization method to enhance unsupervised bilingual lexicon induction, by encouraging symmetry in lexical mapping between a pair of word embedding spaces. Results show that strengthening bi-directional mapping consistency significantly improves the effectiveness over the state-of-the-art method, leading to the best results on a standard benchmark.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,503
|
Q: Using TScreen in Delphi 7 My Delphi-7 application displays :
Screen.DesktopWidth
Screen.DesktopHeight
Screen.Monitors[0].Width
Screen.Monitors[0].Height
and , if there's a second monitor selected , also :
Screen.Monitors[1].Width
Screen.Monitors[1].Height
With the application running on my WinXP-Pro PC , I go to Control Panel / Display / Settings , and change the settings for the second monitor (either add or remove it) .
I then click on a Refresh button to display the new values of the 4 (or 6) parameters , and something unexpected happens : Screen.DesktopWidth and Screen.DesktopHeight show the correct new values , but the values of the other 2 (or 4) parameters are very wrong .
Like Screen.Monitors[0].Width = 5586935 , while it should be 1680 .
Are there some special rules for using TScreen in Delphi 7 ?
A: Came here because of refresh problem (bug) of TScreen when connect or disconnect a monitor or USB display device. The answer of @Dave82 doesn't work for me. The result of the function MonitorFromWindow must return another value (unknown/invalid value) to force an update of the TScreen object.
This cheat below does the trick:
Be sure multimon is in the uses clause:
uses
multimon;
Add this to the interface part (of the form)
protected
procedure WMDeviceChange(var Msg: TMessage); message WM_DEVICECHANGE;
Add this to implementation part (of the form)
function cheatMonitorFromWindow(hWnd: HWND; dwFlags: DWORD): HMONITOR; stdcall;
begin
// Does nothing, returns zero to force invalidate
Result:=0;
end;
procedure TForm1.WMDeviceChange(var Msg: TMessage);
var
iCurrDisplayCount : LongInt;
iNewDisplayCount : LongInt;
pMonitorFromWinProc : TMonitorFromWindow;
begin
iCurrDisplayCount:=Screen.MonitorCount;
// Force monitor update, fix bug in customform, won't update at display change.
// This a hack/cheat to multimon MonitorFromWindow func, it's fakes the result.
// This is required to tell customform.getMonitor() to update the TScreen object.
pMonitorFromWinProc:=MonitorFromWindow; // Backup pointer to dynamic assigned DLL func
MonitorFromWindow:=cheatMonitorFromWindow; // Assign cheat func
monitor; // call the monitor property that calls customform.getMonitor and cheatfunc
MonitorFromWindow:=pMonitorFromWinProc; // restore the original func
// ==========
iNewDisplayCount:=Screen.MonitorCount;
if( iCurrDisplayCount <> iNewDisplayCount ) then
begin
// Display count change!
end;
end;
What happen inside customform (code in Forms.pas)?
function TCustomForm.GetMonitor: TMonitor;
var
HM: HMonitor;
I: Integer;
begin
Result := nil;
HM := MonitorFromWindow(Handle, MONITOR_DEFAULTTONEAREST);
for I := 0 to Screen.MonitorCount - 1 do
if Screen.Monitors[I].Handle = HM then
begin
Result := Screen.Monitors[I];
Exit;
end;
//if we get here, the Monitors array has changed, so we need to clear and reinitialize it
for i := 0 to Screen.MonitorCount-1 do
TMonitor(Screen.FMonitors[i]).Free;
Screen.FMonitors.Clear;
EnumDisplayMonitors(0, nil, @EnumMonitorsProc, LongInt(Screen.FMonitors));
for I := 0 to Screen.MonitorCount - 1 do
if Screen.Monitors[I].Handle = HM then
begin
Result := Screen.Monitors[I];
Exit;
end;
end;
Hopes it helps when somebody is looking for this. When you want to detect display device settings changes (resolution and orientation), catch the WM_DISPLAYCHANGE event instead.
A: Screen.Monitors array contain invalid values if you switch user while your program is running. We use this line of code to force the Screen object to update lists:
Screen.MonitorFromWindow(0, mdNull);
A: Thanks to TLama , I found a workaround for the TScreen problem in Delphi 7 .
The original code that 'caused' the problem :
LabMon1.Caption := ' Mon 1: ' + IntToStr (Screen.Monitors[0].Width) +
' x ' + IntToStr (Screen.Monitors[0].Height);
if (Screen.MonitorCount = 1)
then LabMon2.Caption := ' Mon 2: -'
else LabMon2.Caption := ' Mon 2: ' + IntToStr (Screen.Monitors[1].Width) +
' x ' + IntToStr (Screen.Monitors[1].Height);
I only had to add 1 line of code to solve it :
LabMon1.Caption := ' Mon 1: ' + IntToStr (Monitor.Width) +
' x ' + IntToStr (Monitor.Height) ;
LabMon1.Caption := ' Mon 1: ' + IntToStr (Screen.Monitors[0].Width) +
' x ' + IntToStr (Screen.Monitors[0].Height);
if (Screen.MonitorCount = 1)
then LabMon2.Caption := ' Mon 2: -'
else LabMon2.Caption := ' Mon 2: ' + IntToStr (Screen.Monitors[1].Width) +
' x ' + IntToStr (Screen.Monitors[1].Height);
So thanks again TLama , for your great contributions to this Question-thread !
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,751
|
De Deinonychosauria behoren tot Eumaniraptora, een onderverdeling van de Maniraptora, een groep uit de Theropoda, vleesetende dinosauriërs.
Een infraorde Deinonychosauria werd voor het eerst benoemd door Edwin Harris Colbert in 1969.
De klade is in 1997 voor het eerst gedefinieerd door Padian als de groep bestaande uit alle Maniraptora die nauwer verwant zijn aan Deinonychus dan aan de Aves. In 1999 wijzigde Padian de formulering door Neornithes in plaats van Aves te gebruiken.
Paul Sereno gaf in 1998 een afwijkende definitie als nodusklade: de groep bestaande uit de laatste gemeenschappelijke voorouder van Troodon en Velociraptor en al zijn afstammelingen.
In de jaren negentig had men nog een vrij simplistisch beeld van de verwantschappen binnen de Eumaniraptora. Beide onderzoekers gingen ervan uit dat Dromaeosauridae en Troodontidae elkaars zustergroepen waren met uitsluiting van de vogels. Recenter onderzoek toonde echter aan dat vogels zeer wel nauwer verwant aan Deinonychus zouden kunnen zijn dan aan de troodontiden — of omgekeerd. Daar de vogels afstammen van een groep die men losjes met "dromaeosauriërs" kan aanduiden en die in morfologie erg op Deinychosaurus leek, is het erg onduidelijk welke vormen nu wel of niet tot de Deinonychosauria behoorden en wanneer de groep zich precies heeft afgesplitst. Dit zou voor de definitie van Padian kunnen betekenen dat Deinonychosauria een synoniem wordt van Dromaeosauridae en voor de definitie van Sereno dat het begrip samenvalt met Eumaniraptora. Sereno gaf daarom in 2005 een nieuwe definitie: de groep bevattende Troodon formosus en Velociraptor mongoliensis en alle soorten nauwer verwant aan Velociraptor en Troodon dan aan Ornithomimus edmontonicus of de huismus Passer domesticus. Deze definitie garandeert dat het hele begrip inhoudsloos wordt mochten de vogels of van de Dromaeosauridae of van de Troodontidae de zustergroep vormen en de naam onbruikbaar wordt in plaats van een ongewilde inhoud te krijgen.
Alle leden van de groep — mocht zij bestaan — waren vermoedelijk warmbloedig, bevederd en droegen een vergrote teenklauw. Ze hebben vele zeer vogelachtige kenmerken en het is mogelijk dat ze alle afstammen van een vliegende voorouder.
Maniraptora
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,964
|
'use strict';
const rewire = require('rewire');
const Q = require('q');
const ecr = rewire('./ecr');
const C = require('../../chai');
const aws = {};
function initData() {
aws.ecr = {
createRepository: () => Q.resolve({ repository: {} }),
describeRepositories: () => Q.resolve({ repositories: [
{ repositoryArn: '1', repositoryName: 'test' }] }),
deleteRepository: () => Q.resolve()
};
}
/*global describe, it, expect, beforeEach, afterEach */
describe('AWS: ECR', () => {
beforeEach(initData);
describe('create function', () => {
it('should throw without a name', () => {
expect(() => ecr.create()).to.throw(Error);
});
it('should resolve if AWS ecr resolves', (done) => {
ecr.create(aws, 'test')
.then((r) => C.check(done, () => expect(r).to.be.ok), C.getFail(done));
});
it('should resolve if AWS ecr rejects', (done) => {
aws.ecr.createRepository = () => Q.reject(new Error('test'));
ecr.create(aws, 'test')
.then((r) => C.check(done, () => expect(r).to.be.ok), C.getFail(done));
});
it('should reject if AWS ecr rejects *and* AWS ecr describe rejects',
(done) => {
aws.ecr.createRepository = () => Q.reject(new Error('test'));
aws.ecr.describeRepositories = () => Q.reject(new Error('test 2'));
ecr.create(aws, 'test')
.then(C.getFail(done), (err) => C
.check(done, () => expect(err instanceof Error).to.be.ok));
});
});
describe('destroy function', () => {
it('should throw without a name', () => {
expect(() => ecr.destroy()).to.throw(Error);
});
it('should resolve if AWS ecr resolves', (done) => {
ecr.destroy(aws, 'test')
.then(() => C
.check(done, () => expect(true).to.be.ok), C.getFail(done));
});
it('should reject if AWS ecr rejects *and* AWS ecr describe resolves',
(done) => {
aws.ecr.deleteRepository = () => Q.reject(new Error('test'));
ecr.destroy(aws, 'test')
.then(C.getFail(done), (err) => C
.check(done, () => expect(err instanceof Error).to.be.ok));
});
});
describe('list function', () => {
it('should resolve an array of strings', (done) => {
ecr.list(aws)
.then((r) => C
.check(done, () => expect(Array
.isArray(r))).to.be.ok, C.getFail(done));
});
});
describe('arn function', () => {
it('should throw without a name', () => {
expect(() => ecr.arn()).to.throw(Error);
});
it('should resolve a string', (done) => {
ecr.arn(aws, 'test')
.then((r) => C
.check(done, () => expect(typeof r === 'string').to.be.ok),
C.getFail(done));
});
});
describe('bindAws function', () => {
it('should return a copy of the api', () => {
expect(ecr.bindAws(aws)).to.be.ok;
});
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,149
|
The Winnipeg iF... Improv Festival is going to be hitting Winnipeg September 20th - 24th.
The festival is produced with generous help from the Gas Station Arts Centre and from the blood sweat and tears from the Winnipeg Improv community.
Keep your ear to the wind as we keep out noses to the grindstone. We will be releasing information as it gets leaked to us from out spies... Well, as the information becomes confirmed, we will release it.
Until then, enjoy the hell out of the Winnipeg Fringe Festivalwhich runs July 13th until July 25th.
CLICK HERE for venues and maps.
CLICK HERE for buying advanced tickets.
Thank you, and keep improvising.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,152
|
Dominick and Eugene is a 1988 American drama film directed by Robert M. Young about twin brothers, Dominick and Eugene. Dominick has an intellectual disability due to an accident in his youth. The film stars Ray Liotta, Tom Hulce and Jamie Lee Curtis.
Plot
Dominick "Nicky" and Eugene "Gino" Luciano are twin brothers living together in Pittsburgh, Pennsylvania. Nicky has a learning disability, and Gino cares for him. Gino, who is studying to become a doctor at a local hospital, receives an offer to complete his education at Stanford University but fears that Nicky will not be able to take care of himself. Nicky is a trash collector, a job which finances Gino's education. He and his best friend, Larry, work for Mr. Jesse Johnson. Larry tells Nicky that Gino will leave him for a better life.
Gino helps Jennifer Reston, a medical school student at his hospital, to study for her exams and becomes fond of her. Nicky mentions this to Larry, who tells him that they may get married and abandon him. Although Jennifer is from an affluent family, she is charmed by the brothers' relationship and their humble surroundings. A drug dealer pays the unknowing Nicky ten dollars to deliver an illegal drug, wrapped in newspaper, to a drug user in a rough neighborhood. Nicky forgets the delivery but tells his brother, who worries about his naivete and gullibility.
The night of their birthday, Gino must work late at the hospital and calls a disappointed Nicky to tell him. Nicky wants to take Larry to a Wrestlemania event he and Eugene were going to see, but Larry brings him to visit Mrs. Vinson, with whom Larry occasionally has sex. Nicky goes outside while Larry and Mrs. Vinson are busy, and is surprised to see Mikey Chernak, Mrs. Vinson's neighbor and Nicky's acquaintance, with bruises on his face. When Nicky asks where the bruises are from, Mikey says that he fell. Not knowing that Mikey's father, Martin, abuses him, Nicky believes him.
Larry and Nicky get drunk, and Larry taunts Nicky about Gino's relationship with Jennifer. Nicky goes home, finds Jennifer and Gino talking and tells Gino he knows he is "screwing" Jennifer. Eugene, angry, shoves Nicky before sending him to bed and an embarrassed Jennifer leaves.
Nicky, his dog Fred, Gino, Jennifer, and next-door neighbor Mrs. Gianelli go on a picnic and Fred is hit by a car. Several days later, Nicky is collecting trash at Mikey's house and sees Martin hitting Mikey and shoving him down a flight of stairs. Martin calls 911, saying that Mikey fell. He sees Nicky, who is inconsolable at what he has seen. An ambulance takes Mikey to the hospital and Nicky runs after it, leaving his co-workers.
At the hospital, Martin tells Nicky that Mikey is dead and threatens to kill him if he tells anybody that he pushed the boy; Nicky flees. He takes a gun from Mr. Johnson's truck and returns to Martin's house, where family and friends are grieving. Nicky takes Mikey's baby brother, Joey, from Martin and his wife Theresa at gunpoint, believing that he is protecting Joey from Martin, and is cornered by a SWAT Team in an empty building.
Gino, Jennifer, Martin, and Theresa race to the building, and Gino is sent in to retrieve Nicky and Joey. He confronts Nicky, whose sight of Mikey's abuse had triggered memories that their father had beaten him about the head. Gino breaks down, admitting that Nicky is right; he had protected Gino from their father, taking blows meant for his twin. Sobbing, he tells Nicky that he is sometimes afraid of his anger at him and does not want to become violent like their father. Nicky comforts Gino, telling him he is not like their father and he loves him.
They leave the building and give Joey to Theresa; Eugene protests when the Pittsburgh police handcuff Nicky. Martin pulls a gun aimed at Nicky. Gino and two police officers subdue him, and Nicky tells everyone that Martin killed Mikey. Theresa is horrified; Nicky is released, and Martin is arrested.
Gino kisses Jennifer when he leaves for Stanford for his residency, and she promises to give Nicky her phone number when she starts her residency at Cornell. The twins embrace, and Gino leaves. As the credits roll, Nicky is on his garbage route with a new understanding of himself.
Cast
Tom Hulce as Dominick "Nicky" Luciano
Ray Liotta as Eugene Luciano
Jamie Lee Curtis as Jennifer Reston
Todd Graff as Larry Higgins
Bill Cobbs as Jesse Johnson
David Strathairn as Martin Chernak
Bingo O'Malley as Abe
Reception
The film received positive reviews, holding a 77% rating on the film-review aggregator Rotten Tomatoes based on 22 reviews. The consensus summarizes: "Thanks to strong performances and a steady directorial hand, Dominick and Eugene successfully navigates potentially tricky themes in thoughtful, compelling fashion without resorting to trite sentimentality." Hulce received a Golden Globe nomination for his performance (Best Actor – Motion Picture Drama).
References
External links
1988 films
American drama films
1988 drama films
Films about brothers
Films set in Pittsburgh
Fictional portrayals of the Pittsburgh Bureau of Police
Films directed by Robert M. Young
Films about domestic violence
Films scored by Trevor Jones
Films with screenplays by Alvin Sargent
1980s English-language films
1980s American films
Films about disability
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,428
|
package messagepacket;
import java.io.Serializable;
/**
* Class representing a message.
* Can be of different types - depends on
* the purpose.
*/
public class MessagePacket implements Serializable {
private static final long serialVersionUID = 6774447527662045877L;
private byte msg_id;
private Object message;
/**
* To send data package with coordinates
*/
public MessagePacket(byte player_id, short path_id,
short[] coordinates, long timeStamp ) {
this.msg_id = IdCode.DATA_MESSAGE;
this.message = new DataMessage( player_id, path_id, coordinates, timeStamp );
}
/**
* To send if brush properties changes
*/
public MessagePacket(byte player_id, byte r, byte g,
byte b, byte alpha, byte brush_size,
boolean[] effects ) {
this.msg_id = IdCode.BRUSH_MESSAGE;
this.message = new BrushMessage( player_id, r, g, b, alpha, brush_size, effects );
}
/**
* Info message from client or server
*/
public MessagePacket( byte msg_id, byte player_id, byte members ) {
this.msg_id = msg_id;
this.message = new InfoMessage( player_id, members );
}
/**
* Message to send if removal of specific path is required
*/
public MessagePacket( short path_id ) {
this.msg_id = IdCode.UNDO_PATH;
this.message = new UndoMessage( path_id );
}
/**
* Message to send if close down connection is required
*/
public MessagePacket() {
this.msg_id = IdCode.QUIT_MESSAGE;
}
/**
* Message to send if Redo of drawingpath is required
*/
public MessagePacket(short[] coords, short pID, long timeStamp,
byte alpha, byte r, byte g, byte b, byte brush_size,
boolean[] effects )
{
this.msg_id = IdCode.REDO_PATH;
this.message = new RedoMessage(coords, pID, timeStamp,
alpha, r, g, b, brush_size,
effects );
}
/**
* Get ID code
*/
public byte getMsgId() {
return msg_id;
}
/**
* Return message object
*/
public Object getMsg() {
return message;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,216
|
\section{Quasimonochromatic Signal and Idler Field Operators}
We have taken the positive-frequency field operators $\hat{E}_S(t)$ and $\hat{E}_I(t)$ for the signal and idler outputs of Alice's spontaneous parametric downconverter to be quasimonochromatic, photon-units operators \cite{Blow,YuenShapiro,Shapiro}. As such this gives them the delta-function commutator, $[\hat{E}_J(t),\hat{E}_K^\dagger(u)] = \delta_{JK}\delta(t-u)$, for $J,K = S,I$. Our security proof relies on time and frequency operators whose own delta-function commutator, proved in the next section, depends on the quasimonochromatic condition's validity. The 200\,GHz phase-matching bandwidth assumed in the main text's system example corresponds to 0.1\% fractional bandwidth at the 1.55\,$\upmu$m fiber telecom wavelength. Thus we are justified in assuming that our field operators obey the quasimonochromatic condition.
\section{Proof of $[\hat{\omega}_J,\hat{t}_K] = i\epsilon_J\delta_{JK}$ for $J,K=S,I$}
The operators for arrival-time and frequency-detuning measurements of the signal ($J=S$) and idler ($J=I$) are defined as follows
\begin{subequations}
\label{eq_Def_operators}
\begin{align}
\hat{t}_J &= \int\!dt\, t \hat{E}_J^\dag(t)\hat{E}_J(t)\\
\hat{\omega}_J &= \int\!\frac{d\omega}{2\pi}\, \omega \hat{A}_J^\dag(\omega)\hat{A}_J(\omega),
\end{align}
\end{subequations}
where the field operators are restricted to the Hilbert space spanned by the vacuum state and the single-photon time-domain states $\{\,|t\rangle_J : -\infty < t <\infty\,\}$, or, equivalently, the vacuum state and the single-photon frequency-domain states $\{\,|\omega\rangle_J : -\infty < \omega < \infty\,\}$ \cite{footnote1}. In this Hilbert space the field operators reduce to
\begin{equation}
\hat{E}_J(t) = |0\rangle_J{}_J\langle t|,
\label{Esingle}
\end{equation}
and
\begin{equation}
\hat{A}_J(\omega) = |0\rangle_J{}_J\langle \omega|.
\label{Asingle}
\end{equation}
With the preceding restricted expressions for the field operators in time and frequency, we can easily derive the commutation relation $[\hat{\omega}_J,\hat{t}_K] = i\epsilon_J\delta_{JK}$ for $J,K=S,I$. For $J\neq K$, the frequency-time commutator vanishes because $\hat{E}_J(t)$ commutes with $\hat{E}_K(t)$ and $\hat{E}_K^\dagger(t)$ and hence so too does $\hat{E}_J(t)$ with $\hat{A}_K(\omega)$ and $\hat{A}_K^\dagger(\omega)$. For $J=K=S$ we have that
\begin{align}
\hat{\omega}_S\hat{t}_S - \hat{t}_S\hat{\omega}_S &=
\int\!dt\,\int\!\frac{d\omega}{2\pi}\,\omega t[({}_S\langle \omega|t\rangle_S)|\omega\rangle_S{}_S\langle t| -({}_S\langle t|\omega\rangle_S)|t\rangle_S{}_S\langle \omega| ]
\\
&= \int\!dt\,\int\!\frac{d\omega}{2\pi}\,\omega t[e^{i\omega t}|\omega\rangle_S{}_S\langle t| -
e^{-i\omega t}|t\rangle_S{}_S\langle \omega|],
\label{comm1}
\end{align}
where the second equality follows from
\begin{equation}
|\omega\rangle_S = \int\!dt\,e^{-i\omega t}|t\rangle_S
\label{freqstateS}
\end{equation}
and
\begin{equation}
{}_S\langle t_1|t_2\rangle_S = \delta(t_1-t_2).
\label{comm2}
\end{equation}
Using Eqs.~(\ref{comm1}) and (\ref{comm2}), we then obtain the time-domain matrix elements
\begin{equation}
{}_S\langle t_1|[\hat{\omega}_S,\hat{t}_S]|t_2\rangle_S = (t_2-t_1)\int\!\frac{d\omega}{2\pi}\,\omega e^{-i\omega (t_1-t_2)}.
\label{matrixelement}
\end{equation}
Now, let $f(\tau)$ be any square-integrable function on $-\infty < \tau < \infty$ that is continuous at $\tau = 0$, and has Fourier transform
\begin{equation}
F(\omega) = \int\!d\tau\,f(\tau)e^{-i\omega\tau}.
\end{equation}
We then have that
\begin{equation}
-\int\!d\tau\int\!\frac{d\omega}{2\pi}\,f(\tau)\tau \omega e^{-i\omega\tau} =
-i\int\!\frac{d\omega}{2\pi}\,\omega\frac{dF(\omega)}{d\omega} = -i\frac{\omega}{2\pi} F(\omega)\vert_{-\infty}^\infty + i\int\!\frac{d\omega}{2\pi}\,F(\omega) = i\int\!\frac{d\omega}{2\pi}\,F(\omega) = if(0),
\end{equation}
where the first equality is a Fourier-transform property, the second uses integration by parts, the third results from $f(\tau)$'s being square-integrable, and the fourth by inverse Fourier transformation. Applying this result to Eq.~(\ref{matrixelement}) yields
\begin{equation}
{}_S\langle t_1|[\hat{\omega}_S,\hat{t}_S]|t_2\rangle_S = i\delta(t_1-t_2),
\end{equation}
which is equivalent to $[\hat{\omega}_S,\hat{t}_S] = i$, thus completing our proof for $J=K=S$. For $J=K=I$ the proof follows the same steps using
\begin{equation}
|\omega\rangle_I = \int\!dt\,e^{i\omega t}|t\rangle_I
\label{freqstateI}
\end{equation}
instead of Eq.~(\ref{freqstateS}).
\section{Proof of Lemma~1}
The coincidence probability for our Franson interferometer when only a single photon-pair has been emitted by Alice's source is \cite{footnote2}
\begin{align}
P_{C_{\rm FI}}(\phi_S,\phi_I) &= \frac{\eta}{16} \int\!dt\,\int_{t-T_g/2}^{t+T_g/2}\!du\, \left\langle\left[\hat{E}^\dag_S(t)+e^{i\phi_S}\hat{E}^\dag_S(t-\Delta T)\right]\left[\hat{E}^\dag_I(u)+e^{i\phi_I}\hat{E}^\dag_I(u-\Delta T)\right]\right.\nonumber\\[.05in]
&\times \left.\left[\hat{E}_S(t)+e^{-i\phi_S}\hat{E}_S(t-\Delta T)\right]\left[\hat{E}_I(u)+e^{-i\phi_I}\hat{E}_I(u-\Delta T)\right]\right\rangle,
\end{align}
where $\eta$ accounts for propagation losses \cite{footnote3} and detection efficiencies, and $T_g > \delta T$, with $\delta T$ being the detectors' full-width at half-maximum (FWHM) timing jitter, is the duration of the coincidence gate. Because only a single photon-pair has been emitted, we can employ Eq.~(\ref{Esingle}) and rewrite the coincidence probability as
\begin{align}
P_{C_{\rm FI}}(\phi_S,\phi_I) &= \frac{\eta}{16} \int\!dt\,\int_{t-T_g/2}^{t+T_g/2}\!du\, \left\langle\left[|t\rangle_S{}_S\langle 0|+e^{i\phi_S}|t-\Delta T\rangle_S{}_S\langle 0|\right]\left[|u\rangle_I{}_I\langle 0|+e^{i\phi_I}|u-\Delta T\rangle_I{}_I\langle 0|\right]\right.\nonumber\\[.05in]
&\times \left.\left[|0\rangle_S{}_S\langle t|+e^{-i\phi_S}|0\rangle_S{}_S\langle t-\Delta T|\right]\left[|0\rangle_I{}_I\langle u|+e^{-i\phi_I}|0\rangle_I{}_I\langle u-\Delta T|\right]\right\rangle.
\label{PcFIsingle}
\end{align}
Multiplying out the bracketed expressions inside the averaging leads to a sum of sixteen terms, but only four of them make nonzero contributions when $\Delta T > T_g > \delta T \gg \sigma_{\rm cor}$. To see that this is so, let us first take the undisturbed (no eavesdropping) biphoton wave function \cite{footnote4}
\begin{equation}
\psi_{SI}(t_S,t_I) = \frac{e^{-(t_S+t_I)^2/16\sigma^2_{\rm coh}-(t_S-t_I)^2/4\sigma^2_{\rm cor}-i\omega_P(t_S+t_I)/2}}{\sqrt{2\pi\sigma_{\rm coh}\sigma_{\rm cor}}}.
\label{biphotonTime}
\end{equation}
Averaging a term from Eq.~(\ref{PcFIsingle}) that contains $|t_A\rangle_S{}_S\langle t_B|\otimes|u_A\rangle_I{}_I\langle u_B|$ yields a result that contains $\psi_{SI}^*(t_A,u_A)\psi_{SI}(t_B,u_B)$. Thus, unless $|t_A-u_A| \le T_g/2$ \em and\/\rm\ $|t_B-u_B|\le T_g/2$, this term will make a negligible contribution to $P_{C_{\rm FI}}$. Indeed, the magnitude of each of these noncontributor terms is at least $\exp(-\Delta T^2/8\sigma^2_{\rm cor})$ times smaller than the terms we will retain. For our paper's system example, which assumes $\Delta T = 152.7$\,ps and $\sigma_{\rm cor} = 0.937$\,ps, this attenuation factor is $\exp(-3320)$. These numbers apply to detectors without timing jitter. With timing jitter, the preceding attenuation factor becomes $\exp(-\Delta T^2\ln(2)/2\delta T^2)$, which, because $\delta T = 30\,$ps in our system example, equals $\exp(-8.98)$. Now suppose there is eavesdropping. Then, unless Eve's intrusion makes the root-mean-square arrival-time difference exceed the coincidence gate, such terms will still fail to contribute to the coincidence probability. An Eve who makes that strong a disturbance will easily be detected and accounted for during reconciliation, so we will neglect that possibility in what follows, as we evaluate the coincidence probability.
After eliminating the twelve noncontributors to $P_{C_{\rm FI}}$, we get
\begin{align}
P_{C_{\rm FI}}(\phi_S,\phi_I) &= \frac{\eta}{16} \int\!dt\,\int_{t-T_g/2}^{t+T_g/2}\!du\, \left\langle\left[|t\rangle_S|u\rangle_I{}_I\langle u|{}_S\langle t| + e^{-i(\phi_S + \phi_I)}|t\rangle_S|u\rangle_I{}_I\langle u-\Delta T|{}_S\langle t-\Delta T|\right.\right. \nonumber \\[.05in]
&+ \left.\left.e^{i(\phi_S+\phi_I)}|t-\Delta T\rangle_S|u-\Delta T\rangle_I{}_I\langle u|{}_S\langle t| +
|t-\Delta T\rangle_S|u-\Delta T\rangle_I{}_I\langle u-\Delta T|{}_S\langle t-\Delta T|\right]\right\rangle.
\label{PcFIreduced}
\end{align}
At this point the integration limits for $u$ can be extended to $-\infty < u < \infty$, because the integrand vanishes for $|t-u| > T_g/2$ by an argument similar to what we just gave to go from Eq.~(\ref{PcFIsingle}) to Eq.~(\ref{PcFIreduced}). Thus, we can write
\begin{align}
P_{C_{\rm FI}}(\phi_S,\phi_I) &= \frac{\eta}{16} \int\!dt\,\int\!du\, \left\langle\left[|t\rangle_S|u\rangle_I{}_I\langle u|{}_S\langle t| + e^{-i(\phi_S + \phi_I)}|t\rangle_S|u\rangle_I{}_I\langle u-\Delta T|{}_S\langle t-\Delta T|\right.\right. \nonumber \\[.05in]
&+ \left.\left.e^{i(\phi_S+\phi_I)}|t-\Delta T\rangle_S|u-\Delta T\rangle_I{}_I\langle u|{}_S\langle t| +
|t-\Delta T\rangle_S|u-\Delta T\rangle_I{}_I\langle u-\Delta T|{}_S\langle t-\Delta T|\right]\right\rangle. \\[.05in]
&= \frac{\eta}{16}\int\!dt\int\!du\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\int\!\frac{d\omega_S'}{2\pi}\int\!\frac{d\omega_I'}{2\pi}\,e^{i[(\omega_S-\omega_S')t - (\omega_I-\omega_I')u]}\left[1 + e^{-i[(\phi_S+\phi_I)-(\omega_S'-\omega_I')\Delta T]}\right. \nonumber \\[.05in]
&+ \left. e^{i[(\phi_S + \phi_I) - (\omega_S-\omega_I)\Delta T]} + e^{-i(\omega_S-\omega_I+\omega_I'-\omega_S')\Delta T} \right]\left\langle |\omega_S\rangle_S|\omega_I\rangle_I{}_I\langle \omega_I'|{}_S\langle \omega_S'|\right\rangle \\[.05in]
&= \frac{\eta}{8}\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\, \left[1+{\rm Re}\!\left(e^{i[(\phi_S+\phi_I)-(\omega_S-\omega_I)\Delta T]}\right)\right]\left\langle|\omega_S\rangle_S|\omega_I\rangle_I{}_I\langle \omega_I|{}_S\langle \omega_S|\right\rangle \\[.05in]
&= \frac{\eta}{8}\left[1 + {\rm Re}\!\left(e^{i(\phi_S+\phi_I)}\left\langle e^{-i(\hat{\omega}_S-\hat{\omega}_I)\Delta T}\right\rangle\right)\right],
\end{align}
where the second equality follows from Eqs.~(\ref{freqstateS}) and (\ref{freqstateI}), and the last equality follows from the source's emitting a single photon-pair. Because the coincidence probability only depends on $\phi\equiv \phi_S+\phi_I$, we shall use the notation $P_{C_{\rm FI}}(\phi)$ in what follows.
The $0$-$\pi$ fringe visibility of a Franson interferometer with delay $\Delta T$ is defined as
\begin{equation}
\label{eqFransonVisibility}
V_{\rm FI}(\Delta T) = \frac{P_{C_{\rm FI}}(0)-P_{C_{\rm FI}}(\pi)}{P_{C_{\rm FI}}(0)+P_{C_{\rm FI}}(\pi)} = {\rm Re}\!\left(\left\langle e^{-i(\hat{\omega}_S-\hat{\omega}_I)\Delta T}\right\rangle\right)
= \langle \cos[(\hat{\omega}_S-\hat{\omega}_I)\Delta T]\rangle.
\end{equation}
The frequency-domain biphoton wave function associated with Eq.~(\ref{biphotonTime}),\begin{equation}
\Psi(\omega_S,\omega_I) = \frac{e^{-\omega_+^2\sigma^2_{\rm cor} - \omega_-^2\sigma^2_{\rm coh}}}{\sqrt{\pi/2\sigma_{\rm cor}\sigma_{\rm coh}}}
\label{biphotonFreq}
\end{equation}
where $\omega_+ \equiv (\omega_S+\omega_I)/2$ and $\omega_- = \omega_S -\omega_I$, makes
\begin{align}
\langle\cos[(\hat{\omega}_S-\hat{\omega}_I)\Delta T]\rangle &=
e^{-\langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle\Delta T^2/2} \\
& \le
1- \langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle\Delta T^2/2 +
\langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle^2\Delta T^4/8\\
&= 1- \langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle\Delta T^2/2 +
\langle(\hat{\omega}_S-\hat{\omega}_I)^4\rangle\Delta T^4/24,
\end{align}
with $\langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle = 1/4\sigma_{\rm coh}^2$ and the last equality following from Gaussian moment factoring \cite{Haykin}.
In the presence of eavesdropping we use a three-term Taylor-series expansion to show that
\begin{equation}
\langle\cos[(\hat{\omega}_S-\hat{\omega}_I)\Delta T]\rangle \le 1 - \frac{\langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle\Delta T^2}{2}
+ \frac{\langle(\hat{\omega}_S-\hat{\omega}_I)^4\rangle\Delta T^4}{24}.
\label{FIexpansion}
\end{equation}
Note that (\ref{FIexpansion}) does \em not\/\rm\ assume the biphoton has a Gaussian wave function, so it applies for any biphoton state---pure or mixed---of the signal and idler detected by Alice and Bob.
The third term in this inequality satisfies
\begin{equation}
\langle(\hat{\omega}_S-\hat{\omega}_I)^4\rangle\Delta T^4/24 \le
\langle(\widetilde{\omega}_S-\widetilde{\omega}_I)^4\rangle\Delta T^4/24,
\label{omega4bound}
\end{equation}
where $\widetilde{\omega}_S$ and $\widetilde{\omega}_I$ are classical random variables obtained from frequency measurements at Alice and Bob's terminals respectively, i.e., by disabling the frequency-shifted arms in their conjugate Franson interferometers and relying on those interferometers' dispersive elements to make frequency information manifest in the observed arrival times \cite{footnote5}. Using (\ref{omega4bound}) in (\ref{FIexpansion}) we get
\begin{equation}
\left<(\hat{\omega}_S-\hat{\omega}_I)^2\right> \leq \frac{2[1-V_\text{FI}(\Delta T)]}{\Delta T^2}+\frac{\langle (\widetilde{\omega}_S-\widetilde{\omega}_I)^4 \rangle\Delta T^2}{12}.
\label{Lemma1culm}
\end{equation}
Together with $\langle(\Delta\hat{\omega}_S-\Delta\hat{\omega}_I)^2\rangle \le \langle(\hat{\omega}_S-\hat{\omega}_I)^2\rangle$, (\ref{Lemma1culm}) completes the proof of Lemma~1.
\section{Proof of Lemma 2}
The coincidence probability for our conjugate Franson interferometer when only a single photon-pair has been emitted by Alice's source is \cite{footnote6}
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I) &= \frac{\eta}{16}\int\!dt\int_{t-T_g/2}^{t+T_g/2}\!du\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\int\!\frac{d\omega_S'}{2\pi}\int\!\frac{d\omega_I'}{2\pi}\, \left\langle\left[\hat{A}_S^\dagger(\omega_S) + e^{i\phi_S}\hat{A}_S^\dagger(\omega_S-\Delta\Omega) \right]\right.\nonumber\\
&\times \left[\hat{A}_I^\dagger(\omega_I) + e^{i\phi_I}\hat{A}_I^\dagger(\omega_I-\Delta\Omega) \right]\left[\hat{A}_S(\omega_S') + e^{-i\phi_S}\hat{A}_S(\omega_S'-\Delta\Omega) \right] \nonumber\\
&\times \left.\left[\hat{A}_I(\omega_I') + e^{-i\phi_I}\hat{A}_I(\omega_I'-\Delta\Omega) \right]\right\rangle e^{-i[\beta_2(\omega_S^2-\omega_I^2-\omega_S'^2+\omega_I'^2)/2 - (\omega_S-\omega_S')t +(\omega_I-\omega_I')u]}.
\label{CFI1}
\end{align}
Using Eq.~(\ref{Asingle}), this expression becomes
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I) &= \frac{\eta}{16}\int\!dt\int_{t-T_g/2}^{t+T_g/2}\!du\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\int\!\frac{d\omega_S'}{2\pi}\int\!\frac{d\omega_I'}{2\pi}\, \left\langle\left[|\omega_S\rangle_S{}_S\langle 0| + e^{i\phi_S}|\omega_S-\Delta\Omega\rangle_S{}_S\langle 0| \right]\right.\nonumber\\
&\times \left[|\omega_I\rangle_I{}_I\langle 0| + e^{i\phi_I}|\omega_I-\Delta\Omega\rangle_I{}_I\langle 0| \right]\left[|0\rangle_S{}_S\langle\omega_S'| + e^{-i\phi_S}|0\rangle_S{}_S\langle\omega_S'-\Delta\Omega| \right] \nonumber\\
&\times \left.\left[|0\rangle_I{}_I\langle \omega_I'| + e^{-i\phi_I}|0\rangle_I{}_I\langle\omega_I'-\Delta\Omega| \right]\right\rangle e^{-i[\beta_2(\omega_S^2-\omega_I^2-\omega_S'^2+\omega_I'^2)/2 -(\omega_S-\omega_S')t + (\omega_I-\omega_I')u]}.
\label{CFI2}
\end{align}
Multiplying out the bracketed expressions inside the averaging leads to a sum of sixteen terms, but only four of them make nonzero contributions when $\Delta\Omega > T_g/2\beta_2 \gg 3/\sigma_{\rm coh}$. To show that this is so, we begin by rewriting Eq.~(\ref{CFI2}) in terms of the sum and difference variables $t_+ \equiv (t+u)/2$, $t_-\equiv t-u$, $\omega_+\equiv (\omega_S+\omega_I)/2$, $\omega_-\equiv \omega_S-\omega_I$, $\omega'_+\equiv (\omega'_S+\omega'_I)/2$, and $\omega'_-\equiv \omega'_S-\omega'_I$, obtaining
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I) &= \frac{\eta}{16}\int\!dt_+\int_{-T_g/2}^{T_g/2}\!dt_-\int\!\frac{d\omega_+}{2\pi}\int\!\frac{d\omega_-}{2\pi}\int\!\frac{d\omega'_+}{2\pi}\int\!\frac{d\omega'_-}{2\pi}\, \left\langle\left[|\omega_++\omega_-/2\rangle_S{}_S\langle 0| \right.\right.\nonumber \\
&\left.+\,\, e^{i\phi_S}|\omega_++\omega_-/2-\Delta\Omega\rangle_S{}_S\langle 0| \right]\ \left[|\omega_+-\omega_-/2\rangle_I{}_I\langle 0| + e^{i\phi_I}|\omega_+-\omega_-/2-\Delta\Omega\rangle_I{}_I\langle 0| \right] \nonumber \\
&\times \left[|0\rangle_S{}_S\langle\omega'_++\omega'_-/2| + e^{-i\phi_S}|0\rangle_S{}_S\langle\omega'_++\omega'_-/2-\Delta\Omega| \right] \nonumber\\
&\times \left.\left[|0\rangle_I{}_I\langle \omega'_+-\omega'_-/2| + e^{-i\phi_I}|0\rangle_I{}_I\langle\omega'_+-\omega'_-/2-\Delta\Omega| \right]\right\rangle \nonumber \\
&\times e^{-i[\beta_2(\omega_+\omega_--\omega'_+\omega'_-) -(\omega_+-\omega'_+)t_- - (\omega_--\omega'_-)t_+]}.
\label{CFI3}
\end{align}
Next, we perform the $t_-$, $t_+$, and $\omega'_-$ integrals and get
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I) &= \frac{\eta}{16}\int\!\frac{d\omega_+}{2\pi}\int\!\frac{d\omega_-}{2\pi}\int\!\frac{d\omega'_+}{2\pi}\, \left\langle\left[|\omega_++\omega_-/2\rangle_S{}_S\langle 0| + e^{i\phi_S}|\omega_++\omega_-/2-\Delta\Omega\rangle_S{}_S\langle 0| \right]\right.\nonumber \\
& \times \left[|\omega_+-\omega_-/2\rangle_I{}_I\langle 0| + e^{i\phi_I}|\omega_+-\omega_-/2-\Delta\Omega\rangle_I{}_I\langle 0| \right] \nonumber \\
&\times \left[|0\rangle_S{}_S\langle\omega'_++\omega_-/2| + e^{-i\phi_S}|0\rangle_S{}_S\langle\omega'_++\omega_-/2-\Delta\Omega| \right] \nonumber\\
&\times \left.\left[|0\rangle_I{}_I\langle \omega'_+-\omega_-/2| + e^{-i\phi_I}|0\rangle_I{}_I\langle\omega'_+-\omega_-/2-\Delta\Omega| \right]\right\rangle \nonumber \\
&\times T_g\frac{\sin[(\omega_+-\omega'_+)T_g/2]}{(\omega_+-\omega'_+)T_g/2}
e^{-i\beta_2(\omega_+-\omega'_+)\omega_-}.
\label{CFI4}
\end{align}
Because the phase-matching bandwidth $B_{\rm PM}$ greatly exceeds $1/T_g$, the state-average term in
Eq.~(\ref{CFI4}) is essentially unchanged for $\omega\equiv (\omega_++\omega'_+)/2$ excursions on the order of $10/T_g$. Thus Eq.~(\ref{CFI4}) simplifies to
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I) &= \frac{\eta}{16}\int\!\frac{d\omega}{2\pi}\int\!\frac{d\omega_-}{2\pi}\int\!\frac{d\omega'}{2\pi}\, \left\langle\left[|\omega+\omega_-/2\rangle_S{}_S\langle 0| + e^{i\phi_S}|\omega+\omega_-/2-\Delta\Omega\rangle_S{}_S\langle 0| \right]\right.\nonumber \\
& \times \left[|\omega-\omega_-/2\rangle_I{}_I\langle 0| + e^{i\phi_I}|\omega-\omega_-/2-\Delta\Omega\rangle_I{}_I\langle 0| \right] \nonumber \\
&\times \left[|0\rangle_S{}_S\langle\omega+\omega_-/2| + e^{-i\phi_S}|0\rangle_S{}_S\langle\omega+\omega_-/2-\Delta\Omega| \right] \nonumber\\
&\times \left.\left[|0\rangle_I{}_I\langle \omega-\omega_-/2| + e^{-i\phi_I}|0\rangle_I{}_I\langle\omega-\omega_-/2-\Delta\Omega| \right]\right\rangle \nonumber \\
&\times T_g\frac{\sin(\omega'T_g/2)}{\omega'T_g/2}
e^{-i\beta_2\omega'\omega_-},
\label{CFI5}
\end{align}
where $\omega'\equiv \omega_+-\omega'_+$.
Performing the $\omega'$ integral then gives us
\begin{align}
P_{C_{\rm CFI}}(\phi_S,\phi_I)&= \frac{\eta}{16}\int\!\frac{d\omega}{2\pi}\int_{-T_g/2\beta_2}^{T_g/2\beta_2}\!\frac{d\omega_-}{2\pi}\, \left\langle\left[|\omega+\omega_-/2\rangle_S{}_S\langle 0| + e^{i\phi_S}|\omega+\omega_-/2-\Delta\Omega\rangle_S{}_S\langle 0| \right]\right.\nonumber \\
& \times \left[|\omega-\omega_-/2\rangle_I{}_I\langle 0| + e^{i\phi_I}|\omega-\omega_-/2-\Delta\Omega\rangle_I{}_I\langle 0| \right] \nonumber \\
&\times \left[|0\rangle_S{}_S\langle\omega+\omega_-/2| + e^{-i\phi_S}|0\rangle_S{}_S\langle\omega+\omega_-/2-\Delta\Omega| \right] \nonumber\\
&\times \left.\left[|0\rangle_I{}_I\langle \omega-\omega_-/2| + e^{-i\phi_I}|0\rangle_I{}_I\langle\omega-\omega_-/2-\Delta\Omega| \right]\right\rangle.
\label{CFI6}
\end{align}
At this point we can quickly eliminate twelve terms from Eq.~(\ref{CFI6}). In Eve's absence, any term containing $\langle |\omega_A\rangle_S{}_S\langle \omega_B|\otimes |\omega_A'\rangle_I{}_\langle \omega_B'|\rangle$ will only contribute if $|\omega_A-\omega_A'|\le 3/\sigma_{\rm coh}$ \em and\/\rm\ $|\omega_B-\omega_B'| \le 3/\sigma_{\rm coh}$. Indeed, the magnitude of each of these noncontributor terms is at least $\exp(-\Delta \Omega^2\sigma^2_{\rm coh}/2)$ times smaller than the terms we will retain. For our paper's system example, which assumes $\Delta \Omega/2\pi = 5$\,GHz, this attenuation factor is $\exp(-20)$ for $\sigma_{\rm coh} = 0.20\,$ns (corresponding to $T_f = 16\delta T$) and $\exp(-82)$ for $\sigma_{\rm coh} = 0.41\,$ns (corresponding to $T_f = 32\delta T$). These numbers apply to detectors without timing jitter. With timing jitter, the preceding attenuation factor becomes $\exp(-\Delta \Omega^2\beta_2^2\ln(2)/2\delta T^2)$, which equals $\exp(-8.98)$ for our $\Delta\Omega\beta_2 = \sqrt{2}\,T_g$, $T_g = 108\,$ps , $\delta T = 30\,$ps system example. It follows that the only terms which survive in $P_{C_{\rm CFI}}$ are as given below, where we have reverted to $\omega_S$, $\omega_I$ notation,
\begin{eqnarray}
\lefteqn{P_{C_{\rm CFI}}(\phi_S,\phi_I) = }\nonumber \\
&&\frac{\eta}{16}\int\!\frac{d\omega_S}{2\pi}\int_{\omega_S-T_g/2\beta_2}^{\omega_S+T_g/2\beta_2}\!\frac{d\omega_I}{2\pi}\, \left\langle\left[|\omega_S\rangle_S|\omega_I\rangle_I{}_I\langle \omega_I|{}_S\langle\omega_S|
+e^{i(\phi_S + \phi_I)}|\omega_S-\Delta \Omega\rangle_S|\omega_I-\Delta\Omega\rangle_I{}_I\langle\omega_I|{}_S\langle \omega_S| \right.\right. \nonumber \\
&& \hspace{-.15in}\, + \left.\left. e^{-i(\phi_S+\phi_I)}|\omega_S\rangle_S|\omega_I\rangle_I{}_I\langle \omega_I-\Delta\Omega|{}_S\langle \omega_S-\Delta \Omega| + |\omega_S-\Delta\Omega\rangle_S|\omega_I-\Delta\Omega\rangle_I{}_I\langle \omega_I-\Delta\Omega|{}_S\langle \omega_S-\Delta\Omega|\right]\right\rangle.
\end{eqnarray}
The limits in the $\omega_I$ integral can be extended to $-\infty < \omega_-<\infty$, if no eavesdropping has occurred, because $\Delta\Omega > T_g/2\beta_2 \gg 3/\sigma_{\rm coh}$, in which case
\begin{eqnarray}
\lefteqn{P_{C_{\rm CFI}} = \frac{\eta}{8}\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\,\left[1+{\rm Re}\!\left(e^{i(\phi_S+\phi_I)}\langle|\omega_S-\Delta \Omega\rangle_S|\omega_I-\Delta\Omega\rangle_I{}_I\langle\omega_I|{}_S\langle \omega_S|\rangle\right)\right] }\\
&=& \frac{\eta}{8}\int\!\frac{d\omega_S}{2\pi}\int\!\frac{d\omega_I}{2\pi}\int\!dt_S\int\!dt_I\,\left[1+ {\rm Re}\!\left(e^{i(\phi_S+\phi_I)}\langle|\omega_S-\Delta \Omega\rangle_S|\omega_I-\Delta\Omega\rangle_I{}_I\langle t_I|{}_S\langle t_S|\rangle e^{i(\omega_St_S-\omega_It_I)} \right)\right] \hspace*{.2in}\\
&=& \frac{\eta}{8}\int\!dt_S\int\!dt_I\,\left[1+ {\rm Re}\!\left(e^{i(\phi_S+\phi_I)}e^{i(t_S-t_I)\Delta\Omega} \langle |t_S\rangle_S|t_I\rangle_I{}_I\langle t_I|{}_S\langle t_S|\rangle \right)\right] \\
&=& \frac{\eta}{8}\left[1+{\rm Re}\!\left(e^{i(\phi_S+\phi_I)}\left\langle e^{i(\hat{t}_S-\hat{t}_I)\Delta\Omega}\right\rangle\right)\right].\label{CFI7}
\end{eqnarray}
Now suppose that there is eavesdropping. Any intrusion by Eve that degrades the frequency correlations to the point that the suppressed terms do contribute to $P_{C_{\rm CFI}}$ will be detected by the Franson interferometer and accounted for via Lemma~1, so we will complete our Lemma~2 proof using Eq.~(\ref{CFI7}). Because the coincidence probability only depends on $\phi\equiv \phi_S + \phi_I$, we shall use the notation $P_{C_{\rm CFI}}(\phi)$ in what follows.
The conjugate-Franson interferometer's $0$-$\pi$ fringe visibility is defined to be
\begin{equation}
V_{\rm CFI}(\Delta \Omega) = \frac{P_{C_{\rm CFI}}(0) - P_{C_{\rm CFI}}(\pi)}{P_{C_{\rm CFI}}(0) + P_{C_{\rm CFI}}(\pi)} = {\rm Re}\!\left(\left\langle e^{i(\hat{t}_S-\hat{t}_I)\Delta \Omega}\right\rangle\right) =
\langle \cos[(\hat{t}_S-\hat{t}_I)\Delta \Omega]\rangle.
\end{equation}
The time-domain biphoton wave function, Eq.~(\ref{biphotonTime}), makes
\begin{align}
\langle \cos[(\hat{t}_S-\hat{t}_I)\Delta \Omega]\rangle &= e^{-\langle (\hat{t}_S-\hat{t}_I)^2\rangle\Delta\Omega^2/2} \\
& \le 1- \langle (\hat{t}_S-\hat{t}_I)^2\rangle\Delta\Omega^2/2 + \langle(\hat{t}_S-\hat{t}_I)^2\rangle^2\Delta \Omega^4/8 \\
& = 1- \langle(\hat{t}_S-\hat{t}_I)^2\rangle\Delta\Omega^2/2 +
\langle (\hat{t}_S-\hat{t}_I)^4\rangle\Delta \Omega^4/24,
\end{align}
with $\langle(\hat{t}_S-\hat{t}_I)^2\rangle = \sigma^2_{\rm cor}$ and the last equality again following from Gaussian moment factoring. In the presence of eavesdropping we use a three-term Taylor-series expansion to show that
\begin{equation}
\langle \cos[(\hat{t}_S-\hat{t}_I)\Delta \Omega]\rangle
\le 1 - \langle(\hat{t}_S-\hat{t}_I)^2\rangle\Delta\Omega^2/2 +
\langle (\hat{t}_S-\hat{t}_I)^4\rangle\Delta \Omega^4/24.
\label{CFIexpansion}
\end{equation}
As noted in Lemma~1, (\ref{CFIexpansion}) does \em not\/\rm\ assume the biphoton has a Gaussian wave function, so it applies for any biphoton state---pure or mixed---of the signal and idler detected by Alice and Bob.
The third term in this inequality satisfies
\begin{equation}
\langle(\hat{t}_S-\hat{t}_I)^4\rangle\Delta \Omega^4/24 \le
\langle(\widetilde{t}_S-\widetilde{t}_I)^4\rangle\Delta \Omega^4/24,
\label{t4bound}
\end{equation}
where $\widetilde{t}_S$ and $\widetilde{t}_I$ are classical random variables obtained from arrival-time measurements at Alice and Bob's terminals respectively, i.e., by disabling the long arms in their Franson interferometer's \cite{footnote6a}. Using (\ref{t4bound}) in (\ref{CFIexpansion}) we get
\begin{equation}
\left<(\hat{t}_S-\hat{t}_I)^2\right> \leq \frac{2[1-V_\text{CFI}(\Delta \Omega)]}{\Delta \Omega^2}+\frac{\langle (\widetilde{t}_S-\widetilde{t}_I)^4 \rangle\Delta \Omega^2}{12}.
\label{Lemma2culm}
\end{equation}
Together with $\langle(\Delta\hat{t}_S-\Delta\hat{t}_I)^2\rangle \le \langle(\hat{t}_S-\hat{t}_I)^2\rangle$, (\ref{Lemma2culm}) completes the proof of Lemma~2.
\section{Security calculations}
\subsection{The time-frequency covariance matrix}
The TFCM for the biphoton state from Eq.~(\ref{biphotonTime}) is
\begin{equation}
\Gamma_0 = \left[ \begin{array}{ccc}
\gamma^0_{SS} & &\gamma^0_{SI} \\[.1in]
\gamma^0_{IS} && \gamma^0_{II}
\end{array} \right],
\end{equation}
where
\begin{subequations}
\label{EqTFCM0}
\begin{align}
& \gamma^0_{SS} = \gamma^0_{II} = \left[\begin{array}{ccc}
\sigma^2_\text{cor}/4+\sigma^2_\text{coh} & & 0\\[.1in]
0 & & 1/4\sigma^2_\text{cor}+ 1/16\sigma^2_\text{coh}
\end{array}\right]\\[.1in]
& \gamma^0_{SI} = \gamma^0_{IS} = \left[\begin{array}{ccc}
-\sigma^2_\text{cor}/4+\sigma^2_\text{coh} & & 0\\[.1in]
0 & & 1/4\sigma^2_\text{cor} - 1/16\sigma^2_\text{coh}
\end{array}\right].
\end{align}
\end{subequations}
The root-mean-square two-photon correlation time is given by $\sigma_\text{cor} = \sqrt{2\ln(2)}/(2\pi B_{\rm PM})$ in terms of the FWHM phase-matching bandwidth (in Hz). The root-mean-square two-photon pulse duration is given by $\sigma_\text{coh} = T_f/\sqrt{8\ln(2)}$ in terms of our protocol's frame duration---which is taken to be the FWHM coherence time---and is set by choice of the pump pulse's duration. To achieve multiple secure bits per coincidence, we require $ \sigma_\text{coh} \gg \delta T/\sqrt{8\ln(2)} \gg \sigma_\text{cor} $. Under this condition, we have $ \langle \Delta \hat{t}_{S_0}^2\rangle = \langle \Delta \hat{t}_{I_0}^2 \rangle = \sigma^2_\text{coh} $, $ \langle ( \Delta \hat{t}_{S_0}-\Delta \hat{t}_{I_0})^2\rangle = \sigma^2_\text{cor} $, $ \langle \Delta\hat{\omega}_{S_0}^2\rangle = \langle \Delta\hat{\omega}_{I_0}^2\rangle = 1/4\sigma_\text{cor}^2 $, and $ \langle (\Delta \hat{\omega}_{S_0}-\Delta\hat{\omega}_{I_0})^2\rangle = 1/4\sigma_\text{coh}^2$, where the subscript 0 denotes the initial state produced by the source. The Franson and conjugate-Franson interferometer's allow us to upper bound the variances of Alice and Bob's arrival-time and frequency differences, which we denote $ \langle (\Delta\hat{t}_S - \Delta\hat{t}_I)^2\rangle = (1+\xi_t) \langle (\Delta\hat{t}_{S_0}-\Delta\hat{t}_{I_0})^2\rangle$ and $ \langle (\Delta\hat{\omega}_S-\Delta\hat{\omega}_I)^2 \rangle = (1+\xi_\omega) \langle (\Delta\hat{\omega}_{S_0}-\Delta\hat{\omega}_{I_0})^2\rangle $, where $ \xi_t $ and $ \xi_\omega $ quantify the amount of excess noise.
In an operational system, $\xi_\omega $ and $ \xi_t $ will be bounded from the measured Franson and conjugate-Franson's $0$-$\pi$ visibilities, plus the fourth moments of the arrival-time and frequency differences obtained, respectively, from Franson and conjugate-Franson count records with one arm disabled at Alice and Bob's terminals. For the theoretical assessment of the secure-key rate presented in the paper, we used assumed values for the $0$-$\pi$ visibilities and the following jitter-limited values for the fourth moments appearing in Lemmas~1 and 2:
\begin{equation}
\langle (\widetilde{\omega}_S-\widetilde{\omega}_I)^4\rangle = 3(\delta T/2\sqrt{\ln(2)}\beta_2)^4,
\end{equation}
and
\begin{equation}
\langle (\widetilde{t}_S-\widetilde{t}_I)^4\rangle = 3(\delta T/2\sqrt{\ln(2)})^4.
\end{equation}
Here, we have assumed that our detectors have statistically independent, identically distributed Gaussian timing jitters whose FWHM $\delta T$ satisfies $ \delta T^2 \gg \beta_2^2\langle (\hat{\omega}_S-\hat{\omega}_I)^2\rangle$ and $ \delta T^2 \gg \langle (\hat{t}_S-\hat{t}_I)^2\rangle$.
\subsection{Off-diagonal elements in covariance sub-matrices}
The off-diagonal elements in Eq.~(\ref{EqTFCM0}) are all zero, because Alice's source does not produce any time-frequency cross correlations. However, Eve's intrusion could result in a TFCM
\begin{equation}
\Gamma = \left[ \begin{array}{ccc}
\gamma_{SS} & &\gamma_{SI} \\[.1in]
\gamma_{IS} && \gamma_{II}
\end{array} \right],
\label{postpropTFCM}
\end{equation}
whose sub-matrices have nonzero off-diagonal elements.
Inasmuch as the Franson and conjugate-Franson interferometers do \em not\/\rm\ probe those off-diagonal elements, we now prove that the information they do provide---through Lemmas~1 and 2---suffices to upper bound Eve's Holevo information.
We begin by introducing dimensionless time and frequency operators defined as follows
\begin{subequations}
\begin{align}
\hat{\tilde{t}}_J &= \frac{\hat{t}_J}{T}\\
\hat{\tilde{\omega}}_J &= \hat{\omega}_J T,
\end{align}
\end{subequations}
for $J = S,I$, where $ T \equiv \sqrt{2\sigma_\mathrm{coh}\sigma_\mathrm{cor}}$ is a normalization time that symmetrizes these conjugate observables, i.e., for Alice's SPDC source we have that $\langle \Delta\hat{\tilde{t}}_{J_0}^2\rangle = \langle \Delta\hat{\tilde{\omega}}_{J_0}^2\rangle$. The biphoton wave function from Eq.~(\ref{biphotonTime}) now becomes a two-mode squeezed-vacuum state of the modes associated with the effective annihilation and creation operators
\begin{subequations}
\begin{align}
\hat{\tilde{a}}_J &= (\hat{\tilde{t}}_J - i\epsilon_J\hat{\tilde{\omega}}_J)/\sqrt{2} \\
\hat{\tilde{a}}^\dag_J &= (\hat{\tilde{t}}_J + i\epsilon_J\hat{\tilde{\omega}}_J)/\sqrt{2},
\end{align}
\end{subequations}
because $[\hat{\tilde{\omega}}_J,\hat{\tilde{t}}_K] = i\epsilon_J\delta_{JK}$ implies that
$[\hat{\tilde{a}}_J,\hat{\tilde{a}}^\dagger_K] = \delta_{JK}$.
As a result, we can now apply security results of entanglement-based CVQKD to our TEE-based HDQKD protocol.
Eve's arbitrary Gaussian attack can be decomposed into the following two steps: Step~1, she interacts her ancillary state with the idler beam---while it is en route from Alice to Bob---in a manner that does not introduce any time-frequency off-diagonal elements in Alice and Bob's TFCM. Step~2, she applies individual symplectic transformations to the idler state and her ancillary state, a process which creates such time-frequency off-diagonal elements \cite{serafini04}. Step~2 consists of a rotation followed by quadrature squeezing. Because it only involves local unitary operations, it does not affect Eve's Holevo information. Eve's goal here is to minimize $\delta\tilde{\omega}^2 \equiv \langle (\hat{\tilde{\omega}}_S-\hat{\tilde{\omega}}_I)^2 \rangle$, and thus maximize Alice and Bob's 0-$\pi$ Franson-interferometer visibility so that they will underestimate the information her interaction has yielded \cite{footnote8}. More specifically, Eve's goal in Step~2 is to make $\delta\tilde{\omega}^2$ smaller than $\delta\tilde{\omega}_{\rm diag}^2$, the
$\langle (\hat{\tilde{\omega}}_S-\hat{\tilde{\omega}}_I)^2 \rangle$ value when she \em only\/\rm\ employs Step~1, i.e., when Alice and Bob's TFCM has no off-diagonal terms representing time-frequency cross correlations.
To see that Eve's effort in this regard is futile we proceed as follows, where, for notational simplicity, we have assumed that mean values have been subtracted out. Let the effective idler annihilation operator after Step~1 be $\hat{\tilde{b}}$. The effective idler annihilation operator after Step~2 is then
\begin{equation}
\hat{\tilde{a}}_I = e^{i\varphi}\cosh(r) \hat{\tilde{b}} -e^{i(\theta-\varphi)}\sinh(r) \hat{\tilde{b}}^\dag,
\end{equation}
where $\varphi$ is the phase used in the rotation, $ r $ is the squeezing parameter, and $ \theta $ is the phase used in the squeezing operation. We now find that
\begin{equation}
\hat{\tilde{\omega}}_I = \left[\cos(\varphi) \cosh(r) + \cos(\theta-\varphi)\sinh(r)\right]\hat{\tilde{\omega}}_b + \left[\sin(\varphi) \cosh(r) - \sin(\theta-\varphi)\sinh(r)\right]\hat{\tilde{t}}_b,
\label{step2}
\end{equation}
from which the mean-squared frequency error is seen to obey
\begin{eqnarray}
\label{eqdeltaomega}
\delta\tilde{\omega}^2 &=& \langle \hat{\tilde{\omega}}_S^2\rangle - 2\left[\cos(\varphi) \cosh(r)+\cos(\theta-\varphi)\sinh(r)\right]\langle \hat{\tilde{\omega}}_S\hat{\tilde{\omega}}_b\rangle \notag\\
&+& \left[\cos(\varphi) \cosh(r)+\cos(\theta-\varphi)\sinh(r)\right]^2\langle \hat{\tilde{\omega}}_b^2 \rangle+\left[\sin(\varphi)\cosh(r)-\sin(\theta-\varphi)\sinh(r)\right]^2\langle\hat{\tilde{t}}_b^2\rangle,
\end{eqnarray}
where we have used the fact that Eve's Step~1 does not create any time-frequency cross correlations.
Setting $\varphi=0$, $r=0$, and $\theta=0$ in this expression eliminates Eve's Step~2 and leads us to the benchmark value
\begin{equation}
\delta\tilde{\omega}^2_{\rm diag} = \langle (\hat{\tilde{\omega}}_S - \hat{\tilde{\omega}}_b)^2\rangle,
\end{equation}
that she is trying to beat with her symplectic transformation. Without loss of generality, we can assume that $\delta\tilde{\omega}^2_{\rm diag} \ll \langle \hat{\tilde{\omega}}_S^2\rangle$, because violating this condition makes Eve's presence extremely obvious to Alice and Bob.
To minimize $\delta\tilde{\omega}^2$, we set its partial derivatives with respect to $\varphi, r,$ and $\theta$ to zero. For the partial derivative with respect to $\varphi$ we have
\begin{eqnarray}
\label{eqpartialphi}
\frac{\partial (\delta\tilde{\omega}^2)}{\partial \varphi} &=& 2\left[\sin(\varphi)\cosh(r) -\sin(\theta-\varphi)\sinh(r)\right]\langle \hat{\tilde{\omega}}_S\hat{\tilde{\omega}}_b\rangle \notag \\
&-& 2\left[\sin(\varphi) \cosh(r) - \sin(\theta -\varphi)\sinh(r)\right]\left[\cos(\varphi)\cosh(r) + \cos(\theta-\varphi)\sinh(r)\right](\langle \hat{\tilde{\omega}}_b^2 \rangle
-\langle \hat{\tilde{t}}_b^2\rangle),
\end{eqnarray}
which vanishes if
\begin{equation}
\label{eqCondphi1}
\left[\sin(\varphi)\cosh(r) -\sin(\theta-\varphi)\sinh(r)\right] = 0 \tag{C.1}
\end{equation}
or
\begin{equation}
\label{eqCondphi2}
\langle \hat{\tilde{\omega}}_S\hat{\tilde{\omega}}_b\rangle = \left[\cos(\varphi)\cosh(r) + \cos(\theta-\varphi)\sinh(r) \right](\langle\hat{\tilde{\omega}}^2_b\rangle-\langle \hat{\tilde{t}}^2_b\rangle) \tag{C.2}.
\end{equation}
Condition~(\ref{eqCondphi1}), combined with the fact that Eve's Step~1 does not create any time-frequency cross correlations, implies that the TFCM will not have any time-frequency off-diagonal terms. Hence we will only carry forward Condition~(\ref{eqCondphi2}) in trying to see if Eve's Step~2 helps her.
For the partial derivative with respect to $\theta$ we have
\begin{eqnarray}
\label{eqpartialtheta}
\frac{\partial (\delta\tilde{\omega}^2)}{\partial \theta} &=& 2\sin(\theta-\varphi)\sinh(r)\langle \hat{\tilde{\omega}}_S\hat{\tilde{\omega}}_b\rangle-2\left[\cos(\varphi)\cosh(r) +\cos(\theta-\varphi)\sinh(r)\right]\left[\sin(\theta-\varphi)\sinh(r)\right]\langle\hat{\tilde{\omega}}^2_b\rangle\notag\\
&-& 2\left[\sin(\varphi) \cosh(r)-\sin(\theta-\varphi)\sinh(r)\right]\cos(\theta-\varphi)\sinh(r)\langle \hat{\tilde{t}}^2_b\rangle.
\end{eqnarray}
If the $\varphi$ partial derivative vanishes because (\ref{eqCondphi2}) is satisfied, we need
\begin{equation}
\label{eqCondphi2theta1}
\sin(\theta) = 0 \tag{C.2.1}
\end{equation}
or
\begin{equation}
\label{eqCondphi2theta2}
r = 0 \tag{C.2.2}
\end{equation}
to make the $\theta$ partial derivative vanish.
For the partial derivative with respect to $r$ we have
\begin{eqnarray}
\label{eqpartialr}
\frac{\partial (\delta\tilde{\omega}^2)}{\partial r} &=& -2\left[\cos(\varphi) \sinh(r) + \cos(\theta - \varphi)\cosh(r)\right]\langle\hat{\tilde{\omega}}_S\hat{\tilde{\omega}}_b \rangle\notag\\
&+&2\left[\cos(\varphi)\cosh(r) + \cos(\theta-\varphi)\sinh(r)\right]\left[\cos(\varphi)\sinh(r)+\cos(\theta-\varphi)\cosh(r)\right]\langle\hat{\tilde{\omega}}^2_b\rangle\notag\\
&+&2\left[\sin(\varphi)\cosh(r)-\sin(\theta-\varphi)\sinh(r)\right]\left[\sin(\varphi) \sinh(r) - \sin(\theta-\varphi)\cosh(r)\right]\langle\hat{\tilde{t}}^2_b \rangle.
\end{eqnarray}
If the $\varphi$ and $\theta$ partial derivatives vanish because Conditions~(\ref{eqCondphi2}) and (\ref{eqCondphi2theta1}) are satisfied, then the $r$ partial derivative equals zero when $\sinh(r) = \pm \cosh(r)$, where the plus sign applies for $\cos(\theta) = -1$ and the minus sign for $\cos(\theta) = 1$. We then find that $\delta\tilde{\omega}^2 = \langle \hat{\tilde{\omega}}_S^2\rangle \gg \delta\tilde{\omega}^2_{\rm diag}$, making Eve's presence very detectable if she elects to perform Step~2 with these parameter values.
At this point, the only remaining case that might provide Eve some utility from her Step~2 is when Conditions~(\ref{eqCondphi2}) and (\ref{eqCondphi2theta2}) are satisfied and the $r$ partial derivative vanishes. Here, Step~2 corresponds to rotation without squeezing, in which case we need $\cos(\theta) = 0$ to make the $r$ partial derivative vanish, yielding
\begin{equation}
\delta\tilde{\omega}^2 = \langle \hat{\tilde{\omega}}_S^2\rangle - \cos^2(\varphi)(\langle \hat{\tilde{\omega}}_b^2\rangle - \langle \hat{\tilde{t}}_b^2\rangle) + \langle \hat{\tilde{t}}_b^2\rangle.
\end{equation}
If $\langle \hat{\tilde{\omega}}_b^2\rangle = \langle \hat{\tilde{t}}_b^2\rangle$, we get $\delta\tilde{\omega}^2 \gg \delta\tilde{\omega}^2_{\rm diag}$, so Eve shouldn't use a symplectic transformation after her Step~1. If $\langle \hat{\tilde{\omega}}_b^2\rangle > \langle \hat{\tilde{t}}_b^2\rangle$, then her $\delta\tilde{\omega}^2$ is minimized by choosing $\cos^2(\varphi) = 1$, and Eq.~(\ref{step2}) shows that no off-diagonal TFCM elements arise from time-frequency cross correlations. If $\langle \hat{\tilde{\omega}}_b^2\rangle < \langle \hat{\tilde{t}}_b^2\rangle$, then Eve's $\delta\tilde{\omega}^2$ is minimized by choosing $\cos(\varphi) = 0$, and we get $\delta\tilde{\omega}^2 = \langle \hat{\tilde{\omega}}_S^2\rangle + \langle \hat{\tilde{t}}_b^2\rangle \gg \delta\tilde{\omega}^2_{\rm diag}$, making her presence very detectable if she elects to perform Step~2 with these parameter values.
To summarize, we have shown that for any value of Eve's Holevo information, the minimum value of $\delta\tilde{\omega}^2$ can be achieved \em without\/\rm\ introducing any off-diagonal elements into the TFCM sub-matrices \cite{footnote8}. Hence Lemmas~1 and 2 suffice to bound the information that Eve obtains from an optimized collective attack.
\subsection{Alice and Bob's Shannon Information}
Alice and Bob postselect frames in which each of them had \textit{at least} one detection, which may be either a photon detection or a dark count. A fraction $q$ of these postselected frames are reconciled to generate key, while the rest are used for the Franson and conjugate-Franson measurements needed to estimate $\xi_t$ and $\xi_\omega$ and the decoy-state measurements needed for parameter estimation. The probability for Alice and Bob to postselect a frame is therefore
\begin{equation}
p_r = \sum_{n = 0}^\infty p_s(n)\left[1-(1-\eta_A)^n(1-p_d)\right]\left[1-(1-\eta_B\eta_P)^n(1-p_d)\right],
\end{equation}
where: $p_s(n) = \mu^ne^{-\mu}/n!$ gives the probability that Alice's source emits an $n$-pair state in terms of the average number, $\mu$, of pairs emitted per frame \cite{footnote9}; $\eta_A$ and $\eta_B$ are Alice and Bob's detection efficiencies; $\eta_P$ is the transmissivity of the fiber-propagation link from Alice's source to Bob' terminal; and $p_d$ is the probability of one dark-count occurring in a frame. Note that we are neglecting the possibility of multiple dark counts occurring in a frame, because the product of the frame duration and the dark-count rate (for typical SNSPDs) is much smaller than one.
When either Alice or Bob register more than one detection in a frame, they discard those data and randomly choose an arrival time from a Gaussian distribution whose variance equals their terminal's TFCM entry for arrival time plus the timing-jitter variance \cite{footnote10}. It follows that there are five possibilities for Alice and Bob's joint arrival-time distribution, from which their Shannon information can be found.
\begin{enumerate}
\item Their arrival times are jointly Gaussian random variables with a covariance matrix that is a submatrix of the TFCM augmented to account for their detectors' timing jitters. This case is a postselected frame in which Alice's source emitted one photon-pair and neither Alice nor Bob had a dark count.
\item Their arrival times are independent Gaussian random variables with variances equal to the values from the TFCM plus the variance of their detectors' timing jitters. This case is a postselected frame in which one of two situations occurred: (1) Alice's source emitted multiple photon-pairs and both Alice and Bob registered at least one photon detection; or (2) Alice's source emitted one photon-pair and both Alice and Bob registered photon detections with at least one of them also having a dark count. (Strictly speaking, there could be some correlation between Alice and Bob's arrival times in this case. By neglecting such a possibility, we are underestimating its contribution to their Shannon information.)
\item Alice's arrival time is a Gaussian random variable with variance equal to the value from the TFCM plus the variance of her detector's timing jitter, and Bob's is uniformly distributed over the frame. This case is a postselected frame in which Alice registered at least one photon detection and Bob had a dark count without a photon detection.
\item Alice's arrival time is uniformly distributed over the frame, and Bob's is a Gaussian random variable with variance equal to the value from the TFCM plus the variance of his detector's timing jitter. This case is a postselected frame in which Alice had a dark count without a photon detection and Bob registered at least one photon detection.
\item Both Alice's and Bob's arrival times are uniformly distributed over the frame. This case is a postselected frame in which Alice and Bob both had dark counts and neither had a photon detection.
\end{enumerate}
Given that a particular frame has been postselected, the conditional occurrence probabilities for the preceding five events are
\begin{subequations}
\begin{align}
P_1 &= p_s(1)\eta_A\eta_B\eta_P(1-p_d)^2/p_r\\
P_2 &= \sum_{n = 2}^\infty p_s(n)\left[1-(1-\eta_A)^n\right]\left[1-(1-\eta_B\eta_P)^n\right]/p_r + p_s(1)\eta_A\eta_B\eta_P(2p_d-p_d^2)/p_r\\
P_3 &= \sum_{n = 1}^\infty p_s(n)\left[1-(1-\eta_A)^n\right]\left[p_d(1-\eta_B\eta_P)^n\right]/p_r\\
P_4 &= \sum_{n = 1}^\infty p_s(n)\left[p_d(1-\eta_A)^n\right]\left[1-(1-\eta_B\eta_P)^n\right]/p_r\\
P_5 &= \sum_{n = 0}^\infty p_s(n)p_d^2(1-\eta_A)^n(1-\eta_B\eta_P)^n/p_r.
\end{align}
\end{subequations}
Alice and Bob's Shannon information is given by
\begin{equation}
I(A;B) = \int\!dt_Adt_B\,p_{T_A,T_B}(t_A,t_B)\log_2\left(\frac{p_{T_A,T_B}(t_A,t_B)}{p_{T_A}(t_A)p_{T_B}(t_B)}\right),
\end{equation}
for which we need the joint probability density function, $p_{T_A,T_B}(t_A,t_B)$, for their measured arrival times, $T_A$ and $T_B$, from which the marginal densities, $p_{T_A}(t_A)$ and $p_{T_B}(t_B)$, are easily obtained. The mean values of $T_A$ and $T_B$ are invariant to which of the preceding five postselection events has occurred. Thus they do not affect Alice and Bob's Shannon information, and so we will set them to zero.
The joint density function we need can be found from
\begin{equation}
p_{T_A,T_B}(t_A,t_B) = \sum_{i=1}^5P_i \, p_{T_A,T_B\mid i}(\,t_A,t_B\mid i\,),
\end{equation}
where $p_{T_A,T_B\mid i}(\,t_A,t_B\mid i\,)$ is the joint probability density for $T_A$ and $T_B$ given that event $i$ has occurred. The conditional probability densities for events 2 through 5 are easily found, because $T_A$ and $T_B$ are statistically independent given that one of these events has occurred. We have that
\begin{subequations}
\begin{align}
p_{T_A,T_B\mid 2}(\,t_A,t_B\mid 2\,) &= p_G(t_A; \sigma_A^2)\,p_G(t_B;\sigma_B^2)\\
p_{T_A,T_B\mid 3}(\,t_A,t_B\mid 3\,) &= p_G(t_A;\sigma_A^2)\,p_U(t_B; T_f)\\
p_{T_A,T_B\mid 4}(\,t_A,t_B\mid 4\,) &= p_U(t_A;T_f)\,p_G(t_B;\sigma_B^2)\\
p_{T_A,T_B\mid 2}(\,t_A,t_B\mid 5\,) &= p_U(t_A;T_f)\,p_U(t_B;T_f),
\end{align}
\end{subequations}
where $p_G(t;\sigma^2)$ is a Gaussian probability density with zero mean and variance $\sigma^2$, $p_U(t;T_f)$ is a uniform probability density over the interval of $[-T_f/2,T_f/2]$, and
\begin{subequations}
\begin{align}
\sigma_A^2 &= \langle \Delta \hat{t}_S^2\rangle + (\delta T/2.35)^2\\
\sigma_B^2 &= \langle \Delta \hat{t}_I^2\rangle + (\delta T/2.35)^2.
\end{align}
\end{subequations}
When event~1 has occurred, $T_A$ and $T_B$ are jointly-Gaussian random variables with zero means and covariance matrix
\begin{equation}
\Lambda = \left[\begin{array}{ccc}
\sigma_A^2 & & \langle \Delta \hat{t}_S\Delta \hat{t}_I\rangle \\[.05in]
\langle \Delta \hat{t}_S\Delta \hat{t}_I\rangle && \sigma_B^2\end{array}
\right].
\end{equation}
\subsection{Eve's Holevo information}
Here we describe how Lemmas~1 and 2 permit us to place an upper bound on Eve's Holevo information. Alice and Bob's TFCM has the form given in Eq.~(\ref{postpropTFCM}), with
\begin{subequations}
\label{submatTFCM}
\begin{align}
& \gamma_{SS} =\gamma_{SS}^0\\[.1in]
& \gamma_{SI} = \gamma_{IS} = \left[\begin{array}{ccc}
1-\eta_t & & 0\\[.1in]
0 & & 1-\eta_\omega\end{array}\right]\gamma^0_{SI}\\[.1in]
&\gamma_{II} = \left[\begin{array}{ccc}
1+\epsilon_t & & 0\\[.1in]
0 & & 1+\epsilon_\omega\end{array}\right]\gamma^0_{II},
\end{align}
\end{subequations}
where $\{\eta_t$, $\eta_\omega\}$ quantifies signal-idler correlation loss, and $\{\epsilon_t,\epsilon_\omega\}$ quantifies idler excess noise. By means of Lemmas~1 and 2, the Franson and conjugate-Franson measurements provide upper bounds on the mean-squared arrival-time and frequency differences giving us values for the the excess-noise factors $ \xi_t $ and $ \xi_\omega $. These values do not, however, determine $\{\eta_t,\eta_\omega\}$ and $\{\epsilon_t,\epsilon_\omega\}$. Nevertheless, knowing $ \xi_t $ and $ \xi_\omega $, restricts the set, $\mathcal{M} $, of physically-allowed TFCMs. Our upper bound on Eve's Holevo information is the maximum of her Holevo information over the TFCMs in $\mathcal{M}$.
For a given a TFCM a Gaussian attack maximizes Eve's Holevo information. Thus, we can assume that the Alice, Bob, and Eve share a pure Gaussian state in evaluating that Holevo information. We have that her Holevo information for covariance matrix $ \Gamma $ is
\begin{equation}
\label{eqChiAE}
\chi_\Gamma(A;E) = S(\hat{\rho}_E) - \int\!dt\, p_{T_A}(t_A) S(\hat{\rho}_{E|T_A = t_A}),
\end{equation}
where $S(\hat{\rho}) = - {\rm Tr}[\hat{\rho}\log_2(\hat{\rho})]$ is the von Neumann entropy of the state $\hat{\rho}$. Because Alice, Bob, and Eve's joint quantum state is pure, we have $S(\hat{\rho}_E) = S(\hat{\rho}_{AB})$. Conditioned on Alice's measurement, the quantum state shared by Bob and Eve is also pure, so that $S(\hat{\rho}_{E|T_A = t_A}) = S(\hat{\rho}_{B|T_A = t_A})$. Furthermore, because all these states are Gaussian, the von Neumann entropy of Bob and Eve's conditional quantum state is independent of Alice's measurement result. Thus, we can drop the integral in Eq.~(\ref{eqChiAE}) and get
\begin{equation}
\label{eqChiAE_final}
\chi_\Gamma(A;E) = S(\hat{\rho}_{AB})-S(\hat{\rho}_{B|T_A}).
\end{equation}
To evaluate $\chi_\Gamma(A;E)$ from Eq.~(\ref{eqChiAE_final}), we define
\begin{subequations}
\begin{align}
I_1 &= \langle\Delta\hat{t}_S^2\rangle\langle \Delta\hat{\omega}_S^2\rangle\\
I_2 &= \langle\Delta\hat{t}_I^2\rangle\langle \Delta\hat{\omega}_I^2\rangle\\
I_3 &= \langle\Delta\hat{t}_S\Delta\hat{t}_I\rangle \langle\Delta\hat{\omega}_S\Delta\hat{\omega}_I\rangle \\
I_4 &= (\langle\Delta\hat{t}_S^2\rangle\langle \Delta\hat{t}_I^2\rangle - \langle\Delta\hat{t}_S\Delta\hat{t}_I\rangle^2)(\langle\Delta\hat{\omega}_S^2\rangle\langle \Delta\hat{\omega}_I^2\rangle - \langle\Delta\hat{\omega}_S\Delta\hat{\omega}_I\rangle^2)\\
d_{\pm} &= \frac{1}{\sqrt{2}}\sqrt{I_1+I_2+2I_3\pm\sqrt{(I_1+I_2+2I_3)^2-4I_4}}.
\end{align}
\end{subequations}
We then have that $S(\hat{\rho}_{AB}) = f(d_+)+f(d_-) $, where
\begin{equation}
f(d) = (d+1/2)\log_2(d+1/2)-(d-1/2)\log_2(d-1/2).
\end{equation}
and
\begin{equation}
S(\rho_{B|t}) = f\!\left(\sqrt{\det[\gamma_{I|T_A}]}\right),
\end{equation}
where Bob's conditional covariance matrix is
\begin{equation}
\gamma_{I|T_A} = \left[\begin{array}{ccc}
\langle\Delta\hat{t}_I^2\rangle - \langle\Delta\hat{t}_S\Delta\hat{t}_I\rangle^2/\langle\Delta\hat{t}_S^2\rangle & & 0\\[.1in]
0 & & \langle \Delta\hat{\omega}_I^2\rangle
\end{array}
\right].
\end{equation}
Our upper bound on Eve's Holevo information is now found from
\begin{equation}
\chi^\text{UB}_{\xi_t,\xi_\omega}(A;E) = \sup_{\Gamma\in \mathcal{M} }\{\chi_{\Gamma}(A;E)\}.
\end{equation}
\subsection{Secure-key rate}
For a postselected frame that originated from emission of one photon-pair, Eve's Holevo information about Alice's measurement result is at most $ \chi^\text{UB}_{\xi_t,\xi_\omega}(A;E) $. For a postselected frame that originated from emission of multiple photon-pairs, we grant Eve perfect information about Alice's measurement result. The fraction of the postselected frames that originated from emission of one photon-pair is
\begin{equation}
F = p_s(1)\left[1-(1-\eta_A)(1-p_d)\right]\left[1-(1-\eta_B\eta_P)(1-p_d)\right]/p_r.
\end{equation}
The error-correcting code used during reconciliation to establish $n_R$ shared random bits between Alice and Bob employs, on average, $ n_\text{ECC} $ syndrome bits, where
\begin{equation}
n_R = \beta I(A;B) + n_\text{ECC},
\end{equation}
with $0 < \beta \le 1 $ being the code's reconciliation efficiency \cite{footnote11}. In total, therefore, Eve will have captured at most $ (1-F)n_R+n_\text{ECC}+F\chi^\text{UB}_{\xi_t,\xi_\omega}(A;E) $ bits of information per postselected frame. The secure-key rate, in bits per postselected frame, is the difference between $n_R$ and Eve's total information, hence it satisfies
\begin{equation}
\Delta I(A;B) \ge \beta I(A;B) - (1-F)n_R - F\chi^\text{UB}_{\xi_t,\xi_\omega}(A;E).
\end{equation}
When all postselected frames originated from emissions of one photon-pair, i.e., $F=1$, we recover CVQKD's secure-key rate bound, $ \Delta I(A;B) \ge \beta I(A;B) - \chi^{UB}_{\xi_t,\xi_\omega}(A;E)$, showing that, as expected, the emission of multiple photon-pairs reduces the secure-key rate of time-energy entanglement based HDQKD. The secure-key rate in bits per second for our time-energy entanglement based HDQKD protocol is the product of the frame postselection rate and the secure-key rate in bits per postselected frame, viz.,
\begin{equation}
\text{SKR} \geq \frac{q p_r}{3T_f}\left[\beta I(A;B)-(1-F)n_R- F \chi^\text{UB}_{\xi_t,\xi_\omega}(A;E)\right],
\end{equation}
where we have accounted for frames occurring once every $3T_f$ seconds.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,202
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\section{Introduction}\label{s1}
\noindent
Symmetry renders purely hard scattering processes in QED and gravity impossible \cite{StromingerLectures,StromingerIRrevisited,StromingerBMS,HMPS,LPS,KMS,Banks1,SZ,Weinberg}. Rather the asymptotic particles must be accompanied by infinite clouds of soft photons or gravitons, in addition to the soft radiation emitted during the process. The hard and soft particles are highly correlated. As the resolution of particle detectors is limited, an infinite number of soft particles evades detection in a typical experiment. It is therefore important to understand the nature of the entanglement between the hard and soft degrees of freedom in the final state, and to quantify the information carried by the soft particles. A measure of this information is provided by the entanglement entropy. In \cite{StromingerBHinfo} it was argued that soft quanta emitted during the process of formation/evaporation of a four-dimensional black hole could play an important role in the resolution of the black hole information paradox. See also \cite{Banks2} for related discussions as well as extensions to higher dimensions.
Indeed as shown in \cite{Carney1,Carney2,Carney3}, tracing over the soft particles in the final state can lead to decoherence, revealing strong entanglement between the hard and soft degrees of freedom. See also \cite{Gomez1, Gomez2}. In this paper we consider typical scattering processes in QED in order to study the reduced density matrix and calculate the entanglement entropy perturbatively. We focus on the example of electron-electron scattering to illustrate our results. To regulate the entanglement entropy, we discretize the system by putting the process in a large box of size $L$ and impose an infrared cutoff $\lambda$ of order $1/L$. At the end of the calculation, we take the continuum, $\lambda \to 0$ limit. We would like to investigate if infrared divergences in the entanglement entropy appear, and whether they cancel order by order in perturbation theory. We discuss both a Fock basis computation where we take a state of two bare electrons for the initial state and a proper asymptotic state where the electrons are ``dressed'' with infinite clouds of soft photons, in accordance with the Faddeev-Kulish construction \cite{FK,Chung}. Other pertinent work on entanglement after scattering includes \cite{Bala,Seki,Seki2,Grignani,Petruccione,Calucci,Asorey}.
First we trace over the entire soft part of the Hilbert space, comprised of photon states with total energy less than an infrared energy scale $E$ (smaller than the mass of the electron). This energy scale is set by the sensitivity of the detector. The reduced density matrix is an operator acting on the hard part of the Hilbert space, and (to all orders in perturbation theory) it exhibits decoherence in the continuum limit \cite{Carney2,Carney3}. We find logarithmic infrared divergences in the perturbative expansion for the entanglement entropy, for both the dressed and the Fock basis computations. In both cases and to leading order in perturbation theory, the entanglement entropy is proportional to the conventional Fock basis rate for the two initial electrons to scatter and emit at the same time a single soft photon with frequency in the range $\lambda<\omega_\gamma <E$. This rate diverges logarithmically in the continuum, $\lambda \to 0$ limit at tree level. For the Fock basis calculation the infrared divergence can be attributed to the soft part of the emitted radiation. For the case of Faddeev-Kulish electrons, the divergence can be traced in the overlap of the coherent states describing the soft photon clouds dressing the final state charged particles. Despite the fact that the Faddeev-Kulish $S$-matrix is infrared finite order by order in perturbation theory \cite{FK, Chung}, the dressing does not alleviate logarithmic divergences in the entanglement entropy at the perturbative level. In fact the leading perturbative entanglement entropy is a fraction of the maximal possible value, as set by the dimensionality of the subspace of single soft photon states. We argue that the structure of the singular part is universal, applicable to generic scattering processes, and show that the coefficient of the IR logarithmic singularity is related to the cusp anomalous dimension in QED. We also argue that infrared logarithmic divergences in the entanglement entropy (per particle flux per unit time) persist to all orders in the infinite volume limit.
On the other hand, the Faddeev-Kulish cross-section for the emission of soft photons of energy less than $E_d$, the scale characterizing the photons in the clouds, is suppressed (and likewise for gravitons) \cite{Choi2}. Thus we may distinguish between soft cloud photons and radiated ones in the final state. These observations motivate us to consider a second type of partial trace, over soft photons with frequencies in the range $E_d < \omega_\gamma < E$, comprising in the dressed case the soft part of the emitted radiation. This type of tracing was also advocated in \cite{Gomez1,Gomez2} in order to alleviate the amount of decoherence in the continuum limit. The reduced density matrix is now an operator acting on the space of asymptotic states. Both the diagonal and off diagonal elements are given in terms of Faddeev-Kulish amplitudes, which are free of any infrared divergences in the continuum $\lambda \to 0$ limit (order by order in perturbation theory). The entanglement entropy is finite order by order in perturbation theory. The leading entanglement entropy can be expressed in terms of the Fock basis rate for the emission of a single soft photon with frequency in the range $E_d<\omega_\gamma <E$. When the IR scales $E_d$ and $E$ are sufficiently small, this rate is proportional to the logarithm of the ratio of the infrared scales $E/E_d$, which remains finite in the $\lambda \to 0$ limit. The perturbative analysis is now valid and can be trusted in the continuum limit. In effect the dressing provides an infrared cutoff of order the cloud energy scale $E_d$, curing the singular behavior associated with the previous tracing. As $E \to E_d$, the entanglement entropy becomes very small, and therefore we conclude that a small amount of information is carried by the extra soft radiated photons. These results are consistent with estimates of the amount of decoherence obtained in \cite{Gomez2}.
The plan of the paper is as follows. In \ref{s2} we describe the various infrared energy scales and decompose the Hilbert space into soft and hard factors. We then review the Faddeev-Kulish construction of asymptotic states in QED and exhibit the finiteness of the $S$ matrix. The reader familiar with this construction may omit most material in this section. In \ref{s3} we describe the discretization of the system by replacing infinite space with a large box of finite size and impose an infrared cutoff. We construct dressed states for the discrete system reproducing the Faddeev-Kulish states in the continuum limit. We also explain how to compute various partial traces, which will be useful for the following calculations. In \ref{s4} we consider a two electron scattering process and construct the reduced density matrices after taking the partial tracings over the final state, as outlined above. Keeping the infrared cutoff $\lambda$ finite, we compute the Renyi entropies (for integer $m$) and the entanglement entropy to leading order in perturbation theory. For the dressed case, restricting the trace over the soft part of the emitted radiation, yields a finite entanglement entropy (per unit flux per unit time), free of any infrared divergences in the continuum, $\lambda \to 0$ limit. We summarize our results and discuss implications and open problems in \ref{s5}.
\section{QED scattering, soft photons and entanglement}\label{s2}
\noindent
Scattering processes in QED are constrained by an infinite set of conservation laws associated with large gauge transformations (LGT) \cite{StromingerLectures,StromingerIRrevisited,HMPS,LPS,KMS,Campiglia}. These are transformations that do not vanish at infinity, but instead approach angle dependent constants. Transitions between conventional Fock states, where only a finite number of photons are present in the initial and final states, fail to satisfy the conservation laws associated with LGT. As a result the corresponding $S$-matrix elements vanish \cite{StromingerIRrevisited}. An infinite number of soft photons must be present in the final state. The vanishing of the Fock basis transition amplitudes is more commonly attributed to the exponentiation of virtual infrared divergences, see e.g. \cite{Weinberg}, but it can also be understood as a consequence of symmetry.
On the other hand, the conventional Fock basis states do not diagonalize the asymptotic Hamiltonian, which includes the slowly decaying parts of the interaction Hamiltonian (written in the interaction picture). As shown by Faddeev and Kulish, physical asymptotic states can be constructed by dressing the Fock charged particle states with clouds of soft photons \cite{FK}. The $S$-matrix elements between these dressed states are nonvanishing and free of infrared divergences \cite{FK, Chung, Kibble1, Kibble2}.
See e.g. \cite{StromingerLectures,StromingerIRrevisited,Gomez2,Sever,Porrati} for recent discussions and reviews. The soft photon clouds render the LGT charges of Faddeev-Kulish (FK) states independent of the momenta of the bare charged particles \cite{Sever}. The LGT charges depend only on the net electric charge of the bare particles (and the angle dependent constants at infinity), and so the conservation laws can be trivially satisfied \cite{StromingerIRrevisited,Sever}.
Thus any scattering process in QED inevitably leads to a final state with an infinite number of soft photons. Our goal is to study the entanglement of the hard particles with the soft photons produced in a typical process such as electron-electron scattering, and calculate the entanglement entropy perturbatively. Even though we focus on a particular process, we expect the main conclusions to be applicable to other (perturbative) scattering processes in quantum electrodynamics as well as gravity.
Depending on the sensitivity of the detector, we impose an energy cutoff $E < m_e$, where $m_e$ is the electron mass, in terms of which we decompose the {\it incoming} and {\it outgoing} Hilbert spaces into hard and soft factors
\begin{equation}
{\cal{H}}= {\cal{H}}_H \times {\cal{H}}_S
\end{equation}
where ${\cal{H}}_H$ comprises hard electron, positron and photon states with energy greater than $E$, and ${\cal{H}}_S$ of soft photon states with total energy less than $E$. The initial state is taken to be a two-electron dressed FK state, but we also discuss and compare with perturbative Fock basis computations. The final state will be an entangled state in ${\cal{H}}_H \times {\cal{H}}_S$, as determined by the $S$-matrix. By restricting the incoming energy, we may exclude the possibility of having more than two charged particles in the final state. In addition, we shall distinguish between photons produced as a result of radiation and photons present in the clouds accompanying the outgoing charged particles.
Note that apart from the infrared reference scale $E$ used to decompose the Hilbert space into soft and hard factors, we also have the following infrared energy scales: i)$\lambda$ is the infrared cutoff scale, eventually to be taken to zero. Any logarithmic IR divergences in physical quantities will be displayed as powers of $\log\lambda$. ii) $E_d$ characterizes the energy of the soft photons present in the clouds accompanying the incoming and outgoing charged particles. We set $\lambda < E_d < E$, taking $E_d$ to be sufficiently small so that the leading soft photon theorems can be applied to simplify various dressed amplitudes (see below). iii) $\Lambda$ (which can be taken to be of order $E_d$) is an infrared scale characterizing soft virtual photons. Eventually we take the limit $\lambda \to 0$, keeping the ratios $E_d/E$ and $E/\Lambda$ fixed. We would like to investigate the behavior of the entanglement entropy as the reference scale $E$ approaches the lower infrared scales $E_d$ and $\lambda$, in perturbation theory, as well as the $\lambda \to 0$ limit.
In the rest of this section we review some properties of the FK construction, which will be useful for the entanglement entropy computations among the soft and hard particles. Throughout we work in the Lorenz gauge. For notation and conventions, see \ref{A1}.
\subsection{Faddeev-Kulish states}
\noindent
The FK dressing is effected via the action of $e^{R_f}$,
where
\begin{equation}
R_f=\int~\frac{d^3\vec{p}}{(2\pi)^3}~\hat{\rho}(\vec{p})~\int_\lambda^{E_d}\frac{d^3\vec{k}}{(2\pi)^3}~\frac{1}{(2\omega_{\vec{k}})^{1/2}}~\left(f(\vec{k},\,\vec{p})\cdot a^{\dagger}(\vec{k}) - h.c.\right) \label{FKdressing}
\end{equation}
and
\begin{equation}
\hat{\rho}(\vec{p})=\sum_s b^{s\, \dagger}(\vec{p})b^s(\vec{p}) - d^{s\, \dagger}(\vec{p})d^s(\vec{p})
\end{equation}
is the charge density operator; $b^{s\, \dagger}(\vec{p})$ and $d^{s\, \dagger}(\vec{p})$ are electron and positron creation operators, respectively -- $\vec{p}$ is the momentum and $s$ the spin polarization; $a_r^{\dagger}(\vec{k})$ create photons with momentum $\vec{k}$ and polarization vector $\epsilon_r^{\mu}(\vec{k})$, $r=0,\dots,3$,
and
\begin{equation}
f(\vec{k},\,\vec{p})\cdot a^{\dagger}(\vec{k})=\sum_r f^{\mu}(\vec{k},\,\vec{p})\epsilon^*_{r\mu}(\vec{k})a^{\dagger}_r(\vec{k}) \label{f1}
\end{equation}
with
\begin{equation}
f^{\mu}(\vec{k},\,\vec{p})= e\left(\frac{p^{\mu}}{pk} - c^{\mu}\right)e^{-ipk\, t_0/p^0},\,\,\, c^{\mu}=\left(-\frac{1}{2k^0},\,\frac{\vec{k}}{2(k^0)^2}\right) \label{f2}
\end{equation}
The FK operator is unitary. Notice that the dressing function $f^{\mu}(\vec{k},\,\vec{p})$ is singular as the photon momentum $\vec{k}$ vanishes. We will carry all computations keeping the infrared cutoff scale $\lambda$ finite, taking the $\lambda \to 0$ limit at the end. Here also, $t_0$ is a time reference scale and $c^{\mu}$ is a null vector, $c^2=0$, satisfying $ck=1$. Because of the latter property, the function $f^{\mu}(\vec{k},\,\vec{p})$ is transverse, $fk=0$. So only allowable admixtures of timelike and longitudinal photons are present, in accordance with the Lorenz gauge condition. In particular, these do not contribute to the $S$-matrix elements (as well as to expectation values of gauge invariant quantities)\footnote{Dressed states satisfy the Gupta-Bleuler condition: $\left[a_0(\vec{k})-a_3(\vec{k})\right]\ket{\Psi}=0$.}, and thus we may also restrict the sum in \ref{f1} to transversely polarized photons. The limits of integration in \ref{FKdressing} insure that only soft photons, with energies below the infrared reference scale $E_d$, are present in the cloud. As the integrand is dominated by low momenta, taking $E_d < 1/t_0$, we may approximate the phase $e^{-ipk\, t_0/p^0}$ in \ref{f2} with unity.
\subsection{Dressed electron}
\noindent
For example, consider a bare single electron particle state
\begin{equation}
|\vec{p},\, s \rangle= \sqrt{2 E_{\vec{p}}}\, b^{s\, \dagger}(\vec{p})|0\rangle
\end{equation}
The corresponding dressed state takes a product form
\begin{equation}
|\vec{p},\,s\rangle_{\rm dressed}= |\vec{p},\,s\rangle\times e^{\int_\lambda^{E_d}\frac{d^3\vec{k}}{(2\pi)^3}~\frac{1}{(2\omega_{\vec{k}})^{1/2}}~\left(f(\vec{k},\,\vec{p})\cdot a^{\dagger}(\vec{k}) - h.c.\right)}|0\rangle\label{coherent1}
\end{equation}
Thus the charged particle is accompanied by a photon cloud described by a normalized coherent state. For finite nonzero $\lambda$, the coherent state can also be written in the following useful form
\begin{equation}
|f_{\vec{p}}\rangle = {\cal{N}}_{\vec{p}}~\,e^{\int_\lambda^{E_d}\frac{d^3\vec{k}}{(2\pi)^3}~\frac{1}{(2\omega_{\vec{k}})^{1/2}}~f(\vec{k},\,\vec{p})\cdot a^{\dagger}(\vec{k})}|0\rangle \label{coherent2}
\end{equation}
The normalization factor ${\cal{N}}_{\vec{p}}$ is given by
\begin{equation}
{\cal{N}}_{\vec{p}}=e^{-\frac{1}{2} \int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f^{\mu}(\vec{q},\,\vec{p})f^{*}_{\mu}(\vec{q},\,\vec{p}) } \label{normalization}
\end{equation}
The exponent can easily be computed
\begin{equation}
\frac{1}{2} \int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f^{\mu}(\vec{q},\,\vec{p})f^{*}_{\mu}(\vec{q},\,\vec{p}) = \frac{e^2}{8\pi^2}~\ln{\left(\frac{E_d}{\lambda}\right)}~I(v)
\end{equation}
where $v= |\vec{p}|/p^0$ is the velocity of the electron and
\begin{equation}
I(v) =-2~+~v^{-1}\ln \left(\frac{1+ v}{1-v}\right) \label{kinematical1}
\end{equation}
is a non-negative kinematical factor. In particular, for small $v$, $I(v)=2v^2/3+\dots$\, . As $v \to 1$, $I(v)$ grows logarithmically. Setting
\begin{equation}
{\cal{A}}_{\vec{p}} = \frac{e^2}{8\pi^2}~I(v)
\end{equation}
we get
\begin{equation}
{\cal{N}}_{\vec{p}}=\left(\frac{\lambda}{E_d}\right)^{{\cal{A}}_{\vec{p}}}
\end{equation}
and so ${\cal{N}}_{\vec{p}}$ vanishes in the limit $\lambda \to 0$ (in which case \ref{coherent2} cannot be used).
Let us compute the number of photons in this state. Using standard coherent state algebra, this is given by
\begin{equation}
\langle f_{\vec{p}}| N_{ph} |f_{\vec{p}}\rangle = \int_{\lambda}^{E_d} \frac{d^3 \vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~ f^{\mu}(\vec{q},\, \vec{p})f^*_{\mu}(\vec{q},\, \vec{p})=\frac{e^2}{4\pi^2} \ln{\left(\frac{E_d}{\lambda}\right)}~I(v)
\end{equation}
Thus the cloud contains an infinite number of soft photons in the limit $\lambda \to 0$. On the other hand, the energy of the state is given by
\begin{equation}
\langle f_{\vec{p}}| H_{ph} |f_{\vec{p}}\rangle = \frac{1}{2}~ \int_{\lambda}^{E_d} \frac{d^3 \vec{q}}{(2\pi)^3}~ f^{\mu}(\vec{q},\, \vec{p})f^*_{\mu}(\vec{q},\, \vec{p})=\frac{e^2}{4\pi^2}~I(v)~(E_d-\lambda)
\end{equation}
For generic values of the electron velocity, this is a small fraction of the infrared scale $E_d$. Therefore, the coherent cloud of photons is in the soft part of the Hilbert space ${\cal{H}}_S$.
The mean value of the cloud momentum is also interesting. It is given by
\begin{equation}
\langle f_{\vec{p}}| \vec{P}_{ph} |f_{\vec{p}}\rangle = \int_{\lambda}^{E_d} \frac{d^3 \vec{q}}{(2\pi)^3}~\frac{\vec{q}}{2\omega_{\vec{q}}}~ f^{\mu}(\vec{q},\, \vec{p})f^*_{\mu}(\vec{q},\, \vec{p})=\frac{e^2}{8\pi^2}~(E_d-\lambda)~\left[\frac{3}{v}~I(v)-v~I(v)-2~v\right]~\hat{p}
\end{equation}
As the electron velocity approaches the speed of light, the energy and the magnitude of the cloud momentum grow logarithmically and become equal. Notice that both the energy and the momentum remain appreciably much smaller than the energy and the momentum of the electron. As $v\to 0$, they become vanishingly small, albeit the momentum approaches zero faster.
Finally let us compute the electromagnetic field associated with the cloud.
The expectation value of the gauge potential in the coherent state is
\begin{equation}
\bar{A}_\mu(x)=\langle f_{\vec{p}}| A_{\mu}(x) |f_{\vec{p}}\rangle = \int_{\lambda}^{E_d} \frac{d^3 \vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~\left(f_{\mu}(\vec{q},\, \vec{p})~e^{iqx}~+~f^*_{\mu}(\vec{q},\, \vec{p})~e^{-iqx}\right)
\end{equation}
As noted in \cite{Sotaro}, the terms in $\bar{A}_\mu(t_0,\,\vec{x})$ that are independent of the the null vector $c^{\mu}$ reproduce asymptotically the Lienard Wiechert gauge potential associated with the moving electron.
The electric field is
\begin{equation}
\vec{E}= \int_{\lambda}^{E_d} \frac{d^3 \vec{q}}{(2\pi)^3}~\frac{ie}{2\omega_{\vec{q}}}~\left(\frac{\hat{q}-\vec{v}}{1-\hat{q}\cdot\vec{v}}~+~\vec{v}~-~(1+\hat{q}\cdot\vec{v})~\frac{\hat{q}}{2}\right)~e^{iq(x-pt_0/p^0)}~+~h.c.
\end{equation}
The last term in the parentheses is the contribution of the null vector $c^{\mu}$.
For the case of multielectron/positron states, $\alpha=\{e_i,\, \vec{p}_i,\, s_i\}$, the resulting coherent state $|f_\alpha\rangle$ can be obtained if we replace the function $f^{\mu}(\vec{k},\,\vec{p})$ in expressions
\ref{coherent1} and \ref{coherent2} with
\begin{equation}
f_\alpha^{\mu}(\vec{k})=\sum_{i \in \alpha}~e_i~\left(\frac{p_i^{\mu}}{p_ik} - c^{\mu}\right)~e^{-ip_ik\, t_0/p_i^0}
\end{equation}
where $e_i$ is the charge and $p_i$ is the momentum of the $i$th particle. When $t_0=0$, the second term is equal to $Q_\alpha c^{\mu}$, where $Q_\alpha$ is the total charge of $\alpha$. In particular, the terms proportional to $c^{\mu}$ vanish for states with zero net charge. For simplicity, we choose to set the phases $e^{-ip_ik\, t_0/p_i^0}$ to unity for the following calculations \cite{Gomez2,Chung}.
In \ref{A2} we compute the normalization factor ${\cal{N}}_\alpha$ for the photon coherent state associated with the state $\alpha$. We also compute the overlap between coherent photon states, corresponding to generic charged states $\alpha=\{e_i,\, \vec{p}_i,\,s_i\}$ and $\beta=\{e_i^{\prime}, \,\vec{p}_i^{\,\prime},\,s_i^{\,\prime}\}$. Let us call the $\beta$ particles outgoing and the $\alpha$ particles incoming, and define $\eta_i$ to be $+1$ for all outgoing particles and $-1$ for all incoming particles. Then for the cases of interest $Q_\alpha=Q_\beta$ and to all orders in the electron charge, we find
\begin{equation}
\langle f_\beta|f_\alpha\rangle = \left(\frac{\lambda}{E_d}\right)^{{\cal{B}}_{\beta\alpha}}\label{braketf}
\end{equation}
where
\begin{equation}
{\cal{B}}_{\beta\alpha}=-\frac{1}{16\pi^2}~\sum_{ij}~\eta_i\,\eta_j\,e_i\,e_j~v_{ij}^{-1}~ \ln \left(\frac{1+ v_{ij}}{1-v_{ij}}\right)\label{B}
\end{equation}
and
\begin{equation}
v_{ij}=\left[1-\frac{m_i^2\,m_j^2}{(p_i\cdot p_j)^2}\right]^{1/2}
\end{equation}
is (the magnitude of) the relative velocity of particle $j$ with respect to $i$.
The sums are over all outgoing and incoming particles. When the momenta of the multicharged particle states $\beta$ and $\alpha$ differ, ${\cal{B}}_{\beta\alpha}$ is nonzero and positive \cite{Weinberg}. Then to all orders in the electron charge, the overlap $\langle f_\beta|f_\alpha\rangle$ vanishes in the $\lambda \to 0$ limit.
\subsection{The Faddeev-Kulish $S$-matrix}
\noindent
Next consider a scattering process $\alpha \to \beta$. We first consider cases for which there are no soft photons with energy less than $E_d$ in the initial and final states (beyond the ones specified by the dressing operator). We compute the $S$-matrix element between the incoming/outgoing dressed states, following \cite{Chung}:
\begin{equation}
{\tilde S}_{\beta\alpha}=_{d}\langle\beta|S|\alpha\rangle_d
\end{equation}
We also write
\begin{equation}
S_{\beta\alpha}=\langle\beta|S|\alpha\rangle
\end{equation}
for the $S$-matrix element between the corresponding undressed states. Expanding the exponential operators of the coherent photon states, we obtain
\begin{equation}
{\tilde S}_{\beta\alpha}=
{\cal{N}}_\beta~{\cal{N}}_\alpha
\sum_{m,\,n\,=0}^\infty~\frac{1}{m!\,n!}~\langle\beta|~\prod_{l=1}^m~\int_{\lambda}^{E_d} \frac{d^3\vec{q}_l}{(2\pi)^3}~\frac{f^*_\beta(\vec{q}_l)\cdot a(\vec{q}_l)}{(2\omega_{\vec{q}_l})^{1/2}}~S~\prod_{s=1}^n~\int_{\lambda}^{E_d} \frac{d^3\vec{k}_s}{(2\pi)^3}~\frac{f_\alpha(\vec{k}_s)\cdot a^{\dagger}(\vec{k}_s)}{(2\omega_{\vec{k}_s})^{1/2}}~|\alpha\rangle \label{Sdressed}
\end{equation}
Each term in \ref{Sdressed} is given in terms of scattering amplitudes with $n$ incoming soft photons and $m$ outgoing soft photons. These amplitudes are weighted by $1/m!\,n!$. It is always possible that a number $l$, $0\leq l \leq \rm{min}(m,\,n)$, of these soft photons do not interact with the electrons. Then $n^\prime=n-l$ soft photons are absorbed by an external electron line, and $m^{\prime}=m-l$ are emitted by an external electron line.
The $l$ noninteracting soft photons contribute a factor given by
\begin{equation}
l! ~ \left(\int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f_\alpha^{\mu}(\vec{q})f^{*}_{\beta\,\mu}(\vec{q})\right)^l
\end{equation}
Notice that the sum over photon polarizations -- we restrict the sum over transversely polarized photons ($r=1,2$) -- yields
\begin{equation}
\sum_r \epsilon_{r\mu}(\vec{q})\epsilon^*_{r\nu}(\vec{q})=\eta_{\mu\nu}-q_{\mu}c_{\nu}-q_{\nu}c_{\mu}
\end{equation}
and we have used the fact that the dressing functions are transverse $f^*_{\beta}q=f_{\alpha}q=0$.
Letting the energy scale $E_d$ to be sufficiently small, we can
obtain the contributions of the $n^\prime$ and $m^\prime$ interacting soft photons by using the following leading soft theorems \cite{Bloch,Low1,GellMann,Low2,Yennie,BK,Weinberg}
\begin{equation}
\lim_{|\vec{q}|\to 0}~(2\omega_{\vec{q}})^{1/2}~\langle\beta|a_r(\vec{q})~S~|\alpha\rangle=\left(\sum_{i\in\beta}~ \frac{e_i\,p_i \cdot \epsilon_r^*(\vec{q})}{p_i \cdot q}~-~\sum_{i\in \alpha}~ \frac{e_i\, p_i \cdot \epsilon_r^*(\vec{q})}{p_i \cdot q}\right)~\langle\beta|~S~|\alpha\rangle
\end{equation}
and (by CPT invariance)
\begin{equation}
\lim_{|\vec{k}|\to 0}~(2\omega_{\vec{k}})^{1/2}~\langle\beta|~S~a^{\dagger}_r(\vec{k})~|\alpha\rangle=-\left(\sum_{i\in\beta}~ \frac{e_i\,p_i \cdot \epsilon_r(\vec{k})}{p_i \cdot k}~-~\sum_{i\in \alpha}~ \frac{e_i\, p_i \cdot \epsilon_r(\vec{k})}{p_i \cdot k}\right)~\langle\beta|~S~|\alpha\rangle
\end{equation}
Then $\tilde{S}_{\beta\alpha}$ can be expressed as a sum over all possible $(l, m^{\prime}, n^{\prime})$ configurations, after taking into account all weight factors, including the fact that there are $(n^{\prime}+l)!/n^\prime!\,l!$ ways to choose $l$ photons from the initial $n$ soft photons, and likewise $(m^{\prime}+l)!/m^\prime!\,l!$ ways to choose $l$ photons from the final $m$ soft photons. In all we get
$$
{\tilde S}_{\beta\alpha}=
{{\cal{N}}_\beta~{\cal{N}}_\alpha}
~\sum_{l,\,m^{\prime},\,n^{\prime}\,=\,0}^\infty ~\frac{1}{(m^\prime +l)!\,(n^{\prime}+l)!}~\frac{(m^{\prime}+l)!}{m^\prime!\,l!}~\frac{(n^{\prime}+l)!}{n^\prime!\,l!}~l! ~ \left(\int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f_\alpha^{\mu}(\vec{q})f^{*}_{\beta\,\mu}(\vec{q})\right)^l
$$
$$
\times~\left[\int_{\lambda}^{E_d} \frac{d^3\vec{q}}{(2\pi)^32\omega_{\vec{q}}}~\sum_{i\in{\{\beta,\,\alpha\}}}~\eta_i\,e_i~\left(\frac{f^*_\beta(\vec{q})\cdot p_i}{p_i\cdot q}-f^*_\beta(\vec{q})\cdot c\right)\right]^{m^{\prime}}
$$
\begin{equation}
\times~\left[-\int_{\lambda}^{E_d} \frac{d^3\vec{k}}{(2\pi)^32\omega_{\vec{k}}}~\sum_{i\in{\{\beta,\,\alpha\}}}~\eta_i\,e_i~\left(\frac{f_\alpha(\vec{k})\cdot p_i}{p_i\cdot k}-f_\alpha(\vec{k})\cdot c\right)\right]^{n^{\prime}}~S_{\beta\alpha}
\end{equation}
The last two lines are the contributions of the $n^\prime$ and $m^\prime$ interacting soft photons. It is easy to see that the terms proportional to $c$ vanish by charge conservation ($Q_\alpha=Q_\beta$). After canceling combinatorial factors, it is easy to see that all three series exponentiate to give
\begin{equation}
{\tilde S}_{\beta\alpha}=~
{{\cal{N}}_\beta~{\cal{N}}_\alpha}~e^{\int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f_\alpha^{\mu}(\vec{q})f^{*}_{\beta\,\mu}(\vec{q})}~
~e^{\int_{\lambda}^{E_d} \frac{d^3\vec{k}}{(2\pi)^32\omega_{\vec{k}}}\sum_{ij}~\eta_i\,\eta_j\,e_i\,e_j~\frac{p_i~p_j}{(p_ik)~(p_jk)}}~S_{\beta\alpha}
\end{equation}
The first three factors combine to produce $\langle f_\beta|f_\alpha\rangle$ given by \ref{braketf}. In the second exponential, the $ij$ sums are over all outgoing and incoming particles. The exponent is given by
\begin{equation}
\int_{\lambda}^{E_d} \frac{d^3\vec{k}}{(2\pi)^32\omega_{\vec{k}}}\sum_{ij}~\eta_i\,\eta_j\,e_i\,e_j~\frac{p_j~p_i}{(p_jk)~(p_ik)}=\ln\left(\frac{E_d}{\lambda}\right)~(2{\cal{B}}_{\beta\alpha})
\end{equation}
where ${\cal{B}}_{\beta\alpha}$ is the positive kinematical factor given by \ref{B}. Therefore
\begin{equation}
{\tilde S}_{\beta\alpha}=
~\langle f_\beta|f_\alpha\rangle~
~\left(\frac{E_d}{\lambda}\right)^{2{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}
\end{equation}
On the other hand, as shown in e.g. \cite{Weinberg}, exponentiation of virtual infrared divergences gives
\begin{equation}
S_{\beta\alpha}=\left(\frac{\lambda}{\Lambda}\right)^{{\cal{B}}_{\beta\alpha}}~e^{i\phi_{\beta\alpha}}~S^{(\Lambda)}_{\beta\alpha}
\end{equation}
with $S^{(\Lambda)}_{\beta\alpha}$ the usual $S$-matrix amplitude without virtual soft photons with momentum below the infrared scale $\Lambda$. The phase $\phi_{\alpha\beta}$ is real \cite{Weinberg} and does not contribute to the square of the amplitudes or the corresponding rates. So
\begin{equation}
{\tilde S}_{\beta\alpha}=~\left(\frac{E_d}{\Lambda}\right)^{{\cal{B}}_{\beta\alpha}}~e^{i\phi_{\beta\alpha}}~S^{(\Lambda)}_{\beta\alpha}
\end{equation}
is finite (generically nonzero and free of infrared divergences). In the limit $\lambda \to 0$, we keep the ratio $E_d/\Lambda$ finite. (We may also choose to set $\Lambda=E_d$).
\subsubsection{Single soft photon production}
Now let us add a single soft photon $\gamma$, of momentum $\vec{q}_\gamma$ and polarization vector $\epsilon_{r\mu}(\vec{q}_\gamma)$ ($|\vec{q}_\gamma|<E_d$), in the final state:
\begin{equation}
{\tilde S}_{\beta\gamma,\,\alpha}=_{d}\langle\beta\gamma|S|\alpha\rangle_d
\end{equation}
The case $|\vec{q}_\gamma|>E_d$ is covered by the previous analysis. Such amplitudes in QED and gravity were recently studied in \cite{Choi1,Choi2}.
To calculate the $S$-matrix element, we first note that
\begin{equation}
|\beta\gamma\rangle_d = \left(|\beta\gamma\rangle~-~f_\beta^{*\,\mu}(\vec{q}_\gamma)\epsilon_{r\mu}(\vec{q}_\gamma)~|\beta\rangle\right)\times|f_\beta\rangle
\end{equation}
as obtained by acting with the FK operator $e^{R_f}$ on the undressed state $|\beta\gamma\rangle=a_r^\dagger(\vec{q}_\gamma)\ket{\beta}$. Notice that the trivial part of the $S$-matrix element, given by the overlap $\,_{d}\langle\beta\gamma|\alpha\rangle_d$, vanishes:
\begin{equation}
\,_{d}\langle\beta\gamma|\alpha\rangle_d=\left(f_\alpha(\vec{q}_\gamma)-f_\beta(\vec{q}_\gamma)\right)\cdot\epsilon_{r}^*(\vec{q}_\gamma)~\langle f_\beta|f_\alpha\rangle~\langle\beta|\alpha\rangle=\left(f_\alpha(\vec{q}_\gamma)-f_\alpha(\vec{q}_\gamma)\right)\cdot\epsilon_{r}^*(\vec{q}_\gamma)=0
\end{equation}
(since $\langle\beta|\alpha\rangle=\delta_{\beta\alpha}$). So two states which differ by an extra photon with energy less than $E_d$ (beyond the ones in the dressing) are orthogonal, and so distinguishable. As a result, only the nontrivial part of the $S$-matrix contributes to this matrix element.
In all, ${\tilde S}_{\beta\gamma,\,\alpha}$ can be written as a sum of two parts
\begin{equation}
{\tilde S}_{\beta\gamma,\,\alpha}~=~{\tilde S}_{\beta\gamma,\,\alpha}^{(1)}~+~{\tilde S}_{\beta\gamma,\,\alpha}^{(2)}
\end{equation}
where
\begin{equation}
{\tilde S}_{\beta\gamma,\,\alpha}^{(1)}~=~-f_\beta(\vec{q}_\gamma)\cdot\epsilon_{r}^*(\vec{q}_\gamma)~\tilde{S}_{\beta\alpha}~=~-f_\beta(\vec{q}_\gamma)\cdot\epsilon_{r}^*(\vec{q}_\gamma)~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}
\end{equation}
and
$$
{\tilde S}_{\beta\gamma,\,\alpha}^{(2)}~=~{\cal{N}}_\beta~{\cal{N}}_\alpha~\sum_{m,\,n\,=0}^\infty~\frac{(2\omega_\gamma)^{1/2}}{m!\,n!}
$$
\begin{equation}
\times~\langle\beta|~a_r(\vec{q}_\gamma)~\prod_{l=1}^m~\int_{\lambda}^{E_d} \frac{d^3\vec{q}_l}{(2\pi)^3}~\frac{f^*_\beta(\vec{q}_l)\cdot a(\vec{q}_l)}{(2\omega_{\vec{q}_l})^{1/2}}~S~\prod_{s=1}^n~\int_{\lambda}^{E_d} \frac{d^3\vec{k}_s}{(2\pi)^3}~\frac{f_\alpha(\vec{k}_s)\cdot a^{\dagger}(\vec{k}_s)}{(2\omega_{\vec{k}_s})^{1/2}}~|\alpha\rangle
\end{equation}
For the second part, we note that there are two contributions, depending on whether the extra outgoing soft photon (annihilated by $a_r(\vec{q}_\gamma)$) is interacting. {\it Feynman diagrams in which this extra soft photon is joined to an external electron line yield a net contribution}
\begin{equation}
S^{(2)}_1=\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\gamma,\,\alpha}=\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~\left(\sum_{i\in\beta}~ \frac{e_i\,p_i \cdot \epsilon_r^*(\vec{q}_\gamma)}{p_i \cdot q_\gamma}~-~\sum_{i\in \alpha}~ \frac{e_i\, p_i \cdot \epsilon_r^*(\vec{q}_\gamma)}{p_i \cdot q_\gamma}\right)~+~\dots
\end{equation}
We have used the soft theorem for sufficiently small $|\vec{q}_\gamma|$. The ellipses stand for smooth, nonsingular terms in the limits $\lambda,\,|\vec{q}_\gamma|\to 0$ \footnote{The subleading ${\cal{O}}(\omega_\gamma^0)$ terms obey a universal relation \cite{Low1,GellMann,Low2,BK,Duca,LPS}. At the one loop level, corrections that are logarithmic in the photon frequency can arise \cite{HHW,Bern,BHHW,Sahoo}. These corrections do not affect the leading perturbative computation of the entanglement entropy in \ref{s4}. Notice also that such a logarithmic singularity in the amplitude would be integrable. In particular, it leads to suppressed contributions, of the order $E_d\log E_d$, in various physical quantities, where we integrate over the soft photon momentum.}. Since $Q_\alpha = Q_\beta$, this gives
\begin{equation}
\tilde{S}^{(2)}_1~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~\left(f_\beta(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma)~-~f_\alpha(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma)\right)~+~\dots
\end{equation}
{\it Let us now consider the case for which the extra soft photon is not interacting}. Let the total number of outgoing noninteracting soft photons be $1+l$, and likewise for the incoming ones. Then $n^\prime=n-l-1$ soft photons are absorbed by an external electron line, and $m^{\prime}=m-l$ are emitted by an external electron line. Now the noninteracting soft photons contribute a factor given by
\begin{equation}
(1+l)~l! ~f_\alpha(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma)~ \left(\int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f_\alpha^{\mu}(\vec{q})f^{*}_{\beta\,\mu}(\vec{q})\right)^l
\end{equation}
For the interacting soft photons we must apply the soft theorems as before.
We then sum over all possible $(l, m^{\prime}, n^{\prime})$ configurations, after taking into account all weight factors. Notice that there are $(n^{\prime}+l+1)!/n^\prime!\,(l+1)!$ ways to choose $l+1$ photons from the initial $n$ soft photons, and likewise $(m^{\prime}+l)!/m^\prime!\,l!$ ways to choose $l$ photons from the final $m$ soft photons. In all we get
$$
\tilde{S}^{(2)}_2=
{{\cal{N}}_\beta~{\cal{N}}_\alpha}
~\sum_{l,\,m^{\prime},\,n^{\prime}\,=\,0}^\infty ~\frac{1}{(m^\prime +l)!\,(n^{\prime}+l+1)!}~\frac{(m^{\prime}+l)!}{m^\prime!\,l!}~\frac{(n^{\prime}+l+1)!}{n^\prime!\,(l+1)!}
$$
$$
\times~(1+l)~l! ~f_\alpha(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma) ~ \left(\int_\lambda^{E_d}\frac{d^3\vec{q}}{(2\pi)^3}~\frac{1}{2\omega_{\vec{q}}}~f_\alpha^{\mu}(\vec{q})f^{*}_{\beta\,\mu}(\vec{q})\right)^l
$$
$$
\times~\left[\int_{\lambda}^{E_d} \frac{d^3\vec{q}}{(2\pi)^32\omega_{\vec{q}}}~\sum_{i\in{\{\beta,\,\alpha\}}}~\eta_i\,e_i~\left(\frac{f^*_\beta(\vec{q})\cdot p_i}{p_i\cdot q}-f^*_\beta(\vec{q})\cdot c\right)\right]^{m^{\prime}}
$$
\begin{equation}
\times~\left[-\int_{\lambda}^{E_d} \frac{d^3\vec{k}}{(2\pi)^32\omega_{\vec{k}}}~\sum_{i\in{\{\beta,\,\alpha\}}}~\eta_i\,e_i~\left(\frac{f_\alpha(\vec{k})\cdot p_i}{p_i\cdot k}-f_\alpha(\vec{k})\cdot c\right)\right]^{n^{\prime}}~S_{\beta\alpha}
\end{equation}
After canceling combinatorial factors as before, it is easy to see that all three series exponentiate to give
\begin{equation}
\tilde{S}^{(2)}_2~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~f_\alpha(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma)
\end{equation}
and so
\begin{equation}
{\tilde S}_{\beta\gamma,\,\alpha}^{(2)}~=~ \tilde{S}^{(2)}_1 ~+~\tilde{S}^{(2)}_2~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~f_\beta(\vec{q}_\gamma) \cdot \epsilon_r^*(\vec{q}_\gamma)~+~\dots
\end{equation}
Therefore, adding the two parts together, we find that all singular terms, in the limits $\lambda,\,|\vec{q}_\gamma| \to 0$, cancel:
\begin{equation}
{\tilde S}_{\beta\gamma,\,\alpha}~=~F_{\beta\alpha}(\vec{q}_\gamma,\,\epsilon_r(\vec{q}_\gamma))
\end{equation}
Here $F_{\beta\alpha}(\vec{q}_\gamma,\,\epsilon_r(\vec{q}_\gamma))$ is a smooth function as $\lambda,\,|\vec{q}_\gamma| \to 0$. In fact, it has been shown that by appropriately correcting the dressing function to subleading order in the soft photon momentum (and to leading order in the electron charge), this function is of order $E_d$ \cite{Choi2}. So the dressing suppresses the emission of soft photons with energy $\omega_\gamma < E_d$, at least at tree level. We conclude that the dressed amplitude ${\tilde S}_{\beta\gamma,\,\alpha}$ is nonsingular, and suppressed when $\omega_\gamma < E_d$. {\it This motivates us to distinguish between low frequency photons with frequencies in the range $E_d<\omega_\gamma<E$, comprising the soft part of the emitted radiation, and soft photons present in the clouds accompanying the outgoing charged particles}. It would be interesting to see if the suppression of ${\tilde S}_{\beta\gamma,\,\alpha}$ persists at the one loop level \cite{Choi2}, since then corrections logarithmic in the soft photon frequency appear. One would need to consider $e^2$ corrections to the dressing function for this task.
\section{Discretization}\label{s3}
\noindent
For the entanglement entropy computation, we replace infinite space with a large box of size $L$ (volume $V=L^3$) and impose periodic boundary conditions for the fields. The momenta are quantized as
\begin{equation}
\vec{k} = \frac{2\pi}{L}(n_1,\,n_2,\,n_3)
\end{equation}
We also rescale the annihilation/creation operators
\begin{equation}
a_r(\vec{k}) \to V^{1/2}~\tilde{a}_r(\vec{k})
\end{equation}
so that for the discrete system, the commutation relations read
\begin{equation}
[\tilde{a}_r(\vec{k}),\,\tilde{a}_{r^\prime}^\dagger(\vec{k^\prime})]=\delta_{rr^\prime}\delta_{\vec{k}\,\vec{k}^\prime}
\end{equation}
Here $\delta_{rr^\prime}$ and $\delta_{\vec{k}\,\vec{k}^\prime}$ are Kronecker deltas. We restrict to transversely polarized photons. The single particle states
\begin{equation}
\tilde{a}_{r^\prime}^\dagger(\vec{k})|0\rangle
\end{equation}
are unit normalized. The IR cutoff scale $\lambda$ is naturally taken to be equal to $2\pi/L$. We will drop the tildes for simplicity.
Consider now an initially undressed two electron state $|\beta\rangle=|e_me_n\rangle$. The indices stand for both momentum and polarization.
The effect of dressing yields
\begin{equation}
|\beta\rangle_d=|e_me_n\rangle_H \times |f_\beta\rangle_S
\end{equation}
where $|f_\beta\rangle_S$ is the
coherent state describing the cloud of soft photons. For the discrete system, this is given by
\begin{equation}
|f_{\beta}\rangle_S = U_{\beta}|0\rangle_S={\cal{N}}_{\beta}~e^{A_\beta^\dagger}~|0\rangle_S
\end{equation}
with
\begin{equation}
U_{\beta} = e^{(A_\beta^\dagger - A_\beta)},\,\,\,\,\, A_\beta=\sum_{\omega_{\vec{k}}<{E_d}}~\frac{1}{(2\,V\,\omega_{\vec{k}})^{1/2}}~f^*_{\beta}(\vec{k})\cdot a(\vec{k})
\end{equation}
and
\begin{equation}
{\cal{N}}_{\beta}=e^{-\frac{1}{2} \sum_{\omega_{\vec{k}}<{E_d}}~\frac{1}{2\,V\,\omega_{\vec{k}}}~f_{\beta}^{\mu}(\vec{k})f^{*}_{\beta\,\mu}(\vec{k}) }
\end{equation}
As shown before, $|f_\beta\rangle$ is in the soft part of the Hilbert space ${\cal{H}}_S$.
Next we form the ket-bra operator
\begin{equation}
|\beta\rangle_{d}\langle\beta^\prime|_{d}
\end{equation}
(with $|\beta^\prime\rangle$ a different two electron state). Tracing over the soft part of the Hilbert space gives
\begin{equation}
{\rm Tr}_{{\cal{H}}_S}\left(|\beta\rangle_{d}\langle\beta^\prime|_{d}\right)=|\beta\rangle_{H}\langle\beta^\prime|_{H}~\langle f_{\beta^\prime}|f_\beta\rangle\label{trace1}
\end{equation}
The overlap $\langle f_{\beta^\prime}|f_\beta\rangle$ has been computed in \ref{braketf}, in the continuum limit. In particular, when $\beta\ne\beta^\prime$ and to all orders in the electron charge, the overlap vanishes in the strict $\lambda \to 0$ limit. For any superposition of dressed states, tracing over the soft part of the Hilbert space leads to decoherence and an almost diagonal density matrix \cite{Carney3}.
Now suppose that we add a single soft photon to the undressed state $|\beta\rangle$. Let the soft photon momentum be $\vec{q}$ ($|\vec{q}|<E$) and denote the polarization vector by $\epsilon_{r\mu}(\vec{q})$:
\begin{equation}
|\beta\gamma(r,\,\vec{q})\rangle=a^\dagger_r(\vec{q})|\beta\rangle
\end{equation}
Decomposing the corresponding dressed state in ${\cal{H}}_H \times {\cal{H}}_S$ yields
\begin{equation}
|\beta\gamma\rangle_d=|\beta\rangle_H \times \left(U_{\beta}~a^\dagger_r(\vec{q})~|0\rangle_S\right)=|\beta\rangle_H \times
\left(a^\dagger_r(\vec{q})-[A_\beta,\,a^\dagger_r(\vec{q})]\right)~|f_\beta\rangle_S
\end{equation}
Using this expression, we can readily calculate the partial traces:
\begin{equation}
{\rm Tr}_{{\cal{H}}_S}\left(|\beta\gamma\rangle_{d}\langle\beta^\prime|_{d}\right)=|\beta\rangle_{H}\langle\beta^\prime|_{H}~\langle f_{\beta^\prime}|f_\beta\rangle~\left([A_{\beta^\prime},\,a^\dagger_r(\vec{q})] - [A_\beta,\,a^\dagger_r(\vec{q})]\right)
\end{equation}
If $E_d<|\vec{q}|<E$, the commutators vanish. On the other hand, if $|\vec{q}|<E_d$, the commutators are nontrivial and give
\begin{equation}
{\rm Tr}_{{\cal{H}}_S}\left(|\beta\gamma\rangle_{d}\langle\beta^\prime|_{d}\right)=|\beta\rangle_{H}\langle\beta^\prime|_{H}~\langle f_{\beta^\prime}|f_\beta\rangle~\frac{1}{(2V\omega_{\vec{q}})^{1/2}}~\left(f_{\beta^\prime}^*(\vec{q})-f_\beta^*(\vec{q})\right)\cdot \epsilon_r(\vec{q}) \label{trace2}
\end{equation}
Notice that this vanishes for the diagonal cases $\beta=\beta^\prime$. Also, the function $f_\beta^{\mu}(\vec{q})$ is of order $e$. Next we compute
$$
{\rm Tr}_{{\cal{H}}_S}\left(|\beta\gamma\rangle_{d}\langle\beta^\prime\gamma^\prime|_{d}\right)=|\beta\rangle_{H}\langle\beta^\prime|_{H}~\langle f_{\beta^\prime}|f_\beta\rangle
$$
\begin{equation}
\times ~ \left\{\delta_{rr^\prime}\delta_{\vec{q}\vec{q}^\prime}+\left([a_{r^\prime}(\vec{q}^\prime),\, A_\beta^\dagger]-[a_{r^\prime}(\vec{q}^\prime),\, A_{\beta^\prime}^\dagger]\right)~\left([A_{\beta^\prime},\,a^\dagger_r(\vec{q})] - [A_\beta,\,a^\dagger_r(\vec{q})]\right)\right\}
\end{equation}
If both $|\vec{q}|,\,|\vec{q}^\prime|<E_d$, we obtain
$$
{\rm Tr}_{{\cal{H}}_S}\left(|\beta\gamma\rangle_{d}\langle\beta^\prime\gamma^\prime|_{d}\right)~=~|\beta\rangle_{H}\langle \beta^\prime|_{H}~\langle f_{\beta^\prime}|f_\beta\rangle
$$
\begin{equation}
\times~\left\{\delta_{rr^\prime}\delta_{\vec{q}\vec{q}^\prime}+\frac{1}{(2V\omega_{\vec{q}^\prime})^{1/2}(2V\omega_{\vec{q}})^{1/2}}~\left(f_\beta(\vec{q}^\prime)-f_{\beta^\prime}(\vec{q}^\prime)\right)\cdot \epsilon_{r^\prime}(\vec{q}^\prime)~\left(f_{\beta^\prime}(\vec{q})-f_\beta(\vec{q})\right)\cdot \epsilon_{r}(\vec{q})\right\} \label{trace3}
\end{equation}
In a similar way, we can compute partial traces for the cases in which two or more soft photons are present in the initially undressed states. There will be contributions that are higher order in the function $f_\beta^{\mu}(\vec{q})$.
\section{Scattering with dressed states and entanglement entropy}\label{s4}
\noindent
The incoming state is taken to be
\begin{equation}
|\psi\rangle_{in}=|e_ie_j\rangle_d=|e_ie_j\rangle_H \times |f_\alpha\rangle_S
\end{equation}
We will also adopt the notation $|\alpha\rangle=|e_ie_j\rangle$.
Notice that this is a product state and so there is no entanglement between the soft and hard degrees of freedom \footnote{Had we started with a superposition of dressed states, there would be entanglement between the soft and hard degrees of freedom \cite{Carney4}.}. Entanglement occurs as a result
of scattering. In particular the initial density matrix, including tracing over the undetectable soft photon clouds, is pure:
\begin{equation}
{\rm Tr}_{{\cal{H}}_S}\left(|\psi\rangle_{in}\langle\psi|_{in}\right)=|\alpha\rangle_H\langle\alpha|_H
\end{equation}
The out state is given in terms of the $S$-matrix by
\begin{equation}
|\psi\rangle_{out}=S|\psi\rangle_{in}=(1+iT)|\alpha\rangle_d
\end{equation}
For simplicity, we restrict the incoming energy so that electron/positron pair production is forbidden, and so only two charged particles are present in the final state. Since the $S$-matrix is unitary, we have
\begin{equation}
i(T-T^{\dagger})=-T^{\dagger}T \label{unitarity}
\end{equation}
Inserting a complete basis of dressed states, $\ket{\psi}_{out}$ can be written as
\begin{equation}
|\psi\rangle_{out}=|\alpha\rangle_d~+~\tilde{A}_{\beta\alpha}|\beta\rangle_d~+~\tilde{B}_{\beta\gamma,\,\alpha}|\beta\gamma\rangle_d+\dots
\end{equation}
where $\tilde{A}_{\beta\alpha}=_d\bra{\beta}iT\ket{\alpha}_d$ and $\tilde{B}_{\beta\gamma,\,\alpha}=_d\bra{\beta\gamma}iT\ket{\alpha}_d$ are $S$-matrix elements between dressed states. Summation over the final state electron and photon indices $\beta$ and $\gamma$, respectively, is implied. The leading contributions in $\tilde{A}_{\beta\alpha}$ are of order $e^2$ and in $\tilde{B}_{\beta\gamma,\,\alpha}$ of order $e^3$. The ellipses stand for higher order contributions, arising from states with two or more photons. The associated density matrix is (no sum over $\alpha$)
$$
\ket{\psi}_{out}\bra{\psi}_{out}=~\ket{\alpha}_d\bra{\alpha}_d~+~ \left(\tilde{A}_{\beta\alpha}\ket{\beta}_d~+~\tilde{B}_{\beta\gamma,\,\alpha}\ket{\beta\gamma}_d~+~\dots\right)\bra{\alpha}_d
$$
$$
+~\ket{\alpha}_d\left(\tilde{A}^*_{\beta^\prime\alpha}\bra{\beta^\prime}_d~+~\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\bra{\beta^\prime\gamma^\prime}_d~+~\dots\right)
$$
$$
+~\tilde{A}_{\beta\alpha}\tilde{A}^*_{\beta^\prime\alpha}\ket{\beta}_d\bra{\beta^\prime}_d~+~\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\ket{\beta\gamma}_d\bra{\beta^\prime\gamma^\prime}_d
$$
\begin{equation}
~+~\tilde{B}_{\beta\gamma,\,\alpha}\tilde{A}^*_{\beta^\prime\alpha}\ket{\beta\gamma}_d\bra{\beta^\prime}_d~+~\tilde{A}_{\beta\alpha}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\ket{\beta}_d\bra{\beta^\prime\gamma^\prime}_d ~+~\dots
\end{equation}
As we already remarked, we will discuss two partial traces and the associated density matrices: 1) over all soft photons in ${\cal{H}}_S$ and 2) over soft photons with frequencies in the range $E_d<\omega_\gamma<E$, comprising the soft part of the emitted radiation. The latter is motivated by the fact that the amplitude for the emission of soft photons with energy less than $E_d$ is suppressed \cite{Choi2} -- see the discussion at the end of \ref{s2}. In the first case, the reduced density matrix is an operator in ${\cal{H}}_H$. Since the second case does not prescribe tracing over soft cloud photons, we obtain an operator acting on the space of physical asymptotic states.
\subsection{Tracing over all soft photons}
First we trace over all soft photons, including those in the clouds. The reduced density matrix
\begin{equation}
\rho_H={\rm Tr}_{{\cal{H}}_S}\left(\ket{\psi}_{out}\bra{\psi}_{out}\right)
\end{equation}
can be readily obtained using expressions \ref{trace1}, \ref{trace2} and \ref{trace3}. It takes the following form
$$
\rho_H=\ket{\alpha}_H\bra{\alpha}_H + \left(C_{\beta}\ket{\beta}_H~+~\sum_{\omega_\gamma>E}C_{\beta\gamma}\ket{\beta\gamma}_H+\dots\right)\bra{\alpha}_H
$$
$$
+\ket{\alpha}_H\left(C^*_{\beta^\prime}\bra{\beta^\prime}_H~+~\sum_{\omega_{\gamma^\prime}>E}C^*_{\beta^\prime\gamma^\prime}\bra{\beta^\prime\gamma^\prime}_H+\dots\right)
$$
$$
+D_{\beta,\,\beta^\prime}\ket{\beta}_H\bra{\beta^\prime}_H~+~\sum_{\omega_\gamma,\omega_{\gamma^\prime}>E}D_{\beta\gamma,\,\beta^\prime\gamma^\prime} \ket{\beta\gamma}_H\bra{\beta^\prime\gamma^\prime}_H
$$
\begin{equation}
+ \sum_{\omega_\gamma>E}D_{\beta\gamma,\,\beta^\prime}\ket{\beta\gamma}_H\bra{\beta^\prime}_H~+~\sum_{\omega_{\gamma^\prime}>E}D^*_{\beta^\prime\gamma^\prime,\,\beta}\ket{\beta}_H\bra{\beta^\prime\gamma^\prime}_H ~+~ \dots
\end{equation}
where
\begin{equation}
C_{\beta} = \bra{f_\alpha}\ket{f_\beta}~\left(\tilde{A}_{\beta\alpha}~+~\sum_{\omega_\gamma<{E_d}}~\frac{1}{(2V\omega_\gamma)^{1/2}}~\tilde{B}_{\beta\gamma,\,\alpha}\left(f_\alpha^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot\epsilon(\gamma)~+~\dots\right)\label{C1}
\end{equation}
$$
$$
\begin{equation}
C_{\beta\gamma}=\bra{f_\alpha}\ket{f_\beta}~\tilde{B}_{\beta\gamma,\,\alpha}~+~\dots \label{C2}
\end{equation}
$$
$$
$$
D_{\beta,\,\beta^{\prime}}/\bra{f_{\beta^\prime}}\ket{f_\beta}=\tilde{A}_{\beta\alpha}\tilde{A}^*_{\beta^\prime\alpha} ~+~\sum_{\omega_\gamma<{E_d}}~\frac{1}{(2V\omega_\gamma)^{1/2}}~\tilde{B}_{\beta\gamma,\,\alpha}\tilde{A}^*_{\beta^\prime\alpha}\left(f_{\beta^\prime}^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot\epsilon(\gamma)
$$
\begin{equation}
+~\sum_{\omega_{\gamma^\prime}<{E_d}}\frac{1}{(2V\omega_{\gamma^\prime})^{1/2}}~\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\tilde{A}_{\beta\alpha}\left(f_{\beta}(\vec{q}_{\gamma^\prime})-f_{\beta^\prime}(\vec{q}_{\gamma^\prime})\right)\cdot\epsilon^*(\gamma^\prime)+\sum_{\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta^\prime\gamma,\,\alpha}~+~\dots \label{C3}
\end{equation}
$$
$$
\begin{equation}
D_{\beta\gamma,\,\beta^\prime\gamma^\prime}=\bra{f_{\beta^\prime}}\ket{f_\beta}~\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}~+~\dots\label{C4}
\end{equation}
$$
$$
\begin{equation}
D_{\beta\gamma,\,\beta^\prime}=\bra{f_{\beta^\prime}}\ket{f_\beta}~\tilde{B}_{\beta\gamma,\,\alpha}\tilde{A}^*_{\beta^\prime\alpha}~+~\dots\label{C5}
\end{equation}
The matrix elements of $\rho_H$ are given in terms of dressed amplitudes, which are free of any IR divergences (at least perturbatively),
as well as overlaps of coherent photon states describing the soft clouds. The diagonal elements are proportional to inclusive Bloch-Nordsieck type rates associated with dressed box states, and they are free of any IR divergences in $\lambda$, order by order in perturbation theory \footnote{As we discuss in \ref{continuum}, they scale inversely with (powers of) the volume in the continuum (large volume) limit.}.
For example
\begin{equation}
D_{\beta,\,\beta}=\tilde{A}_{\beta\alpha}\tilde{A}^*_{\beta\alpha}~+~\sum_{\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}~+~\dots \label{Ddiagonal}
\end{equation}
is proportional to the rate for the transition of the initial dressed state $\ket{\alpha}_d$ to $\ket{\beta}_d$, and any number of photons with total energy less than $E$.
The off diagonal elements, e.g. $D_{\beta,\,\beta^\prime}$ ($\beta\ne\beta^\prime$), are proportional to the overlap $\bra{f_{\beta^\prime}}\ket{f_\beta}$, which, at any finite order in perturbation theory, induces logarithmic divergences in $\lambda$ (via its perturbative expansion at finite $\lambda$ -- see \ref{bracketf2} below). Thus generically, at any finite order in perturbation theory, the off diagonal elements are nonzero and must be taken into account. {\it To all orders in the electron charge}, these IR logarithmic terms exponentiate. As a result, when the momenta of the two-electron particle states $\beta$ and $\beta^\prime$ differ, the overlap $\bra{f_{\beta^\prime}}\ket{f_\beta}$ to all orders vanishes in the strict $\lambda \to 0$ limit. {\it Therefore, to all orders in perturbation theory, the density matrix assumes an almost diagonal form in the continuum limit, exhibiting decoherence} \cite{Carney3}. In the following, we keep the volume of the box and the infrared cutoff $\lambda$ finite in order to regularize the entanglement entropy, {\it working at finite order in perturbation theory and taking into account the contributions of the off diagonal elements.}
We would like to investigate if in the continuum limit, the entanglement entropy is free of any IR logarithmic divergences in $\lambda$, order by order in perturbation theory.
This behavior of the off diagonal elements is reminiscent of the behavior of the conventional Fock basis amplitudes (with a finite number of photons in the initial and final states). At any finite order in perturbation theory, these amplitudes are nonzero, containing logarithmic divergences in $\lambda$ due to virtual soft photons. Their contributions must be taken into account in the perturbative calculation of the inclusive cross sections. Notice, however, that to all orders in perturbation theory, the virtual infrared divergences exponentiate, causing the individual Fock basis amplitudes to vanish. IR logarithmic divergences in $\lambda$ appear also due to real soft photon emission. The inclusive cross sections are free of any IR divergences, order by order in perturbation theory. The IR divergences due to virtual soft photons and real soft photon emission cancel against each other in this case \cite{Weinberg}.
We can extract the analogous Fock basis computation, where the initial state is taken to be a state of two bare electrons, by setting the function $f_\beta^{\mu}(\vec{q})$ to be zero and replacing the dressed amplitudes with conventional Fock basis amplitudes. At any finite order in perturbation theory, the off diagonal elements are nonzero and contain logarithmic divergences in $\lambda$. {\it To all orders in perturbation theory however,} the IR divergences exponentiate, leading to the vanishing of the off diagonal elements of the corresponding density matrix in the continuum, $\lambda \to 0$ limit \cite{Carney2}. The diagonal elements are given in terms of Bloch-Nordsieck rates, and so they are free of IR divergences order by order in perturbation theory. As in the dressed case, we fix $\lambda$ and work at finite order in perturbation theory, taking into account the contributions of the off diagonal elements to the entanglement entropy. The continuum $\lambda \to 0$ limit is taken at the end.
Some more comments are in order:
\begin{itemize}
\item
For the Fock basis case, it is clear that the entanglement between the soft and hard parts of the Hilbert space arises due to soft photon emission. The entanglement entropy is of order $e^6$ \footnote{More precisely, the leading entanglement entropy is of order $e^6\ln{e^6}$. The entanglement entropy is not analytic in $e$ at $e=0$.}, with Feynman diagrams involving the emission of a single soft photon contributing at leading order. Likewise for the dressed case, the entanglement entropy is of order $e^6$. At lower orders, the density matrix assumes a product form, and so it is pure.
\item
The leading contributions in $C_{\beta}$ are of order $e^2$, in $C_{\beta\gamma}$ of order $e^3$, in $D_{\beta,\,\beta^\prime}$ of order $e^4$, in $D_{\beta\gamma,\,\beta^\prime}$ of order $e^5$ and in $D_{\beta\gamma,\,\beta^\prime\gamma^\prime}$ of order $e^6$.
\item
The last three terms in $D_{\beta,\,\beta^\prime}$, see equation \ref{C3}, are of order $e^6$. The ellipses include terms of higher order than $e^6$, which do not contribute to the entanglement entropy at leading order. Likewise, the ellipses in $D_{\beta\gamma,\,\beta^\prime\gamma^\prime}$, see \ref{C4}, include terms of higher order than $e^6$, which can be ignored at leading order.
\item
The second term in $C_{\beta}$ (equation \ref{C1}) is of order $e^4$ and vanishes when $\beta=\alpha$. Contributions from two or more photon states are proportional to the matrix elements $\tilde{B}_{\beta{\gamma_1}{\gamma_2},\dots,\gamma_i\dots,\,\alpha}$ and products of the differences $f_{\alpha}^\mu(\vec{q}_i)-f_{\beta}^\mu(\vec{q}_i)$, and so they also vanish when $\beta=\alpha$. Only the first term contributes in $C_{\alpha}$: $C_{\alpha}=\tilde{A}_{\alpha\alpha}$ to all orders.
\item
As we will see, $C_{\alpha}+C_{\alpha}^*$ is of order $e^4$ by unitarity. This result considerably simplifies the leading order computation of the entanglement entropy.
\end{itemize}
\subsection{Perturbative analysis to order $e^6$}
We proceed to compute the Renyi entropies
\begin{equation}
S_{m}=\frac{1}{1-m}\log {\rm Tr} (\rho_H)^m
\end{equation}
for integer $m \ge 2$, to leading order in perturbation theory ($e^6$). Had the density matrix $\rho_H$ corresponded to a pure state (all eigenvalues zero but one eigenvalue equal to one), the Renyi entropies would vanish. So they measure the degree of entanglement and the information carried by the soft photons. The entanglement entropy
\begin{equation}
S_{ent}=-{\rm Tr} \rho_H \log \rho_H
\end{equation}
can be written as an infinite series of the Renyi entropies for integer $m$ \footnote{We can also obtain the entanglement entropy in the limit $m \to 1$: $\lim_{m \to 1} S_{m} = S_{ent}$.}:
\begin{equation}
S_{ent}=\sum_{n=1}^{\infty}\sum_{m=0}^{n}\frac{(n-1)!}{(n-m)!~m!}~(-1)^m~e^{-mS_{m+1}} \label{entR}
\end{equation}
Let us set
\begin{equation}
\rho_H = \rho_0 + {\varepsilon}
\end{equation}
where $\rho_0 = \ket{\alpha}_H\bra{\alpha}_H$.
Then since ${\rm Tr} \rho_H={\rm Tr} \rho_0=1$, ${\rm Tr} \varepsilon =0$. Indeed computing the trace explicitly, we obtain
$$
{\rm Tr} \varepsilon = C_\alpha + C_\alpha^* + \sum_{\beta}\left(D_{\beta,\,\beta} + \sum_{\omega_\gamma>E} D_{\beta\gamma,\, \beta\gamma}\right) + \dots
$$
\begin{equation}
=\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^* + \sum_{\beta}\tilde{A}_{\beta\alpha}\tilde{A}_{\beta\alpha}^* + \sum_{\beta\gamma}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}_{\beta\gamma,\,\alpha}^* + \dots=_d\bra{\alpha}i(T-T^\dagger)+T^\dagger T\ket{\alpha}_d=0 \label{unitarity2}
\end{equation}
The last equation follows from unitarity (see \ref{unitarity}).
Now $\varepsilon$ is of order $e^2$. So to obtain the leading contribution to the trace of $\rho_H^m$ ($m>2$) and to the corresponding Renyi entropy (which is of order $e^6$), we need to expand $\rho_H^m$ to the cubic order in $\varepsilon$. The fact that $\rho_0^2=\rho_0$ and the cyclic property of the trace limit the number of structures we need to consider.
At the linear level, we need only compute $\varepsilon \rho_0$ and its trace:
$$
\varepsilon \rho_0~=~C_{\beta}\ket{\beta}_H\bra{\alpha}_H ~+~ \sum_{\omega_\gamma>E}C_{\beta\gamma}\ket{\beta\gamma}_H\bra{\alpha}_H ~+~ C_\alpha^*\ket{\alpha}_H\bra{\alpha}_H
$$
\begin{equation}
+~D_{\beta,\,\alpha}\ket{\beta}_H\bra{\alpha}_H~+~\sum_{\omega_\gamma>E}D_{\beta\gamma,\,\alpha}\ket{\beta\gamma}_H\bra{\alpha}_H~+~\dots
\end{equation}
$$
$$
\begin{equation}
{\rm Tr}(\varepsilon \rho_0)~=~C_{\alpha}~+~C_{\alpha}^*~+~D_{\alpha,\,\alpha}~=~\tilde{A}_{\alpha\alpha}~+~\tilde{A}_{\alpha\alpha}^* ~+~\tilde{A}_{\alpha\alpha}\tilde{A}_{\alpha\alpha}^*~+~\left(\sum_{\omega_{\gamma}<E}\tilde{B}_{\alpha\gamma,\,\alpha}\tilde{B}^*_{\alpha\gamma,\,\alpha}\right)~+~\dots
\end{equation}
The ellipses in the trace include terms of order higher than $e^6$ and can be dropped to leading order in the entanglement entropy.
At the quadratic level, it suffices to consider the following structures: $\varepsilon^2$, $\varepsilon^2\rho_0$ and $\varepsilon\rho_0\varepsilon\rho_0$. First we get:
$$
\varepsilon^2 ~=~ \left(C_\beta C_\beta^*~+~\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\beta\gamma}^* \right)~\ket{\alpha}_H\bra{\alpha}_H
$$
$$
+~\left(C_\beta C_\alpha~+~D_{\beta,\, \beta^\prime}C_{\beta^\prime}\right) ~\ket{\beta}_H\bra{\alpha}_H ~+~ \left(C_\beta^* C_\alpha^*~+~C_{\beta^\prime}^*D_{\beta^\prime,\,\beta}\right) ~\ket{\alpha}_H\bra{\beta}_H
$$
$$
+~\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\alpha}~\ket{\beta\gamma}_H\bra{\alpha}_H~+~C_{\beta\gamma}^*C_{\alpha}^*~\ket{\alpha}_H\bra{\beta\gamma}_H
$$
$$
+~\left(C_\beta C_{\beta^\prime}^*~+~D_{\beta,\,\alpha}C_{\beta^\prime}^*~+~D_{\alpha,\,\beta^\prime}C_{\beta} \right)~\ket{\beta}_H\bra{\beta^\prime}_H ~+~ \sum_{\omega_\gamma, \omega_{\gamma^\prime}>E}C_{\beta\gamma}C_{\beta^\prime\gamma^\prime}^*~\ket{\beta\gamma}_H\bra{\beta^\prime\gamma^\prime}_H
$$
\begin{equation}
+~\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\beta^\prime}^*~\ket{\beta\gamma}_H\bra{\beta^\prime}_H~+~C_{\beta\gamma}^*C_{\beta^\prime}~\ket{\beta^\prime}_H\bra{\beta\gamma}_H~+~\dots
\end{equation}
Taking the trace yields
$$
{\rm Tr} \varepsilon^2 = C_\alpha^2 + C_\alpha^{*\, 2} + \sum_\beta \left(2D_{\alpha,\, \beta}C_\beta +
2D_{\beta,\,\alpha}C_{\beta}^*+ 2C_\beta C_\beta^*~+~2\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\beta\gamma}^*\right) + \dots
$$
$$
= \tilde{A}_{\alpha\alpha}^2 + \tilde{A}_{\alpha\alpha}^{*\, 2}+2\sum_{\beta} |\bra{f_\beta}\ket{f_\alpha}|^2\left(1+\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^*\right)\tilde{A}_{\beta\alpha}\tilde{A}_{\beta\alpha}^*
$$
$$
+2\sum_{\beta}\sum_{\omega_\gamma<{E_d}}\frac{|\bra{f_\beta}\ket{f_\alpha}|^2}{(2V\omega_\gamma)^{1/2}} \left[ \tilde{A}_{\beta\alpha}\tilde{B}_{\beta\gamma,\,\alpha}^*\left(f_\alpha(\vec{q}_\gamma)-f_\beta(\vec{q}_\gamma)\right)\cdot \epsilon^*(\gamma)+\tilde{A}_{\beta\alpha}^*\tilde{B}_{\beta\gamma,\,\alpha}\left(f_\alpha^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot \epsilon(\gamma)\right]
$$
\begin{equation}
+2\sum_\beta\sum_{\omega_\gamma>E}|\bra{f_\beta}\ket{f_\alpha}|^2\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}_{\beta\gamma,\,\alpha}^* +\dots
\end{equation}
In the last line, the ellipses stand for terms of higher order than $e^6$. The second line vanishes when the dressing function is set to zero, and so it is absent in the Fock basis computation.
Next we calculate $\varepsilon^2 \rho_0$:
$$
\varepsilon^2\rho_0 ~=~ \left(C_{\alpha}^{*\, 2}~+~C_\beta C_\beta^*~+~\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\beta\gamma}^* ~+~C_{\beta}^*D_{\beta,\,\alpha}\right)~\ket{\alpha}_H\bra{\alpha}_H
$$
$$
+~\left(C_\beta C_\alpha~+~D_{\beta,\, \beta^\prime}C_{\beta^\prime}~+~C_\beta C_{\alpha}^*~+~D_{\beta,\,\alpha}C_{\alpha}^*~+~D_{\alpha,\,\alpha}C_{\beta}\right) ~\ket{\beta}_H\bra{\alpha}_H
$$
\begin{equation}
+~\sum_{\omega_\gamma>E}\left(C_{\beta\gamma}C_{\alpha}~+~C_{\beta\gamma}C_{\alpha}^*\right)~\ket{\beta\gamma}_H\bra{\alpha}_H~+~\dots
\end{equation}
For the trace we obtain
$$
{\rm Tr} (\varepsilon^2\rho_0) = C_\alpha^2 + C_\alpha^{*\, 2} + C_{\alpha}C_{\alpha}^*+ D_{\alpha,\,\alpha} (C_\alpha + C_{\alpha}^*)
$$
$$
+ \sum_\beta \left(D_{\alpha,\, \beta}C_\beta +
D_{\beta,\,\alpha}C_{\beta}^*+ C_\beta C_\beta^*~+~\sum_{\omega_\gamma>E}C_{\beta\gamma}C_{\beta\gamma}^*\right) + \dots
$$
$$
= \tilde{A}_{\alpha\alpha}^2 + \tilde{A}_{\alpha\alpha}^{*\, 2}+\left(1+\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^*\right) \tilde{A}_{\alpha\alpha}\tilde{A}_{\alpha\alpha}^*+\sum_{\beta} |\bra{f_\beta}\ket{f_\alpha}|^2\left(1+\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^*\right)\tilde{A}_{\beta\alpha}\tilde{A}_{\beta\alpha}^*
$$
$$
+\sum_{\beta}\sum_{\omega_\gamma<{E_d}}\frac{|\bra{f_\beta}\ket{f_\alpha}|^2}{(2V\omega_\gamma)^{1/2}} \left[ \tilde{A}_{\beta\alpha}\tilde{B}_{\beta\gamma,\,\alpha}^*\left(f_\alpha(\vec{q}_\gamma)-f_\beta(\vec{q}_\gamma)\right)\cdot \epsilon^*(\gamma)+\tilde{A}_{\beta\alpha}^*\tilde{B}_{\beta\gamma,\,\alpha}\left(f_\alpha^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot \epsilon(\gamma)\right]
$$
\begin{equation}
+\sum_\beta\sum_{\omega_\gamma>E}|\bra{f_\beta}\ket{f_\alpha}|^2\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}_{\beta\gamma,\,\alpha}^* +\dots
\end{equation}
Notice the appearance of off diagonal elements in the perturbative expansion for both ${\rm Tr} \varepsilon^2$ and ${\rm Tr} \varepsilon^2\rho_0$. As we remarked before, at any finite order in perturbation theory, the off diagonal elements are nonzero and contain IR logarithmic divergences in $\lambda$. Both of these traces contribute to the Renyi and the entanglement entropies to leading order ($e^6$), and we would like to investigate whether the logarithmic divergences cancel.
Finally for $\varepsilon\rho_0\varepsilon\rho_0$ and its trace we get:
$$
\varepsilon \rho_0\varepsilon\rho_0~=~\left[(C_{\alpha}+C_{\alpha}^*+D_{\alpha,\,\alpha})C_{\beta}+(C_{\alpha}+C_{\alpha}^*)D_{\beta,\,\alpha}\right]\ket{\beta}_H\bra{\alpha}_H
$$
\begin{equation}
~+~ \sum_{\omega_\gamma>E}(C_{\alpha}+C_{\alpha}^*)C_{\beta\gamma}\ket{\beta\gamma}_H\bra{\alpha}_H~+~C_\alpha^*(C_\alpha + C_\alpha^* + D_{\alpha,\,\alpha})\ket{\alpha}_H\bra{\alpha}_H ~+~\dots
\end{equation}
$$
$$
$$
{\rm Tr}(\varepsilon \rho_0\varepsilon\rho_0)=(C_\alpha + C_\alpha^*)(C_\alpha + C_\alpha^*+2D_{\alpha,\,\alpha})+\dots
$$
\begin{equation}
=(\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^*)(\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^* +2\tilde{A}_{\alpha\alpha}\tilde{A}_{\alpha\alpha}^*)+\dots
\end{equation}
This trace vanishes to order $e^6$ by unitarity.
To cubic order, it is sufficient to compute $\varepsilon^3$, $\varepsilon^3\rho_0$, $\varepsilon^2\rho_0\varepsilon\rho_0$ and
$\varepsilon\rho_0\varepsilon\rho_0\varepsilon\rho_0$.
As we will show, these traces are vanishing to order $e^6$, and so they do not contribute to the entanglement entropy at leading order. Indeed, we find
$$
\varepsilon^3 ~=~ (C_\alpha + C_\alpha^*) C_\beta C_\beta^* ~\ket{\alpha}_H\bra{\alpha}_H~+~C_{\beta^\prime} (C_\beta C_\beta^* + C_\alpha^2) \ket{\beta^\prime}_H\bra{\alpha}_H
$$
\begin{equation}
+~ ~C_{\beta^\prime}^* (C_\beta C_\beta^* + C_\alpha^{*\,2})\ket{\alpha}_H\bra{\beta^\prime}_H ~+~ (C_\alpha^*+C_{\alpha}) C_{\beta^\prime}C_{\beta}^* ~\ket{\beta^\prime}_H\bra{\beta}_H~+~\dots
\end{equation}
$$
$$
\begin{equation}
{\rm Tr} \varepsilon^3 ~=~C_{\alpha}^3 + C_{\alpha}^{*\,3}~+~3 (C_\alpha + C_\alpha^*) \sum_{\beta} C_\beta C_\beta^* +\dots=~\tilde{A}_{\alpha\alpha}^3 + \tilde{A}_{\alpha\alpha}^{*\,3}~+~3 (\tilde{A}_{\alpha\alpha} + \tilde{A}_{\alpha\alpha}^*) \sum_{\beta} C_\beta C_\beta^*~+~\dots
\end{equation}
The trace vanishes to order $e^6$ by unitarity.
Similarly, we obtain
\begin{equation}
\varepsilon^3\rho_0 ~=~ \left[(C_\alpha + 2C_\alpha^*) C_\beta C_\beta^* ~+~ C_\alpha^{*\,3}\right] ~\ket{\alpha}_H\bra{\alpha}_H~+~C_{\beta^\prime} \left[(C_\beta C_\beta^* + C_\alpha^2)~+~(C_\alpha^*+C_{\alpha})C_{\alpha}^*\right]~ \ket{\beta^\prime}_H\bra{\alpha}_H +\dots
\end{equation}
$$
$$
\begin{equation}
{\rm Tr}(\varepsilon^3\rho_0)~=~C_{\alpha}^3 + C_{\alpha}^{*\,3}~+~ (C_\alpha + C_\alpha^*) \left(C_{\alpha}C_{\alpha}^*~+~2\sum_{\beta} C_\beta C_\beta^*\right)~+~\dots
\end{equation}
This trace does not contribute to the entanglement entropy at leading order.
Next we get
\begin{equation}
\varepsilon^2\rho_0\varepsilon\rho_0 ~=~ \left(C_{\alpha}~+~C_{\alpha}^*\right)\left(C_{\alpha}^{*\, 2}~+~C_\beta C_\beta^*\right)~\ket{\alpha}_H\bra{\alpha}_H
~+~\left(C_{\alpha}~+~C_{\alpha}^*\right)\left(C_\beta C_\alpha~+~C_\beta C_{\alpha}^*\right) ~\ket{\beta}_H\bra{\alpha}_H +\dots
\end{equation}
and
\begin{equation}
{\rm Tr}(\varepsilon^2\rho_0\varepsilon\rho_0)~=~\left(C_{\alpha}~+~C_{\alpha}^*\right)\left(C_{\alpha}^2~+~C_{\alpha}^{*\, 2}~+~C_{\alpha}C_{\alpha}^*~+~\sum_{\beta}C_\beta C_\beta^*\right)+\dots
\end{equation}
which vanishes to order $e^6$.
Finally to cubic order, we have
\begin{equation}
\varepsilon\rho_0\varepsilon\rho_0\varepsilon\rho_0~=~C_{\alpha}^*\left(C_{\alpha}~+~C_{\alpha}^*\right)^2~\ket{\alpha}_H\bra{\alpha}_H
~+~C_{\beta}^*\left(C_{\alpha}~+~C_{\alpha}^*\right)^2~\ket{\beta}_H\bra{\alpha}_H +\dots
\end{equation}
and
\begin{equation}
{\rm Tr}(\varepsilon\rho_0\varepsilon\rho_0\varepsilon\rho_0)=\left(C_{\alpha}~+~C_{\alpha}^*\right)^3+\dots
\end{equation}
with vanishing contributions to the entanglement entropy at leading order.
So
\begin{equation}
{\rm Tr}\varepsilon,\,\, {\rm Tr} \varepsilon\rho_0 \varepsilon \rho_0 \varepsilon \rho_0, \,\, {\rm Tr} \varepsilon^2 \rho_0 \varepsilon \rho_0,\,\, {\rm Tr}\varepsilon^3\rho_0, \,\,\ {\rm Tr}\varepsilon^3, \,\, {\rm Tr} \varepsilon\rho_0\varepsilon\rho_0\,\, \rightarrow 0
\end{equation}
to order $e^6$ (${\rm Tr}\varepsilon =0$ to all orders). The nonzero traces, which further simplify (by unitarity), are ${\rm Tr} \varepsilon \rho_0$,
${\rm Tr} \varepsilon^2 $ and ${\rm Tr} \varepsilon^2 \rho_0$.
Recall from the previous section that
\begin{equation}
\bra{f_\beta}\ket{f_\alpha}= \left(\frac{\lambda}{E_d}\right)^{{\cal{B}}_{\beta\alpha}}=e^{{\cal{B}}_{\beta\alpha}\ln(\lambda/E_d)}=1+{\cal{B}}_{\beta\alpha}\ln(\lambda/E_d)+\dots \label{bracketf2}
\end{equation}
with ${\cal{B}_{\beta\alpha}}$ of order $e^2$ (given in \ref{B}), ${\cal{B}_{\alpha\alpha}}=0$, and
$$
{\tilde S}_{\beta\alpha}=
~\bra{f_\beta}\ket{f_\alpha}~
~\left(\frac{E_d}{\lambda}\right)^{2{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\alpha}
$$
$$
{\tilde S}_{\beta\gamma,\,\alpha}=
~\bra{f_\beta}\ket{f_\alpha}~
~\left(\frac{E_d}{\lambda}\right)^{2{\cal{B}}_{\beta\alpha}}~S_{\beta\gamma,\,\alpha}~=~\left(\frac{E_d}{\lambda}\right)^{{\cal{B}}_{\beta\alpha}}~S_{\beta\gamma,\,\alpha}~~~~{\rm{if}}~~~\omega_\gamma>E_d
$$
$$
{\tilde B}_{\beta\gamma,\,\alpha}~=~\frac{1}{(2V^5\omega_\gamma)^{1/2}}~F_{\beta\alpha}(\vec{q}_\gamma,\,\epsilon_r(\vec{q}_\gamma))~~~~{\rm{if}}~~~\omega_\gamma<E_d
$$
To leading order ($e^3$), the function $F_{\beta\alpha}(\vec{q}_\gamma,\,\epsilon_r(\vec{q}_\gamma))$ is smooth and nonsingular in the limits $\lambda,\,|\vec{q}_\gamma| \to 0$ (and of order the dressing scale $E_d$ upon suitably modifying the dressing function $f_\beta^\mu(\vec{q})$ to subleading order in $\vec{q}$ \cite{Choi2}). The volume factors are due to the relative normalization between box and continuum states -- see below. (Some energy factors of the initial and final electron states can be absorbed in the definition of $F_{\beta\alpha}(\gamma)$).
From these, it is easy to deduce the relations
\begin{equation}
{\tilde A}_{\beta\alpha}=A_{\beta\alpha}/\bra{f_\beta}\ket{f_\alpha}
\end{equation}
and
\begin{equation}
{\tilde B}_{\beta\gamma,\,\alpha}=B_{\beta\gamma,\,\alpha}/\bra{f_\beta}\ket{f_\alpha}~~~~{\rm{if}}~~~\omega_\gamma>E_d
\end{equation}
For the purposes of perturbation theory it will be more convenient to express the traces in terms of Fock basis amplitudes, which are easier to compute via Feynman diagrams. Since the dressed amplitudes are free of IR divergences order by order in perturbation theory, any logarithmic divergence in the IR cutoff $\lambda$ at the perturbative level can be attributed to the soft clouds of photons via the coherent state overlaps.
We incorporate the results above and collect the terms contributing to the nonzero traces to order $e^6$. First we find
\begin{equation}
{\rm Tr} \varepsilon \rho_0 = A_{\alpha\alpha} + A^*_{\alpha\alpha} + A_{\alpha\alpha}A^*_{\alpha\alpha}~+~\left(\sum_{\omega_{\gamma}<E_d}\frac{1}{2V^5\omega_\gamma}F_{\alpha\alpha}(\gamma)F^*_{\alpha\alpha}(\gamma)~+~\sum_{E_d<\omega_{\gamma}<E}{B}_{\alpha\gamma,\,\alpha}{B}^*_{\alpha\gamma,\,\alpha}\right) \label{Finaltrace1}
\end{equation}
Let us discuss the two terms in the parentheses. The last term vanishes by energy conservation. In the continuum limit, the first term in the parentheses gives, up to $\lambda$ independent multiplicative factors,
\begin{equation}
\int_\lambda^{E_d}~\frac{d^3\vec{q}}{(2\pi)^32\omega_{\vec{q}}}~\sum_r|F_{\alpha\alpha}(\vec{q},\,\epsilon_r(\vec{q}))|^2
\end{equation}
Since the function $F_{\beta\alpha}(\gamma)$ is smooth in the limits $\lambda,\, |\vec{q}| \to 0$, the integral is of order $E_d^2$ (at most). Therefore, this contribution is suppressed and can be dropped.
For the quadratic traces we get
$$
{\rm Tr} \varepsilon^2 = A_{\alpha\alpha}^2 + A_{\alpha\alpha}^{*\, 2}+2\sum_{\beta} \left(A_{\beta\alpha}A_{\beta\alpha}^* +\sum_{\omega_\gamma>E}B_{\beta\gamma,\,\alpha}B_{\beta\gamma,\,\alpha}^* \right)
$$
\begin{equation}
+2\sum_{\beta}\sum_{\omega_\gamma<{E_d}}\frac{1}{2V^3\omega_\gamma} \left[ A_{\beta\alpha}F_{\beta\alpha}^*(\gamma)\left(f_\alpha(\vec{q}_\gamma)-f_\beta(\vec{q}_\gamma)\right)\cdot \epsilon^*(\gamma)+A_{\beta\alpha}^*F_{\beta\alpha}(\gamma)\left(f_\alpha^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot \epsilon(\gamma)\right] \label{delta2}
\end{equation}
$$
$$
$$
{\rm Tr} \varepsilon^2 \rho_0 = A_{\alpha\alpha}^2 + A_{\alpha\alpha}^{*\, 2}+ A_{\alpha\alpha}A_{\alpha\alpha}^*+\sum_{\beta}\left(A_{\beta\alpha}A_{\beta\alpha}^*+\sum_{\omega_\gamma>E}B_{\beta\gamma,\,\alpha}B_{\beta\gamma,\,\alpha}^* \right)
$$
\begin{equation}
+\sum_{\beta}\sum_{\omega_\gamma<{E_d}}\frac{1}{2V^3\omega_\gamma} \left[ A_{\beta\alpha}F_{\beta\alpha}^*(\gamma)\left(f_\alpha(\vec{q}_\gamma)-f_\beta(\vec{q}_\gamma)\right)\cdot \epsilon^*(\gamma)+A_{\beta\alpha}^*F_{\beta\alpha}(\gamma)\left(f_\alpha^*(\vec{q}_\gamma)-f_\beta^*(\vec{q}_\gamma)\right)\cdot \epsilon(\gamma)\right] \label{delta2rho}
\end{equation}
The last lines in \ref{delta2} and \ref{delta2rho} arise due to the dressing. Notice that since $F_{\beta\alpha}(\gamma)$ is of order $e^3$ and the dressing function of order $e$, the amplitude $A_{\beta\alpha}$ must be computed at tree level, and so it does not exhibit any IR divergences as $\lambda \to 0$. In the continuum limit, these lines give rise to the following integral (up to smooth, nonsingular factors as $\lambda \to 0$ and volume factors):
\begin{equation}
\int_\lambda^{E_d}~\frac{d^3\vec{q}}{(2\pi)^32\omega_{\vec{q}}}~\sum_rF_{\beta\alpha}^*(\vec{q},\,\epsilon_r(\vec{q}))~\sum_{s\in\{\alpha,\,\beta\}}~\frac{e_s\eta_s~p_s\cdot\epsilon_r^*(\vec{q})}{p_s\cdot q} ~+~ h.c.
\end{equation}
Taking into account the measure of integration, the integrand is smooth in the $|\vec{q}| \to 0$ limit. So the integral is of order $E_d$. The last lines in \ref{delta2} and \ref{delta2rho} give negligible contributions to the entanglement entropy.
Now let us compute ${\rm Tr} (\rho_H)^2$ to order $e^6$. It is given by
$$
{\rm Tr} (\rho_H)^2 = {\rm Tr} \rho_0^2 + 2 {\rm Tr} \varepsilon \rho_0 + {\rm Tr} \varepsilon^2
$$
\begin{equation}
= 1 + 2(A_{\alpha\alpha} + A^*_{\alpha\alpha}) + (A_{\alpha\alpha} + A^*_{\alpha\alpha})^2 + 2\sum_{\beta} \left(A_{\beta\alpha}A_{\beta\alpha}^* +\sum_{\omega_\gamma>E}B_{\beta\gamma,\,\alpha}B_{\beta\gamma,\,\alpha}^* \right)
\end{equation}
Using the unitarity relation, \ref{unitarity2}, this simplifies further to
\begin{equation}
{\rm Tr} (\rho_H)^2 = 1 - 2 \Delta
\end{equation}
where
\begin{equation}
\Delta=\sum_{\beta} \sum_{\omega_\gamma<E}B_{\beta\gamma,\,\alpha}B_{\beta\gamma,\,\alpha}^*
\end{equation}
is an order $e^6$ quantity, which depends crucially on the {\it undressed} amplitude to emit a single soft photon with energy $\lambda<\omega_{\gamma}<E$.
Next we consider ${\rm Tr} (\rho_H)^m$ for $m\ge 3$. To order $e^6$, only two structures contribute: ${\rm Tr} \varepsilon \rho_0$ and ${\rm Tr} \varepsilon^2 \rho_0$ with both coefficients being equal to $m$. We get
\begin{equation}
{\rm Tr} (\rho_H)^m = 1 + m {\rm Tr} \varepsilon \rho_0 + m {\rm Tr} \varepsilon^2 \rho_0
\end{equation}
Using \ref{delta2rho} and \ref{unitarity2}, it is easy to see that
\begin{equation}
{\rm Tr} (\rho_H)^m = 1 - m \Delta
\end{equation}
This result is in accordance with the fact that $\rho_H$ has one large eigenvalue, which, to order $e^6$, is equal to $1-\Delta$. All of the rest nonvanishing eigenvalues are of order $e^6$ (or higher), and their sum is equal to $\Delta$. This sum sets the behavior of the leading entanglement entropy.
\subsection{Entanglement entropy}
\noindent
We proceed now to compute the Renyi entropies to leading order in perturbation theory.
For any $m\ge 1$, we obtain
\begin{equation}
S_{m+1} = -\frac{1}{m}\log\left[1 - (m+1) \Delta\right]=\frac{m+1}{m}\Delta
\end{equation}
Using \ref{entR},
the perturbative result for the Renyi entropies and the identity
$$
\sum_{m=0}^n (^{\, n}_{\, m}) (-1)^m =0
$$
we obtain for the leading entanglement entropy
\begin{equation}
S_{ent}=-\Delta~\ln{e^6}
\end{equation}
(The leading entanglement entropy is of order $e^6\ln{e^6}$).
Now $\Delta$ is singular in the limit $\lambda \to 0$. Let us examine the singular part. We have
\begin{equation}
\Delta_{sing}=\sum_{\beta} \sum_{\omega_\gamma<{E_d}}B_{\beta\gamma,\,\alpha}B_{\beta\gamma,\,\alpha}^*
\end{equation}
Using soft photon theorems, we find
\begin{equation}
\Delta_{sing}=\sum_{\beta} e^2 (A_{\beta\alpha}A_{\beta\alpha}^*) \left[\sum_{\omega_\gamma<{E_d}}\frac{1}{(2V\omega_\gamma)}
\sum_{ss^\prime\in\{\alpha,\,\beta\}} \eta_s \eta_{s^\prime} ~\frac{p_sp_{s^\prime}}{(p_sq_\gamma)(p_{s^\prime}q_\gamma)}\right] \label{DeltaSingular}
\end{equation}
where the undressed amplitude $A_{\beta\alpha}$ is computed at tree level.
{\it The same result is obtained in the Fock basis case, in the absence of dressing}. As we have seen, the dressing adds
negligible contributions of order $E_d$ to the entanglement entropy and does not alleviate logarithmic singularities at the leading perturbative level -- see \ref{Finaltrace1}, \ref{delta2} and \ref{delta2rho} and the discussions around them. It would be
interesting to verify this result to all orders in perturbation theory.
\subsection{Continuum limit}\label{continuum}
\noindent
To take the continuum limit, recall that a box single particle state (which is normalizable) is related to a continuum single particle state
(which is $\delta$-function normalizable) by the factor
\begin{equation}
\ket{\vec{p}}_{Box} \to \frac{1}{(2E_{\vec{p}}~V)^{1/2}}\ket{\vec{p}}
\end{equation}
So in the continuum limit we obtain for the singular part of the entanglement entropy
$$
S_{ent,\,sing} = -\frac{e^2}{2\,V^2} \int\frac{d^3\vec{p}_k}{(2\pi)^32E_k}~\int\frac{d^3\vec{p}_l}{(2\pi)^32E_l}~\frac{\ln{e^6}}{2E_i~2E_j}~\left|i{\cal{M}}_{kl}^{ij}\right|^2~\left[(2\pi)^4\delta^4(p_k + p_l - p_i -p_j)\right]^2
$$
\begin{equation}
\times ~\int_{\lambda}^{E_d}\frac{d^3\vec{q}}{(2\pi)^32\omega_{\vec{q}}}~\sum_{ss^\prime\in\{i,j,k,l\}} \eta_s \eta_{s^\prime} ~\frac{p_sp_{s^\prime}}{(p_sq)(p_{s^\prime}q)} \label{EntSingular}
\end{equation}
where $i{\cal{M}}_{kl}^{ij}$ is the invariant amplitude for the process $e_i + e_j \to e_k + e_l$ (Moller scattering), given in terms of tree level Feynman diagrams. Integration over $\vec{p}_l$ imposes momentum conservation, $\vec{p}_l=\vec{p}_i+\vec{p}_j-\vec{p}_k$, and yields an additional volume factor in the numerator ($(2\pi)^3 \delta^3(0)=V$). Integrating in addition over the soft photon momentum yields the logarithmically divergent factor
\begin{equation}
S_{ent,\,sing} = -\frac{1}{V}~\ln\left(\frac{E_d}{\lambda}\right)~ \int\frac{d^3\vec{p}_k}{(2\pi)^32E_k}~\frac{\ln{e^6}}{8E_iE_jE_l}~\left|i{\cal{M}}_{kl}^{ij}\right|^2~{\cal{B}}_{kl,\,ij}~\left[(2\pi)\delta(E_k + E_l - E_i -E_j)\right]^2
\end{equation}
where ${\cal{B}}_{kl,\,ij}={\cal{B}}_{\beta\alpha}$ is given by \ref{B}.
We let the incoming electrons have opposite momenta along the $z$-axis, $\vec{p}_i=-\vec{p}_j=p_0 \hat{z}$, working in the center of mass frame. Without loss of generality we take $p_0$ to be positive. The center of mass energy is $E_{cm} =2E_i= 2\sqrt{p_0^2 + m^2}$.
Thus in this frame, we may set $\vec{p}_k=-\vec{p}_l=p^{\prime} \hat{k}$ and $E_k=E_l =\sqrt{p^{\prime\, 2}+m^2}$. Integration over the magnitude of $\vec{p}_k$ imposes energy conservation, $|p^{\prime}|=p_0$ (or $E_k = E_l = E_{cm}/2$), and yields a factor of $2\pi \delta (E_i - E_i)=T$, with $T$ the timescale of the experiment. We finally obtain
\begin{equation}
S_{ent,\, sing} = -\frac{T\,v_{rel}}{16\,V}~\ln\left(\frac{E_d}{\lambda}\right)~ \int\frac{d^2\hat{k}}{(2\pi)^2}~\frac{\ln{e^6}}{E_{cm}^2}\left|i{\cal{M}}_{kl}^{ij}\right|^2~{\cal{B}}_{kl,\,ij} \label{Entd}
\end{equation}
where $v_{rel}=2p_0/E_i=4p_0/E_{cm}$ is the relative velocity of the particles.
Now $v_{rel}/V$ is the flux of particle $j$ with respect to particle $i$ (and vice versa). We define the entanglement entropy per flux per unit time, $s_{ent}$, to find
\begin{equation}
s_{ent,\, sing} = -\frac{1}{16}~\ln\left(\frac{E_d}{\lambda}\right)~ \int\frac{d^2\hat{k}}{(2\pi)^2}~\frac{\ln{e^6}}{E_{cm}^2}\left|i{\cal{M}}_{kl}^{ij}\right|^2~{\cal{B}}_{kl,\,ij}
\end{equation}
Notice that the integrand is a function of the scattering angle $\theta$ ($\cos\theta = \hat{k}\cdot \hat{z}$). For slowly moving particles, the Moller amplitude squared (averaged over spin polarizations) scales as $\left|i{\cal{M}}_{kl}^{ij}\right|^2 \sim e^4 m^4/p_0^4\sin^4\theta$. Likewise ${\cal{B}}_{kl,\,ij}$ scales as $\sin^2\theta$. So the integrand diverges for forward ($\theta=0$) and backward ($\theta=\pi$) scattering. However, scattering at $\theta = 0$ or $\theta = \pi$ cannot be distinguished from no scattering. This introduces an effective lower and upper cutoff $\theta_0\le \theta \le \pi -\theta_0$ on the scattering angle, which regularizes the above integral.
Recall that to leading order only single photon particle states contribute to the traces and the entanglement entropy -- see \ref{DeltaSingular}. The dimensionality of the subspace of single photon states with frequency less than $E_d$ scales as $D\sim (E_dL)^3$, where $L$ is the size of the box (and becomes infinite in the continuum limit). In fact the entanglement entropy between the soft and the hard particles cannot exceed $\log D$. Taking $\lambda$ to be of order $1/L$, we see that the dominant contribution to the entanglement entropy \ref{Entd} is a fraction of the maximum possible value.
Thus the perturbative calculation of the entanglement entropy associated with tracing over all soft photons in ${\cal{H}}_S$ breaks down in the strict $\lambda \to 0$ limit. The logarithmic divergences in $\lambda$ do not cancel order by order in perturbation theory. One may wonder if the entanglement entropy to all orders is finite in the continuum limit, since the reduced density matrix is dominated by the diagonal elements which are free of any IR divergences. Notice that the diagonal element \ref{Ddiagonal} scales inversely proportional with a power of the volume $V$ in the continuum limit. So the entanglement entropy per flux per unit time is expected to diverge logarithmically in the volume: $s_{ent} \sim \log V$. At the intuitive level, this behavior can be understood as follows. The density matrix becomes very incoherent in this limit. We expect the number of its nonzero eigenvalues to be of order the dimensionality $D_S$ of ${\cal{H}}_S$ \footnote{Suppose a quantum system consists of two subsystems $A$ and $B$, with dimensionalities $D_B > D_A$. Let the whole system be in a pure state. Then the density matrices describing the two subsystems have equal non-zero eigenvalues. When maximal disorder is reached, the number of non-zero eigenvalues attains its maximal possible value, set by the smaller dimensionality $D_A$. Moreover, the non-zero eigenvalues become equal to each other, and so equal to $1/D_A$. The entanglement entropy is equal to $\log D_A$.}, and each to scale with $1/D_S \sim 1/V$. The entanglement entropy scales with $\log V$.
We emphasize that the singular part of the entanglement entropy, \ref{DeltaSingular} (and \ref{EntSingular} in the continuum limit), does not depend on the details of the Faddeev-Kulish dressing. As we have already explained, precisely the same expression is obtained in the Fock basis computation, where the initial state is taken to be undressed. The structure of the expression is suggestive of a universal applicability to generic scattering processes. Namely, the leading entanglement entropy is given as a sum over transition probabilities (for the initial state $\alpha$ to scatter to a final hard state $\beta$), with each probability weighted by a soft photon factor. Integration over the soft photon momentum gives rise to the logarithmic singularity in $\lambda$. It would be interesting to see if and how higher order corrections in the electron coupling modify this structure.
It is interesting to contrast our findings concerning the entanglement between soft and hard degrees of freedom after scattering with other examples of entanglement in quantum field theory, such as the entanglement between the local degrees of freedom associated with a region of space and the degrees of freedom of its complement -- See e.g. \cite{Rangamani}. The entanglement entropy in this case is UV divergent (unlike the case studied in this work, where the divergence is infrared in nature), with the divergences arising from local effects. As the case at hand, the coefficients of the singular terms are universal and contain physical information. For example, the coefficient of the leading quadratic term in the UV cutoff is proportional to the area of the boundary of the region and the number of degrees of freedom of the field theory. Likewise the coefficient of the logarithmic singular term scales with the number of degrees of freedom and depends on the shape of the boundary via an integral of $K^2$, where $K$ is the trace of the second fundamental form of the induced metric on the boundary surface \cite{MaldacenaENT,Solodukhin}.
The coefficient of the IR logarithmic singularity in our case also contains physical information. The soft photon factor in the last line of \ref{EntSingular} gives
${\cal{B}}_{kl,\,ij}~\ln\left(E_d/\lambda\right)$,
with ${\cal{B}}_{kl,\,ij}$ given by \ref{B}. In terms of Mandelstam variables we obtain
\begin{equation}
{\cal{B}}_{kl,\,ij}~=~\frac{e^2}{4\pi^2}~\left[~\frac{1-\frac{2m^2}{t}}{\sqrt{1-\frac{4m^2}{t}}}\ln\left(\frac{1-\frac{2m^2}{t}+\sqrt{1-\frac{4m^2}{t}}}{1-\frac{2m^2}{t}-\sqrt{1-\frac{4m^2}{t}}}\right)~+~\left(t \leftrightarrow u\right)~-~\left(t \leftrightarrow s\right)~-~2~\right]
\end{equation}
Let us consider high energy and small angle scattering, keeping $t$ to be large and fixed. In this limit, we get
\begin{equation}
{\cal{B}}_{kl,\,ij}~\simeq~\frac{e^2}{4\pi^2}\ln\left(\frac{|t|}{4m^2}\right)
\end{equation}
and so the soft photon factor gives rise to a double log contribution. The coefficient then becomes equal to the cusp anomalous dimension in QED, $\Gamma({\varphi},\;\alpha)$ via the relation $|t|=2m^2 (\cosh{\varphi} - 1)$, controlling the vacuum expectation value of a Wilson loop with a cusp of angle $\varphi$ -- See e.g. \cite{Korchemsky} and \cite{Sever2} for discussions.
\subsection{Soft radiation and entanglement}
\noindent
We proceed now to study the reduced density matrix obtained by tracing over soft radiation photons with frequencies $E_d<\omega_\gamma<E$, as advocated also in \cite{Gomez2}. This tracing is motivated by the fact that starting with initial dressed states, the amplitude to emit a photon with energy below the dressing scale $E_d$ is suppressed. The density matrix takes the following form
$$
\rho_{asym}=\ket{\alpha}_d\bra{\alpha}_d + \left(\tilde{A}_{\beta\alpha}\ket{\beta}_d~+~\sum_{\omega_\gamma <E_d, \,\omega_\gamma>E}\tilde{B}_{\beta\gamma,\,\alpha}\ket{\beta\gamma}_d~+~\dots\right)\bra{\alpha}_d
$$
$$
+\ket{\alpha}_d\left(\tilde{A}^*_{\beta^\prime\alpha}\bra{\beta^\prime}_d~+~\sum_{\omega_{\gamma^\prime}<E_d,\,\,\omega_{\gamma^\prime}>E}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\bra{\beta^\prime\gamma^\prime}_d~+~\dots\right)
$$
$$
+\left(\tilde{A}_{\beta\alpha}\tilde{A}^*_{\beta^\prime\alpha}~+~\sum_{E_d<\omega_\gamma<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta^\prime\gamma,\,\alpha}\right)\ket{\beta}_d\bra{\beta^\prime}_d~+~\sum_{\omega_\gamma,\omega_{\gamma^\prime}<E_d,\,\,\omega_\gamma,\omega_{\gamma^\prime}>E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha} \ket{\beta\gamma}_d\bra{\beta^\prime\gamma^\prime}_d
$$
\begin{equation}
+ \sum_{\omega_\gamma<E_d,\,\omega_\gamma>E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{A}^*_{\beta^\prime\alpha}\ket{\beta\gamma}_d\bra{\beta^\prime}_d~+~\sum_{\omega_{\gamma^\prime}<E_d,\,\omega_{\gamma^\prime}>E}\tilde{B}^*_{\beta^\prime\gamma^\prime,\,\alpha}\tilde{A}_{\beta\alpha}\ket{\beta}_d\bra{\beta^\prime\gamma^\prime}_d ~+~ \dots
\end{equation}
This density matrix is an operator acting on the space of asymptotic states. The matrix elements are given exclusively in terms of dressed amplitudes -- the overlaps $\bra{f_{\beta^\prime}}\ket{f_\beta}$ are absent. Thus, no IR divergences appear in the $\lambda \to 0$ limit at any finite order in perturbation theory. Moreover, the off-diagonal elements remain nonvanishing in the $\lambda \to 0$ limit (as compared with the diagonal elements), to all orders in perturbation theory. The density matrix does not exhibit decoherence. Since the dressed amplitude $\tilde{B}_{\beta\gamma,\,\alpha}$ to emit a photon of energy less than $E_d$ is suppressed, the contributions of various sums over photon frequencies smaller than $E_d$ can be neglected.
Now let us compute the entanglement entropy to leading order in perturbation theory. As before we set $\varepsilon=\rho_{asym}-\rho_0$, where now $\rho_0=\ket{\alpha}_d\bra{\alpha}_d$, $\rho_0^2=\rho_0$ and ${\rm Tr} \varepsilon =0$ by unitarity. At leading order ($e^6$), the only nonvanishing traces are
\begin{equation}
{\rm Tr} \varepsilon\rho_0 ~=~ \tilde{A}_{\alpha\alpha} ~+~ \tilde{A}_{\alpha\alpha}^* ~+~ \tilde{A}_{\alpha\alpha}\tilde{A}_{\alpha\alpha}^* ~+~ \sum_{E_d<\omega_\gamma<E}\tilde{B}_{\alpha\gamma,\,\alpha}\tilde{B}^*_{\alpha\gamma,\,\alpha}
\end{equation}
$$
$$
\begin{equation}
{\rm Tr} \varepsilon^2 ~=~ \tilde{A}_{\alpha\alpha}^2 ~+~ \tilde{A}_{\alpha\alpha}^{*\,2}~+~2~\sum_{\beta}\left[(1 + \tilde{A}_{\alpha\alpha} + \tilde{A}_{\alpha\alpha}^*)\tilde{A}_{\beta\alpha}\tilde{A}_{\beta\alpha}^* ~+~ \sum_{\omega_\gamma<E_d,\,\omega_{\gamma}>E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}\right]
\end{equation}
$$
$$
$$
{\rm Tr} \varepsilon^2\rho_0 ~=~\tilde{A}_{\alpha\alpha}^2~+~\tilde{A}_{\alpha\alpha}^{*\,2}~+~\tilde{A}_{\alpha\alpha}\tilde{A}_{\alpha\alpha}^*(1+\tilde{A}_{\alpha\alpha}+\tilde{A}_{\alpha\alpha}^*)
$$
\begin{equation}
+~\sum_{\beta}\left[(1+ \tilde{A}_{\alpha\alpha} + \tilde{A}_{\alpha\alpha}^*) \tilde{A}_{\beta\alpha} \tilde{A}_{\beta\alpha}^* ~+~ \sum_{\omega_\gamma<E_d,\,\omega_{\gamma}>E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}\right]
\end{equation}
These can be further simplified using \ref{unitarity2}. Also to order $e^6$
$$
\sum_{E_d<\omega_{\gamma}<E}\tilde{B}_{\alpha\gamma,\,\alpha}\tilde{B}^*_{\alpha\gamma,\,\alpha}~=~\sum_{E_d<\omega_{\gamma}<E}B_{\alpha\gamma,\,\alpha}B^*_{\alpha\gamma,\,\alpha}=0
$$
the latter vanishing by energy conservation.
Thus
$$
{\rm Tr}(\rho_H)^2 ~=~ 1~+~2{\rm Tr}\varepsilon\rho_0~+~{\rm Tr}\varepsilon^2~=~1~+~2~(\tilde{A}_{\alpha\alpha} + \tilde{A}_{\alpha\alpha})
$$
\begin{equation}
+~2~\sum_{\beta}\left( \tilde{A}_{\beta\alpha} \tilde{A}_{\beta\alpha}^* ~+~ \sum_{\omega_\gamma<E_d,\,\omega_{\gamma}>E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}\right)~=~1~-~2~\sum_{\beta}\sum_{E_d<\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}
\end{equation}
We used \ref{unitarity2} and we dropped terms of order $e^6$. Likewise we can show
\begin{equation}
{\rm Tr}(\rho_H)^m ~=~ 1~+~m{\rm Tr}\varepsilon\rho_0~+~{\rm Tr}\varepsilon^2\rho_0~=~1~-~m~\sum_{\beta}\sum_{E_d<\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}
\end{equation}
The Renyi entropies and the entanglement entropy are given by
\begin{equation}
S_{m+1} =\frac{m+1}{m}\sum_{\beta}\sum_{E_d<\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha},\,\,\, m\ge 1
\end{equation}
and
\begin{equation}
S_{ent}= -\ln{e^6}~\sum_{\beta}\sum_{E_d<\omega_{\gamma}<E}\tilde{B}_{\beta\gamma,\,\alpha}\tilde{B}^*_{\beta\gamma,\,\alpha}
\end{equation}
To order $e^6\ln{e^6}$, this is given by
\begin{equation}
S_{ent}= -\ln{e^6}~\sum_{\beta}\sum_{E_d<\omega_{\gamma}<E}{B}_{\beta\gamma,\,\alpha}{B}^*_{\beta\gamma,\,\alpha}
\end{equation}
where the dressing scale $E_d$ provides the lower cutoff. In particular, $E_d$ is kept finite in the continuum, $\lambda \to 0$ limit, and so the leading perturbative entanglement entropy is finite.
Letting the energy scale $E$ to be sufficiently small and repeating steps as in the previous section, we obtain for the entanglement entropy per unit flux per unit time in the continuum limit:
\begin{equation}
s_{ent} = -\frac{1}{16}~\ln\left(\frac{E}{E_d}\right)~ \int\frac{d^2\hat{k}}{(2\pi)^2}~\frac{\ln{e^6}}{E_{cm}^2}\left|i{\cal{M}}_{kl}^{ij}\right|^2~{\cal{B}}_{kl,\,ij} \label{entfinite}
\end{equation}
This quantity is finite in the $\lambda \to 0$ limit. Notice that as $E \to E_d$, the entanglement entropy becomes vanishingly small. In particular the radiated soft photons carry little information. We should emphasize that expression \ref{entfinite} arises when the IR scales $E$ and $E_d$ are sufficiently small. In general the expression for the entanglement entropy per unit flux per unit time in the continuum limit will be more complicated, depending on the scales $E_d$ and $E$.
\section{Conclusions}\label{s5}
\noindent
In this paper we studied the entanglement between the hard and soft particles produced during a typical scattering process of Faddeev-Kulish electrons in QED. Tracing over the entire spectrum of soft photons leads to decoherence and infrared divergences in the perturbative expansion for the entanglement entropy. To leading order, the entanglement entropy is set by the conventional Fock basis amplitude squared for real single soft photon emission, leading to a logarithmic infrared divergence when integrated over the soft momentum. The same result is obtained in a Fock basis computation, where the initial state consists of two bare electrons. In particular, the singular part of the entanglement entropy does not depend on the details of the Faddeev-Kulish dressing. The expression is suggestive for a universal applicability to generic scattering processes. For the case of Faddeev-Kulish electrons though the divergence can be traced in the overlap of the coherent states describing the soft photon clouds that accompany the asymptotic charged particles. Thus there is strong entanglement between the final state hard charged particles and the photons in the clouds.
By suitably modifying the dressing function to subleading order in the soft momentum, one can show that the Faddeev-Kulish
amplitudes for the emission of soft photons with energies less than $E_d$, the characteristic energy of photons in the clouds, are suppressed (of order $E_d$), at least at tree level \cite{Choi2}. This suggests that the soft part of the emitted radiation consists of low energy photons with energy greater than the dressing scale $E_d$. Taking a partial trace over these soft radiative photons produces a well defined density matrix, free of any infrared divergences order by order in perturbation theory. The reduced density matrix is now an operator acting on the space of asymptotic states, and does not exhibit large amount of decoherence \cite{Gomez2}. The entanglement entropy is free of any infrared divergences at any order in the perturbative expansion. As the energy set by the resolution of the detector approaches the effective cutoff scale $E_d$, provided by the dressing, the leading entanglement entropy becomes vanishingly small, suggesting that a small amount of information is carried by the soft radiated photons. It would be interesting to see if the suppression of the Faddeev-Kulish amplitudes for the emission of soft photons with energies less than $E_d$ persists at the one loop level, since then logarithmic corrections in the soft photon frequency appear. One would need to consider higher order corrections to the dressing function to implement this task.
It would also be interesting to investigate the applicability of our results to the case of gravity. At least the perturbative analysis in this work suggests strong correlations between the hard particles produced in a scattering process and the soft gravitons present in the clouds accompanying them. Conservation laws associated with large gauge transformations (supertranslations and superrotations) require the hard Hawking quanta produced during the process of formation/evaporation of a black hole, to be accompanied by clouds of soft gravitons and photons \cite{StromingerBHinfo,HPS}. Despite the entanglement between these hard and soft degrees of freedom, it is difficult to see how black hole evaporation would result in a pure state of properly dressed, asymptotic particles, without invoking correlations between early and late time Hawking quanta \cite{Page}. Arguments suggesting the decoupling of soft variables from the hard dynamics seem to support this point of view \cite{Porrati2}.
\section*{Acknowledgements} We thank C. Bachas and E. Kiritsis for useful discussions. N.T. wishes to thank the ITCP and the Department of Physics of the University of Crete where parts of this work were done for hospitality.
\noindent
\bigskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,068
|
//
// ViewController.m
// Overlay-Graphics
//
// Created by Sridhar on 28/02/14.
// Copyright (c) 2014 Tokbox. All rights reserved.
//
#import "ViewController.h"
#import <OpenTok/OpenTok.h>
#import "TBExampleVideoView.h"
#import "TBExampleOverlayView.h"
#import "TBExampleVideoCapture.h"
@interface ViewController () <OTSessionDelegate, OTSubscriberKitDelegate, OTPublisherDelegate, TBExampleVideoViewDelegate>
@property (nonatomic) OTSession *session;
@property (nonatomic) OTPublisher *publisher;
@property (nonatomic) OTSubscriber *subscriber;
@property (nonatomic) TBExampleVideoView *publisherVideoView;
@property (nonatomic) TBExampleVideoView *subscriberVideoView;
@end
@implementation ViewController
static double widgetHeight = 240;
static double widgetWidth = 320;
// *** Fill the following variables using your own Project info ***
// *** https://dashboard.tokbox.com/projects ***
// Replace with your OpenTok API key
static NSString *const kApiKey = @"";
// Replace with your generated session ID
static NSString *const kSessionId = @"";
// Replace with your generated token
static NSString *const kToken = @"";
#pragma mark - View lifecycle
- (void)viewDidLoad
{
[super viewDidLoad];
// Step 1: As the view comes into the foreground, initialize a new instance
// of OTSession and begin the connection process.
_session = [[OTSession alloc] initWithApiKey:kApiKey
sessionId:kSessionId
delegate:self];
[self doConnect];
}
- (BOOL)prefersStatusBarHidden
{
return YES;
}
- (BOOL)shouldAutorotate {
return UIUserInterfaceIdiomPhone != [[UIDevice currentDevice] userInterfaceIdiom];
}
#pragma mark - OpenTok methods
/**
* Asynchronously begins the session connect process. Some time later, we will
* expect a delegate method to call us back with the results of this action.
*/
- (void)doConnect
{
OTError *error = nil;
[_session connectWithToken:kToken error:&error];
if (error)
{
[self showAlert:[error localizedDescription]];
}
}
/**
* Sets up an instance of OTPublisher to use with this session. OTPubilsher
* binds to the device camera and microphone, and will provide A/V streams
* to the OpenTok session.
*/
- (void)doPublish
{
OTPublisherSettings *pubSettings = [[OTPublisherSettings alloc] init];
pubSettings.name = [[UIDevice currentDevice] name];
_publisher = [[OTPublisher alloc]
initWithDelegate:self
settings:pubSettings];
TBExampleVideoCapture* videoCapture =
[[[TBExampleVideoCapture alloc] init] autorelease];
[_publisher setVideoCapture:videoCapture];
_publisherVideoView =
[[TBExampleVideoView alloc] initWithFrame:CGRectMake(0,0,1,1)
delegate:self
type:OTVideoViewTypePublisher
displayName:nil];
// Set mirroring only if the front camera is being used.
[_publisherVideoView.videoView setMirroring:
(AVCaptureDevicePositionFront == videoCapture.cameraPosition)];
[_publisher setVideoRender:_publisherVideoView];
OTError *error = nil;
[_session publish:_publisher error:&error];
if (error)
{
[self showAlert:[error localizedDescription]];
}
[_publisherVideoView setFrame:CGRectMake(0, 0, widgetWidth, widgetHeight)];
[self.view addSubview:_publisherVideoView];
}
/**
* Cleans up the publisher and its view. At this point, the publisher should not
* be attached to the session any more.
*/
- (void)cleanupPublisher {
[_publisher.view removeFromSuperview];
_publisher = nil;
// this is a good place to notify the end-user that publishing has stopped.
}
/**
* Instantiates a subscriber for the given stream and asynchronously begins the
* process to begin receiving A/V content for this stream. Unlike doPublish,
* this method does not add the subscriber to the view hierarchy. Instead, we
* add the subscriber only after it has connected and begins receiving data.
*/
- (void)doSubscribe:(OTStream*)stream
{
_subscriber = [[OTSubscriber alloc] initWithStream:stream
delegate:self];
_subscriberVideoView =
[[TBExampleVideoView alloc] initWithFrame:CGRectMake(0,0,1,1)
delegate:self
type:OTVideoViewTypeSubscriber
displayName:nil];
[_subscriber setVideoRender:_subscriberVideoView];
OTError *error = nil;
[_session subscribe:_subscriber error:&error];
if (error)
{
[self showAlert:[error localizedDescription]];
}
_subscriber.audioLevelDelegate = self;
}
/**
* Cleans the subscriber from the view hierarchy, if any.
* NB: You do *not* have to call unsubscribe in your controller in response to
* a streamDestroyed event. Any subscribers (or the publisher) for a stream will
* be automatically removed from the session during cleanup of the stream.
*/
- (void)cleanupSubscriber
{
[_subscriber.view removeFromSuperview];
_subscriber = nil;
}
# pragma mark - OTSession delegate callbacks
- (void)sessionDidConnect:(OTSession*)session
{
NSLog(@"sessionDidConnect (%@)", session.sessionId);
// Step 2: We have successfully connected, now instantiate a publisher and
// begin pushing A/V streams into OpenTok.
[self doPublish];
}
- (void)sessionDidDisconnect:(OTSession*)session
{
NSString* alertMessage =
[NSString stringWithFormat:@"Session disconnected: (%@)",
session.sessionId];
NSLog(@"sessionDidDisconnect (%@)", alertMessage);
}
- (void)session:(OTSession*)mySession
streamCreated:(OTStream *)stream
{
NSLog(@"session streamCreated (%@)", stream.streamId);
// Step 3a: Begin subscribing to a stream we
// have seen on the OpenTok session.
if (nil == _subscriber)
{
[self doSubscribe:stream];
}
}
- (void)session:(OTSession*)session
streamDestroyed:(OTStream *)stream
{
NSLog(@"session streamDestroyed (%@)", stream.streamId);
if ([_subscriber.stream.streamId isEqualToString:stream.streamId])
{
[self cleanupSubscriber];
}
}
- (void) session:(OTSession *)session
connectionCreated:(OTConnection *)connection
{
NSLog(@"session connectionCreated (%@)", connection.connectionId);
}
- (void) session:(OTSession *)session
connectionDestroyed:(OTConnection *)connection
{
NSLog(@"session connectionDestroyed (%@)", connection.connectionId);
if ([_subscriber.stream.connection.connectionId
isEqualToString:connection.connectionId])
{
[self cleanupSubscriber];
}
}
- (void) session:(OTSession*)session
didFailWithError:(OTError*)error
{
NSLog(@"didFailWithError: (%@)", error);
}
# pragma mark - OTSubscriber delegate callbacks
- (void)subscriberDidConnectToStream:(OTSubscriberKit*)subscriber
{
NSLog(@"subscriberDidConnectToStream (%@)",
subscriber.stream.connection.connectionId);
[_subscriberVideoView setFrame:CGRectMake(0, widgetHeight, widgetWidth,
widgetHeight)];
[self.view addSubview:_subscriberVideoView];
}
- (void)subscriber:(OTSubscriberKit*)subscriber
didFailWithError:(OTError*)error
{
NSLog(@"subscriber %@ didFailWithError %@",
subscriber.stream.streamId,
error);
}
# pragma mark - OTPublisher delegate callbacks
- (void)publisher:(OTPublisherKit *)publisher
streamCreated:(OTStream *)stream
{
NSLog(@"Publishing");
}
- (void)publisher:(OTPublisherKit*)publisher
streamDestroyed:(OTStream *)stream
{
if ([_subscriber.stream.streamId isEqualToString:stream.streamId])
{
[self cleanupSubscriber];
}
[self cleanupPublisher];
}
- (void)publisher:(OTPublisherKit*)publisher
didFailWithError:(OTError*) error
{
NSLog(@"publisher didFailWithError %@", error);
[self cleanupPublisher];
}
- (void)showAlert:(NSString *)string
{
// show alertview on main UI
dispatch_async(dispatch_get_main_queue(), ^{
UIAlertController *alertVC = [UIAlertController alertControllerWithTitle:@"Message from video session"
message:string
preferredStyle:UIAlertControllerStyleAlert];
[self presentViewController:alertVC animated:YES completion:nil];
});
}
- (void) session:(OTSession*)session
archiveStartedWithId:(NSString *)archiveId
name:(NSString *)name
{
NSLog(@"session archiving started with id:%@ name:%@", archiveId, name);
TBExampleOverlayView *overlayView =
[(TBExampleVideoView *)[_publisher view] overlayView];
[overlayView startArchiveAnimation];
}
- (void) session:(OTSession*)session
archiveStoppedWithId:(NSString *)archiveId
{
NSLog(@"session archiving stopped with id:%@", archiveId);
TBExampleOverlayView *overlayView =
[(TBExampleVideoView *)[_publisher view] overlayView];
[overlayView stopArchiveAnimation];
}
- (void)subscriberVideoDisabled:(OTSubscriberKit*)subscriber
reason:(OTSubscriberVideoEventReason)reason
{
[(TBExampleVideoView*)subscriber.videoRender audioOnlyView].hidden = NO;
if (reason == OTSubscriberVideoEventQualityChanged)
[[(TBExampleVideoView*)subscriber.videoRender overlayView]
showVideoDisabled];
_subscriber.audioLevelDelegate = self;
}
- (void)subscriberVideoEnabled:(OTSubscriberKit*)subscriber
reason:(OTSubscriberVideoEventReason)reason
{
[(TBExampleVideoView*)subscriber.videoRender audioOnlyView].hidden = YES;
if (reason == OTSubscriberVideoEventQualityChanged)
[[(TBExampleVideoView*)subscriber.videoRender overlayView] resetView];
_subscriber.audioLevelDelegate = nil;
}
- (void)subscriberVideoDisableWarning:(OTSubscriberKit*)subscriber
{
NSLog(@"subscriberVideoDisableWarning");
[[(TBExampleVideoView*)subscriber.videoRender overlayView]
showVideoMayDisableWarning];
}
- (void)subscriberVideoDisableWarningLifted:(OTSubscriberKit*)subscriber
{
NSLog(@"subscriberVideoDisableWarningLifted");
[[(TBExampleVideoView*)subscriber.videoRender overlayView] resetView];
}
- (void)subscriber:(OTSubscriberKit *)subscriber
audioLevelUpdated:(float)audioLevel{
float db = 20 * log10(audioLevel);
float floor = -40;
float level = 0;
if (db > floor) {
level = db + fabsf(floor);
level /= fabsf(floor);
}
_subscriberVideoView.audioLevelMeter.level = level;
}
#pragma mark - OTVideoViewDelegate
- (void)videoViewDidToggleCamera:(TBExampleVideoView*)videoView {
if (videoView == _publisherVideoView) {
[((TBExampleVideoCapture*)_publisher.videoCapture) toggleCameraPosition];
}
}
- (void)videoView:(TBExampleVideoView*)videoView
publisherWasMuted:(BOOL)publisherMuted
{
[_publisher setPublishAudio:!publisherMuted];
}
- (void)videoView:(TBExampleVideoView*)videoView
subscriberVolumeWasMuted:(BOOL)subscriberMuted
{
[_subscriber setSubscribeToAudio:!subscriberMuted];
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,491
|
Asceticists 2006 is the eighteenth studio album by power electronics group Whitehouse, released in 2006 through Susan Lawly. It was reissued on vinyl format through Very Friendly in October 2008.
Track listing
Personnel
William Bennett - vocals, synthesizers, production
Philip Best - vocals, synthesizers
Denis Blackham - mastering
References
External links
2006 albums
Whitehouse (band) albums
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,297
|
package pa5_EC;
/**
* Project 5: Starbucks Simulation
*
* Purpose: Store the fields relevant to a customer
*
* @author MichaelSetteducati
*
*/
public class Customer {
/**
* Instance variable stores name of Customer
*/
private String name;
/**
* Instance variable stores arrival time of Customer
*/
private int arrivalTime;
/**
* Instance variable stores serviceTime required for customer
*/
private int serviceTime;
/**
* Constructor for Customer
*
* @param name Name of customer
* @param arrivalTime ArrivalTime of customer
* @param serviceTime Time required to prepare customer's drink
*/
public Customer(String name, int serviceTime, int arrivalTime) {
this.name = name;
this.arrivalTime = arrivalTime;
this.serviceTime = serviceTime;
}
/**
* Getter method for Name
*
* @return Name of customer
*/
public String getName() {
return name;
}
/**
* Getter method for arrivalTime
*
* @return ArrivalTime of customer
*/
public int getArrivalTime() {
return arrivalTime;
}
/**
* Getter method for serviceTime
*
* @return ServiceTime required for this customer
*/
public int getServiceTime() {
return serviceTime;
}
/**
* toString method for customer returns a String representation of Customer in the following format:
* <Name> - <ServiceTime> (<ArrivalTime>)
*
* @return String representation of Customer
*/
public String toString(){
return this.name + " - (serviceTime: " + this.serviceTime + ") (arrivalTime: " + this.getArrivalTime() + ")";
}
/**
* Equals method overrides the default equals method for Object
*
* @return True if this customer equals o, false otherwise
*/
@Override
public boolean equals(Object o) {
if(!(o instanceof Customer))
return false;
else if(this.name.equals( ( (Customer) o).getName() ) ) //If names are the same, customer is the same
return true;
return false;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 799
|
LuCI - Lua Configuration Interface
Copyright 2008 Steven Barth <steven@midlink.org>
Copyright 2008 Jo-Philipp Wich <xm@leipzig.freifunk.net>
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
$Id: wifi.lua 8953 2012-08-08 20:07:36Z jow $
]]--
-- Data init --
local fs = require "nixio.fs"
local sys = require "luci.sys"
local uci = require "luci.model.uci".cursor()
if not uci:get("network", "wan") then
uci:section("network", "interface", "wan", {proto="none", ifname=" "})
uci:save("network")
uci:commit("network")
end
local wlcursor = luci.model.uci.cursor_state()
local wireless = wlcursor:get_all("wireless")
local wifidevs = {}
local ifaces = {}
for k, v in pairs(wireless) do
if v[".type"] == "wifi-iface" then
table.insert(ifaces, v)
end
end
wlcursor:foreach("wireless", "wifi-device",
function(section)
table.insert(wifidevs, section[".name"])
end)
-- Main Map --
m = Map("wireless", translate("Wifi"), translate("Here you can configure installed wifi devices."))
m:chain("network")
-- Status Table --
s = m:section(Table, ifaces, translate("Networks"))
link = s:option(DummyValue, "_link", translate("Link"))
function link.cfgvalue(self, section)
local ifname = self.map:get(section, "ifname")
local iwinfo = sys.wifi.getiwinfo(ifname)
return iwinfo and "%d/%d" %{ iwinfo.quality, iwinfo.quality_max } or "-"
end
essid = s:option(DummyValue, "ssid", "ESSID")
bssid = s:option(DummyValue, "_bsiid", "BSSID")
function bssid.cfgvalue(self, section)
local ifname = self.map:get(section, "ifname")
local iwinfo = sys.wifi.getiwinfo(ifname)
return iwinfo and iwinfo.bssid or "-"
end
channel = s:option(DummyValue, "channel", translate("Channel"))
function channel.cfgvalue(self, section)
return wireless[self.map:get(section, "device")].channel
end
protocol = s:option(DummyValue, "_mode", translate("Protocol"))
function protocol.cfgvalue(self, section)
local mode = wireless[self.map:get(section, "device")].mode
return mode and "802." .. mode
end
mode = s:option(DummyValue, "mode", translate("Mode"))
encryption = s:option(DummyValue, "encryption", translate("<abbr title=\"Encrypted\">Encr.</abbr>"))
power = s:option(DummyValue, "_power", translate("Power"))
function power.cfgvalue(self, section)
local ifname = self.map:get(section, "ifname")
local iwinfo = sys.wifi.getiwinfo(ifname)
return iwinfo and "%d dBm" % iwinfo.txpower or "-"
end
scan = s:option(Button, "_scan", translate("Scan"))
scan.inputstyle = "find"
function scan.cfgvalue(self, section)
return self.map:get(section, "ifname") or false
end
-- WLAN-Scan-Table --
t2 = m:section(Table, {}, translate("<abbr title=\"Wireless Local Area Network\">WLAN</abbr>-Scan"), translate("Wifi networks in your local environment"))
function scan.write(self, section)
m.autoapply = false
t2.render = t2._render
local ifname = self.map:get(section, "ifname")
local iwinfo = sys.wifi.getiwinfo(ifname)
if iwinfo then
local _, cell
for _, cell in ipairs(iwinfo.scanlist) do
t2.data[#t2.data+1] = {
Quality = "%d/%d" %{ cell.quality, cell.quality_max },
ESSID = cell.ssid,
Address = cell.bssid,
Mode = cell.mode,
["Encryption key"] = cell.encryption.enabled and "On" or "Off",
["Signal level"] = "%d dBm" % cell.signal,
["Noise level"] = "%d dBm" % iwinfo.noise
}
end
end
end
t2._render = t2.render
t2.render = function() end
t2:option(DummyValue, "Quality", translate("Link"))
essid = t2:option(DummyValue, "ESSID", "ESSID")
function essid.cfgvalue(self, section)
return self.map:get(section, "ESSID")
end
t2:option(DummyValue, "Address", "BSSID")
t2:option(DummyValue, "Mode", translate("Mode"))
chan = t2:option(DummyValue, "channel", translate("Channel"))
function chan.cfgvalue(self, section)
return self.map:get(section, "Channel")
or self.map:get(section, "Frequency")
or "-"
end
t2:option(DummyValue, "Encryption key", translate("<abbr title=\"Encrypted\">Encr.</abbr>"))
t2:option(DummyValue, "Signal level", translate("Signal"))
t2:option(DummyValue, "Noise level", translate("Noise"))
if #wifidevs < 1 then
return m
end
-- Config Section --
s = m:section(NamedSection, wifidevs[1], "wifi-device", translate("Devices"))
s.addremove = false
en = s:option(Flag, "disabled", translate("enable"))
en.rmempty = false
en.enabled = "0"
en.disabled = "1"
function en.cfgvalue(self, section)
return Flag.cfgvalue(self, section) or "0"
end
local hwtype = m:get(wifidevs[1], "type")
if hwtype == "atheros" then
mode = s:option(ListValue, "hwmode", translate("Mode"))
mode.override_values = true
mode:value("", "auto")
mode:value("11b", "802.11b")
mode:value("11g", "802.11g")
mode:value("11a", "802.11a")
mode:value("11bg", "802.11b+g")
mode.rmempty = true
end
ch = s:option(Value, "channel", translate("Channel"))
for i=1, 14 do
ch:value(i, i .. " (2.4 GHz)")
end
s = m:section(TypedSection, "wifi-iface", translate("Local Network"))
s.anonymous = true
s.addremove = false
s:option(Value, "ssid", translate("Network Name (<abbr title=\"Extended Service Set Identifier\">ESSID</abbr>)"))
bssid = s:option(Value, "bssid", translate("<abbr title=\"Basic Service Set Identifier\">BSSID</abbr>"))
local devs = {}
luci.model.uci.cursor():foreach("wireless", "wifi-device",
function (section)
table.insert(devs, section[".name"])
end)
if #devs > 1 then
device = s:option(DummyValue, "device", translate("Device"))
else
s.defaults.device = devs[1]
end
mode = s:option(ListValue, "mode", translate("Mode"))
mode.override_values = true
mode:value("ap", translate("Provide (Access Point)"))
mode:value("adhoc", translate("Independent (Ad-Hoc)"))
mode:value("sta", translate("Join (Client)"))
function mode.write(self, section, value)
if value == "sta" then
local oldif = m.uci:get("network", "wan", "ifname")
if oldif and oldif ~= " " then
m.uci:set("network", "wan", "_ifname", oldif)
end
m.uci:set("network", "wan", "ifname", " ")
self.map:set(section, "network", "wan")
else
if m.uci:get("network", "wan", "_ifname") then
m.uci:set("network", "wan", "ifname", m.uci:get("network", "wan", "_ifname"))
end
self.map:set(section, "network", "lan")
end
return ListValue.write(self, section, value)
end
encr = s:option(ListValue, "encryption", translate("Encryption"))
encr.override_values = true
encr:value("none", "No Encryption")
encr:value("wep", "WEP")
if hwtype == "atheros" or hwtype == "mac80211" then
local supplicant = fs.access("/usr/sbin/wpa_supplicant")
local hostapd = fs.access("/usr/sbin/hostapd")
if hostapd and supplicant then
encr:value("psk", "WPA-PSK")
encr:value("psk2", "WPA2-PSK")
encr:value("psk-mixed", "WPA-PSK/WPA2-PSK Mixed Mode")
encr:value("wpa", "WPA-Radius", {mode="ap"}, {mode="sta"})
encr:value("wpa2", "WPA2-Radius", {mode="ap"}, {mode="sta"})
elseif hostapd and not supplicant then
encr:value("psk", "WPA-PSK", {mode="ap"}, {mode="adhoc"})
encr:value("psk2", "WPA2-PSK", {mode="ap"}, {mode="adhoc"})
encr:value("psk-mixed", "WPA-PSK/WPA2-PSK Mixed Mode", {mode="ap"}, {mode="adhoc"})
encr:value("wpa", "WPA-Radius", {mode="ap"})
encr:value("wpa2", "WPA2-Radius", {mode="ap"})
encr.description = translate(
"WPA-Encryption requires wpa_supplicant (for client mode) or hostapd (for AP " ..
"and ad-hoc mode) to be installed."
)
elseif not hostapd and supplicant then
encr:value("psk", "WPA-PSK", {mode="sta"})
encr:value("psk2", "WPA2-PSK", {mode="sta"})
encr:value("psk-mixed", "WPA-PSK/WPA2-PSK Mixed Mode", {mode="sta"})
encr:value("wpa", "WPA-EAP", {mode="sta"})
encr:value("wpa2", "WPA2-EAP", {mode="sta"})
encr.description = translate(
"WPA-Encryption requires wpa_supplicant (for client mode) or hostapd (for AP " ..
"and ad-hoc mode) to be installed."
)
else
encr.description = translate(
"WPA-Encryption requires wpa_supplicant (for client mode) or hostapd (for AP " ..
"and ad-hoc mode) to be installed."
)
end
elseif hwtype == "broadcom" then
encr:value("psk", "WPA-PSK")
encr:value("psk2", "WPA2-PSK")
encr:value("psk+psk2", "WPA-PSK/WPA2-PSK Mixed Mode")
end
key = s:option(Value, "key", translate("Key"))
key:depends("encryption", "wep")
key:depends("encryption", "psk")
key:depends("encryption", "psk2")
key:depends("encryption", "psk+psk2")
key:depends("encryption", "psk-mixed")
key:depends({mode="ap", encryption="wpa"})
key:depends({mode="ap", encryption="wpa2"})
key.rmempty = true
key.password = true
server = s:option(Value, "server", translate("Radius-Server"))
server:depends({mode="ap", encryption="wpa"})
server:depends({mode="ap", encryption="wpa2"})
server.rmempty = true
port = s:option(Value, "port", translate("Radius-Port"))
port:depends({mode="ap", encryption="wpa"})
port:depends({mode="ap", encryption="wpa2"})
port.rmempty = true
if hwtype == "atheros" or hwtype == "mac80211" then
nasid = s:option(Value, "nasid", translate("NAS ID"))
nasid:depends({mode="ap", encryption="wpa"})
nasid:depends({mode="ap", encryption="wpa2"})
nasid.rmempty = true
eaptype = s:option(ListValue, "eap_type", translate("EAP-Method"))
eaptype:value("TLS")
eaptype:value("TTLS")
eaptype:value("PEAP")
eaptype:depends({mode="sta", encryption="wpa"})
eaptype:depends({mode="sta", encryption="wpa2"})
cacert = s:option(FileUpload, "ca_cert", translate("Path to CA-Certificate"))
cacert:depends({mode="sta", encryption="wpa"})
cacert:depends({mode="sta", encryption="wpa2"})
privkey = s:option(FileUpload, "priv_key", translate("Path to Private Key"))
privkey:depends({mode="sta", eap_type="TLS", encryption="wpa2"})
privkey:depends({mode="sta", eap_type="TLS", encryption="wpa"})
privkeypwd = s:option(Value, "priv_key_pwd", translate("Password of Private Key"))
privkeypwd:depends({mode="sta", eap_type="TLS", encryption="wpa2"})
privkeypwd:depends({mode="sta", eap_type="TLS", encryption="wpa"})
auth = s:option(Value, "auth", translate("Authentication"))
auth:value("PAP")
auth:value("CHAP")
auth:value("MSCHAP")
auth:value("MSCHAPV2")
auth:depends({mode="sta", eap_type="PEAP", encryption="wpa2"})
auth:depends({mode="sta", eap_type="PEAP", encryption="wpa"})
auth:depends({mode="sta", eap_type="TTLS", encryption="wpa2"})
auth:depends({mode="sta", eap_type="TTLS", encryption="wpa"})
identity = s:option(Value, "identity", translate("Identity"))
identity:depends({mode="sta", eap_type="PEAP", encryption="wpa2"})
identity:depends({mode="sta", eap_type="PEAP", encryption="wpa"})
identity:depends({mode="sta", eap_type="TTLS", encryption="wpa2"})
identity:depends({mode="sta", eap_type="TTLS", encryption="wpa"})
password = s:option(Value, "password", translate("Password"))
password:depends({mode="sta", eap_type="PEAP", encryption="wpa2"})
password:depends({mode="sta", eap_type="PEAP", encryption="wpa"})
password:depends({mode="sta", eap_type="TTLS", encryption="wpa2"})
password:depends({mode="sta", eap_type="TTLS", encryption="wpa"})
end
if hwtype == "atheros" or hwtype == "broadcom" then
iso = s:option(Flag, "isolate", translate("AP-Isolation"), translate("Prevents Client to Client communication"))
iso.rmempty = true
iso:depends("mode", "ap")
hide = s:option(Flag, "hidden", translate("Hide <abbr title=\"Extended Service Set Identifier\">ESSID</abbr>"))
hide.rmempty = true
hide:depends("mode", "ap")
end
if hwtype == "mac80211" or hwtype == "atheros" then
bssid:depends({mode="adhoc"})
end
if hwtype == "broadcom" then
bssid:depends({mode="wds"})
bssid:depends({mode="adhoc"})
end
return m
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,337
|
\section{Introduction}
The radio frequency spectrum is a valuable but congested natural resource because it is shared by an increasing number of users. Cognitive radio (CR) \cite{Mitola-99PCM,Letaief-09ProcIEEE,Goldsmith-09IProc,Liang-11TVT} is viewed as an effective means to improve the utilization of the
radio frequency spectrum by introducing dynamic spectrum access technology. Such technology allows secondary users (SUs, also known as CR users) to access the
radio spectrum originally allocated to primary users (PUs). In the CR literature, two cognitive spectrum access models have been widely adopted
\cite{Liang-11TVT}: 1) the \emph{opportunistic spectrum access} model and 2) the \emph{concurrent spectrum access} model. In the opportunistic spectrum access
model, SUs carry out spectrum sensing to detect spectrum holes and reconfigure their transmission to operate only in the identified holes
\cite{Mitola-99PCM,Kim-11TVT}. Meanwhile, in the concurrent spectrum access model, SUs transmit simultaneously with PUs as long as interference to PUs is limited
\cite{Zheng-09TSP,Huang-11JSAC}.
In this paper, we focus on the concurrent spectrum access model particularly when the secondary base station (BS) is equipped with multiple antennas. A desirable condition in the concurrent spectrum access model is for SUs to maximize their own performance while minimizing the interference caused to the PUs.
Several transmit schemes have been studied to balance the transmissions to the SUs and the PUs \cite{ZhangR-08JSTSP,Hamdi-09TWC,HeYY-13TWC,HeYY-13ICASSP,Chen-13TVT}. In \cite{ZhangR-08JSTSP}, a transmit algorithm has been proposed based on the singular value decomposition of the secondary channel after the projection into the null
space of the channel from the secondary BS to the PUs. A spectrum sharing scheme has been designed for a large number of SUs \cite{Hamdi-09TWC}, in which the
SUs are pre-selected so that their channels are nearly orthogonal to the channels of the PUs. Doing so ensures that the SUs cause the lowest interference to the PUs.
In multi-antenna and multiuser downlink systems, a common technique to mitigate the multiuser interference is a zero-forcing (ZF) precoding
\cite{Caire-03TIT,Samardzija-07ICC,Irmer-09COMMag,Suraweera-13ICC}, which is computationally more efficient than its non-linear alternatives. However, the achievable rates of the ZF precoding are severely compromised when the channel matrix is ill conditioned. Then, regularized ZF (RZF) precoding \cite{Joham-02ISSSTA,Peel-05Tcom} is proposed to mitigate the ill-conditioned problem by employing a regularization parameter in the channel inversion. The regularization parameter can control the amount of introduced interference. Several applications based on the RZF framework have been developed, such as transmitter designs for \emph{non}-CR broadcast systems \cite{Nguyen-08GLCOM,Muharar-11ICC,Wagner-12IT,Muharar-13TCom}, security systems \cite{Geraci-13JSAC,ZhangJun-14CL}, and multi-cell cooperative systems
\cite{Muharar-12ISIT,Huang-13TWC,ZhangJun-13TWC,WenCK-14TWC}.
While directly applying RZF to CR networks, the secondary BS can \emph{only} control the interference in inter-SUs. A \emph{partially-projected} RZF (PP-RZF)
precoding has been proposed \cite{HeYY-13TWC,HeYY-13ICASSP}, which limits the interference from the SUs to the PUs by combining the RZF
\cite{Joham-02ISSSTA,Peel-05Tcom} with the channel projection idea \cite{ZhangR-08JSTSP}. The PP-RZF precoding follows the classical RZF technique, although the
former is based on the partially-projected channel, which is obtained by partially projecting the channel matrix into the null space of the channel from the
secondary BS to the PUs. The amount of interference to the PUs decreases with increasing amounts of projection into the null space of the PUs, which can be
achieved by tuning the projection control parameter. However, the search for the optimal regularization parameter and projection control parameter is a demanding
process because it involves Monte-Carlo averaging. Therefore, a deterministic (or large-system) approximation of the signal-to-interference-plus-noise ratio (SINR)
for the PP-RZF scheme has been derived \cite{HeYY-13TWC,HeYY-13ICASSP}. Unfortunately, only the CR channel with a \emph{single} PU has been studied and the scenario where \emph{multiple} PUs are present remains unsolved \cite{HeYY-13TWC}.
To apply the PP-RZF precoding scheme in a CR network with \emph{multiple} PUs, a new analytical technique that deals with a \emph{multi}-dimensional random projection matrix, which is generated by partially projecting the channel matrix into the null spaces of \emph{multiple} PUs, is required. This paper aims to address the above mentioned challenge by providing analytical results in a more general setting than that in \cite{HeYY-13TWC,HeYY-13ICASSP}. Specifically, we focus on a downlink multiuser CR network (Fig.~\ref{fig:1}), which consists of a secondary BS with multiple antennas, SUs, and PUs as well as different channel gains. Our main contributions are summarized below.
\begin{itemize}
\item We derive deterministic equivalents for the SINR and the ergodic sum-rate achieved by the PP-RZF precoding under the general CR network. Unlike previous works \cite{HeYY-13TWC,HeYY-13ICASSP}, our model considers \emph{multiple} PUs and allows different channel gains from the secondary BS to each user. Owing to recent advancements in large dimensional random matrix theory (RMT) with respect to complex combinations of different types of
independent random matrices \cite{Couillet-11BOOK}, we identify the large system distribution of the Stieltjes transform for a new class of random matrix. Therefore, our extension becomes non trivial and novel.
\item In the PP-RZF precoding, the regularization parameter and the projection control parameter can regulate the amount of interference to the SUs and the PUs, but a wrong choice of parameters can considerably degrade the performance of the CR network. However, the search for the optimal parameters is a demanding process because Monte-Carlo averaging is required. We overcome the fundamental difficulty of applying PP-RZF precoding in the CR network. The deterministic equivalent for the ergodic sum-rate provides an efficient way of finding the asymptotically optimal regularization parameter and the asymptotically optimal projection control parameter. Simulation results indicate good agreement with the optimum in terms of the ergodic sum-rate.
\item We provide several useful observations on the condition that the regularization parameter and the projection control parameter can achieve the optimal sum-rate. We also reveal the relationship between the parameters and the signal-to-noise ratio (SNR).
\end{itemize}
\emph{Notations}---We use uppercase and lowercase boldface letters to denote matrices and vectors, respectively.
An $N \times N$ identity matrix is denoted by ${\bf{I}}_N$, an all-zero matrix by ${\bf 0}$, and an all-one matrix by ${\bf 1}$.
The superscripts $(\cdot)^{H}$, $(\cdot)^{T}$, and $(\cdot)^{*}$ denote the conjugate transpose, transpose, and conjugate operations, respectively. ${\sf E}\{\cdot\}$ returns the expectation with respect to all random variables within the bracket, and $\log(\cdot)$ is the natural logarithm.
We use $[{\bf A}]_{kl}$, $[{\bf A}]_{l,k}$, or $A_{kl}$ to denote the ($k$,$l$)-th entry of the matrix $\bf A$, and $a_k$ denotes the $k$-th entry of the column vector $\bf{a}$.
The operators $(\cdot)^{\frac{1}{2}}$, $(\cdot)^{-1}$, ${{\sf tr}}(\cdot)$, and $\det(\cdot)$ represent the matrix principal square root, inverse, trace, and determinant, respectively, $\|\cdot\|$ represents the Euclidean norm of an input vector or the spectral norm of an input matrix, and ${\sf diag}(\bf{x})$ denotes a diagonal matrix with $\bf{x}$ along its main diagonal. The notation ``$\xrightarrow{a.s.}$'' denotes the almost sure (a.s.) convergence.
\section{System Model and Problem Formulation}
\subsection{System Model}
\begin{figure}
\begin{center}
\resizebox{4.5in}{!}{%
\includegraphics*{systemModel_New.eps} }%
\caption{A downlink multiuser cognitive radio network.}\label{fig:1}
\end{center}
\end{figure}
As illustrated in Fig.~\ref{fig:1}, we consider a downlink multiuser CR network that consists of a secondary BS with $N$ antennas (labeled as ${\sf BS}$). The ${\sf BS}$ simultaneously transmits $K$ independent messages to $K$ single antenna SUs (labeled as ${\sf SU}_1, \dots, {\sf SU}_K$). We assume that
all the SUs share the same spectrum with $L$ single antenna PUs (labeled as ${\sf PU}_1, \dots, {\sf PU}_L$). Let ${\bf h}_k^H \in {\mathbb C}^{1 \times N}$ be the
fading channel vector between ${\sf BS}$ and ${\sf SU}_k$, ${\bf f}_{l}^H \in {\mathbb C}^{1 \times N}$ be the fading channel vector between ${\sf BS}$ and ${\sf PU}_l$, and
${\bf g}_{k} \in {\mathbb C}^{N \times 1}$ be the precoding vector of ${\sf SU}_k$. The received signal at ${\sf SU}_k$ can therefore be expressed as
\begin{equation}\label{eq:the received signal of SUEk}
y_k = {\bf h}_k^H {\bf g}_k s_k + \sum_{j=1, j \ne k}^K {\bf h}_k^H {\bf g}_j s_j + z_k,
\end{equation}
where $s_k$ is the data symbol of ${\sf SU}_k$, $s_j$'s are independent and identically distributed (i.i.d.) data symbols with zero mean and unit variance, respectively,
and $z_k$ is the additive Gaussian noise with zero mean and variance of $\sigma^2$. For ease of exposition, we define ${\bf H} \triangleq \left[ {\bf h}_1,
\dots, {\bf h}_K \right]^H \in {\mathbb C}^{K\times N}$, ${\bf F} \triangleq \left[ {\bf f}_1, \dots, {\bf f}_L \right]^H \in {\mathbb C}^{L\times N}$, ${\bf G} \triangleq \left[ {\bf g}_1, \dots,
{\bf g}_K \right] \in {\mathbb C}^{N \times K}$, ${\bf y}\triangleq[ {y_1}, \dots ,{y_K}]^T \in {\mathbb C}^K$, ${\bf s} \triangleq \left[{s_1}, \dots ,{s_K} \right]^T \in {\mathbb C}^K$, and ${\bf z}
\triangleq \left[ {z_1}, \dots ,{z_K} \right]^T \in {\mathbb C}^K$. The received signal of all the SUs in vector form is given by
\begin{equation}\label{eq:the concatenated received signal vector}
{\bf y}= {\bf H} {\bf G} {\bf s} + {\bf z}.
\end{equation}
We also assume that ${\sf BS}$ satisfies the average total transmit power constraint
\begin{equation}\label{eq:base station transmitted power constrain}
{\sf E} \left\{ {\sf tr}\left( {\bf G} {\bf G}^H \right) \right\} \leq {N P_T},
\end{equation}
where $P_T > 0$ is the parameter that determines the power budget of ${\sf BS}$. Notably, if we consider the instantaneous transmit power constraint, i.e., ${\sf tr}( {\bf G} {\bf G}^H ) \leq {N P_T}$, we can obtain the same constraint in a large-system regime, as shown in Appendix B-III.
The peak received interference power constraint or the average received interference power constraint is used to protect the PUs. Given that
the latter is more flexible for dynamically allocating transmission powers over different fading states than the former
\cite{ZhangR-09TWC,Wang-09TWC}, we employ the average received interference power constraint and consider two cases: Case I---the average received
interference power constraint at each PU and Case II---the total average received interference power constraint at all PUs\footnote{Notably, multiple single-antenna PUs exist. These PUs can also be considered a single equivalent PU with multiple receive antennas.}. These cases are respectively given by
\begin{subequations} \label{eq:the total received interference constrain}
\begin{align}
&\mbox{Case I (Per PU power constraint):~~~~} {\sf E} \left\{ {\bf f}_l^H {\bf G} {\bf G}^H {\bf f}_l \right\} \leq P_l, \mbox{~~for}~~ l = 1,\ldots,L, \label{eq:the total received interference constrain 1}\\
&\mbox{Case II (Sum power constraint):~~~~}{\sf E} \left\{ {\sf tr}\left( {\bf F} {\bf G} {\bf G}^H {\bf F}^H \right) \right\} \leq P_{\rm all},\label{eq:the total received interference constrain 2}
\end{align}
\end{subequations}
where $P_l > 0$ denotes the interference power threshold of ${\sf PU}_l$, and $P_{\rm all} > 0$ represents the total interference power threshold of all PUs.
We then set $P_l = \theta_l P_T$ and $P_{\rm all} = \theta_{\rm all} P_T$ with $\theta_l, \theta_{\rm all}$ being positive scalar parameters to make a
connection with the transmit power. Although we only consider equal power allocation for simplicity in this paper, our framework can be easily extended to
arbitrary power allocation by replacing ${\bf G}$ with ${\bf G}{\bf P}^{\frac{1}{2}}$, where ${\bf P} = {\sf diag}(p_1,\ldots, p_K)$ with $p_k\geq 0$ being the signal power of
${\sf SU}_k$ (see \cite{Wagner-12IT,Muharar-13TCom} for a similar application).
Next, to incorporate path loss and other large-scale fading effects, we model the channel vectors by
\begin{align}
{\bf h}_k^H & = \sqrt{r_{1,k}} \, {\tilde{\qh}}_k^H \mbox{~~and}~~ {\bf f}_l^H = \sqrt{r_{2,l}} \, {\tilde{\qf}}_l^H,
\end{align}
where ${\tilde{\qh}}_k^H$ and ${\tilde{\qf}}_l^H$ are the small-scale (or fast) fading vectors, and $r_{1,k}$ and $r_{2,l}$ denote the large-scale fading coefficients (or
channel path gains), including the geometric attenuation and shadow effect. Using the above notations, the concerned channel matrices can be rewritten as
\begin{align} \label{eq:def_tilde_HandF}
{\bf H} & = {\bf R}_{1}^{\frac{1}{2}} {\tilde{\qH}} \mbox{~~and}~~{\bf F} = {\bf R}_{2}^{\frac{1}{2}} {\tilde{\qF}},
\end{align}
where ${\tilde{\qH}} \equiv [\frac{1}{\sqrt{N}} \tilde{h}_{ij}] \in {\mathbb C}^{K \times N}$ and ${\tilde{\qF}} \equiv [\frac{1}{\sqrt{N}} \tilde{f}_{ij}] \in {\mathbb C}^{L \times N}$ consist
of the random components of the channel in which $\tilde{h}_{ij}$'s and $\tilde{f}_{ij}$'s are i.i.d.~complex random variables with zero mean and unit variance, respectively, and ${\bf R}_1 \in {\mathbb C}^{K \times K}$ and ${\bf R}_2 \in {\mathbb C}^{L \times L}$ are diagonal matrices whose diagonal elements are given by $[{\bf R}_1]_{kk} = r_{1,k}$ and $[{\bf R}_2]_{ll} = r_{2,l}$, respectively. In line with \cite{HeYY-13TWC,HeYY-13ICASSP}, we assume that ${\bf H}$ is perfectly known to ${\sf BS}$ in this paper. Since ${\sf BS}$ needs to predict the interference power in \eqref{eq:the total received interference constrain}, we further assume that perfect knowledge of ${\bf F}$ is available at ${\sf BS}$ \cite{LZhang-09TWC,HeYY-13TWC,HeYY-13ICASSP}. To acquire perfect channel state information (CSI) for ${\bf H}$ and ${\bf F}$, transmission protocols need to incorporate certain cooperation among the PUs, the SUs, and ${\sf BS}$ \cite{LZhang-09TWC}. Further research can focus on the case with imperfect CSI or estimation of channel \cite{Dai-13JSAC,Gao-14CL}.
In the downlink CR network \eqref{eq:the concatenated received signal vector}, we consider the RZF precoding because this precoding's relatively low complexity compared with dirty paper coding \cite{Joham-02ISSSTA,Peel-05Tcom,Wagner-12IT,ZhangJun-13TWC}. However, a direct application of the conventional RZF to the secondary
BS will result in a very inefficient transmission because a large power back-off at the secondary BS is required to satisfy the interference power constraint \eqref{eq:the total received interference constrain}. Therefore, following \cite{HeYY-13TWC,HeYY-13ICASSP}, we adopt the RZF precoding based on the \emph{partially-projected} channel matrix
\begin{equation} \label{eq:checkH}
\check{{\bf H}} ={\bf H}({\bf I}_N-\beta{\bf W}^H{\bf W}),
\end{equation}
where ${\bf W} \triangleq ({\bf F}\qF^H)^{-\frac{1}{2}} {\bf F} \in {\mathbb C}^{L \times N}$, and $\beta \in [0,1]$ is the projection control parameter. Note that the projected
channel matrix $\check{{\bf H}}$ is obtained by \emph{partially} projecting ${\bf H}$ into the null space of ${\bf F}$. Specifically, the RZF precoding matrix is given by
\begin{equation}\label{eq:the RZF precoding}
{\bf G} = \xi \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H,
\end{equation}
where $\xi$ is a normalization parameter that fulfills the BS transmit power constraint \eqref{eq:base station transmitted power constrain} and the
interference power constraint \eqref{eq:the total received interference constrain}, and $\alpha > 0$ represents the regularization parameter. We refer to this
precoding as PP-RZF precoding.
Before setting each of the parameters in \eqref{eq:the RZF precoding}, two special cases of the PP-RZF precoding are considered first. On the one hand, if
$\beta=0$ then ${\bf G}$ degrades to the conventional RZF precoding. On the other hand, if $\beta=1$ then $\check{{\bf H}}$ is completely orthogonal to ${\bf F}$ and we have ${\bf F}\check{{\bf H}}^H = {\bf 0}$, i.e., no interference signal from the secondary BS will leak to the PUs. Therefore, the interference power constraint (\ref{eq:the total
received interference constrain}) is naturally guaranteed. Furthermore, the amount of the interference to the PUs decreases as the projection control
parameter increases.
Now we return to the setting of the normalization parameter in \eqref{eq:the RZF precoding}. Considering Case I, from \eqref{eq:base station transmitted power
constrain} and \eqref{eq:the total received interference constrain 1}, we have
\begin{subequations} \label{eq:xi_i}
\begin{align}
\xi^2 \leq & \xi_0^2 \triangleq \frac{P_T}{ {\sf E} \left\{\frac{1}{N} {\sf tr}\left( \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \right) \right\} }, \\
\xi^2 \leq & \xi_l^2 \triangleq \frac{\theta_l P_T}{ {\sf E} \left\{ {\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \right\} }, \mbox{~~for}~ l = 1,\ldots,L.
\end{align}
\end{subequations}
To satisfy \eqref{eq:base station transmitted power constrain} and \eqref{eq:the total received interference constrain 1} simultaneously, we set $\xi^2 = \min \{\xi_0^2, \xi_l^2, l = 1,\ldots,L\}$. Then, the SINR of secondary user ${\sf SU}_k$ is given by
\begin{align}\label{eq:SINR}
\gamma_{k} & = \frac{ \left| {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \right|^2} { {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k + \frac{\sigma^2}{\xi^2} } \nonumber\\
&= \frac{ \rho \left| {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \right|^2} { \rho {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k + \nu},
\end{align}
where $\check{{\bf H}}_{[k]} \triangleq [ \check{{\bf h}}_1, \ldots, \check{{\bf h}}_{k-1}, \check{{\bf h}}_{k+1}, \ldots, \check{{\bf h}}_K ]^H \in {\mathbb C}^{(K-1)\times N}$,
$\check{{\bf h}}_k \triangleq ({\bf I}_N-\beta{\bf W}^H{\bf W}){\bf h}_k$, $\rho \triangleq P_T/\sigma^2$, and
\begin{align}
\nu \triangleq \frac{P_T}{\xi^2} = \max &\bigg\{ {\sf E} \left\{ \frac{1}{N} {\sf tr}\left( \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \right) \right\} , \nonumber\\
&~~~~\frac{1}{\theta_l} {\sf E} \left\{ {\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \right\}, l = 1,\ldots,L \bigg\}. \label{eq:nu}
\end{align}
Here, the equality of (\ref{eq:nu}) follows from (\ref{eq:xi_i}). For Case II, we have
\begin{align}
\nu = \max &\bigg\{ {\sf E} \left\{ \frac{1}{N} {\sf tr}\left( \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \right) \right\} , \nonumber\\
&~~~~\frac{1}{\theta_{\rm all}} {\sf E} \left\{ {\sf tr}\left( {\bf F} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf F}^H \right) \right\} \bigg\}. \label{eq:nu2}
\end{align}
Consequently, under the assumption of perfect CSI at both transmitter and receivers, the ergodic sum-rate of the CR network with Gaussian signaling can be defined as
\begin{align}\label{eq:the ergodic sum-rate}
R_{\rm{sum}} \triangleq \sum_{k=1}^K {\sf E} \left\{\log \left( 1 + \gamma_{k} \right)\right\}.
\end{align}
Note that $\gamma_k$ in the ergodic sum-rate is subject to the BS transmit power constraint in (3) and the interference power constraint (to the primary users) in (4).
\subsection{Problem Formulation}
The SINR $\gamma_{k}$ in \eqref{eq:SINR} is a function of the regularization parameter $\alpha$ and the projection control parameter $\beta$. In the
literature, adopting incorrect regularization parameter would degrade performance significantly \cite{Peel-05Tcom,Wagner-12IT,ZhangJun-13TWC}. In light of the discussion in the previous subsection, one can realize that a proper projection control parameter can assist in decreasing the interference to the
PUs. As a result, using the PP-RZF precoding effectively requires obtaining appropriate values of $\alpha$ and $\beta$ to optimize certain
performance metrics. In this paper, we are interested in finding $(\alpha,\beta)$, which maximizes the ergodic sum-rate (\ref{eq:the ergodic sum-rate}).
Formally, we have
\begin{align}\label{eq:optimal ergodic sum-rate}
\left\{\alpha^{\rm opt}, \beta^{\rm opt} \right\} = \operatornamewithlimits{arg\, max}_{\alpha > 0, 1 \geq \beta \geq 0} & ~R_{\rm{sum}}.
\end{align}
The above problem does not admit a simple closed-form solution and the solution must be computed via a two-dimensional line search. Monte-Carlo
averaging over the channels is required to evaluate the ergodic sum-rate \eqref{eq:the ergodic sum-rate} for each choice of $\alpha$ and $\beta$, which,
unfortunately, makes the overall computational complexity prohibitive. This drawback hinders the development of the PP-RZF precoding. To address this problem,
we resort to an asymptotic expression of \eqref{eq:the ergodic sum-rate} in the large-system regime in the next section.
\section{Performance Analysis of Large Systems}
This section presents the main results of the paper. First, we derive deterministic equivalents for the SINR $\gamma_{k}$ and the ergodic sum-rate
$R_{\rm{sum}}$ in a large-system regime. Then, we identify the asymptotically optimal regularization parameter and the asymptotically optimal projection control parameter
to achieve the optimal deterministic equivalent for the ergodic sum-rate.
\subsection{Deterministic Equivalents for the SINR and the Ergodic Sum-Rate}
We present a deterministic equivalent for the SINR $\gamma_{k}$ by considering the
large-system regime, where $N$, $K$, and $L$ approach infinity, whereas
\begin{equation*}
c_1 = \frac{N}{K} \mbox{~~and~~} c_2 = \frac{L}{N}
\end{equation*}
are fixed ratios, such that $0 < \lim \inf_N c_1 \leq \lim
\sup_N c_1 < \infty, 0 < \lim \inf_N c_2 \leq \lim \sup_N c_2 \leq 1$. For brevity, we simply use ${\mathcal N} \rightarrow \infty$ to represent the quantity in such
limit. In addition, we impose the assumptions below in our derivations.
\begin{assumption} \label{Assum: 1}
For the channel matrices ${\bf H}$ and ${\bf G}$ in \eqref{eq:def_tilde_HandF}, we have the following hypotheses:
\begin{itemize}
\item[1)] ${\tilde{\qH}} = [\frac{1}{\sqrt{N}} \tilde{h}_{ij}] \in {\mathbb C}^{K \times N}$, where $\tilde{h}_{ij}$'s
are i.i.d. standard Gaussian.
\item[2)] ${\tilde{\qF}} = [\frac{1}{\sqrt{N}} \tilde{f}_{ij}] \in {\mathbb C}^{L \times N}$, where $\tilde{f}_{ij}$'s have the same statistical
properties as $\tilde{h}_{ij}$'s.
\item[3)] ${\bf R}_1 = {\sf diag}(r_{1,1},\ldots, r_{1,K}) \in {\mathbb C}^{K \times K}$ and ${\bf R}_2 = {\sf diag}(r_{2,1},\ldots, r_{2,L}) \in {\mathbb C}^{L \times L}$ are diagonal matrices with uniformly bounded spectral norm\footnote{\cite{Horn-90BOOK}: The spectral norm $|\!|\!|\bullet|\!|\!|_2$ is defined on ${\mathbb C}^{n\times n}$ by $|\!|\!|{\bf A}|\!|\!|_2 \equiv \max \{\sqrt{\lambda}: \lambda \mbox{~~is an eigenvalue of~~} {\bf A}^*{\bf A}\}$.} with respect to $K$ and $L$, respectively.
\end{itemize}
\end{assumption}
Based on the definition of ${\bf W}$ in \eqref{eq:checkH}, ${\bf W}^H{\bf W} = {\bf F}^H ({\bf F}\qF^H)^{-1}{\bf F} = {\tilde{\qF}}^H ({\tilde{\qF}}\tqF^H)^{-1}{\tilde{\qF}}$ $= {\tilde{\qW}}^H{\tilde{\qW}}$, where
${\tilde{\qW}} \triangleq ({\tilde{\qF}}\tqF^H)^{-\frac{1}{2}}{\tilde{\qF}}$. Therefore, ${\tilde{\qW}}$ is $L \leq N$ rows of an $N \times N$ Haar-distributed unitary random matrix \cite[Definition 4.6]{Couillet-11BOOK}. The partially-projected channel matrix $\check{{\bf H}}$ is clearly composed of the product of two different types of independent
random matrices. Owing to recent advancements in large dimensional RMT \cite{Couillet-11BOOK}, we arrive at the following theorem, and the
details are given in Appendix A.
\begin{theorem}\label{Th: 2}
Under Assumption \ref{Assum: 1}, in Case I (per PU power constraint), as ${\mathcal N} \rightarrow \infty$, we have $\gamma_k - \overline{\gamma}_k \xrightarrow{a.s.} 0$, for $k = 1,\dots,K$,
where
\begin{align}
\overline{\gamma}_{k} &= \frac{ \rho \overline{a}_k^2} { \rho \overline{b}_k + \overline{\nu}}, \label{eq:gamma1 deterministic equivalent}
\end{align}
with
\begin{subequations} \label{eq:u_all}
\begin{align}
\overline{a}_k&= \frac{r_{1,k} (t_1 + t_2(1-\beta)) }{\alpha+r_{1,k}(t_1 + t_2(1-\beta)^2)}, \label{eq:Dea} \\
\overline{b}_k &= r_{1,k} \left( \frac{\left(1-\overline{a}_k\right)^2 t_1}{1+e} + \frac{\left(1-(1-\beta)\overline{a}_k\right)^2 (1-\beta)^2 t_2}{1+(1-\beta)^2 e} \right)\frac{\partial e}{\partial \alpha} , \label{eq:Deb}\\
\overline{\nu} & = \max \bigg\{ \left(\frac{t_1}{1+e}+\frac{(1-\beta)^2 t_2}{1+(1-\beta)^2 e}\right)\frac{\partial e}{\partial \alpha}, \frac{r_{2,l}}{\theta_l c_2} \frac{(1-\beta)^2 t_2}{1+(1-\beta)^2 e} \frac{\partial e}{\partial \alpha}, l = 1,\ldots,L \bigg\}, \label{eq:Dev} \\
\frac{\partial e}{\partial \alpha} &= \frac{\frac{1}{N} {\sf tr} {\bf R}_1 \left(\alpha {\bf I}_K + \left(t_1 + t_2 (1-\beta)^2\right) {\bf R}_1 \right)^{-2}}{1 - \left(\frac{t_1}{1+e}+\frac{(1-\beta)^4 t_2}{1+(1-\beta)^2 e}\right) \frac{1}{N} {\sf tr} \left( {\bf R}_1 \left(\alpha {\bf I}_K + \left(t_1 + t_2 (1-\beta)^2\right) {\bf R}_1 \right)^{-1}\right)^2}, \label{eq:partial e} \\
t_1 &= \frac{1-c_2}{1+e},
~~~~~~~~t_2 = \frac{c_2}{1+(1-\beta)^2 e}, \label{eq:t1_f
\end{align}
\end{subequations}
and $e$ is given as the unique solution to the fixed point equation
\begin{align}
e =& \frac{1}{N} {\sf tr} {\bf R}_1 \left(\alpha {\bf I}_K + \left(t_1 + t_2 (1-\beta)^2\right) {\bf R}_1 \right)^{-1}. \label{eq:e1}
\end{align}
Meanwhile, in Case II (sum power constraint), all asymptotic expressions remain, except for $\overline{\nu}$, which should be changed to
\begin{align}
\overline{\nu} & = \max \bigg\{ \left(\frac{t_1}{1+e}+\frac{(1-\beta)^2 t_2}{1+(1-\beta)^2 e}\right)\frac{\partial e}{\partial \alpha}, \frac{{\sf tr} {\bf R}_2}{\theta_{\rm all} c_2} \frac{(1-\beta)^2 t_2}{1+(1-\beta)^2 e} \frac{\partial e}{\partial \alpha} \bigg\}. \label{eq:Dev2}
\end{align}
\hfill $\blacksquare$
\end{theorem}
An intuitive application of Theorem \ref{Th: 2} is that $\gamma_{k}$ can be approximated by its deterministic equivalent\footnote{\cite[Definition 6.1]{Couillet-11BOOK} (also see \cite{Hachem-07AAP}): Consider a series of Hermitian random matrices ${\bf B}_1, \,{\bf B}_2, \, \ldots,$ with ${\bf B}_N \in {\mathbb C}^{N \times N}$ and a series $f_1,f_2,\ldots$ of functionals of $1\times 1, 2\times 2, \ldots$ matrices. A deterministic equivalent of ${\bf B}_N$ for functional $f_N$ is a series ${\bf B}_1^{\circ}, \, {\bf B}_2^{\circ}, \, \ldots$, where ${\bf B}_N^{\circ}\in {\mathbb C}^{N\times N}$, of deterministic matrices, such that $\lim_{N\rightarrow\infty} f_N ({\bf B}_N)-f_N({\bf B}_N^{\circ}) \rightarrow 0$. In this case, the convergence often be with probability one. Similarly, we term $g_N \triangleq f_N({\bf B}_N^{\circ})$ the deterministic equivalent of $f_N ({\bf B}_N)$, that is, the deterministic series $g_1, g_2, \ldots$, such that $f_N ({\bf B}_N) - g_N \rightarrow 0$ in some sense. \\
Note that the deterministic equivalent of the Hermitian random matrix ${\bf B}_N$ is a \emph{deterministic} and a \emph{finite dimensional} matrix ${\bf B}_N^{\circ}$. In addition, the deterministic equivalent of $f_N ({\bf B}_N)$ is $g_N \triangleq f_N({\bf B}_N^{\circ})$, which is a function of ${\bf B}_N^{\circ}$.} $\overline{\gamma}_{k}$, which can be determined based only on statistical channel knowledge, that is, ${\bf R}_1$, ${\bf R}_2$, and $\sigma^2$. Note that, according to the definition of the deterministic equivalent (see footnote 3), in the expression of the deterministic equivalent $\overline{\gamma}_{k}$, the parameters $N$, $K$, $L$, as well as the matrix dimensions of ${\bf R}_1$ and ${\bf R}_2$, are \emph{finite}.
Combining Theorem 1 with the continuous mapping theorem\footnote{\cite[Theorem 25.7-Corollary 2]{Billingsley-11BOOK}: If $x_n \xrightarrow{a.s.} a$ and $h$ is continuous function at $a$, then $h(x_n) \xrightarrow{a.s.} h(a)$.}, we have $\log \left( 1 + \gamma_{k} \right) - \log \left( 1 + \overline{\gamma}_{k} \right)\xrightarrow{a.s.} 0 $. An approximation $\overline{R}_{\rm{sum}}$ of the ergodic sum-rate $R_{\rm{sum}}$ in \eqref{eq:the ergodic sum-rate} is obtained by replacing the instantaneous SINR $\gamma_{k}$ with its large system approximation $\overline{\gamma}_{k}$, that is,
\begin{equation} \label{eq:deterministic equivalent sum-rate}
\overline{R}_{\rm{sum}} =\sum_{k=1}^K \log \left( 1 + \overline{\gamma}_{k} \right).
\end{equation}
Therefore, when ${\mathcal N} \rightarrow\infty, \frac{1}{K} \left( R_{\rm{sum}} - \overline{R}_{\rm{sum}} \right) \xrightarrow{a.s.} 0$
holds true almost surely.
To facilitate our understanding of Theorem \ref{Th: 2}, we look at it from the two special cases as follows:
\begin{enumerate}
\item In Theorem \ref{Th: 2}, we introduce the two variables $t_1$ and $t_2$ to reflect the effects of the projection control parameter $\beta$.
If $\beta = 1$, from \eqref{eq:u_all}, then the deterministic equivalent $\overline{\gamma}_{k}$ does not depend on $t_2$. Substituting $\beta = 1$ into
\eqref{eq:gamma1 deterministic equivalent} and letting ${\bf R}_1 = {\bf I}_K$, we have
\begin{align} \label{eq:gammabeta1}
\overline{\gamma}_{k} &= \frac{\rho\Big(c_1(1-c_2)(1+\zeta(\mu,\eta,\alpha))^2-\zeta(\mu,\eta,\alpha)^2 \Big)}{\rho + \Big(1 +\zeta(\mu,\eta,\alpha)\Big)^2},
\end{align}
where $\zeta(\mu,\eta,\alpha) \triangleq t_1/\alpha$, $\mu \triangleq 1-c_2$, and $\eta \triangleq 1/c_1$. Combining \eqref{eq:t1_f} and \eqref{eq:e1}, we
obtain
\begin{equation} \label{eq:t1_SpecCase}
\zeta(\mu,\eta,\alpha) \triangleq \frac{t_1}{\alpha}
= \frac{1}{2} \left( \frac{\mu-\eta}{\alpha} - 1 + \sqrt{\frac{(\mu-\eta)^2}{\alpha^2} + \frac{2(\mu+\eta)}{\alpha} +1 } \, \right).
\end{equation}
Before providing an observation based on the above, we briefly review a well-known result from the large dimensional RMT. First, we consider the
definition of ${\bf H}$ from (\ref{eq:def_tilde_HandF}). If ${\bf R}_1 = {\bf I}_K$, the entries of the $K \times N$ matrix ${\bf H}$ are zero mean i.i.d. with variance $1/N$.
Following \cite[Chapter 3]{Couillet-11BOOK}, we see that as $N, \, K \rightarrow \infty$ with $N/K \rightarrow c_1$, ${\bf h}_k^H \left( {\bf H}^H {\bf H} + \alpha {\bf I}_N
\right)^{-1} {\bf h}_k$ converges almost surely to
\begin{equation} \label{eq:t0_def}
\int_{a}^{b} \frac{1}{\lambda+\alpha} f(\lambda) \,d\lambda,
\end{equation}
where
\begin{equation}
f(\lambda) = \left(1- \eta \right)^{+}\delta(\lambda) + \frac{\sqrt{(\lambda-a)^{+}(b-\lambda)^{+}}}{2\pi \lambda}
\end{equation}
with $(x)^{+} \triangleq \max\{x,\,0\}$, $a \triangleq (1- \sqrt{\eta})^2$, and $b \triangleq (1+ \sqrt{\eta})^2$. In fact, $f(u)$ is the limiting empirical
distribution of the eigenvalues of ${\bf H}^H {\bf H}$ and is known as the Mar\v{c}cenko-Pastur law \cite{Mar-67}. The integral of (\ref{eq:t0_def}) can be evaluated
in closed form
\begin{equation} \label{eq:t0}
\frac{1}{2} \left( \frac{1-\eta}{\alpha} - 1 + \sqrt{\frac{(1-\eta)^2}{\alpha^2} + \frac{2(1+\eta)}{\alpha} +1 } \, \right).
\end{equation}
Note that (\ref{eq:t1_SpecCase}) is equal to (\ref{eq:t0}) when $\mu$ is replaced with $1$, i.e., \eqref{eq:t0} is equal to $\zeta(1,\eta,\alpha)$.
In fact, following the similar derivations of Theorem \ref{Th: 2}, we can show that \eqref{eq:gammabeta1} and the SINR of the conventional RZF precoding share
the same formulation by replacing $\zeta(\mu,\eta,\alpha)$ in \eqref{eq:gammabeta1} with $\zeta(1,\eta,\alpha)$. Substituting the definitions of $c_1,
c_2$ into $\mu$ and $\eta$, we have $\mu-\eta = 1-c_2-1/c_1 = (N-(L+K))/N$. Comparing this value with $1-\eta = (N-K)/N$ in \eqref{eq:t0}, we thus
conclude that if $\beta =1$, the SINR of the PP-RZF precoding is \emph{similar}\footnote{Notably, when $\beta =1$, the SINRs of the PP-RZF precoding and the conventional RZF precoding are similar but \emph{not} identical because $\zeta(\mu,\eta,\alpha)$ is replaced with $\zeta(1,\eta,\alpha)$.} to that of the conventional RZF precoding but with an increase in the number of active users from $K$ to $K+L$. Hence, the degrees of freedom of the PP-RZF precoding is reduced to $N-(K+L)$ because the additional $L$ degrees of freedom are used to suppress interference to the PUs.
\item For another extreme case with $\beta = 0$ in Theorem \ref{Th: 2}, $t_1+t_2=\frac{1}{1+e}$. Letting ${\bf R}_1 = {\bf I}_K$,
we obtain $\frac{1}{\alpha(1+e)} = \zeta(1,\eta,\alpha) $, such that
\begin{align} \label{eq:gammabeta0}
\overline{\gamma}_{k} &= \frac{\rho\Big(c_1(1+\zeta(1,\eta,\alpha))^2-\zeta(1,\eta,\alpha)^2 \Big)}{\rho + \nu_0\Big(1+\zeta(1,\eta,\alpha)\Big)^2},
\end{align}
where $ \nu_0 = \max \{1, r_{2,l}/\theta_l, l=1,\ldots,L\}$ is for Case I and $ \nu_0 = \max \{1, {\sf tr}{\bf R}_2/\theta_{\rm all}\}$ is for Case II. The received
interference power constraint at the PUs (\ref{eq:the total received interference constrain}) can be controlled only through $\nu_0$, where $\beta$ is
not involved in $\nu_0$. Therefore, the SINR $\overline{\gamma}_{k}$ is significantly degraded if the channel path gains between the BS and the PUs (that is,
$r_{2,l}$'s) are strong. However, if the channel path gains between the BS and the PUs are weak, then $\nu_0 = 1$ and $\overline{\gamma}_{k}$ behave in a manner similar to but \emph{not} identical to that of the conventional RZF precoding because $c_1$ is replaced with $c_1(1-c_2)$.
Comparing (\ref{eq:gammabeta0}) for $\beta = 0$ with (\ref{eq:gammabeta1}) for $\beta = 1$ obtains notable results. First, we note that
(\ref{eq:gammabeta1}) and (\ref{eq:gammabeta0}) share a similar formulation, except the additional $\nu_0$ appears at the denominator of
(\ref{eq:gammabeta0}). When $\beta = 1$, the secondary BS yields zero interference on the PUs, such that the interference power constraint in
(\ref{eq:the total received interference constrain}) is always inactive. Therefore, no additional parameter $\nu_0$ is required to reflect the received
interference power constraint at the PUs. Although $\nu_0 \geq 1$, the SINR performance of the PP-RZF precoding with $\beta = 1$ is not implied to
be always better than that with $\beta = 0$. An additional note should be given on $\zeta(\cdot,\eta,\alpha) $, where the argument $\cdot$ is $\mu$ for
$\beta = 1$ and $1$ for $\beta = 0$. The parameter $\mu = (N-L)/N$ for $\beta = 1$ implies that the additional $L$ degrees of freedom is used to suppress
interference to the PUs. Consequently, if the channel path gains between the BS and the PUs are weak, the SINR performance of the PP-RZF precoding with
$\beta = 1$ shall not be better than that with $\beta = 0$. Thus, we infer that the projection control parameter should be decreased if the received
interference power constraint at the PUs is relaxed.
Finally, we note that $\zeta(1,\eta,\alpha)$ agrees with $z(r,\alpha_0)$ in \cite[Theorem 1]{HeYY-13TWC,HeYY-13ICASSP}. As a result,
(\ref{eq:gammabeta0}) is identical to the deterministic equivalent for the SINR obtained in \cite[Theorem 1]{HeYY-13TWC,HeYY-13ICASSP}, where the PP-RZF precoding
with a \emph{single} PU is considered. The deterministic equivalent for the SINR in \cite[Theorem 1]{HeYY-13TWC,HeYY-13ICASSP} is clearly a special case
of \eqref{eq:gamma1 deterministic equivalent} with $\beta = 0$ even though the case of $\beta \neq 0$ is considered in \cite[Theorem 1]{HeYY-13TWC,HeYY-13ICASSP} because a single PU results only in one-dimensional perturbation, and the effect of such perturbation \emph{vanishes} in a large system. Even if the number of PUs $L$ is finite and only $N$ becomes large, the effect of $\beta$ vanishes. The lack of a relation between $\beta$ and the SINRs will result in a bias when the number of antennas at the BS is not so large. However, our analytical results show the effect of $\beta$ by assuming that $N$, $K$, and $L$ are large, whereas $c_1 = N/K$ and $c_2 = L/N$ are fixed ratios. Thus, our results are clearly more general than those in \cite{HeYY-13TWC,HeYY-13ICASSP}.
\end{enumerate}
\begin{corollary}\label{Co: Th1}
In addition to the assumptions of Theorem \ref{Th: 2}, we suppose further that $c_2 = 1$ (that is, $N = L$), ${\bf R}_1 = r_1 {\bf I}_K$, and $\beta \in [0,1)$.
Then, as ${\mathcal N} \rightarrow \infty$, we have $\gamma_{k} - \overline{\gamma} \xrightarrow{a.s.} 0$ for $k = 1,\dots,K$, where
\begin{align}
\overline{\gamma} &= \frac{ \rho \left( c_1 r_1^2-(c_1 \alpha e - r_1)^2 \right)} { \rho (c_1 \alpha e)^2 + \nu_0}, \label{eq:Degamma1}
\end{align}
and $e$ is given as an unique solution to the fixed point equation
\begin{align}
e = \frac{r_1(1+e(1-\beta)^2)}{c_1 \alpha (1+e(1-\beta)^2) + c_1 r_1 (1-\beta)^2}, \nonumber
\end{align}
and $ \nu_0 = r_1 \max \{1, r_{2,l}/\theta_l, l=1,\ldots,L\}$ for Case I or $ \nu_0 = r_1 \max \{1, {\sf tr}{\bf R}_2/\theta_{\rm all}\}$ for Case II.
\end{corollary}
\begin{proof}
By letting $c_2 = 1$ and ${\bf R}_1 = r_1 {\bf I}_K$, we immediately obtain the result from Theorems \ref{Th: 2} and \ref{Th: 1}.
\end{proof}
For a brief illustration, we consider only Case II of Corollary \ref{Co: Th1} because the same characteristics can be found in Case I. Given that
$\theta_{\rm all} = P_{\rm all}/P_T = P_{\rm all}/(\sigma^2 \rho)$, \eqref{eq:Degamma1} can be rewritten as
\begin{equation}\label{eq:gammaRemark}
\overline{\gamma} = \left\{
\begin{aligned}
&\frac{ c_1 r_1^2-(c_1 \alpha e - r_1)^2 } { (c_1 \alpha e)^2 + 1/\rho}, & & 0<\frac{\rho \sigma^2{\sf tr}{\bf R}_2}{P_{\rm all}} \leq 1; \\
&\frac{ c_1 r_1^2-(c_1 \alpha e - r_1)^2 } { (c_1 \alpha e)^2 + \sigma^2{\sf tr}{\bf R}_2/P_{\rm all} }, & & 1 < \frac{\rho \sigma^2{\sf tr}{\bf R}_2}{P_{\rm all}} .
\end{aligned}
\right.
\end{equation}
We can see that $\overline{\gamma}$ does not depend on the SNR $\rho$ when $1 < \rho \sigma^2{\sf tr}{\bf R}_2/ P_{\rm all}$. In this case, the system performance is
interference-limited. Notably, the assumptions of $c_2 = 1$ and $\beta \neq 1$ are taken in Corollary \ref{Co: Th1}. In the case of $c_2 = 1$ and
$\beta = 1$, from \eqref{eq:Dea}, we have $\overline{a}_k = 0$ and consequently $\overline{\gamma}_{k}=0$, which implies a failure in the transmission. This result
is reasonable because when $c_2 = 1$, the dimension of the null space of ${\bf F}$ is zero with probability one.\footnote{ If $N = L$, we have ${\sf
Rank}({\bf I}-{\bf F}^H({\bf F}\qF^H)^{-1}{\bf F}) = 0$ with probability one because from \cite[Theorem 1.1]{Rudelson-09Arxiv}, ${\bf F}$ is a full rank square matrix with probability
one. } Therefore, the setting of $\beta = 1$ results in transmission failure, \emph{even} when the channel path gains between the BS and the PUs are weak. We thus show that a choice of appropriate $\beta$ significantly affects the successful operation of the CR network, which serves as
motivation for the remainder of this paper.
\subsection{Asymptotically Optimal Parameters}
Our numerical results confirm the high accuracy of the deterministic equivalent for the ergodic sum-rate $\overline{R}_{\rm{sum}}$ in the next section. Therefore, the
deterministic equivalent for the ergodic sum-rate can be used to determine the regularization parameter $\alpha$ and the projection control parameter $\beta$. By replacing
$R_{\rm{sum}}$ with $\overline{R}_{\rm{sum}}$ in \eqref{eq:optimal ergodic sum-rate}, we focus on this particular optimization to maximize the deterministic
equivalent for the ergodic sum-rate
\begin{align}\label{eq:optimal de sum-rate}
\left\{\overline{\alpha}^{\rm opt}, \overline{\beta}^{\rm opt} \right\} = \operatornamewithlimits{arg\, max}_{\alpha > 0, 1 \geq \beta \geq 0} & \overline{R}_{\rm{sum}}.
\end{align}
Similar to the problem in (\ref{eq:optimal ergodic sum-rate}), the asymptotically optimal solutions $\overline{\alpha}^{\rm opt}$ and $\overline{\beta}^{\rm
opt}$ do not permit closed-form solutions. However, the asymptotically optimal solution can be computed efficiently via the following methods without the need for Monte-Carlo averaging because $\overline{\gamma}_{k}$ is deterministic. First, given that $\beta$ is fixed, the optimal $\overline{\alpha}^{\rm opt}(\beta) := \operatornamewithlimits{arg\, max}_{\alpha > 0} \overline{R}_{\rm{sum}}(\beta)$ can be obtained efficiently via one-dimensional line search \cite{Wagner-12IT,ZhangJun-13TWC}, which performs the simple gradient method. The complexity in this part is linear. Then, we obtain the optimal $\overline{\beta}^{\rm opt} := \operatornamewithlimits{arg\, max}_{0 \leq \beta \leq 1} \overline{R}_{\rm{sum}}(\overline{\alpha}^{\rm opt}(\beta),\beta)$ through the one-dimensional exhaustive search\footnote{Although the one-dimensional exhaustive search seems burdensome, the case in question here is easy because the search is only over a closed set $0 \leq \beta \leq 1$.}. Finally, the optimal parameters are given by $\{\overline{\alpha}^{\rm opt}(\overline{\beta}^{\rm opt}), \overline{\beta}^{\rm opt} \}$. For a special case, we obtain a condition of the optimal solutions in the following proposition:
\begin{proposition}\label{proposition: opt alphabeta}
Under the assumptions of Corollary \ref{Co: Th1}, the asymptotically optimal parameters $\overline{\alpha}^{\rm opt}$ and $\overline{\beta}^{\rm opt}$ satisfy the equation
\begin{align}
\overline{\alpha}^{\rm opt} = \frac{\nu_0 \left(1-\overline{\beta}^{\rm opt}\right)^2}{\rho c_1 r_1}. \label{eq:optimal alphabeta}
\end{align}
where $\overline{\beta}^{\rm opt} \in [0,1)$.
\end{proposition}
\begin{proof}
By differentiating $\overline{R}_{\rm{sum}}$ with respect to $\alpha$ and $\beta$, we immediately obtain the result from Corollary \ref{Co: Th1}.
\end{proof}
From Proposition \ref{proposition: opt alphabeta}, we note that the number of asymptotically optimal solutions is infinite. All $\alpha$'s and $\beta$'s
that satisfy \eqref{eq:optimal alphabeta} are optimal. This condition will be confirmed in the next section.
Similar to \eqref{eq:gammaRemark}, we consider Case II for brief illustration. In this case, \eqref{eq:optimal alphabeta} can be rewritten as
\begin{equation}\label{eq:alphaRemark}
\overline{\alpha}^{\rm opt} = \left\{
\begin{aligned}
&\frac{ (1-\overline{\beta}^{\rm opt})^2}{\rho c_1 r_1}, & & 0<\frac{\rho \sigma^2{\sf tr}{\bf R}_2}{P_{\rm all}} \leq 1; \\
&\frac{ (1-\overline{\beta}^{\rm opt})^2 \sigma^2{\sf tr}{\bf R}_2 }{ c_1 r_1 P_{\rm all}}, & & 1 < \frac{\rho \sigma^2{\sf tr}{\bf R}_2}{P_{\rm all}} .
\end{aligned}
\right.
\end{equation}
From \eqref{eq:gammaRemark}, when $0< \rho \sigma^2{\sf tr}{\bf R}_2/P_{\rm all} \leq 1$, the system performance is unaffected by the average received
interference power constraint. In this case, $\overline{\beta}^{\rm opt}$ is expected to be close to $0$ because the weak interference at all the PUs is negligible. This condition is combined with the first term of \eqref{eq:alphaRemark} to reveal that $\overline{\alpha}^{\rm opt}$ decreases with
increasing $\rho$, where $\rho = P_T/\sigma^2$ is the same as previously defined. However, when $\rho \sigma^2{\sf tr}{\bf R}_2/P_{\rm all} >1$, the system
performance is limited by the average received interference power constraint. To decrease the interference, $\overline{\beta}^{\rm opt}$ is expected to be
close to $1$. Therefore, the second term of \eqref{eq:alphaRemark} reveals that $\alpha$ decreases to $0$ with an increase in $\overline{\beta}^{\rm opt}$.
We end this section by observing two additional extreme cases in Theorem \ref{Th: 2} for ${\bf R}_1 = {\bf I}_K$: If $\beta = 0$, by means of some
algebraic manipulations, we obtain $\overline{\alpha}^{\rm opt} = \nu_0/(c_1\rho)$. By contrast, if $\beta = 1$ and $c_2 \neq 1$, we obtain
$\overline{\alpha}^{\rm opt} = 1/(c_1 \rho)$. We find that the optimal regularization parameter tends to decrease monotonically with increasing $\rho$,
as expected. This characteristic is similar to that of the conventional RZF precoding in \cite{Peel-05Tcom,Nguyen-08GLCOM}, where $r_1= 1$ is assumed and
the asymptotically optimal regularization parameter $\overline{\alpha}^{\rm opt} = 1/(c_1 \rho )$ is derived.
\section{Simulations}\label{Se:simulations}
In this section, we conduct simulations to confirm our analytical results. First, we compare the analytical results \eqref{eq:deterministic equivalent
sum-rate} in Theorem \ref{Th: 2} and the Monte-Carlo simulation results \eqref{eq:the ergodic sum-rate} obtained from averaging over a large number of i.i.d. Rayleigh
fading channels. In the simulations, we set channel path gains $r_{1,k} = 1$ and $r_{2,l} = 0.6$ for all $k$ and $l$ and assume that $P_l = P$ for all $l$ in Case
I and $P_{\rm all} = LP$ in Case II. Several characteristics of Cases I and II are similar. Thus, without loss of generality, we provide the numerical results of Case I only.
\begin{figure}
\begin{center}
\resizebox{4.5 in}{!}{%
\includegraphics*{Compare_SumrateMandD.eps} }%
\caption{Ergodic sum-rate and the deterministic equivalent results under different interference power threshold and and two different antenna configuration cases.}\label{fig:2}
\end{center}
\end{figure}
Fig. \ref{fig:2} compares the ergodic sum-rate and its deterministic equivalent result under different interference power thresholds $P\in \{-10 {\rm dB} ,
\, 0{\rm dB} \}$ and two different antenna configuration cases: $\{ N=10,~K=8,~L=6\}$ and $\{N=16,~K=8,~L=6\}$. In the simulation, $\{\alpha^{\rm opt}, \beta^{\rm
opt} \}$ is obtained by using the two-dimensional line search in \eqref{eq:optimal ergodic sum-rate}. We find that the deterministic
equivalent is accurate under various settings even for systems with a not-so-large number of antennas. In addition, Fig. \ref{fig:2} illustrates that for the case
with $\{ N=10,~K=8,~L=6\}$, the sum-rate of the SUs cannot increase linearly in SNR and becomes interference-limited because the sum-rate of the SUs is
easily restricted by the average received interference power at each PU, particularly when the number of active users is larger than the number of
antennas at the BS, that is, $ L+K \geq N$.
\begin{figure}
\begin{center}
\resizebox{4.5 in}{!}{%
\includegraphics*{Compare_Sumrate_paramter.eps} }%
\caption{Ergodic sum-rate results under various parameters with $P=0$dB and $\{N=16,~K=8,~L=6\}$.}\label{fig:3}
\end{center}
\end{figure}
In the above simulations, the best solutions of $\{\alpha^{\rm opt}, \beta^{\rm opt} \}$ are calculated by Monte-Carlo averaging over $10^4$ independent trials; doing so
which clearly results in a high computational cost. To confirm that the optimization based on the deterministic equivalent is not only more
computationally efficient but also near-optimal, we compare the ergodic sum-rate of the PP-RZF precoding with $P=0$dB and $\{ N=16,~K=8,~L=6\}$ in Fig.
\ref{fig:3} for the following four cases: 1) $\{\overline{\alpha}^{\rm opt}, \overline{\beta}^{\rm opt} \}$, 2) $\{\alpha^{\rm opt}, \beta^{\rm opt} \}$, 3)
$\{\alpha^{\rm opt}, \beta=0\}$, and 4) $\{\alpha^{\rm opt}, \beta=1\}$. The solution of $\{\overline{\alpha}^{\rm opt}, \overline{\beta}^{\rm opt} \}$ is obtained by
using the two-dimensional line search in \eqref{eq:optimal de sum-rate}. $\{\overline{\alpha}^{\rm opt}, \overline{\beta}^{\rm opt} \}$ provides
results that are indistinguishable from those achieved by $\{\alpha^{\rm opt}, \beta^{\rm opt} \}$, which demonstrates that the optimization based on the deterministic
equivalent is promising. Moreover, the performance is significantly improved if the PP-RZF precoding with an appropriate choice of $\{\alpha, \beta \}$
is employed. In the low-SNR regime, the optimal transmission becomes the conventional RZF precoding, whereas the optimal transmission is the PP-RZF
precoding with $\beta=1$ in the high-SNR regime.
\begin{figure}
\begin{center}
\resizebox{4.5 in}{!}{%
\includegraphics*{Compare_alpha.eps} }%
\caption{Optimal $\alpha$ under different the interference power threshold for $\{N=16,~K=8,~L=6\}$.}\label{fig:4}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{4.5 in}{!}{%
\includegraphics*{Compare_beta.eps} }%
\caption{Optimal $\beta$ under different interference power threshold for $\{N=16,~K=8,~L=6\}$.}\label{fig:5}
\end{center}
\end{figure}
To provide further results on the optimal solutions of $\{\alpha, \beta \}$, Figs. \ref{fig:4} and \ref{fig:5} show the values of $\{\overline{\alpha}^{\rm
opt}, \overline{\beta}^{\rm opt} \}$, $\{\alpha^{\rm opt}, \beta^{\rm opt} \}$ under various settings. We have observed that the optimal parameter
$\{\overline{\alpha}^{\rm opt}, \overline{\beta}^{\rm opt} \}$ based on the deterministic equivalent result is almost consistent with $\{\alpha^{\rm opt},
\beta^{\rm opt} \}$ based on the ergodic sum-rate. Moreover, we have observed that with increasing $\rho$, $\alpha^{\rm opt}$ (or $\overline{\alpha}^{\rm opt}$)
tends to monotonically decrease to $0$, whereas $\beta^{\rm opt}$ (or $\overline{\beta}^{\rm opt}$) tends to monotonically increase from $0$ to $1$. These
characteristics are expected based on the analysis in Section III.
Finally, we confirm the result in Proposition \ref{proposition: opt alphabeta}. Fig. \ref{fig:6} displays the ergodic sum-rate under various parameter settings
with $P=0$dB and $\{N=10,~K=8,~L=10\}$. We find that when $c_2=1$, the parameters that satisfy \eqref{eq:optimal alphabeta} can achieve the
asymptotically optimal sum-rate for any $\beta \in [0,1)$, such that infinitely many asymptotically optimal solutions exist.
\begin{figure}
\begin{center}
\resizebox{4.5 in}{!}{%
\includegraphics*{Compare_Sumrate_paramter_C2=1.eps} }%
\caption{Ergodic sum-rate results under various parameters for $P=0$dB and $\{N=10,~K=8,~L=10\}$.}\label{fig:6}
\end{center}
\end{figure}
\section{Conclusion}
By exploiting the recent advancements in large dimensional RMT, we investigated downlink multiuser CR networks that consist of multiple SUs and multiple PUs.
The deterministic equivalent of the ergodic sum-rate based on the PP-RZF precoding was derived. Numerical results revealed that the deterministic equivalent sum-rate provides reliable performance predictions even for systems with a not-so-large number of
antennas. We thus used the deterministic equivalent result to identify the asymptotically optimal regularization parameter and the asymptotically optimal projection
control parameter. In addition, we provided the condition that the regularization parameter and the projection control parameter are asymptotically optimal. Several
insights have been gained into the optimal PP-RZF precoding design. A natural extension of this is to consider the PP-RZF precoding under various scenarios, such as spatial correlations and imperfect CSI at the transmitter. However, such development is still ongoing because of mathematical difficulties.
\section*{Appendix A: Proof of Theorem \ref{Th: 2}} \label{Appendix:Th2proof}
To complete this proof, we first introduce the limiting distribution for a new class of random Hermitian matrix in Theorem \ref{Th: 1}. Such distribution serves as the mathematical basis for the latter derivation. We recall the definition of the Stieltjes transform (see, e.g., \cite{SilversteinBai-95}). For a Hermitian matrix ${\bf B}_N \in {\mathbb C}^{N\times N}$, the Stieltjes transform of ${\bf B}_N$, is defined as
\begin{equation}
m_{{\bf B}_N}(\alpha) =\frac{1}{N}{\sf tr} \left({\bf B}_N + \alpha {\bf I}_N\right)^{-1} ~~ \mbox{for}~ \alpha \in {{\mathbb R}^+}. \nonumber
\end{equation}
For ease of explanation, we also define the matrix product Stieltjes transform of ${\bf B}_N$ as
\begin{equation}
m_{{\bf B}_N, {\bf Q}}(\alpha)=\frac{1}{N}{\sf tr} {\bf Q} \left({\bf B}_N + \alpha {\bf I}_N\right)^{-1}, \nonumber
\end{equation}
where ${\bf Q}$ is any matrix with bounded spectrum norm (with respect to $N$).
Notably, both $m_{{\bf B}_N}(\alpha)$ and $m_{{\bf B}_N, {\bf Q}}(\alpha)$ are functions of $\alpha$, but for ease of notation, $\alpha$ is dropped. In addition, all the
subsequent approximations will be performed under the limit ${\mathcal N} \rightarrow \infty$, and for ease of expression, $a \asymp b$ denotes that $a-b \xrightarrow{a.s.} 0$ as ${\mathcal N} \rightarrow \infty$.
\begin{theorem}\label{Th: 1}
Consider an $N \times N$ matrix of the following form:
\begin{equation}\label{eq:B}
{\bf B}_N = \check{{\bf H}}^H \check{{\bf H}} = \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}} \big) {\tilde{\qH}}^H {\bf R}_{1} {\tilde{\qH}} \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}} \big),
\end{equation}
where ${\tilde{\qW}}$, ${\tilde{\qH}}$, and ${\bf R}_{1}$ follow the restrictions given by Assumption \ref{Assum: 1}. Then, as ${\mathcal N} \rightarrow \infty$, we have
\begin{equation}
m_{{\bf B}_N, {\bf Q}} \asymp \frac{ t_1 + t_2}{\alpha} \frac{1}{N} {\sf tr} {\bf Q}, \label{eq:trQB}
\end{equation}
where $t_1 = \frac{1-c_2}{1+e}$ and $t_2 = \frac{c_2}{1+e(1-\beta)^2}$ with $e$ being the unique solution to the fixed point equation
\begin{align}
e =& \frac{1}{N} {\sf tr} {\bf R}_1 \left(\alpha {\bf I}_K + \left(t_1 + t_2 (1-\beta)^2\right) {\bf R}_1 \right)^{-1}. \label{eq:e}
\end{align}
\end{theorem}
\begin{proof}
If $c_2 = 1$ (i.e., $N = L$), ${\tilde{\qW}}^H {\tilde{\qW}} = {\bf I}_L$, the result is directly obtained by Lemma \ref{Lemma:TXXT} (see Appendix C).
We consider the case with $c_2 < 1$. Given that $m_{{\bf B}_N, {\bf Q}}$ is a function of two random matrices ${\tilde{\qW}}$ and ${\tilde{\qH}}$, we aim to derive an iterative deterministic equivalent \cite{Hoydis-11} of $m_{{\bf B}_N, {\bf Q}}$. In particular, we first find a function $\tilde{g}_N({\tilde{\qW}},\alpha)$, such that $f_N(({\tilde{\qH}},{\tilde{\qW}}),\alpha) \asymp \tilde{g}_N({\tilde{\qW}},\alpha)$, where
$f_N(({\tilde{\qH}},{\tilde{\qW}}),\alpha)\triangleq m_{{\bf B}_N, {\bf Q}}$, and $\tilde{g}_N({\tilde{\qW}},\alpha)$ is a function of ${\tilde{\qW}}$ and is independent of $\{{\tilde{\qH}}\}_{N\geq 1}$. Notably,
$\tilde{g}_N({\tilde{\qW}},\alpha)$ is a deterministic equivalent of $f_N(({\tilde{\qH}},{\tilde{\qW}}),\alpha)$ with respect to random matrix sequences $\{{\tilde{\qH}}\}_{N\geq 1}$. Second, we further find a function $g_N(\alpha)$, such that $\tilde{g}_N({\tilde{\qW}},\alpha) \asymp g_N(\alpha)$. Thus, we obtain an iterative deterministic equivalent
$g_N(\alpha)$ of $f_N(({\tilde{\qH}},{\tilde{\qW}}),\alpha)$, i.e., $f_N(({\tilde{\qH}},{\tilde{\qW}}),\alpha) \asymp g_N(\alpha)$.
When ${\tilde{\qW}}$ is treated as a deterministic matrix, applying Lemma \ref{Lemma:TXXT} (see Appendix C), we have
\begin{equation}
\frac{1}{N}{\sf tr} {\bf Q} \left({\bf B}_N + \alpha {\bf I}_N\right)^{-1} \asymp \frac{1}{N} {\sf tr} {\bf Q} \left(\alpha {\bf I}_N + \alpha e \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}} \big)^2 \right)^{-1}, \label{eq:trQB1}
\end{equation}
where
\begin{align}
e & = \frac{1}{N} {\sf tr} {\bf R}_1 \left(\alpha {\bf I}_K + {\tilde{e}} {\bf R}_1 \right)^{-1}, \label{eq:e2} \\
{\tilde{e}} & = \frac{1}{N} {\sf tr} \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}} \big)^2 \left( {\bf I}_N + e \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}} \big)^2 \right)^{-1}. \label{eq:te1}
\end{align}
Notice the fact that $({\tilde{\qW}}^H{\tilde{\qW}})^2 = {\tilde{\qW}}^H{\tilde{\qW}}$ so \eqref{eq:trQB1} and \eqref{eq:te1} can be written respectively as
\begin{align}
\frac{1}{N}& {\sf tr} {\bf Q} \left(\alpha {\bf I}_{N} + \alpha e \big({\bf I}_N-\beta{\tilde{\qW}}^H{\tilde{\qW}}\big)^2 \right)^{-1} = \frac{1}{\alpha(\beta^2-2\beta)e} \frac{1}{N} {\sf tr} {\bf Q} \big(\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1}, \label{eq:trQB2}
\end{align}
and
\begin{align}
{\tilde{e}} = &\frac{1}{(\beta^2-2\beta)e} \frac{1}{N}{\sf tr} \big(\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} + \frac{1}{e} \frac{1}{N}\sum_{l=1}^L {\tilde{\qw}}_l^H \big(\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} {\tilde{\qw}}_l, \label{eq:te2}
\end{align}
where $\omega \triangleq \frac{1 + e}{(\beta^2-2\beta)e}$ and ${\tilde{\qw}}_l$ denotes the $l-$th row of ${\tilde{\qW}}$.
Next, we aim to derive the deterministic equivalents of the terms $\frac{1}{N} {\sf tr} {\bf Q} (\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} )^{-1}$ and ${\tilde{\qw}}_l^H (\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} )^{-1} {\tilde{\qw}}_l$. Applying a result of the Haar matrix in Lemma \ref{Lemma:WW} (see Appendix C) to \eqref{eq:trQB2} and combing \eqref{eq:trQB1}, we immediately get \eqref{eq:trQB}. Then, we deal with the deterministic equivalent of ${\tilde{\qw}}_l^H \big(\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} {\tilde{\qw}}_l$. According to the matrix inverse lemma (see, e.g., \cite[Lemma 2.1]{Bai-09}\footnote{\cite[Lemma 2.1]{Bai-09}: For any ${\bf A}\in {\mathbb C}^{n\times n}$ and ${\bf q} \in {\mathbb C}^n$ with ${\bf A}$ and ${\bf A}+{\bf q}\qq^H$ invertible, we have $${\bf q}^H \left( {\bf A}+{\bf q}\qq^H \right)^{-1} = \frac{1}{1+{\bf q}^H{\bf A}^{-1}{\bf q}} {\bf q}^H{\bf A}^{-1}.$$}), we find
\begin{align}
{\tilde{\qw}}_l^H \big( \omega {\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} {\tilde{\qw}}_l = \frac{{\tilde{\qw}}_l^H \big( \omega {\bf I}_{N} + {\tilde{\qW}}_{[l]}^H{\tilde{\qW}}_{[l]} \big)^{-1} {\tilde{\qw}}_l}{1 + {\bf w}_l^H \big( \omega {\bf I}_{N} + {\tilde{\qW}}_{[l]}^H{\tilde{\qW}}_{[l]} \big)^{-1} {\tilde{\qw}}_l},
\end{align}
where ${\tilde{\qW}}_{[l]} \triangleq [ {\tilde{\qw}}_1, \ldots, {\tilde{\qw}}_{l-1}, {\tilde{\qw}}_{l+1}, \ldots, {\tilde{\qw}}_L ]^H \in {\mathbb C}^{(L-1)\times N}$. Then, the trace lemma for isometric matrices
\cite{Debbah-03TIT,Couillet-12IT} gives us
\begin{align}
{\tilde{\qw}}_l^H \big(\omega{\bf I}_{N} + {\tilde{\qW}}_{[l]}^H{\tilde{\qW}}_{[l]} \big)^{-1} {\tilde{\qw}}_l \asymp & \frac{1}{N-L} {\sf tr} \big({\bf I}_N-{\tilde{\qW}}_{[l]}^H{\tilde{\qW}}_{[l]}\big) \big(\omega{\bf I}_{N} + {\tilde{\qW}}_{[l]}^H{\tilde{\qW}}_{[l]} \big)^{-1} \nonumber \\
= & \frac{1+ \omega}{N-L} {\sf tr} \big(\omega{\bf I}_{N} + {\tilde{\qW}}_{[l]}^H {\tilde{\qW}}_{[l]} \big)^{-1} - \frac{N}{N-L}. \label{eq:wWlWlw}
\end{align}
Now, applying \cite[Lemma 2.2]{Bai-09} and \eqref{eq:lemmaQI} to \eqref{eq:wWlWlw}, we get
\begin{align}
{\tilde{\qw}}_l^H \big( \omega {\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} {\tilde{\qw}}_l \asymp \frac{1}{\omega+1}. \label{eq:wWWw}
\end{align}
Substituting \eqref{eq:wWWw} into \eqref{eq:te2} and using \eqref{eq:lemmaQI} and \eqref{eq:e2}, we obtain \eqref{eq:e}.
\end{proof}
Note that $m_{{\bf B}_N, {\bf Q}}$, $e$, $t_1$, and $t_2$ are all functions of $\alpha$ and $\beta$, but for ease of expression, $\alpha$ and $\beta$ are dropped.
Theorem \ref{Th: 1} indicates that $m_{{\bf B},{\bf Q}}$ can be approximated by its deterministic equivalent $ \frac{ t_1 + t_2}{\alpha} \frac{1}{N} {\sf tr} {\bf Q}$ without knowing the actual realization of channel random components. The deterministic equivalent is analytical and is much easier to compute than ${\sf E}_{{\bf B}}\{ m_{{\bf B},{\bf Q}} \}$, which requires time-consuming Monte-Carlo simulations. Motivated by this result in the large system limit, we aim to derive the deterministic equivalent of $\gamma_{k}$.
The SINR $\gamma_k$ in \eqref{eq:SINR} consists of three terms: $(\textrm{i})$ the signal power $| {\bf h}_k^H ( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N)^{-1} \check{{\bf h}}_k |^2$, $(\textrm{ii})$ the interference power ${\bf h}_k^H ( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N )^{-1} \check{{\bf H}}_{[k]}^H {\bf P}_{[k]} \check{{\bf H}}_{[k]} (\check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N )^{-1} {\bf h}_k$, and $(\textrm{iii})$ the noise power $\nu$. Using Theorem \ref{Th: 1}, we establish the following three lemmas to derive the deterministic equivalent of each term, whose proofs are detailed in Appendices B-I, B-II, and B-III, successively.
\begin{lemma}\label{Lemma:Deterministic of signal power}
Under the assumption of Theorem \ref{Th: 1}, as ${\mathcal N} \rightarrow \infty$, we have
\begin{equation}\label{eq:Deterministic of signal power}
{\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \asymp \overline{a}_k,
\end{equation}
where $\overline{a}_k$ has been obtained by \eqref{eq:Dea}.
\end{lemma}
\begin{lemma}\label{Lemma:Deterministic of interference power}
Under the assumption of Theorem \ref{Th: 1}, as ${\mathcal N} \rightarrow \infty$, we have
\begin{equation}\label{eq:Deterministic of interference power}
{\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \asymp \overline{b}_k,
\end{equation}
where $\overline{b}_k$ has been obtained by \eqref{eq:Deb}.
\end{lemma}
\begin{lemma}\label{Lemma:Deterministic of noise power}
Under the assumption of Theorem \ref{Th: 1}, as ${\mathcal N} \rightarrow \infty$, we have
\begin{equation}\label{eq:Deterministic of noise power}
\nu \asymp \overline{\nu} ,
\end{equation}
where $\overline{\nu}$ can be obtained by \eqref{eq:Dev} for Case I and by \eqref{eq:Dev2} for Case II.
\end{lemma}
According to Lemma \ref{Lemma:Deterministic of signal power}, Lemma \ref{Lemma:Deterministic of interference power}, and Lemma \ref{Lemma:Deterministic of noise
power}, we obtain the deterministic equivalent $\overline{\gamma}_k$ of $\gamma_k$ in \eqref{eq:gamma1 deterministic equivalent}. The proof is then completed.
\section*{Appendix B: Proofs of Lemma \ref{Lemma:Deterministic of signal power}, Lemma \ref{Lemma:Deterministic of interference power}, and Lemma \ref{Lemma:Deterministic of noise power}}
\subsection*{B-I: Proof of Lemma \ref{Lemma:Deterministic of signal power}}
We start from an application of the matrix inverse lemma \cite[Lemma 2.1]{Bai-09} to the signal term, which results in
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k = \frac{{\bf h}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k}{1+\check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k}. \label{eq:hcHcHch}
\end{align}
Using \cite[Lemma 2.3 and Lemma 2.2]{Bai-09}, we obtain
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k
\asymp & r_{1,k}\frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} - r_{1,k} \beta \frac{1}{N} {\sf tr} {\bf W}^H {\bf W} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1}. \label{eq:hBch}
\end{align}
Similarly,
\begin{align}
& \check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \nonumber \\
\asymp & r_{1,k} \frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} + r_{1,k} (\beta^2-2\beta) \frac{1}{N} {\sf tr} {\bf W}^H {\bf W} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1}. \label{eq:chBch}
\end{align}
According to Theorem \ref{Th: 1}, we have
\begin{align}
\frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \asymp \frac{t_1 + t_2 }{\alpha}. \label{eq:cHcH}
\end{align}
Noticing that ${\bf W}^H{\bf W} = {\tilde{\qW}}^H{\tilde{\qW}}$ and by using the same approach as \eqref{eq:te2}, we obtain
\begin{align}
\frac{1}{N} {\sf tr} {\bf W}^H {\bf W} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1}
\asymp & \frac{1}{\alpha(\beta^2-2\beta)e} \frac{1}{N} \sum_{l=1}^L {\tilde{\qw}}_l^H \big(\omega{\bf I}_{N} + {\tilde{\qW}}^H{\tilde{\qW}} \big)^{-1} {\tilde{\qw}}_l \asymp \frac{t_2}{\alpha}. \label{eq:WWcHcHI}
\end{align}
Substituting \eqref{eq:cHcH} and \eqref{eq:WWcHcHI} into \eqref{eq:hBch} and \eqref{eq:chBch}, we obtain
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k &\asymp \frac{r_{1,k} \left( t_1 + t_2(1-\beta) \right)}{\alpha}, \label{eq:DehcHcHch}\\
\check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k &\asymp \frac{r_{1,k} \left( t_1 + t_2(1-\beta)^2 \right)}{\alpha}. \label{eq:DechcHcHch}
\end{align}
Consequently, the expression of \eqref{eq:hcHcHch}, together with \eqref{eq:DehcHcHch} and \eqref{eq:DechcHcHch}, yields \eqref{eq:Deterministic of signal power}.
\subsection*{B-II: Proof of Lemma \ref{Lemma:Deterministic of interference power}}
Using the fact that ${\bf A}^{-1}-{\bf D}^{-1}=-{\bf A}^{-1}({\bf A}-{\bf D}){\bf D}^{-1}$, we have
\begin{align}
&{\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \nonumber\\
=&{\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k - \alpha {\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k\nonumber\\
&- {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \check{{\bf h}}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \nonumber\\
&+ \alpha {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf h}}_k \check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k. \label{eq:hchHIchHkchHIh}
\end{align}
Applying the matrix inverse lemma, we obtain
\begin{align}
& {\bf h}_k^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \nonumber\\
= & {\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k - \frac{{\bf h}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} {\bf h}_k}{1+ \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k}. \label{eq:hcHcHh}
\end{align}
Similarly,
\begin{align}
&{\bf h}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \nonumber\\
= & {\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} {\bf h}_k - \frac{{\bf h}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-2} \check{{\bf h}}_k \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} {\bf h}_k}{1+ \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k}, \label{eq:hcHcHcHcHh}
\end{align}
and
\begin{align}
&\check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} {\bf h}_k \nonumber\\
= & \check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} {\bf h}_k - \frac{\check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-2} \check{{\bf h}}_k \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} {\bf h}_k}{1+ \check{{\bf h}}_k^H \big( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \big)^{-1} \check{{\bf h}}_k}.\label{eq:chcHcHcHcHh}
\end{align}
According to Theorem \ref{Th: 1}, we have
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k \asymp \frac{r_{1,k} \left( t_1 + t_2 \right)}{\alpha}. \label{eq:DehcHcHh}
\end{align}
Noticing that
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} {\bf h}_k = - \frac{\partial}{\partial\alpha} {\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-1} {\bf h}_k,\nonumber
\end{align}
we thus obtain
\begin{align}
{\bf h}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} {\bf h}_k \asymp - r_{1,k} \frac{\partial }{\partial\alpha} \left(\frac{ t_1 + t_2 }{\alpha}\right). \label{eq:DehcHcH2h}
\end{align}
Similarly, combining \eqref{eq:DehcHcHch} and \eqref{eq:DechcHcHch} yields
\begin{align}
\check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} {\bf h}_k &\asymp - r_{1,k} \frac{\partial }{\partial\alpha} \left(\frac{ t_1 + (1-\beta)t_2 }{\alpha}\right), \label{eq:DechcHcH2h} \\
\check{{\bf h}}_k^H \left( \check{{\bf H}}_{[k]}^H \check{{\bf H}}_{[k]} + \alpha {\bf I}_N \right)^{-2} \check{{\bf h}}_k &\asymp - r_{1,k} \frac{\partial }{\partial\alpha} \left(\frac{ t_1 + (1-\beta)^2 t_2 }{\alpha}\right). \label{eq:DechcHcH2ch}
\end{align}
Substituting \eqref{eq:DehcHcHch}, \eqref{eq:DechcHcHch}, \eqref{eq:DehcHcHh}, \eqref{eq:DehcHcH2h}, \eqref{eq:DechcHcH2h}, and \eqref{eq:DechcHcH2ch} into \eqref{eq:hcHcHh}, \eqref{eq:hcHcHcHcHh}, and \eqref{eq:chcHcHcHcHh}, and combining \eqref{eq:Deterministic of signal power} and \eqref{eq:hchHIchHkchHIh}, we obtain \eqref{eq:Deterministic of interference power}.
\subsection*{B-III: Proof of Lemma \ref{Lemma:Deterministic of noise power}}
From \eqref{eq:nu}, we first have
\begin{align}
& \frac{1}{N} {\sf tr}\left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \nonumber \\
= & \frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} - \alpha \frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-2},
\end{align}
which, together with Theorem \ref{Th: 1}, yields
\begin{align}
\frac{1}{N} {\sf tr} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \asymp \frac{\partial t_1}{\partial \alpha}+\frac{\partial t_2}{\partial \alpha} . \label{eq:cHcHcHcH}
\end{align}
For Case I, we have
\begin{align}
&{\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \nonumber \\
= & {\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l - \alpha {\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-2} {\bf f}_l, \label{eq:fcHcHcHcHf}
\end{align}
where $l = 1,\ldots,L$. From \eqref{eq:trQB1} and \eqref{eq:trQB2}, we obtain
\begin{align}
{\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \asymp \frac{1}{\alpha(\beta^2-2\beta)e} {\sf tr} {\bf f}_l {\bf f}_l^H \left(\omega{\bf I}_{N} + {\bf W}^H{\bf W} \right)^{-1}.
\end{align}
Noticing the fact that ${\bf W} = ({\bf F}\qF^H)^{-\frac{1}{2}}{\bf F}$, by using the matrix inversion formula\footnote{
For invertible ${\bf A},{\bf B}$ and ${\bf R}$ matrices, suppose that ${\bf B}={\bf A}+{\bf X}{\bf R}{\bf Y}$, then ${\bf B}^{-1} = {\bf A}^{-1}-{\bf A}^{-1}{\bf X}({\bf R}^{-1}+{\bf Y}{\bf A}^{-1}{\bf X})^{-1}{\bf Y}{\bf A}^{-1}$.}, we obtain
\begin{align}
{\sf tr} {\bf f}_l {\bf f}_l^H \left(\omega{\bf I}_{N} + {\bf W}^H{\bf W} \right)^{-1}
= & {\sf tr} {\bf f}_l {\bf f}_l^H \left(\omega^{-1}{\bf I}_{N} - \omega^{-1} {\bf F}^H \left({\bf F}\qF^H + \omega^{-1} {\bf F} {\bf F}^H\right)^{-1} {\bf F} \omega^{-1} \right) \nonumber \\
= & \frac{1}{\omega+1} {\sf tr} {\bf f}_l {\bf f}_l^H \asymp \frac{r_{2,l}}{\omega+1}.
\end{align}
As a result,
\begin{align}
{\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \asymp \frac{r_{2,l} t_2}{c_2\alpha}.
\end{align}
Combining this with \eqref{eq:fcHcHcHcHf}, we obtain
\begin{align}
&{\bf f}_l^H \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} \check{{\bf H}}^H \check{{\bf H}} \left( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \right)^{-1} {\bf f}_l \asymp \frac{r_{2,l}}{c_2} \frac{\partial t_2}{\partial \alpha}. \label{eq:fcHcHcHcHf2}
\end{align}
From \eqref{eq:cHcHcHcH} and \eqref{eq:fcHcHcHcHf2}, the proof of \eqref{eq:Dev} can be accomplished using
\begin{subequations}\label{eq:m_Exm}
\begin{align}
&\frac{1}{N} {\sf tr} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \check{{\bf H}}^H \check{{\bf H}} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1}\nonumber \\
\asymp & {\sf E} \left\{ \frac{1}{N} {\sf tr} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \check{{\bf H}}^H \check{{\bf H}} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \right\}, \\
&{\bf f}_l^H \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \check{{\bf H}}^H \check{{\bf H}} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} {\bf f}_l \nonumber \\
\asymp & {\sf E} \left\{ {\bf f}_l^H \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \check{{\bf H}}^H \check{{\bf H}} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} {\bf f}_l \right\}.
\end{align}
\end{subequations}
By using the martingale approach, we can prove \eqref{eq:m_Exm} (See \cite{WenCK-13TIT} for a similar application).
Similarly, for Case II, we have
\begin{align}
{\sf E} \left\{ {\sf tr} {\bf F} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} \check{{\bf H}}^H \check{{\bf H}} \big( \check{{\bf H}}^H \check{{\bf H}} + \alpha {\bf I}_N \big)^{-1} {\bf F}^H \right\} \asymp \frac{{\sf tr} {\bf R}_2}{c_2} \frac{\partial t_2}{\partial \alpha}. \label{eq:FcHcHcHcHF}
\end{align}
Therefore, we obtain \eqref{eq:Dev2}.
\section*{Appendix C: Related Lemmas}\label{Appendix:Related Lemmas}
In this appendix, we provide some lemmas needed in the proof of Appendix A.
\begin{lemma}\label{Lemma:TXXT}
Let ${\bf X} \equiv [\frac{1}{\sqrt{N}}X_{ij}] \in {\mathbb C}^{N \times K}$, where $X_{ij}$'s are i.i.d. with zero mean, unit variance and finite $4$-th order moment. In addition, let ${\bf Q} \in {\mathbb C}^{N
\times N}$, ${\bf T} \in {\mathbb C}^{N\times N}$, and ${\bf R} \in {\mathbb C}^{K\times K}$ be nonnegative definite matrices with uniformly bounded spectral norm (with respect to $N$, $N$, and $K$, respectively).
Consider an $N \times N$ matrix of the form ${\bf B}_N = {\bf T}^\frac{1}{2} {\bf X} {\bf R} {\bf X}^H {\bf T}^\frac{1}{2}$. Define $c_1 = N/K$. Then, as $K, N \rightarrow \infty$ such that $0<\lim\inf_N c_1
\leq \lim\sup_N c_1 <\infty$, the following holds for any $\omega \in {\mathbb R}^+$:
\begin{equation}
\frac{1}{N} {\sf tr} {\bf Q} \big( {\bf B}_N + \omega {\bf I}_{N} \big)^{-1} \asymp \frac{1}{N} {\sf tr} {\bf Q} \left(\omega {\bf I}_{N} + \omega e {\bf T}\right)^{-1},
\end{equation}
where $e$ is given as the unique solution to the fixed-point equations
\begin{align}
e &= \frac{1}{N} {\sf tr} {\bf R} (\omega {\bf I}_K + {\tilde{e}} {\bf R} )^{-1}, \nonumber \\
{\tilde{e}} &= \frac{1}{N} {\sf tr} {\bf T} ( {\bf I}_N + e {\bf T} )^{-1}. \nonumber
\end{align}
\end{lemma}
\begin{proof}
As a special case of \cite[Theorem 1]{ZhangJun-13JSAC} or \cite[Theorem 1]{Wagner-12IT}, the result can be obtained immediately.
\end{proof}
\begin{lemma}\label{Lemma:WW}
Let ${\bf Q} \in {\mathbb C}^{N \times N}$ be a nonnegative definite matrix with uniformly bounded spectral norm (with respect to $N$) and ${\tilde{\qW}} \in {\mathbb C}^{L\times N}$ be
$L \leq N$ rows of an $N \times N$ Haar-distributed unitary random matrix. Define $c_2 = L/N$. Then, as $L, N \rightarrow \infty$ such that $0<\lim\inf_N c_2 \leq
\lim\sup_N c_2 \leq 1$, the following holds for any $\omega \in {\mathbb R}^+$:
\begin{equation}
\frac{1}{N} {\sf tr} {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_{N} \big)^{-1} \asymp \left( \frac{c_2}{\omega+1} + \frac{1-c_2}{\omega} \right) \frac{1}{N} {\sf tr} {\bf Q}. \label{eq:trQWW}
\end{equation}
\end{lemma}
\begin{proof}
Since ${\tilde{\qW}}^H {\tilde{\qW}} = {\bf I}_L$ for $c_2 = 1$ (i.e., $N = L$), \eqref{eq:trQWW} evidently holds. We assume $c_2 < 1$ in the following proof.
Firstly, we consider a special case with ${\bf Q} = {\bf I}$. Using the identity of the Stieltjes transform \cite[Lemma 3.1]{Couillet-11BOOK} \footnote{\cite[Lemma 3.1]{Couillet-11BOOK}: Let ${\bf A} \in {\mathbb C}^{N\times n}$, ${\bf B} \in {\mathbb C}^{n\times N}$, such that ${\bf A}{\bf B}$ is Hermitian. Then, for $z \in {\mathbb C} \backslash {\mathbb R}$ $$\frac{n}{N} m_{{\bf B}{\bf A}}(z) = m_{{\bf A}{\bf B}}(z) + \frac{N-n}{N} \frac{1}{z}.$$}, we have
\begin{align}
\frac{1}{N} {\sf tr} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_{N} \big)^{-1} = \frac{c_2}{L} {\sf tr} \big( {\tilde{\qW}} {\tilde{\qW}}^H + \omega {\bf I}_L \big)^{-1} + \frac{1-c_2}{\omega}.
\end{align}
Notice that the rows of ${\tilde{\qW}}$ are orthogonal and hence ${\tilde{\qW}} {\tilde{\qW}}^H = {\bf I}_L$. Therefore,
\begin{align}
\frac{1}{N} {\sf tr} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_{N} \big)^{-1} \asymp \delta \triangleq \frac{c_2}{\omega+1} + \frac{1-c_2}{\omega}. \label{eq:lemmaQI}
\end{align}
Next, for any nonnegative definite matrix with uniformly bounded spectral norm (with respect to $N$) ${\bf Q}$, we have
\begin{align}
&\frac{1}{N} {\sf tr} {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} - \delta \frac{1}{N} {\sf tr} {\bf Q} \nonumber \\
= & \left( 1 - \delta \omega \right) \frac{1}{N} {\sf tr} {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} - \delta \sum_{l=1}^L {\bf w}_l^H {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} {\bf w}_l, \label{eq:lemmaQ1}
\end{align}
where the first equality follows from the resolvent identity: ${\bf A}^{-1}-{\bf B}^{-1} = {\bf A}^{-1} ({\bf B} - {\bf A}) {\bf B}^{-1}$ for invertible matrices ${\bf A}$ and ${\bf B}$. Using the matrix inverse lemma \cite[Lemma 2.1]{Bai-09}, the trace lemma for isometric matrices \cite{Debbah-03TIT,Couillet-12IT}, and the fact that ${\bf Q}$ has uniformly bounded spectral norm (with respect to $N$), we obtain
\begin{align}
\sum_{l=1}^L {\bf w}_l^H {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} {\bf w}_l
= & \sum_{l=1}^L \frac{{\bf w}_l^H {\bf Q} \big( {\tilde{\qW}}^H_{[l]} {\tilde{\qW}}_{[l]} + \omega {\bf I}_N \big)^{-1} {\bf w}_l}{1+{\bf w}_l^H \big( {\tilde{\qW}}^H_{[l]} {\tilde{\qW}}_{[l]} + \omega {\bf I}_N \big)^{-1} {\bf w}_l} \nonumber \\
\asymp & c_2 \frac{ (1+\omega)\frac{1}{N} {\sf tr} {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} - \frac{1}{N} {\sf tr} {\bf Q}}{ (1+\omega)\frac{1}{N} {\sf tr} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} - c_2}. \label{eq:sumwQw}
\end{align}
Substituting \eqref{eq:sumwQw} into \eqref{eq:lemmaQ1}, and combining \eqref{eq:lemmaQI}, yields
\begin{align}
\frac{(1+\omega)\delta \left( \frac{1}{N} {\sf tr} {\bf Q} \big( {\tilde{\qW}}^H {\tilde{\qW}} + \omega {\bf I}_N \big)^{-1} - \delta \frac{1}{N} {\sf tr} {\bf Q} \right)}{(1+\omega)\delta -c_2} \asymp 0.
\end{align}
Therefore, we get \eqref{eq:trQWW}.
\end{proof}
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{
\begin{lemma}[$S_i \succeq 0, S \succ 0$ and $S_i^\top=S_i$] \label{lem:SposDef}
The matrices $S_i$ are positive semidefinite, $S=\sum_{i=1}^N S_i$ is positive definite and $S_i=S_i^\top$.
\end{lemma}
\begin{proof}
By Assumption \ref{ass:LICQ} we have that all $\bar H_i$s are positive definite, i.e. $x^\top \bar H_i x > 0$ for all $x \in \mathbb{R}^{n_i}$. With $x:=H_i^{-1}y$ we have $x^\top \bar H_i x=y^\top (\bar H_i^{-1})^\top \bar H_i \bar H_i^{-1}y=y^\top \bar H_i^{-1} y > 0$. Furthermore let $y:=\bar A_i z$. Then $z^\top \bar A_i^\top \bar H_i^{-1} \bar A_i z = z^\top S_i z \geq 0$ for all $z \in \mathbb{R}^{n_c}$ as $\bar A_i$ may be rank deficient.
$S\succ 0:$ We know from Assumption \ref{ass:LICQ} that $x^\top \bar Hx > 0$. By defining $y:=Ax$ we have $x^\top A^\top \bar H A x=x^\top Sx >0$ as $A$ has full rank by LICQ.
As $H_i=H_i^\top$, we have $\bar H_i^\top = (Z_i^\top H_i Z_i)^\top=(H_i Z_i)^\top Z_i = Z_i^\top H_i^\top Z_i= \bar H_i$ and by the same argument $S_i^\top=(\bar A_i \bar H_i^{-1}\bar A_i^\top)^\top=S_i$.
To obtain $\tilde S_i$ we add elements to the main diagonal only yielding $\tilde S_i=\tilde S_i^\top$.
\end{proof}
\section{Conclusion}
\appendices
\input{appendix.tex}
\printbibliography
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\end{document}
\section{Introduction}
\emph{Distributed} optimization algorithms are of interest in many engineering applications due to their ability to solve large-scale problems efficiently and enable solution to optimization problems with limited information exchange \cite{Boyd2011}.\footnote{Note that there is no unified notion of \emph{distributed optimization}---while the classical optimization literature allows for (preferably small) central coordination \cite{Bertsekas1989}, in the power systems community optimization with any kind of centralized coordination is called \emph{hierarchical} or hierarchically distributed \cite{Molzahn2017}.}
These algorithms often employ a (usually simple) global coordination step while computationally expensive operations are executed in parallel or decentralized by local agents.
Some algorithms avoid any kind of central coordination and communicate on a neighborhood basis only; they are commonly denoted as \emph{decentralized} \cite{Nedic2018a}.
Decentralized algorithms are of significant application interest; yet they are difficult to design and to analyze.
The majority of results on distributed optimization investigates \emph{convex} problems \cite{Boyd2011,Bertsekas1989,Gabay1976}.
Many practically relevant problems, however, are inherently \emph{non-convex}; examples range from non-linear model predictive control \cite{Stewart2011,Mehrez2017} to power systems \cite{Erseghe2014a,Engelmann18a,Kekatos2013} and wireless sensor networks \cite{Lee2006}.
An approach to unconstrained non-convex problems via a push-sum algorithm can be found in \cite{Tatarenko2017}; \cite{Hours2017} employs an alternating trust-region method with convergence guarantees for general non-convex problems.
Algorithms based on distributing steps of centralized algorithms like Sequential Quadratic Programming (SQP) can be found in \cite{Tran-Dinh2013,Necoara2009a}.
A decomposition method of the linear algebra subproblems of an interior point method using the Schur complement is presented in \cite{Kang2014}.
Moreover, for special classes of non-convex problems, the Augmented Direction of Multipliers Method (ADMM) has convergence guarantees \cite{Hong2016,Wang2019}.
Note, however, that only a few algorithms for \emph{decentralized} non-convex optimization exist; to the best of our knowledge the only currently available algorithms are decentralized variants of the before mentioned algorithms \cite{Hong2016} and \cite{Tatarenko2017}.
The present paper proposes a design framework for general purpose decentralized algorithms applicable to constrained non-convex optimization defined over networks with generic topology.
To this end, we build upon the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method \cite{Houska2016} which solves general non-convex problems to local optimality with guarantees.
ALADIN exhibits advantageous local quadratic convergence under mild technical assumptions; however it requires solving a centralized Quadratic Program (QP) as coordination step.
Specifically, we propose to \emph{decentralize} ALADIN by solving the coordination QP---which is the only centralized step---in a decentralized fashion.\footnote{Note that the globalization routine in ALADIN also requires central coordination. However, as our goal for the present paper is developing a local algorithm, the globalization routine is not considered. Hence decentralizing the solution of the coordination QP provides an avenue towards a fully decentralized algorithm.}
To this end, we apply condensing techniques similar to \cite{Frasch2013,Kouzoupis2016} to reduce the dimension of the coordination QP.
Moreover, we prove that this coordination QP inherits the sparsity pattern from the original problem.
We use this insight in the key step of our developments:
the introduction of a second (inner) level of problem distribution to ALADIN. In other words, we show how the coordination QP can be solved efficiently in a decentralized fashion.
To the latter end, we propose a decentralized variant of the Conjugate Gradient (CG) method.
We also investigate the application of decentralized ADMM.
The proposed framework is based on two consecutive layers of problem distribution: the general outer ALADIN structure is combined with a second inner layer. Hence we refer to it as \emph{bi-level distribution}.
As iterative methods (such as CG and ADMM) typically return inexact solutions, the original local convergence analysis for ALADIN \cite{Houska2016} is not directly applicable.
Accounting for this fact, we show that local convergence properties of ALADIN are preserved by enforcing bounds on the accuracy of the inner decentralized methods.
These bounds are derived using arguments from inexact Newton methods \cite{Dembo1982}.
This way we obtain---to the best of our knowledge---one of the first \emph{decentralized} algorithms with local convergence guarantees for constrained non-convex problems.
The remainder is structured as follows: Section \ref{sec:Prelim} recalls ALADIN and condensing techniques for the coordination QP.
Section \ref{sec:convAnal} shows how the local convergence properties of ALADIN while solving inexactly the coordination QP.
Section \ref{sec:distrSol} provides details on how to solve the reduced system in a decentralized fashion using decentralized ADMM \cite{Gabay1976,Boyd2011} and decentralized conjugate gradient.
Finally, examples from power systems and from robotics are presented in Section \ref{sec:NumRes}.
\emph{Notation.}
If not explicitly stated differently, we use superscripts $(\cdot)^k$ for inner iterations and omit outer iteration indexes for simplicity. In optimization problems the Lagrange multiplier $\kappa$ associated to constraint $h$ is denoted as $h(x)\leq 0 \;|\; \kappa$. Given a matrix $S\in \mathbb{R}^{n\times m}$, $S_{ij}$ denotes its $ij$th entry.
\section{Preliminaries \& Problem Statement } \label{sec:Prelim}
\subsection{Recalling ALADIN}
Distributed optimization aims at solving problems of the form\footnote{Ovserve that \eqref{eq:sepProb} can be interpreted as a generalization of a consensus problem \cite{Boyd2011} in the sense that any consensus problem can be expressed in form of \eqref{eq:sepProb} by appropriately choosing $A_i$.}${}^,$\footnote{
\label{footnote:constr}
For the sake of simplified notation we consider only inequality constraints $h_i$ here. Including equality constraints $g_i:\mathbb{R}^{n_{x_i}}\mapsto \mathbb{R}^{n_{g_i}}$ does not pose any difficulty as $g_i$ can be reformulated as $0\leq g_i(x_i)\leq 0$.}
\begin{subequations}\label{eq:sepProb}
\begin{align}
\min_{x\in\mathbb{R}^{n_x}} \,& \sum_{i\in \mathcal{R}} f_i(x_i) \\
\text{subject to} \qquad h_i(x_i)&\leq 0 \quad \forall\, i \in \mathcal{R} \label{eq:IneqConstr}\\
\sum_{i\in \mathcal{R}}A_i x_i&=0, \label{eq:ConsConstr}
\end{align}
\end{subequations}
with objective functions $f_i:\mathbb{R}^{n_{x_i}}\rightarrow\mathbb{R}$ and constraints $h_i:\mathbb{R}^{n_{x_i}}\rightarrow\mathbb{R}^{n_{h_i}}$. In all subproblems $i\in \mathcal{R}=\{1,\dots,N\}$ the functions $f_i$ and $h_i$ are assumed to be twice continuously differentiable and possibly non-convex. The overall decision vector is
$x:={(x_1^\top,\dots,x^\top_{N})}^\top \in \mathbb{R}^{n_x}$ and the matrices $A_i \in \mathbb{R}^{n_c\times n_{x_i}}$ describe couplings between subproblems.
Standard ALADIN is summarized in Algorithm~\ref{alg:ALADIN2}; we refer to \cite{Houska2016,Engelmann2019,Jiang2017} for details including convergence proofs and application examples.
\begin{algorithm}[t]
\caption{Standard ALADIN (full-step variant)}
\small
\textbf{Initialization:} Initial guess $(z^0,\lambda^0)$, parameters $\Sigma_i\succ 0,\rho,\mu$. \\
\textbf{Repeat} (until convergence):
\begin{enumerate}
\item \textit{Parallelizable Step:} Solve for all $i\in \mathcal{R}$ locally
\begin{equation}
\begin{aligned} \label{eq:locStep}
x_i^k=\underset{x_i}{\text{arg}\text{min}}&\quad f_i(x_i) + (\lambda^k)^\top A_i x_i + \frac{\rho}{2}\left\|x_i-z_i^k\right\|_{\Sigma_i}^2 \\
&\text{s.t.}\quad h_i(x_i) \leq 0 \quad \mid \kappa_i^k,
\end{aligned}
\end{equation}
and compute sensitivities $H_i^k$, $g_i^k$ and $C_i^k$, cf. \cite{Houska2016}.
\item \textit{Coordination Step:} Solve the coordination problem \label{step:QPstep}
\begin{align} \label{eq:globStep}
\notag
&\underset{\Delta x,s}{\text{min}}\;\sum_{i\in \mathcal{R}} \displaystyle \frac{1}{2}\Delta x_i^\top H^k_i\Delta x_i + {g_i^k}^\top \Delta x_i \hspace{-0.1em} + \hspace{-0.1em} {\lambda^k}^\top \hspace{-0.2em} s + \hspace{-0.1em} \textstyle \frac{\mu}{2}\|s\|^2_2 \\
&\begin{aligned}
\text{s.t.} &\;\;\, \sum_{i\in \mathcal{R}}A_i(x^k_i+\Delta x_i) = s &&|\, \lambda^\text{QP},\\
&\;\;\; C^{\mathrm{act}\,k}_i \Delta x_i = 0 &&\forall i\in \mathcal{R}.
\end{aligned}
\end{align}
\item \textit{Broadcast and Update Variables:} \label{step:update} \noindent \\
$ \qquad
z^{k+1} \leftarrow x^k + \Delta x^k\quad\text{and} \quad \lambda^{k+1} \leftarrow \lambda^\mathrm{QP}.
$
\end{enumerate}
\label{alg:ALADIN2}
\end{algorithm}
Two main steps in ALADIN require central coordination and thus render ALADIN distributed instead of decentralized: (i) the coordination QP in Step~\ref{step:QPstep}) and (ii) an additional globalization strategy which is neglected (for the sake of simplicity) in Step~\ref{step:update}). Here, we focus on designing a \emph{local} optimization algorithm. Hence, we use the full-step variant of ALADIN and focus on issue (i). Note that---upon solving Step~\ref{step:QPstep}) exactly in a decentralized fashion and modulo technical subtleties---one directly obtains a decentralized algorithm for constrained non-convex problems \eqref{eq:sepProb}.
\subsection{Condensing the coordination QP}
In ALADIN (Algorithm \ref{alg:ALADIN2}) the coordination QP \eqref{eq:globStep} directly scales with the number of decision variables and constraints $(n_x + n_h + n_c)$, which may be prohibitive in many applications.
Hence we aim at reducing the size of \eqref{eq:globStep} to the number of coupling constraints $n_c$ which is typically much smaller than $(n_x + n_h)$.
In context of direct methods for numerical optimal control, a similar approach has been used in \cite{Kouzoupis2016}.
Subsequently, we derive the reduced QP based on the Schur-complement whereas the analysis in \cite{Kouzoupis2016} is based on dualization. In contrast to \cite{Kouzoupis2016}, we consider slack variables $s$ as they are important in practice.
In Step 2) of ALADIN one solves the coordination QP
\begin{align}
\label{eq::conqp}\notag
&\min_{\Delta x,s}& \sum_{i\in \mathcal{R}} \frac{1}{2}\Delta x_i^\top H_i\Delta x_i + &{g_i}^\top \Delta x_i
+ \lambda^\top s + \frac{\mu}{2}\|s\|^2_2\\
&\quad\text{s.t.} & C^\mathrm{act}_i \Delta x_i &= 0, \qquad \; \,\forall i\in \mathcal{R}\\ \notag
&&\sum_{i\in \mathcal{R}}A_i(x_i+\Delta x_i) &= s \quad|\; \lambda^\mathrm{QP},
\end{align}
where $H_i \in \mathbb{R}^{n_{x_i} \times n_{x_i}}$ are positive definite Hessian approximations of the local Lagrangians, $g_i\in \mathbb{R}^{n_{x_i}}$ and the gradients, $\lambda \in \mathbb{R}^{n_c}$ are Lagrange multiplier estimates for the consensus constraint, $C^\mathrm{act}_i\in \mathbb{R}^{n^k_{h_i}\times n_{x_i}}$ are constraint linearizations of the active constraints with $n^k_{h_i}$ being the number of active constraints in the $k$th ALADIN iteration in subproblem $i \in \mathcal{R}$. $A_i \in \mathbb{R}^{n_c \times n_{x_i}}$ describes linear coupling between the subproblems.
The slack $s \in \mathbb{R}^{n_c}$ in combination with a sufficiently large penalty parameter $\mu \in \mathbb{R}_{+}$ fosters numerical stability.\footnote{Neglecting the slack variables $s$ simplifies condensing. However, these variables are essential for handling inconsistent constraint linearizations \cite{Houska2016}. The examples of Section \ref{sec:NumRes} fail to converge in absence of them.} For the sake of readability, we suppress the ALADIN iteration superscripts $(\cdot)^k$ whenever possible without ambiguity.
\begin{assumption}[Strong regularity] \label{ass:LICQ}
For all ALADIN iterates $k \in \mathbb{N}$, for all $i\in \mathcal{R}$, and for all local minimizers of \eqref{eq:sepProb}, linear independence constraint qualification (LICQ), strict complementarity condition (SCC) and second-order sufficient conditions (SOSC) are satisfied on the nullspace of $C_i$, cf. \Cite{Nocedal2006}.\footnote{Note that the SOSC assumption made here is slightly stronger than the general SOSC from \cite{Nocedal2006}. Here we require positive definiteness of $H_i$ on the tangent space of the nonlinear constraints and do not consider the nullspace of the consensus constraints \eqref{eq:ConsConstr}.}
\end{assumption}
We employ the nullspace method \cite{Nocedal2006} to project \eqref{eq::conqp} onto the subspace spanned by $C_i^\mathrm{act}$.
Assumption~\ref{ass:LICQ} implies that $C^\mathrm{act}:=\operatorname{diag}_{i\in \mathcal{R}} C_i^\mathrm{act} \in \mathbb{R}^{n^k_{h}\times n_x}$ has full row-rank. Hence parametrizing $\operatorname{null}(C^\mathrm{act})$ in terms of $v\in \mathbb{R}^{n_x-n^k_{h}}$ yields
\[
\operatorname{null}(C^\mathrm{act})=\{ x \in \mathbb{R}^{n_x} \; | \; x=Zv,\; v \in \mathbb{R}^{n_x-n^k_{h}}\}
\]
where the columns of $Z \in \mathbb{R}^{n_x \times (n_x-n^k_{h})}$ form a basis of $\operatorname{null}(C^\mathrm{act})$. With $x_i:=Z_iv_i$ for all $i \in \mathcal{R}$, we write \eqref{eq::conqp} as
\begin{align} \notag
&\min_{\Delta v,s}&& \sum_{i\in \mathcal{R}} \frac{1}{2}\Delta v_i^\top\bar H_i\Delta v_i + {\bar g_i}^{\top} \Delta v_i + \lambda^{\top} s + \frac{\mu}{2}\|s\|^2_2\\
\label{eq::redProb}
&\quad\text{s.t.} && \sum_{i\in \mathcal{R}}\bar A_i(v_i+\Delta v_i) = s \quad|\; \lambda^\mathrm{QP}.
\end{align}
where $\bar H_i := Z_i^\top H_i Z_i$, $\bar g_i := Z_i g_i$, and $\bar A_i := A_i Z_i$.
Upon eliminating the equation for $s$, the KKT conditions of \eqref{eq::redProb} read
\begin{equation} \label{eq:KKTsystem}
\begin{pmatrix}
\bar H & \bar A^\top \\
\bar A & -\frac{1}{\mu} I
\end{pmatrix}
\begin{pmatrix}
\Delta v \\ \lambda^{\text{QP}}
\end{pmatrix}
=
\begin{pmatrix}
-\bar g \\ - \bar Av - \frac{1}{\mu} \lambda
\end{pmatrix},
\end{equation}
where $\bar H := \operatorname{diag}\left (\{\bar H_i\}_{i \in \mathcal{R}}\right )$, $\bar g := \left (\,\bar g_1^\top,\; \dots,\; \bar g_N^\top\, \right )^\top$ and ${\bar A:= \left (\,\bar A_1,\; \dots\;, \bar A_N \,\right )}$.
We use the \emph{Schur-complement} \cite[Chap. 16]{Nocedal2006} to further reduce \eqref{eq:KKTsystem}. SOSC implies that $\bar H$ is positive definite and therefore invertible. Hence, we solve the first row of \eqref{eq:KKTsystem} as $\Delta v = \bar H^{-1}(-{\bar A}^\top \lambda^{\text{QP}} -\bar g)$ and obtain
\begin{equation} \label{eq:redRedSystem}
( \mu^{-1}I + \bar A \bar H ^{-1} \bar A^\top) \lambda^{\text{QP}} = \bar A (v - \bar H ^{-1}\bar g) + \mu^{-1} \lambda
\end{equation}
which is a linear system of equations of dimension ${n_c}$.
After solving \eqref{eq:redRedSystem}, the solution to \eqref{eq::conqp} $\Delta x$ can be obtained by backwards substitution.
Exploiting the block structure of $\bar H$, $\bar g$ and $\bar A$ we write \eqref{eq:redRedSystem} as
\begin{equation} \label{eq:redRedSystem2}
\left(\mu^{-1}I + \sum_{i\in \mathcal{R}} S_i \right) \lambda^{\text{QP}} = \mu^{-1} \lambda + \sum_{i\in \mathcal{R}} s_i
\end{equation}
where $S_i=\bar A_i \bar H_i^{-1} \bar A_i^\top$ and $s_i=\bar A_i (v_i - \bar H_i ^{-1} \bar g_i)$. Observe that the matrices $S_i$ and the vectors $s_i$ can be computed entirely locally.
Furthermore, reverse application of the above formulas shows that the increments $\Delta x_i$ can be computed locally via
\begin{equation} \label{eq:backSub}
\Delta x_i = Z_i \bar H_i^{-1}\left ( -\bar A_i^\top \lambda^{\text{QP}}-\bar g_i\right ).
\end{equation}
Doing so, we arrive at a variant of ALADIN requiring less communication compared to the standard one in Algorithm~\ref{alg:ALADIN2}.
\subsection{Bi-level distributed ALADIN}
\begin{algorithm}[t]
\caption{Bi-level distributed ALADIN}
\small
\textbf{Initialization:} Initial guess $(z^0,\lambda^0)$, parameters $\Sigma_i\succ 0,\rho,\mu$. \\
\textbf{Repeat} (until convergence):
\begin{enumerate}
\item \textit{Parallelizable Step:} Solve for all $i\in \mathcal{R}$ locally \label{step:LocStepBil}
\begin{equation}
\begin{aligned} \label{eq:locStepbil}
x_i^k=\underset{x_i}{\text{arg}\text{min}}&\quad f_i(x_i) + (\lambda^k)^\top A_i x_i + \frac{\rho}{2}\left\|x_i-z_i^k\right\|_{\Sigma_i}^2 \\
&\text{s.t.}\quad h_i(x_i) \leq 0 \quad \mid \kappa_i^k,
\end{aligned}
\end{equation}
and compute \emph{condensed} sensitivities $\tilde S_i$ and $\tilde s_i$.
\item \textit{Coordination Step:} Solve decentralized/distributed \label{step:GlobStepBil}
\begin{equation} \label{eq:redRedSystemAlg}
\left(\mu^{-1}I + \sum_{i\in \mathcal{R}} S_i \right) \lambda^{\text{QP}} = \mu^{-1} \lambda + \sum_{i\in \mathcal{R}} s_i
\end{equation}
with residuum $ \|r_\lambda^k\|$ \eqref{eq:lamResidual} small enough according to \eqref{eq:pRes}.
\item \textit{Broadcast and
Update Variables:} \\
$\lambda^k\leftarrow\lambda^{\text{QP}}$ and $z_i^{k+1}\leftarrow x_i^k + \Delta x_i^k$ using \eqref{eq:backSub}.
\end{enumerate}
\label{alg:ALADINbil}
\end{algorithm}
Algorithm \ref{alg:ALADINbil} summarizes the general algorithmic framework for bi-level distributed ALADIN.
Note that the condensing all iterates $k$ for \eqref{eq::conqp} can be performed locally and that a coordination QP of reduced dimension is used for coordination. This distributed algorithm can in principle be applied as is. However, it still requires solving a (less complex) hierarchical coordination problem \eqref{eq:redRedSystemAlg}.
Observe that solving \eqref{eq:redRedSystemAlg} by a decentralized algorithm, one obtains a decentralized variant of ALADIN. In Section \ref{sec:distrSol} we propose two variants for doing so: one based on conjugate gradient and one based on ADMM.
As, for conceptual and numerical reasons, these iterative algorithms do not yield an exact values of $\lambda^{\text{QP}}$, the next section presents a convergence analysis of ALADIN for inexact solutions to \eqref{eq:redRedSystemAlg}.
\section{Numerical Case Studies} \label{sec:NumRes}
\subsection{AC Optimal Power Flow}
Non-convex AC optimal power flow problems are of crucial interest in control of power systems.
Specifically, we investigate the IEEE 30-bus system shown in Figure \ref{fig:ieee30bussystem} with data from \cite{Zimmerman2011}.
For details on how to formulate OPF problems in form of \eqref{eq:sepProb}
see \cite{Molzahn2017, Erseghe2015,Engelmann2019}.
Here we use the problem formulation and partitioning $\mathcal{P}$ from \cite{Engelmann2019} with ALADIN parameters $\rho = 10^6$, $\mu = 10^7$ and the step size for the lower-level ADMM $\rho= 2\cdot10^{-2}$.
In all cases we use warm-starting for CG and ADMM to accelerate convergence.
\begin{figure}[t]
\centering
\includegraphics[trim={7em 45em 7em 7em},clip,width=1\linewidth]{30busGraph}
\caption{IEEE 30-bus system with induced connectivity graph $\mathcal{G}$ and paritioning $\mathcal{P}= \{\{1\text{-}8, 28\} $, $ \{9\text{-}11, 17, 21, 22\}$, $\{24\text{-}27, 29, 30\}$, $ \{12\text{-}16, 18\text{-}20, 23\}\}$.}
\label{fig:ieee30bussystem}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[trim={1.4em 0.5em 0 0},clip,width=0.9\linewidth]{sparsityIpe2}
\caption{Sparsity patterns of the Schur complements for subproblem 1 and 2 with $\mathcal{C}(1)\cap\mathcal{C}(2)=\{5,\dots,12\}$ and the dashed interconnection from Figure \ref{fig:ieee30bussystem}.}
\label{fig:Si}
\end{figure}
\begin{figure*}[t]
\centering
\begin{subfigure}{0.49\textwidth}
\includegraphics[trim={2.5em 2em 0em 3em},width=1\linewidth]{convAL}
\captionsetup{width=.95\linewidth}
\subcaption{Convergence of ALADIN with exact linear algebra and distributed conjugate gradient with $80$ inner iterations. \phantom{BlindtextBlindtextBlindtextBlindtext}}
\label{fig:fullALCG}
\end{subfigure}
\begin{subfigure}{0.485\textwidth}
\includegraphics[trim={2em 1em 0 3em},clip,width=1\linewidth]{differentInnerIters}
\captionsetup{width=1\linewidth}
\subcaption{Convergence of ALADIN for $\{80,100,200,400,1000\}$ inner iterations with ADMM and for $80$ inner iterations with distributed conjugate gradient.}
\label{fig:diffIters}
\end{subfigure}
\caption{Convergence behavior of different decentralized ALADIN variants.}
\label{fig:convALCG}
\end{figure*}
The 30-bus example has two physical interconnections between subproblems 1 and 2 shown in Figure \ref{fig:ieee30bussystem}.
This leads to eight consensus constraints jointly assigned to subproblem 1 and 2 \cite{Engelmann2019}.
Figure \ref{fig:Si} shows the resulting sparsity patterns of the corresponding Schur-complements $ \tilde S_1\in \mathbb{R}^{32\times 32}$ and $ \tilde S_2\in \mathbb{R}^{32\times 32}$.
One can observe an overlap in the corresponding rows/columns of $\tilde S_1$ and $\tilde S_2$ predicted by Lemma \ref{lem:sparsity}. The rows/columns of the remaining Schur-complements $\tilde S_3$ and $\tilde S_4$ are zero respectively.
Figure \ref{fig:fullALCG} shows the behavior of standard ALADIN (exactly solved coordination QP) and for ALADIN CG.
Figure \ref{fig:diffIters} depicts the results for inexactly solved coordination QP with different fixed numbers of inner iterations for ALADIN CG and ALADIN ADMM.
Observe that there is almost no difference in the convergence rate of standard ALADIN compared with ALADIN CG with 80 inner iterations.
In contrast, different numbers of inner iterations influence the total convergence behavior of ALADIN ADMM, cf. Figure \ref{fig:diffIters}. Indeed the convergence speed varies greatly with $n^{\text{AD}} \in \{80,100,200,400,1000\}$; also the achievable accuracy of ALADIN ADMM seems to be limited by different numbers of inner ADMM iterations.
Whereas for ALADIN CG a fixed number of inner iterations yields good performance, the number of inner iterations necessary for ALADIN ADMM depends on the desired solution accuracy and it effects the overall convergence speed (i.e. the number of outer ALADIN iterations).
This behavior is underpinned by the total number of inner iterations (\# of inner iterations times \# of outer iterations) shown in Table \ref{tab::totInnerIter}.
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.7}
\caption{Total iterations versus inner iterations for OPF.}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{crllllll}
\toprule
$n^{inner}$ & $\epsilon$ & 80 & 100 & 200 & 400 & 1000
\\ \midrule
CG & $10^{-4}$ & 800 & 800 & 800 & 800 & 800 \\
ADMM & $10^{-2}$ & - & 7 000 & 7 000 & 7 600 & 11 000 \\
& $10^{-3}$ & - & - & 10 800 & 10 800 & 13 000 \\
& $10^{-4}$ & - & - & - & 14 800 & 16 000 \\
\bottomrule
\end{tabular}
\label{tab::totInnerIter}
\end{table}
Figure \ref{fig:convLAlate} depicts the convergence behavior of distributed conjugate gradient and ADMM for two different instances of \eqref{eq:redRedSystemAlg}.
The left-hand side shows the results for ALADIN CG and ALADIN ADMM at one of the first iterations of ALADIN where $\tilde S$ is quite ill-conditioned.
The right-hand side depicts the convergence of both algorithms when ALADIN is almost converged and therefore the condition number of $\tilde S$ is smaller.
Observe the sublinear convergence rate of ADMM versus the practically superlinear convergence rate of conjugate gradient (cf. Table \ref{tab::convPropCGAL}) in both cases.
Furthermore, note that the theoretical finite convergence of CG (here this would be 32 iterations) is not realized due to the conditioning of $\tilde S$.
However, the practical convergence rate of centralized CG appears to be superior to most other available decentralized
methods \cite{Nedic2018}.
\begin{figure}[t]
\centering
\includegraphics[trim={0.5em 1em 0em 1em},width=0.48\linewidth]{convLA}
\includegraphics[trim={0.5em 1em 0em 1em},width=0.48\linewidth]{convLAlate}
\caption{Convergence of ADMM and CG for typical $\tilde S$ occuring in OPF. On the left we have a $\tilde S$ with $\operatorname{cond(\tilde S)=8 \cdot 10^9}$ in the beginning of the ALADIN iterations and on the right we have $\operatorname{cond(\tilde S)=5 \cdot 10^6}$ when ALADIN is almost converged.}
\label{fig:convLAlate}
\end{figure}
\subsection{Distributed control of mobile robots}
As a second example we consider an Optimal Control Problem (OCP) where two mobile robots should reach their final position while keeping a minimum distance to each other, cf. \cite{Mehrez2017}.
The centralized OCP reads
\begin{subequations} \label{eq:robotOCP}
\begin{align}
&\hspace{-2em}\min_{z_i(\cdot),u_i(\cdot), \forall i \in \mathcal{R}} \int_0^T \sum_{i\in \mathcal{R}} \|z_i-z^e_i\|_{Q_i}^2 + \|u_i\|_{R_i}^2\, dt \label{eq:stageCost} \\
\quad \text{s.t.} \;\;\; & \dot z_i(t) = f_i(z_i(t),u_i(t)),\;\; z_i(0)=z_{i0}, && \hspace{-0.6cm}\forall i \in \mathcal{R} \label{eq:dynamics} \\ \label{eq:zeroTermConstr}
&(x,y)_i^\top(T)=(x^e,y^e)_i^\top, &&\hspace{-0.6cm} \forall i \in \mathcal{R} \\
& \|(x, y)_i^\top(t)-(x, y)_j^\top(t)\|_2^2\geq d^2, && \hspace{-0.6cm} i \neq j
\end{align}
\end{subequations}
where $z_i=(x_i\; y_i\; \theta_i)^\top$ is the state of each robot $i \in \mathcal{R}$, $x_i$ and $y_i$ describe the robots position in the $x$-$y$-plane, and $\theta_i$ is the yaw angle with respect to the $x$-axis (Fig. \ref{fig:robots}).
The stage cost \eqref{eq:stageCost} is the sum of quadratic tracking cost with respect to the desired end position $z^e_i \in \mathbb{R}^3$ for all robots. Constraint \eqref{eq:dynamics} are the continuous-time dynamics
\[
\dot z_i= f_i(z_i,u_i) :=
\begin{pmatrix}
v_i\cos (\theta_i) & v_i\sin(\theta_i) & \omega_i \end{pmatrix}^\top.
\forall i \in \{1,2\}.
\]
The inputs $u_i=(v_i\;\omega_i)^\top$ are the velocity $v_i$ the turning rate $\omega_i$.
The terminal constraint \eqref{eq:zeroTermConstr} and the stage cost \eqref{eq:stageCost} are chosen having
a distributed NMPC setting in mind \cite{Rawlings2017}.
In order to convert \eqref{eq:robotOCP} into a partially separable NLP \eqref{eq:sepProb}, we introduce auxiliary variables duplicating the predicted $(x$-$y)$ trajectories of each robot pair and enforce consensus by the constraint \eqref{eq:ConsConstr}.
Due to space limitations we do not elaborate this in detail.
We employ a direct solution approach and discretize \eqref{eq:robotOCP} via Euler-backward; the sampling period is $0.1\,$ seconds and the horizon is $T=10\,$ seconds. We consider $|\mathcal{R}|=2$ robots which should keep a distance of $d=5\,$m with $Q=0.1\cdot \operatorname{diag}\large ((10\;\;10\;\;1))$ and $R=\operatorname{diag}\large ((1\;\;1))$.
We use $\rho=10^2$, $\mu = 10^6$ and $\rho^{\text{AD}}=10^{-1}$ as tuning parameters for ALADIN.
Figure \ref{fig:openLoppTraj} shows the optimal open-loop trajectories for \eqref{eq:robotOCP}.
One can observe that the goal of collision avoidance is satisfied
while the robots move to their target positions.
Interestingly, Problem \eqref{eq:robotOCP} seems to be numerically quite different to the OPF problem.
Here, $n^{\text{CG}}=30$ inner iterations for CG suffice for local convergence although the problem size is ($n_x=1\,200$) much larger.
At the same time, at least $n^{\text{AD}}=2\,400$ inner iterations were needed for ADMM to achieve an accuracy of $\epsilon=10^{-4}$.
\begin{figure}[t]
\centering
\captionsetup[subfigure]{position=b}
\subcaptionbox{Robot models. \label{fig:robots}}{\includegraphics[width=0.49\linewidth]{robot2}}
\centering
\subcaptionbox{Open-loop trajectories. \label{fig:openLoppTraj}}{\includegraphics[trim={0em 0em 0em 0em},clip,width=0.49\linewidth]{robotPath2}}
\caption{Distributed control of mobile robots.}
\end{figure}
\subsection{Numerical communication analysis}
Finally, we evaluate forward communication as introduced in Section \ref{se:commAnal} practically.
Table \ref{tab::comparison} summarizes the forward communication for both examples.
In addition the last two rows in both parts of Table~\ref{tab::comparison} depict the total communication (per step-communication times outer \# of ALADIN iterations) for a termination tolerance of $\epsilon=10^{-4}$.
\begin{table}
\centering
\renewcommand{\arraystretch}{1.7}
\caption{Forward comm. to $\epsilon = 10^{-4}$ with $n^{\text{AD}}=400$, $n^{\text{CG}}=80$ for OPF and $n^{\text{AD}}=2\,400$, $n^{\text{CG}}=30$ for robot control.}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{crcccc}
\toprule
& variant & standard & \hspace{-1em} condensed & ADMM & CG \\ \midrule
\multirow{5}{*}{{\rotatebox[origin=c]{90}{OPF}}}
& local prep. & - & - & - & $2\,048$ \\
& local iter. & - & - & $25\, 600$ & $5\,120$ \\
& global & \hspace{-1em} $>\hspace{-0.3em}9\,858$ & \hspace{-1em} $1\,056$ & - & $960$ \\
\cmidrule{2-6} & local tot. & - & - & $691\,200$ & $53\,248$ \\
& global tot. & \hspace{-1em} $>\hspace{-0.3em} 98\,580$ & \hspace{-1em} $10\,560$ & - & $9\,600$ \\ \midrule
\multirow{5}{*}{{\rotatebox[origin=c]{90}{robots}}}
& local prep. & - & - & - & $80\,000$ \\
& local iter. & - & - & $960\,000$ & $12\,000$ \\
& global & \hspace{-1em} $>\hspace{-0.3em}824\,506$ & $40\,200$ & - & $120$ \\
\cmidrule{2-6} & local tot. & - & - & $9\,600\,$k & $200\,$k \\
& global tot. & \hspace{-1em} $>\hspace{-0.3em} 20\,613\,$k & $1\,005\,$k & - & $1\,$k \\ \bottomrule
\end{tabular}
\label{tab::comparison}
\end{table}
As expected, ALADIN with condensing (Algorithm \ref{alg:ALADINbil}) needs much less communication compared to standard ALADIN variant (Algorithm \ref{alg:ALADIN2}).
Solving \eqref{eq:redRedSystemAlg} with the decentralized variants of conjugate gradient or ADMM increases total communication compared to the condensed ALADIN variant.
Furthermore, the total communication of ALADIN CG is smaller compared to standard ALADIN.
The comparably large local communication burden of ALADIN ADMM stems from the increased number of inner iterations, cf. Figure \ref{fig:diffIters} and Table \ref{tab::totInnerIter}.
Finally, it is worth to be noted investing the very limited global coordination and communication effort required by ALADIN CG one can achieve much better performance compared with entirely decentralized coordination, cf. right-hand side columns of Table \ref{tab::comparison}.
\section{Summary \& Outlook}
This paper has proposed a framework for designing decentralized algorithms for non-convex constrained optimization problems via bi-level distribution of the ALADIN algorithm. The core idea is to add a second (inner) layer of distributed/decentralized computation to ALADIN, whereby the coordination QP is first condensed (as a post-processing step of solving the local non-convex subproblems) and then solved in decentralized fashion.
We have presented sufficient conditions on the numerical solution accuracy necessary to preserve local quadratic convergence properties of ALADIN. Moreover, we have shown how this bound can be enforced by means of decentralized inner algorithms. Specifically, we have proposed a decentralized variant of the conjugate gradient method, which shows promising performance. We also compared it to using ADMM at the inner level.
Simulation studies from power systems and robotics underpin the efficacy of the proposed scheme. These studies also indicate that decentralized conjugate gradient outperforms ADMM in terms of convergence speed and in terms of total communication effort.
We expect that the proposed bi-level distribution framework opens new avenues for future research, e.g., on decentralizing globalization strategies or on tailored decentralized algorithms for distributed non-linear model predictive control.
\section{Local Convergence Analysis} \label{sec:convAnal}
Usually decentralized algorithms solving \eqref{eq:redRedSystemAlg} achieve a finite precision only.
Hence it is fair to ask whether it is possible to preserve local convergence guarantees under inexact solutions.
We answer this question by combining properties of ALADIN \cite{Houska2016} with classical results from inexact Newton methods \cite{Dembo1982}.
Bi-level distributed ALADIN is composed of two main steps: the parallelizable Step~\ref{step:LocStepBil}) solving local NLPs and computing (condensed) sensitivities as well as the coordination Step~\ref{step:GlobStepBil}) solving \eqref{eq:redRedSystemAlg}. In order to establish local convergence properties of bi-level distributed ALADIN, we aim at ensuring progress towards a local minimizer in both steps.
From \cite[Lemma 3]{Houska2018} we have that the mapping formed by Step~\ref{step:LocStepBil}) is locally Lipschitz, i.e.
\begin{equation} \label{eq:minLip}
\|p^{k}-p^\star\| \leq \chi \|q^{k}-p^\star\|
\end{equation}
with $q^k=(z^k, \lambda^k,\kappa^{k-1})$ and $p^k:=(x^k,\lambda^k,\kappa^k)$ for some $\chi < \infty$.
The superscript $(\cdot)^\star$ denotes optimal primal and dual variables of
\eqref{eq:sepProb}
It remains to analyze the progress in the coordination problem~\eqref{eq:redRedSystemAlg}.
Eliminating $s$, the optimality conditions of \eqref{eq:globStep} read
\begin{equation} \label{eq:redKKT-QPdelta}
\underbrace{\begin{pmatrix}
H & A^\top & C^{a^\top} \\
A & -\frac{1}{\mu}I & 0 \\
C^{a} & 0 & 0
\end{pmatrix}}
_
{=:M(p^k)}
\Delta q^k \hspace{-1mm}
= \hspace{-1mm}
\underbrace{\begin{pmatrix}
-g -C^{a\top} \kappa^k -A^\top \lambda^k \\ -Ax^k +b \\ 0
\end{pmatrix}}
_
{=:m(p^k)},
\end{equation}
with $\Delta q^k=q^{k+1}-p^k$.
Apart from the entry $-\frac{1}{\mu}I$, \eqref{eq:redKKT-QPdelta} is equivalent to a Newton step for \eqref{eq:sepProb} if exact Hessians and Jacobians are used.
Hence, we have the typical progress in Step~\ref{step:QPstep}) known from Newton-type methods \cite{Nocedal2006}
\[
\|q^{k+1}-p^\star\|\leq \gamma \|p^k-p^\star\| + \frac{\omega}{2}\|p^k-p^\star\|^2_2
\]
where $\gamma=\|I-M(p^k)^{-1}\nabla^2\mathcal{L}(p^k)\|<1$ can be seen as a bound on the error of $\nabla^2\mathcal{L}(p^k)$ with $\mathcal{L}(x,\lambda,\kappa):=f(x)+\lambda^\top Ax + \kappa^\top h(x)$ being the Lagrangian to \eqref{eq:sepProb}.
Yet this only holds for an exact solution to \eqref{eq:redKKT-QPdelta}.
Denote the approximate solution by $\bar q^{k+1}$ and $ \Delta \bar q^k= \bar q^{k+1}-p^k$.
We define the residual for \eqref{eq:redKKT-QPdelta} similar to inexact Newton methods as
\[
r_p^k:= M(p^k)\Delta \bar q^k - m(p^k).
\]
We assume that the residual is bounded by
\begin{equation} \label{eq:pRes}
\|r_p^k\|\leq \eta^k\|m(p^k)\|,
\end{equation}
which we have to guarantee during the ALADIN iterations.
Now we have all the ingredients to prove the main result of this section.
\begin{theorem}[Conv. of Bi-level decentralized ALADIN]~\\
Consider bi-level distributed ALADIN (Algorithm \ref{alg:ALADINbil}).
Suppose Assumption~\ref{ass:LICQ} holds. Let $\frac{1}{\mu^k}=O(\|q^k-p^\star\|)$, let $\nabla^2\mathcal{L}$ and $\nabla\mathcal{L}$ be Lipschitz, and let the solution to the condensed QP \eqref{eq:redRedSystemAlg} satisfy \eqref{eq:pRes} in each iteration $k\in \mathbb{N}_+$.
Then there exists $\eta\in (\eta^k,\,\infty)$ such that bi-level distributed ALADIN converges locally to $(x^\star,\lambda^\star,\kappa^\star)$
\begin{itemize}
\item at linear rate; and
\item at quadratic rate if $\eta^k=O(\|q^{k}-p^\star\|)$.
\end{itemize}
\end{theorem}
\begin{proof}
The inequalities \eqref{eq:minLip}, \eqref{eq:pRes} and the Lipschitz property of $m$ with $\frac{1}{\mu^k}=O(\|q^k-p^\star\|)$ imply
\begin{align*}
\|\bar q^{k+1}-p^\star\|&\leq \|q^{k+1}-p^\star\|+\|\bar q^{k+1} - q^{k+1}\| \\
&\leq \frac{\omega}{2}\|q^k-p^\star\|^2_2 + \alpha\cdot \eta^k\|m(p^k)-m(p^\star)\| \\
&\leq \frac{\omega}{2}\|q^k-p^\star\|^2_2 + \alpha\cdot \beta\cdot \eta^k\|p^k-p^\star\|\\
&\leq \frac{\omega}{2} \|q^k-p^\star\|^2_2 + \alpha\cdot \beta\cdot\chi \cdot \eta^k\|q^k-p^\star\|,
\end{align*}
where $\beta$ is the Lipschitz-constant of $m$.
The finiteness of $\alpha,\beta$ and $\chi$ shows linear convergence if ${\alpha\cdot\beta\cdot\chi\cdot\eta^k<1}$.
Quadratic convergence follows immediately from the above inequality if $\eta^k=O(\|q^k-p^\star\|)$.
\end{proof}
The above result shows that inexact solutions to \eqref{eq:redRedSystemAlg} do not jeopardize linear or even quadratic local convergence of bi-level distributed ALADIN.
However, the question of how to evaluate \eqref{eq:pRes} in a decentralized setting arises.
To this end, we draw upon $r_p^k$ the residual of \eqref{eq:redRedSystemAlg}
\begin{align} \label{eq:lamResidual}
r_\lambda^k:=\left(\mu^{-1}I + \sum_{i\in \mathcal{R}} S_i \right) \lambda^k - \mu^{-1} \lambda - \sum_{i\in \mathcal{R}} s_i.
\end{align}
The structure of $ S$, $s$, and \eqref{eq:lamResidual} imply that
$r_\lambda^k = \bar A\bar H^{-1}(-\bar g-\bar A^\top \lambda^k) = \bar A \Delta v= \nabla_{\lambda} \mathcal{L}(p^k)$.
As we enforce $\nabla_{\Delta x} \mathcal{L}(p^k)=0$ and $\nabla_{\kappa} \mathcal{L}(p^k)=0$ by virtue of the nullspace method and the first row of \eqref{eq:KKTsystem}, we obtain $r_p^k={(0^\top \ \ 0^\top \ \ r_\lambda^{k\top})}^\top$ and $\|r_p^k\|=\|r_\lambda^k\|$.
Hence, note that one can evaluate \eqref{eq:pRes} using only the residual of the reduced system $\|r^k_\lambda\|$.
\section{Decentralized Solution of the Coord. QP \eqref{eq:redRedSystemAlg}}
\label{sec:distrSol}
Observe that the QP \eqref{eq:redRedSystemAlg} inherits structural properties of problem \eqref{eq:sepProb}; i.e. the Schur-complements $S_i$ inherit the sparsity pattern induced by the coupling matrices $A_i$.
This sparsity can be exploited---either to further reduce communication by using sparse matrix storage formats or to design decentralized algorithms.
Here we focus on the latter. We first analyze the sparsity of the matrices $S_i$s and then we propose two decentralized algorithms exploiting this sparsity.
\subsection{Sparsity of the Schur-complements}
Usually, the consensus constraint \eqref{eq:ConsConstr} describes couplings between two neighboring subproblems $i, j \in \mathcal{R}$. This means that in the matrices $A_i$ and $A_j$ the $i$th and $j$th rows are nonzero.
\begin{definition}[Assigned consensus constraints] \label{def:nonAss}
A subproblem $i\in \mathcal{R}$ is called assigned to consensus constraint $j \in{ \mathcal{C} = \{1,\dots,n_c\}}$, if the $j$th row of $A_i$ is non-zero.
Furthermore, all subproblems assigned to consensus constraint $j$ are collected in $\mathcal{R}(j):=\{i\in \mathcal{R}\;|\; i \; \text{assigned to}\; {j \in \mathcal{C}}\}$.
A consensus constraint $j \in \mathcal{C}$ is called $n$-assigned, if $|\mathcal{R}(j)|=n.$ Furthermore, if $|\mathcal{R}(j)|\leq n$ for all $j\in \mathcal{C}$, problem \eqref{eq:sepProb} is called $n$-assigned.
\end{definition}
Observe that assigned consensus constraints generalize the usual consensus setting \cite{Boyd2011}. Moreover, they provide an effective framework to analyze the sparsity pattern of the Schur-complements.
We remark that any generic consensus problem can be expressed in this form via appropriate choice of $A_i$ and using additional local variables.
\begin{remark}[Reformulation as 2-assigned problem]
Without loss of generality any $n$-assigned problem can be reformulated as $2$-assigned problem by introduction of auxiliary decision variables.
For example consider a $3$-assigned problem with consensus constraint
$ A_1x_1 + A_2 x_2 + A_3x_3=0$
where $A_1,A_2,$$A_3 \neq 0$.
Introduce a copy of $x_2$ in subproblem $1$ as $\tilde x_2 := x_2$ and define an augmented decision vector $\tilde x_1:=(x_1 \; \tilde x_2)^\top$. This yields a $2$-assigned problem in terms of the augmented decision vectors $(\tilde x_1, x_2, x_3)$
\begin{align*}
(A_1 \quad A_2)\, \tilde x_1 + A_3x_3 = 0, \quad
(0\quad I) \, \tilde x_1 - I x_2 = 0.
\end{align*}
\end{remark}
\begin{lemma}[Sparsity of $S_i$] \label{lem:sparsity}
The rows and columns of $S_i$ and entries of $s_i$, $i \in \mathcal{R}$, which are not assigned to consensus constraint $j$, (i.e. all $j \notin \mathcal{C}(i):= \{j \in \mathcal{C}\; |\; i \in \mathcal{R}(j) \}$) are zero.
\end{lemma}
\begin{proof}
We have $S_i= \bar A_i \bar H_i^{-1} \bar A_i^\top= A_i(Z_i \bar H_i^{-1} Z_i^\top) A_i^\top$. All columns of $A_i$ with $j \notin \mathcal{C}(i)$ are zero by Definition \ref{def:nonAss}. It follows immediately that the rows and columns of $S_i$ with $j \notin \mathcal{C}(i)$ are zero. The sparsity of $s_i=\bar A_i (v_i - \bar H_i ^{-1} \bar g_i)$ follows analogously.
\end{proof}
Lemma \ref{lem:sparsity} shows that the matrices $S_i$ and vectors $s_i$ have non-zero entries only for neighboring subproblems.
\vspace*{-3mm}
\subsection{Consensus reformulation}
Now, we reformulate \eqref{eq:redRedSystemAlg} as a strictly convex consensus problem such that the conjugate gradient method and ADMM are applicable.
Specifically, we reformulate \eqref{eq:redRedSystemAlg} as
\begin{equation} \label{eq:sumKKTsystem}
\left (\sum_{i\in \mathcal{R}} \tilde S_i \right ) \lambda^{\text{QP}} = \sum_{i\in \mathcal{R}} \tilde s_i
\end{equation}
where each $\tilde S_i$ and $\tilde s_i$ is constructed by local information only.
Equation \eqref{eq:redRedSystemAlg} implies that the reduced QP is separable as it involves sums of $S_i$ and $s_i$.
However, the terms $\mu^{-1} I$ and $\mu^{-1}\lambda$ can not directly be assigned to any of the subproblems.
One possibility is to introduce an additional subproblem which would serve as a coordinator. However, here we are interested in relying on neighborhood communication only.
Hence we distribute $\mu^{-1} I$ and $\mu^{-1}\lambda$ uniformly to all subproblems assigned to the corresponding consensus constraint. This yields
\begin{equation} \label{eq:sparsDef}
\tilde S_i := S_i + \sum_{j=1}^{n_c} \frac{\delta_{ij}}{|\mathcal{R}(j)|\, \mu } I_j, \; \tilde s_i := s_i + \sum_{j=1}^{n_c} \frac{\delta_{ij}}{|\mathcal{R}(j)|\, \mu } \lambda I_j,
\end{equation}
where $I_j$ contains only zeros except for $I_{jj}=1$ and $\delta_{ij} := 1$ if $j \in \mathcal{C}(i)$ and $0$ else.
This way \eqref{eq:redRedSystemAlg} is expressed in the form of \eqref{eq:sumKKTsystem} without destroying its sparsity pattern.
The next result reformulates \eqref{eq:sumKKTsystem} as strictly convex QP.
\begin{lemma}[Minimization to solve \eqref{eq:redRedSystem2}] \label{lem:SposDef}
The minimizer of
\begin{align} \label{eq:minLam}
&\min_{\lambda}\; \frac{1}{2}\lambda^\top \sum_{i=1}^{N} \tilde S_i \lambda - \sum_{i=1}^{N} \tilde s_i^\top \lambda.
\end{align}
is unique and solves \eqref{eq:sumKKTsystem}.
Furthermore, \eqref{eq:minLam} is strictly convex.
\end{lemma}
\begin{proof}
The first-order necessary condition for \eqref{eq:minLam} reads $\frac{1}{2}(\sum_{i=1}^{N} \tilde S_i + \sum_{i=1}^{N} \tilde S_i^\top) \lambda - \sum_{i=1}^{N} \tilde s_i =0$. From Lemma \ref{lem:SposDef} (given in the Appendix) one has that $\tilde S_i= \tilde S_i^\top$. This proves the first assertion. Moreover, Lemma \ref{lem:SposDef} gives $\sum_{i=1}^N \tilde S_i \succ 0$ which implies strict convexity of \eqref{eq:minLam}.
\end{proof}
\subsection{Decentralized conjugate gradient}
\begin{algorithm}[t]
\small
\caption{Decentralized conjugate gradient}
\textbf{Initialization:} Initial guess $r^0=p^0= \tilde s-\tilde S\lambda^0 $.\\
\textbf{Preparation:} Exchange $\tilde S_{i}e_j$ and $\tilde s_i^\top e_j$ between neighboring regions $\mathcal{R}(j)$ locally for all $j \in \mathcal{C}$.\\
\textbf{Repeat:}
\begin{enumerate}
\item \textit{Compute locally}
\[
\mathsf{r}^{\mathsf{S}k}_j= \sum_{j\in \mathcal{C}} r^{k\top} \tilde Se_j r^k_j \text{ and } \mathsf{r}^{\mathsf{2}k}_j=(r_j^k)^2,
\]
between all $\mathcal{R}(j)$ for all $j \in \mathcal{C}$ using \eqref{eq:sparsityExploitS}.
\item \textit{Sum up globally} $\alpha^k=\left({\sum_{j\in \mathcal{C}} \mathsf{r}_j^{\mathsf{2}k}}\right)\Huge /\left({\sum_{j\in \mathcal{C}} \mathsf{r}_j^{\mathsf{S}k}}\right )$. \label{step:globSum1}
\item \textit{Compute locally for all $j \in \mathcal{C}$}
\begin{align*}
\lambda_j^{k+1} &= \lambda^k_j + \alpha^k p^k_j,\\
r^{k+1}_j &= r^k_j-\alpha^k \sum_{i \in \mathcal{C}} \tilde S_{ji} p^{k}_i,\\% and $
\mathsf{r}_j^{\mathsf{2}k+1}&=(r_j^{k+1})^2.
\end{align*}
\item \textit{Sum up globally} $\beta^k=\dfrac{1}{\mathsf{r}_j^{\mathsf{2}k}
\displaystyle\sum_{j\in \mathcal{C}} \mathsf{r}_j^{\mathsf{2}k+1}$
\label{step:globSum2}
\item \textit{Compute locally} $p_j^{k+1} = r^k_j + \beta^k p^k_j$ for all $j \in \mathcal{C}$.
\end{enumerate}
\label{alg:dCG}
\end{algorithm}
Next, we propose a sparsity exploiting variant of the conjugate gradient algorithm.
The usual centralized conjugate gradient method with $r^0=p^0=\tilde s-\tilde S\lambda^0$ reads \cite{Nocedal2006}
\begin{subequations} \label{eq:CGiter}
\begin{align}
\alpha^k &= \frac{r^{k\top}r^k}{r^{k\top}\tilde Sr^k}, \label{eq:CGalpha}\\
\lambda^{k+1} &= \lambda^k + \alpha^k p^k, \label{eq:CGlam
\end{align}
\begin{align}
r^{k+1} &= r^k-\alpha^k\tilde Sp^k, \label{eq:CGr}\\
\beta^k &= \frac{r^{k+1 \top} r^{k+1}}{r^{k^\top}r^k}, \label{eq:CGbeta} \\
p^{k+1} &= r^{k+1} + \beta^k p^k. \label{eq:CGp}
\end{align}
\end{subequations}
Recall that $\tilde S=\sum_{i \in \mathcal{R}}\tilde S_i$ and let $e_j$ be the $j$th unit vector. Then, from Lemma \ref{lem:sparsity} and \eqref{eq:sparsDef} we have
\begin{align} \label{eq:sumSi}
\tilde S e_j = \sum_{i\in \mathcal{R}}\tilde S_i e_j = \sum_{i\in \mathcal{R}(j)} \tilde S_i e_j,
\end{align}
i.e. the $j$th column of $\tilde S$ belonging to consensus constraint $j$ is the sum \emph{only} of the respective rows of $\tilde S_i$ of the subproblems $i\in \mathcal{R}(j)$ assigned to consensus constraint $j \in \mathcal{C}$.
Therefore the rows of $\tilde S$ can be constructed locally based on neighborhood communication between the assigned subproblems.
Furthermore, in \eqref{eq:CGalpha} we have to compute
\begin{align} \label{eq:rTopSi}
r^{k\top}\tilde S= \left (r^{k\top} \tilde Se_1,\;\dots\;,r^{k\top}\tilde S e_{n_c} \right ).
\end{align}
From \eqref{eq:sumSi} and Lemma \ref{lem:sparsity} we know that $\tilde S_{i}e_j=0$ for $i \in \mathcal{C}\setminus \cup_{i \in \mathcal{R}(j)} \mathcal{C}(i)$. Hence, the components of \eqref{eq:rTopSi} are
\begin{align} \label{eq:sparsityExploitS}
r^{k\top} \tilde Se_j = \sum_{i \in \mathcal{C}} r^{k}_i \tilde S_{ij} = \hspace{-0.1em} \sum_{i \in \cup_{l \in \mathcal{R}(j)} \mathcal{C}(l)} \hspace{-0.1em} r^{k}_i \tilde S_{ij}, \quad \forall j \in \mathcal{C},
\end{align}
where $\tilde S_{ij}$ denotes the $ij$th element of $\tilde S$.
Observe that it suffices to exchange $r^{k}_i$ and $\tilde S_{ji}$ locally between all $i\in \mathcal{R}(j)$. As
\begin{equation} \label{eq:summands_rSr}
r^{k\top}\tilde Sr^k = \sum_{j\in \mathcal{C}}\left ( r^{k\top} \tilde Se_j \right )r^k_j = \sum_{j\in \mathcal{C}} r^{k\top} \tilde Se_j r^k_j
\end{equation}
and $r^k_j$ is also known locally, all summands in \eqref{eq:summands_rSr} can be computed locally. The only centralized operation is evaluating one global sum. The same applies to
\begin{align*}
r^{k\top}r^k = \sum_{j\in \mathcal{C}} (r_j^k)^2,
\end{align*}
where $(r_j^k)^2$ can be computed locally. Similar analysis applies to \eqref{eq:CGlam}-\eqref{eq:CGp}, where in \eqref{eq:CGbeta} an additional global sum is needed and therefore the conjugate gradient needs two global sums in each iteration.\footnote{Note that although the sum is global, it can easily be decentralized by computing the sum via ``hopping" (i.e. a round-robin protocol) from neighbor to neighbor.} Algorithm \ref{alg:dCG} summarizes the proposed decentralized variant of the conjugate gradient method.
Note that the decentralized conjugate gradient algorithm requires communication between all subproblems assigned to a specific consensus constraint. In other words, this algorithm can be executed in decentralized fashion if the coupling described in the $A_i$s refer to two subproblems only, i.e. if Problem~\eqref{eq:sepProb} is \emph{$2$-assigned}. The same holds for ADMM as we will see in the next section.
\subsection{Decentralized ADMM}
The above proposed decentralized conjugate gradient method still requires (very little) central coordination using the global sums in Step~\ref{step:globSum1}) and Step~\ref{step:globSum2}) of Algorithm \ref{alg:dCG}. As an alternative, we consider a decentralized variant of ADMM for solving \eqref{eq:redRedSystemAlg} without these centralized steps.
We rely on decentralized ADMM in so-called consensus form to \eqref{eq:minLam} \cite{Boyd2011,Bertsekas1989}. To this end, we introduce variable copies of $\lambda$, $\lambda_1,\dots,\lambda_N$ and write \eqref{eq:minLam} as
\begin{equation} \label{eq:consProb}
\begin{aligned}
\min_{\lambda_1,\dots,\lambda_N,\bar \lambda} \quad&\sum_{i=1}^{N} f_i(\lambda_i) \\
\;\text{s.t.} \quad&\lambda_i=\bar \lambda\;\; | \;\; \gamma_i, \qquad i=1 ,\dots,N,
\end{aligned}
\end{equation}
with $f_i(\lambda_i) := \lambda_i^\top S_i \lambda_i - s_i^\top \lambda_i$.
The ADMM iteration rules can be derived from the method of multipliers combined with coordinate descent \cite{Bertsekas1989}.
Decentralized ADMM is summarized in Algorithm~\ref{alg:dADM}.
Observe that \eqref{eq:lamUp} is an entirely local step, \eqref{eq:lamBarUp} is a simple averaging step based on neighborhood communication, and \eqref{eq:gamUp} is again a local step.
Furthermore \eqref{eq:lamUp} requires solving a linear system with changing right-hand sides, which means that $(\tilde S_i + \rho I)$ has to be factorized once only and can be reused in all ADMM iterations.
\begin{algorithm}[b]
\small
\caption{Decentralized ADMM}
\textbf{Initialization:} Initial guess $\lambda^0$ and parameter $\rho=\rho^\text{ADM}$.\\
\textbf{Repeat:}
\begin{enumerate}
\item \textit{Compute locally} for all $i\in \mathcal{R}$
\begin{equation} \label{eq:lamUp}
\lambda_i^{k+1} \hspace{-1mm}= \hspace{-0.5mm} \underset{{\lambda_i}}{\operatorname{argmin}} \; \lambda_i^\top \hspace{-1mm} \left ( \tilde S_i +\rho I \right) \hspace{-0.5mm} \lambda_i
+ \left (\gamma^k_i - \tilde s_i - \rho \bar \lambda^k \right )^\top \hspace{-1.5mm} \lambda_i.
\end{equation}
\item \textit{Compute locally} for all consensus constraints $j\in \mathcal{C}$
\begin{align} \label{eq:lamBarUp}
e_j^\top \bar \lambda^{k+1} &=\frac{1}{|\mathcal{R}(j)|} \sum_{i \in \mathcal{R}(j)} e_j^\top \lambda_i^{k+1}.
\end{align}
\item \textit{Compute locally} for all $ i \in \mathcal{R}$
\begin{align} \label{eq:gamUp}
\gamma_i^{k+1} = \gamma_i^k + \rho\left (\lambda_i^{k+1}-\bar \lambda_i^{k+1}\right ).
\end{align} \vspace*{-3mm}
\end{enumerate}
\label{alg:dADM
\end{algorithm}
\subsection{Comparison of CG and ADMM}
The convergence properties of CG and ADMM are summarized in Table \ref{tab::convPropCGAL}. In theory, CG yields the \emph{exact} solution in at most $n_c$ steps \cite[Thm 5.1]{Nocedal2006}.
However, in practice the convergence is typically slower as conjugate gradient is sensitive to errors caused by finite precision arithmetic. Practically one observes superlinear convergence \cite{Beckermann2001}.
The recent paper \cite{Makhdoumi2017} shows sublinear convergence of ADMM for convex objectives $f_i$.\footnote{For strongly convex $f_i$, linear convergence of ADMM can be shown \cite{Yang2016,Nedic2018a,Makhdoumi2017}. In the present paper the $f_i$ of \eqref{eq:consProb} are only convex but not strictly convex.}
In case of \eqref{alg:ALADINbil}, the $f_i$s are only convex, hence at least sublinear convergence can be expected which is in line with our later numerical observations.
Thus conjugate gradient is expected to outperform ADMM.
An advantage of CG compared to ADMM is that no tuning of the step size is needed, as this is done ``automatically'' in Step~2) and Step~4) of CG.
As discussed in the previous section, satisfying \eqref{eq:pRes} preserves the convergence properties in bi-level distributed ALADIN.
Note that criterion \eqref{eq:pRes} can be evaluated locally by computing $e_j^\top r^k_\lambda$ for each $j \in \mathcal{C}$ and calculating one additional global sum.
However, in implementations it turns out that a fixed number of iterations for the coordination step combined with warm starting often suffices to ensure $0<\eta_k<0$.
\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.2}
\caption{Convergence properties of decentralized CG and decentralized ADMM for \eqref{eq:redRedSystemAlg}.}
\begin{tabular}{rlll}
\toprule
conv. rate & CG & ADMM \\
\midrule
theoretical& $n_c$-step & sublinear$^{\footnotesize 8}$ \\
practical & linear/superlinear\footnote{Analyzing the convergence rate of conjugate gradient methods seems quite complex as there are different phases with different convergence rates during the iteration cf. \cite{Axelsson2014,Beckermann2001}. However, the practically observed convergence rate often is superlinear \cite{Axelsson2014}.} & sublinear \\
tuning & no & yes \\
\bottomrule
\end{tabular}
\label{tab::convPropCGAL}
\end{table}
\begin{remark}[Related works on optimization over networks]
Related results to our above developments can be found in the context of distributed optimization over networks, see \cite{Nedic2018,Nedic2018a} for recent overviews. The problems considered therein are in general more difficult.
Frequently, communication delays, a time-varying network topology and asynchronous operation might be considered.
Prominent algorithms tailored to distributed optimization over networks are, for example, EXTRA \cite{Shi2015}, NEXT and also the widely used decentralized variant of ADMM \cite{Shi2014}. Linear systems of equations are considered in \cite{Lu2018b,Mou2015}, gradient and subgradient-based algorithms can be found in \cite{Nedic2010,Jakovetic2014}.
Indeed most of the algorithms cited above can in principle be used to solve \eqref{eq:minLam} in decentralized fashion. A potential pitfall might be that the convergence rate of these algorithms is at most linear, in many cases merely sublinear.
\end{remark}
\subsection{Communication analysis} \label{se:commAnal}
We turn to analyze the forward communication need in all ALADIN variants for 2-assigned problems.
Forward means that, for the sake of simplicity, we consider the communication in Step~2) of the different ALADIN variants where local sensitivities are communicated to the coordination QP. The backward communication in Step~3) is negligible compared to forward one. Our analysis evaluates communication by counting the number of floating point numbers.
Moreover, we distinguish two different kinds of communication: The first one is global communication, i.e. the information sent to any central (coordinating) entity.
The second kind is local communication between neighbors.
We assess the local preparation steps, which are done only once per outer ALADIN iteration in a preprocessing phase between neighboring subproblems.\footnote{Note that we analyze the communication under symmetric conditions; i.e. both regions assigned to a consensus constraint send and receive the values corresponding to the respective consensus constraint. In general, it would suffice to choose one of these two participating regions to take care of the computations. However, this would render the algorithm somehow asymmetric.}
\begin{table}[t]
\centering
\caption{Per-step forward communication (number of floats) for 2-assigned problems and different ALADIN variants. }
\begin{tabular}{rcccc}
\toprule
variant & standard & cond. & ADMM & CG
\\ \midrule
local prep.& - & - & -& $2 n_c^2$ $\phantom{\displaystyle \sum}$ \\
local iter. & - & - & $2n_c n^{\mathrm{AD}}$ &$ 2n_c n^{\mathrm{CG}}$ $\phantom{\displaystyle \sum^N}$\\
global& $>\hspace{-0.3em}\displaystyle \sum_{i=1}^N\frac{(n_{x_i}+n_{g_i})^2}{2}$ & ${{n_c^2 + n_c}}$ & - & $2N n^{\mathrm{CG}}$ $\phantom{\displaystyle \sum_{i=1}^N}$
\\ \bottomrule
\end{tabular}
\label{tab:forwComm}
\end{table}
The forward communication for solving the coordination problem~\eqref{eq:redRedSystemAlg} of bi-level distributed ALADIN once is shown in Table \ref{tab:forwComm}.
In its full variant, ALADIN communicates the first and second-order sensitivities of the objective and the first-order sensitivity of the constraints to the coordinator.
Let the constraints $h_i$ \eqref{eq:IneqConstr} consist of $n_{gi}$ equalities (handled as per Footnote \ref{footnote:constr}) and $n_{h_i} - n_{g_i}$ inequalities.
Neglecting sparsity and counting the number of all entries of the sensitivity matrices/vectors yields the following lower bound
$
\sum_{i=1}^N\frac{(n_{x_i}+n_{g_i})(n_{x_i}+n_{g_i}+1)}{2}.
$
Note that we do not count the communication of the $A_i$s here as they have to be communication only once and do not change during iterations
In case of active inequality constraints, $n_{g_i}$ is enlarged by the number of active inequality constraints which is bounded by $n_{h_i} - n_{g_i}$.
Hence, the above is a lower bound on the per-step communication which may vary during the ALADIN outer iterations.
For a detailed application-specific communication analysis for the standard ALADIN see \cite{Engelmann2019}.
In the condensed and sparsity exploiting variant of ALADIN---i.e. Algorithm \ref{alg:ALADINbil} without decentralization of \eqref{eq:redRedSystemAlg}---the global forward communication is
$
{n_c(n_c+1)}
$
where $n_c$ is the number of coupling constraints. The number of coupling constraints is typically much smaller than the total number of decision variables thus reducing the necessary communication effectively.
Note that the $2$ in the denominator disappears due to 2-assignment and therefore each row of $\tilde S$ is composed of the rows of exactly two $S_i$.
The bi-level distributed ALADIN ADMM variant (ALADIN ADMM) relies on purely local communication; i.e.
in each iteration, the respective $\lambda_i$'s between two neighboring regions are exchanged. Hence, in ALADIN ADMM one communicates
$
2n_c \cdot n^{\mathrm{AD}}
$
floats locally, where $n^{\mathrm{AD}}$ is the number of inner ADMM iterations.
Similarly, in the bi-level distributed ALADIN with conjugate gradient (ALADIN CG) one communicates
$
2n_c \cdot n^{\mathrm{CG}}
$
floats locally and additionally $2\cdot n_c^2$ in the local preparation phase (the rows of the Schur-complements $e_j^\top S_i$).
Finally, the global communication for computing $\alpha$ and $\beta$ is $2N\cdot n^{\mathrm{CG}}$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 381
|
{"url":"https:\/\/motls.blogspot.com\/2007\/06\/may-2007-was-035-c-cooler-than-january.html","text":"## Friday, June 08, 2007 ... \/\/\n\n### May 2007 was 0.35 C cooler than January 2007\n\nUAH data show that May 2007 was 0.35 Celsius degrees cooler than January 2007. This temperature difference equals to the hypothetical global warming trend predicted for two or three decades. Nature is nevertheless able to make such a change within four months. Have you noticed that the Earth was 0.35 C cooler than in January? I have not. Moreover, I can show you that you couldn't have either. May 2007 was also cooler than September 1980 and many other months when you were much younger.\n\nThe picture above doesn't describe the current state but the state of Earth in a few thousand years. We are approaching the end of an interglacial. Click to see more comments on ice ages.\n\nThe Northern Hemisphere reveals an even more impressive recent cooling trend. May 2007 was 0.47 Celsius degrees cooler than February 2007; it was the coolest month since the beginning of 2005. Also, May 2007 was 0.84 Celsius degrees cooler than April 1998. The same magnitude of warming as the cooling that occurred on the Northern Hemisphere within 3 months is predicted for the future period of 30-50 years. Analogously, the cooling since April 1998 equals the predicted artificial warming for most of the 21st century and some people, including many people who don't yet live in asylum, are ready to sacrifice trillions of dollars (from the pockets of other people, of course) to avoid this \"catastrophic\" hypothetical prediction. The world is just mad, isn't it?\n\nRSS data make the picture even more complete although they only talk about latitudes between -70 (South) and +82.5 (North). May 2007 was the coolest month after July 2004. The most drastic cooling between April 2007 and May 2007 appeared between latitudes +60 and +82.5 (the Northern polar areas), namely by 1.91 degrees Celsius in one month! You should note that one month is enough to cool the Arctic region by two Celsius degrees - the same temperature change that is routinely presented as a catastrophe even if it takes one century!\n\nYour humble correspondent doesn't trust the surface data too much. But according to GISS, May 2007 was 0.37 Celsius below January 2007.","date":"2019-11-19 12:40:37","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\": 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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.42741498351097107, \"perplexity\": 1585.8597748038555}, \"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\/1573496670151.97\/warc\/CC-MAIN-20191119121339-20191119145339-00123.warc.gz\"}"}
| null | null |
We stand in solidarity with the Black Lives Matter movement, with our Black colleagues and customers, and with the entire Black community.
The Black Lives Matter movement has sparked some crucial conversations within Octopus Energy about what we can do to affect meaningful change, both within our business and beyond. Some of these conversations should've been had a long time ago, and this movement has provided a crucial opportunity to self reflect – something we think is more important and more beneficial than just a simple post of support for the movement on social media.
We strongly encourage other companies to do the same work internally.
We haven't donated to any specific BLM organisations – rather, we've started an internal fund to combat inequality and racism in energy.
Here's some words from our founder Greg on the steps we're taking towards meaningful change at Octopus and beyond.
The brutal killing of George Floyd has triggered hurt across the world. The footage was sickening. This act has become a potent symbol of the racism that continues to take a painful toll on the Black community. Although I cannot begin to understand the hurt and anguish many are feeling right now I want to be clear: we stand with the Black community and we stand with our Black colleagues.
This oppression must end.
Our team have told me that they think Octopus should say something (and I could not agree more), but I did not want it to be a token. So many voices speak up at these times, but fade without lasting effect.
We've listened really hard - to the comments in our internal #BlackLivesMatter Slack channel but also to many other people who have approached me and many others directly with thoughts. And let's be clear - it has been hard maintaining my own silence publicly so I could hear what people think, without crowding out views with my own. But I am white and I needed to hear your voices.
As a company, our DNA is that we do things that will actually enable and drive meaningful change. We invest in what we believe in, and where the topic is so critically important, we need to be more confident than ever in sticking to that DNA even when our lack of statement starts to be painful.
So thank you for sharing your views, experience, ideas and priorities. We have listened, and this is what we are going to do.
1. Today, we announce an internal Octopus Energy Black Lives Matter fund – with over £100,000 initial funding
Rather than donating to any specific Black Lives Matter organisations externally, we'll be starting an internal fund to support our team and oppose racism, which the £100,000 initial funding will go towards. They will bring their own talent and training to bear in a truly meaningful way, and will also have access to Octopus skills and resources to drive change and share learnings back with Octopus.
Team members can nominate themselves or others to be the secondee(s) or to help oversee the fund.
As well as the initial funding, any employee can donate or raise money and the company will match every pound raised or donated
To kickstart the fund, I've donated £54,486 (this is all of the interest I received from loans I made to the company in the early days – I'd intended to use this for good, and I can't think of anything more important). The company is matching that so the fund starts at £108,972
If all employers did the same - on a pro-rata basis - the UK would generate a fund of around £3bn. We want to lead corporations in not paying lip service to such an endemic issue, but enabling real change.
2. We will work hard to recruit, at all levels, people from BAME (Black, Asian and Minority Ethnic) backgrounds and to ensure we're the workplace we should be for everyone.
We'll do this Octopus style - no tokenism or targets but by working relentlessly to do what is right and what is solid.
At a recent "new joiners" training session, two people of colour in the group remarked that this was the first time they'd got a job where they were sure it was because of how great they were – not to tick a box. They were right, and it must always stay that way. So we need to make sure our recruitment reaches into all communities and works for all great candidates.
Our senior operations managers have been leading on ways to deliver this, including unconscious bias training, moving away from CVs and a whole load of other measures. We'll work fast to expand what works across functions, locations and levels quickly – and to take it beyond recruitment to ensure that as a company we have greater understanding of language and unconscious bias.
We recently started to sponsor Generating Genius to help BAME talent into STEM roles, and had recommendations from the team for other organisations we should look at working with. Anyone reading this – if you've had personal experience of any others (or run one yourself!) please contact us.
For perspective, 13% of our middle and senior leaders identify as BAME versus 14% of the population. But our diversity is not evenly distributed across the business and locations and we'll ensure that we take steps to close gaps.
We pride ourselves on our progressive nature – on building a company that is better for the world, that champions social justice and creates better careers and internal community. But I hugely welcome the voices which tell me where we've got things wrong. Thank you to those who've shone a light on these things.
I was embarrassed that we inadvertently planned a big party to clash with Ramadan, for example. We can clearly do more to help team members deal with things like customers who use racist language and we'll fix this. And too many people in the company didn't know that one of the Senior Leadership Team is a woman who identifies as black.
It's no coincidence that we chose office locations in Soho, Brighton and Leicester – each of them a watchword for diversity – because we've wanted to build a rainbow business (and I'm delighted that Warwick also taps into diverse pools). Our determination that all permanent employees have equity means that we're already able to share the benefits of ownership with our team. By being based in diverse areas, and investing in growing in those communities, we are putting back where others extract.
Thank you for being an incredible team – those who shared difficult and personal views, who debated when we don't all agree with each other, who've shown kindness and understanding to those who seek to learn more, and for increasingly offering support and recognition.
There's still so much more to say, and do, but I hope that what we are announcing today will make meaningful change, and is the basis for really positive discussions and actions going forward.
Further reading to educate yourself and your loved ones
This small list is just a starting point of recommendations from our team, born out of this week's internal discussions.
So You Want to Talk About Race – Ijeoma Oluo
How To Be An Antiracist – Ibra X. Kendi
Conversations in Black – Ed Gordon
Brown Girl Dreaming – Jacqueline Woodman
Natives – Akala
Taking Up Space – Chelsea Kwakye & Ore Ogunbiyi
Why I'm No Longer Speaking to White People about Race – Reni Eddo-Lodge
Don't Touch My Hair – Emma Dabiri
Me and White Supremacy – Layla F Saad
I Am Not Your Baby Mother – Candice Brathwaite
Black Feminist Thought – Patricia Hill Collins
Ain't I A Woman – bell hooks
The New Black Vanguard – Antwaun Sargent
Decolonising the Camera – Mark Sealy
Reflections in Black: A History of Black Photographers 1840 to Present – Deborah Willis
Robin DiAngelo - White Fragility
Back to Black – Kehinde Andrews
Women, Race, Class – Angela Davis
The Fire Next Time – James Baldwin
You can also check out Ibram X. Kendi's anti-racism reading list from the New York Times.
There are many resources available for parents looking to educate their kids on diversity, inclusion and equality – here's a few lists from PopSugar and the New York Times.
About Race with Reni Eddo-Lodge: A podcast that features key voices from the last few decades of anti-racist activism.
Code Switch: A weekly podcast by journalists of colour who discuss race and all the ways it interacts with society and culture
Witness Black History: The BBC's podcast interviews key figures from black and civil rights history.
Time: The Kalief Browder story
Other useful starters
From MTV's Decoded, If Microaggressions Happened to White People
Business Insider's piece on What is a Microaggression.
Kimberlé Crenshaw's brilliant TedTalk, The Urgency of Intersectionality
A reflection on the history of racism and black hair.
Published on 5th June 2020 by:
Greg Jackson
Back to Blog More on Our community
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,767
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Federation of Horsepower
Spotlight: Federation of Horsepower 10th Anniversary Show
This Friday, Federation of Horsepower will be celebrating its 10th anniversary of rocking the hell out of Kansas City and all over the country. The Friday show will feature an all-star lineup of every official member of the band, and is the farewell show for drummer Kriss Ward before his move to Austin. In our feature, we talk to founding member and frontman Gregg Todt all about FOHP. Where they started, what we can expect for Friday's show, and what lies ahead. You can check it out right here, and be sure to hit The Brick Friday for what's sure to be one of the biggest rock shows of the year.
For 10 years, Federation of Horsepower has made a name for itself for being one of loudest, toughest bands in Kansas City. This Friday at The Brick, this rock n' roll machine will celebrating 10 years together and every current and former member will take to the stage. We sit down with lead vocalist and founding member Gregg Todt to tell us a little bit about the band.
The Deli: Give me a little history about the band.
Gregg Todt: Mason Fann (original guitarist) and I had been batting about the idea of starting a band in early 2002. Sometime in late February of that year, our friends from Seattle, The Gloryholes were heading out on tour. In a nutshell, it all went down like this:
We're coming to KC in 3 weeks, we need an opener, start a band.
FOHP made its live debut on St. Patrick's Day 2002 at El Torreon.
It's all been downhill since!
The Deli: Who had the initial idea for the band?
Gregg: Me and Mason started it. I blame the booze.
The Deli: You guys have always played straight-up rock n' roll, even when it hasn't been the "cool" or "indie" thing to do. Why do you continue to rock harder than most other bands in KC and beyond?
Gregg: Trends come and go. Styles change. But AC/DC and Motorhead are still around and kicking large ass... my guess is they're doing something right.
Here's the deal... People are always going to hate their job. They're always going to hate their life. They are always going to need that visceral release that only a guitar plugged into a loud ass amplifier can provide.
We provide a service.
This is what we were built to do.
The Deli: You guys have been on a ton of huge bills with groups like Judas Priest, Motorhead, etc. What shows have been your favorites over the years?
Gregg: Honestly, most of those shows are a complete blur. Talking to Lemmy for a few minutes was cool. Duff McKagan telling us how much he liked us was pretty damn awesome.
But it's the little shows that make it cool... Chipping a tooth on the mic at The News Room. Playing a coming home party for a soldier in San Antonio. Playing in Austin and seeing some of your musical heroes watching you. Every show, good or bad, holds a special place in my memory banks.
The Deli: Even though you've had a few lineup changes, you've been a part of the KC scene for 10 years. That's a huge accomplishment. Why do you think you have the staying power that others don't?
Gregg: It's simple: You're never too old to play music that went out of style 30 years ago. That, and there's no pretense to what we do (at least not that I know of)... We're loud, we sweat, we fall down. We're honest.
I write about what I know. Sometimes those subjects are painful. But it's real.
That and I am obviously too stupid to quit.
The Deli: Are there any other local bands that you think can deliver a similar high-octane dose of rock n' roll that you guys do?
Gregg: I love love love Cherokee Rock Rifle and Radkey. But there's a ton of great music that isn't just like us that I love too. The Latenight Callers, The Quivers, Deco Auto, The Cave Girls... Kansas City is brimming with amazing music these days and I find that very exciting.
The Deli: Tell me about the Friday show. What can we expect? Will there be fire?
Gregg: No fire, but lots of fun. Pretty much everyone who was an official member of the band will be onstage that night. It's going to be a long ass night for me because I'm the only constant member.
But hey, if it kills me, at least it'll be doing something I love.
The Deli: This will also be drummer Kriss Ward's last show before he moves to Austin. What's next for the band?
Gregg: I had considered kicking it in the head when Kriss told us he was moving. But then we found out that Chris Fugitt was moving back and it just seemed right. Chris had replaced Kriss, and then Kriss replaced Chris, so now Chris is replacing Kriss again. Makes it easier on my old brain.
We're working on recording a new CD. It's going to feature the current line up as well as some tracks we recorded with Troy and Kriss. It's tentatively going to be titled Hermanos de Sangre (Blood Brothers)... because once you're in FOHP, you're it it for life. This is my family.
It's amazing to me that we've lasted this long, but it makes me happy that I'm still friends with all of the former members of the band. Like I said, this is my family. It makes me incredibly sad that Kriss is moving away, but he is my brother and he has to do what's best for him and his family. In the end, this is just a band, it's temporary. Family and brotherhood are forever.
The Deli: Do you see FOHP being a band for another 10 years?
Gregg: Well, considering I am going to be 50 next year... if I am still doing this at 60, we are going to have one hell of a party.
Federation of Horsepower is:
Gregg Todt: lead vocals/guitar
Johnny Catfish: bass guitar
Christian Liljequist: guitar
Kriss Ward: drums (last show is Friday, September 14)
Chris Fugitt: drums (first show back is Saturday, September 22)
Past members:
Mason Fann: guitar
Paul Clark: bass
Kristin Thompson: bass
Troy Van Horn: guitar
And guest appearances from:
Chip Sage: drums
Brock Ginther: bass
Mike Rooney: guitar
Mike Myers: drums
Make sure you don't miss a moment of the 10th anniversary Federation of Horsepower show at The Brick this Friday. The lineup will feature The Buddy Lush Phenomenon, followed by an evening of songs by every former and current member of the group. You should not and cannot miss this show with one of Kansas City's most electrifying rock bands. It will leave you banging your head and moving your body and wanting more afterwards.
Michelle is editor-in-chief of The Deli - Kansas City. She also has a weekly column with The Kansas City Star and reviews music for Ink. She plays with Deco Auto, Drew Black and Dirty Electric, and Dolls on Fire. Unfortunately, she didn't begin eating Fruity Pebbles until too late in life.
On The Beat with Kriss Ward
This week's On The Beat features one of Kansas City's most longstanding and voracious drummers, Kriss Ward of Federation of Horsepower. Kriss will be playing his last show with Federation of Horsepower this Friday before he relocates to Austin. We sit down with him and find out where he's been and what's coming next. Catch the beat right here!
On The Beat is typically brought to you by Sergio Moreno, but has been overtaken this week by drummer and The Deli - Kansas City editor-in-chief Michelle Bacon. This weekly interview features some of the many talented drummers in the area.
Kriss Ward
Kriss Ward has been one of the toughest, most animated drummers in Kansas City for several years. This Friday, September 14, he'll be playing his last show with Federation of Horsepower before he moves to Austin. We talk to Kriss about his rock n roll career in town and see what's ahead.
The Deli: How did the drums find you?
Kriss Ward: My first instrument was bass guitar, then I dedicated a lot of time to vocals before finally doing what I always wanted to do. I started playing almost 20 years ago and went right to work.
The Deli: Besides Federation of Horsepower, what other bands have you drummed in?
Kriss: I've played with The Last of the V8s, The Lustertones, Ramalamas, Phaze II, Savage 7, Bloodfeast (Misfits tribute), Funhouse (Stooges tribute) and various cover bands.
The Deli: You're one of the most powerful, emotive drummers I've ever seen around here. Where does all that energy come from, and what's your approach?
Kriss: I'm a show drummer. I want people to get off on the beat! I just play every show like it's my last, and at my age I never know! Haha! My approach to drumming is to let fucking loose!
The Deli: What type of kits do you use?
Kriss: I play Pearl drums, always have since I started. They're durable and tough-sounding drums.
The Deli: Obligatory question: favorite drummers?
Kriss: Eric Singer, Tommy Aldridge, Cozy Powell, Bill Ward: old-school drummers that swing hard.
The Deli: Do you have any big surprises in store for the big Federation show on Friday?
Kriss: FOHP has been really busy this summer, probably busier than ever. Even though the band has released only one studio release in 10 years, we have been asked to play all over the Midwest and Texas so we don't have a lot of time to bust out many surprises. But if you've followed the band over the last 10 years, you will not be disappointed this Friday.
The Deli: What will you miss the most about playing with the guys in Federation?
Kriss: We are one big family and that's what I'll miss most.
The Deli: Do you plan to join other projects once you're settled in your new home in Austin?
Kriss: I always find myself in a project no matter where I go, it's just natural. There's no doubt that I will play in Austin, but nothing lined up right now.
The Deli: Say Gregg [Todt] calls you up in 10 years for a 20-year Federation reunion. You up for it?
Kriss: Gregg is my big bro. We have the same rock heroes and take the same approach to the stage. The connection between the front man and the drummer is what the "rock show" is. We will play together again, I know we will.
The Deli: Is there anything you'll miss about KC music?
Kriss: I will miss Davey's Uptown. The KC scene has always been modern and artsy; Federation of Horsepower is neither one of those.
Kriss will be pounding out the beat with Federation of Horsepower at The Brick this Friday. The lineup will feature The Buddy Lush Phenomenon, followed by an evening of songs by every former and current member of Federation. This show and Kriss's brilliant drumming is definitely not to be missed.
Michelle is editor-in-chief of The Deli - Kansas City. She also has a weekly column with The Kansas City Star and reviews music for Ink. She plays with Deco Auto, Drew Black and Dirty Electric, and Dolls on Fire. Her favorite food is Laksa. Ever had it? Probably not.
Photo by Todd Zimmer
Federation of Horsepower - Stay Down
|
{
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| 1,660
|
Q: Why does Kotlin change a list of type List> to List if a list is added with the "+" operator? Given the following two lists
val listA = listOf<List<Int>>()
val listB = listOf(1, 2, 3)
and the following operation
val listC = listA + listB
I am adding a list with the type List<Int> to a list with the type List<List<Int>>. However, my IDE is telling me that the type of the resulting list C is List<Any> and not, as I expected, List<List<Int>>. Could someone explain to me why?
A: Have a look at the list of all plus operators in kotlin.collections, and you'll see that the one you are calling resolves to this one:
operator fun <T> Collection<T>.plus(
elements: Iterable<T>
): List<T>
The compiler tries very hard to infer a type for T, so that the call is valid, and it finds Any as such a type. After all, a List<List<Int>> is a Collection<Any> and a List<Int> is also a Iterable<Any>.
Note that there is also:
operator fun <T> Collection<T>.plus(element: T): List<T>
However, overload resolution doesn't pick this one because the one that takes a Iterable<T> is considered "more specific". After all, it takes specifically an Iterable, rather than just any type (T), including Iterable. See also the rationale section of the spec.
You should probably use plusElement instead.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,273
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'use strict';
import { Injectable } from '@angular/core';
import {
API,
Application,
Business,
Configuration,
IConfigService,
IEnvironment
} from './interfaces/config.service.interface';
import { WindowRef } from './window.service';
import { BehaviorSubject } from 'rxjs/BehaviorSubject';
let config: Configuration = require('configuration'),
pkg: any = require('package'),
configService: ConfigService;
@Injectable()
export class ConfigService implements IConfigService {
public static ENVIRONMENT: string = 'ENVIRONMENT';
private configuration: Configuration = null;
public environment: BehaviorSubject<IEnvironment> = new BehaviorSubject<IEnvironment>(null);
private pkg: any = null;
constructor (
private window: WindowRef
) {
if (!configService) {
configService = this;
this
.init();
} else {
return configService;
}
return this;
}
private init (): ConfigService {
let window = this.window.nativeWindow;
if (window && window.hasOwnProperty(ConfigService.ENVIRONMENT)) {
this.environment.next(
JSON.parse(
window[
ConfigService.ENVIRONMENT
]
.replace(/"/g, '"')
)
);
}
this.configuration = config;
this.pkg = pkg;
return this;
}
public getEnvironment (): IEnvironment {
return this.environment.value;
}
public updateEnvironment (env: IEnvironment): void {
this
.environment
.next(env);
}
public getApplication (): Application {
return this
.configuration
.application;
}
public get (): Configuration {
return this.configuration;
}
public getPkg (): any {
return this.pkg;
}
public getBusiness (): Business {
return this
.configuration
.business;
}
public getAPIs (): API {
return this
.configuration
.api;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,070
|
{"url":"https:\/\/en.wikiquote.org\/wiki\/User:Henry_Delforn~enwikiquote","text":"# User:Henry Delforn~enwikiquote\n\n## On What must be shall be\n\nTchaikovsky Romeo and Juliet Fantasy Overture [10]\n\n\"Oh Venus, goddess of love in Woman's heart, make so that we can enjoy together that which is most sublime and divine in life\".\nHenry Delforn 12:21, 17 August (UTC)\n\n## On life and death\n\n${\\displaystyle G_{\\mu \\nu }+\\Lambda g_{\\mu \\nu }={8\\pi G \\over c^{4}}T_{\\mu \\nu }\\,}$ Aether solutions: ${\\displaystyle R_{\\mu \\nu }={1 \\over 2}R\\,g_{\\mu \\nu }\\,}$\n\n## On Time\n\n\"Time is the hand of God passing by at the speed of light giving us space and motion\" Henry Delforn 16:24, 11 August (UTC)\n\nThus...\nwe're talking about a material\/wave aether-like field in very fast (c) motion relative to matter and constantly bombarding our brain cells since before birth, which would give rise to our common concept of Time. It's obvious that Time is what makes motion possible...but not only motion, matter & space too. i'll give you an example how screwed up and under developed our common concept of Time is, take spacetime. Here's an idea that links 3d with common Time, that is, the two concepts are unseperable in reality. But one can still think (as we often do) or conceptualize about 3d without time (such as in a photograph). But God forbid we should think about Time without 3d space. Can you even picture this in your mind? Can you really picture Time without space? Henry Delforn\n\ni feel we're talking about a material\/wave aether-like field in very fast (c) motion relative to matter and constantly bombarding our brain cells since before birth, which would give rise to our common concept of Time. Time makes motion possible...but not only motion, matter & space too.\n\nBut the concept is all screwed up. i'll give you an example how screwed up the concept of Time is...take spacetime. Here's an idea that links 3d with Time, the two concepts are unseperable in reality. But one can still think (and often do) or conceptualize about 3d without Time (such as in a photograph). But we can't think about Time without 3d space. Can you even picture this in your mind? Can you picture Time without space? The asymmetry of our concept of Time is all screwed up. Henry Delforn\n\n## On Espa\u00f1a\n\n\"Tu vino es rojo y dulce...como tus labios con que sue\u00f1o\" Henry Delforn 00:22, 8 June (UTC)\n\n## Beauties by Jerome Thompson.jpg On Beauty\n\n\"God has already made thee beautiful, heroine, elegant, and powerful all within\", Henry Delforn 06:04, 17 July (UTC)\n\n## On Free Will\n\nThe Standard Argument Against free will: First, if determinism is the case, then the will is not free. Second, if indeterminism is the case, then randomness exist and the will is not in control. [22]\n\nThe Argument Against The Standard Argument:\n\"The duality of nature (per quantum mechanics) is both deterministic or random depending on the scale of the space and time in question relative to agent's will.\" Henry Delforn 22:43, 27 August (UTC)\n\n## On Aether\n\n\"With the new theory of electrodynamics we are rather forced to have an aether.\", P.A.M. Dirac\n\n## On Cuba\n\nCuba Dreams Awake\n\nDream, that any day now, my new world will arrive, that no borders exist, dream of love without barriers.\n\nPlant, along my path, a new destiny, where the sun will shine, where my soul unites, where kindness and love is reborn in me, and the day i find that dream, i will be changed, and there will be no one, who can destroy the truth in my soul.\n\nLive, with the emotion of once again feeling peace in life in Cuba.\n\n## On Celos\n\nPrimero vino amor despu\u00e9s celos. Sin amor no hay celos. Pero que se dice de este precio de amor? Digo, pagalo si tienes. Si no tienes, pidelo prestado. Si no te prestan, r\u00f3balo. Y si fallas en robar, entonces no vale la pena porque ese amor no era tuyo. Henry Delforn 10:28, 5 October 2010 (UTC)\n\n## On Cruelty\n\nSilence Says More Than A Thousand Words and One Cruel Word Says Too Much. Henry Delforn 20:39, 8 October 2010 (UTC)","date":"2019-06-19 05:02:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"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\": 0.390830934047699, \"perplexity\": 6210.640020533875}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2019-26\/segments\/1560627998913.66\/warc\/CC-MAIN-20190619043625-20190619065625-00492.warc.gz\"}"}
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Q: Using Theme.Sherlock.Light, my layout preview is white, but it's gray on physical devices I'm using ActionBarSherlock, but I'm not using any kind of of other theming. My application looks fine in the layout editor, because I'm using a certain color scheme that goes well with the white background shown. Although, when I run my application on a device 2.x, 3.x or 4.x, I get a very light gray color as the background, but it's definitely not white. Am I missing something? I thought the Light theme was a light gray action bar with a white background.
A: The Sherlock light theme is a copy of Holo.Light, which uses a very light grey as the default background colour.
You can override it to white it you like. In your application theme (create one if you need to which extends Sherlock Light), set the following attribute:
<item name="android:windowBackground">@android:color/white</item>
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"redpajama_set_name": "RedPajamaStackExchange"
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Conquests of Camelot: The Search for the Grail
Sierra On-Line 1989
Christy Marx
Amiga, Atari ST, PC
Medieval, Myths and legends
You take on the role of king Arthur. Several of your most prominent knights have gone missing in their search for the Holy Grail, so you decide to go on this quest yourself.
Like most Sierra games from this era, you control your character onscreen using normal text input.
The game is notable for its extensive research into Arthurian legends.
The second (and last) game in the Conquest series, the point'n'click controlled Conquests of the Longbow: The Legend of Robin Hood, was released in 1992.
Solution (Amiga)
by ?
Search at Atari Legend
Search at Lemon Amiga
Users who like this also enjoyed:
Mordon's Quest
Himalayan Odyssey
Gateway to Karos
Average User Rating: 7 (1 rating)
Gunness (20-11-2014 09:31)
One of my favourite Sierra games - it really nails the atmosphere of the legends.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,048
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\section{Introduction.}\label{intro}
In this paper, we construct an invariant $\widetilde z_n$ of rational homology 3-spheres via vector fields.
As an application, we prove that the Kontsevich-Kuperberg-Thurston invariant $z^{\rm KKT}=\{z_n^{\rm KKT}\}_{n\in\mathbb N}$ coincides with Watanabe's Morse homotopy invariants $z^{\rm FW}=\{z^{\rm FW}_{2n,3n}\}_{n\in \mathbb N}$ for any rational homology 3-sphere.
Note that both $z^{\rm KKT}_n$ and $z_n^{\rm FW}$ are topological invariants which take values in the real vector space $\mathcal A_n(\emptyset)$ of Jacobi diagrams.
M.~Kontsevich~\cite{Kon}, S.~Axelrod and I. M.~Singer~\cite{AS} proposed the Chern-Simons perturbation theory and gave a topological invariant of 3-manifolds.
Based on Kontsevich' s work, G.~Kuperberg and D.~Thurston constructed in \cite{KKT} a topological invariant $z^{\rm KKT}$ of rational homology 3-spheres.
Kuperberg and Thurston proved that $z^{\rm KKT}$ is a universal finite type invariant for homology 3-spheres by showing surgery formulas.
C.~Lescop obtained surgery formulas of other types in \cite{Splitting} and \cite{Surgery}.
Lescop reviewed $z^{\rm KKT}$ and gave a more direct proof of well-definedness of this invariant in \cite{Lescop}.
K.~Fukaya~\cite{Fukaya} constructed a topological invariant of 3-manifolds with local coefficients using Morse functions.
Fukaya's invariant is closely related to the theta graph $\theta$. His invariant essentially takes values in $\mathcal A_1(\emptyset)$.
M. Futaki~\cite{Futaki} pointed out that Fukaya's invariant depends on the choice of Morse functions.
T.~Watanabe~\cite{Watanabe} gave an invariant of rational homology 3-spheres without local coefficients using Morse functions.
He also investigated higher loop graphs (and broken graphs) and then he defined a topological invariant $z_{2n,3n}^{\rm FW}$ of (rational) homology 3-spheres taking values in $\mathcal A_n(\emptyset)$ for each $n\in \mathbb N$.
The construction of $z_{2,3}^{\rm FW}$ is related to the construction of a Morse propagator constructed by Lescop~\cite{LescopHeegaard}.
Fukaya's construction is inspired by the construction of the 2-loop term of the Chern-Simons perturbation theory
and he conjectured in \S 8 in \cite{Fukaya} that his invariant is related to the 2-loop term of the Chern-Simons perturbation theory.
Watanabe also conjectured in Conjecture 1.2 in \cite{Watanabe} that his invariants is related to Axelrod and Singer's invariant~\cite{AS} or Kontsevich's invariant~\cite{Kon}.
The main theorem of this paper is the following.
\begin{theorem}\label{theorem1}
$z_n^{\rm KKT}(Y)=z_{2n,3n}^{\rm FW}(Y)$ for any rational homology 3-sphere $Y$, for any $n\in \mathbb N$.
\end{theorem}
The idea of the proof of Theorem~\ref{theorem1} is the following.
We construct an invariant $\widetilde z_n$ of rational homology 3-spheres using vector fields.
Let $Y$ be a rational homology 3-sphere and let $\infty\in Y$ be a base point.
$z_n^{\rm KKT}(Y)$, $z^{\rm FW}_{2n,3n}(Y)$ and $\widetilde z_n$ are defined by using an extra information of $Y$.
The extra information used in definition of $z_n^{\rm KKT}$, $z_{2n,3n}^{\rm FW}$ and $\widetilde z_n$ are a framing of $Y\setminus\infty$,
a family of Morse functions on $Y\setminus \infty$ and a family of vector fields on $Y\setminus \infty$, respectively.
We prove that it is possible to regard the constructions of $z^{\rm FW}_{2n,3n}$ and $z_n^{\rm KKT}$ as special cases of the construction of $\widetilde z_n$.
In fact a framing gives us a non-vanishing vector field and a Morse function gives us a gradient vector field.
The principal term of $\widetilde z_1$ is related to Lescop's invariant \cite{comb} for rational homology 3-spheres with non-vanishing vector fields.
The organization of this paper is as follows.
In Section 2 we prepare some notations.
In Section 3 we review notions and facts about configuration spaces and graphs discussed by Lescop \cite{Lescop} and Watanabe \cite{Watanabe}.
In Section 4 we define the invariants $\widetilde z_n$ using vector fields and prove the independence of the choice of vector fields.
In Section 5 we review the construction in Lescop \cite{Lescop} of $z^{\rm KKT}$.
In Section 6 we review the construction of $z^{\rm FW}$ in Watanabe \cite{Watanabe} with a little modification.
In Section 7 we prove Theorem~\ref{theorem1}.
In Section 8 we prove some Lemmas for a compactification of the moduli space of flow graphs used in Sections 6 and 7.
In Appendix A we give a more direct proof of Theorem~\ref{theorem1} in the case of $n=1$.
\subsection*{Acknowledgments.}
The author would like to thank Professor Mikio Furuta for his encouragement and
for helpful comments and suggestions in particular about Morse functions
on punctured manifolds.
The author would also like to thank Professor Tadayuki Watanabe for his
helpful comments and suggestions for an earlier draft and his patient explanation of the
detail of the construction of his invariant.
The author also expresses his appreciation to Professor Christine Lescop for her
kind and helpful comments and suggestions to improve an earlier
draft. The last part of the proof of
Lemma~\ref{keylemma} is due to her ideas.
\section{Notation and some remarks.}
In this article, all manifolds are smooth and oriented.
Homology and cohomology are with rational coefficients.
Let $c$ be a $\QQ$-linear sum of finitely many maps from compact $k$-dimensional manifolds with corners to a topological space $X$.
We consider $c$ as a $k$-chain of $X$ via appropriate (not unique) triangulations of each $k$-manifold.
Let $Y$ be a submanifold of a manifold $X$.
Let $c=\sum_ia_i(f_i:\Sigma_i\to X)$ be a chain of $X$, where $f_i:\Sigma_i\to X$ are smooth maps from compact manifolds with corners and $a_i$ are rational numbers.
If $f_i$ is transverse to $Y$ for each $i$, then we say that $c$ is transverse to $f$.
When $B$ is a submanifold of a manifold $A$, We denote by $A(B)$ the manifold given by real blowing up of $A$ along $B$.
Namely $A(B)=(A\setminus B)\cup S\nu_{B}$ where $\nu_B$ is the normal bundle of $B\subset A$ and $S\nu_B$ is the sphere bundle of $\nu_B$
(see \cite{KKT} for more details of real blow up).
Note that if a submanifold $C\subset A$ is transverse to $B$, then $C(A\cap B)$ is a proper embedded submanifold of $A(B)$.
Let us denote by $\Delta\subset A\times \cdots \times A$ the fat diagonal of the direct sum of a manifold $A$.
Let us denote by $\underline{\RR^k}$ the trivial vector bundle over an appropriate base space with rank $k\in \mathbb N$.
For a real vector space $X$, we denote by $SX$ or $S(X)$ the unit sphere of $X$
and for a real vector bundle $E\to B$ over a manifold $B$, we denote by $SE$ or $S(E)$ the unit sphere bundle of $E$.
\subsection{Notations about 3-manifolds and Morse functions.}
Let $f:Y\to \RR$ be a Morse function on a 3-dimensional manifold $Y$ with a metric satisfying the Morse-Smale condition.
Let $\grad f$ be the gradient vector field of $f$ and the metric of $Y$.
Let us denote by $\Crit (f)$ the set of all critical points of $f$.
Let $\{\Phi_f^t\}_{t\in\RR}:Y\to Y$ be the 1-parameter group of diffeomorphisms associated to $-\grad~f$.
We denote by
$$\mathcal A_p=\{x\in Y\mid \lim_{t\to\infty}\Phi_f^t(x)=p\}~{\rm and}$$
$$\mathcal D_p=\{x\in Y\mid \lim_{t\to-\infty}\Phi_f^t(x)=p\}$$
the ascending manifold and descending manifold at $p\in \Crit(f)$ respectively.
\subsection{Conventions on orientations.}
Boundaries are oriented by the outward normal first convention.
Products are oriented by the order of the factors.
Let $y\in B$ be a regular point of a smooth map $f:A\to B$ between smooth manifolds $A$ and $B$.
Let us orient $f^{-1}(y)$ by the following rules:
$T_xf^{-1}(y)\oplus f^*T_{f(x)}B=T_xA$, for any $x\in f^{-1}(C)$
where $f^*:T_{f(x)}B\to T_xA$ is a linear map satisfying $f_*\circ f^*={\rm id}_{T_{f(x)}B}$.
We denote by $-X$ the orientation reversed manifold of oriented manifold $X$.
Suppose that $Y,f$ and $\grad f$ are as above.
Let us orient ascending manifolds and descending manifolds by imposing the condition:
$T_p\mathcal A_p\oplus T_p\mathcal D_p\cong T_pY$ for any $p\in \Crit(f)$.
Let $p,q\in \Crit(f)$ be the critical points of index $2$ and $1$ respectively.
By the Morse-Smale condition, $\mathcal D_q\cap \mathcal A_p$ is a 1-manifold.
Let us orient $\mathcal D_q\cap \mathcal A_p$ by the following rule:
$$T_{q'}(\mathcal D_q\cap \mathcal A_p)\oplus T_{q'}\mathcal D_q\cong T_{q'}\mathcal D_p,$$
where $q'\in \mathcal D_q\cap \mathcal A_p$ is a point near $q$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=140pt]{notationpic1.eps}
\caption{The orientation of $\mathcal D_q\cap \mathcal A_p$.}
\end{center}
\end{figure}
\section{Configuration space and Jacobi diagrams.}
In this section, we introduce some notations about configuration spaces and Jacobi diagrams.
Most of this section depends on Lescop \cite{Lescop}.
\subsection{The configuration space $C_{2n}(Y)$.}\label{c_2(Y)}
The reference here is Lescop \cite[\S 1.1,1.2,2.1]{Lescop}.
Let $Y$ be a homology 3-sphere with a base point $\infty$.
Let $N(\infty;Y)$ be a regular neighborhood (that is diffeomorphic to an open ball) of $\infty$ in $Y$
and let $N(\infty;S^3)$ be a regular neighborhood of $\infty$ in $S^3=\RR^3\cup \infty$.
We fix a diffeomorphism $\tau^{\infty}:(N(\infty;Y),\infty)\cong (N(\infty;S^3),\infty)$
between $N(\infty;Y)$ and $N(\infty;S^3)$.
We identify $N(\infty;Y)$ with $N(\infty;S^3)$ under $\tau^{\infty}$.
Let $\breve C_{2n}(Y)=(Y\setminus \infty)^{2n}\setminus \Delta=\{\{1,\cdots,2n\}\hookrightarrow Y\setminus \infty\}$
and let $C_{2n}(Y)$ the compactification of $\breve C_{2n}(Y)$ given by Lescop \cite[\S 3]{Lescop}.
(This compactification is similar to Fulton-MacPherson compactification~\cite{FM}).
Roughly speaking, $C_{2n}(Y)$ is obtained from $Y^{2n}$ by real blowing up along all diagonals and
$\{(x_1,\cdots,x_{2n}\mid \exists i~\mbox{such that}~x_i=\infty\}$. See \S 3 in \cite{Lescop} for the complete definition.)
Note that $C_2(Y)$ is given by real blowing up $Y^2$ along
$(\infty,\infty)$, $\infty\times (Y\setminus\infty), (Y\setminus \infty)\times \infty$ and $\Delta$ in turn.
Let us denote by $q:C_2(Y)\to (Y\setminus \infty)^2$ the composition of the blow down maps.
Then $\partial C_2(Y)=ST_{\infty}Y\times (Y\setminus \infty)\cup (Y\setminus \infty)\times ST_{\infty}Y\cup S\nu_{\Delta(Y\setminus \infty)}\cup q^{-1}(\infty^2)$.
We identify $S\nu_{\Delta(Y\setminus \infty)}$ with $STY|_{Y\setminus \infty}$ by the canonical isomorphism $S\nu_{\Delta Y}\cong STY$.
The involution $Y^2\to Y^2, (x,y)\mapsto (y,x)$ induces an involution of $C_2(Y)$.
We denote by $\iota:C_2(Y)\to C_2(Y)$ this involution.
Let
$p_1:(\partial C_2(Y)\supset) ST_{\infty}Y\times (Y\setminus \infty)\to ST_{\infty}Y\stackrel{\tau^{\infty}}{=}ST_{\infty}S^3=S^2$ and
$p_2:(\partial C_2(Y)\supset) (Y\setminus \infty)\times ST_{\infty}Y\to ST_{\infty}Y=ST_{\infty}S^3=S^2$
be the projections.
We denote by $\iota_{S^2}:S^2\to S^2$ the involution induced by $\times (-1):\RR^3\to \RR^3$.
Let $p_c:C_2(S^3)\to S^2$ be the extension of the map ${\rm int}C_2(S^3)=(\RR^3\times \RR^3)\setminus\Delta\to S^2$, $(x,y)\mapsto (y-x)/\|y-x\|$.
Since it is possible to identify $q^{-1}(N(\infty;Y)^2)\subset \partial C_2(Y)$ with $q^{-1}(N(\infty;S^3)^2)\subset \partial C_2(S^3)$ by $\tau^{\infty}$,
we get a map $\partial C_2(Y)\supset q^{-1}((N(\infty;Y)\setminus \infty)^2)\stackrel{p_c}{\to}S^2$.
Since $p_1, \iota_{S^2}\circ p_2$ and $p_c$ are compatible on boundary,
these maps define the map $$p_Y:\partial C_2(Y)\setminus S\nu_{\Delta(Y\setminus N(\infty;Y))}\to S^2.$$
(Here we note that $\partial C_2(Y)\setminus S\nu_{\Delta(Y\setminus N(\infty;Y))}=ST_{\infty}Y\times (Y\setminus \infty)\cup (Y\setminus \infty)\times ST_{\infty}Y\cup q^{-1}(N(\infty;Y)^2)$.)
\subsection{More on the boundary $\partial C_{2n}(Y)$.}
The reference here is Lescop \cite[\S 2.2, \S 3]{Lescop}.
For $B\subset \{1,\cdots,2n\}$, we set
$$F(\infty;B)=q^{-1}(\{(x_1,\cdots, x_{2n})\mid x_i=\infty~\mbox{iff}~i\in B,~\mbox{if}~i,j\not\in B~\mbox{then}~ x_i\not=x_j \}) ,$$
and for $B\subset \{1,\cdots,2n\}$($\sharp B\ge 2$), we set
$$F(B)=q^{-1}(\{(x_1,\cdots, x_{2n})\in(Y\setminus \infty)^{2n}\mid\exists y, x_i=y~\mbox{iff}~i\in B,~\mbox{if}~i,j\not\in B~\mbox{then}~ x_i\not=x_j \}).$$
Under these notations, $\partial C_{2n}(Y)=\bigcup_B F(\infty;B)\cup \bigcup_{\sharp B\ge 2}F(B)$.
We remark that $\partial C_{2n}(Y)$ has smooth structure (See \cite[\S~3]{Lescop}).
Let $X$ be a 3-dimensional real vector space. Let $V$ be a finite set.
we define $\breve S_V(X)$ to be the set of injective maps from $V$ to $X$ up to translations and dilations.
Set $k=\{1,\cdots,k\}$. Note that $\breve S_2(X)=S(X)$.
For an $\RR^3$ vector bundle $E\to M$, we denote by $\breve S_V(E)\to M$ the fiber bundle where the fiber over $x\in M$ is $\breve S_V(E_x)$.
Under these notations, $F(2n)=\breve S_{2n}(T(Y\setminus \infty))$.
We remark that $F(B)$ has a fiber bundle structure where the typical fiber is $\breve S_B(\RR^3)$.
Lescop gave a compactification $S_V(X), S_V(E)$ of $\breve S_V(X), \breve S_V(E)$ respectively in \cite{Lescop}.
Let $f(B)(X)=\breve S_B(X)\times \breve S_{\{b\}\cup B}(X)$, for $B\subset V$ with $B\not=V$ and $\sharp B\ge 2$.
Let $f(B)(E)\to M$ be the fiber bundle where the fiber over $x\in M$ is $f(B)(E_x)$.
Under this notation,
$$\partial S_V(X)=\bigcup_{\sharp B\ge 2}f(B)(X), \partial S_V(E)=\bigcup_{\sharp B\ge 2}f(B)(E)$$
(See Proposition 2.8 in \cite{Lescop}).
We remark that $f(B)(E)$ has a fiber bundle structure where the typical fiber is $\breve S_B(\RR^3)$.
\subsection{Jacobi diagrams.}
The reference here is Lescop \cite[\S 1.3, 2.3]{Lescop}.
A {\it Jacobi diagram} of degree $n$ is defined to be a trivalent graph with $2n$ vertices and $3n$ edges without simple loops.
For a Jacobi diagram $\overline{\Gamma}$, we denote by $H(\overline \Gamma), E(\overline \Gamma)$ and $V(\overline\Gamma)$
the set of half edges, the set of edges and the set of vertices respectively.
An {\it orientation} of a vertex of $\overline{\Gamma}$ is a cyclic order of the three half-edges that meet at the vertex.
A Jacobi diagram is {\it oriented} if all its vertices are oriented.
Let
$$\mathcal A_n(\emptyset)=\{\mbox{degree}~n~\mbox{oriented Jacobi diagrams}\}\RR/{\rm AS,IHX},$$
where the relations AS and IHX are locally represented by the following pictures.
\begin{figure}[h]
\begin{center}
\includegraphics[width=280pt]{PICn1.eps}
\end{center}
Here the orientation of each vertex is given by counterclockwise order of the half edges.
\end{figure}
Let
$$\mathcal E_n=\{\Gamma=(\overline{\Gamma},\varphi_E,\varphi_V,{\rm ori}_E)\}$$
Here $\overline{\Gamma}$ is a Jacobi diagram of degree $n$, $\varphi_E:E(\overline\Gamma)\cong \{1,2,\cdots,3n\}$ and $\varphi_V:V(\overline\Gamma)\cong \{1,2,\cdots,2n\}$ are labels of edges and vertices respectively, and ${\rm ori}_E$ is a collection of orientations of each edge.
These data and an orientation of $\overline{\Gamma}$ induce two orientations of $H(\overline \Gamma)$.
The first one is the edge-orientation induced by $\varphi_E$ and ${\rm ori}_E$.
The second one is the vertex-orientation induced by $\varphi_V$ and orientation of $\overline \Gamma$.
We choose the orientation of $\overline{\Gamma}$ so that the edge-orientation coincides with the vertex-orientation.
Let us denote by $[\Gamma]\in \mathcal A_n(\emptyset)$ the oriented Jacobi diagram given by $\Gamma$ in such a way.
\begin{remark}
The notation $\mathcal A_{2n,3n}$ used by Watanabe \cite{Watanabe} coincides with the notation $\mathcal A_n(\emptyset)$ used by Lescop \cite{Lescop}
as $\RR$-vector spaces.
\end{remark}
\section{Construction of an invariant of rational homology 3-sphere via vector fields.}\label{newinv}
Let $n$ be a natural number.
In this section, we define an invariant $\widetilde z_n$ using vector fields.
The idea of construction of $\widetilde z_n$ is based on Kuperberg, Thurston \cite{KKT}, Lescop \cite{Lescop} and
the construction of the anomaly part of Watanabe's Morse homotopy invariant~\cite{Watanabe}.
Let $Y$ be a rational homology 3-sphere with a base point $\infty$.
In Subsection~\ref{ad.form}, we will define the notion of admissible vector fields on $T(Y\setminus \infty)$.
In Subsections~\ref{mainterm},~\ref{anomaly term}, we will define $\widetilde z_n(Y;\vec\gamma)$ and $\widetilde z_n^{\rm anomaly}(\vec\gamma)$
using a family of admissible vector fields $\vec\gamma$.
Thus we obtain a topological invariant $\widetilde z_n(Y)=\widetilde z_n(Y;\vec\gamma)-\widetilde z_n^{\rm anomaly}(\vec\gamma)$ of $Y$
in Subsection~\ref{25}.
We will prove well-definedness of $\widetilde z_n$ in Subsection~\ref{s46}.
\subsection{Admissible vector fields on $T(Y\setminus \infty)$.}\label{ad.form}
For $a\in S^2\subset\RR^3$, the map $q_a:\RR^3\to \RR$ is defined by $q_a(x)=\langle x,a\rangle$ where $\langle,\rangle$ is the standard inner product on $\RR^3$.
Write $\pm a=\{a,-a\}$.
\begin{defini}
A vector field $\gamma\in \Gamma T(Y\setminus\infty)$ is an {\it admissible vector field (with respect to $a$)} if the following conditions hold.
\begin{itemize}
\item $\gamma|_{N(\infty;Y)\setminus \infty}=-\grad~q_a|_{N(\infty;S^3)\setminus \infty}$,
\item $\gamma$ is transverse to the zero section in $T(Y\setminus \infty)$.
\end{itemize}
\end{defini}
\begin{example}
We give two important examples of admissible vector fields with respect to $a$.
\begin{itemize}
\item[{\rm (1)}] Let $\tau_{\RR^3}:T\RR^3\cong \underline{\RR^3}$ be the standard framing of $T\RR^3$.
We regard $a\in \RR^3$ as a constant section of the trivial bundle $\underline{\RR^3}$.
For a framing $\tau:T(Y\setminus \infty)\cong \underline{\RR^3}$ such that $\tau|_{N(\infty;Y)\setminus\infty}=\tau_{\RR^3}|_{N(\infty;S^3)\setminus \infty}$, the pull-back vector field $\tau^*a$ is an admissible vector field with respect to $-a$.
\item[{\rm (2)}] For a Morse function $f:Y\setminus\infty\to\RR$ such that $f|_{N(\infty;Y)\setminus\infty}=q_a|_{N(\infty;S)\setminus \infty}$
, $\grad f$ is an admissible vector field with respect to $a$.
\end{itemize}
\end{example}
The following lemma plays an important role in the next subsection.
For an admissible vector field $\gamma$,
let
$$\overline c_{\gamma}=\overline{\left\{\frac{\gamma(x)}{\|\gamma(x)\|}\in ST_xY~\Big|~ x\in Y\setminus(\infty\cup \gamma^{-1}(0))\right\}}^{\rm closure}\subset ST(Y\setminus\infty).$$
Here we choose the orientation of $\overline c_{\gamma}$ such that the restriction of the projection $STY\to Y$ to $\overline c_{\gamma}$ is
orientation preserving.
\begin{lemma}\label{keylemma}
$$c_0(\gamma)=\overline c_{\gamma}\cup \overline c_{-\gamma}$$
is a submanifold of $ST(Y\setminus \infty)$ without boundary.
\end{lemma}
To prove this lemma, we first remark the following lemma.
Let $n,k\ge 0$ be integral numbers.
Let $s:(\RR^{n+k},0)\to (\RR^{n},0)$ be a $\CC$ map which is transverse to the origin $0\in\RR^{n}$.
\begin{lemma}\label{lemmaA}
There is a diffeomorphism $\varphi:(\RR^{n+k},0)\to (\RR^{n+k},0)$
such that $s\circ \varphi$ coincides with $p_{\RR^n}$ as germs at $0\in \RR^{n+k}$.
Here $p_{\RR^n}:\RR^{n+k}=\RR^n\times \RR^k\to \RR^n$ is the orthogonal projection.
\end{lemma}
\begin{proof}
This is a consequence of the implicit function theorem.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{keylemma}]
It is sufficient to check this claim near $\gamma^{-1}(0)$.
Let $x\in \gamma^{-1}(0)$.
We fix a trivialization $\psi:T(Y\setminus \infty)|_{U_0}\cong U_0\times \RR^3$ on a neighborhood $U_0$ of $x$ in $Y$.
By the above Lemma~\ref{lemmaA}, there is a neighborhood $U\subset U_0$ of $x$ and local coordinates $\varphi:\RR^3\cong U$(which is independent of $\psi$) such that $(\varphi^{-1}\times {\rm id})\circ\psi\circ\gamma\circ\varphi:\RR^3\to \RR^3\times \RR^3$ is represented by $(\varphi^{-1}\times {\rm id})\circ\psi\circ\gamma\circ\varphi(x)=(x,x)$.
We fix these local trivialization and coordinates and we write $\gamma$ instead of $(\varphi^{-1}\times {\rm id})\circ\psi\circ\gamma\circ\varphi$.
We first show that $\partial\overline c_{\gamma}\cap STU=-(\partial\overline c_{-\gamma}\cap STU)$ as oriented manifolds.
Let $D_+=\overline c_{\gamma}\cap STU$ and $D_-=\overline c_{-\gamma}\cap STU$.
Under the above local coordinates, $D_+=\{(tx,x/\|x\|)\mid x\in S^2, t\in [0,\infty)\}\subset (S^2\times [0,\infty)/(S^2\times 0))\times S^2=\RR^3\times S^2$ and
$D_-=\{(tx,-x/\|x\|)\mid x\in S^2, t\in [0,\infty)\}$.
Both projection $\pi:D_{+}\to \RR^3$ and the projection $\pi:D_{-}\to\RR^3$
are orientation preserving (or reversing).
Let $g:\RR^3\times S^2\to \RR^3\times S^2$ be the bundle map defined by $(x,v)\mapsto (x,-v)$.
So $g:\partial D_+\cong \partial D_-$ is orientation preserving.
On the other hand, $g|_{\{0\}\times S^2}:\{0\}\times S^2\to \{0\}\times S^2$ is orientation reversing.
Hence, the identity map ${\rm id}:\{0\}\times S^2\to \{0\}\times S^2$ is orientation reversing map as a map between $\partial D_+$ and $\partial D_-$.
Therefore $\partial \overline c_{\gamma}=\partial D_+=-\partial D_-=-\partial \overline c_{-\gamma}$.
We next prove that $c_0(\gamma)\cap STU$ is a submanifold of $STU\cong \RR^3\times S^2$.
Let $p_2:\RR^3\times S^2\to S^2$ be the projection.
For each $v\in S^2$, we have $(p_2|_{c_0(\gamma)})^{-1}(v)=\RR v\times \{v\}\subset \RR^3\times S^2$.
The set $\bigcup_{v\in S^2}\RR v\times \{v\}$ is a submanifold of $\RR^3\times S^2$.
In fact, for any $v_0\in S^2$ and for any sufficiently small neighborhood $B_{v_0}\subset S^2$ of $v_0$ we can take a diffeomorphism
$$\Phi_{v_0}:(\RR^3\times B_{v_0},\bigcup_{v\in B_{v_0}}\RR v\times \{v\})\stackrel{\cong}{\to}(\RR^3\times B_{v_0}, \RR w_0\times B_{v_0})$$
as follow.\footnote{The author is indebted to Professor Christine Lescop for this construction.}
Here $w_0\in S^2\subset\RR^3$ is a point orthogonal to $v_0$ in $\RR^3$ and $\RR w_0$ is the 1-dimensional vector subspace of $\RR^3$ spanned by $w_0$.
For each $v\in B_{v_0}$, let $m(v,w_0)\in S^2$ be the middle point of the geodesic segment from $v$ to $w_0$.
Let $\rho(v,w_0)\in SO(3)$ be the rotation with axis directed by $m(v,w_0)$ and with angle $\pi$.
So $\rho(v,w_0)$ exchanges $v$ and $w_0$.
Then we can define $\Phi_{v_0}:\RR^3\times B_{v_0}\to \RR^3\times B_{v_0}$ by $\Phi_{v_0}(x,v)=(\rho(v,w_0)(x),v)$
for each $(x,v)\in \RR^3\times B_{v_0}$.
Therefore $c_0(\gamma)\cap (\RR^3\times S^2)=\bigcup_{v\in S^2}\RR v\times\{v\}$ is a submanifold of $\RR^3\times S^2$.
\end{proof}
\subsection{The principal term $\widetilde z(Y;\vec\gamma)$.}\label{mainterm}
In this subsection, we define the principal term $\widetilde z(Y;\vec\gamma)$ of the invariant $\widetilde z(Y)$.
We define $$c(\gamma)=p_Y^{-1}(\pm a)\cup c_0(\gamma)\subset \partial C_2(Y).$$
By the definition of $\gamma$ and Lemma~\ref{keylemma},
$c(\gamma)$ is a closed 3-manifold.
Therefore $[c(\gamma)]\in H_3(\partial C_2(Y);\mathbb R)$.
Let $\omega_{S^2}^a$ be an anti-symmetric closed 2-form on $S^2$ such that
$\omega_{S^2}^a$ represents the Poincar\'e dual of $[\pm a]$ and the support of $\omega_{S^2}^a$ is concentrated in near $\pm a$.
Let $\omega_{\partial}(\gamma)$ be a closed 2-form on $\partial C_2(Y)$ satisfying the following conditions.
\begin{itemize}
\item $2\omega_{\partial}(\gamma)$ represents the Poincar\'e dual of $[c(\gamma)]$,
\item The support of $\omega_{\partial}(\gamma)$ is concentrated in near $c(\gamma)$,
\item $\iota^*\omega_{\partial}(\gamma)=-\omega_{\partial}(\gamma)$ and
\item $\omega_{\partial}(\gamma)|_{\partial C_2(Y)\setminus S\nu_{\Delta(Y\setminus N(\infty;Y))}}=\frac{1}{2}p_Y^*\omega_{S^2}^a$.
\end{itemize}
Since $Y$ is a rational homology 3-sphere, the restriction $H^2(C_2(Y);\RR)\to H^2(\partial C_2(Y);\RR)$ is an isomorphism.
Thus there is a closed 2-form $\omega(\gamma)$ on $C_2(Y)$ satisfying the following conditions.
\begin{itemize}
\item $\omega(\gamma)|_{\partial C_2(Y)}=\omega_{\partial}(\gamma)$ and
\item $\iota^*\omega(\gamma)=-\omega(\gamma)$.
\end{itemize}
\begin{defini}[propagator]
We call $\omega(\gamma)$ a {\it propagator with respect to} $\gamma$.
\end{defini}
Take $a_1,\cdots ,a_{3n}\in S^2$ (we may take, for example, $a_1=\cdots=a_{3n}$).
Let $\gamma_i$ be an admissible vector field with respect to $a_i$ and let
$\omega(\gamma_i)$ be a propagator with respect to $\gamma_i$ for each $i\in \{1,\cdots,3n\}$.
To simplify notation, we write $\vec\gamma$ instead of $(\gamma_1,\cdots, \gamma_{3n})$.
For each $\Gamma=(\overline \Gamma, \varphi_E,\varphi_V,{\rm ori}_E)\in \mathcal E_n$
and for each $\varphi_E^{-1}(i)\in E(\overline \Gamma)$, let $s(i),t(i)\in \{1,\cdots,2n\}$ denote the labels of the initial vertex and the terminal
vertex of $\varphi_E^{-1}(i)$ respectively.
The embedding $\{1,2\}\cong\{s(i),t(i)\}\hookrightarrow \{1,\cdots, 2n\}$ induces
the projection $\pi_{\breve C_{2n}(Y)}:\breve C_{2n}(Y)\to\breve C_2(Y)$.
Furthermore it is possible to extend $\pi_{\breve C_{2n}(Y)}$ to $C_{2n}(Y)$ by the definition of $C_{2n}(Y)$. We denote by $P_i(\Gamma):C_{2n}(Y)\to C_2(Y)$ such the extended map
(see \cite{Lescop}~\S 2.3 for more detail).
\begin{defini}
$$\widetilde z_n(Y;\vec\gamma)=\sum_{\Gamma\in\mathcal E_n}\left(\int_{C_{2n}(Y)}\bigwedge_{i}P_i(\Gamma)^*\omega(\gamma_i)\right)[\Gamma]\in \mathcal A_n(\emptyset).$$
\end{defini}
\begin{remark}
By the above definition, the value $\widetilde z_n(Y;\vec\gamma)$ often depends on the choices of $\omega(\gamma_i)$ even if we fix $\vec\gamma$.
We will prove in Subsection~\ref{s46} that $\widetilde z_n(Y;\vec\gamma)$, however, depends only on the choice of $\vec \gamma$ for generic $\vec\gamma$.
\end{remark}
\subsection{Alternative description of $\widetilde z_n(Y;\vec\gamma)$.}
In this subsection, we give an alternative description of $\widetilde z_n(Y;\vec\gamma)$ using cohomologies of simplicial complexes with coefficients in $\RR$.
This description will be needed in Section~\ref{proof}.
The admissible vector field $\gamma_i$ with respect to $a_i$ and the 3-cycle $c(\gamma_i)\subset \partial C_2(Y)$ are as above.
Let $T_{C_2(Y)}$ be the simplicial decomposition of $C_2(Y)$ given by pulling back a simplicial decomposition of $C_2(Y)/\iota$.
So the simplicial decomposition $T_{C_2(Y)}$ is compatible with the action of $\iota$.
By replacing such a simplicial decomposition if necessary, we may assume that each simplex of $T_{C_2(Y)}$ is transverse to $c(\gamma_i)$.
Let $\omega_{\partial}^s(\gamma_i)\in S^2(\partial C_2(Y))$ be the 2-cocycle defined by
$\omega_{\partial}^s(\gamma_i)(\sigma)=\frac{1}{2}\sharp (\sigma\cap c(\gamma_i))$ for each 2-cycle $\sigma$ in $T_{C_2(Y)}|_{\partial C_2(Y)}$.
Thus $\omega_{\partial}^s(\gamma_i)$ is anti-symmetric under the involution $\iota$.
Let $\omega^s(\gamma_i)$ be an extension of $\omega_{\partial}^s(\gamma_i)$ to $C_2(Y)=|T_{C_2(Y)}|$ satisfying the following conditions.
\begin{itemize}
\item $\omega^s(\gamma_i)|_{\partial C_2(Y)}=\omega^s_{\partial}(\gamma_i)$ and
\item $\iota^*\omega^s(\gamma_i)=-\omega^s(\gamma_i)$.
\end{itemize}
We call it a {\it simplicial propagator}.
Take an appropriate simplicial decomposition of $C_{2n}(Y)$.
Then we have the 2-cocycle $P_i(\Gamma)^*\omega^s(\gamma_i)\in S^{2n}(C_{2n}(Y))$.
By the construction, $\bigwedge_i P_i(\Gamma)^*\omega^s(\gamma_i)$ is a cocycle in $(C_{2n}(Y),\partial C_{2n}(Y))$.
If necessary we replace the simplicial decompositions with a smaller one, we have the following lemma via the intersection theory.
\begin{lemma}[Alternative description of $\widetilde z_n(Y;\gamma)$]\label{alt}
If $(\bigcap_i P_i(\Gamma)^{-1}{\rm support}(\omega^s(\gamma_i)))\\
\cap\partial C_{2n}(Y)=\emptyset$ for any $\Gamma$,
$$\widetilde z_n(Y;\vec\gamma)=\sum_{\Gamma\in\mathcal E_n}\langle\bigwedge_{i}P_i(\Gamma)^*\omega^s(\gamma_i), [C_{2n}(Y),\partial C_{2n}(Y)]\rangle [\Gamma]
\in\mathcal A_n(\emptyset).$$
Here $ [C_{2n}(Y),\partial C_{2n}(Y)]$ denotes the fundamental homology class and $\langle,\rangle$ denotes the Kronecker product.
\end{lemma}
\subsection{The anomaly term $\widetilde z_n^{\rm anomaly}(\vec\gamma)$.}\label{anomaly term}
In this subsection, we define the anomaly term $\widetilde z_n^{\rm anomaly}(Y;\vec\gamma)$ of the invariant $\widetilde z_n(Y)$.
The idea of the construction of this anomaly term is based on the construction of the anomaly term of Watanabe's invariant~\cite{Watanabe}.
Let $Y$, $\infty$,
$a_1,\cdots,a_{3n}\in S^2$,
$\gamma_1,\cdots,\gamma_{3n}$ (admissible vector fields with respect to $a_1,\cdots,a_{3n}$ respectively) and
$\omega(\gamma_1),\cdots,\omega(\gamma_{3n})$
be the same as above.
Let $X$ be a connected oriented 4-manifold with $\partial X=Y$ and $\chi (X)=0$.
For example, we can take $X=(T^4\sharp \mathbb CP^2)\setminus B^4$ when $Y=S^3$.
For a framing $\tau'$ of $TY$ or $\underline{\RR}\oplus TY$, we denote by $\sigma_Y(\tau')\in \ZZ$ the signature defect of $\tau'$.
Let $\tau_{S^3}$ be a framing\footnote{There is such a framing. For example, the Lie framing $\tau_{SU(2)}$ of $S^3=SU(2)$ satisfies
$\sigma_{S^3}(\tau_{SU(2)})=2$. See R.~Kirby and P.~Melvin~\cite{KM} for more details. We can get $\tau_{S^3}$ by modifying $\tau_{SU(2)}$.} of $TS^3$ satisfying the following two conditions:
\begin{itemize}
\item $\sigma_{S^3}(\tau_{S^3})=2$,
\item $\tau_{S^3}|_{S^3\setminus N'(\infty;S^3)}=\tau_{\RR^3}|_{S^3\setminus N'(\infty;S^3)}$.
\end{itemize}
Here $N'(\infty;S^3)$ is a neighborhood of $\infty$ smaller than $N(\infty;S^3)$, i.e., $\infty\in N'(\infty;S^3)\subset N(\infty;S^3)$.
\begin{remark}
There is no special meaning in the number "$2$" in the condition $\sigma_{S^3}(\tau_{S^3})=2$.
The anomaly term $\widetilde z^{\rm anomaly}_n(\vec\gamma)$ is independent of the choice
of $\tau_{S^3}$ even if $\sigma_{S^3}(\tau_{S^3})$ not be $2$.
We remark that there is no framing $\tau$ on $S^3$ such that $\sigma_{S^3}(\tau)=0$.
\end{remark}
Let $\eta_Y$ be the outward unit vector field of $TY=T(\partial X)\subset TX|_Y$ in $TX$.
Since $\chi (X)=0$, it is possible to extend $\eta_Y$ to a unit vector field of $TX$.
We denote by $\eta_X\in \Gamma TX$ such an extended vector field.
Let $T^vX$ be the normal bundle of $\eta_X$. We remark that $T^vX|_Y=TY$.
The vector field $\tau_{S^3}^*a_i$ of $TY|_{N(\infty;Y)}$ is the pull-back of $a_i\in S^2\subset \RR^3$ along
$\tau_{S^3}|_{N(\infty;Y)}$
\footnote{We sometimes regard a framing as a bundle map to the trivial bundle over a point.}.
Since $\gamma_i|_{Y\setminus N(\infty;Y)}\in \Gamma T(Y\setminus N(\infty;Y))$ and $\tau_{S^3}^*a_i|_{N(\infty;Y)}\in \Gamma TY|_{N(\infty;Y)}$
are compatible, these vector fields define the vector field $\gamma'_i\in \Gamma TY$.
Let $\beta_i\in \Gamma T^vX$ be a vector field of $T^vX$ transverse to the zero section in $T^vX$ satisfying $\beta_i|_{Y}=\gamma'_i$.
By a similar argument of Lemma~\ref{keylemma},
$$c_0(\beta_i)=\overline{\left\{\frac{\beta(x)}{\|\beta(x)\|}, \frac{-\beta(x)}{\|\beta(x)\|}\in S(T^vX)_x~\Big|~ x\in X\setminus \beta^{-1}(0)\right\}}^{\rm closure}\subset ST^vX$$
is a submanifold of $ST^vX$ satisfying $\partial c_0(\beta_i)\subset STY$.
Hence $c_0(\beta_i)$ is a cycle of $(ST^vX,\partial ST^vX)$.
Here we choose the orientation of $c_0(\beta_i)$ such that the restriction of the projection $ST^vX\to X$ to $c_0(\beta_i)$ is orientation preserving.
We note that $c_0(\beta_i)$ satisfies $c_0(\beta_i)\cap S\nu_{\Delta(Y\setminus N(\infty;Y))}=c_0(\gamma_i)$.
Let $W(\gamma_i)$ be a closed 2-form on $ST^vX$ satisfying the following conditions.
\begin{itemize}
\item $2W(\gamma_i)$ represents the Poincar\'e dual of $[c_0(\beta_i),\partial c_0(\beta_i)]$,
\item The support of $W(\gamma_i)$ is concentrated in near $c_0(\beta_i)$,
\item $W(\gamma_i)|_{ST(Y\setminus N(\infty;Y))}=\omega_{\partial}(\gamma_i)|_{S\nu_{\Delta(Y\setminus N(\infty;Y))}}$ and
\item $W(\gamma_i)|_{STN(\infty;Y)}=\frac{1}{2}\tau_{S^3}^*\omega_{S^2}^{a_i}$.
\end{itemize}
For $i\in \{1,2,\cdots, 3n\}$, let $\phi_i^0(\Gamma):\breve S_{2n}(T^vX)\to S_2(T^vX)$ be the map induced by $\{1,2\}\cong\{s(i),t(i)\}\hookrightarrow \{1,\cdots, 2n\}$. It is possible to extend $\phi_i^0(\Gamma)$ to $S_{2n}(T^vX)$. We denote by $\phi_i(\Gamma):S_{2n}(T^vX)\to S(T^vX)$ such the extended map.
By an argument similar to Proposition 4.17 in \cite{Watanabe}, the following lemma holds.
\begin{lemma}\label{lem48}
There exists $\mu_n\in\mathcal A_n(\emptyset)$ such that
$$-\mu_n\Sign X+\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge _i \phi_i(\Gamma)^*W(\gamma_i)[\Gamma]\in \mathcal A_n(\emptyset)$$
does not depend on the choice of $X$, $\beta_i$, and $W(\gamma_i)$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem48}.]
Let $X$ be a closed 4-manifold with $\Sign X=0$ and $\chi(X)=0$.
When $X$ is not connected, we assume that the Euler number of each component of $X$ is zero.
Let $\eta_X$ be an unit vector field of $TX$ and let $T^vX$ be the normal bundle of $\eta_X$ in $TX$.
Let $\beta_1,\cdots, \beta_{3n}$ be a family of sections of $T^vX$ that are transverse to the zero section in $T^vX$.
Let $W_i$ be a closed 2-form that represents the Poincar\'e dual of $c_0(\beta_i)$ in $ST^vX$, for $i=1,\cdots,3n$.
By a cobordism argument, it is sufficient to show that $\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W_i[\Gamma]=0$.
We first prove that there exist an oriented compact 5-manifold $Z$ and there exist unit vector fields $\eta_Z^1,\eta_Z^2\in \Gamma TZ$ such that:
\begin{itemize}
\item $\partial Z=X\sqcup X$,
\item $\eta_Z^1,\eta_Z^2$ are linearly independent at any point in $Z$, i.e., $(\eta_Z^1,\eta_Z^2)$ is a 2-framing of $TZ$,
\item $\eta_Z^1|_{\partial Z}$ is the outward unit vector field of $X=\partial Z$, and
\item $\eta_Z^2|_{\partial Z}=\eta_X\sqcup \eta_X$.
\end{itemize}
Since $\Sign X=0$, there exists a connected compact oriented 5-manifold $Z_0$ such that $\partial Z_0=X$.
Let $\eta_{Z_0}\in \Gamma TZ_0|_{X}$ be the outward unit vector field of $X=\partial Z_0$.
By attaching 2-handles along the knots generating $H_1(Z_0;\mathbb Z/2)$ if necessary,
we may assume that $H_1(Z_0;\mathbb Z/2)\cong H^4(Z_0;\partial Z_0;\mathbb Z/2)=0$.
Thus the primary obstruction $o_{Z_0}$ to extending the 2-framing $(\eta_{Z_0}, \eta_X)$ of $TZ_0|_{X}$ into $Z_0$ is in $H^5(Z_0,\partial Z_0;\pi_4(V_{5,2}))
=H^5(Z_0,\partial Z_0;\ZZ/2)$.
Let $Z=Z_0\sharp Z_0$. Then the obstruction to extending the 2-framing $(\eta_{Z_0}\sqcup \eta_{Z_0}, \eta_X\sqcup \eta_X)$ of $TZ|_{X\sqcup X}$ into $Z$ is $o_{Z_0}+o_{Z_0}=0\in H^5(Z,\partial Z;\ZZ/2)$.
So we can take $\eta_Z^1, \eta_Z^2$ satisfying the above conditions.
Let $T^vZ$ be the normal bundle of $\langle \eta^1_Z,\eta^2_Z\rangle$ in $TZ$.
Then $T^vZ$ is a rank 3 sub-bundle of $TZ$ satisfying $T^vZ|_{X}=T^vX$.
Let $\widetilde\beta_i\in \Gamma T^vZ$ be a vector field transverse to the zero section in $T^vZ$ satisfying $T^vZ|_{X}=\beta_i$.
Then $c_0(\widetilde\beta_i)=\overline{\left\{\frac{\widetilde\beta_i(x)}{\|\widetilde\beta_i(x)\|}, \frac{-\widetilde\beta_i(x)}{\|\widetilde\beta_i(x)\|}\in S(T^vZ)_x~\Big|~ x\in Z\setminus \widetilde\beta_i^{-1}(0)\right\}}^{\rm closure}\subset ST^vZ$
is a submanifold of $ST^vZ$ satisfying $\partial c_0(\widetilde\beta_i)=c_0(\beta_i)$.
Let $W(\widetilde\beta_i)$ be a closed 2-form on $ST^vZ$ that represents the Poincar\'e dual of $[c_0(\widetilde\beta_i),\partial c_0(\widetilde\beta_i)]$
and satisfying $W(\widetilde\beta_i)|_{ST^vX}=W_i$.
By Stokes' theorem, we have
\begin{eqnarray*}
0&=&\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vZ)}d\left(\bigwedge_i\phi_i(\Gamma)^*W(\widetilde\beta_i)\right)[\Gamma]\\
&=&2\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W_i[\Gamma]+
\sum_{\Gamma\in\mathcal E_n}\int_{\partial S_{2n}(T^vZ)}\bigwedge_i\phi_i(\Gamma)^*W(\widetilde\beta_i)[\Gamma]\\
&=&2\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W_i[\Gamma]+
\sum_{\Gamma\in\mathcal E_n}\sum_{2\le \sharp B<2n}\int_{f(B)(T^vZ)}\bigwedge_i\phi_i(\Gamma)^*W(\widetilde\beta_i)[\Gamma]\\
&=&2\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W_i[\Gamma].
\end{eqnarray*}
The lase equation is given by Lemma~\ref{lem419}.
\end{proof}
Let $\tau_Y$ be a framing of $T(Y\setminus \infty)$ satisfying $\tau_Y|_{N(\infty;Y)\setminus\infty}=\tau_{\RR^3}|_{N(\infty;S^3)\setminus \infty}$.
Then $\tau_Y^*\vec a=(\tau_Y^*a_1,\cdots,\tau_Y^*a_{3n})$ is a family of admissible vector fields.
Let $\tau_Y'=\tau_Y|_{Y\setminus N(\infty;Y)}\cup \tau_{S^3}|_{N(\infty;S^3)}$. So $\tau_Y'$ is a framing of $TY$.
Take $W(\tau_Y^*{a_i})|_{STY}=\frac{1}{2}(\tau'_Y)^*\omega_{S^2}^{a_i}$.
\begin{lemma}
$\int_{S_{2n}(T^vX)}\bigwedge _i \phi_i(\Gamma)^*W(\tau_Y^*a_i)$ is independent of the choice of $a_1,\cdots, a_{3n}$.
\end{lemma}
\begin{proof}
Let $a_i'$ be an alternative choice of $a_i$ for any $i$.
Let $\widetilde\omega_{S^2}^{i}$ be a closed 2-form on $S^2\times [0,1]$ satisfying
$\widetilde\omega_{S^2}^{i}|_{S^2\times\{0\}}=\omega_{S^2}^{a_i}$ and
$\widetilde\omega_{S^2}^{i}|_{S^2\times\{1\}}=\omega_{S^2}^{a_i'}$.
Let $STY\times [0,1]\subset ST^vX$ be the collar of $STY$ such that $STY\times \{0\}=\partial ST^vX$.
We take $W(\tau_Y^*a_i)|_{STY\times [0,1]}=\frac{1}{2}(\tau_Y')^*\widetilde\omega_{S^2}^i$.
Thus $W(\tau_Y^*a_i)|_{STY\times \{1\}}=W(\tau_Y^*a_i')$.
Since Lemma~\ref{lem48} {\rm (1)}, we have
\begin{eqnarray*}
&&\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W(\tau_Y^*a_i)-
\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W(\tau_Y^*a_i')\\
&=&\int_{S_{2n}(TY)\times [0,1]}\bigwedge_i\phi_i(\Gamma)^*W(\tau_Y^*a_i)\\
&=&\frac{1}{2^{3n}}\int_{S_{2n}(TY)\times [0,1]}\bigwedge_i\phi_i(\Gamma)^*(\tau_Y'\times {\rm id})^*\widetilde\omega_{S^2}^i\\
\end{eqnarray*}
The map $S_{2n}(TY)\times[0,1] \stackrel{\prod_i\phi_i(\Gamma)}{\to}(STY\times [0,1])^{3n}\stackrel{(\tau_Y'\times{\rm id})^{3n}}{\to}(S^2\times [0,1])^{3n}$ factors through $S_{2n}(\RR^3)\times [0,1]$.
Hence we have $((\tau_Y'\times{\rm id})^{3n}\circ\prod_i\phi_i(\Gamma))^*\widetilde\omega_{S^2}^i\in {\rm Im}(\Omega^{6n}(S_{2n}(\RR^3)\times[0,1])\to
\Omega^{6n}(S_{2n}(TY)\times [0,1]))$.
Since $\dim S_{2n}(\RR^3)\times [0,1]=6n-3<6n=\dim \bigwedge_i\phi_i(\Gamma)^*(\tau_Y'\times{\rm id})^*\widetilde\omega_{S^2}^i$,
we have $\int_{S_{2n}(TY)\times [0,1]}\bigwedge_i\phi_i(\Gamma)^*(\tau_Y')^*\widetilde\omega_{S^2}^i=0$.
\end{proof}
Because of the above two lemmas,
$-\mu_n\Sign X+\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge _i \phi_i(\Gamma)^*W(\tau_{\RR^3}^*a_i)[\Gamma]$
is independent of the choice of a 4-manifold $X$ bounded by $S^3$ and a family $a_1,\cdots, a_{3n}$.
We define $$c_n=-\mu_n\Sign X+\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge _i \phi_i(\Gamma)^*W(\tau_{\RR^3}^*a_i)[\Gamma]\in \mathcal A_n(\emptyset).$$
\begin{defini}
$$\widetilde{z_n}^{\rm anomaly}(\vec\gamma)=-\mu_n\Sign X+\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge _i \phi_i(\Gamma)^*W(\gamma_i)[\Gamma]-c_n\in \mathcal A_n(\emptyset).$$
\end{defini}
\begin{remark}
We will show that $\mu_n=\frac{3}{2}c_n$ in Lemma~\ref{Lem74} and Lemma~\ref{Lem75}.
We can show that $\mu_1=72[\theta]\in \QQ[\theta]=\mathcal A_1(\emptyset)$ by explicit computation (cf. the proof of Proposition~\ref{rationalprop}).
\end{remark}
\subsection{Definition of the invariant.}\label{25}
\begin{theorem}\label{mainprop}
$$\widetilde z_n(Y)=\widetilde z_n(Y;\vec\gamma)-\widetilde z_n^{\rm anomaly}(\vec\gamma)\in \mathcal A_n(\emptyset)$$
does not depend on the choice of $\vec\gamma$. Thus $\widetilde z_n(Y)$ is a topological invariant of $Y$.
\end{theorem}
\begin{defini}
$$\widetilde z_n(Y)=\widetilde z_n(Y;\vec\gamma)-\widetilde z_n^{\rm anomaly}(\vec\gamma)\in \mathcal A_n(\emptyset).$$
\end{defini}
\subsection{Well-definedness of $\widetilde z_n(Y)$ (proof of Theorem~\ref{mainprop}).}\label{s46}
In this section we give the sketch of the proof of well-definedness of $\widetilde z_n(Y)$, i.e., Theorem~\ref{mainprop}.
The proof of well-definedness of $\widetilde z_n$ is almost parallel to that of $z^{\rm KKT}_n$ by Lescop \cite{Lescop}.
Fix $i\in\{1,\cdots,3n\}$.
For any $j\in \{1,\cdots, 3n\}$, let $a_j'$, $\gamma_j'$, $\beta_j'$, $\omega(\gamma_j')$ and $W(\gamma_j')$ be alternative choices of $a_j$, $\gamma_j$, $\beta_j$, $\omega(\gamma_j)$ and $W(\gamma_j)$ respectively.
Here $a_j'=a_j, \gamma'_j=\gamma_j$, $\omega(\gamma_j')=\omega(\gamma_j)$, $\beta'_j=\beta_j$ and $W(\omega'_j)=W(\omega_j)$ for $j\not=i$.
By the same argument of Proposition 2.15 in \cite{Lescop}, we have the following lemma.
\begin{lemma}
There exists a one-form $\eta_{S^2}\in \Omega^1(S^2)$ such that $d\eta_{S^2}=\omega^{a_j'}_{S^2}-\omega^{a_j}_{S^2}$,
and a one-form $\eta\in \Omega^1(C_2(Y))$ such that
\begin{itemize}
\item $d\eta=\omega(\gamma_i')-\omega(\gamma_i)$,
\item $\eta|_{\partial C_2(Y)\setminus S\nu_{\Delta(Y\setminus N(\infty;Y))}}=p_Y^*\eta_{S^2}$.
\end{itemize}
\end{lemma}
Similarly, the following lemma holds.
\begin{lemma}
There exists a one-form $\eta_X\in \Omega^1(ST^vX)$ such that
\begin{itemize}
\item $d\eta_{X}=W(\gamma_i')-W(\gamma_i)$,
\item $\eta_X|_{ST(Y\setminus N(\infty;Y))}=\eta|_{S\nu_{\Delta(Y\setminus N(\infty;Y))}}$,
\item $\eta_X|_{ST^vX|_{N(\infty;Y)}}=\tau_{S^3}^*\eta_{S^2}$.
\end{itemize}
\end{lemma}
\begin{proof}
Set $\eta_X^0=\eta|_{ST(Y\setminus N(\infty;Y))}\cup \tau_{S^3}^*\eta_{S^2}$.
By the construction of $c_0(\beta_i), c_0(\beta_i')$, we have $[W(\gamma_i)]=[W(\gamma_i')]\in H^2(ST^vX)$ (cf. Lemma~\ref{lemma1}).
Thus there is a one-form $\eta_X^1\in\Omega^1(ST^vY)$ such that $d\eta^1_X=W(\gamma_i)-W(\gamma_i')$.
Since $H^1(ST^vX)=0$, there is a function $\mu_X\in \Omega^0(ST^vY)$ such that $d\mu_X=\eta_X^1|_{ST^vY}-\eta_X^0$.
Let $h:ST^vX\to \RR$ be a $C^{\infty}$ function such that $h\equiv 1$ near $ST^vY(=\partial ST^vX)$ and $h\equiv 0$ far from $ST^vY$.
We can take $\eta_X=\eta_X^1-d(h\mu_X)$ using collar of $ST^vY$ in $ST^vX$.
\end{proof}
Set
\[ \widetilde\omega_j=
\begin{cases}
\omega(\gamma_j) (=\omega(\gamma_j'))& j\not=i,\\
\eta & j=i.
\end{cases} \]
Set
\[ \widetilde W_j=
\begin{cases}
W(\gamma_j) (=W(\gamma_j'))& j\not=i,\\
\eta_X & j=i.
\end{cases} \]
By Stokes' theorem,
\begin{eqnarray*}
&&\int_{C_{2n}(Y)}\bigwedge_jP_j(\Gamma)^*\omega(\gamma_j)-
\int_{C_{2n}(Y)}\bigwedge_jP_j(\Gamma)^*\omega(\gamma'_j)\\
&=&\int_{\partial C_{2n}(Y)} \bigwedge_jP_j(\Gamma)^*\widetilde\omega_j\\
&=&\sum_{F\subset \partial C_{2n}(Y): \mbox{face}}\int_F\bigwedge_jP_j(\Gamma)^*\widetilde\omega_j.
\end{eqnarray*}
\begin{lemma}[{Lescop \cite[Lemma 2.17]{Lescop}}]\label{lem417}
For any non-empty subset $B$ of $2n=\{1,\cdots, 2n\}$, for any $\Gamma\in\mathcal E_n$,
$$\int_{F(\infty;B)}\bigwedge_jP_j(\Gamma)^*\widetilde\omega_j=0.$$
\end{lemma}
\begin{lemma}[{Lescop \cite[Lemma 2.18, 2.19, 2.20, and 2.21]{Lescop}}]\label{lem418}
For any $B\subset \{1,\cdots, 2n\}$ with $\sharp B\ge 2$ and $B\not=\{1,\cdots 2n\}$
$$\sum_{\Gamma\in\mathcal E_n}\left(\int_{F(B)}\bigwedge_jP_j(\Gamma)^*\widetilde\omega_j\right)[\Gamma]=0.$$
\end{lemma}
The proofs of these two lemmas are completely same as the proof in \cite{Lescop}.
The following lemma is proved as Lemma 2.18, 2.19, 2.20, and 2.21 in \cite{Lescop}
(See also the proof of Proposition 2.10 in \cite{Lescop}).
\begin{lemma}\label{lem419}
For any $B\subset \{1,\cdots,2n\}$ with $2\le \sharp B<2n$,
\begin{itemize}
\item[{\rm (1)}] $\sum_{\Gamma\in\mathcal E_n}\int_{f(B)(T^vX)}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]=0$,
\item[{\rm (2)}] $\sum_{\Gamma\in\mathcal E_n}\int_{f(B)(T^vZ)}\bigwedge_j\phi_j(\Gamma)^*W(\widetilde\beta_j)[\Gamma]=0$
(See the proof of Lemma~\ref{lem48} for the notation $Z, W(\widetilde\beta_j)$).
\end{itemize}
\end{lemma}
By Lemma~\ref{lem417} and Lemma~\ref{lem418},
\begin{eqnarray*}
&&\widetilde z_n(Y;\vec \gamma)-\widetilde z_n(Y;\vec \gamma')\\
&=&\sum_{\Gamma\in\mathcal E_n}\left(\int_{C_{2n}(Y)}\bigwedge_jP_j(\Gamma)^*\omega(\gamma_j)\right)[\Gamma]
-\sum_{\Gamma\in\mathcal E_n}\left(\int_{C_{2n}(Y)}\bigwedge_jP_j(\Gamma)^*\omega(\gamma'_j)\right)[\Gamma]\\
&=&\sum_{\Gamma\in\mathcal E_n}\left(\int_{F(2n)}\bigwedge_jP_j(\Gamma)^*\widetilde\omega_j\right)[\Gamma].
\end{eqnarray*}
Since $F(2n)=\breve S(T(Y\setminus \infty))$, the restriction of $P_j(\Gamma)$ to $F(2n)$ coincides with $\phi_j^0(\Gamma):\breve S_{2n}(T(Y\setminus \infty))\to S\nu_{\Delta(Y\setminus \infty)}\subset \partial C_2(Y)$.
Therefore
\begin{eqnarray*}
&&\sum_{\Gamma\in\mathcal E_n}\int_{F(2n)}\bigwedge_jP_j(\Gamma)^*\widetilde\omega_j[\Gamma]\\
&=&\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(Y\setminus \infty))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]\\
&=&\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(Y\setminus N(\infty;Y)))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]
+\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(N(\infty;Y)\setminus\infty))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]\\
&=&\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(Y\setminus N(\infty;Y)))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma].
\end{eqnarray*}
The last equation comes from the following lemma.
\begin{lemma}\label{dimreason}
$\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(N(\infty;Y)\setminus\infty))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]=0$.
\end{lemma}
\begin{proof}
Since $\breve S_{2n}(T(N(\infty;Y)\setminus\infty))=(N(\infty;Y)\setminus\infty)\times \breve S_{2n}(\RR^3)$ and
$\widetilde\omega_j|_{ST(N(\infty;Y)\setminus \infty)}=\tau_{S^3}^*\omega_{S^2}$ (or $\tau_{S^3}^*\eta_{S^2}$),
the form $\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j|_{\breve S_{2n}(T(N(\infty;Y)\setminus \infty))}$ is in the image of the map $(\tau_{S^3})^{3n}\circ \prod_j\phi^0_j(\Gamma)$.
The map $(\tau_{S^3})^{3n}\circ \prod_j\phi^0_j(\Gamma)|_{\breve S_{2n}(T(N(\infty;Y)\setminus\infty))}:\breve S_{2n}(T(N(\infty;Y)\setminus\infty))\to (ST(N(\infty;Y)\setminus\infty))^{3n}\to (S^2)^{3n}$
factors through $\breve S_{2n}(\RR^3)$.
Since $\dim\breve S_{2n}(\RR^3)=6n-4<6n-1=\dim \bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j$,
we have $\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T^vY|_{N(\infty;Y)})}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]=0$.
\end{proof}
On the other hand, by Stokes' theorem,
\begin{eqnarray*}
&&\widetilde z_n^{\rm anomaly}(\vec\gamma)-\widetilde z_n^{\rm anomaly}(\vec\gamma')\\
&=& \sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vY)}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]
+\sum_{\Gamma\in\mathcal E_n}\int_{\partial S_{2n}(T^vX)}\bigwedge_j\phi_j(\Gamma)^*\widetilde{W}_j[\Gamma]\\
&\stackrel{(*)}{=}& \sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vY)}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]\\
&=& \sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T(Y\setminus N(\infty;Y))}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]+
\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vY|_{N(\infty;Y)})}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]\\
&=& \sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T(Y\setminus N(\infty;Y))}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma].
\end{eqnarray*}
The equation (*) is given by Lemma~\ref{lem419}{\rm (1)} and
the last equation comes from the following lemma.
\begin{lemma}
$\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vY|_{N(\infty;Y)})}\bigwedge_j\phi_j(\Gamma)^*\widetilde W(\gamma_j')[\Gamma]=0$.
\end{lemma}
The proof of this lemma is parallel to the proof of Lemma~\ref{dimreason}.
Since $\widetilde W_j|_{S\nu_{\Delta(Y\setminus N(\infty;Y))}}=\widetilde\omega_j|_{S\nu_{\Delta(Y\setminus N(\infty;Y))}}$
for any $j$, we have
$$\widetilde z_n(Y;\vec \gamma)-\widetilde z_n(Y;\vec \gamma')=
\sum_{\Gamma\in\mathcal E_n}\int_{\breve S_{2n}(T(Y\setminus N(\infty;Y)))}\bigwedge_j\phi^0_j(\Gamma)^*\widetilde\omega_j[\Gamma]~~~~~~~~~~$$
$$~~~~~~~~~~~~~~~=\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T(Y\setminus N(\infty;Y)))}\bigwedge_j\phi_j(\Gamma)^*\widetilde W_j[\Gamma]
=\widetilde z_n^{\rm anomaly}(\vec\gamma)-\widetilde z_n^{\rm anomaly}(\vec\gamma').$$
Now we finish the proof of Theorem~\ref{mainprop}.
\section{Review of $z_n^{\rm KKT}$.}\label{KKT}
In this section, we review the construction of $z_n^{\rm KKT}$ for rational homology 3-spheres.
This section is based on Lescop \cite{Lescop}.
Let $\tau_Y:T(Y\setminus \infty)\cong \underline{\RR^3}$ be a framing satisfying $\tau_Y|_{N(\infty;Y)\setminus\infty}=\tau_{\RR^3}$.
$\tau_Y|_{Y\setminus N(\infty;Y)}\cup \tau_{S^3}|_{N(\infty;S^3)}$ is a framing of $TY$ by the assumption of $\tau_Y$.
We define
$$\sigma_{Y\setminus\infty}(\tau_Y)=\sigma_Y(\tau_Y|_{Y\setminus N(\infty;Y)}\cup \tau_{S^3}|_{N(\infty;S^3)})-\sigma_{S^3}(\tau_{S^3})$$
$$~~~~=\sigma_Y(\tau_Y|_{Y\setminus N(\infty;Y)}\cup \tau_{S^3}|_{N(\infty;S^3)})-2$$
and call it the {\it signature defect} of $\tau_Y$ of a framing of
$Y\setminus \infty$. For example $\sigma_{\RR^3}(\tau_{\RR^3})=0$.
The canonical isomorphism $S\nu_{\Delta(Y\setminus \infty)}\cong T(Y\setminus \infty)$ and the framing $\tau_Y$ induce the map
$p_{\Delta}(\tau_Y):S\nu_{\Delta(Y\setminus \infty)}\to S^2$.
Since the assumption of $\tau_Y$, maps $p_{\Delta}(\tau_Y)$ and $p_Y:\partial C_2(Y)\setminus S\nu\to S^2$ are compatible.
So we get the map $p(\tau_Y)=p_Y\cup p_{\Delta}(\tau_Y):\partial C_2(Y)\to S^2$.
Let $\omega_{S^2}\in\Omega^2(S^2)$ be an anti-symmetric 2-form satisfying $\int_{S^2}\omega_{S^2}=1$.
Let $\omega(\tau_Y)$ be an anti-symmetric closed 2-from on $C_2(Y)$ satisfying
$\omega(\tau_Y)|_{\partial C_2(Y)}=p(\tau_Y)^*\omega_{S^2}\in \Omega^2(\partial C_2(Y))$.
\begin{prop}[{Lescop \cite[Theorem 1.9 and Proposition 2.11]{Lescop}}]
There exists constants $\delta_n\in\mathcal A_n(\emptyset)$ such that
$$\sum_{\Gamma\in \mathcal E_n}\int_{C_{2n}(Y)}
\left(\bigwedge_iP_i(\Gamma)^*\omega(\tau_Y)\right)[\Gamma]-\frac{\sigma_{Y\setminus \infty}(\tau_Y)}{4}\delta_n\in \mathcal A_n(\emptyset)$$
does not depend on the choice of $\tau_Y$.
\end{prop}
\begin{defini}[{Kuperberg and Thurston~\cite{KKT}, Lescop~\cite{Lescop}}]
$$z_n^{\rm KKT}(Y;\tau_Y)=\sum_{\Gamma\in \mathcal E_n}\int_{C_{2n}(Y)}
\left(\bigwedge_iP_i(\Gamma)^*\omega(\tau_Y)\right)[\Gamma],$$
$$z_n^{\rm KKT}(Y)=z_n^{\rm KKT}(Y;\tau_Y)-\frac{\sigma_{Y\setminus \infty}(\tau_Y)}{4}\delta_n\in \mathcal A_n(\emptyset).$$
\end{defini}
We remark that $\delta_n$ is given by explicit formula in Proposition 2.10 in \cite{Lescop}.
\begin{remark}
The universal finite type invariant $Z_n^{\rm KKT}$ described in \cite{Lescop} equals to the degree $n$ part of
$\exp(\sum_n \frac{1}{2^{3n}(3n)!(2n)!}z^{\rm KKT}_n)$.
See before Lemma 2.12 in \cite{Lescop} for more detail.
\end{remark}
\begin{remark}
We will show that $\delta_n=\frac{4}{3}\mu_n$ in Lemma~\ref{Lem74}.
\end{remark}
\section{Review of Watanabe's Morse homotopy invariants $z^{\rm FW}_n$.}\label{Morse}
In this section we give a modified construction of Watanabe's Morse homotopy invariant \cite{Watanabe} $z_{2n,3n}^{\rm FW}$ for rational homology 3-spheres.
We will remark the differences between our modified construction and Watanabe's original construction after the definition of $z_{2n,3n}^{\rm FW}(Y)$.
The invariant $z^{\rm FW}_{2n,3n}(Y)$ is a sum of the principal term $z_{2n,3n}^{\rm FW}(Y;\vec f)$ and the anomaly term
$z_{2n,3n}^{\rm anomaly}(\vec f)$ of $\vec f$
where $\vec f=(f_1,f_2,\cdots,f_{3n})$ is a family of Morse functions on $Y\setminus \infty$.
Fix a point $a\in S^2$.
\begin{defini}
A Morse function $f:Y\setminus \infty\to\RR$ is an {\it admissible Morse function with respect to} $a$ if it satisfies the following conditions.
\begin{itemize}
\item $f|_{N(\infty;Y)\setminus \infty}=q_a|_{N(\infty;S^3)\setminus \infty}$ and
\item $f$ has no critical point of index $0$ or $3$.
\end{itemize}
\end{defini}
Let $\Crit (f)=\{p_1,\cdots,p_k,q_1,\cdots,q_k\}$ be the set of critical points of $f$ where ${\rm ind}(p_i)=2,{\rm ind}(q_i)=1$.
Let $$0\to C_2(Y\setminus \infty;f)\stackrel{\partial}{\to} C_1(Y\setminus \infty;f)\to 0$$
be the Morse complex of $f$ with rational coefficients.
Let $g:C_1(Y\setminus \infty;f)\to C_2(Y\setminus \infty;f)$, $g([q_i])=\sum_jg_{ij}[p_j]$ be the inverse map of the boundary map
$\partial :C_2(Y\setminus \infty;f)\to C_1(Y\setminus \infty;f), \partial[p_i]=\sum_j\partial_{ij}[q_j]$.
($g$ is called a combinatorial propagator in \cite{Watanabe}.)
We now construct $\mathcal M(f)$ which is the weighted sum of (non-compact) 4-manifold in $Y^2\setminus \Delta$.
Let $M_{\to}(f)={\rm pr}(\varphi^{-1}(\Delta))$ where $\varphi: Y\times Y\times (0,\infty) \to Y\times Y$ is the map defined by $(x,y)\mapsto (y,\Phi^t_f(x))$
and ${\rm pr}:Y\times Y\times (0,\infty)\to Y\times Y$ is the projection.
We choose the orientation of $M_{\to}(f)$ such that the inclusion $Y\times (0,\ep)\hookrightarrow M_{\to}(f), (x,t)\mapsto (x,\Phi_f^t(x))$
preserves orientations.
We define $$\mathcal M(f)=M_{\to}(f)-\sum_{i,j}g_{ij}(\mathcal A_{q_i}\times \mathcal D_{p_j})\setminus \Delta.$$
We remark that the orientation of $\mathcal M(f)$ does not depend on the choice of orientations of $\mathcal A_{q_i},\mathcal D_{p_j}$.
Let $a_1,\cdots ,a_{3n}\in S^2\subset \RR^3$ be the points such that any different three points of them are linearly independent in $\RR^3$.
Let $f_i:Y\setminus \infty\to \RR$ be a sufficiently generic admissible Morse function with respect to $a_i$ for each $i=1,\cdots,3n$.
We write $\vec f=(f_1,\cdots,f_{3n})$ to simplify notation.
We replace a metric of $Y$ such that the Morse-Smale condition holds for each $f_i$ if necessary.
Set $\mathcal M(\pm f_i)=\mathcal M(f_i)+\mathcal M(-f_i)$.
\begin{defini}\label{defini:pt}
For generic $\vec f$,
$$z_{2n,3n}^{\rm FW}(Y;\vec f)=\sum_{\Gamma\in \mathcal E_n}
\frac{1}{2^{3n}}\sharp\left( \bigcap_{i=1}^{3n}P_i(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))
\right) [\Gamma]\in\mathcal A_n(\emptyset).$$
\end{defini}
We next define the anomaly part.
Set $\grad~\vec f=(\grad~f_1,\cdots,\grad~f_{3n})$.
\begin{defini}
$$z_{2n,3n}^{\rm anomaly}(\vec f)=\tilde z^{\rm anomaly}_n(\grad~\vec f).$$
\end{defini}
\begin{defini}[{Watanabe~\cite{Watanabe}}]
$$z^{\rm FW}_{2n,3n}(Y)=z^{\rm FW}_{2n,3n}(Y;\vec f)-z_{2n,3n}^{\rm anomaly}(\vec f).$$
\end{defini}
\begin{remark}\label{differ}
A difference between our modified construction of $z_{2n,3n}^{\rm FW}$ and
Watanabe's original construction in \cite{Watanabe} is the conditions for Morse functions.
Our Morse function is on $Y\setminus \infty$ and explicitly written on $N(\infty;Y)\setminus \infty$.
On the other hand, Watanabe uses any Morse functions on $Y$.
We note that $Y\setminus\infty\subset Y\sharp S^3$ where $Y\sharp S^3$ is the connected sum of $Y$ and $S^3$ at $\infty\in Y$ and $0\in S^3$.
Then it is possible to extend $f:Y\setminus\infty\to \RR$ to $Y\sharp S^3\cong Y$ in standard way.
\begin{figure}[h]
\begin{center}
\includegraphics[width=240pt]{PICn2.eps}
\caption{The extension of $f$ to $Y\sharp S^3$}
\end{center}
\end{figure}
Then we can show that the difference between the value $z_{2n,3n}^{\rm FW}(Y)$ described in this Section and
the value of Watanabe's original invariant of $Y$ is a constant which is independent of $Y$.
\end{remark}
We must prove that $\sharp\left( \bigcap_{i}P_i(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))
\right)$ is well defined for generic $\vec f$, because Morse functions used in the above definition differ from Morse functions used in the original definition in \cite{Watanabe} near $N(\infty;Y)\setminus\infty$ (See Remark~\ref{differ} for more details).
\begin{lemma}
$P_1(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))$
, $\cdots, P_{3n}(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))$
transversally intersect at finitely many points, for generic $f_1,\cdots,f_{3n}$ and $a_1,\cdots, a_{3n}$, for any $\Gamma\in \mathcal E_n$.
\end{lemma}
\begin{proof}
Let $x=(x_1,\cdots,x_{2n})\in \bigcap_{i}P_i(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))
\subset (Y\setminus \infty)^{2n}\setminus \Delta$.
\vskip2mm
\underline{The case of $x\in (Y\setminus N(\infty;Y))^{2n}$.}\\
Thanks to \S 2.4 of \cite{Watanabe}, the transversality at $x$ is given by generic $\vec f$ .
\vskip2mm
\underline{The case of $x\not\in (Y\setminus N(\infty;Y))^{2n}$.}\\
We show that for generic $a_1,\cdots, a_{3n}$, there are no such $x$. (Then, in particular, $\bigcap_{i}P_i(\Gamma)|_{(Y\setminus \infty)^{2n}\setminus \Delta}^{-1}(\mathcal M(\pm f_i))$ is a $0$-dimensional compact manifold).
Let $B=\{ i\in \{1,\cdots,2n\}\mid x_i\in Y\setminus N(\infty;Y)\}$. Let
$$E_B=\{i\in \{1,\cdots,3n\}\cong E(\Gamma)\mid \{s(i),t(i)\}\subset B\},$$
$$E^{\partial}_B=\{i\in \{1,\cdots,3n\}\cong E(\Gamma)\mid \{s(i),t(i)\}\cap B\not=\emptyset\}\setminus E_B.$$
Let $\Gamma/B$ be the labelled graph obtained from $\Gamma$ by collapsing $B$ to a point $b_0$ and removing all edges in $E_B$.
Here the label of edges and vertices of $\Gamma/B$ are $\{1,\cdots, 3n\}\setminus E_B$, $\{0,1,\cdots,2n\}\setminus B$ respectively (the label of $b_0$ is 0).
Note that $\sharp(V(\Gamma/B)-\{b_0\})=2n-\sharp B$ and $\sharp E(\Gamma/B)\geq 3n-\frac{3\sharp B}{2}$.
Let $\pi:Y\setminus \infty\to Y/(Y\setminus N(\infty;Y))\stackrel{\tau_{\infty}}{=}\RR^3$ be the map obtained by collapsing $Y\setminus N(\infty;Y)$ to the point $0\in\RR^3$.
Let $\pi'_i:\RR\to \RR$ be the map obtained by collapsing ${\rm Im}(f_i:Y\setminus N(\infty;Y)\to \RR)$ to $0\in\RR$.
Then $\pi'_i\circ f_i=q_{a_i}\circ\pi:Y\setminus\infty\to \RR$.
Let $x':V(\Gamma/B)-\{b_0\}\hookrightarrow \RR^3$ be the restriction of $\pi\circ x:V(\Gamma)\hookrightarrow \RR^3$ to $V(\Gamma/B)-\{b_0\}\subset V(\Gamma)$.
Let $a'\in (S^2)^{E(\Gamma/B)}$ be the points obtained from $a=(a_1,\cdots,a_{3n})$ removing all $a_i$, $i\in E_B$.
We define the map
$$\varphi:(\RR^3)^{V(\Gamma/B)-\{b_0\}}\setminus\Delta\to (S^2)^{E(\Gamma/B)}$$
as $$\varphi(y)=\left(\frac{y_{s(i)}-y_{t(i)}}{\| y_{s(i)}-y_{t(i)}\|}\right)_{i\in E(\Gamma/B)}.$$
Here if $i\in E^{\partial}_B$ then either $s(i)$ or $t(i)$ is $0$.
Then $x'\in \varphi^{-1}(a')$.
By the following lemma, there is no $x'$ for a generic $a'$.
Therefore there is no $x$ for a generic $a$.
\begin{lemma}
For a generic $a'$ we have $\varphi^{-1}(a')=\emptyset$.
\end{lemma}
\begin{proof}
For any $y\in\varphi^{-1}(a')$ and for any $t\in (0,\infty)$, we have $ty\in\varphi^{-1}(a')$.
Thus if $\varphi^{^1}(a')\not=\emptyset$, we have $\dim \varphi^{-1}(a')\ge 1$.
On the other hand, $\dim ((\RR^3)^{V(\Gamma/B)-\{b_0\}})=6n-3\sharp B\le 2\sharp E(\Gamma/B)=\dim((S^2)^{E(\Gamma/B)})$.
Hence we have $\dim \varphi^{-1}(a')\le 0$ for a generic $a'$. This is contradiction.
\end{proof}
\end{proof}
\section{Proof of Theorem~\ref{theorem1}.}\label{proof}
In this section we prove Theorem~\ref{theorem1} in Section~\ref{intro}.
\subsection{Proof of $\widetilde{z_n}(Y)=z_n^{\rm KKT}(Y)$.}
We follow the notations used in Section~\ref{KKT}. For example, $Y$ is a rational homology 3-sphere and $\infty\in Y$ is a base point, and so on.
Let $\tau_Y:T(Y\setminus \infty)\cong \underline{\RR^3}$ be a framing of $Y\setminus \infty$ satisfying $\tau_Y|_{N(\infty;Y)\setminus \infty}=\tau_{\RR^3}|_{N(\infty;S^3)\setminus \infty}$.
We denote $\tau_Y^*\vec a=(\tau_Y^*a,\cdots,\tau_Y^*a)$ for $a\in S^2$.
We take $\omega_{S^2}=\frac{1}{2}\omega_{S^2}^a$ in the definition of $z_n^{\rm KKT}(Y;\tau_Y)$,
and we take $\omega(\tau_Y^*a)=\omega(\tau_Y)$ in the definition of $\widetilde z_n(Y;\tau_Y^*\vec a)$.
Thus
$$\widetilde z_n(Y;\tau_Y^*\vec a)=\sum_{\Gamma\in\mathcal E_n}\int_{C_{2n}(Y)}\bigwedge_iP_i(\Gamma)^*\omega(\tau_Y)[\Gamma]
=z_n^{\rm KKT}(Y;\tau_Y).$$
Then we only need show that
$$\widetilde z_n^{\rm anomaly}(Y;\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n$$
in this condition.
The idea of the proof of $\widetilde z_n^{\rm anomaly}(Y;\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n$
is as follows. We first prove this equation in the case of $Y=S^3$.
The well-definedness of $\widetilde z_n^{\rm anomaly}(Y)$ implies that $\widetilde z_n^{\rm anomaly}(S^3;\tau^*\vec a)=\frac{1}{4}\sigma_{\RR^3}(\tau)\delta_n$ for any framing $\tau$ of $S^3\setminus\infty$.
The general case is reduced to the case of $Y=S^3$ by a cobordism argument.
We introduce notation.
For a compact 4-manifold $X$ such that $\partial X=Y$ and $\chi(X)=0$,
we denote $\widetilde z^{\rm anomaly}(\vec\gamma;X)=\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^vX)}\bigwedge_i\phi_i(\Gamma)^*W(\gamma_i)[\Gamma]=\widetilde z_n^{\rm anomaly}(\vec \gamma)+\mu_n\Sign X +c_n$.
Then $\widetilde z^{\rm anomaly}(\vec\gamma)=\widetilde z^{\rm anomaly}(\vec\gamma;X)-\mu_n\Sign X-c_n$ by the definition.
\begin{lemma}
$\widetilde z_n(S^3)=z^{\rm KKT}(S^3)$.
\end{lemma}
\begin{proof}
Let $X$ be a compact 4-manifold with $\partial X=S^3$ and $\chi(X)=0$.
\begin{eqnarray*}
\widetilde z_n(S^3)&=&\widetilde z_n(S^3;\tau_{\RR^3}^*\vec a)-\widetilde z_n^{\rm anomaly}(\tau_{\RR^3}^*\vec a;X)+\mu_n\Sign X+c_n\\
&=&\widetilde z_n(S^3;\tau_{\RR^3}^*\vec a)\\
&=&z_n^{\rm KKT}(S^3;\tau_{\RR^3}).
\end{eqnarray*}
Since $\sigma_{\RR^3}(\tau_{\RR^3})=0$,
we have $z_n^{\rm KKT}(S^3;\tau_{\RR^3})=z_n^{\rm KKT}(S^3)$.
Therefore $\widetilde z_n(S^3)=z_n^{\rm KKT}(S^3;\tau_{\RR^3})=z_n^{\rm KKT}(S^3)$.
\end{proof}
Since $\widetilde z_n^{\rm anomaly}(S^3)$ is independent of the choice of framing on $\RR^3=S^3\setminus \infty$,
we have the following corollary.
\begin{cor}\label{Lem711}
For any framing $\tau$ on $\RR^3=S^3\setminus\infty$ such that $\tau|_{N(\infty;S^3)\setminus \infty}=\tau_{\RR^3}|_{N(\infty;S^3)\setminus \infty}$,
the equation $\widetilde z_n^{\rm anomaly}(S^3;\tau^*\vec a)=\frac{1}{4}\sigma_{\RR^3}(\tau)\delta_n$ holds.
\end{cor}
Recall that the framing $\tau_Y$ of $T(Y\setminus \infty)$ gives the framing $\tau_Y\cup \tau_{S^3}=\tau_Y|_{Y\setminus N(\infty;Y)}\cup \tau_{S^3}|_{N(\infty;S^3)}$ of $TY$ and $\sigma_{Y\setminus\infty}(\tau_Y)=\sigma_{Y}(\tau_Y\cup\tau_{S^3})-\sigma(\tau_{S^3})
=\sigma_{Y}(\tau_Y\cup \tau_{S^3})-2$.
We give the spin structure on $Y$ using $\tau_Y\cup \tau_{S^3}$.
\begin{lemma}
There exists a positive integer $k$ and a spin 4-manifold $X_0$ such that $\chi(X_0)=0$ and $\partial X_0=Y\sqcup k(-S^3)$ as spin manifolds.
Here $-S^3$ is $S^3$ with the opposite orientation.
\end{lemma}
\begin{proof}
Since the 3-dimensional spin cobordism group equals to zero, there exists a spin 4-manifold $\widetilde X$ such that $\partial \widetilde X=Y$.
Let $k=\chi(\widetilde X)$.
We may assume that $k\ge 0$, by replacing $\widetilde X$ by $\widetilde X\sharp nK3$ for sufficiently large integer $n$ if necessary.
Let $X_0$ be the spin 4-manifold obtained by removing $k$ disjoint 4-balls, i.e., $X_0=\widetilde X\setminus kB^4$.
Then $\chi (X_0)=0$ and $\partial X_0=Y\sqcup k(-S^3)$.
\end{proof}
\begin{remark}
Since $\chi(X_0\sharp T^4)=\chi(X_0)-2, \chi(X_0\sharp K3)=\chi(X_0)+22$ and $T^4$, $K3$ are spin,
it is possible to choose $k+2n$ instead of $k$ for any $n\in\ZZ$.
\end{remark}
\begin{remark}
Since the Euler number of a closed spin 4-manifold is even, the number $k(Y)=k \mod 2\in\ZZ/2$ is an invariant of a spin 3-manifold $Y$.
It is known that $k(Y)={\rm rk} H_1(Y;\ZZ/2)+1$ (See Theorem 2.6 in \cite{KM}).
We also remark that $k(Y)\equiv \sigma_{Y\setminus\infty}(\tau_Y)+1 \mod 2$.
\end{remark}
Let $X_0$ be a spin 4-manifold such that $\chi(X_0)=0$ and $\partial X_0=Y\sqcup k(-S^3)$ for some $k\ge 1$.
We denote $S^3_i$ the $i$-th $S^3$-boundary of $X_0$.
Then $\partial X_0=Y\sqcup -S^3_1\sqcup\cdots\sqcup -S^3_k$.
By the obstruction theory, it is possible to extend the framing $\eta_Y\oplus(\tau_Y\cup\tau_{S^3})$ of $TX_0|_{Y}$ to $X_0$
where $\eta_Y$ is the outward unit vector field on $Y\subset \partial X_0$
(see \cite{KM} for more details).
We choose such a extended framing $\widetilde\tau_X$ such that $\widetilde\tau_X^*\null^t(1,0,0,0)|_{k(-S^3)}$ is the inward unit vector field on $k(-S^3)\subset \partial X_0\subset X_0$.
If necessary we modify $\widetilde\tau_X$ by using homotopy, we may assume that there exists a framing $\tau_i$ of $S^3_i\setminus \infty$
such that $\tau_i|_{N(\infty;S^3_i)\setminus \infty}=\tau_{\RR^3}|_{N(\infty;S^3)\setminus \infty}$ and $-\eta_i\oplus(\tau_i\cup \tau_{S^3})=\widetilde\tau_X|_{-S^3_i}$. Here $-\eta_i$ is the inward unit vector field on $-S^3_i\subset X_0$.
Let $X'$ be a compact oriented 4-manifold with $\chi(X')=0$ and $\partial X'=S^3$.
Then $X_0\cup kX'$ is a compact 4-manifold with $\chi(X_0\cup kX')=0$ and $\partial(X_0\cup kX')=Y$.
\begin{lemma}\label{Lem73}
The following three equations hold.
\begin{itemize}
\item[{\rm (1)}] $\widetilde z^{\rm anomaly}_n(\tau_Y^*\vec a;X_0\cup kX')
=\sum_{i=1}^k\widetilde z^{\rm anomaly}_n(\tau_i^*\vec a;X')$.
\item[{\rm (2)}] $\sigma_{Y\setminus \infty}(\tau_Y)=\sum_{i=1}^k\sigma_{\RR^3}(\tau_i)+2(k-1)-3\Sign X_0$.
\item[{\rm (3)}] $\widetilde z^{\rm anomaly}_n(\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n+\left(\frac{3}{4}\delta_n-\mu_n\right)\Sign X_0+\frac{k-1}{2}\delta_n+(k-1)c_n$.
\end{itemize}
\end{lemma}
\begin{proof}
{\rm (1)} We take a 3-bundle $T^v(X_0\sqcup kX')\subset T(X_0\sqcup kX')$ over $X_0\cup kX'$ such that $T^v(X_0\sqcup kX')|_{X_0}$ is the normal bundle
of $\widetilde\tau_X^*\null^t(1,0,0,0)$.
We denote $T^vX_0=T^v(X_0\sqcup kX')|_{X_0}$, $T^v(kX')=T^v(X_0\sqcup kX')|_{kX'}$.
Let $\beta$ be a section of $T^v(X_0\sqcup kX')$ such that $\beta|_{X_0}=\widetilde\tau_X^*a$ and $\beta$ is transverse to the zero section in $T^v(X_0\sqcup kX')$.
In this setting, we can take $W(\tau_Y^*a)|_{ST^vX_0}=\widetilde\tau_X^*\omega_{S^2}$.
Then $\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a;X_0\sqcup kX')=
\sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^v(X_0\sqcup kX'))}\bigwedge_i \phi_i(\Gamma)^*W(\tau_Y^*a)[\Gamma]
=\sum_{\Gamma}\int_{S_{2n}(T^vX_0)}\bigwedge_i\phi_i(\Gamma)^*\widetilde\tau_X^*\omega_{S^2}[\Gamma]\\
+\sum_{\Gamma}\int_{S_{2n}(T^v(kX'))}\bigwedge_i \phi_i(\Gamma)^*W(\tau_Y^*a)[\Gamma]$.
We show that $\int_{S_{2n}(T^vX_0)}\bigwedge_i \phi_i(\Gamma)^*\widetilde\tau_X^*\omega_{S^2}=0$ for any $\Gamma\in\mathcal E_n$.
The map $(\widetilde\tau_X)^{3n}\circ(\prod_i \phi_i(\Gamma)):S_{2n}(T^vX_0)\to (S^2)^{3n}$ factors through
$S_{2n}(\RR^3)$:
\[\xymatrix{S_{2n}(T^vX_0) \ar[d]_{\widetilde\tau_X} \ar[r]^(0.5){\prod_i\phi_i(\Gamma)} \ar@{}[dr]^(0.4)\circlearrowleft & (ST^vX_0)^{3n} \ar[d]_{(\widetilde\tau_X)^{3n}} & \\
S_{2n}(\RR^3) \ar[r] & (S^2)^{3n}. &\\
}\]\
Hence we have $\bigwedge_i\phi_i(\Gamma)^*\widetilde\tau_X^*\omega_{S^2}|_{ST^vX_0}=((\prod \widetilde\tau_X)^{3n}\circ \bigwedge_i\phi_i(\Gamma))^*(\omega_{S^2})^{3n}\\
\in {\rm Im}(\Omega^{6n}(S_{2n}(\RR^3))\to\Omega^{6n}(S_{2n}(T^vX_0)))$.
Since $\dim \breve S_{2n}(\RR^3)=6n-4<6n=\dim \bigwedge_i\phi_i(\Gamma)^*\widetilde\tau_X^*\omega_{S^2}$, we have
$\bigwedge_i \phi_i(\Gamma)^*\widetilde\tau_X^*\omega_{S^2}=0$.
Therefore
\begin{eqnarray*}
\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a;X_0\sqcup kX')
&=& \sum_{\Gamma\in\mathcal E_n}\int_{S_{2n}(T^v kX')}\bigwedge_i \phi_i(\Gamma)^*W(\tau_Y^*a)[\Gamma]\\
&=&\sum_{i=1}^k\widetilde z_n^{\rm anomaly}(\tau_i^*\vec a;X').
\end{eqnarray*}
{\rm (2)} By the obstruction theory and the definition of the signature defect, we have
$\sigma_Y(\tau_Y\cup\tau_{S^3})+3\Sign X_0=\sum_{i=1}^k\sigma_{S^3}(\tau_i\cup \tau_{S^3})$.
Since $\sigma_{Y\setminus \infty}(\tau_Y)=\sigma_Y(\tau_Y\cup\tau_{S^3})-2$ and
$\sigma_{\RR^3}(\tau_i)=\sigma_{S^3}(\tau_i\cup\tau_{S^3})-2$, the equation $(2)$ holds.
{\rm (3)}
\begin{eqnarray*}
\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a)&&=
\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a;X_0\sqcup kX')-\mu_n\Sign (X_0\sqcup kX')-c_n\\
&&\stackrel{{\rm (1)}}{=}\widetilde z_n^{\rm anomaly}(\tau_1^*\vec a;X')-\mu_n\Sign X'-c_n\\
&&+\cdots+\widetilde z_n^{\rm anomaly}(\tau_k^*\vec a;X')-\mu_n\Sign X'-c_n\\
&&-\mu_n\Sign X_0+(k-1)c_n\\
&&=\sum_i\widetilde z_n^{\rm anomaly}(\tau_i^*\vec a)-\mu_n\Sign X_0+(k-1)c_n\\
&&\stackrel{{\rm Corollary}~\ref{Lem711}}{=}\sum_i\frac{1}{4}\sigma_{\RR^3}(\tau_i)\delta_n-\mu_n\Sign X_0+(k-1)c_n\\
&&\stackrel{{\rm (2)}}{=}\frac{1}{4}(\sigma_{Y\setminus\infty}(\tau_Y)-2(k-1)+3\Sign X_0)\delta_n-\mu_n\Sign X_0+(k-1)c_n.
\end{eqnarray*}
\end{proof}
We next compute $\mu_n,c_n$ and prove that $\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus\infty}(\tau_Y)\delta_n$ by using the above lemma.
\begin{lemma}\label{Lem74}
$\mu_n=\frac{3}{4}\delta_n$.
\end{lemma}
\begin{proof}
Let $X_0=K3\sharp 11T^4\setminus (B^4\sqcup B^4)$.
Then $X_0$ is a spin 4-manifold satisfying $\chi(X_0)=0$ and $\Sign X_0=16$.
It is possible to deal with $\partial X_0=S^3\sqcup -S^3$.
By Lemma~\ref{Lem73} (3), we have
$0=\widetilde z_n^{\rm anomaly}(\tau_{\RR^3}^*\vec a)
=(\frac{3}{4}\delta_n-\mu_n)\Sign X_0$.
Since $\Sign X_0=16\not=0$, we have $\mu_n=\frac{3}{4}\delta_n$.
\end{proof}
\begin{lemma}\label{Lem75}
$c_n=\frac{1}{2}\delta_n$.
\end{lemma}
\begin{proof}
Let $X_0=K3\sharp 10T^4\setminus (B^4\sqcup 3B^4)$.
Then $X_0$ is a spin 4-manifold satisfying $\chi(X_0)=0$ and $\Sign X_0=16$.
It is possible to deal with $\partial X_0=S^3\sqcup 3(-S^3)$.
By Lemma~\ref{Lem73} (3) and Lemma~\ref{Lem74}, we have
$0=\widetilde z_n^{\rm anomaly}(\tau_{\RR^3}^*\vec a)
=-\delta_n+2c_n$.
Then $c_n=\frac{1}{2}\delta_n$.
\end{proof}
\begin{prop}\label{77}
$\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n.$
\end{prop}
\begin{proof}
Take $X_0, k, \widetilde\tau_X$ as in Lemma~\ref{Lem73}.
By Lemma~\ref{Lem73} (3), Lemma~\ref{Lem74} and Lemma~\ref{Lem75}, we have
$\widetilde z_n^{\rm anomaly}(\tau_Y^*\vec a)=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n-\frac{k-1}{2}\delta_n+(k-1)c_n
=\frac{1}{4}\sigma_{Y\setminus \infty}(\tau_Y)\delta_n$.
\end{proof}
\subsection{Proof of $\widetilde z_n(Y)=z^{\rm FW}_{2n,3n}(Y)$.}
Let $f$ be an admissible Morse function with respect to $a\in S^2$.
The weighted sum $\mathcal M(f)+\mathcal M(-f)$ consists of weighted pairs of two distinct points on a gradient trajectory.
There is a compactification $\mathcal M_S(\pm f)$ of $\mathcal M(f)+\mathcal M(-f)$ by adding pairs of points on broken trajectories as the Morse theory. Then $\mathcal M_S(\pm f)$ becomes a 4-cycle in $(C_2(Y),\partial C_2(Y))$ (Lemma~\ref{64}).
See for Section~\ref{sect6} for the detail of the above argument.
\begin{lemma}
$\partial \mathcal M_S(\pm f)=c(\grad f)$ for any admissible Morse function $f$.
\end{lemma}
\begin{proof}
Since $\grad f|_{N(\infty;Y)}=\grad q_a$,
if $(x,u)\in \partial \mathcal M_S(\pm f)\cap((Y\setminus \infty)\times ST_{\infty}Y)$ then $u=\pm a$.
On the other hand, $\partial\mathcal M_S(\pm f)\cap(\{x\}\times ST_{\infty}Y)=\{(x,a),(x,-a)\}$ for any $x\not\in \Crit(f)$.
Since $\partial \mathcal M_S(\pm f)$ is a 3-cycle, we have $\partial \mathcal M_S(f)\cap((Y\setminus \infty)\times ST_{\infty}Y)=(Y\setminus\infty)\times (\pm a)$.
With a similar argument, we have $\partial C_2(Y)\setminus S\nu_{\Delta(Y\setminus \infty)}=p_Y^{-1}(\pm a)$.
Since this fact and Lemma~\ref{65} we conclude the proof.
\end{proof}
We follow the notations $a_1,\cdots, a_{3n}$, $f_1,\cdots, f_{3n}$ as in Section~\ref{Morse}.
In the following proposition, the notion "generic $\vec f$" means that $\bigcap_iP_i(\Gamma)^{-1}\mathcal M_S(\pm f_i)=\emptyset$ for any $\Gamma\in\mathcal E_n$.
We remark that there exists such a $\vec f$ (See Remark~\ref{rmk79}).
\begin{prop}
For generic $\vec f$,
$z_{2n,3n}^{\rm FW}(Y;\vec f)=
\widetilde z_n(Y;\grad \vec f)$.
\end{prop}
\begin{proof}
We define the 2-cocycle $\omega^s_i(\grad~f_i)\in S^2(|T_{C_2(Y)}|)$ by
$\omega^s(\grad~f_i)(\sigma)=\frac{1}{2}\sharp(\sigma\cap \mathcal M_S(f_i))$ for each 2-cycle $\sigma$ of $T_{C_2(Y)}$.
By the construction, $\omega^s(\grad~f_i)$ is simplicial propagator for each $i$.
By the intersection theory and Lemma~\ref{alt}, we have
$$z^{\rm FW}_{2n,3n}(Y;\vec f)=\langle \bigwedge_i P_i(\Gamma)^*\omega^s(\grad~f_i),[C_{2n}(Y),\partial C_{2n}(Y)]\rangle
=\frac{1}{2^{3n}}\sharp \left(\bigcap_i P_i(\Gamma)^{-1}\mathcal M_S(f_i)\right) $$
for any $\Gamma\in\mathcal E_n$.
\end{proof}
\begin{remark}\label{rmk79}
We can show that $\partial C_{2n}(Y)\cap (\bigcap_iP_i(\Gamma)^{-1}\mathcal M_S(\pm f_i))=\emptyset$ for generic $\vec f$ by an argument similar to Lemma 2.7 in Watanabe \cite{Watanabe}.
For example,
we take the following $\Phi'_{\Gamma}$ instead of $\Phi$ in Lemma 2.7 in \cite{Watanabe} when we prove $F(\{1,2,4\})\cap(\bigcap_{i=1}^6P_i({\rm Smooth}(\Gamma))^{-1}\mathcal M_S(\pm f_i))=\emptyset$ for the graph $\Gamma$ in the picture (2.2) in \cite{Watanabe} (See Example 2.6 in \cite{Watanabe} and see \S 3.4 of \cite{Watanabe} for the definition of the operator $\rm Smooth$).
$$\phi'_{\Gamma}:F(\{1,2,4\})\times \left(\bigcup_{f_1\in\mathcal U_1} \mathcal A_p(f_1)\cap \mathcal D_q(f_1)\right) \times (\RR_{>0})^3\times \prod_{i=2}^4\mathcal U_i~~~~~~~~~~~~~~$$
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~\to Y^3\times (TY)^2\times (TY)^2\times Y^3,$$
\begin{eqnarray*}
&&\Phi'_{\Gamma}(((x_1,[w_1,w_2,w_4]), x_3),u,t_2,t_3,t_4,f_2,f_3,f_4)\\
&&=((x_1,u,\Phi_{f_6}^{t_6}(x_3)), (\grad_{x_1}f_2, \frac{w_2-w_1}{\|w_2-w_1\|}),\\
&&~~(\grad_{x_1}f_3, \frac{w_4-w_2}{\|w_4-w_2\|}), (x_3,\Phi_{f_4}^{t_4}(x_1),\Phi_{f_5}^{t_5}(x_1))).
\end{eqnarray*}
Here $x_1\in Y\setminus \infty$, $[w_1,w_2,w_3]\in \breve S_{\{1,2,4\}}T_{x_1}Y$, $x_3\in Y\setminus \{x_1,\infty\}$.
Let
$$\Delta'_{\Gamma}=\{((y_1,y_1,y_1), ((y_2,s_2v_2),(y_2,t_2v_2)), ((y_3,s_3v_3),(y_3,t_3v_3)),(y_4,y_4,y_4))$$
$$~~~~~\mid (y_1,y_2,y_3,y_4)\in (Y\setminus \infty)^4, t_i,s_i\ge 0, v_i\in T_{y_i}Y\}.$$
Then $\Phi'_{\Gamma}$ is transverse to $\Delta'_{\Gamma}$ as Lemma~2.7 in \cite{Watanabe}.
\end{remark}
It is obvious that
$z_{2n,3n}^{\rm anomaly}(Y;\vec f)=\widetilde z_n^{\rm anomaly}(Y;\grad\vec f)$ by the definitions of the anomaly parts.
\section{Compactification of moduli space $\mathcal M(f)$}\label{sect6}
In this section we give a compactification $\mathcal M_S(\pm f)$ of
$\mathcal M(f)\cup \mathcal M(-f)$ and then show that $\mathcal M_S(\pm f)$ is a 4-cycle in $(C_2(Y),\partial C_2(Y))$.
Let $M_{\to}(f)=\varphi^{-1}|_{Y^2\times (0,\infty)}(\Delta)$ where $\varphi:Y^2\times (-\infty,\infty)\to Y^2, (x,y)\mapsto (y,\Phi_f^t(x))$.
\begin{lemma}[{Watanabe \cite[Proposition 2.12]{Watanabe} (cf. \cite{BH})}]\label{Lem71}
There is a manifold with corners $\overline{M}_{\to}(f)$ satisfying the following conditions.
\begin{itemize}
\item[{\rm (1)}] $\overline{M}_{\to}(f)=\{g:I\to Y\mid I\subset -\RR,\\
g~\mbox{\rm is a piecewise smooth map}, f(g(t))=t, \frac{dg(t)}{dt}=\frac{\grad_{g(t)} f}{\|\grad_{g(t)}f\|^2}~\mbox{\rm for any}~ t \}$ as sets,
\item[{\rm (2)}] ${\rm int}\overline M_{\to}(f)=M_{\to}(f)$, and
\item[{\rm (3)}] $\partial \overline M_{\to}(f)=\sum_{i}\mathcal A_{p_i}\times \mathcal D_{p_i}+\sum_{j}\mathcal A_{q_j}\times \mathcal D_{q_j}$.
\end{itemize}
\end{lemma}
Note that ${\rm int}(\overline M_{\to}(f)+\overline M_{\to}(-f))=\varphi^{-1}(\Delta)$.
We denote by $\overline M_{\to}(f)\to (Y\setminus \infty)^2$ the continuous map that is the extension of the embedding
$M_{\to}(f)\to (Y\setminus \infty)^2$ to $\overline M_{\to}(f)$.
For simplicity of notation, we write $\overline M_{\to}(f)$ instead of $\overline{M}_{\to}(f)\to (Y\setminus\infty)^2$.
Similarly we denote by $\overline{\mathcal A_{p_i}}\to Y$ the extension of $B^1(1)\cong \mathcal A_{p_i}\to Y$ to $\overline{B^1(1)}$
and we write $\overline{\mathcal A_{p_i}}$ instead of $\overline{\mathcal A_{p_i}}\to Y$
(We remark that $\mathcal A_{p_i}$ is diffeomorphic to $B^1(1)$ the interior of unit disk in $\RR^1$).
We also define $\overline{\mathcal D_{p_i}}, \overline{\mathcal A_{q_j}}$, and so on.
\begin{lemma}
\begin{itemize}
\item[{\rm (1)}] $\overline{M}_{\to}(f)+\overline{M}_{\to}(-f)$ is transverse to $\Delta$.
\item[{\rm (2)}] $\overline{\mathcal A_{q_j}}\times \overline{\mathcal D_{p_i}}$ is transverse to $\Delta$.
\end{itemize}
\end{lemma}
\begin{proof}
{\rm (1)}
$\grad f$(which is the section of $\nu_{\Delta(Y\setminus \infty)}$) is transverse to the zero section in $\nu_{\Delta(Y\setminus \infty)}$.
$\mathcal A_p\times \mathcal D_p\subset Y^2$ is transverse to $\Delta$ for any critical point $p\in \Crit(f)=\Crit(-f)$.
Thanks to Lemma~\ref{Lem71}~{\rm (2),(3)}, this finishes the proof of {\rm (1)}.
{\rm (2)} is immediate from the Morse-Smale condition.
\end{proof}
By this Lemma, $ (\overline{M}_{\to}(f)+\overline{M}_{\to}(-f))(\Delta)$ and
$(\overline{\mathcal A_{q_j}}\times \overline{\mathcal D_{p_i}})(\Delta)$ are well-defined.
It is clear that $ (\overline{M}_{\to}(f)+\overline{M}_{\to}(-f))(\Delta)=\\
(\overline{M}_{\to}(f)+\overline{M}_{\to}(-f))\setminus \Delta\cup\{(x,\frac{\pm\grad_xf}{\|\grad_xf\|})\mid x\in Y\setminus (\infty\cup \Crit(f))\}$ by the construction.
\begin{defini}
$\mathcal M_S^0(\pm f)= (\overline{M}_{\to}(f)+\overline{M}_{\to}(-f))(\Delta)
+\sum_{i,j}g_{ij}(\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{p_j}})(\Delta)
+\sum_{i,j}(-g_{ij})(\overline{\mathcal D_{p_j}}\times \overline{\mathcal A_{q_i}})(\Delta).$
\end{defini}
Let $\mathcal M_S(\pm f)$ be the extension of $\mathcal M_S^0(\pm f)$ to $C_2(Y)$.
\begin{lemma}\label{64}
$\mathcal M_S(\pm f)$ is a 4-cycle in $(C_2(Y),\partial C_2(Y))$.
\end{lemma}
\begin{proof}
Since
${\rm Im}(\partial (\overline{\mathcal A_{q_i}}\times\overline{\mathcal D_{p_j}})\to Y^2)=
\sum_k \partial_{ki}\overline{\mathcal A_{p_k}}\times \overline{\mathcal D_{p_j}}
+\sum_k\partial_{jk}\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{q_k}}$,
\begin{eqnarray*}
&&{\rm Im}(\sum_{i,j}g_{ij}\partial (\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{p_j}}\to Y^2))\\
&=&\sum_{i,j,k} g_{ij}\partial_{ki}\overline{\mathcal A_{p_k}}\times \overline{\mathcal D_{p_j}}
+\sum_{i,j,k}g_{ij}\partial_{jk}\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{q_k}}\\
&=&\sum_{i,j,k}\delta_{kj}\overline{\mathcal A_{p_k}}\times \overline{\mathcal D_{p_j}}
+\sum_{i,j,k}\delta_{ik}\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{q_k}}\\
&=&\sum_{j}\overline{\mathcal A_{p_j}}\times \overline{\mathcal D_{p_j}}
+\sum_{j}\overline{\mathcal A_{q_j}}\times \overline{\mathcal D_{q_j}}\\
&=& \partial \overline M_{\to}(f)\setminus \Delta .
\end{eqnarray*}
Therefore $\partial \mathcal M_S(\pm f)\setminus \partial C_2(Y)=\emptyset$.
\end{proof}
Under the identification $S\nu_{\Delta (Y\setminus\infty)}\cong ST(Y\setminus \infty)$, we have the following description.
\begin{lemma}\label{65}
$\partial \mathcal M_S(\pm f)\cap ST(Y\setminus \infty)=\overline{\{ (x,\frac{\pm \grad_xf}{\|\grad_xf\|})\mid x\in Y\setminus (\infty\cup \Crit(f)\}}$.
\end{lemma}
\begin{proof}
Note that $(\overline{\mathcal A_{q_i}}\times \overline{\mathcal D_{p_j}})\cap\Delta=\overline{\mathcal A_{q_i}\cap \mathcal D_{p_j}}$.
By the definition of blow up, we have
$\partial \mathcal M_S(\pm f)\cap S\nu_{\Delta(Y\setminus \infty)}$
$$=\overline{\left\{ \left(x,\frac{\pm \grad_xf}{\|\grad_xf\|}\right)\right\}}+\sum_{i,j}g_{ij}\pi^{-1}(\overline{\mathcal A_{q_i}\cap \mathcal D_{p_j}}) +\sum_{i,j}(-g_{ij})\pi^{-1}(\overline{\mathcal D_{p_j}\cap \mathcal A_{q_i}})$$ where $\pi:STY\to Y$ is the projection.
Since $\sum_{i,j}g_{ij}\pi^{-1}(\overline{\mathcal A_{q_i}\cap \mathcal D_{p_j}})+\sum_{i,j}(-g_{ij})\pi^{-1}(\overline{\mathcal D_{p_j}\cap \mathcal A_{q_i}})=0$ as chains, we conclude the proof.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,978
|
Variety Plus Icon Read Next: U.K. Culture Minister to Speak at Creative Coalition Festival 2022 Amid Row Over BBC Funding
May 9, 2014 3:23pm PT
TV Networks Clean House, Cancel Slew of Shows in Advance of Fall Presentations
By Nikara Johns
Nikara Johns
@NikaraJohns
Political Documentaries Put a Human Face on Controversial Issues 7 years ago
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Edie Falco Remembers 9/11 on Charity Day 7 years ago
As the major networks prepare for upfront presentations in New York next week, the cancellation ax is swinging.
CBS has cut the cord on five first-year shows: "The Crazy Ones," "Hostages," "Intelligence," "Friends with Better Lives" and "Bad Teacher." "Crazy Ones," starring Robin Williams and Sarah Michelle Gellar, had the much-coveted post-"Big Bang Theory" Thursday time slot, but was unable to capitalize on the lead-in.
"Hostages" was an experiment for the Eye with a shorter 15-episode order that allowed it to alternate in the Monday 10 p.m. slot with sci-fi actioner "Intelligence."
A number of Fox's freshman series were tabled including "Rake," "Dads," "Enlisted," "Surviving Jack" and "Almost Human" (which was cancelled April 29). It was announced in February that Simon Cowell's "The X Factor" would not be returning, and that "Raising Hope" would come to an end after its fourth season.
"Dads," the first live-action series from Seth MacFarlane's shop, failed to catch fire and was not given a second season, even with ratings that were on par with renewed comedies "The Mindy Project," "New Girl" and "Brooklyn Nine-Nine." "Enlisted" and "Surviving Jack," despite earning critical support, failed to find a foothold with viewers.
"Almost Human," a high-concept sci-fi series starring Karl Urban and Michael Ealy, drew 5.6 million viewers and a 1.5 in the adults 18-49 demo for its finale — which matched renewed thriller "The Following's" finale rating in the demo and beat it in total viewers. But Fox's competitive drama slate for the 2014-2015 season — which includes newly ordered "Gotham," "Red Band Society" and "Empire" — ultimately left the Bad Robot production shut out.
"Rake," based on an Australian format and starring Greg Kinnear, debuted to a 1.7 rating despite a powerhouse "American Idol" lead-in, and was quickly shunted to a Friday slot before its final two episodes were burned off on Saturday, April 5.
R.I.P. Cancelled TV Shows: What's Not Returning Next Year
NBC's cult favorite "Community" (pictured, above) was cancelled after five seasons, but it's expected that producers Sony TV will shop the comedy elsewhere. Post-apocalyptic drama "Revolution" was also cancelled on Friday, having floundered without "The Voice" as a lead-in during its sophomore season. "Dracula," starring Jonathan Rhys Meyers, was sucked dry after a single season.
NBC's midseason dramas "Believe" and "Crisis," and singlecam comedy "Growing Up Fisher" also failed to make the cut. The two dramas were already believed to be in danger after NBC pulled them from their Sunday timeslots for the last weekend of May sweeps — replacing them with a "Women of SNL" special — while "Fisher" has averaged a 1.7 among adults 18-49.
Previously, NBC announced the cancellations of "Ironside," "Welcome to the Family," "The Michael J. Fox Show" and "Sean Saves the World," all of which were pulled from the schedule with unaired episodes remaining. The lackluster performance of "The Michael J. Fox Show" was particularly disappointing, as NBC banked on Fox's star power early, giving the show a straight-to-series order of 22 episodes.
Freshman series "Trophy Wife," starring Malin Akerman, was cancelled by ABC, along with Rebel Wilson's "Super Fun Night," and "Mixology." Though "Trophy Wife" had critical acclaim, the comedy failed to show any discernible growth on other platforms, and an expensive ensemble cast likely contributed to its demise.
Although, "Once Upon A Time" was renewed, spinoff "Once Upon A Time in Wonderland" was canceled back in March.
ABC's "Suburgatory" was canceled after three seasons. It was two seasons and out for alien comedy "The Neighbors," which has been drawing a 0.9 demo rating in its spot between "Last Man Standing" and "Shark Tank."
Despite airing between solid performers "The Middle" and "Modern Family," "Suburgatory" never matched the success of its timeslot companions, and creator Emily Kapnek was already set to depart the comedy in order to oversee the newly picked up "Selfie" for the network.
Over at The CW, young Carrie Bradshaw will not return to the screen as "The Carrie Diaries" was axed. Rookie series "The Tomorrow People" and "Star-Crossed" will also not return, as their final installments averaged a paltry 0.3 and 0.4 respectively in adults under 50.
Tv Cancellations
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,149
|
\section{Introduction}
\label{introduction}
Identifying planar regions is an important task, that can be used in the context of segmentation and reconstruction for 3D scenes.
Recent developments in the area of 3D plane detection from single RGB images \cite{liuPlaneNetPiecewisePlanar2018}, \cite{liuPlaneRCNN3DPlane2019} open up opportunities to employ such approaches for various applications.
\gls{ar} is one such application that can be used in numerous domains, ranging from logistics, manufacturing and military to education and entertainment \cite{yuUsefulVisualizationTechnique2010}.
Identified planes can be used to inpaint information and to place and simulate objects in a scene \cite{karschAutomaticSceneInference2014}.
For example in manufacturing, \gls{ar} can be used to simulate, assist and improve processes \cite{ongAugmentedRealityApplications2008}.
In addition to that, robotics is a broad area of application, where plane segmentation can help to navigate, grasp and perform various other tasks.
A common domain, where objects are confined to cuboid shapes is logistics.
One benefit of using plane segmentation for object detection in such environments is its robustness compared to \gls{sota} segmentation techniques \cite{heMaskRCNN2017} that rely on knowing instances of the object categories beforehand.
Thus, an accurate plane segmentation could be used for reconstruction or damage and tampering detection \cite{nocetiMulticameraSystemDamage2018} in logistics contexts.
Moreover, \gls{ar} can assist the packaging process to reduce error rates and document the process \cite{hochsteinPackassistentAssistenzsystemFuer2016}.
Using \gls{sota} plane segmentation techniques alone is suitable for many applications, such as partial inpainting, however, these techniques still have difficulties in delivering fine-grained segmentation results.
To overcome this issue, we propose a post-processing technique to refine the results of plane segmentation approaches such as the PlaneRCNN \cite{liuPlaneRCNN3DPlane2019}, to accurately detect the planes of cuboid-shaped objects in an image.
Plane segmentation masks are refined separately and thus, not only cuboid-shaped objects, such as packages, but any plane with a rhombic shape can be rectified.
In addition to that, the surfaces of those planes do not need to be completely flat, as in the case of a parcel, since also small 3D structures on the plane can be handled.
Our approach uses different edge segmentation techniques \cite{cannyComputationalApproachEdge1986}, \cite{soriaDenseExtremeInception2020} to align the masks resulting from the plane segmentation with the edges in the image.
For an overview of the pipeline, see \fig{fig:overview}.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/overview1.png}
\caption{Input image}
\label{fig:overview:a}
\vspace{2ex}
\end{subfigure
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/overview2.png}
\caption{Edge detection}
\label{fig:overview:b}
\vspace{2ex}
\end{subfigure}
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/overview3.png}
\caption{Plane segmentation}
\label{fig:overview:c}
\end{subfigure
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/overview4.png}
\caption{Refined segmentation}
\label{fig:overview:d}
\end{subfigure}
\caption{Overview of the segmentation pipeline. We consider an image (a) and combine edge detection (b) and plane segmentation (c) to obtain fine-grained plane segmentation results (d) close to the ground truth (blue contours).}
\label{fig:overview}
\end{figure}
We collected a dataset of 34 images in a logistics context with ground truth plane segmentations, which we made publicly available\footnote{Dataset available under \href{https://url.fzi.de/dataset_planeseg}{https://url.fzi.de/dataset_planeseg}.}.
The performance of our approach, compared to the PlaneRCNN baseline \cite{liuPlaneRCNN3DPlane2019} and a fallback routine which is also introduced in this work, is evaluated on this dataset.
We show an overall improvement of the averaged \gls{iou} of over $4.25$ percentage points compared to the baseline.
The quality of the refinement for individual images depends on the quality of the prior segmentation and the edge detection in the image.
Several examples are provided to illustrate the capabilities of the approach even in difficult settings, but also to point out areas for further improvements.
We will make our code publicly available under \href{https://url.fzi.de/refined_planeseg}{https://url.fzi.de/refined_planeseg}.
This work is structured as follows.
In \secr{sec:related_work} we will present an overview over related literature.
\secr{sec:approach} outlines our plane segmentation mask refinement approach and \secr{sec:results_and_evaluation} evaluates our approach by comparing it to two baselines on our newly collected dataset.
\secr{sec:conclusion} concludes the paper.
\section{Related Work}
\label{sec:related_work}
To the best of our knowledge, there has been no approach yet to improve the performance of \gls{sota} plane segmentation techniques by leveraging additional information from edge detection techniques.
Edge detection is a field thoroughly studied in computer vision \cite{oskoeiSurveyEdgeDetection2010}.
\citeauthor{sobelCameraModelsMachine1972}'s work \cite{sobelCameraModelsMachine1972} was one of the early contributions, that inspired numerous edge detection techniques.
The Canny Edge Detector \cite{cannyComputationalApproachEdge1986} is one of those techniques, that was developed in \citeyear{cannyComputationalApproachEdge1986} and is still a very common choice to date.
Due to the popularity of the Canny Edge Detector, there has been a lot of research on improving it, for example by using an adaptive thresholds \cite{fangStudyApplicationOtsu2009}.
Recently, also \glspl{cnn} have been used to detect edges \cite{xieHolisticallyNestedEdgeDetection2015},
\cite{soriaDenseExtremeInception2020} in images.
These approaches often provide the advantage of being less sensitive to noise and are a promising alternative to classical techniques.
Image segmentation is a field intensively studied in computer vision that reached remarkable performance on closed-set configurations, where the objects of interests are known beforehand \cite{garcia-garciaReviewDeepLearning2017}.
In the area of plane segmentation,
End-to-End trained \glspl{nn} have only been presented recently \cite{liuPlaneNetPiecewisePlanar2018}, \cite{liuPlaneRCNN3DPlane2019}.
In addition to improving on the \gls{sota} in plane segmentation, these approaches also improved the \gls{sota} in single-image depth estimation.
\section{Plane Segmentation Refinement}
\label{sec:approach}
Our approach aims to improve the granularity of the plane segmentation by exploiting additional information from edge detection.
We assume a prior segmentation by plane segmentation techniques, however, the procedure is independent of the source of these prior masks.
In the following, we introduce the edge detection techniques we consider in \secr{sec:approach:edge}.
Afterwards, we present the plane segmentation approach whose segmentation masks are the input for our post-processing in \secr{sec:approach:plane}.
Finally, we propose a method that leverages clustering and regression to find line segments along a rhombus in \secr{sec:approach:refine} and combine these ideas for the final refinement in \secr{sec:approach:approach}.
\subsection{Edge Detection}
\label{sec:approach:edge}
We consider two different techniques for edge segmentation.
A recent work building upon the Canny Edge Detector \cite{cannyComputationalApproachEdge1986} by automating the thresholding process \cite{fangStudyApplicationOtsu2009}, which we call Adaptive Canny and a machine learning based approach called DexiNed by \citeauthor{soriaDenseExtremeInception2020} \cite{soriaDenseExtremeInception2020}.
We use two different forms of the latter approach, by applying it to the full resolution image (1280x960) and to a downsized image (640x480), since this seems to trigger a focus on important edges.
\subsection{Plane Segmentation}
\label{sec:approach:plane}
The PlaneRCNN deep neural architecture was introduced by \citeauthor{liuPlaneRCNN3DPlane2019} in \citeyear{liuPlaneRCNN3DPlane2019} \cite{liuPlaneRCNN3DPlane2019}.
It improves upon earlier models \cite{liuPlaneNetPiecewisePlanar2018} \cite{yangRecovering3DPlanes2018} by not requiring the maximum number of planes a priori and generalizing better to unseen scenes.
The input to the model is a single RGB image, which is processed by three components.
The first component is a Mask R-CNN \cite{heMaskRCNN2017} based model for plane detection.
In addition to the plane segmentation, also the plane normal and depth values for each pixel are estimated.
The second and third component are responsible for a refinement and the enforcement of consistency of the reconstructions.
In this work, we use only the plane segmentations retrieved by the PlaneRCNN.
\subsection{Line Segmentation}
\label{sec:approach:refine}
Given a binary contour image and a segmentation of this image in planes, we consider the process of refining the given segmentation for each mask separately.
We present two approaches for detecting line segments on the bounding edges of a plane belonging to a cuboid-shaped object, which will be combined in the final solution.
The starting point for both approaches is to overlay the binary image with a widened contour line of the mask (See \fig{fig:line_detection:a}).
For a reasonable prior segmentation, this contour should contain all or at least some of the bounding edges for the respective plane.
The first approach relies on \gls{dbscan} \cite{esterDensitybasedAlgorithmDiscovering1996} to identify connected structures on the extract of the binary image.
For a good prior segmentation, this extract of the binary image might comprise a completely connected cluster or similarly, only two or three clusters for rhombus as seen in \fig{fig:line_detection:a}.
Since we try to estimate all line segments separately, we perform a corner detection \cite{harrisCombinedCornerEdge1988} and break the clusters up by removing the detected corners from the binary mask.
Thereafter, we are left with line-shaped clusters only as in \fig{fig:line_detection:b}, if all corners are detected correctly.
In addition to that, we omit very small clusters and clusters with big variance in two directions, since they most likely constitute areas of noise.
This leaves us with mainly line-shaped clusters of pixels.
We use a RANSAC \cite{fischlerRandomSampleConsensus1987} linear regression to find a two point description for each of those lines.
For each line, this first estimation is used to search for extensions of the line that where not captured by the widened contour line of the mask.
This becomes necessary when the widened contour line only overlays with parts of the current edge, since non-overlapping contours were previously ignored.
Hence, we create a mask along the estimated line across the whole image and repeat the former process.
We apply clustering onto the new mask and perform a RANSAC linear regression on the dominant cluster to obtain new end point estimations for the current line.
\scomment{Finally, to get the starting and end point of the line we use the minimum and maximum $x$ values.}
\begin{figure}[ht!]
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/line_detection1.jpg}
\caption{Boundary of input mask}
\label{fig:line_detection:a}
\vspace{2ex}
\end{subfigure
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/line_detection2.jpg}
\caption{Relevant segments after split}
\label{fig:line_detection:b}
\vspace{2ex}
\end{subfigure}
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/line_detection3.jpg}
\caption{Fitted lines}
\label{fig:line_detection:c}
\end{subfigure
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/line_detection4.jpg}
\caption{Endpoints of lines}
\label{fig:line_detection:d}
\end{subfigure}
\caption{Overview of the pipeline for the segmentation refinement. We use the widened contour of the mask (a) to identify clusters of line segments (b). We try to fit a line for each segment (c), which can be used for the estimation of the final segmentation mask (d).}
\label{fig:line_detection}
\end{figure}
The second approach is based on the Hough Transform \cite{houghMethodMeansRecognizing1962}.
We cluster and average the lines resulting from applying the Hough Transform to the extract of the binary image.
This approach is complementary to the first approach, since it does not rely on the connectedness within edge segments.
Using the resulting lines in normal form, we estimate start and end points by leveraging their points of intersection.
More precisely, for each line we compute its points of intersection with all other lines and assume that valid endpoints lie in the vicinity of the considered mask.
The check for vicinity is performed by examining if a circle around the point of intersection with a 40 pixel radius intersects with the extract of the binary image.
\subsection{Combined Approach}
\label{sec:approach:approach}
Still considering each plane separately, we combine the results from the first and the second approach from \secr{sec:approach:refine}.
The result from each approach is a list of line segments described by a start and an end point, which represent approximations of the edges.
We cluster the line segments\scomment{in normal form} to identify lines describing the same edge.
By using a k-means algorithm, we group start points into a set $A$ and end points into a set $B$.
These groups are each complemented by a set of ten randomly chosen points on the binary image in the vicinity of the centroids resulting from the k-means algorithm.
We then identify the start point $P_a \in A$ and the end point $P_b \in B$ best fitting the considered edge $e$, for each cluster of line segments.
Our cost function $C$ describing the quality of the fit incorporates the overlap with the underlying mask $m_e$ and the normalized length of the line with equal weights
\[
C(P_a, P_b) = 0.5 \cdot \text{I}(\overline{P_aP_b}, m_e) + 0.5 \cdot \frac{\norm{P_a - P_b}}{\text{max}_{k \in A, l \in B}(\norm{P_k - P_l})},
\]
where $\text{I}(\overline{P_aP_b}, m_e)$ is the normalized intersection
\[
\text{I}(\overline{P_aP_b}, m_e) = \frac{\overline{P_aP_b} \cap m_e}{\overline{P_aP_b}}.
\]
The area $\overline{P_aP_b}$ is defined by a line with one pixel thickness between $P_a$ and $P_b$.
Note that focusing on maximum overlap only might lead to very accurate, however, contracted line segments that do not represent the edges of the plane well.
Since we are aware of the dependence of our approach on the quality of the input information, we check the consistency of the refined mask with the prior mask by calculating their \gls{iou}.
For large deviations, i.e. an \gls{iou} of less than $0.75$, we resort back to a simple baseline approach.
This baseline consist of calculating the prior mask's convex hull and iteratively reducing the set of points describing the mask \cite{douglasAlgorithmsReductionNumber1973} to 20 points or less.
Our approach is based on considering one mask at a time. Note that the dependencies between masks belonging to the same object can be used to further refine the segmentation results.
\section{Results and Evaluation}
\label{sec:results_and_evaluation}
We first describe the dataset we collected in \secr{sec:eval:dataset} and comment on its separation into different classes.
Subsequently, we will shortly present the baseline approaches used in the evaluation and finally discuss the results of our approach.
\subsection{Dataset}
\label{sec:eval:dataset}
We collected a set of 34 images, each picturing one cuboid object with different backgrounds.
The backgrounds include carpet, table surfaces, floor tiles and the inside of a container.
The cuboid-shaped objects include different types of parcels made from carton and plastic.
The surfaces of the parcels range from almost plain to complex structures.
During the process of collecting the images, we manually discarded all images where the PlaneRCNN was not able to grasp the scene, i.e. where it did not detect all visible planes of the cuboid object of interest.
This was mostly the case for flat objects and difficult camera angles, however, it also happened in some simpler scenes.
We split up the dataset in three groups by manually assessing the complexity of the scenes, i.e. the quality of the prior segmentation masks and the edge detection.
Of the 34 images, 7 were grouped into the category easy, 16 were grouped into the category medium and the remaining 11 were assigned the difficulty hard.
Note that even the easy category contains diverse backgrounds and different types of packages.
The data was labeled manually using the VGG Image Annotator \cite{duttaAnnotationSoftwareImages2019}.
\subsection{Baseline Approaches}
\label{sec:eval:baseline}
The results from the PlaneRCNN \cite{liuPlaneRCNN3DPlane2019} are used as input for our approach and thus, constitute a first baseline.
In addition to that, we present the results of using our fallback solution that was described in \secr{sec:approach:approach}.
Results separated by categories are presented in Table \ref{table:results}.
We use the \gls{iou} as metric for comparison as common for segmentation tasks.
The \gls{iou} over all masks in an image is averaged and subsequently the average over all images in the dataset is taken.
As mentioned above, we removed images from the dataset, where the PlaneRCNN was not able to grasp the scene.
If the PlaneRCNN is able to grasp the scene, it reaches 81.94\% \gls{iou} with the ground truth masks.
Our fallback solution shows a slight improvement on the PlaneRCNN by $0.23$ percentage points for the \gls{iou}.
Since it rectifies the PlaneRCNN masks, their shape is more consistent with the ground truth masks.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Dataset & PlaneRCNN & Fallback & Dexi LR & Dexi FR & Canny\\
\hline
Easy & 83.85\% & 84.00\% & \textbf{90.74\%} & 87.16\% & 82.03\% \\
Medium & 82.52\% & 82.67\% & \textbf{84.75\%} & 82.66\% & 81.91\% \\
Hard & 79.88\% & 80.29\% & \textbf{82.38\%} & 80.43\% & 79.42\% \\
\hline
All & 81.94\% & 82.17\% & \textbf{85.20\%} & 83.10\% & 81.07\% \\
\hline
\end{tabular}
\caption{Average \gls{iou} over all masks in each dataset for the PlaneRCNN, the fallback solution and our suggested approach with different edge segmentation techniques (Dexi LR = DexiNed with low resolution, Dexi FR = DexiNed with full resolution, Canny = Adaptive Canny).}
\label{table:results}
\end{table}
\subsection{Our Approach}
\label{sec:eval:our}
The evaluation results for our approach using different edge detection techniques are reported in Table \ref{table:results}.
The evaluation shows an improvement of over $4.25$ percentage points compared to the PlaneRCNN when edge detection is performed by DexiNed with a low resolution image as input.
The results on the dataset classified as easy show an improvement of almost $7$ percentage points.
Thus, especially for reasonable prior segmentation masks and edge detections a considerable improvement over the baseline can be achieved.
We exemplarily show such segmentation results in \fig{fig:overview} and \fig{fig:good}.
Note that, even for complex backgrounds and feature-rich objects, our approach achieves good accuracy.
\begin{figure}[ht!]
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.45\linewidth]{figures/good1.png}
\includegraphics[width=0.45\linewidth]{figures/good2.png}
\caption{Classified as easy.}
\label{fig:good:a}
\end{subfigure
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.45\linewidth]{figures/good3.png}
\includegraphics[width=0.45\linewidth]{figures/good4.png}
\caption{Classified as hard.}
\label{fig:good:b}
\end{subfigure}
\caption{Exemplary results of the plane segmentation refinement. The blue contours constitute the ground truth.}
\label{fig:good}
\end{figure}
The importance of the edge detection technique is seen by its strong influence on the evaluation results.
We observe a consistent decline over all datasets moving from DexiNed with low resolution to DexiNed with full resolution and from DexiNed with full resolution to the Adaptive Canny algorithm.
Using the adaptive Canny Edge Detector even yields results slightly worse than the baseline, as can be seen in Table \ref{table:results}.
This is due to feature-rich backgrounds and parcels where the bounding edges are not dominant.
By manually assessing the images, we identified reasons for bad segmentation results.
The approach performs well when the prior mask and the edge detection yield reasonable inputs.
Strong edges in the background as in \fig{fig:issues:a} and dominant lines on the parcel as the black label in \fig{fig:issues:d} can sometimes mislead the algorithm and cause deviations from the desired mask.
In addition to that, the ability to break up the clustered segments around the mask by detecting and removing corners does affect the accuracy.
In \fig{fig:issues:b}, for example, the vertical lines of the blue plane were not detected since they were clustered with the more dominant horizontal lines.
Finally, imprecise prior masks can cause wrongfully merging two planes of an object as in \fig{fig:issues:d} or prohibit the algorithm from detecting the full surface of the plane as in \fig{fig:issues:c}.
\begin{figure}[ht!]
\begin{subfigure}[b]{0.24\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/issues1.png}
\caption{
\label{fig:issues:a}
\end{subfigure
\begin{subfigure}[b]{0.24\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/issues2.png}
\caption{
\label{fig:issues:b}
\end{subfigure
\begin{subfigure}[b]{0.24\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/issues3.png}
\caption{
\label{fig:issues:c}
\end{subfigure
\begin{subfigure}[b]{0.24\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/issues4.png}
\caption{
\label{fig:issues:d}
\end{subfigure}
\caption{Visualization of the dependence of our approach on accurate prior information.
Examples with strong background edges (a), failed corner detection (b), too small prior segmentation (c) and too big prior segmentation (d).}
\label{fig:issues}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work, we presented an approach for the refinement of segmentation masks of cuboid-shaped objects.
Existing \gls{sota} plane segmentation methods generalize well for the logistics environment we considered exemplarily and often yielded reasonable scene segmentations.
However, those segmentation masks lack accuracy for fine-grained details such as corners.
To enable the use of plane segmentation techniques for a wider range of applications, where this accuracy is necessary, we propose a post-processing technique.
We combine edge detection and plane segmentation techniques with clustering approaches to perform a mask refinement along the edges of the object.
To achieve robustness, we complement these techniques with a simple, yet effective fallback solution.
Our approach improves the accuracy of \gls{sota} plane segmentation techniques by over $4.25$ percentage points and generates masks that are more consistent in shape with the ground truth masks.
The refined segmentation masks have several applications, for instance inpainting for \gls{ar} or object detection for cuboid-shaped objects.
The further improvement of the segmentation by exploiting the fact that planes belong to the same object is left for future research.
\section*{Acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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|
package com.amazonaws.services.emrserverless.model;
import javax.annotation.Generated;
/**
* <p>
* Request processing failed because of an error or failure with the service.
* </p>
*/
@Generated("com.amazonaws:aws-java-sdk-code-generator")
public class InternalServerException extends com.amazonaws.services.emrserverless.model.AWSEMRServerlessException {
private static final long serialVersionUID = 1L;
/**
* Constructs a new InternalServerException with the specified error message.
*
* @param message
* Describes the error encountered.
*/
public InternalServerException(String message) {
super(message);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,245
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,043
|
import template from './resetPassword.html!text';
import 'angular-ui-router';
import '../../services/security/authFactory';
var app = angular.module('cn.resetPassword', [
'ngMaterial',
'cn.auth',
'ui.router'
])
.directive('cnResetPassword', function() {
return {
restrict: 'E',
template: template,
controllerAs: 'ctrl',
bindToController: true,
controller: /*@ngInject*/function controller($state, authService, $location) {
let code = encodeURIComponent($location.search().code),
userId = $location.search().email;
this.setTouched = () => {
angular.forEach(this.form.$error.required, function(field) {
field.$setTouched();
});
};
this.resetPassword = (data) => {
this.loading = true;
return authService.confirmResetPassword(userId, code, data.password).then((response) => {
this.loading = false;
$state.go('login');
},
(err) => {
this.loading = false;
this.formInvalid = true;
});
};
}
};
});
export default app;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,716
|
Q: Sharing a SAN LUN across 2 clients I have an ISCSI SAN that we purchased to store images for our cluster (2TB of images). After reading, we probably should have gone with a NAS rather than a SAN, but back to the point.
I'm going to need two servers to share a single LUN on the SAN for failover. We have haproxy and nginx setup on these two centos 6.5 machines. We're using keepalived to share a virtual ip between these two machines in case one crashes. Similarly, we need the iscsi lun to be available on whichever machine is active so we can serve images to our cluster.
Is there a "simple" way to make this happen?
A: A clustered filesystem is not a requirement.
The "simplest" way (but not at all simple) is to use HA LVM. See High Availability LVM
A: As was mentioned in your other question, you need a clustered filesystem on the shared LUN.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,500
|
Your ultimate traditional Conference Satchel at its finest! Built with high quality materials with subtle featured panels. Large branding area for your logo with a document storage capacity of 2 litres.
Product specifications: 600D/300D polyester with PVC backing, zippered main compartment, webbing carry handles, Business card holder on back and can fit an A4 notepad.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,377
|
{-# LANGUAGE OverloadedStrings, OverloadedLists #-}
module Network.IRC.Client.Amphibian.Utility
(Response,
Error(..),
getResponse,
tryGetResponse,
byteOfChar,
splitOnSpaces,
userTypePrefix,
ourDecodeUtf8,
asyncHandleResponse,
syncHandleResponse,
displayError,
filterMessageText,
matchByteStringGlob)
where
import Network.IRC.Client.Amphibian.Types
import qualified Data.Text as T
import qualified Data.ByteString as B
import qualified Data.Sequence as S
import Data.Sequence (ViewL(..),
ViewR(..))
import Data.Text.Encoding (encodeUtf8,
decodeUtf8With)
import Data.Text.Encoding.Error (lenientDecode)
import Data.Word (Word8)
import Text.Printf (printf)
import Control.Concurrent.Async (Async,
async)
import Control.Concurrent.STM (STM,
atomically)
import Control.Concurrent.STM.TMVar (readTMVar,
tryReadTMVar)
import System.IO (stderr)
import Data.Text.IO (hPutStr)
import Data.Char (isSpace)
-- | Get a response.
getResponse :: Response a -> STM (Either Error a)
getResponse (Response response) = readTMVar response
-- | Try to get a response.
tryGetResponse :: Response a -> STM (Maybe (Either Error a))
tryGetResponse (Response response) = tryReadTMVar response
-- | Get byte of char.
byteOfChar :: Char -> Word8
byteOfChar char = B.head . encodeUtf8 $ T.pack [char]
-- | Split a bytestring on one or more spaces.
splitOnSpaces :: B.ByteString -> (B.ByteString, Maybe B.ByteString)
splitOnSpaces bytes =
let (part, rest) = B.break (== byteOfChar ' ') bytes
rest' = B.dropWhile (== byteOfChar ' ') rest
in if B.length rest' > 0
then (part, Just rest')
else (part, Nothing)
-- | Get prefix character for user type.
userTypePrefix :: UserType -> T.Text
userTypePrefix OwnerUser = "~"
userTypePrefix AdminUser = "&"
userTypePrefix OpUser = "@"
userTypePrefix HalfOpUser = "%"
userTypePrefix VoiceUser = "+"
userTypePrefix NormalUser = ""
-- | Our UTF-8 decoder.
ourDecodeUtf8 :: B.ByteString -> T.Text
ourDecodeUtf8 = decodeUtf8With lenientDecode
-- | Asynchronously handle response.
asyncHandleResponse :: Response a -> IO ()
asyncHandleResponse response = do
async $ do
result <- atomically $ getResponse response
case result of
Right _ -> return ()
Left (Error errorText) -> displayError errorText
return ()
-- | Synchronously handle response.
syncHandleResponse :: Response a -> IO ()
syncHandleResponse response = do
result <- atomically $ getResponse response
case result of
Right _ -> return ()
Left (Error errorText) -> displayError errorText
-- | Display an error.
displayError :: T.Text -> IO ()
displayError = hPutStr stderr . T.pack . printf "%s\n"
-- | Filter message text to eliminate passwords and like.
filterMessageText :: B.ByteString -> T.Text -> T.Text
filterMessageText nickOrName text =
if nickOrName == encodeUtf8 "NickServ"
then
let (part0, part1) = T.span isSpace text
(part1', part2) = T.span (not . isSpace) part1
(part2', part3) = T.span isSpace part2
(part3', part4) = T.span (not . isSpace) part3
(part4', part5) = T.span isSpace part4
in if part1' == "identify"
then T.concat [part0, part1', part2', "****"]
else if part1' == "release"
then T.concat [part0, part1', part2', part3', part4', "****"]
else text
else text
-- | Match a bytestring against a glob.
matchByteStringGlob :: B.ByteString -> B.ByteString -> Bool
matchByteStringGlob glob matched =
let parts = S.fromList $ B.split (byteOfChar '*') glob
in case S.viewl parts of
first :< parts ->
case B.stripPrefix first matched of
Just matched ->
case S.viewr parts of
parts :> last ->
case B.stripSuffix last matched of
Just matched -> matchParts parts matched
Nothing -> False
EmptyR -> B.null matched
Nothing -> False
EmptyL -> error "impossible"
where matchParts parts matched =
case S.viewl parts of
first :< parts ->
let (_, matchedPart) = B.breakSubstring first matched
in case B.stripPrefix first matchedPart of
Just matched -> matchParts parts matched
Nothing -> False
EmptyL -> True
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,060
|
\section{Introduction}
One of the manifestations of the Aharonov-Bohm (AB) effect
\cite{AhaB59} in the ring geometry \cite{ByeY61,ButIL83} is the
periodic dependence of the transmission coefficient for an electron
traversing the ring on the magnetic flux $\Phi$ through the
ring.\cite{GefIA84,ButILP85} The period of oscillations is equal to
$\Phi_0=hc/e$ --- the universal flux quantum.
For one-dimensional (1D) continuum interacting quantum systems with
translational invariance there is also a periodicity of many-particle
states as a functions of flux.\cite{SutS90,ShaS90,MulWL93,Kus94a} In
1D lattice systems, the lifting of Galilean invariance allows for
various periodicities of the states.\cite{SutS90,ShaS90} For the
ground state, this behavior can be interpreted, according to the above
definition of $\Phi_0$, as a signature of the existence of elementary
excitations with multiple --- sometimes even fractional --- charges
.\cite{SutS90,RomS94b,RomS94c,Kus95,KriSSJ96} In the case of strong
electron-electron interaction the adequate description of the
many-body states is based on excitations of the Wigner-crystal
.\cite{Los92,KriSSG95} Furthermore, the absence of sensitivity to the
flux in such systems is an indication of the onset of the Mott
transition. \cite{ShaS90,Koh64,RomP95} Similarly, the sensitivity of
single-particle energies to the flux \cite{Tho74} can be used as a
criterion of the Anderson-type metal-insulator transition in
disordered systems.\cite{And58} Combined effects of interactions and
disorder in 1D have received much attention in the last decade
.\cite{RomP95,Dor90,She94,LeaRS99,RomLS99b} Numerical studies of
pairing effects for two particles with repulsive interaction in a
disordered environment were carried out using the AB setting
.\cite{WeiMPF95} Other physical manifestations of the AB effect in
the ring geometry considered in the literature include the evolution
of electron states for a time-dependent flux,\cite{GorKGS97} and a
flux-dependent equilibrium distortion of the lattice caused by
electron-phonon interactions. \cite{Kus92}
The physical origin of the flux sensitivity of an electron on the ring
is its charge which couples to the vector potential. Correspondingly,
the coupling to the flux has the opposite sign for an electron and a
hole. For this reason an {\em exciton}, being a bound state of
electron and hole and thus a {\em neutral}\ entity, should not be
sensitive to the flux. However, due to the finite size of the
exciton, such a sensitivity will emerge. This effect is demonstrated
in the present paper.
Below we study the AB-oscillations both in the binding energy and in
the oscillator strength of the exciton absorption. We choose as a
model a short-range attraction potential between electron and hole,
which allows to solve the three-body problem (electron, hole, and a
ring) exactly. From this exact solution, we trace the behavior of the
AB oscillations when increasing the radius of the ring or the strength
of the electron-hole attraction.
Denote with $\varphi_e$ and $\varphi_h$ the azimuthal coordinates of
the electron and hole, respectively. In the absence of interaction the
wave functions of electrons and holes are given by
\begin{equation}
\label{eq-eigen}
\Psi_N^{(e)}(\varphi_e) =\frac{1}{\sqrt{2\pi}}e^{iN\varphi_e}, \quad
\Psi_{N^{\prime}}^{(h)}(\varphi_h)
=\frac{1}{\sqrt{2\pi}}e^{iN^{\prime}\varphi_h},
\end{equation}
where $N$ and $N^{\prime}$ are integers. The corresponding energies
are
\begin{equation}
\label{eq-energies}
E_N^{(e)}=\frac{\hbar^2}{2m_e\rho^2}\Biggl(N-\frac{\Phi}{\Phi_0}\Biggr)^2,
\quad E_{N^{\prime}}^{(h)}
=\frac{\hbar^2}{2m_h\rho^2}\Biggl(N^{\prime}+\frac{\Phi}{\Phi_0}
\Biggr)^2.
\end{equation}
Here $\rho$ is the radius of the ring, and $m_e$, $m_h$ stand for the
effective masses of electron and hole, respectively. In the presence
of an interaction $V\Bigl[R(\varphi_e-\varphi_h)\Bigr]$, where
$R(\varphi_e-\varphi_h)=2\rho\sin(\frac{\varphi_e-\varphi_h}{2})$ is
the distance between electron and hole, we search for the wave
function of the exciton in the form
\begin{equation}
\label{eq-wave}
\Psi(\varphi_e,\varphi_h)= \sum_{N,N^{\prime}}A_{N,N^{\prime}}
\Psi_N^{(e)}(\varphi_e)\Psi_{N^{\prime}}^{(h)}(\varphi_h).
\end{equation}
The coefficients $A_{N,N^{\prime}}$ are to be found from the equation
\begin{equation}
\label{eq-A}
\sum_{N,N^{\prime}}A_{N,N^{\prime}}
\Bigl[E_N^{(e)}+E_{N^{\prime}}^{(h)} - \Delta\Bigr]
\Psi_N^{(e)}(\varphi_e)\Psi_{N^{\prime}}^{(h)}(\varphi_h) +
V\Bigl[R(\varphi_e-\varphi_h)\Bigr]\Psi(\varphi_e,\varphi_h)=0,
\end{equation}
where $\Delta$ is the energy of the exciton. The formal expression for
$A_{N,N^{\prime}}$ follows from Eq.\ (\ref{eq-A}) after multiplying it
by
$\Bigl[\Psi_N^{(e)}(\varphi_e)
\Psi_{N^{\prime}}^{(h)}(\varphi_h)\Bigr]^{\dagger}$
and integrating over $\varphi_e$ and $\varphi_h$
\begin{equation}
\label{eq-formal}
A_{N,N^{\prime}}=
-\frac{1}{2\pi}\int_0^{2\pi}d\varphi_e\int_0^{2\pi}d\varphi_h
\frac{V\Bigl[R(\varphi_e-\varphi_h)\Bigr]\Psi(\varphi_e,\varphi_h)}
{E_N^{(e)}+E_{N^{\prime}}^{(h)} - \Delta}
e^{-i(N\varphi_e+N^{\prime}\varphi_h)}.
\end{equation}
At this point we make use of the assumption that the potential
$V\Bigl[R(\varphi_e-\varphi_h)\Bigr]$ is short-ranged. This implies
that the integral over $\varphi_h$ is determined by a narrow interval
of $\varphi_h$ close to $\varphi_e$. Then we can replace $\varphi_h$
by $\varphi_e$ in the rest of the integrand. As a result, Eq.\
(\ref{eq-formal}) simplifies to
\begin{equation}
\label{eq-simplified}
A_{N,N^{\prime}}= -\frac{V_0}{E_N^{(e)}+E_{N^{\prime}}^{(h)} - \Delta}
\int_0^{2\pi}d\varphi_e\Psi(\varphi_e,\varphi_e)e^{-i(N+N^{\prime})\varphi_e},
\end{equation}
where the constant $V_0<0$ is defined as
\begin{equation}
\label{eq-not}
V_0=\frac{1}{2\pi}\int d\varphi V\Bigl[R(\varphi)\Bigr].
\end{equation}
Finally we derive a closed equation, which determines the exciton
energies. This equation follows from Eqs.\ (\ref{eq-wave}) and
(\ref{eq-simplified}) as a self-consistency condition. Indeed, by
setting in Eq.\ (\ref{eq-wave}) $\varphi_e=\varphi_h$, multiplying
both sides by $\exp(-iN_0\varphi_e)$, and integrating over
$\varphi_e$, we obtain
\begin{equation}
\label{eq-integral}
\int_0^{2\pi}d\varphi_e\Psi(\varphi_e,\varphi_e)e^{-iN_0\varphi_e}
=\sum_N A_{N,N_0-N}.
\end{equation}
Substituting (\ref{eq-simplified}) into (\ref{eq-integral}) we arrive
at the desired condition
\begin{equation}
\label{eq-condition}
1+V_0\sum_N\frac{1}{E_N^{(e)}+E_{N_0-N}^{(h)} - \Delta_{N_0}}=0.
\end{equation}
For each integer $N_0$ the solutions of Eq.\ (\ref{eq-condition}) form
a discrete set, $\Delta_{N_0}^m$. The corresponding (non-normalized)
wave functions have the form
\begin{equation}
\label{eq-wf}
\Psi_{N_0}^m\propto
e^{iN_0\varphi_h}
\sum_N\frac{e^{iN(\varphi_e-\varphi_h)}}{E_N^{(e)}+E_{N_0-N}^{(h)}
- \Delta_{N_0}^m}.
\end{equation}
The exponential factor in front of the sum insures that in the dipole
approximation only the excitons with $N_0=0$ can be created by light.
The frequency dependence of the exciton absorption, $\alpha(\omega)$,
can be presented as
\begin{equation}
\label{eq-spectral}
\alpha(\omega)\propto\sum_mF_m\delta(\hbar\omega-E_g-\Delta_0^m),
\end{equation}
where $E_g$ is the band-gap of the material of the ring; the
coefficients $F_m$ stand for the oscillator strengths of the
corresponding transitions. A general expression for $F_m$ through the
eigenfunction, $\Psi_0^m$, of the excitonic state reads
\begin{equation}
\label{eq-strength}
F_m=\frac{|\int_0^{2\pi}d\varphi_e\int_0^{2\pi}d\varphi_h
\Psi_0^m(\varphi_e,\varphi_h)
\delta(\varphi_e-\varphi_h)|^2}
{\int_0^{2\pi}d\varphi_e\int_0^{2\pi}d\varphi_h
|\Psi_0^m(\varphi_e,\varphi_h)|^2}.
\end{equation}
Upon substituting Eq.\ (\ref{eq-wf}) into Eq.\ (\ref{eq-strength}) and
making use of Eq.\ (\ref{eq-condition}), we obtain
\begin{equation}
\label{eq-new}
F_m=\Biggl[V_0^2\sum_N\frac{1}{(E_N^{(e)}+E_{-N}^{(h)} -
\Delta_0^m)^2}\Biggr]^{-1}.
\end{equation}
The latter expression can be presented in a more compact form by
introducing the rate of change of the exciton energy with the
interaction parameter $V_0$. Indeed, taking the differential of Eq.\
(\ref{eq-condition}), yields
\begin{equation}
\label{eq-last}
F_m=-\frac{\partial\Delta_0^m}{\partial V_0}.
\end{equation}
We note that the summation in Eq.\ (\ref{eq-condition}) can be carried
out in a closed form by using the identity
\begin{equation}
\label{eq-identity}
\sum_{N=-\infty}^{\infty}\frac{1}{(\pi N-a_1)(\pi N-a_2)}=
\frac{1}{(a_1-a_2)}\Biggl(\frac{1}{\tan a_2}-\frac{1}{\tan
a_1}\Biggr).
\end{equation}
For the most interesting case $N_0=0$ the parameters $a_1$, $a_2$ are
equal to
\begin{equation}
\label{eq-12}
a_{1,2}=-\pi\Biggl[\frac{\Phi}{\Phi_0}\pm
\Bigl(\frac{\Delta_0^m}{\varepsilon_0} \Bigr)^{1/2}\Biggr],
\end{equation}
where
\begin{equation}
\label{eq-eps}
\varepsilon_0=\frac{\hbar^2}{2\rho^2}\Bigl(\frac{1}{m_e}+\frac{1}{m_h}\Bigr)=
\frac{\hbar^2}{2\mu \rho^2},
\end{equation}
and $\mu=m_em_h/(m_e+m_h)$ denotes the reduced mass of electron and
hole. Then the equation (\ref{eq-condition}) for the exciton energies
takes the form
\begin{equation}
\label{eq-form}
\Biggl(\frac{\Delta_0^m}{\varepsilon_0}\Biggr)^{1/2}=-\Biggl(\frac{\pi
V_0}{\varepsilon_0}\Biggr)\frac{\sin
\Bigl(2\pi(\Delta_0^m/\varepsilon_0)^{1/2}\Bigr)} {\cos
\Bigl(2\pi(\Delta_0^m/\varepsilon_0)^{1/2}\Bigr)-\cos\Bigl(2\pi(\Phi/\Phi_0)
\Bigr)}.
\end{equation}
This equation is our main result. It is seen from Eq.\ (\ref{eq-form})
that the structure of the excitonic spectrum is determined by a
dimensionless ratio $|V_0|/\varepsilon_0$. From the definition
(\ref{eq-not}) it follows that, with increasing the radius $\rho$ of
the ring, $V_0$ falls off as $1/\rho$. Thus, $|V_0|/\varepsilon_0$ is
proportional to $\rho$. In the limit of large $\rho$, when $|V_0|\gg
\varepsilon_0$, the spectrum can be found analytically. The ground
state corresponds to negative energy and is given by
\begin{equation}
\label{eq-correction}
\Delta_0^0=-\frac{\pi^2V_0^2}{\varepsilon_0}\Biggl[1+4\cos\Bigl(\frac{2\pi\Phi}
{\Phi_0}\Bigr)\exp\Bigl(-\frac{2\pi^2|V_0|}{\varepsilon_0}\Bigr)\Biggr].
\end{equation}
We note that the prefactor $\pi^2V_0^2/\varepsilon_0$ is independent
of $\rho$. It is equal to the binding energy of an exciton on a
straight line. It is easy to see that in the limit under
consideration we have $|\Delta_0^0|\gg|V_0|\gg\varepsilon_0$.
The second term in the brackets of Eq.\ (\ref{eq-correction})
describes the AB effect for the exciton. In the limit of large $\rho$
its magnitude is exponentially small. The physical meaning of the
exponential prefactor can be understood after rewriting it in the form
$\exp(-2\pi\rho\gamma)$, where
$\gamma=\pi|V_0|\Bigl(2\mu/\hbar^2\varepsilon_0\Bigr)^{1/2}$ is the
inverse decay length of the wave function of the internal motion of
electron and hole in the limit $\rho\rightarrow\infty$. Thus, the
magnitude of the AB effect in the limit of large $\rho$ represents the
amplitude for bound electron and hole to tunnel in the opposite
directions and meet each other ``on the opposite side of the ring''
(opposite with respect to the point where they were created by a
photon). This qualitative consideration allows to specify the
condition that the interaction potential is short-ranged. Namely, for
Eq.\ (\ref{eq-correction}) to apply, the radius of potential should be
much smaller than $\gamma^{-1}$. It is also clear from the above
consideration that, within a prefactor, the magnitude of the AB effect
is given by $\exp(-2\pi\rho\gamma)$ for arbitrary attractive
potential, as long as the decay length $\gamma^{-1}$ is smaller than
the perimeter of the ring. In Figs.\ \ref{fig-dbe-x} and
\ref{fig-be-x} we plot the numerical solution of Eq.\ (\ref{eq-form})
for various values of $\Phi$ together with the asymptotic solution
(\ref{eq-correction}) valid in the limit of large $\gamma\rho$. We see
that the maximum possible change in exciton energy by threading the
ring with a flux $\Phi_0/2$ is $25\%$ of the size-quantization energy
$\varepsilon_0$. The asymptotic expression of (\ref{eq-correction})
is good down to $\gamma\rho\approx \pi^{-1}$. In Fig.\
\ref{fig-be-phi-x}, we show the variation of the exciton energy with
$\Phi$ within one period. As expected, the AB oscillations are close
to sinusoidal for large values of $2\pi\gamma\rho$, whereas for
$2\pi\gamma\rho = 1$, unharmonicity is already quite pronounced. The
increase of the exciton energy as the flux is switched on has a simple
physical interpretation. If the single-electron energy
(\ref{eq-energies}) {\em grows} with $\Phi$ then the single-hole
energy is {\em reduced} with $\Phi$ and vice versa. This suppresses
the electron-hole binding. Fig.\ \ref{fig-be-phi-x} illustrates how
the amplitudes of the AB oscillations decrease with increasing ring
perimeter $2\pi \gamma\rho$ as described by Eq.\
(\ref{eq-correction}). The AB oscillations in the oscillator strength
are plotted in Fig.\ \ref{fig-os-phi-x}. As expected, the shift is
most pronounced for $\Phi= \Phi_0/2$, and the relative magnitude is
nearly $80\%$ for the smallest value of $2\pi \gamma\rho$. For larger
values of $2\pi\gamma\rho$, the oscillations in $F_0(\Phi)$ become
increasingly sinusoidal as can be seen by differentiating Eq.\
(\ref{eq-correction}) with respect to $V_0$.
In the consideration above we assumed the width of the ring to be
zero. In fact, if the width is finite but smaller than the radius of
the exciton, $\gamma^{-1}$, it can be taken into account in a similar
fashion as in \cite{WenF95} by adding $\hbar^2 \pi^2 / 2 m_e W^2$ and
$\hbar^2 \pi^2 / 2 m_h W^2$ to the single-electron and single-hole
energies (\ref{eq-energies}), respectively. Here, $W$ stands for the
width of the ring and a hard-wall confinement in the radial direction
is assumed. This would leave the AB oscillations unchanged. In the
opposite case $W \gg \gamma^{-1}$ the oscillations are suppressed. The
precise form of the suppression factor as a function of
$(W\gamma)^{-1}$ is unknown and depends on the details of the
confinement.
Let us briefly address the excited states of the exciton corresponding
to $m>0$. In the limit $|V_0| \gg \varepsilon_0$ for the energies with
numbers $m < |V_0|/\varepsilon_0$ we get from Eq.\ (\ref{eq-form})
\begin{equation}
\label{eq-excited}
\Delta_0^m = \frac{\varepsilon_0}{4} \Bigl[ m^2 +
(-1)^m(m+\frac{1}{2}) \frac{\varepsilon_0}{\pi^2 V_0}
\cos\Bigl(\frac{2\pi\Phi}{\Phi_0}\Bigr) \Bigr].
\end{equation}
In contrast to the ground state as in (\ref{eq-correction}) the AB
contribution to the energy $\Delta_0^m$ is not exponentially small.
Still the AB term is small (in parameter $\varepsilon_{0} / |V_0| \ll
1$) compared to the level spacing at $\Phi=0$.
An alternative way to derive Eq.\ (\ref{eq-form}) is to follow the
Bethe ansatz approach.\cite{Mat93} The intimate relation between Eq.\
(\ref{eq-form}) and a Bethe ansatz equation becomes most apparent in
the absence of magnetic flux, $\Phi=0$, when (\ref{eq-form}) can be
rewritten as
\begin{equation}
\label{eq-bethe}
2\pi\rho k_m= 2 \pi m + 2 \arctan \Bigl(\frac{\rho k_m}{c}\Bigr),
\end{equation}
where $k_m=(2\Delta_0^m\mu)^{1/2}/\hbar$ is the wave vector and $c= 2
\pi \mu V_0 \rho^2/\hbar^2$ parameterizes the strength of the
attraction analogously to the well-known $\delta$-function gas
.\cite{LieL63,Lie63,Mcg64} At finite flux, the structure of the Bethe
ansatz equations will be very similar to the equations for a 1D
Hubbard model \cite{LieW68} in the presence of a spin flux coupling to
the spin-up and spin-down degrees of freedom of the electrons
.\cite{RomS94b,RomP95} We emphasize that in such discrete models the
periodicity will also be influenced by whether the number of sites in
the ring is even or odd \cite{WuM91} in addition to the continuous
situation considered in the present manuscript.
First experimental studies of the AB effect were carried out on
metallic rings.\cite{Was91} The next generation of rings were based on
GaAs/AlGaAs hetereostructures as in Refs.\ \onlinecite{MaiCB93} and
\onlinecite{YacHMS95} and had a circumference of $\sim 6000$nm and
$3000$nm, respectively. For such rings the magnitude of the excitonic
AB oscillations will be very small. However, quite recently much more
compact ring-shaped dots of InAs in GaAs with a circumference of $\sim
250$nm were demonstrated to exist.\cite{LorLFK99,LorLGK99} This was
achieved by modification of a standard growth procedure
\cite{LeoKRD93} used for the fabrication of arrays of self-assembled
InAs quantum dots in GaAs. Recent light absorption experiments on
nano-rings reveal an excitonic structure.\cite{PetWLK00} However, it
is much more advantageous to search for the AB oscillations proposed
in the present paper not in absorption, but in luminescence studies.
This is because near-field techniques developed in the last decade
allow to "see" a single quantum dot and thus avoid the inhomogeneous
broadening. This technique was applied to many structures containing
ensembles of quantum dots ({\em e.g.},
GaAs/AlGaAs,\cite{BruBAW92,BruABT94,BruABT94b,HesBHP94,ZreBHA94,BocRFH96,GamSSK96,GamSSK96b,GamBSK97,WegSAR97,BonEGP98}
ZnSe\cite{KumWBF98}). In particular, extremely narrow and temperature
insensitive (up to $50$K) luminescence lines from a single InAs
quantum dot in GaAs were recorded in Refs.\
\onlinecite{MarGIB94,GruCLB95,DekGES98}.
In conclusion, we have demonstrated the AB oscillations for a {\em
neutral} object. This constitutes the main qualitative difference
between our paper and previous considerations \cite{WenFC95} for two
interacting {\em electrons} on a ring. Lastly, we note that the
possibility of the related effect of Aharonov-Casher oscillations for
an exciton was considered previously in Ref.\ \onlinecite{KriK94}. The
underlying physics in Ref.\ \onlinecite{KriK94} is that even a {\em
zero-size} exciton having zero charge can still have a finite {\em
magnetic moment}.
Upon completion of this work we have been made aware of Ref.\
\onlinecite{Cha95} in which the underlying physics of the AB
oscillations of excitonic levels was uncovered. Although the
analytical approach employed in Ref.\ \onlinecite{Cha95} is different
from ours, the result obtained for the ground state energy is similar
to Eq.\ (\ref{eq-correction}).
\acknowledgements
This work was supported by the NSF-DAAD collaborative research grant
INT-9815194. MER was supported in part by NSF grant DMR 9732820. RAR
also gratefully acknowledges the support of DFG under
Sonderforschungsbereich~393. We are grateful to M.\ B\"{u}ttiker, A.\
Lorke, T.\ V.\ Shahbazyan, R.\ Warburton, and J.\ Worlock for useful
discussions. We thank A.\ V.\ Chaplik for pointing out Ref.\
\onlinecite{Cha95} to us.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,790
|
Having the top spot at a hotel chain undergoing change, particularly one that's a decades-old, formerly privately held family firm whose founding patriarch's name still defines the company's identity, is likely not the easiest place to be. But that's just where John M. Kidd finds himself some seven months after being named CEO/COO of Carlson Hotels.
The Minnetonka, MN-based company, parent to such well-known brands as Radisson, Radisson Red, Radisson Blu, Country Inns & Suites, Park Inn by Radisson, Park Plaza and Quorvus Collection, was acquired last year by China-based HNA Hospitality Group, where Kidd served as president/COO.
Kidd is the third executive to occupy the C-suite since Carlson Hotels was acquired. Initially, incumbent CEO David Berg was slated to remain in place; however, he was replaced this past January by Federico González Tejera. Tejera resigned a few months later to become president/CEO of Rezidor Hotel Group, the Brussels-based master franchisor of Carlson Hotels outside of the United States.
Kidd indicated the overall plan for the chain is to become a high-profile player.
After 20 years with Hilton, Kidd spent five years with HNA in Beijing with several "terrific assignments," most recently spending 10 months in French Polynesia where he put together a common platform for HNA's hotels, airlines and travel agencies, before being tapped for the CEO post.
Also on tap is accelerating the transformation of Country Inns & Suites to a Gen 4 product, which has proven successful, based on owner validation and increases in ADR and NPS, according to the company. "We strongly believe our Country Inns & Suites portfolio can be three to four times larger than it is today," said the CEO.
Purely organic growth and an asset-light strategy in the Americas with select managed contracts (when it make sense) will be the order of the new day. An aggressive development approach also is on the agenda.
On his personal agenda, Kidd enjoys sharing quality time at home on the weekends with his wife, entertaining family and friends. "I have particularly enjoyed American football since I was in high school in California, as well as watching the mighty All Blacks rugby team of my home country, New Zealand. I also enjoy researching assorted topics—usually related to business—and in between, trying to fit in a session or two in the gym, although the latter activity is sadly sometimes difficult to maintain consistently," said Kidd.
As its new leader, the CEO said he understands the reason Carlson Hotels is a great company is due to its legacy of outstanding family entrepreneurship.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 474
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Competitive Trampolining
We accept all applicants to trial at City of Salford Trampoline and DMT Club with levels to suit all gymnasts from basic recreational to International performance.
(All members of the squad must be a silver member of British Gymnastics at the cost of £42 and member of the North West at the cost of £5. These can be purchased at the British Gymnastic website at: www.british-gymnastics.org)
We will always suggest the best course of action for the gymnast and advise a level which will provide the best environment for their progress. The levels below will allow you to see what we look for in the gymnasts that are joining the squad.
This is a development squad and is the very start of their squad career. We look for those who wish to trial to have very good basics with all body landings and twisting to and from the body landings. We also would require the gymnast to be able to salto forward and backwards with or without twist, these should all be shown with good control.
NDP Squad
This is the final level where the gymnasts is at a development stage. At this stage the gymnasts will be competing at a low Regional NDP level with all basic skills in good order and being able to complete around 6-8 salto's within a 10 bounce routine.
FIG Development
The first stage of being on full squad where a gymnast has shown improvements in the skills on and off the trampoline. Dedication to stretching and core and body fitness is equally as important to a gymnast as being able to perform multiple salto's with and without twist. At this stage the gymnasts should be performing at a high NDP level Regionally or Nationally.
The highest level of squad we do. This is at the stage where a gymnast should be performing at the highest National level, (FIG). They should be committed and at this stage performing double salto's with and without twist. This is an International squad standard that has produced Olympic Gymnasts and countless National Champions.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,997
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Fairphone launches second edition of its ethical smartphone
Emma Tucker | 29 September 2015 1 comment
London Design Festival 2015: Amsterdam tech company Fairphone has launched a new version of its ethically produced smartphone.
The Fairphone 2 has been created in partnership with London design and branding agency Seymourpowell, and features a five-inch (12.7 centimetre) LCD display covered in damage-resistant Gorilla Glass.
The handset runs on the Android 5.1 Lollipop operating system, and offers 32 gigabytes (GB) of storage. It's bigger in size and memory than its predecessor, which had a 4.3-inch (10.9 centimetre) screen, and just 16GB of memory.
The phone is designed to be easily taken apart and repaired to increase its lifespan – similar to Google's Project Ara prototype – as an alternative to current fast-moving smartphone trends.
Health-monitoring components designed for Google's Project Ara modular smartphone
Fairphone hopes this will reduce the enormous amount of electronic waste currently contributed by discarded consumer electronics – reported to be 41 million tonnes worldwide.
The handset also continues the company's focus on transparent and ethical production, and a complete breakdown of costs is available on the Fairphone site. The report details everything from taxes and reseller margin, to product and investment costs.
The original Fairphone – purchased by 60,000 people – was launched in 2013 and included on the Designs of the Year 2014 shortlist.
The company used ethical material sources – such as conflict-free mines in the Democratic Republic of Congo – to produce the phone, in addition to setting up a worker welfare fund in the Chinese factory the phones were built in. A portion of every Fairphone 2 sale will go directly to this fund.
Fairphone's Amsterdam offices built inside an old warehouse using reclaimed materials
The company reflected its ethical focus in the design of its Amsterdam warehouse offices, which left many of the building's original features intact and used sustainable materials for furnishing.
"Almost 14,000 Fairphone 2 buyers have already voted with their wallets for a fairer, more sustainable economy and with the phone starting delivery in the next month, and becoming more widely available through partners like The Phone Co-op, we believe the movement will grow," said Fairphone founder and CEO Bas van Abel.
Fairphone 2 was launched on 25 September 2015 during this year's London Design Festival, which took place from 19 to 27 September. Also at this year's festival, Swiss technology company Punkt collaborated with designer Jasper Morrison to develop a stripped-back basic mobile handset that would be a "liberating" alternative to smartphones.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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{"url":"https:\/\/www.groundai.com\/project\/generalised-cp-and-a_4-family-symmetry\/","text":"A Group theory of A_{4}\n\n# Generalised CP and A4 Family Symmetry\n\n## Abstract\n\nWe perform a comprehensive study of family symmetry models based on combined with the generalised CP symmetry . We investigate the lepton mixing parameters which can be obtained from the original symmetry breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely in the neutrino sector and in the charged lepton sector, leading to the prediction of no CP violation, namely and the Majorana phases and are all equal to either zero or . We then propose an effective supersymmetric model based on the symmetry in which trimaximal lepton mixing is predicted together with either zero CP violation or with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with or maximally broken with and sharp predictions for the Majorana phases and neutrinoless double beta decay.\n\n## 1 Introduction\n\nAfter the measurement of the reactor mixing angle by the Daya Bay\u00a0[1], RENO\u00a0[2], and Double Chooz\u00a0[3] reactor neutrino experiments, all three lepton mixing angles , , and both mass-squared differences and have been measured to reasonably good accuracy. Yet within the standard framework of three-neutrino oscillations, the Dirac CP phase and neutrino mass ordering still elude measurement so far. Furthermore, if neutrinos are Majorana particles, there exist two more unknown Majorana CP phases which may play a role in neutrinoless double-beta decay searches. Thus, determining the exact neutrino mass ordering and measuring the Dirac and Majorana CP violating phases are the primary goals of future neutrino oscillation experiments. The CP violation has been firmly established in the quark sector and it is natural to expect that CP violation occurs in the lepton sector as well. It is insightful to note that hints of a nonzero have begun to show up in global analysis of neutrino oscillation data\u00a0[4, 5, 6].\n\nWhat would we learn from the measurements of the lepton CP violating phases? What is the underlying physics? These questions are particularly imperative in view of foreseeable future experimental programs to measure the CP-violation in the neutrino oscillations sector. In the past years, much effort has been devoted to explaining the structure of the lepton mixing angles through the introduction of family symmetries. In this scheme, one generally assumes a non-abelian discrete flavour group which is broken to different subgroups in the neutrino and charged lepton sectors. The mismatch between these two subgroups leads to particular predictions for the lepton mixing angles. For recent reviews, see Ref.\u00a0[7] and Ref.\u00a0[8] for the model building and relevant group theory aspects, respectively. Motivated by this approach one can extend the family symmetry to include a generalised CP symmetry \u00a0[9] which will allow the prediction of both CP phases and mixing angles.\n\nThe possibility of combining a family symmetry with a generalised CP symmetry has already been discussed in the literature. For example, the simple reflection symmetry, which is a combination of the canonical CP transformation and the exchange symmetry, has been discussed and successfully implemented in a number of models where both atmospheric mixing angle and Dirac CP phase were predicted to be maximal[10, 11, 12]. Additionally in Ref.\u00a0[13], the phenomenological consequences of imposing both an flavour symmetry and a generalised CP symmetry have been analysed in a model-independent way. They found that all lepton mixing angles and CP phases depend on one free parameter for the symmetry breaking of to in the neutrino sector and to some abelian subgroup of in the charged lepton sector. Concrete family models with a generalised CP symmetry have been constructed in Refs.\u00a0[14, 15, 16] where the spontaneous breaking of the down to in the neutrino sector was implemented. Other models with a family symmetry and a generalised CP symmetry can also be found in Refs.\u00a0[17, 18, 19]. In addition, there are other theoretical frameworks comprising both family symmetry and CP violation\u00a0[20, 21, 22].\n\nIn this work, we study generalised CP symmetry in the context of the most popular family symmetry 4 (please see Ref.\u00a0[25, 26] for a classification of the models on the market). The generalised CP transformation compatible with an family symmetry is clarified, and a model-independent analysis of the lepton mixing matrix is performed by scanning all of the possible remnant subgroups in the neutrino and charged lepton sectors. We construct an effective model, where non-renormalisable operators are involved. The lepton mixing is predicted to be trimaximal pattern in the model, and the Dirac phase is trivial or nearly maximal. Furthermore, this effective model is promoted to a renormalisable one in which the higher order operators are under control.\n\nThe remainder of this paper is organised as follows. In Section\u00a02, we present the general CP transformations consistent with the family symmetry. In Section\u00a03, we perform a thorough scan of leptonic mixing parameters which can be obtained from the remnant symmetries of the underlying combined symmetry group . We find that only one case out of all possibilities is phenomenologically viable. This case predicts both Dirac and Majorana phases to be trivial. In Section\u00a04 we specify the structure of the model at leading order, and the required vacuum alignment is justified. In subsection 4.3, we analyse the subleading Next-to-Leading-Order (NLO) corrections induced by higher dimensional operators and phenomenological predictions of the model are presented. In Section\u00a05, we address the ultraviolet completion of the model which significantly increases the predictability of the theory such that all the mixing angles, CP phases and the absolute neutrino mass scale are fixed. We conclude in Section\u00a06. The details of the group theory of are collected in Appendix\u00a0A and Appendices B-D contain the implications of preserving other subgroups of different than and . Finally, Appendix\u00a0E describes the diagonalisation of a general symmetric complex matrix.\n\n## 2 Generalised CP transformations with family symmetry\n\n### 2.1 General family symmetry group\n\nIn general, it is nontrivial to combine the family symmetry and the generalised CP symmetry together because the definition of the generalised CP transformations must be compatible with the family symmetry. Thus, the generalised CP transformations are subject to certain consistency conditions\u00a0[27, 13, 28]. Namely, for a set of fields in a generic irreducible representation of , it transforms under the action of as\n\n \u03c6(x)\\lx@stackrelGf\u27f6\u03c1r(g)\u03c6(x),g\u2208Gf, (2.1)\n\nwhere denotes the representation matrix for the element in the irreducible representation , the generalised CP transformation is of the form\n\n \u03c6(x)\\lx@stackrelCP\u27f6Xr\u03c6\u2217(x\u2032), (2.2)\n\nwhere and the obvious action of CP on the spinor indices is omitted for the case of being spinor. Here we are considering the \u201cminimal\u201d theory in which the generalised CP transforms the field into its complex conjugate , and the transformation into another field with is beyond the present scope since both and would be required to be present in pair and correlated with each other in that case. Notice that should be a unitary matrix to keep the kinetic term invariant. Now if we first perform a CP transformation, then apply a family symmetry transformation, and finally an inverse CP transformation is followed, i.e.\n\n \u03c6(x)\\lx@stackrelCP\u27f6Xr\u03c6\u2217(x\u2032)\\lx@stackrelGf\u27f6Xr\u03c1\u2217r(g)\u03c6\u2217(x\u2032)\\lx@stackrelCP\u22121\u27f6Xr\u03c1\u2217r(g)X\u22121r\u03c6(x), (2.3)\n\nthe theory should still be invariant since it is invariant under each transformation individually. To make the theory consistent the resulting net transformation should be equivalent to a family symmetry transformation of some family group element , i.e.\n\n Xr\u03c1\u2217r(g)X\u22121r=\u03c1r(g\u2032),g\u2032\u2208Gf, (2.4)\n\nwhere the elements and must be the same for all irreducible representations of . Eq. (2.4) is the important consistency condition which has to be fulfilled in order to impose both generalised CP and family symmetry invariance simultaneously. It also implies that the generalised CP transformation maps the group element into and that the family group structure is preserved under this mapping. Therefore Eq. (2.4) defines a homomorphism of the family symmetry group . Notice that in the case where is a faithful representation, the elements and have the same order, the mapping defined in Eq. (2.4) is bijective, and thus the associated CP transformation becomes an automorphism [28]. It is notable that both and also satisfy the consistency equation of Eq.\u00a0(2.4) for a generalised CP transformation , where is real and is any element of . Therefore the possible form of the CP transformation is only determined by the consistency equation up to an overall arbitrary phase and family symmetry transformation for a given irreducible representation . In the following, we investigate the generalised CP transformations consistent with an family symmetry for different irreducible representations, i.e. .\n\n### 2.2 A4 family symmetry\n\nThe group can be generated by two generators and , which are of orders two and three, respectively (see Appendix\u00a0A for the details of the group theory of ). To include a generalised CP symmetry consistent with an family symmetry, it is sufficient to only impose the consistency condition in Eq.\u00a0(2.4) on the group generators:\n\n Xr\u03c1\u2217r(S)X\u22121r=\u03c1r(S\u2032),Xr\u03c1\u2217r(T)X\u22121r=\u03c1r(T\u2032). (2.5)\n\nTo do this, we start with the faithful triplet representation . Then the order of and will be 2 and 3, respectively. Therefore and can only belong to certain conjugacy classes of . Namely,\n\n S\u2032\u22083C2,T\u2032\u22084C3\u222a4C23 (2.6)\n\nIt is remarkable that the consistency condition of Eq.\u00a0(2.4) must hold for all representations simultaneously. However, because of the models constructed in later sections, we assume that our theory contains only one of the nontrivial singlet irreducible representations (either or ) in the flavon sector and further restrict ourselves to a minimal case where there exists only one flavon transforming under that nontrivial singlet irreducible representation (in addition to other flavons transforming under the and representations). However, in these models there does exist a and in the matter sector. Yet, additional symmetry forbids the interchanging of these fields under the generalised CP symmetry. Therefore we have chosen to define a generalised CP symmetry without the interchanging of fields transforming under conjugate representations, e.g. fields transforming under and representations. Then, the element can further be constrained by these nontrivial singlet representations and , where the corresponding generalised CP transformations are numbers with absolute value equal to 1, and then we have\n\n \u03c11\u2032,1\u2032\u2032(T\u2032)=X1\u2032,1\u2032\u2032\u03c1\u22171\u2032,1\u2032\u2032(T)X\u221211\u2032,1\u2032\u2032=\u03c1\u22171\u2032,1\u2032\u2032(T)=\u03c9\u22132 (2.7)\n\nConsequently, the element can only be in the conjugacy class . In summary, the consistency equation applied to our \u201cminimal\u201d case restricts and to\n\n S\u2032\u22083C2,T\u2032\u22084C23. (2.8)\n\nFor the simple case of and in the -dimensional representation, the associated CP transformation satisfying Eq.\u00a0(2.4) can be found straightforwardly:\n\n X0=\u239b\u239c\u239d100010001\u239e\u239f\u23a0\u2261\\mathbbm13, (2.9)\n\nwhich is the canonical CP transformation. The remaining eleven possible choices for and lead to different solutions for . These solutions are listed in Table\u00a01 and can be neatly summarised in a compact way:\n\n X3=\u03c13(g),g\u2208A4. (2.10)\n\nFor the singlet representations , and , we take\n\n X1,1\u2032,1\u2032\u2032=\u03c11,1\u2032,1\u2032\u2032(g),g\u2208A4. (2.11)\n\nTherefore the generalised CP transformation consistent with an family symmetry is of the same form as the family group transformation, i.e.\n\n Xr=\u03c1r(g),g\u2208A4. (2.12)\n\nNow that we have found all generalised CP transformations consistent with the family symmetry,5 we proceed by investigating their implications on lepton masses and mixings.\n\n## 3 General analysis of lepton mixing from preserved family and CP symmetries\n\n### 3.1 General family symmetry\n\nTo obtain definite predictions for both the lepton mixing angles and CP violating phases from symmetry, we impose the family symmetry and the generalised CP symmetry simultaneously at high energies. Then the family symmetry is spontaneously broken to the and subgroups in the neutrino and the charged lepton sector respectively, and the remnant CP symmetries from the breaking of are and , respectively. The mismatch between the remnant symmetry groups and gives rise to particular values for both mixing angles and CP phases. As usual, the three generations of the left-handed (LH) lepton doublets are unified into a three-dimensional representation of . The invariance under the residual family symmetries and implies that the neutrino mass matrix and the charged lepton mass matrix satisfy\n\n \u03c1T3(g\u03bdi)m\u03bd\u03c13(g\u03bdi)=m\u03bd,g\u03bdi\u2208G\u03bd, \u03c1\u20203(gli)mlm\u2020l\u03c13(gli)=mlm\u2020l,gli\u2208Gl. (3.1)\n\nwhere the charged lepton mass matrix is given in the convention in which the left-handed (right-handed) fields are on the left-hand (right-hand) side of . Moreover, the neutrino and the charged lepton mass matrices are constrained by the residual CP symmetry via\n\n XT3\u03bdm\u03bdX3\u03bd=m\u2217\u03bd,X3\u03bd\u2208H\u03bdCP, X\u20203lmlm\u2020lX3l=(mlm\u2020l)\u2217,X3l\u2208HlCP. (3.2)\n\nSince there are both remnant family and CP symmetries, the corresponding consistency equation similar to Eq.\u00a0(2.4) has to be satisfied. Namely, the elements of and of should satisfy\n\n Xr\u03bd\u03c1\u2217r(g\u03bdi)X\u22121r\u03bd=\u03c1r(g\u03bdj),g\u03bdi,g\u03bdj\u2208G\u03bd, Xrl\u03c1\u2217r(gli)X\u22121rl=\u03c1r(glj),gli,glj\u2208Gl. (3.3)\n\nGiven a set of solutions and , we can straightforwardly check that and are solutions as well. The invariance conditions of Eqs.\u00a0(3.1)-(3.2) allow us to reconstruct the mass matrices and , and eventually determine the lepton mixing matrix . Furthermore, if two other residual family symmetries and are conjugate to and under the element , i.e.\n\n G\u2032\u03bd=hG\u03bdh\u22121,G\u2032l=hGlh\u22121, (3.4)\n\nthen the associated residual CP symmetries and are related to and as\n\n H\u03bd\u2032CP=\u03c1r(h)H\u03bdCP\u03c1Tr(h),Hl\u2032CP=\u03c1r(h)HlCP\u03c1Tr(h), (3.5)\n\nand the corresponding neutrino and charged lepton mass matrices are of the form\n\n m\u2032\u03bd=\u03c1\u22173(h)m\u03bd\u03c1\u20203(h),m\u2032lm\u2032\u2020l=\u03c13(h)mlm\u2020l\u03c1\u20203(h). (3.6)\n\nTherefore, the remnant subgroups and lead to the same mixing matrix as and do.\n\nHaving completed a general discussion of the implementation of a generalised CP symmetry with a family symmetry, we now concentrate on the case of interest in which the family symmetry and a generalised CP symmetry consistent with is imposed. Thus, the theory respects the full symmetry . In the following, we perform a model independent study of the constraints that these symmetries impose on the neutrino mass matrix, the charged lepton mass matrix and the PMNS matrix by scanning all the possible remnant symmetries and . We begin this study with an analysis of the neutrino sector.\n\n### 3.2 Neutrino sector from a subgroup of A4\u22caHCP\n\nAs shown in Appendix\u00a0B, the case is not phenomenologically viable. To resolve this issue, we assume that the underlying symmetry is broken into 6 in the neutrino sector\u00a0[13]. Since the three subgroups in Eq.\u00a0(A.6) are related by conjugation as and , it is sufficient to only consider , where the element of should satisfy\n\n Xr\u03bd\u03c1\u2217r(S)X\u22121r\u03bd=\u03c1r(S). (3.7)\n\nIt is found that only 4 of the 12 non-trivial CP transformations are acceptable7,\n\n H\u03bdCP={\u03c1r(1),\u03c1r(S),\u03c1r(T2ST),\u03c1r(TST2)}. (3.8)\n\nThus, the neutrino mass matrix is constrained by\n\n \u03c1T3(S)m\u03bd\u03c13(S)=m\u03bd, (3.9) XT3\u03bdm\u03bdX3\u03bd=m\u2217\u03bd, (3.10)\n\nwhere Eq.\u00a0(3.9) is the invariance condition under , and it implies that the neutrino mass matrix is of the form\n\n m\u03bd=\u03b1\u239b\u239c\u239d2\u22121\u22121\u221212\u22121\u22121\u221212\u239e\u239f\u23a0+\u03b2\u239b\u239c\u239d100001010\u239e\u239f\u23a0+\u03b3\u239b\u239c\u239d011110101\u239e\u239f\u23a0+\u03f5\u239b\u239c\u239d01\u221211\u221210\u2212101\u239e\u239f\u23a0, (3.11)\n\nwhere , , and are complex parameters, and they are further constrained by the remnant CP symmetry shown in Eq.\u00a0(3.10). In order to diagonalise the neutrino mass matrix in Eq. (3.11), we first apply the tri-bimaximal transformation to yield\n\n m\u2032\u03bd=UTTBm\u03bdUTB=\u239b\u239c\u239d3\u03b1+\u03b2\u2212\u03b30\u2212\u221a3\u03f50\u03b2+2\u03b30\u2212\u221a3\u03f503\u03b1\u2212\u03b2+\u03b3\u239e\u239f\u23a0, (3.12)\n\nwhere\n\n UTB=\u239b\u239c \u239c \u239c \u239c \u239c\u239d\u221a231\u221a30\u22121\u221a61\u221a3\u22121\u221a2\u22121\u221a61\u221a31\u221a2\u239e\u239f \u239f \u239f \u239f \u239f\u23a0. (3.13)\n\nNow we return to the investigation of the residual CP symmetry constraint of Eq.\u00a0(3.10). Two distinct phenomenological predictions arise for the different choices of :\n\n\u2022 For this case, we see that we can straightforwardly solve Eq.\u00a0(3.10) and find that all four parameters , , and are real. Then can be further diagonalised by\n\n U\u2032T\u03bdm\u2032\u03bdU\u2032\u03bd=diag(m1,m2,m3),U\u2032\u03bd=R(\u03b8)P, (3.14)\n\nwhere is a unitary diagonal matrix with entries or which renders the light neutrino masses positive, and\n\n R(\u03b8)=\u239b\u239c\u239dcos\u03b80sin\u03b8010\u2212sin\u03b80cos\u03b8\u239e\u239f\u23a0 (3.15)\n\nis a rotation matrix with\n\n tan2\u03b8=\u221a3\u03f5\u03b2\u2212\u03b3. (3.16)\n\nThis diagonalisation reveals that the light neutrino masses are given by\n\n m1=\u2223\u2223\u22233\u03b1+sign((\u03b2\u2212\u03b3)cos2\u03b8)\u221a(\u03b2\u2212\u03b3)2+3\u03f52\u2223\u2223\u2223, m2=|\u03b2+2\u03b3|, m3=\u2223\u2223\u22233\u03b1\u2212sign((\u03b2\u2212\u03b3)cos2\u03b8)\u221a(\u03b2\u2212\u03b3)2+3\u03f52\u2223\u2223\u2223. (3.17)\n\nWe conclude that this case is acceptable.\n\n\u2022 In this case, it can be seen that the of Eq.\u00a0(3.11) is purely imaginary, and the remaining parameters , and are real. Then the hermitian combination turns out to be of the form:\n\n m\u2032\u2020\u03bdm\u2032\u03bd=diag(\u22129\u03b12+(\u03b2\u2212\u03b3)2+3\u03f52,(\u03b2+2\u03b3)2,\u22129\u03b12+(\u03b2\u2212\u03b3)2+3\u03f52), (3.18)\n\nwhich implies . Clearly, this is not consistent with the experimental observation that the three light neutrinos have different masses. Note that the generalised CP transformations are not symmetric in the chosen basis, and hence we confirm the argument of Ref.\u00a0[13] that non-symmetric CP transformations consistent with the remnant family symmetry in the neutrino sector lead to partially degenerate neutrino masses.\n\nSince the remaining choices or are related to the discussed case by conjugation, the corresponding remnant CP symmetry is or , respectively, where is given by Eq.\u00a0(3.8). Then their corresponding neutrino mass matrices are of the form or , respectively, with given in Eq.\u00a0(3.11). Now that we have finished a systematic discussion of the effects of the residual flavour and CP symmetries on the neutrino mass matrix, we turn to analyse their effects on the charged lepton mass matrix.\n\n### 3.3 Charged lepton sector from a subgroup of A4\u22caHCP\n\nIn Appendices\u00a0C and D we consider the cases and and show that they are not phenomenologically viable. Here we consider the successful case that is one of the subgroups shown in Eq.\u00a0(A.7). Since the four subgroups are conjugate to each other, i.e.\n\n (TST2)ZT3(TST2)\u22121=ZST3,(T2ST)ZT3(T2ST)\u22121=ZTS3,SZT3S=ZSTS3, SZST3S=ZTSS,(T2ST)ZST3(T2ST)\u22121=ZSTS3,(TST2)ZTS3(TST2)\u22121=ZSTS3, (3.19)\n\nwe choose for demonstration. Then the combined symmetry group is broken to in the charged lepton sector. The element of should satisfy the consistency equation8\n\n Xrl\u03c1\u2217r(T)X\u22121rl=\u03c1r(T2). (3.20)\n\nIt is found that the remnant CP transformation can be\n\n Missing \\left or extra \\right (3.21)\n\nSimilar to the neutrino mass matrix, the charged lepton mass matrix must respect both the residual family symmetry and the generalised CP symmetry , i.e.\n\n \u03c1\u20203(T)mlm\u2020l\u03c13(T)=mlm\u2020l, \u03c1\u20203(1)mlm\u2020l\u03c13(1)=(mlm\u2020l)\u2217, (3.22)\n\nwhere from Eq. (3.21) has been taken. For the value or , the resulting constraint is equivalent to Eq.\u00a0(3.22). One can easily see that is diagonal in this case,\n\n mlm\u2020l=diag(m2e,m2\u03bc,m2\u03c4), (3.23)\n\nwhere , and are the electron, muon and tau masses, respectively. For the other choices and , the corresponding residual CP symmetry and the mass matrix follow from the general relations Eq.\u00a0(3.5) and Eq.\u00a0(3.6) immediately with and , respectively.\n\n### 3.4 Lepton mixing from A4\u22caHCP broken to G\u03bdCP\u2245ZS2\u00d7H\u03bdCP and GlCP\u2245ZT3\u22caHlCP\n\nIn the context of family symmetry and its extension of including generalised CP symmetry, a specific lepton mixing pattern arises from the mismatch between the symmetry breaking in the neutrino and the charged lepton sectors. In this section, we perform a comprehensive analysis of all possible lepton mixing matrices obtainable from the implementation of an family symmetry and its corresponding generalised CP symmetry by considering all possible residual symmetries and discussed in previous sections.\n\nImmediately we can disregard the cases predicting partially degenerate lepton masses. Therefore, breaking to the subgroups or will be neglected in the following. Furthermore, in order that the elements of and give rise to the entire family symmetry group , we take to be one of the subgroups shown in Eq.\u00a0(A.7). Then, there are combinations for and . However, we find that all of these are conjugate to each other9. As a result, all possible symmetry breaking chains of this kind lead to the same lepton mixing matrix . This important point is further confirmed by straightforward calculations which are lengthy and tedious.\n\nWithout loss of generality, it is sufficient to consider the representative values and , and the original symmetry is broken to in the neutrino sector and in the charged lepton sector, where 10 and . In this case, is diagonal as shown in Eq.\u00a0(3.23). Therefore, no rotation of the charged lepton fields is needed to get to the mass eigenstate basis, and the lepton mixing comes completely from the neutrino sector. In the PDG convention\u00a0[29], the PMNS matrix is cast in the form\n\n UPMNS=Vdiag(1,ei\u03b1212,ei\u03b1312), (3.24)\n\nwith\n\n V=\u239b\u239c\u239dc12c13s12c13s13e\u2212i\u03b4CP\u2212s12c23\u2212c12s23s13ei\u03b4CPc12c23\u2212s12s23s13ei\u03b4CPs23c13s12s23\u2212c12c23s13ei\u03b4CP\u2212c12s23\u2212s12c23s13ei\u03b4CPc23c13\u239e\u239f\u23a0. (3.25)\n\nwhere we use the shorthand notation and , is the Dirac CP phase, and are the Majorana CP phases. Using this PDG convention we find that the resulting PMNS matrix is:\n\n UPMNS=UTBR(\u03b8)P=\u239b\u239c \u239c \u239c \u239c \u239c\u239d2\u221a6cos\u03b81\u221a32\u221a6sin\u03b8\u22121\u221a6cos\u03b8+1\u221a2sin\u03b81\u221a3\u22121\u221a6sin\u03b8\u22121\u221a2cos\u03b8\u22121\u221a6cos\u03b8\u22121\u221a2sin\u03b81\u221a3\u22121\u221a6sin\u03b8+1\u221a2cos\u03b8\u239e\u239f \u239f \u239f \u239f \u239f\u23a0P, (3.26)\n\nwhere as shown previously is a unitary diagonal matrix with entries or and and are given in Eq.\u00a0(3.13) and Eq.\u00a0(3.15). Hence, the lepton mixing angles and CP phases are\n\n sin\u03b4CP=sin\u03b121=sin\u03b131=0, sin2\u03b813=23sin2\u03b8,sin2\u03b812=12+cos2\u03b8=13cos2\u03b813,sin2\u03b823=12[1+\u221a3sin2\u03b82+cos2\u03b8], (3.27)\n\nwhich implies the three CP phases , , , and therefore there is no CP violation in this case. Note that the same results are found in Ref.\u00a0[13].\n\nTo summarise the arguments of the preceding section, if one imposes the symmetry , which is spontaneously broken to certain residual family and CP symmetries in order to obtain definite predictions for mixing angles and CP phases, then only the symmetry breaking of to in the neutrino sector and in the charged lepton sector can lead to lepton mixing angles in the experimentally preferred range. However, there is no CP violation in this case. This is consistent with the result found for for the case where with [14]. For it was possible to achieve maximal CP violation for the case with . This case is not directly accessible for since the generator is absent, although it is accidentally present at LO in the models that we now discuss.\n\n## 4 Model with A4 and generalised CP symmetries\n\nGuided by the general analysis of previous sections, we construct an effective model in this section. The predictions of Eq.\u00a0(3.27) are realised if the remnant CP is preserved otherwise the Dirac CP phase is approximately maximal. The model is based on , which is supplemented by the extra symmetries . 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Four killed, 27 injured in Meghalaya storm
Four people, including a six-month-old baby, were killed and at least 27 others injured in a cyclonic storm in Meghalaya's West Khasi Hills district, officials said Thursday.
High velocity winds, accompanied by rain and lightning late Wednesday night rendered hundreds of families homeless and uprooted trees, and telephone and electric poles.
Four people died on the spot when their house was destroyed in the cyclone, West Khasi Hills District Magistrate S. Kharlyngdoh told IANS over phone from Nongstoin, 94 km from Shillong.
Those killed have been identified as Edmund Nongsieg, 20, Pynibor Nongsieg, 12, Phibanroi Marngar, 25, and her six-month-old baby.
He said 27 people have been admitted to a government hospital at Nongstoin. While 15 of them are in critical condition, seven were discharged after first aid.
Officials have rushed to the six affected villages to take stock of the situation, while the government has set up relief camps at four places to accommodate the displaced villagers.
Deputy Chief Minister R.C. Laloo, who is also in charge of revenue and disaster management, announced an ex-gratia payment of Rs.1.5 lakh each to the kin of those killed in the storm.
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Nevro Corp.'s (NVRO - Free Report) fourth-quarter 2018 adjusted loss per share was 32 cents, wider than the Zacks Consensus Estimate of a loss of 17 cents. Notably, the year-ago quarter's loss per share was 15 cents.
For 2019, Nevro expects revenues of $400-$410 million. The mid-point of $405 million of the guided range lies slightly below the Zacks Consensus Estimate of $405.8 million. Following the earnings announcement, shares of this Zacks Rank #4 (Sell) company fell 7.5% to close at $43.85 on Mar 13.
Fourth-quarter revenues totaled $107.9 million, edging past the Zacks Consensus Estimate of $107 million. Revenues improved 10.2% year over year.
Full-year revenues totaled $387.3 million, up 18.6% from 2017.
2018 adjusted loss per share was $1.64 compared with a loss of $1.25 of 2017.
In the quarter under review, international revenues were $16.3, up 1% at constant currency (cc). Per management, results were driven by robust demand for the company's flagship H10 therapy.
U.S. revenues for the quarter totaled $91.6 million, reflecting a 13% year-over-year increase on continued Senza system adoption.
Additionally, the company launched Senza II, a smaller-footprint, advanced battery system, and received FDA approval for Conditional Full Body MRI.
At the 21st Annual Meeting of the North American Neuromodulation Society, Nevro presented positive clinical trial results.
In the fourth quarter, gross profit totaled $76.2 million, up 9.6% year over year. As a percentage of revenues, gross margin in the quarter was 70.5%, down 50 basis points (bps).
Research and development expenses totaled $13.5 million, up 35.6% year over year.
Sales, general and administrative totaled $71.3 million, up 15.4%.
Total operating expenses in the quarter were $84.8 million, up 18.2%.
Operating loss of $8.6 million was significantly wider than the year-ago loss of $2.2 million.
Nevro exited the fourth quarter on a tepid note. Loss per share widened on a year-over-year basis along with the issuance of a downbeat guidance for 2019. In fact, management expects growth in U.S. revenues to be partially offset by declines in international revenues. Contraction in gross margin adds to the woes.
On the bright side, surge in domestic and international revenues is promising. The company continues to gain from flagship platforms like H10 and Senza.
Some better-ranked MedTech stocks that delivered solid quarterly results are Varian Medical Systems (VAR - Free Report) , Stryker Corporation (SYK - Free Report) and CONMED Corporation (CNMD - Free Report) . Notably, each of these stocks carries a Zacks Rank #2 (Buy). You can see the complete list of today's Zacks #1 Rank (Strong Buy) stocks here.
Varian reported first-quarter fiscal 2019 adjusted earnings per share of $1.06, in line with the Zacks Consensus Estimate. Revenues of $741 million outpaced the consensus mark of $717.9 million.
Stryker delivered fourth-quarter 2018 adjusted earnings per share of $2.18, beating the Zacks Consensus Estimate by 1.4%. Revenues of $3.80 billion were well ahead of the Zacks Consensus Estimate of $3.73 billion.
CONMED delivered fourth-quarter 2018 adjusted earnings per share of 73 cents, in line with the Zacks Consensus Estimate. Revenues of $242.4 million outshined the Zacks Consensus Estimate of $229.2 million.
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Bandits Are Erecting Illegal Structures Within And Outside University Premises, Attacking Students – UniAbuja VC , Na'Allah Cries For Help
September 10, 2020 admin Crime/Security Reports, Latest, News 0
The Vice-Chancellor of the University of Abuja, Professor Abdul-Rasheed Na'Allah has raised concerns over persistent efforts by bandits to take over the institution's land, The Whistler reports.
Na'Allah, said that the bandits have been erecting illegal structures within the premises of the federal university and attempts made to stop or recover the land occupied by them had resulted in attacks.
He disclosed this on Tuesday while receiving the Director, Department of Development Control of the Federal Capital Territory, Murkta Galadima, at the university.
The vice chancellor said that the worsening situation had put fear in staff residing within the premises of the university as the bandits had attacked students on several occasions.
He calls for assistance from the FCT minister, while noting that the matter was preventing the management from carrying out developmental projects withing the university.
He said, "The truth is that this great University of ours is in trouble right now. I say this because we have all sorts of people who are living on the campus, who are using the land without respect for the environment. It is worse now, because bandits are coming in and taking over land both within staff residences and outside.
"Our students are being attacked; we are living in fear of the bandits. Some of the indigenes give land to these bandits without really knowing them, some don't even ask for permission they just take over. I can tell you that on many occasions, we have had clashes.
"They are called non state actors. If we fail to act the future will never forgive us, this is serious and dangerous and that is why we are appealing to the Minister of FCT, Mallam Muhammad Bello, to assist us. We need FCT to work with us to the very end to get this campus safe," he said.
Na'Allah said the bandits were fond of demanding money from the university's management whenever it wanted to put its land to use.
"Whenever we want to construct any structure, they attack us and many times we had to negotiate with them sometimes, they collect money from the University just to use the land of the university. I am tired of negotiating with bandits," he said.
The vice chancellor also thanked the federal government for approving N400 million for fencing of the institution's premises. The said the project would commence between now and December.
In his remark, Galadima recommended that due process should be followed in removing the illegal structures erected by the bandits.
"The rate of squatting, to encroachment and invasion so all these things are beyond this office, it has to be done with FCT administration. We shall meet with the university, village Chiefs, Area Council Secretariat and tell them these are the challenges. Resettlement will also come in. We have already started the process. The next line is to come up with action plan.
"What we will do first is to sensitize these villages and map out strategies of the date and time to remove the structures," he said.
Drama As Wedding Was Canceled After Man Stormed Ceremony, Said He's Father To Bride And Groom
Friendship/Relationship Teaching With Pst Mrs. Banke Bello.
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Produced by KD Weeks, Chris Curnow and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)
------------------------------------------------------------------------
Transcriber's Note:
This version of the text cannot represent certain typographical effects.
Italics are delimited with the '_' character as _italic_. =Bold font= is
indicated with the '=' character.
Footnotes are limited to a single quoted passage, and have been
relocated to follow that passage.
Minor errors, attributable to the printer, have been corrected. Please
see the transcriber's note at the end of this text for details regarding
the handling of any textual issues encountered during its preparation.
TOBACCO:
GROWING, CURING, AND MANUFACTURING.
------------------------------------------------------------------------
TOBACCO:
GROWING, CURING, & MANUFACTURING.
A HANDBOOK FOR PLANTERS
IN ALL PARTS OF THE WORLD.
EDITED BY
C. G. WARNFORD LOCK, F.L.S.
[Illustration]
E. & F. N. SPON, 125, STRAND, LONDON.
NEW YORK: 35, MURRAY STREET.
1886.
PREFACE.
Tobacco growing is one of the most profitable branches of tropical and
sub-tropical agriculture; the$1"$2"$3has even been proposed as a
remunerative crop for the British farmer, and is very extensively grown
in continental Europe. The attention recently drawn to the subject has
resulted in many inquiries for information useful to the planter
desirous of starting a tobacco estate. But beyond scattered articles in
newspapers and the proceedings of agricultural societies, there has been
no practical literature available for the English reader. It is a little
remarkable that while our neighbours have been writing extensively about
tobacco growing, of late years, no English book devoted exclusively to
this subject has been published for nearly thirty years. A glance at the
bibliography given at the end of this volume will show that the French,
German, Swiss, Italian, Dutch, Sicilian, and even Scandinavian planter
has a reliable handbook to guide him in this important branch of
agriculture, while British settlers in our numerous tobacco-growing
colonies must glean their information as best they may from periodical
literature.
To supply the want thus indicated, the present volume has been prepared.
The invaluable assistance of tobacco-planters in both the Indies and in
many other tropical countries, has rendered the portion relating to
field operations eminently practical and complete, while the editor's
acquaintance with agricultural chemistry and familiarity with the best
tobacco-growing regions of Asiatic Turkey, have enabled him to exercise
a general supervision over the statements of the various contributors.
CONTENTS.
CHAPTER I.
PAGE
THE PLANT 1
CHAPTER II.
CULTIVATION 7
CHAPTER III.
CURING 67
CHAPTER IV.
PRODUCTION AND COMMERCE 137
CHAPTER V.
PREPARATION AND USE 231
CHAPTER VI.
NATURE AND PROPERTIES 253
CHAPTER VII.
ADULTERATIONS AND SUBSTITUTES 267
CHAPTER VIII.
IMPORTS, DUTIES, VALUES, AND CONSUMPTION 271
CHAPTER IX.
BIBLIOGRAPHY 276
INDEX 281
LIST OF ILLUSTRATIONS.
FIG. PAGE
1. CUBAN TOBACCO PLANT 4
2. MARYLAND TOBACCO PLANT 5
3. AMERSFORT TOBACCO PLANT 6
4. STRAW MAT FOR COVERING SEED-BEDS 47
5. SHADE FRAMES USED IN CUBA 49
6. QUINCUNX PLANTING 52
7. TOBACCO WORM AND MOTH 56
8. SHED FOR SUN-CURING TOBACCO 83
9. HANGING BUNCHES OF LEAVES 95
10. TOBACCO BARN 95
11. INTERIOR OF TOBACCO BARN 96
12. HAND OF TOBACCO 108
13. PACKING HOGSHEAD 133
14 to 17. TOBACCO-CUTTING MACHINE 234
18. MACHINE FOR MAKING PLUG TOBACCO 237
19 to 21. MACHINE FOR MAKING TWIST OR ROLL TOBACCO 238
22, 23. DIAGRAMS OF SEGMENT ROLLERS OF TWIST MACHINE 240
24 to 26. ANDREW'S IMPROVEMENTS IN TWIST MACHINE 243–4
27. MACHINE FOR CUTTING AND SIFTING SCRAP TOBACCO 246
28. MACHINE FOR MAKING CIGARETTES 247
29. RESWEATING APPARATUS 249
30. MACHINE FOR WEIGHING OUT SMALL PARCELS OF TOBACCO 250
31. TOBACCO-CUTTING MACHINE 252
TOBACCO:
GROWING, CURING, AND MANUFACTURING.
CHAPTER I.
THE PLANT.
Next to the most common grains and pulses, probably no plant is so
widely and generally cultivated as tobacco. In what country or at what
date its use originated has little to do with us from a practical point
of view, though interesting enough as a subject for the student of
ethnography and natural history. Suffice it to say that it has been
grown and smoked since pre-historic times in many tropical and
sub-tropical countries, and has assumed an importance in modern daily
life only surpassed by a few prominent food plants and cotton.
This long-continued and widespread cultivation has helped to produce
local varieties or races of the plant which have sometimes been mistaken
for distinct species, and caused a multiplication of scientific names
almost bewildering. The following epitome comprehends the species and
varieties of _Nicotiana_ possessing interest for the cultivator:—
I. _N. Tabacum macrophylla_ [_latifolia_, _lattissima_,
_gigantea_]—Maryland tobacco. Of this, there are two sub-species—(1)
Stalkless Maryland, of the following varieties: (_a_) _N. macrophylla
ovata_—short-leaved Maryland, producing a good smoking-tobacco, (_b_)
_N. macrophylla longifolia_—long-leaved Maryland, yielding a good
smoking-tobacco, and excellent wrappers for cigars, (_c_) _N.
macrophylla pandurata_—broad-leaved, or Amersfort, much cultivated in
Germany and Holland, a heavy cropper, and especially adapted for the
manufacture of good snuff; (2) Stalked Maryland, of the following
varieties: (_a_) _N. macrophylla alata_, (_b_) _N. macrophylla
cordata_—heart-shaped Maryland, producing a very fine leaf, from which
probably the finest Turkish is obtained. Cuban and Manilla are now
attributed to this group.
II. _N. Tabacum angustifolia_—Virginian tobacco. Of this, there are two
sub-species—(1) Stalkless Virginian of the following varieties: (_a_)
_N. angustifolia acuminata_, grown in Germany for snuff, seldom for
smoking, (_b_) _N. angustifolia lanceolata_, affords snuff, (_c_) _N.
angustifolia pendulifolia_, another snuff tobacco, (_d_) _N.
angustifolia latifolia_—broad-leaved Virginian, used chiefly for snuff,
(_e_) _N. angustifolia undulata_—wave-like Virginian, matures quickly,
(_f_) _N. angustifolia pandurata_, furnishes good leaves for smoking,
produces heavily, and is much grown in Germany, and said to be grown at
the Pruth as "tempyki," and highly esteemed there; (2) Stalked
Virginian, of the following varieties: (_a_) _N. angustifolia alata_,
(_b_) _N. angustifolia lanceolata_ [_N. fructiosa_], growing to a height
of 8 ft., (_c_) _N. angustifolia oblonga_, (_d_) _N. angustifolia
cordata_—E. Indian, producing heavily in good soil, and well adapted for
snuff, but not for smoking. Latakia and Turkish are now accredited to
_N. Tabacum_.
III. _N. rustica._—Common, Hungarian, or Turkish tobacco. Of this, there
are two varieties: (_a_) _N. rustica cordata_—large-leaved Hungarian,
Brazilian, Turkish, Asiatic, furnishing leaves for smoking; (_b_) _N.
rustica ovata_—small-leaved Hungarian, affords fine aromatic leaves for
smoking, but the yield is small. Until quite recently, Latakia, Turkish,
and Manilla tobaccos were referred to this species; Latakia is now
proved to belong to _N. Tabacum_, and Manilla is said to be absolutely
identical with Cuban, which latter is now ascribed to _N. Tabacum
macrophylla_.
IV. _N. crispa._—This species is much grown in Syria, Calabria, and
Central Asia, and furnishes leaves for the celebrated cigars of the
Levant.
V. _N. persica._—Hitherto supposed to be a distinct species, affording
the Shiraz tobacco, but now proved to be only a form of _N. Tabacum_.
VI. _N. repanda._—A Mexican plant, with small foliage. Long thought to
be a distinct species peculiar to Cuba, but none such is now to be found
in Cuba, whether wild or cultivated, and all the Cuban tobacco is now
obtained from _N. Tabacum macrophyllum_.
Among the many other forms interesting only to the botanist or
horticulturist, the principal are _N. paniculata_, _N. glutinosa_, _N.
glauca_, attaining a height of 18 ft., and _N. clevelandii_, exceedingly
strong, quite recently discovered in California, and supposed to have
been used by the early natives of that country.
Thus the bulk of the best tobaccos of the world is afforded by the old
well-known species _Nicotiana Tabacum_.
A good idea of the foliage and inflorescence of commonly cultivated
tobaccos may be gained from a study of the accompanying illustrations.
[Illustration: FIG. 1.]
Fig. 1 is a Cuban tobacco, and much grown on the continent of Europe,
notably in Holland, Germany, and Switzerland, and there known as
_goundie_, from the name of an American consul who introduced the plant
into Germany in 1848. It has a broad yet somewhat pointed leaf, with the
ribs not arranged in pairs; it is fine, soft, thin, and esteemed for
smoking in pipes and for wrappers of cigars.
One variety of the Maryland plant is shown in Fig. 2. The leaves spring
from a tall stem at considerable intervals, and are broad and rounded at
the end. This kind is valued for cigar-wrappers, and assumes a fine
light brown colour when well cured.
[Illustration: FIG. 2.]
A broad-leaved Cuban or Maryland growth long naturalized in Germany, and
now familiar as Amersfort, is represented in Fig. 3. It is distinguished
by unusual length of leaf accompanied by a corresponding narrowness. A
stem and flower are shown at _a_, a leaf at _b_, a flower in section at
_c_, a capsule at _d_, a seed at _e_, and a cross-section of a leaflet
at _f_.
[Illustration: FIG. 3.]
These three examples represent the most successful kinds grown in Europe
and at the same time some of the most marked diversities of form of
leaf.
CHAPTER II.
CULTIVATION.
The following observations on the methods of cultivating tobacco have
reference more particularly to the processes as conducted in Cuba,
India, and the United States; this branch of agriculture has been
brought to great perfection in the last-named country, and the
supervision of the operations in India is mostly entrusted to skilled
Americans.
_Climate._—Of the many conditions affecting the quality of tobacco, the
most important is climate. The other conditions that must be fulfilled
in order to succeed in the cultivation of this crop may be modified, or
even sometimes created, to suit the purpose; but cultivators can do
little with reference to climate: the utmost they can do is to change
the cultivating season, and this only in places where tobacco can be
grown nearly throughout the year. The aromatic principles, on the
presence of which the value of a tobacco chiefly depends, can only be
properly developed in the plant by the agency of high temperature and
moisture. The fame that Cuban and Manilla tobaccos enjoy is mostly due
to the climate. The article produced in Cuba is most highly esteemed; up
to this time, no other country has been able to compete successfully
with it. However it cannot be doubted that there are many places whose
climate justifies the assumption that a tobacco could be grown there,
not inferior to that produced in the West Indies. The more closely the
climate of a place corresponds with that of Cuba, the greater chance is
there that a Havana a variety will preserve its peculiar aroma. In such
places, a fine and valuable tobacco may be grown with less expenditure
on labour, &c., than it is necessary to bestow in raising an inferior
article in less suitable climes. In countries where a low temperature
rules, the plants must be raised in hot-beds, and there is also a great
risk that the young plants may be destroyed by frost, or afterwards by
hailstones. When damp weather prevails during the tobacco harvest, it is
often injured; and to give the required flavour, &c., to make the
article marketable, macerating has often to be resorted to, thus
involving great risk and expenditure. But in spite of these drawbacks,
tobacco cultivation is often very remuneratively carried out in
countries possessing an unfavourable climate. The deficient climatic
conditions are here partly compensated for by making the other
conditions affecting the quality of tobacco, and which can be controlled
by the cultivator, the most favourable possible.
_Soil._—The soil affects to a great extent the quality of a tobacco. The
plant thrives best in a soil rich in vegetable mould; this, however, is
not so much required to supply the necessary plant food, as to keep the
soil in a good physical condition. No other plant requires the soil in
such a friable state. A light soil, sand or sandy loam, containing an
average amount of organic matter, and well drained, is considered best
adapted for raising smoking-tobacco; such a soil produces the finest
leaves. The more organic matter a soil contains, the heavier is the
outturn; but the leaves grow thicker, and the aroma becomes less. As, in
tropical climates, the physical properties of the soil play a prominent
part in its productive capabilities generally, and the presence of
organic matter in the soil tends to improve these properties, it will
rarely occur that in such places a soil will contain too much humus. The
more clay in a soil, the less is it adapted to the production of fine
smoking-tobacco, on account of its physical properties being less
favourable to the development of the aromatic principles; the leaf
becomes also generally thick and coarse, but the outturn on such soils
is commonly heavier than on a more sandy one. A clay soil possessing a
great amount of humus may, if properly tilled, produce an ordinary
smoking-tobacco, and may even, if great attention be paid to the
selection of the variety, &c., produce leaves for cigar-wrappers.
Of less importance than the physical properties of the soil is its
chemical composition. By proper tillage and heavy manuring, tobacco is
sometimes grown on comparatively poor soils. From analysis of the plant,
it is clear that it contains a large amount of ash constituents, which
it extracts from the soil; the most important of these are potash and
lime. A soil destitute of these constituents would require a great
quantity of manure to supply the wants of tobacco.
An experienced Ohio planter, Judson Popenoe, speaking of soil, says "A
rich, sandy, second bottom, I believe to be the best for raising
tobacco, although our chocolate- uplands, when very rich and
highly manured, will grow an excellent quality of tobacco, but will not
yield as much to the acre. Black river-bottoms will yield more to the
acre than any other kind of land, but the tobacco is not of so fine a
quality; it grows larger, has coarser stems, and heavier body, and
consequently, in my opinion, is not so good for wrappers or fine cut as
the second bottom or upland tobacco."
On the same subject, an Illinois grower observes, "for us in the West,
and for all the localities that have not an over-amount of heat,
experience has proved, that a dry, warm soil (loam or sandy loam), rich,
deep, and containing lime, is most suitable for tobacco. The more sandy,
to a certain degree, the soil is, the better will be the quality of the
tobacco; the nearer the soil is to clay, the poorer will be the crop
under similar circumstances, although the yield may yet be satisfactory.
Clayey soil will hardly produce tobacco suitable for cigars. Wet and
tough clay soils are under no circumstances suitable to tobacco."
_Situation._—Land intended for tobacco-culture should have good
drainage, and be sheltered from high winds. In Holland, where
tobacco-cultivation is carried out to great perfection, each field is
surrounded by a hedge about 7 ft. high; the fields are divided into
small plots, which are again bordered by rows of plants that are able to
break the force of the wind, which would injure the leaves, and render
them of comparatively little value. To this circumstance must chiefly be
attributed the fact that Dutch growers succeed in getting as much as 50
per cent. of leaves of the first quality, whereas in most other
countries 25 per cent. is considered to be a very good outturn.
In the United States, several rows of pole beans, i. e. scarlet runners,
a few steps apart, are sometimes planted as a wind-screen.
_Manure._—In its natural state, the soil will rarely possess the
elements of plant food in such a form as is most conducive to the
production of a fine tobacco-leaf. Any deficiency must be supplied in
the shape of suitable manure. Schlösing found that a bad burning tobacco
was produced on a soil containing little potash, on unmanured soil, on
soil manured with flesh, humus, calcium chloride, magnesium chloride,
and potassium chloride. A good burning tobacco was produced on a soil
manured with potassium carbonate, saltpetre, and potassium sulphate.
More recent experiments carried out by other investigators tend to
corroborate these conclusions. It is generally assumed that a soil rich
in nitrogenous organic matter produces a strong tobacco that burns
badly.
The results of Nessler's experiments clearly show that it is not
sufficient to apply the element most needed by the plant—potash—in any
form, but that, to produce a good tobacco, it is necessary to apply it
in a particular combination. It was found that potash carbonate applied
as manure produced the best tobacco: it burned for the longest time, and
its ash contained most potash carbonate; whereas potash chloride
produced a much inferior tobacco. The assertion of other experimenters
that chlorides produce a bad tobacco is thus confirmed. Potash sulphate
and lime sulphate produced a good tobacco. It may be noticed here that
tobacco which was manured with gypsum contained a great amount of potash
carbonate in the ash, probably due to the fact that gypsum is a solvent
for the inert potash salts. From the foregoing, it may be concluded that
in tobacco cultivation, the elements potassium and calcium should be
restored to the soil in the form of carbonate, sulphate, or nitrate, but
not as chlorides. Poudrette, or prepared night-soil, generally contains
a considerable amount of chlorides, and is not well suited as manure for
fine tobacco. It has been found that fields manured with chlorides
produced heavily; a small proportion of chlorides may therefore be
applied in this form, whenever quality is of less importance than
quantity. Farmyard manure may suffice when tobacco is cultivated in
proper rotation, but here also, unless the soil be very rich in
potassium and calcium, the application of some special manure will
greatly enhance the value of the outturn. Wood-ashes are a valuable
supplement to stable dung. Gypsum is an excellent dressing for soils in
a good manurial condition: it supplies the lime needed by the tobacco,
and acts as a solvent on the inert potash salts. Gypsum applied on poor
land, however, hastens the exhaustion of the soil. It is said that crops
manured with gypsum suffer less from the effects of drought, and require
less irrigation, than when manured otherwise: the leaves of plants that
had been manured with gypsum exhaling less water than when manured with
other substances. If this assertion be correct, gypsum would be
invaluable to the Indian cultivator.
With regard to the amount of manure to be employed, it may be observed
that, with farmyard manure properly rotted, there is no theoretical
limit, especially when the tobacco is intended for snuff, and is grown
in a hot climate, where the physical properties of the soil are of the
utmost importance. It is said that some Rhenish-Bavarian soils contain
as much as 15 per cent. of organic matter, yet the cultivator considers
it necessary to heavily manure each tobacco crop. Dutch growers apply to
the rich alluvial soil as much as 25 tons an acre of well-rotted
cattle-manure. In America, it is reported that the heaviest crops are
obtained on soil newly taken up, and very rich in vegetable mould. It is
considered nearly everywhere that tobacco will pay best when heavily
manured. The first care of even the poorest peasant in the tobacco
districts of Germany, Holland, &c., as soon as he sells his tobacco, is
to purchase the manure which he considers essential to his success.
The amount of any special manure which can be applied without injury to
the plants depends very much on the solubility of the stuff, and the
manner of applying it. Highly soluble salts, such as soda or potash
nitrate, should be applied in smaller quantities than salts which
dissolve slowly. With regard to the manner of applying concentrated
manures, it is evident that, when a salt is applied in close proximity
to the plant, less will be required than when strewn over the whole
field. When applied in solution, not more than 300 lb. of nitrate per
acre should be used at one time. The amount to be applied varies also
with the soil; a sandy soil, which has little absorptive power, should
receive less than a clay. Salts easily disintegrating should not be
applied before tobacco has been planted, especially not before heavy
rains which would carry off the salt. To supply the potash required by
the tobacco plant, 200 lb. of good saltpetre per acre would be
sufficient in most cases. Lime, although removed from the soil in large
quantities, is rarely applied to tobacco as a special manure. Where
wood-ashes can be had at a moderate price, lime may be applied in this
form. Some ashes are very rich in lime. It has been found that ashes
obtained from beech-wood contain 52 per cent. of lime, and those from
oak-wood as much as 75.
Whilst most growers are agreed that tobacco is a crop demanding a rich
soil, there is a want of uniformity of opinion as to the best method of
manuring. On this point, C. Schneider, a successful Illinois planter,
says "manuring cannot be done too early, or too heavily. The manures are
very different, and equally useful for the different kinds of tobacco.
We may classify them as follows:—
"To be applied shortly before planting, and in equal quantities, for all
kinds of tobacco: 1. Guano, 200 to 300 pounds on the acre; 2.
Poultry-droppings, 400 to 500 pounds; 3. Green manure in any quantity;
4. Sheep-dung, 6 two-horse loads; 5. Cattle manure, 10 two-horse loads.
"For chewing-tobacco and snuff: 1. Sheep-dung, 10 to 12 loads per acre;
2. Cattle manure, 20 to 30 loads; 3. Horse-dung, 15 to 25 loads; 4. Hog
manure, 20 to 30 loads. The last two are useless for smoking-tobacco, or
for that to be used for cigars.
"The first three manures (guano, poultry-droppings, and green manure)
must be followed after the tobacco-crop, by a plentiful supply of
stable-manure. The tobacco-stalks themselves, rotted or burned to ashes,
sown over the field before the transplanting, or in the
planting-furrows, will act as a good manure, but are not sufficient. In
highly-worked farms, that is, where the soil is valuable, and cannot
remain idle, it will pay every way, to sow rye for fodder on the
tobacco-land in the fall; this may be made into hay, or turned under as
manure at the beginning of July, just as may seem most profitable. Deep
ploughing for the rye, and afterward for the tobacco, must not be
forgotten."
R. E. Burton, in the _Sugar Cane_, translating from Mitjen's essay on
tobacco growing in the most renowned district of Cuba, has the following
sensible remarks on the all-important subject of manuring:—
"Each veguero or farmer should make a hole or rotting-bin in which he
should deposit as much muck and leaves as he may be able to accumulate,
and, before giving the last ploughing to prepare his field for planting
the tobacco, he should spread over it all the prepared rotten manure he
can procure. Manure that is not thoroughly rotten injures the plants
more than benefits them. A piece of land, well manured and thoroughly
worked up, will produce four times more tobacco than one badly prepared
would. Consequently no expense or labour is so remunerative as that
which is applied to the soil. This is a very important point which
should fix the attention of every agriculturist who desires to prosper.
"Agriculturists acknowledge the advantage of manuring. In tobacco
cultivation it produces the most brilliant results, but in Vuelta-Abajo
it is very difficult to procure sufficient country manure. Yagues (i.e.
strips of palm bark used as screens, and for baling) and all the refuse
from palm trees are excellent; grass from the savannahs and all kinds of
vegetables in a thoroughly putrid state are very good, but it requires a
great quantity, and the immense labour to collect and prepare these,
frightens the greater number of vegueros, and few have sufficient
constancy to enable them to collect enough properly prepared manure for
their fields.
"The most which some manage to do is to spread refuse over some portions
of land, where it rots and fertilizes the soil; but this system is
inefficacious, because the vegetable substances being very light, the
heavy rains wash away the greater portion of the decomposed matter, and
fully nine-tenths are lost. If the system was adopted of depositing this
manure in holes or trenches, from which it can be removed when
thoroughly rotted and fit for the fields, it would produce much more
with much less labour; for although at first sight the labour appears to
be doubled, by having to carry it twice, it must be remembered that one
load of well-prepared manure is better than ten or twenty of grass or
bush that is not rotten.
"But in every way there is great difficulty in collecting vegetable
manure in sufficient quantities; recently, guano has been tried with the
most brilliant success.
"Peruvian guano is the most compact fertilizer known, and a very small
quantity suffices to manure a tobacco field; its cost is not excessive,
and is very frequently less than the carriage of other manures to the
spot where they are to be used. Its most active results are shown on
light and sandy soil; it quickens vegetation, and experience has shown
that it increases prodigiously the quantity and value of crops; we
therefore recommend the use of guano as a fertilizer of the first order
for tobacco cultivation, and as light and sandy soils possess in
themselves the substances most suitable for the development of the
tobacco plant, on such soils guano acts as a stimulant to the plant.
"Before using Peruvian guano, it should be sifted; all the stones and
lumps remaining should be broken up, and again sifted, so that nothing
may be lost. After this, three or four times its weight of dry sandy
soil should be thoroughly mixed with it, and it should remain thus 6–8
days before being used. This preparation should be made under cover, to
avoid the possibility of rain falling on the mixture, and the heap
should be covered with the empty guano bags, or anything else, to
prevent the evaporation of the volatile alkali which it contains.
"It is better to prepare this mixture in detail, each heap containing
one bag of guano, whose weight is 150–160 lb., so as to facilitate the
calculation of the quantity that should be applied, and prevent
mistakes. We will start, therefore, on this calculation.
"On lands of good quality, but which, nevertheless, require manure, from
having been overworked, one pound of guano should be applied to each
15–20 superficial yards, or, say one heap of compost for each 2500–3000
yards, or, otherwise said, one heap of manure will suffice for a surface
that contains 5000–6000 plants.
"In sandy unproductive soil, and on sterile savannah lands, 1 lb. of
guano to 9–12 yards; or a heap of compost guano to 1500–2000 yards; or
one heap for 3000–4000 plants.
"These are the proportions to be used for the first year; for the
second, and forward, two-thirds of that employed the first year will be
sufficient.
"When crops of tobacco and corn are grown on the same lands, half the
guano should be applied to the corn and the other half to the tobacco;
but then a somewhat larger quantity will be required. The manure should
be applied shortly before transplanting, and after the ground has been
well cross-ploughed and prepared, and the ground should be plotted out
into squares or beds of 50 yards square. The manure should then be
spread and ploughed in, and the land should at once be furrowed and
planted.
"Under this system of applying Peruvian guano as manure for tobacco the
best results have been obtained, and, of all the various trials made,
this is the most simple and the easiest to execute."
The remarks of the last-quoted essayist are good so long as guano is to
be had. But there is a limit to the supply, and in many places it would
be unprocurable.
The necessity for more definite knowledge concerning the actual wants of
the tobacco plant in the matter of food, led to an investigation of the
subject some years ago by Prof. S. W. Johnson on behalf of the
Connecticut State Board of Agriculture, and more recently by Schiffmayer
for the Agricultural Department of the Madras Presidency.
Prof. Johnson aptly observes it to be "a well-established fact that
plants may receive from the soil and retain a larger portion of
ash-ingredients than is needful for nutrition. This is especially marked
in case of the lime, potash, and soda salts. The excess of these
substances thus taken up may either be deposited in the solid state in
the cells of the plant, or may remain dissolved in the juices. In
tobacco, a part of the nitrogen usually exists as a nitrate, in
combination with potash. That is to say, portions of the nitrogenous
food of the plant—the nitrates of the soil—are not completely worked
over into albuminoids, and into nicotine, the nitrogenous constituents
of tobacco, but accumulate and remain in considerable quantity in the
sap. When a dry tobacco-leaf is set on fire, it often burns like 'touch
paper' (paper soaked in a solution of saltpetre and dried) with bright
sparkles of fire, indicating the points where the nitre has gathered in
minute crystals as the juice of the leaf evaporated. The quantity of
superfluous salts in the plant depends upon its succulence, and upon the
supply of them in the soil. Doubtless certain definite amounts of
potash, lime, magnesia, iron, sulphuric acid and phosphoric acid are
absolutely necessary to produce a given weight of tobacco. In case
several or all these substances are superabundant in the soil, the plant
has no power to exclude any unnecessary surplus of one or all of them
from its interior altogether, although there are good reasons known to
prevent their entrance beyond a certain limit. In one soil potash may be
relatively most abundant, and may for that reason be found in the crop
in greater quantity than was necessary for the growth of that crop. In
another soil lime may be in surplus, and there the crop may have the
minimum of potash, and a considerable excess of lime.
"The crop is a result of the working together of a number of causes or
conditions; these are the heat and light of the sun, carbonic acid and
oxygen of the atmosphere, water, nitrates and ammonia, and the
ash-elements enumerated in our table of analyses. The crop is limited in
quantity by that condition of growth, which is presented to it most
sparingly. The richest and best prepared soil without solar warmth, or
without due supplies of rain, cannot give a crop, and if weather be most
favourable, then in one field it may be too little potash, in another
too little phosphoric acid, in another too little nitrogen, which lowers
the yield, or reduces the quality of the product.
"It is usual in tobacco culture to manure very heavily, and in many
cases it is probable that all the various forms of plant food are
present in available abundance. But soils differ in the nature of the
supplies which they are able to yield to crops, and fertilizers even,
when the same in name, may be very unlike in fact. The chief reliance of
the tobacco farmer is stable manure. This, however, is by no means
uniform in origin, appearance, evident quality, or chemical composition.
The manure from bullocks, wintered on hay and roots, is very different
from that of horses maintained chiefly on oats or corn. The yard manure
that contains much strawy litter or much wasted hay, differs again from
that of the city stables, from which the straw is carefully raked out to
be used over and over again for bedding. The farm-made manure is likely
to be much richer in potash and lime, and the city manure is richer in
phosphates and nitrogen. Yet in the reports of the farmer, these two
essentially different fertilizers are designated as stable manure
simply.
"Every one understands that a fertilizer acts upon the plant to supply
it with food, and to favour its growth; everybody is also convinced that
some fertilizers act upon the soil, improving its texture and
composition and increasing its fertility. It is an equally well
ascertained fact that the soil acts upon fertilizers to modify their
effect. A very wet or very dry soil is known to nullify the benefit
which might be expected of a fertilizer in a simply moist soil; but more
than this, more than by the accident of external circumstances, it is a
fact that each kind of soil has a special action of its own on
fertilizers, so that if it were asserted of two soils, which, unmanured,
were of equal fertility, that a given fertilizer applied to both,
greatly improved the crop on one, and had little effect on the other,
such a statement might not only be accepted as a fact, but an
explanation might be given in general terms for such a fact.
"Now experiments have shown that different soils when mixed with like
quantities of various fertilizing elements and then treated with water,
in imitation of rain, manifest very different behaviour toward the
admixed substances. One soil will lay hold of the potash in a
fertilizer, and fix it in a kind of chemical combination so firmly that
water can dissolve it but with extreme slowness; another soil puts its
grasp on the lime of a fertilizer, and at the same time allows potash
which belongs to itself to be dissolved out freely. There is, in fact,
always a complicated series of changes set in operation whenever any
fertilizer is incorporated with the soil, be it animal, vegetable, or
mineral; be it alkali, acid, or saline; be it made on the farm or
imported from abroad; be it natural or artificial. The fertilizer acts
on the soil, and the soil reacts on the fertilizer; but the point we
wish to make prominent is this, that different soils are differently
affected by one and the same application, or in other words, a given
manure fertilizes a given crop unequally in degree, and unlike in kind,
on different soils, by virtue of the different assimilating or fixing
power, which the soil exerts upon its ingredients.
"We know of the existence of these peculiarities of soils, and something
of their causes and of the laws by which they act; but the real
necessities of the tobacco crop, or of any other crop, as respects
soil-ingredients, cannot be arrived at by chemical analysis of a single
sample, nor of a dozen samples." Thus analyses of a dozen New England
tobaccos showed the following highest and lowest percentages of each
ash-ingredient, and of nitrogen:—
Silica 0·05 to 0·30│Magnesia 0·94 " 2·21
Chlorine 0·08 " 2·55│Potash 3·90 " 7·45
Sulphuric acid 0·52 " 1·69│Soda 0·08 " 1·81
Phosphoric acid 0·47 " 0·80│Nitrogen 3·20 " 5·11
Lime 3·17 " 8·22│
"It appears that the percentages of nitrogen, phosphoric acid and potash
are nearly twice as great in some samples as in others; that the
proportions of magnesia and lime are about 2½ times greater in some
samples than in others, and that sulphuric acid is 3 times more in one
case than in another. The variation of silica is still greater, and the
disparity rises to its extreme in case of soda and chlorine, whose
maxima are respectively 20 and 30 times greater than their minima."
The three ingredients chlorine, silica, and soda cannot be considered in
the light of essentials to tobacco culture; but the other substances are
absolutely indispensable to plant growth, and the absence of any one of
these would render a soil incapable of sustaining agricultural
vegetation of any kind. "The variation in the percentage of these
ingredients depends somewhat upon the fact that the leaves of different
crops are unequally developed, and therefore their nutritive needs are
unlike; but it is, no doubt, chiefly connected with the fact that the
plant takes up from a highly fertilized soil more of each or every
element than is essential for growth. The nearly certain conclusion is
that every one of the crops analysed contains more of some elements than
belongs to its nutrition. It is quite certain that the average of the
analyses of the New England tobaccos is fully up to the mark as regards
the necessities of the crop. It is, indeed, not improbable that the
lowest percentages of each ingredient are quantities sufficient for a
perfect crop. Still, it is not proved that lime may not partially take
the place of potash, or the reverse. The probability of such a
substitution is great upon the face of most of the analyses. As a rule,
those which show most potash show least lime and _vice versâ_; but in
one sample both ingredients are considerably below the average. The
practical issue of these considerations is to give great probability to
the view that the tobacco crop is fed unnecessarily (and wastefully?)
high." (Prof. Johnson.)
Tobacco is usually characterized as a very exhausting crop. This is not
true as regards the amount of nutriment taken from the soil, for in this
respect tobacco is less exacting than hay, potatoes, or rye. It demands
chiefly potash and lime, with phosphoric acid and nitrogen. Prof.
Johnson recommends for the manuring of one acre, besides ploughing in
the stalks of the plants, 500 lb. rock guano or 800 lb. fish guano, 500
lb. kainit (potash salts), and 50 lb. quicklime. But surely it cannot be
advisable to mix quicklime with an ammoniacal manure like guano; it
seems to the writer that gypsum, or spent calcium oxide from gasworks,
would be a far preferable medium for conveying lime to the soil.
As observed by Johnson, the "demand made on the soil or on fertilizers
by the tobacco crop, is for certain reasons greater than that made by
other crops which receive more of nearly every kind of plant food. Hay
is more exhausting than tobacco as measured by total export from the
soil, but grass grows the whole year throughout, save when the ground is
frozen or covered with snow, or for more than 8 months. The period of
active growth which is required to mature a hay crop, begins indeed in
April, and is finished by July, a period of 3 months, but during the
year previous, for at least 5 months, in case of the first crop, the
grass plants have been getting a hold upon the soil, filling it with
their roots, and storing up food in their root-stocks or bulbs, for the
more rapid aftergrowth. Tobacco on the other hand cannot be set out in
the field before about the 10th of June, and should be in the shed in
about 3 months. Its growth then must be a very rapid one, and the
supplies of food in the soil must be very abundant so that the
quick-extending roots may be met at every point with their necessary
pabulum. A crop of 1260 lb. dry leaves requires about 1100 lb. of dry
stalks to support the leaves, making a total of 2360 lb. of dry
vegetable matter. As new hay contains not less than one-sixth of
moisture, we increase the above dry weight of the tobacco crop by
one-sixth, to make a fair comparison, and obtain as the yield of an
average tobacco field 2750 lb. of air-dry vegetable matter, or more than
1⅓ tons. The matter stands then thus: An acre of first-rate grass land
yields as the result of 8 months' growth, 2¾ tons of crop, while the
tobacco land must yield 1⅓ tons in 3 months.
"If the above data are correct, the _average_ rate of growth of tobacco
is greater than that of a corresponding hay crop, in the ratio of 9:7.
The real disparity is, however, much greater. The principal growth of
tobacco is accomplished in the hottest summer weather, and in a period
of some 40–50 days. Very heavy manurings are therefore essential to
provide for its nourishment, and the more so because the best tobacco
lands are light in texture, and may suffer great loss by drainage,
evaporation, and decomposition."
From these premises, Prof. Johnson advances to the question of what
should or should not be presented to the plant in the form of manure. He
commences with a caution that, in general, growers must "avoid employing
fertilizers which contain salt or other chlorine compound in raising
wrapping or smoking tobacco. It is evident, also, that there is no
occasion to use any fertilizer for the special object of supplying
phosphoric acid, since the heaviest export of this substance does not
exceed 10 lb. per acre, annually. It may be well to mention here that
phosphates which may be put upon a tobacco field, in guano, &c., cannot
suffer waste by washing out, and will come to use when grain or grass
shall follow in the rotation."
He observes of gypsum (lime sulphate) that it is "a valuable application
to tobacco, not because it is very largely taken up by the crop, for the
greatest export of sulphuric acid, viz. 20 lb. per acre, is restored by
50 lb. of plaster, and the greatest export of lime, 120 lb., is made
good by 400 lb. of the sulphate, but because lime sulphate dissolves in
400 times its weight of water, and may rapidly wash out of the porous
tobacco lands, and especially because the solution of lime sulphate in
the soil is a very effective agent in rendering soluble and accessible
to crops the potash and magnesia, which too often exist in close-locked
combinations. The average annual rainfall (snow included) in our
latitudes, is no less than 10,000,000 lb. per acre. This enormous
quantity of water would be enough to dissolve and wash out of the soil
25,000 lb. of gypsum per acre if it had time to saturate itself, and
then flowed off. In fact, but a small proportion of the rainfall runs
through and out of the soil, not more than 10 to 20 per cent., according
to its porosity and situation; but it is plain that there is nothing to
hinder the waste of a hundred pounds or more of gypsum per acre yearly,
Since all investigations go to show that the soil has no retaining power
for lime sulphate as it has for potash and for phosphoric acid. In
Nessler's experiments, gypsum had an excellent effect on the burning
quality of the tobacco raised under its application, an effect
attributable, he believes, to the fact that this fertilizer often
liberates potash in the soil, as Liebig and Deherain have demonstrated,
and is therefore equivalent to an application of potash, provided the
latter actually exists in the soil.
"Potash is exported in the tobacco crop to the amount of 70–80 lb. per
acre yearly, and is required for the stalks to the extent of some 50
lb., making a total of 120–130 lb. As already intimated, potash does not
commonly waste from the soil by washing. It is seldom found in
appreciable quantity in well or drain water, and most soils absorb it
and fix it so firmly that water can remove it but very slowly. It does,
however, appear in the drain water from very heavily dunged fields,
though in small proportion. Stable or yard manure on the average
contains one-half per cent. of potash, or 10 lb. per ton. Twelve or
thirteen tons of stable manure would therefore contain the potash
needful to produce a crop. The dressing of 20 tons of 10 cords of stable
manure, per acre, which is often employed on tobacco, is doubtless
enough to fully supply the crop, and the application of additional
potash is apparently quite unnecessary. The employment of potash salts
upon tobacco lands would therefore seem to be uncalled for unless the
amount of stable manure is greatly diminished, or its quality is very
inferior. In case potash salts are to be applied, the best form to make
use of is potash sulphate, of which 250 lb. contains 135 of potash. Next
to this is probably potash carbonate, i. e. the ordinary potash of
commerce, which contains some 70 per cent. of potash; 200 lb. of this
would be sufficient for an acre. To apply it I would suggest breaking it
up into small pieces and soaking it in two or three times its weight of
water until the lumps crush easily, and mixing these with so much ground
gypsum as will make a mass dry enough to handle.
"Kainit, which contains some 15 to 20 per cent. of potash, but also 10
per cent. or more of chlorine, is not so good for leaf tobacco, and
least of all to be recommended is potassium chloride (muriate of potash)
which is nearly half chlorine.
"Magnesia is an element which is abundantly provided for in stable
manure, every ton of which, according to analyses on record, contains
some 3 lb. of this substance.
"Lime is supplied in relative abundance in stable manure, the average
ton of which contains some 15 lb. We have seen that 600 lb. of gypsum
contain as much lime as the average tobacco crop: guano, dry fish, and
superphosphate, each contains some 5–10 per cent. of lime. There is,
furthermore, little likelihood that any soil intended for tobacco would
not of itself contain enough lime to support the crop. Lime in the
caustic state has, however, a value independent of its direct nutritive
power, which is well worth the attention of the tobacco raiser. Of this
I shall write briefly in a subsequent paragraph.
"Nitrogen in absolutely dry New England tobacco leaf ranges from 3·2 to
5·1 per cent., or 4·24 as the average. This is a larger proportion than
exists in any of our ordinary field crops, except the seeds of legumes.
The grain of wheat and red clover hay contain when dry scarcely 2½ per
cent., and they exceed all other usually raised vegetable products,
except the leguminous seeds. The pea and bean contain, when dry, 4·5 to
4·7 per cent. of nitrogen. The acreage export of nitrogen is
nevertheless not large according to the data of our tables. It should be
remembered, however, that the average is derived from 5 samples only....
There are reasons to suppose that this result is too low. Furthermore it
is not improbable that tobacco loses nitrogen during the curing
process."
The advantages of artificial manuring have been made manifest in all
branches of agriculture, and there is no doubt that the nitrogenous
qualities of farmyard dung may be replaced by soda nitrate, ammonia
sulphate, &c., only it must be remembered that these have not nearly the
lasting effect of dung, the latter liberating its ammonia but slowly.
Indeed "when a soil has been heavily dunged for a term of years, it
accumulates a large quantity of nitrogen, which is comparatively inert
and therefore nearly useless to crops. Quicklime assists to convert this
nitrogen into the active forms of ammonia or nitrates," hence Prof.
Johnson's suggestion that an "application of lime may sometimes be
advantageously substituted for one of stable manure. In fact, it is not
improbable that moderate doses of lime might be turned under with stable
manure or green crops, with the effect of exalting the action of these
fertilizers, and obtaining from them a larger return of nitrogenous
plant food. Lime, however, gives effect to the nitrogen of the soil by
causing the destruction of the organic matters—_humus_—in which this
nitrogen lies in an inactive state. These organic matters have
themselves a value independent of their nitrogen, which must be taken
account of, and therefore the use of lime must be undertaken cautiously,
and with an intelligent comprehension of the various effects which it
may produce."
_Rotation._—A proper rotation of crops is particularly advantageous for
the cultivation of tobacco, since it requires a great amount of readily
accessible inorganic matter in the soil, especially potash and lime.
Although the importance of cultivating tobacco in rotation is admitted,
there may be circumstances that justify the growth of this crop
consecutively for several years in the same field. In America, tobacco
is grown successively for several years on new land, where the elements
of plant food exist in such abundance that the crop may be thus
cultivated without for a time showing any notable decrease in yield; it
is even said that the outturn of the second year is heavier than that of
the first. In Hungary and Holland, the best tobacco is grown for many
years in succession on the same land. There the plan is adopted partly
out of necessity and partly for convenience. The small landholder is
often obliged to grow tobacco on the same field, because he has only one
properly fitted for it; for convenience, he grows it every year on the
same place near his homestead, to allow of the closest attention to the
crop, but he manures heavily. Nessler, in Carlsruhe, cultivated tobacco
during six consecutive years in the same field, without noticing any
perceptible decrease in yield or quality. To admit of such a system, the
soil must either be very rich in the essential elements, or be heavily
manured, as is the practice in Holland. It is generally assumed that,
when tobacco is grown on the same field in succession, the leaves do not
become so large after the first year, but grow thicker and more gummy,
and contain less water.
From the foregoing, it would appear that, although tobacco may be grown
successfully on the same land uninterruptedly under special
circumstances, the cultivator will find it advantageous to adopt some
plan of rotation. Cereals and pulses are very well adapted for this
purpose, the reason being that tobacco removes but little phosphoric
acid from the soil, and thus leaves it rich in the element most
necessary for the growth of cereals. It has also been found that hemp
thrives particularly well after tobacco.
Judson Popenoe suggests that there "should be a good coat of clover to
plough under; if the ground is naturally rich, this alone will make a
good crop, but hog and stable manure, well rotted, is what the tobacco,
as well as any other crop, delights in, and the more manure the better
the tobacco. The plan that I am now experimenting on is, as soon as I
cut my tobacco in the fall I give the ground a good harrowing, and then
drill in wheat; the ground being well cultivated all the fall, is clear
of weeds and mellow and needs no ploughing. In the spring I sow clover,
after the wheat is off; I keep the stock off until about September, to
give the clover a chance to harden and spread. I then let the stock eat
as low as they want to, which drives the clover to root, and causes the
crown to spread; I do not suffer stock to run on the clover during
winter or spring; about the last of May or first of June I plough the
clover under, which is now in blossom, and so I alternately keep two
fields in tobacco and wheat, at the same time feeding the ground a crop
of clover every two years; in this way I expect my land to increase in
fertility all the time. The clover turned under makes food for the
cut-worms, and they trouble the tobacco-plants but little."
_Selection of Sort._—The cultivator must carefully compare the
requirements of the different sorts, and the means at his disposal to
satisfy them, before making his selection. Though tobacco is a hardy
plant, and grows under varied conditions, yet to become a remunerative
crop, the plant should not be placed under circumstances very dissimilar
from those to which it has been accustomed. By importing seed of a fine
sort directly from its native land, the plants will not retain in the
new habitat all their special qualities, unless climate, soil and
treatment are nearly the same. Climate must first be considered. Fine
and valuable tobacco is a product of tropical countries: in a warm and
humid climate, by employing common means, tobacco may be made to yield a
profit not attainable in less favoured regions. A warm, moist climate
permits the selection of those sorts that command the highest prices; if
to this be added a suitable soil, and proper treatment, the cultivation
of tobacco yields a profit not easily obtainable from any other crop.
As the Havanna tobaccos command the highest prices, the cultivator
nearly everywhere attempts to introduce and cultivate them. There is no
great difficulty in raising plants of these varieties, but they speedily
degenerate and form new varieties, if the climatic conditions, &c., are
not favourable. Virginian tobacco was previously extensively cultivated,
but has of late been frequently replaced by the Maryland kind. It is
still much favoured by cultivators in temperate climates, as it does not
require a high temperature. On account of its botanical characteristics,
it is usually not much liked by manufacturers of cigars; some varieties,
however, that have less of the marked specific characters, yield
tolerably fine leaves for cigars. As the price of this tobacco is rather
low, it is not so well suited for export. Hungarian tobacco is
considered to be very hardy, but is less valuable than the foregoing.
The leaves are generally small, and possess a peculiar aroma.
A high price is generally commanded, irrespective of the species, by
those tobaccos that possess a large, smooth, thin, elastic leaf,
possessing a fine golden colour and a good aroma; the ribs and veins
should be thin, and the former should branch off from the midrib at
nearly right angles, and should be far apart from each other. The lower
the percentage of the weight in ribs, the thinner and broader the leaf,
and the fewer the leaves torn, the more wrappers can be cut out of 1 lb.
of tobacco, other conditions being equal, and consequently the higher is
the price of the article. The cigar-manufacturer often does not
appreciate the aroma so much as the other qualities. He can do nothing
to improve the botanical characters: the finest aromatic leaf would be
of little value to him if it were torn; but he is to a certain extent
able artificially to improve defects in flavour. Of all kinds, Maryland
is considered to possess the qualities that distinguish a good tobacco
in the highest degree. Some of the Havanna tobaccos belong to this sort,
as also the Ohio, Amersfort, Turkish, and Dutten tobaccos. Its
cultivation assumes larger proportions every year, and the number of
varieties and sub-varieties increases accordingly. Perhaps the finest
wrappers for cigars are grown in Manilla.
On this subject, Judson Popenoe remarks that he has "cultivated various
kinds of tobacco, but have come to the conclusion that what we call the
Ohio seed-leaf is the best and most profitable kind for general
cultivation. There are other kinds of tobacco that sometimes are
profitable, and do well, but most of these do not cure out so well, nor
colour so evenly, nor are they so fine and saleable as the seed-leaf.
The Havanna tobacco is too small and has not the fine flavour of the
imported. The Connecticut seed-leaf I believe to be identical with our
Ohio seed-leaf; the difference in the climate may make a slight
variation in the quality, but we plant the Connecticut seed-leaf here in
Ohio, and I do not think they can be told apart."
Schneider recommends the following varieties: "1. Connecticut seed-leaf,
principally for cigar-wrappers; 2. Cuba, for fillers and wrappers; 3.
Maryland; 4. Virginia, the last two principally for smoking and chewing
tobacco. For snuff everything may be used, the refuse and even the
stems. The Connecticut, Maryland, and Virginia yield the largest crops,
the Cuba the smallest but best. The first varieties yield about one
thousand pounds, the latter five hundred pounds. In very favourable
seasons double the amount may be raised. All tobacco-seed, which is
removed from its native clime and soil, will deteriorate, and the seed
must be renewed from its native place, although the seed may, when it
finds favourable soil, &c., yield just as good, if not a better
variety."
In Virginia, remarks Thomas, there are "as many varieties of
tobacco-seed as of corn or wheat. I will name a few: The Big Frederic,
the Little Frederic, the Blue Stalk, the Brittle Stem, the Big Orinoco,
the Little Orinoco, and half-a-dozen others, each having, or supposed to
have, some characteristic distinguishing it from all the others. But the
Brittle Stem and the Orinocos were the varieties mostly cultivated, the
former for its early maturity, the latter for its comparative heaviness.
There are several varieties, also, in this vicinity, such as the Brittle
Stem, the Graham Tobacco, and the Cuban, but the names convey little
certain information, as the same varieties bear different names in
different localities. But some varieties are evidently to be preferred
to others—one noted for early maturity, all things else equal, is
preferable to another that ripens late. One distinguished for fineness
of texture, all things else equal, is better than another of coarser
fibre, &c. Upon the whole, the surest and most profitable variety is
that which ripens earliest, and yields the largest number of pounds,
cured, to a given number of hills planted."
In the opinion of Perry Hull, a grower in Litchfield county,
Connecticut, "the variety best adapted to our purpose is that known in
this State as the Bull Tongue. The leaf is neither too long nor too
short; the length and width being in such good proportion that
manufacturers considered there is less waste than there is to a very
long narrow leaf, or a very broad short leaf. It yields well, and ripens
at least one week earlier than-many of the broader varieties. Almost any
of the seed-leaf varieties will do well; but never patronize any of the
humbugs sent from the Patent Office, under the name of Graham tobacco,
Maryland broad leaf, &c. They are a Southern tobacco, and when grown
upon that soil, make chewing-tobacco; but here it is good for nothing
for that purpose, and is too coarse for cigar-wrappers."
According to Dennis, an Indiana planter, "selection of seed depends upon
the kind of land you have and the quality of tobacco you wish to raise.
Rich, fertile bottom-lands will grow only heavy, strong tobacco, and it
is the interest of the farmer to select that kind of seed that will
produce the plant of the greatest weight; in other words, to make weight
the prominent object in the result of the crop. Thinner, poorer land
will produce tobacco of lighter weight, but of finer and more desirable
quality, and one that will bring a correspondingly higher price. The
Orinoco tobacco is raised extensively in Missouri and Kentucky for heavy
tobacco, and is known in market as Kentucky Leaf. The seed for the finer
qualities passes (as does the other also) under different names, but may
be procured in Pike and Calloway counties, Missouri, and in Virginia;
the Orinoco, and kindred kinds, in Howard and Chariton counties in
Missouri. I should suggest that the seed may be procured through the
agents of express-companies at Glasgow, Brunswick, and Renick for the
Orinoco, and at Louisiana or Fulton for the other qualities. I would
recommend the culture of the coarser, heavier kinds, for the reason that
the finer quality needs much more care and experience in the handling,
in order that it may go into market in a condition to command such a
price as its quality, when well handled, entitles it to."
In the words of Libhart, a Pennsylvanian farmer, the "best variety for
cultivation in a high northern latitude is the Connecticut seed-leaf, as
it ripens two weeks earlier than most any other variety, cures and
colours better, and commands the highest price in the market. The
Pennsylvania seed-leaf outstrips the Connecticut in size and weight, but
owing to its requiring a longer time to mature in, is not so well
adapted to climates north of 41° or 42°."
An experienced Missouri grower, named Pursley, remarks that there "are
more than twenty distinct varieties, of which I will only mention the
most valuable:—The Yellow Prior, Blue Prior, Orinoco, Little Frederic,
Big Frederic, Cuba, and Spanish tobacco. These are considered the most
valuable in this State. The Yellow Prior and Orinoco are the most
profitable.
"I prefer the Yellow Prior, as it is the easiest cultivated and is the
most fine and smooth of the many varieties. Some growers prefer the
Orinoco, on account of it being the heaviest. I do not for various
reasons: it has large stiff fibres and ruffled stalks, which afford
hiding-places for insects; it moulds easier, is harder to cure, and
generally does not bring as good a price as the Yellow Prior."
_Seed._—The best and strongest plants are selected for affording seed.
These are not "topped" like the remainder of the crop, and are left
standing when the crop is gathered. All suckers are carefully removed
from the stems, and sometimes from the leaves also. When the crop is
cut, the seed-stalks should be staked, to prevent their destruction by
the wind. As soon as the seed-pods blacken, the seed is ripe; the heads
are then cut off below the forks of the plant, and are hung in a dry and
safe place to cure. Care must be taken to gather them before frost has
impaired their vitality. During leisure time, the pods are stripped from
the stalks, and the seed is rubbed out by hand, and winnowed. Its
vitality is proved by its crackling when thrown upon a hot stove.
_Seed-beds._—A very light friable soil is necessary for the seed-beds;
to obtain this, it should be broken up to a depth of 1½ ft. some months
before the sowing season. A drain is dug around the beds, and the soil
is utilized in raising the surface. In America, a very warm and
sheltered situation, such as the south end of a barn, is selected for
the seed-beds. It is a common plan there to burn a brush-heap over the
ground, thus supplying potash and killing weeds. The time for sowing in
America is usually from the middle of March to the 10th of April, or as
soon as the ground admits of working in the spring; in India, it depends
upon the locality: when the monsoon rains are very heavy, it should
follow them; in other cases, it may precede them.
Unless the soil be very rich in humus, it should be heavily manured with
well-preserved farmyard manure soon after breaking up. The soil of a
tobacco nursery cannot contain too much organic matter; the presence of
much humus will prevent, to a great extent, the formation of a surface
crust, which is so detrimental to the development of the plants during
their early growth, and will also facilitate the extraction of the
plants when transplanting takes place. After a few weeks have elapsed,
the soil should be dug over a second time, and the whole be reduced to a
fine tilth. The land may now remain untouched until the sowing-time,
unless weeds should spring up: these must be eradicated.
The area required for a nursery depends on the area of ground to be
planted, and on the distance separating the plants in the field. About 1
sq. in. space should be allotted to each of the young plants in the
nursery. Taking the number to be 7260 plants required for an acre (at 3
ft. × 2 ft.), and giving each plant 1 sq. in. of room, an area of 7000
sq. in. or 50 sq. ft. would raise plants sufficient for an acre. But as
some are injured during growth, many rendered useless in lifting them
for transplanting, and more needed to replace those that die after
transplanting, double the number should be raised, or 100 sq. ft. of
nursery bed for an acre.
The amount of seed required for an acre depends chiefly on its vitality.
An ounce contains about 100,000 seeds, or sufficient for nearly 7 acres
if all grew; but as even the best has not a very high percentage of
vitality, ½-1 oz. is generally sown to produce the plants required for
one acre.
Sowing-time having arrived, the nursery is divided into beds, most
conveniently, 10 ft. long and 5 ft. wide, making 50 sq. ft. each, on
which plants for ½ acre can easily be raised. As, even with a small
tobacco plantation, several days are required for transplanting, all the
beds should not be sown at one time, but at intervals of a few days.
This will also lessen the risk of the young plants being all destroyed
by a storm, insects, &c. Before sowing the seed, the soil is dug over to
the depth of 6 inches, and levelled with a rake. The seed must then be
sown evenly on the surface, and beaten down slightly with the hand or
otherwise. The seed being very small, many cultivators mix it with
ashes, or pulverized gypsum, in order to distribute it regularly over
the bed. The seed must be covered only slightly, best done by strewing a
little fine compost manure over it. Ants, which often destroy the seeds,
may be kept off by sprinkling some ashes over the bed. Finally cut straw
may be scattered over the surface. In India, to protect the nursery from
the sun and rain, the whole is covered with a roof made of straw,
leaves, or cloth, supported by poles, at only a few feet above the
ground. The soil must be kept constantly moist, but not wet; weak liquid
manure may be used for watering. Much time is saved by starting the seed
in a warm room before sowing.
The plants, which will appear about a week after sowing, are very tender
during the first stage of their growth, and require frequent watering
through a fine rose. The straw will now prevent the water falling with
any force immediately on the plants, and its tendency to wash the soil
from the fine rootlets. If the plants spring up thickly, they are
thinned out, when about a week or two old, leaving about 1 sq. in. for
each. Those taken out may be used to fill blanks in the nursery bed, or,
if more plants are taken out than are required for this purpose, they
should be planted in a separate bed. It is universally acknowledged that
plants transplanted when very young develop more roots, grow more
vigorously, and become more hardy afterwards, than when not transplanted
at this stage. When the plants are about two weeks old, they require
less attention, and should be watered less frequently, to harden them
before transplanting. Any weeds appearing must be removed, and injurious
insects must be killed. In about 7–8 weeks after sowing, the plants will
be fit for transplanting.
Bowie, a Maryland planter, gives his experience in the following
words:—"After a thorough burning of brush, dig deep, and continue to
dig, rake, and chop until every clod, root, and stone be removed; then
level and pulverize nicely with a rake. As to the variety to plant, I
think the Cuba is a very good kind for our climate. The Connecticut
seed-leaf is the best, but culture has more than anything else to do
with the quality. Mix 1 gill of seed for every 10 square yards with a
quart of plaster or sifted ashes, and sow it regularly in the same
manner that gardeners sow small seeds, only with a heavier hand; roll
with a hand-roller or tramp it with the feet. If the bed is sown early,
it ought to be covered with brush free from leaves; but it is not
necessary to cover it after the middle of March. Tobacco-beds may be
sown at any time during the winter if the ground be not too wet or
frozen. The best time for sowing is from the 10th to the 20th of March,
though it is safest to sow at intervals, whenever the land is in fine
order for working. Never sow unless the land is in good order, for the
work will be thrown away if the land be too moist or be not perfectly
prepared. The beds must be kept free from grass or weeds, which must be
picked out one at a time by the fingers. It is a tedious and troublesome
operation, therefore you should be very careful not to use any manures
on your beds which have grass or weed-seeds in them. After the plants
are up, they should receive a slight top-dressing of manure once a week,
sown broadcast by the hand. This manure should be composed of ½ bushel
of unleached ashes (or 1 bushel of burnt turf), 1 bushel of fresh virgin
woods-earth, 1 gallon of plaster, ½ gallon of soot, 1 quart of salt
dissolved in 2 gallons of liquid from barnyard, and 4 lb. of pulverized
sulphur, the whole well intermixed. Let a large quantity be got together
early in the spring, or winter rather, and put away in barrels for use
when wanted. This, and other such mixtures, have been found efficacious
in arresting the ravages of the fly—both from the frequent dusting of
the plants and the increased vigour which it imparts to them, thereby
enabling the plant the sooner to get out of the tender state in which
the fly is most destructive to it. The fly is a small black insect,
somewhat like the flea, and delights in cold, dry, harsh weather, but
disappears with the mild showers and hot suns of opening summer. If
possible, the plants should stand in the bed from ½ inch to 1 inch
apart, and if they are too thick they must be raked when they have
generally become as large as 5 or 10-cent pieces. The rake proper for
the purpose should be a small common rake, with iron teeth 3 inches
long, curved at the points, teeth flat, and ⅜ inch wide, and set ½ inch
apart."
Schneider, whose success as an Illinois planter has already been
mentioned, expresses himself thus:—"Raising tobacco-plants from seed is
somewhat similar to raising cabbage-plants, but is different in two
important things: It takes considerably more time for the seed to sprout
(six weeks), and, on account of disturbing the roots, cannot well stand
weeding. Therefore the principal care in providing the seed-bed is, to
prepare for the early starting of the seed, and to have the bed free
from all weed-seeds. In the West we prepare the seed-bed in the
following manner: we take a plot of land—newly cleared land is
preferred—sloping southward, and protected against winds. The bed should
be 4 feet broad and 8 feet long; on this we pile brush, wood, and heavy
logs, sufficient to keep up a strong fire for at least one hour, and
burn it. When the coals begin to die out, or before the soil is cold,
the bed is cleared off, and only the fine ashes are left; then it is
hoed thoroughly and as deep as the strongest heat has penetrated, after
which it is raked cross and lengthwise, until the soil is entirely
pulverized. Everything that might hinder the growing of the plants, and
their taking out afterwards, is carefully removed. On this bed a
thimbleful of seed, well mixed with a few handfuls of ashes or earth, is
sown broadcast, and tramped in with the feet, or slapped with the under
side of the spade or any other suitable instrument. After this, the bed
is thoroughly wetted with a weak manure-water, 12 lb. of hen-droppings,
or 1 lb. of soot in 10 gallons of water, and lightly covered with straw.
The seed-bed does not need much attention at first, if the weather
remains mild; but if there is danger of night-frosts, a layer of brush
must be made, and on this a layer of straw 2 to 4 inches thick,
according to the degree of frost. The straw is removed in the morning,
and put on again at evening, leaving it off entirely when the nights are
mild. Although the seed-bed is ready now, it must not be left to itself,
and requires some care. The plants must always have sufficient moisture,
and if timely rains do not fall, they must be watered with weak liquid
manure as often as needed. Should weeds appear, notwithstanding all
precautions, they must be removed with the utmost care. The
above-mentioned quantity of seed is sufficient to raise plants for one
acre.
"Whoever is in possession of a hot-bed can raise the plants much easier;
he can sow later and have plants earlier and with more certainty. But
even the common bed may be made into a kind of hot-bed. The burned and
hoed surface soil is removed and put on one side, then one foot of fresh
horse-dung is laid on the subsoil, and the surface soil put back again.
Boards may be placed around, cross-pieces laid over them, and the straw
covering put on these.
"The earlier the young plants are ready for transplanting the surer the
tobacco crop will be. March is the latest to make the seed-bed in the
open air, and June the latest for transplanting. Some time may be gained
by keeping the seed in damp earth in the room, and sow it in the
seed-bed just before it commences to sprout."
Having selected a suitable location, says White, a Connecticut grower,
"next consider how large a bed you will need. That depends on the
surface you intend to plant out. A bed 2 rods long, by 12 feet wide,
will produce a sufficient number of good plants to set an acre. On such
a bed you should spread a heavy coat of good, fine, well-rotted manure,
at least 2 inches thick; let it be free from straw or other litter.
Then, with a good strong back, and long-handled spade (or other as you
prefer), spade up the bed, mixing in the manure very fine. Have ready
some fine dry brush, or the like, and spread over the whole surface; set
it on fire and burn to ashes. A small quantity will answer better than a
very large one, for if very much is burned, it is apt to do injury by
burning the soil. The less quantity will tend to destroy any foreign
seed turned up, and warm the ground. Having reduced the brush to ashes,
take a fine iron or steel rake, and proceed to pulverize very finely the
whole surface spaded up. After reducing it to as fine a state as
possible, and having made it flat and level, leave it till the next day.
Then, with your rake, carefully rake over the whole bed; it is now ready
for the seed. Sow the seed on broadcast; be careful to sow it even and
true. About two thimblefuls, or a little less, will be sufficient for
such a bed. It is better to have too little than too much, as in the
first instance, the plants will have room to form thick stalky roots and
well-spread leaves, while in the latter they will be crowded with
spindling tops as well as small roots. Having sowed your seed, take a
good heavy garden-roller and roll the surface down hard and smooth. In
the absence of a roll, a very good substitute can be made by taking a
piece of 2-inch plank, say 18 inches long by 14 inches wide; in the
centre, place an upright handle. With this spat the bed over, being
careful to do it evenly, and to leave the surface solid and level, the
reasons for which you will afterward discover in weeding and taking out
plants to set in the field. This should be done in the spring, as soon
as the ground will permit, say first of April, if the frost is out and
the ground settled. The roll or spatter will cover the seed sufficiently
without any other covering. To be able to sow the seed with the least
trouble, mix it in thoroughly with wood-ashes or plaster, before sowing.
To obtain plants earlier, you can mix your seed thoroughly in about a
quart of light chip dirt from under your wood-shed; put it in some
proper vessel, and wet to the consistence of soft putty, with water as
warm as can be well borne by the hand. Set it on the mantle-shelf in the
kitchen, not too near the stove or fire, but where it will keep warm. In
the course of a week or ten days, the seed will have cracked the shell,
and will show the small white germ or sprout. It should now be sowed
broadcast very evenly, and treat as before described. If properly wet at
first, it will need no more water to sprout the seed. Before sowing,
pulverize the mass containing the seed, to facilitate the sowing. Having
thus sown and rolled down your bed very nicely, it is well to have
something to protect it from the encroachment of the fowls. For this
purpose, spread a net of twine or a few brush over the surface, covering
it so that they may not disturb the surface by scratching and wallowing.
It may now be left till the weeds begin to make their appearance; these
you will need to extract by the roots as soon as the plants can be
distinguished; these last may be known by two very small nearly round
leaves opening over flat on the ground. Now procure a plank or some
substitute a little longer than your bed is wide, also two blocks 5 or 6
inches square, as long or longer than your plank is wide; place one on
one side of the bed, the other on the opposite side; on these two blocks
place your plank, and you will have a fine platform on which you can sit
and weed any part, or all, of your bed, by moving it as occasion may
require. To assist in pulling out the weeds, procure a moderately
sharp-pointed knife, and with the same grasped in the hand with the
thumb near the point, pinch out the weeds, being careful not to disturb
the dirt any more than absolutely necessary. The process of weeding must
be repeated as often as necessary, to keep the bed clean from weeds."
[Illustration: FIG. 4.]
Obviously, no frost must be allowed to reach the seedbed when once
sowing has taken place. To prevent this, and for another purpose to be
described presently, Perry Hull advises the construction of a straw mat,
as shown in Fig. 4, which is very light to handle, easily made, and
sufficiently strong to last one season. It is made "by laying a
scantling (6 feet long, 1½ inches wide, ¾ inch thick) upon the barn
floor; place a layer of good straight rye-straw upon it, so that the
scantling will come about in the middle of the straw, then another layer
with the tips the other way, that it may be of uniform thickness in all
its parts (about 1½ inches thick). Place a similar scantling exactly
over it, and with sixpenny nails, nail them tight; with an axe trim both
edges straight, and to a width of 3 feet, and the mat is made. With
these the beds should be covered every night, cold or warm; in the
daytime they should be set up at the north side of the bed, at an angle
of about 65 degrees, by driving crotches just inside of the bed, for the
end of the scantling to rest in, the lower edge of the mat resting on
the ground, outside the bed.
"The plants, as soon as they are out of the ground, which will be in a
few days, require strict attention. The beds should be made high enough,
so that in fair weather a little water can be applied every night. After
the fourth leaf appears, manure-water should be used. Place an old
barrel near the beds, and throw into it ½ bushel of hen-manure, and fill
with water; after it is well soaked, use ½ pailful of it, and fill up
with clear water with the chill taken off. As the plants get larger, the
strength of the infusion can be increased, being careful that it is not
so strong as to turn the plants yellow. As soon as the plants are large
enough to be readily taken hold of by the thumb and point of a knife,
they should be thinned to about 144 per square foot, and kept free from
weeds. This plan is decidedly preferable to raising under glass. It is
less expensive, the plants are more hardy to set out in the field, are
got fully as early, and a little carelessness on a hot day will not ruin
the whole. It has been my method for the past 8 years, and during that
time I have never failed to have good strong plants ready for the field
between the 5th and 10th of June."
Mitjen, whose essay on tobacco-growing in Cuba has been already
mentioned, recommends a system of shade frames borne on small tramway
trucks, as illustrated in Fig. 5—(_a_) seed beds, raised above the
surrounding level; (_b_) light pointed covers of thatch on a wooden
frame, and provided with grooved wheels; (_c_) rails on which the frames
run, facilitating their application or removal as the vicissitudes of
the weather may demand.
[Illustration: FIG. 5.]
_Preparation of the Field._—Land intended to be planted with tobacco
should receive several ploughings not less than 9 inches deep. As a
rule, clay requires to be more deeply ploughed than sandy or loamy soil.
It greatly conduces to success, if the land is allowed to lie fallow for
several months before planting the crop, to admit of the proper
preparation of the soil, by ploughing, rolling, harrowing, &c., and to
allow the attainment of as fine a tilth as is usual in gardens. No crop
will better repay the expense of proper preparation of the soil than
tobacco; the fineness of the leaf and the aroma of the tobacco depend to
a great degree upon this. The land should be ridged immediately before
planting. The distance apart at which to make the ridges is governed by
the quality of the soil and the sort of plant to be raised. With good
soil, the ridges must be farther apart than in a poor one, because of
producing larger leaves. The ridges should allow a passage between the
rows, for the purpose of weeding, hoeing, suckering, &c., without
breaking the leaves. In the lines, the plants may be 6 in.–1 ft. closer
than the ridges. In some places, a plough is run at right angles across
the ridges before planting, at the distance at which the plants have to
stand in the lines, thus forming small hills on which the seedlings are
planted.
_Planting._—Planting should take place only in the evening (or even at
night in India), unless the weather be cloudy, when it may be performed
during the whole day. Some hours before commencing to transplant, the
nursery should be thoroughly watered, to facilitate the removal of the
plants, without tearing their roots. If the plants are of even size, so
that all can be removed, the best plan is to take them out with a spade,
or trowel, leaving a lump of soil on each. But in most cases, it will be
necessary to take up each plant separately; this should be done very
carefully, holding with the thumb and forefinger as near as possible to
the roots, and drawing out the plants, if possible, with a little soil
adhering to their roots., The plants are taken at once in a basket to
the field for planting. An attendant going between two ridges places a
plant on each hill, right and left. One attendant is sufficient for two
planters, who follow immediately. The planting is nearly the same as
with cabbages, but requires more care, the plants being more tender, and
their roots and leaves springing nearly from the same point, they are
more difficult to handle. The plants should be placed in a hollow made
on each hill, which will serve as a reservoir for the water to be
applied, and also afford some shade.
In India, the plants are watered immediately after planting; they should
also by some means be shaded during the first few days, which can easily
be done when only a small area is planted, but is rather difficult to
manage on a large scale. In the latter case, the shade afforded by
planting in a slight cavity must suffice. If the plants have been taken
from the nursery with some soil adhering to their roots, and are kept
sufficiently moist during the first few days, few of them will die. When
the weather is dry, water should be applied at morning and evening, and
after that time, once daily until the plants have taken root, after
which, occasional waterings, varying with soil, weather, and kind of
plant, must be given. In dry weather, and with a soil poor in humus, one
watering every second or third day may be necessary, whereas with a soil
rich in organic matter, and in a moist atmosphere, watering may be
entirely dispensed with. During the first few days, the water is applied
with a watering-pot, held very low, otherwise the soil would be washed
from the plant-roots, and expose them to the direct rays of the sun,
causing death. The arrangement of the plants in what is known as
quincunx order, as shown in Fig. 6, is generally adopted.
[Illustration: FIG. 6.]
This part of the operations connected with tobacco-growing is described
at some length by Mitjen so far as the practice rules in Cuba. His
translator remarks that "as soon as the land has been prepared, it
should be furrowed at a distance of 1 yard between each two furrows.
This operation should be simultaneous with the planting, and should be
done, if possible, after 3 o'clock in the afternoon, and on cloudy days,
so as to prevent the recently set plants from being scorched by the sun.
The furrows should run more or less from north to south, as, by making
them in this direction, the plants are less injured by the sun, or the
strong winds which generally blow about the planting season.
Immediately, and behind the man who is furrowing, another should follow,
placing the plants at every ½ foot all along the furrow, and behind them
another should at once set the plants, the first walking in the
distance, or bank, and the other in the furrow. The one should open the
land with his right hand, behind which, with his left, the other will
place the plant, being careful neither to double the stalk nor the
roots, and, letting the ground fall directly on the roots, should press
it lightly on them with his hand. The plants should be buried half-way
up the stalk, or, if the plant is small, it should be covered to where
the leaves spread. Care should be taken that the plants have no _dry_
mould sticking to their roots, and that no ground from the furrow falls
in the _centre or sprout_, and when the planting is going on, the ground
should not be too wet. The plants should be set on the side of the
furrow, and on that side which is next the setting sun, so that the
rising sun may strike upon them, and they may be somewhat protected from
the rays of the afternoon sun.
"Generally the plants wither after being transplanted, but on the third
or fourth day after they are set they begin to shoot up, and on the
fifth day or the sixth, those that have not taken root can be
distinguished. Then, and without loss of time, others should be
supplied, this operation being repeated at the end of another 5 or 6
days, so that the whole field may be well filled with living plants.
This is one of the most important operations for securing a good crop,
because the fields will require as much cultivation and labour bestowed
on them if they have vacant spots as if they were full and regularly
planted, and, of course, the yield will be less, besides many other
evils well known to practical _vegueros_.
"According to the best opinions admitted among _vegueros_, one man can
take care of 12,000 tobacco plants, and prudence dictates that no more
land should be planted than that which can be well attended to, as
experience shows that in exceeding this number for each man, instead of
proving advantageous to the planter, it is frequently the cause of
considerable loss. Excessive planting produces, at once, an increase of
labour, and if, unfortunately, a hard year should occur, occasioned by
caterpillars or other causes, it almost always happens that the man who
has only planted 12,000 plants, for each labourer he can command,
produces four times as much tobacco, and of a better quality, than he
who may have planted from 25,000 to 30,000 plants per labourer.
"When the plantations are out of proportion to the strength of the
labour which can be counted on, all the work becomes slowly and badly
done, and these faults most sensibly prejudice both the yield and the
quality of the crop, and consequently the interest of the planter.
Immediately after supplying the fields, the tobacco plants should be
carefully inspected, almost daily, in order to exterminate the
caterpillars of every kind that may be found, and this operation should
_always_ be made during the morning, because in the heat of the day the
worms are accustomed to hide themselves from the sun, and the wind
agitates the leaves too strongly to permit them to be handled without
risk of being broken or torn, especially when they are somewhat large."
_After-cultivation._—After the plants have once taken root, they grow
rapidly. They are hoed when about 6–9 in. high, and the soil is drawn
from the furrows to raise the hills, maintaining a depression round the
stems. If the soil is not very rich, a special manure should be applied
at this stage of growth. The best manure generally will be nitre in a
liquid state, which can be applied in the depression around the plants
with a watering-pot. By applying it in solution and close to the plant,
less is required than when spread over the whole field. Some weeks
afterwards, another hoeing and heaping of earth round the plants will be
necessary. It is most difficult to say the number of hoeings which may
be required by a tobacco crop. The general rule to be followed is to
keep the soil loose, friable, and free from weeds. The more organic
matter the soil contains, the more will it remain loose and friable; the
less organic matter, the more waterings will be required, which causes
the soil to crust over, and to assume a close texture, and necessitates
frequent hoeings. As long as the plants have not spread much, the hoeing
may be done by a cultivator, followed by some men to perform the
heaping. Insects which attack the tobacco must be carefully sought for
and killed at once. They can easily be discovered in the mornings; if
not killed, they may destroy the whole crop in a few days. Turkeys are
invaluable for their grub-eating propensities.
[Illustration: FIG. 7.
THE TOBACCO WORM.]
Worms, in the American phraseology, here generally known as
caterpillars, are the _bête noire_ of the tobacco grower. The most
common is highly destructive also to the potato and tomato foliage. The
worm as it comes from the egg is so small as to be unobserved, but
having an enormous appetite, it devours rapidly, and soon grows to a
great size. When not feeding, it lifts up the head and fore-part of the
body, and remains apparently lifeless. From its resemblance in this
position to the Egyptian Sphinx, Linnæus gave the name _Sphinx_ to the
genus. The larva is of a light green colour, with whitish oblique
stripes, and has a horn upon the rear end of the body. Though it is
repulsive in appearance, it is perfectly harmless to touch, and may be
picked off with the hands without fear. After it has reached its full
size, it leaves the scene of its ravages and goes into the earth, where
it throws off its skin and becomes a brown- chrysalis. The
curious projection, like a handle, at the end of the chrysalis, is a
sheath which holds the tongue of the future moth. The moth or perfect
insect is fully 2 in. long in the body and the spread of its wings
reaches 5 in. It is of a grey colour, with orange- spots on each
side of the body. As there are five of these spots on each side, it is
called _Sphinx quinque-maculatus_, or Five-spotted Sphinx. The moths may
be seen towards night flitting about the flowers, from which they suck
the juices by means of their remarkable tongue, which is 5–6 inches
long. When the tongue is not in use, it is closely coiled up and hidden
between the two feelers. From the manner of their flight and feeding,
they are frequently mistaken for humming-birds, and are called
"humming-bird moths," and "horn-blowers." The moths should always be
destroyed if possible; by so doing we prevent the production of several
hundreds of most destructive worms. Naturalists make one or two other
species, which closely resemble the Five-spotted Moth, and are only
distinguished by characters which would not be noticed except by the
entomologist.
Judson Popenoe gives the following advice with regard to these pests.
"As soon as worms appear, which is generally when the leaves are as big
as a man's hand, go over the tobacco, looking carefully at every plant.
The worms usually stay on the under side of the leaf; if you see a hole
in the leaf, no matter how small, raise it up and you will generally
find a worm under it. Worming can not be done too carefully. Miss one or
two worms on a plant, and before you are aware of it the plant is nearly
eaten up. When you find a worm, take hold of it with the thumb and
forefinger, giving your thumb that peculiar twist which none but those
who are practised in it know how to do, and put the proper amount of
pressure on, and my word for it you will render his wormship harmless.
Worming has to be continued until the tobacco is cut; the last worming
to immediately precede cutting and housing."
Schneider remarks that "from the first starting of the tobacco plant, it
has its enemies. First appears a cutworm that works in the soil and eats
the roots off. Then comes a little caterpillar which enjoys itself on
the young leaves, and lastly the beautiful and large tobacco-worm, which
eats into the leaf, and in a short time leaves nothing but the
leaf-stems and stalk. The only remedies against these enemies are the
vigilance and industry of the planter—looking after them, digging up,
picking, and destroying once or twice a day, or as often as there are
any traces of them. Children, to whom premiums are offered, will be very
successful in destroying them. A herd of turkeys, if given access to the
tobacco-field, are a very valuable help. A <DW64> from South Carolina
told me a few days ago, that a solution of blue vitriol in water,
sprinkled over the plants, will kill the worms. The remedy may be worth
trying. Of course the solution must be made weak enough, so that it will
not destroy the plants as well as the worms."
On the same subject, White recommends the planter on the "next, or at
farthest, the second morning after having set your plants, go over to
see that the worms do not eat up one-half of them. You can tell where
they are and have been, by seeing a plant with a single leaf, and
sometimes the whole plant eaten off and drawn down into the hole
occupied by a large brown or black worm; you will see little ant-hills
like, and round holes in the ground; by poking around a little in the
dirt, you will find a worm very near the mouth of these little holes.
Destroy it, and all you can find, and thus save your crop. This
searching for worms must be kept up till they cease to do mischief. All
plants missing in the field should be renewed from the bed at the first
opportunity. The morning is the best time to find the worms, as they are
near the surface of the ground; later, they retire into the ground to
appear again near sundown, and work during the night and early morning."
Thomas describes tobacco worms as "hatched from eggs deposited by what
is called the 'tobacco fly.' It is a large, dusky-brown, winged miller,
nearly as large as a humming-bird. It lays its eggs on fair evenings and
moonlight nights in July and August. It can be seen almost any clear
evening, among what are called 'Jimson-weeds,' sucking the flowers. The
eggs will hatch out in 24 hours, and the worms commence eating when less
than ½ inch long, and continue to eat till they attain the length of 4–5
inches. One worm, in 6 weeks, will destroy a plant so completely as to
render it utterly valueless. This pest is vastly more numerous in some
seasons than in others. Four years ago there were scarcely any; but for
the last three years they have been destructively numerous. The worming
of the crop, when they are numerous, is, by far, the most disagreeable
and tedious labour attending it. Much of the value of the crop depends
upon the care or inattention of performing this part of the work. The
crop may have been planted in good time—ploughed, hoed, primed,
suckered, topped, cut, and cured well; yet it may have been so riddled
by worms as to be comparatively good for nothing in market; hence, they
must be picked off and destroyed, and that promptly."
_Topping and Suckering._—The plants will commence to flower about two
months after planting, when 2–7 feet high. When the flower-buds appear,
they must be broken off, and with them the top and bottom leaves. By
breaking off the flower-buds at an early date, the sap that would be
used in the formation of these organs flows to the leaves, which thereby
increase in size, and the outturn becomes much heavier than when the
plant is allowed to flower. But it is generally admitted that the leaves
lose much in aroma. To what extent the early removal of the flower-buds
impairs the quality has not been properly investigated. It is very
probable that the greater yield does not always compensate for the loss
in quality. The bottom leaves are generally of inferior quality, small,
torn, and dirty. The number of leaves to be left on the plant varies
greatly, according to species, quality of soil, and method of
cultivation. The minimum may be placed at 6, the maximum at 22. The only
rule to be observed is to retain as many leaves as the plants are able
to mature. Soon after the plants have been topped, suckers appear in the
axils of the leaves; these should be broken off as soon as they come, at
least they should not be allowed to grow longer than 4 inches. If the
suckers are not removed soon after their appearance, the size of the
leaves will be seriously impaired. After the plants are half-grown,
great care must be taken when going through the lines, whether for the
purposes of hoeing, watering, or suckering, &c., not to tear the leaves.
In India, hoeing and suckering should be performed only when the leaves
have lost part of their turgescence, attained at night. Insects,
however, must be killed during the morning and evening; at other times,
they are not easily found. Leaves which are torn are not fit for
cigar-wrappers, and must often be thrown on the refuse heap as
valueless, even if well developed and of good colour.
The plants commence to ripen about three months after being planted;
this is indicated by the leaves assuming a marbled appearance, and a
yellowish-green colour. The leaves also generally become gummy, and the
tips bend downwards. It is considered that tobacco intended for snuff
should have attained more maturity than tobacco for smoking. Nessler
found that the less ripe leaves contained more carbonate of potash, and
burnt consequently better, than the more ripe ones, but the total amount
of potash was larger in the latter than in the former; cigars made from
less ripe leaves kept the fire when lighted for a shorter time than
those made from more ripe leaves.
In the words of Judson Popenoe, the "tobacco is ready to top when the
button (as the blossom or top of the stalk is called) has put out
sufficiently to be taken hold of, without injury to the top leaves. As
tobacco is not regular in coming into blossom, it is the usual practice
to let those stalks that blossom first, run a little beyond their time
of topping, and then top all that is in button as you go. There is no
particular height to top at, but as a general thing 16 to 18 leaves are
left; judgment is necessary to determine where to top; if topped too
high, 2 or 3 of the top leaves are so small as not to amount to much; if
topped low, the tobacco spreads better; if just coming out in top, reach
down among the top leaves, and with thumb and forefinger pinch the top
or button off below 2 or 3 leaves; if well out in top, break off several
inches down from the button and 4 or 5 leaves below it. As soon as the
tobacco is topped, the suckers begin to grow; one shoots out from the
stalk at the root of each leaf, on the upper side. When the top suckers
are 3–4 inches long, the suckering should be done; with the right hand
take hold of the top sucker, with the left take hold of the next, close
to the stalk, and break them off, and so proceed, using both hands,
stooping over the stalk, taking care not to injure the leaf. Break the
suckers about half-way down the stalk, the balance being too short to
need removing until the second suckering. In about 2 weeks from topping,
the tobacco is ready to cut; now give it the last worming and suckering,
breaking all suckers off down to the ground, and remove every worm, if
you don't want your tobacco eaten in the sheds."
Another process, called "priming" by Schneider, is thus described by
him. "The object of priming is to break off the leaves that come out too
near the ground, which, when large, lie flat on it, and therefore rot or
get dirty. This work should be done early, the sooner the better, so
that the plant does not lose much strength by their growing. These
leaves must not be torn off, especially not downward, because the plant
would be injured, and instead of throwing the strength gained into the
other leaves, it would be thrown away to heal the wound. The distance
from the ground at which this priming should be done, depends upon the
variety grown and upon the time at which the work is done: 4–6 inches is
the right distance. This priming is not done by every one. One farmer
may practise it, while his neighbour does not; but sorts the lower
leaves separately, and sells them as so-called 'lugs,' for which he gets
a little over half the price of the good upper leaves. Those who do not
prime, must generally top lower, or they must risk that the whole plant,
or at least the upper leaves, will not mature fully.
"Topping is done to throw the strength, which would go to develop seeds,
into the leaves. It must, therefore, be done as early as the seed-buds
show themselves, if not earlier. This work must be done, and the
question is, how to do it. If there are but few leaves on the plant,
even these will not ripen, if it is not topped; if there are many, then
the grower has the choice either to break off the flower-stalk only or
to take off one or more leaves also. This should be done in answer to
the questions: 1st. Is there time enough to ripen even the upper leaves
fully? and, 2nd, Are the plant and the soil strong enough to ripen all
leaves, even the upper ones? The answers to these queries will decide
the way of topping. If yes, he takes off the flower-stalk only; if no,
he tops to 8, 10, 12, 14, or 16 leaves, according to his judgment, that
is, he allows so many leaves to remain as will have a good fair chance
of reaching maturity."
As Bishop remarks, cultivators are not agreed on the time and place for
topping tobacco plants. "Some favour the plan of topping as soon as the
blossom-buds appear, others prefer to wait until in blossom. I think
there is no harm in letting the earliest plants bloom before being
topped, but after once beginning, they should be broken off as soon as
the buds begin to look yellow, and the latest plants as soon as the buds
appear. A new beginner will be apt to top the plants too high. The
object is to ripen and develop as many leaves as the plant can support;
if topped too high, the top leaves are small, and when cured are nearly
worthless, and the other leaves are not as large or heavy, whereas, if
topped too low, then you lose one, two, or three leaves, which the plant
might have supported. As a general rule, a plant just in blossom should
be topped down to where the leaves are full 7 inches wide, leaving on
the stalk from 15 to 18 leaves. This will leave the stalks about 2½ feet
high in good tobacco. Later in the season, top the plants sooner and
lower. Let as many of the earliest plants as will be wanted remain for
seed. One plant will furnish seed enough to put out 5 acres, at least.
These should be wormed and suckered like the rest, only leaving the
suckers above where you would ordinarily break it off, were you to top
it. The piece should now be looked over every other day, to break off
the suckers and catch the worms. This should be done as soon as the dew
is off in the morning, and towards night, as the worms are eating then,
and can be found more readily, while in the heat of the day they remain
hid. Great care should be taken not to break off the leaves while going
through it, as they are nearly all wasted before the crop is ripe. As
soon as the top is broken off, the sap is thrown into the leaves,
causing them to expand rapidly. In the meantime suckers will start out
just above where each leaf joins the stalk; these must be broken off, or
the growth of the leaf will be checked, as the sap will be thrown into
these young sprouts. Those nearest the top will start soonest, and will
require breaking off twice before the plant is ripe; those at the bottom
must all be broken off. This is the hardest and slowest work of all. Not
only will these suckers check the growth of the plants, but if allowed
to grow will soon break or pry off the leaves, or cause them to grow out
at right angles from the stalk, rendering them more liable to be broken
off. It is a good plan to have a piece of corn on the north side of a
piece of tobacco, or, at least, two or three rows, to shield the growing
plants from winds."
Priming is defined by Thomas as "pulling off the bottom leaves to the
number of 4 or 5," and he says that any plant large enough to be topped
ought to be primed first. All conditions being favourable, he considers
that in Ohio, a "tobacco plant will ripen in as many weeks, from the
time of topping it, as there are leaves left on the stalk. Consequently,
if the topping is done early, it can be topped high, if later, it must
be done lower, and if still later, still lower. Planters differ very
much at this point. Some will top as high as 16 leaves, others 10, and a
great many at 8. My own opinion is, that a plant topped at 10 will weigh
as much as one at 16, topped at the same time, and on the same kind of
land. About a week after a plant has been topped the suckers will begin
to grow. A sucker is only an auxiliary branch which shoots out at the
junction of the leaves to the stalk. If not removed, they will grow, and
bloom, and ripen seed, and in doing so they will 'suck' the parent-stem
of much of its vitality. When the crop of suckers are about 1 inch long
they can be pulled or rubbed off, and it should surely be done. In about
a week or 10 days a second crop of them will appear. These must also be
promptly removed, and then the third crop will show itself, which must
be similarly treated. The longer they are permitted to remain on the
plant, the more they <DW44> its development, and delay its maturity."
CHAPTER III.
CURING.
Growing tobacco is only half the battle. Having raised a crop to a state
of perfection, the next object is to cure it for the market. This branch
of the business demands fully as much care and skill as the purely
agricultural part preceding it, and is perhaps equally influenced by the
weather. The best crop ever grown may be completely spoiled by
injudicious conduct during the drying, &c., while a growth of moderate
quality may be made the most of by extra care and trouble.
_Harvesting._—The leaf being matured, it should be harvested only after
the dew is off the plants, and not on a rainy day. There are two modes
of harvesting—gathering the leaves singly, and cutting down the whole
plant. Gathering single leaves admits of removing them from the plant as
they ripen; the bottom leaves are removed first, and the top ones are
left some time longer, until they have attained full maturity. The
cultivator is thereby enabled to gather his crop when it possesses the
greatest value. This plan necessitates, however, a great amount of
labour, and, in a hot climate, the single leaves are apt to dry so
rapidly as not to attain a proper colour, unless stacked early in heaps.
But stacking in heaps involves great risk of the leaves heating too
much, and developing a bad flavour, whereby the tobacco loses more or
less in value. For Indian circumstances generally, cutting the whole
plants is better than gathering the leaves singly.
For cutting down the plants, a long knife or chopper is used. A man
takes the plant with his left hand about 9 inches from the ground, and
with the knife in his right hand, cuts through the stem of the plant
just above the ground. If the plants are sufficiently "wilted," he may
lay them on the ground and proceed to cut down others; if, however, they
are so brittle as to cause the leaves to be injured by laying them down,
he should give them to another person, to carry them at once under
shade. During bright weather, the plants should not be allowed to lie
exposed to the sun on the ground, or they will become sun-burnt, and
lose in value. A temporary shed should be erected; it might be simply a
light roof of palm-leaves or thatched straw, supported by poles; a large
tree standing near will also serve the purpose. Under this shade,
parallel rows of posts are put up, and on the posts, light poles or
strong bamboos are fixed horizontally. The parallel lines should be
about 4½ feet apart and the horizontal poles about 4–5 feet from the
ground, according to the height of the tobacco plants. Rods are cut in
lengths of 5 feet, and laid over the parallel bars, so that they will
project about 3 inches at each end. A very light and convenient shelter
sometimes used for sun-drying in America, consists of rods laid
crosswise, supported on four upright poles, and covered with a sloping
roof of boards. The plants that have been cut are immediately brought
into the shade, tied in pairs, and hung across the rods. They must not
be hung so close as to press each other, and the rods should therefore
be 6–12 inches apart. The framework should be so large as to allow of
one day's cutting being hung. The plants are left thus for one day,
during which time they will be wilted sufficiently to allow handling
without tearing the leaves. In a very dry wind, mats or other cover
should be laid against the plants most exposed to it, or their leaves
will dry rapidly, shrivel up, and remain green. Next day the leaves are
carted to the drying-shed. A cart supplied with a framework, in order
that the plants may be hung as they were hung under the shade, is the
best means. Perpendicular uprights at each corner of a cart or waggon
are fixed together by horizontal poles. The plants may be hung so close
as not to press heavily on each other, 200–400 being brought to the shed
at one time.
As a general rule, Judson Popenoe thinks "tobacco should be cut in about
2 weeks from topping, at which time the leaves assume a spotted
appearance and appear to have fulled up thicker; double up the leaf and
press it together with thumb and finger, and, if ready to cut, the leaf
where pressed will break crisp and short. Do not let your tobacco get
over-ripe, or it will cure up yellow and spotted: it is better to cut
too soon than too late. Take a hatchet or short corn-knife, grasp the
stalk with the left hand, bend it well to the left, so as to expose the
lower part of the stalk, strike with the knife just at the surface of
the ground, let the stalk drop over on the ground without doubling the
leaves under, and leave it to wilt. The usual practice is to worm and
sucker while the dew is on in the morning, and as soon as the dew is off
to commence cutting. There are some who advocate cutting in the
afternoon, say 3 o'clock; let it wilt and lie out until the dew is off
next day, and take it in before the sun gets hot enough to burn it. I
prefer the first plan, because a heavy dew may fall on the tobacco, and
next day be cloudy, leaving the tobacco wet and unpleasant to handle.
After cutting, allow the tobacco to wilt long enough to make the leaves
tough, so that they can be handled without tearing. Great care is now
necessary to keep the tobacco from sun-burning; cutting should be
commenced as soon as the dew is off, and all that is cut should be
housed by 11 o'clock, unless it is cloudy; from 11 to 2 o'clock the
direct rays of the sun on the tobacco, after it is cut, will burn the
leaves in 20 minutes; after 2 P. M., as a general thing, there is no
danger of such burning, the sun's rays not striking direct on the
tobacco. Have a waggon at hand, with stiff boards, 12 feet long, laid on
the running gears; as soon as the tobacco is wilted so that it can be
handled without breaking, commence loading on both sides of the waggon
on the front end, lapping the tobacco the same as loading fodder,
keeping the butts out on both sides—build about 2 feet high, and so on
until loaded."
Any one accustomed to the cultivation of the crop, says Bishop, "knows
when it is ripe,—the veins of the leaves are swollen, the leaves begin
to look spotted and feel thick and gummy. The ends of the leaves will
crack on being doubled up. After it is ripe, the sooner it is cut the
better, as it is liable to injury by frost or hail, and will not
increase in weight as fast as the worms eat it, and the leaves get
broken by catching them. The plants will generally ripen from the 1st to
the 15th of September; they should not be cut immediately after a heavy
rain unless in danger of frost, as a portion of the gum washes out, but
should be allowed to stand 2–3 days. The cutting should not begin until
the dew is off; a cloudy day is best, for when the sun shines hot, they
will not have time to wilt sufficiently before they will sunburn, which
may be known by the leaves turning white and looking puckered. Commence
on one side of the piece, laying the plants all one way, in order to
facilitate loading. The plants may, most of them, be broken off easily,
by gently bending them over one way and another. Small plants, which
will not break, may be sawed off with an old saw or cut with a hatchet.
If the sun shines too hot, the plants should be turned over carefully to
prevent burning. After lying an hour or two to wilt sufficiently, so as
not to break by handling, they may be carted to the barn."
In the words of Schneider, "when the plant begins to yellow, it is time
to put it away. It is cut off close to the ground, by turning up the
bottom leaves and striking with a tobacco-knife, formed of an old
scythe—such knives as are often used for cutting corn. Let it lie on the
ground for a short time to wilt, and then carry it to the tobacco-house,
when it may be put away in three different modes, by 'pegging,'
'spearing,' and 'splitting.' Pegging tobacco is the neatest way and
best, yet the slowest. It is done by driving pegs about 6 inches long
and ½ inch or less square into the stalk, about 4 inches from the big
end of the stalk; and these pegs are driven in with a mallet, in a
slanting direction, so as to hook on to the sticks in the house. It is
then put on to a 'horse,' which, by a rope fixed to one corner, is
pulled up in the house and there hung upon the sticks, which are
regulated at proper distances. A 'tobacco-horse' is nothing more than
three small sticks nailed together so as to form a triangle, each side
being 3–4 feet long. Spearing is the plan I pursue; because it is neat
enough and decidedly the quickest plan. A rough block, with a hole
mortised in it, and a little fork a few inches from the hole for the
tobacco-stick to rest upon, one end being in the hole and a spear on the
other end of the stick, is all the apparatus required; the plant is
then, with both hands, run over the spear and thus strung upon the
sticks, which, when full, are taken to the house and hung up at once.
There are 'dart-spears,' like the Indian dart, and 'round spears.'
Either will do. 'Splitting' tobacco is admired by many, who contend that
it cures brighter, quicker, and is less likely to 'house-burn' or injure
from too thick hanging. This mode is pursued easily by simply splitting,
with a knife made for the purpose, the plant from the top to within a
few inches of the bottom, before it is cut down for housing."
Another planter observes that "when a plant begins to ripen, it will
gradually assume a 'piebald' or spotted appearance. As the ripening
advances, the spots will become more distinct and individualized. When
the spots can be distinguished at the distance of 10 steps, and the
leaves of the plant turn down, become stiff to the touch, and their ends
curl under, the plant is ripe, and should be cut. From the moment it has
arrived at maturity, it begins to decay. Remember that all the plants in
your crop are to be hung after they are cut—hung on something, and by
something. Prepare a knife—a butcher-knife answers well—have it
sharp—enter it at the top of the plant, where the top was broken off.
Enter it centrally; press it downwards, dividing the stalk into two
equal portions. Continue it downwards till within 5 inches of the
ground. Withdraw the knife, and cut off the stalk close to the ground.
The plant is now cut. Lay it on the ground with the lower end towards
the sun. The plants should be placed in rows as they are cut, in order
to facilitate the labour of gathering them. There is one caution to be
heeded in cutting tobacco, and that is, do not let it be burnt or
blistered by the heat of the sun. In some varieties of tobacco this will
be effected in one hour; in others, not so soon. But this danger can be
evaded in two ways: first, by cutting late in the evening; second, by
throwing it in the shade, or covering it so as to weaken the power of
the sun. Some varieties of tobacco will wilt (that is, become soft or
limber) in 2 hours; others, in a longer time, according to the degree of
sun-heat."
Bishop tells us that when "the plant begins to yellow or turn spotted,
it is time to put it away. It is cut off close to the ground, turning up
the leaves, and cutting off close to the roots, by a single stroke of a
hatchet, or tobacco-knife, made of an old scythe, such as are used in
cutting up corn. After cutting, let it lie on the ground a short time to
wilt, when it may be handled without danger of tearing the leaves; it is
then to be taken to the house to be 'hung.'"
The condition of the leaf, according to Pursley, may be judged in the
following manner:—"When the tobacco is ripe, it has a yellow faded
colour, and becomes brittle; the surface of the leaf is rough and
ridged. By bending the leaf short between the fingers, it will break
before it will double. The sticks to hang it on should be in readiness.
The best mode of hanging or stringing is with a V-shaped spear, made of
iron or steel. The spear has a socket, large to admit the end of the
stick. The sticks should be sharpened at one end, to fit the socket;
should be 4 feet 6 inches in length, 2 inches wide, and 1 inch thick. A
stick of these dimensions will hold 8 plants. The tobacco should be cut
off just below the bottom leaf, then turn the plant upside down, and let
it remain so till the sun wilts it. When it is wilted it can be handled
without breaking; then it should be taken up and laid in piles of 8
stalks each, placing the butts of the stalks towards the sun, to prevent
it from sun-burning. When it is sun-burnt it turns black, and it cannot
be cured any other colour than black, which ruins its sale. The sticks
should be strewed along, one stick to a pile; place the spear on the end
of the stick, and set the stick upright; then take up the tobacco, one
stalk at a time, and thrust it on the stick, letting the spear pass
through the stalk, about 6 inches from the butt end; then take the spear
off and take up the stick, and shake the tobacco out straight, and set
the stick up with the butts towards the sun."
Some tobacco-growers, remarks Pursley, "prefer splitting the stalk from
the top down to within about 6 inches of the butt, then hang it on the
sticks. But I cannot agree with them, for it is more difficult to
handle, and is apt to slip off the stick, when moving it; besides, the
tobacco cured in this manner is not so heavy as if it was speared. It
dries out quicker by being split, but the substance evaporates instead
of remaining in the leaf. I am not certain that it injures the taste of
the tobacco, but I am certain that split tobacco is lighter than that
which is speared. Some prefer hanging the tobacco on scaffolds in the
field until it is ready to be put in the barn and cured by fire. But it
is the safest to house it as soon as it is strung on the sticks.
Scaffolding is done by placing poles on forks, about 4 feet apart, and
4–5 feet from the ground; then hang the tobacco between the poles,
letting the ends of the sticks rest on the poles. This procedure is
unsafe, for the rain may come and saturate the tobacco and wash off the
gum, thus making it light and chaffy."
The maturity of tobacco is defined by Schneider as when the leaves,
which have hitherto been green, on holding them "against the sun, show
yellowish, reddish, or brownish spots, feel sticky, and when bent break
off short and clean. Before this period sets in, the _drying-house_
should be in good order. This house is built to give room for the free
hanging up of the tobacco, so that it is protected from the sun, wind,
and rain, and is allowed to dry by the free circulation of the air. Any
building, therefore, will answer which has a good roof, boarded sides,
and enough windows and air-holes (which can be closed at will) to keep
up a mild circulation of air inside, and also to keep out strong and too
quick drying winds. If the tobacco is grown on a large scale, the house
should have large doorways to drive a waggon in and out. There must be
sticks all over the house, either cross or lengthwise, and these sticks
must be ready and in their places. Now the work of harvesting the crop
is commenced on a clear or cloudy but not rainy day. The mature plants
(those not ripe are left longer on the field if not too late in the
season) are cut off near the ground, two of them tied together by the
butt-ends and hung up in the field on riders, which rest on two forks
fastened in the ground, and they are left there until evening to wilt;
then they are brought to the drying-house and hung up. The tobacco is
hung up on the upper sticks first, and the work continued downward; care
is taken that the sticks are 6–8 inches apart, also that the plants are
not too near together on the sticks, because the air should have free
passage among the plants, and when they touch or rub against each other,
unsightly spots are produced. The sticks must be pretty wide, so that
the two plants which are tied together, and one of which hangs on each
side, are held well apart. Later, when the tobacco has dried off
somewhat, the sticks and plants may be moved a little nearer to each
other; but the plants on the upper sticks must not touch those on the
lower; they should be so arranged that one lower stick is just in the
middle of the space between two upper ones."
Another method of harvesting is recommended by Schneider for those "who
cultivate tobacco on a small scale, or who have hands and time enough.
As all the leaves on the plant do not ripen at the same time, but the
under leaves are always a little earlier than the upper ones, they may
gather the crop in the leaf, that is, taking only the matured leaves
from the stalk; this must be done daily, and so long as there are leaves
on the stalk. In this way the crop will be harvested slower, and it will
cost more, but the tobacco will be of more even quality and better. The
leaves are strung on strings instead of being hung up on sticks, with
the same care and precautions as recommended for hanging up the whole
plants. After the leaves are off, the stalks must be cut off or pulled
up, for they would still vegetate, and needlessly take away nourishment
from the soil. No more tobacco, leaves or plants should be cut than can
be taken to the drying-house and hung up the same day."
Perry Hull's instructions commence with a caution that the plant should
never be cut while the dew is on the leaves; "but wait until it is off,
say 10 o'clock, and what tobacco is cut from that time until 2 o'clock,
if the day is hot, will need close attention. In short, the whole
operation, from cutting in the field, to the hanging upon the poles in
the barn, needs care, as a little carelessness or inattention will
damage many dollars' worth. No hand should be allowed to handle it, who
is unwilling to use care, and perform every operation just as directed,
or else by breaking of leaves, or sticking fingers through them, &c., he
may do more damage than his wages amount to. The plant to be cut should
be taken by the left hand, not carelessly by the leaves, but carefully
by the stalk, and as carefully leaned over, to give a chance to use the
axe, which should have a handle about one foot long. Cut the plant with
one blow, laying it carefully down, with the top to the sun; if it is
laid otherwise, the leaf will burn before the main stalk of the leaf
will wilt sufficiently to admit of handling. Even in that position, it
may burn unless attended to, but not as soon. After lying until pretty
well wilted, and before burning, turn it over and wilt the other side.
When so wilted that the main stem has lost most of its brittleness, load
as explained above; taking hold of the butt of the stalk, lay them
carefully upon the arm, and again as carefully upon the load. If the day
be very hot, use expedition in getting to the shed, else, if the
distance be great, the load may heat, which will spoil the leaves for
anything but fillers."
When the plants are carried into the shed, "if quite warm, they should
be left only one plant deep upon the floor and scaffolds. If the day be
cool, and they are to be hung up soon, they may lie much thicker. They
should never be hung upon a pole less than 5 inches in width. If sawed
pieces are used, saw them just that; if poles are used, see that they
are about that; for if anything of less width is used, the plants will
hang so close, that the chances of 'pole-burn' are greatly increased.
They are fastened to the pole by a half hitch. (Their position is
represented by Fig. 9 on p. 95.) It requires two hands to hang them, one
to hand them, another to tie them. The poles should be about 18 inches
apart, and the number hung upon a 12-foot pole will depend upon the
size, from 24 to 30, so regulating them, that when thoroughly wilted,
they will scarcely touch each other. If hung thicker than this, a little
unfavourable weather will cause more or less pole-burn, sweat and mould.
After the tobacco is hung, the building should be so thoroughly
ventilated that there will be a circulation of air through every part.
The ventilators should be kept open during all fair weather, until well
cured down. During storms, shut the doors and exclude as much wet as
possible; being cautious to give it a thorough ventilation again, as
soon as the rain ceases. When it is cured enough to be husky in dry
weather, exclude all hard winds, that will crack and damage the leaves.
When the leaves are so much cured, that there is nothing about them
green but the stem, a moderate quantity of wet weather will not injure
it, but rather improve the colour; as the sap of the stalk works through
the stems into the leaves, during moist weather until the stalk has been
well frozen; after this takes place, the tobacco should be picked."
White estimates that in "the course of 2 or 3 weeks after topping, the
plants will begin to ripen, which may be known by the change in colour
of the leaf. It will look spotted with spots of lighter green, a
yellowish green. When fully ripe the leaf may be folded together, and
moderately pressed without breaking or cracking. Now is the time to
begin to harvest it. All this is supposed to take place before there is
any appearance of frost, as a very light frost often does great damage.
All touched by it is ruined, and good for nothing. The crop must be cut
and hung, even if not fully ripe, before any frosts occur. If there are
strong appearances of a frost, you can secure the crop by cutting it
down, and putting it either under your sheds, or by putting it in piles,
not over 1 foot deep, in the field, and covering with straw. It is well
to let it stand, if not fully ripe, as long as it can safely, for the
cool nights have a tendency to thicken up the leaves. The cutting is
best performed with a hay-knife, with a sharp, rounding point, in the
following way: stand at the right-hand side of the plant or row; with
the left hand grasp the stalk down 2 or 3 leaves from the top and lean
it back on the row; now, with the point of your cutter held in the right
hand 2–3 inches from the stalk, close to the root under the bottom leaf,
with a sudden stroke or dab, sever the same from the root; lay it gently
down back in a line with the row. Proceed in like manner to cut what you
can take care of, and not get injured by sunburn. Have two rows of butts
together, lying the same way for after-convenience. This cutting is done
after the dew is off in the morning, or in the afternoon. Let it remain
until the top side is somewhat wilted; then commence to turn it over.
Step between the two rows with the butts lying toward you, and with each
hand take a plant on either side; raise them from the ground, and by
twisting the hands in or out, turn the plants, laying them either to the
right or left, as most convenient, at right angles to their former
position. Go through with the 2 rows, and you have the next 2 with the
butts the other way; take these and lay the tips directly opposite those
first turned, and you have an alley, with the butts of the plants of two
rows on either side, which will be convenient to drive in to load. When
wilted sufficient to be handled without breaking, if in the forenoon,
you can load it from the rows as they lie; if in the afternoon, it is
best to put in hakes, which is done by putting five plants at the
bottom, and on these four, decreasing one on each layer, and terminating
with one on the top; this will protect it from dew and wet. The best
cart for hauling the tobacco is a one-horse waggon, geared long, with
merely a platform resting on the axles. Such a cart can be driven
between the rows and loaded from either side, having the butts of the
plants uniformly one way, and laid crosswise on the platform. Great care
should be used, in all the handling, not to bruise, break, or tear the
leaves. Having cut all, excepting your seed-plants, strip all the leaves
from these, and set a stake to each to tie it up to; let the stake be a
foot taller than the plant; it will answer to keep a piece of old carpet
from breaking down the stalk when you wish to cover it up on cold
nights. Let the seed-plants stand till the pods or bolls are cured to a
brown, and the seed is ripe; then cut off the top of the seed-stalk, and
hang it up in some dry and safe place, where it will be ready to shell
and use the next season; only the ripest and best pods should be used."
Libhart alludes to the existence of several ways of hanging cut tobacco
plants, but specifies the two following as the best and shortest:
"first, splitting and hanging it upon laths or poles and leaving it to
partially cure in the field; secondly, nailing it to rails with
lathing-nails, at once in the shed. The former method, for high northern
latitudes, is by far the best, as it will cure in a much shorter time
(and thus prevent the destruction of the crop by freezing in the shed),
by the drying of the pith of the stalk, which is the main reservoir of
moisture. It is performed as follows:—Have a chisel about 1 foot long
and 3 inches broad, the sharp end not bevelled on one side, but coming
to an edge by a gradual taper on both sides (a common tenon-saw will do
pretty well); place the edge of the chisel in the centre of the stalk
upon the end where it has been topped, and push it down, guiding it in
its course so as not to break or cut off any leaves, to within 3–4
inches of the ground; the stalk may then be cut off with a hatchet, or
with the chisel if it be made pretty strong. The splitting may be done
in the morning when the leaves are too brittle to admit of the stalk
being cut down, and then when the sun has sufficiently wilted the
leaves, the stalk may be cut and left to lie until it will bear handling
without breaking the leaves. The lath being previously prepared, 4 feet
in length and about 1 inch in thickness on one edge, and ½ inch on the
other, and 2 inches broad (or poles cut in the forest will answer pretty
well); then have trestles prepared high enough to allow the stalks to
hang suspended without touching the ground, and set far enough apart in
the field to admit of the lath reaching from one to another; now place
the stalks of tobacco upon the lath (previously laid across the
trestles), by slipping them over and down until they will hang
perpendicular and 6–8 inches apart, so they will merely touch, without
crowding too much. It may be left hanging thus exposed to the weather
until the leaves are so wilted that the stalks hang apart without
touching, and the lower leaves begin to dry, when it is taken off the
trestles, each lath entire, and laid upon a waggon and hauled to the
drying-shed."
Before the tobacco is ready for harvesting, Hudson suggests the
preparation of "a supply of sticks for hanging. Sticks 4 feet long and 1
inch square are most convenient; 12 sticks to every 100 plants will be
sufficient. For sun-curing, there should be a shed built at one or more
convenient points of the patch. This may be done by placing posts in the
ground to support the poles, as represented in Fig. 8. The poles _a_
being for the support of the smaller poles _c_, upon which the
tobacco-sticks are placed, and _b_ for the cover, when necessary that it
should be shedded."
[Illustration: FIG. 8.]
Mitjen's translator gives the following account of the Cuban practice.
"Tobacco should be cut during the wane of the moon; and although most
_vegueros_ say that it is impossible to do this, because the leaves
commence to ripen both during the new and the full moon, and would be
over-ripe before its wane, we can, nevertheless, assert that we know
persons who never cut their tobacco during the first quarter, or when
rain has made it again green. These persons have never experienced any
difficulty; rather, on the contrary, they are those who always obtain
the best prices and the greatest money results. Cutting tobacco during
the first quarter of the moon, or when vegetation is renewed in the
leaf, is one of the principal reasons why the leaf becomes pricked with
holes, and this very frequently even before it is taken from the
plantation to the market. The system generally observed is, in cutting
tobacco, to take off, at once, all those parts of the plants which may
be really or apparently ripe, and to load up the poles indiscriminately,
without any division between the pairs of leaves (_mancuernas_). This
system is highly prejudicial. The leaves of the same plant are not all
of the same quality, neither do they all at the same time acquire the
same degree of ripeness. Those of the crown, or the pairs at the top of
the plant, immediately next the flower or seed, receive the sun direct
on their upper surface, and are the first to ripen, whereas the lower
ones, being shaded by the upper ones, remain still in an unripe state;
moreover, the lower leaves at the foot of the plant, and even those of
the fifth or fourth pairs (_mancuernas_), compared with those of the
first, second, and third pairs, are inferior in quality, and,
comparatively speaking, may be termed leaves without substance. The
contact of these leaves with the upper ones frequently occasions putrid
fermentation on the poles (_cujes_) and in the packs (this is vulgarly
called _sahorno_), especially if there is much dampness in the
atmosphere. When this misfortune happens in a tobacco curing-house all
the weak leaves will be lost, and the strong ones will be so injured
that the best quality of _capa_ would turn to _tripa_, and that of bad
consistency.
"The cause of this destruction, from which the _veguero_ suffers more or
less in the best of crops, may be easily explained. The curing of
tobacco is nothing more than a series of fermentations. It ferments on
the poles (_cujes_), ferments in the heaps (_pilon_), and ferments in
the bales. All these fermentations are requisite for obtaining a good
colour and smell, but it is better that each quality or consistency of
tobacco should ferment apart. Tobacco of good strong quality, which is
that produced by the upper leaves, naturally suffers a much stronger
fermentation than the weak ones, because the former contain a larger
proportion of juice; as the lower leaves have less substance, the
fermentation is naturally weaker and lasts less time; but if the leaves
are put in contact with those of a stronger quality, the fermentation
would be kept up by the latter, and it would indispensably result that
the weak ones would rot, and their contact be injurious to the stronger
ones. But by separating, in the field, the leaves of different
consistencies which each tobacco stalk produces, this evil is avoided,
and the dry rot is rendered impossible, unless no care whatsoever is
given in the curing-house. Therefore, the mode of reaping should be
reformed. It is best to cut the tobacco when it is thoroughly ripe, and
in the wane of the moon, making this operation in three sections or
cuts, each of which should always be placed on separate poles, in
separate rooms, heaps, and carefully picked.
"The first cut should consist only of the pair of crown leaves, and for
the poles which they are hung on, a special corner in the curing-house
should be set apart. After the first cutting, and 3 or 4 days of sun,
the second and third pairs of leaves will be ripe, and may be cut at one
and the same time, care being taken to place them on separate poles and
rooms; and, lastly, 3 or 4 days after the second cutting, the remainder
of the leaves may be gathered, but the last leaf near the ground should
not be taken, as it has no consistency, and therefore no value as
tobacco, and only serves to increase the work and give discredit to the
class of tobacco.
"Tobacco should be cut during the hottest part of the day; each pair of
leaves should be placed on the ground face downwards, so that the sun
may strike on the under part of the leaf, and in this state it should be
allowed to remain a sufficient length of time to wither, after which the
pairs of leaves (_mancuernas_) should be picked up one by one, placed
evenly on the arm, with the upper side of the leaf inwards, and each
armful should be carried to and placed on the poles (_cujes_), which
should be prepared beforehand near the spot where the tobacco is being
cut. Two forked sticks should be placed strongly in the ground, and on
these the pole should rest. After the tobacco leaves have been placed
carefully on these poles and been allowed to wither, they should be
carried to the curing-house before the sun has time to dry them. This
operation must be performed by two labourers, who can carry each time
two poles, placing the end of each on either shoulder, so that, in
walking, the leaves on one pole may not cut against those on the other.
These poles of leaves, when brought to the curing-house, should be fixed
or hung by the points on the lowest stages, but so high that the points
of the leaves do not touch the ground, and sufficiently apart one from
the other that the leaves may not touch, because, being brought in from
the field warmed by the sun, it is not judicious to allow them to touch.
When the sun is not sufficiently strong to wither the cut leaves,
reaping should not be continued. The tobacco should be so arranged on
the poles that the pieces of stalk should gently touch one with the
other, but without crowding." However, if the weather should be damp,
and the leaves large, space should be left between the pairs.
_Drying._—The drying-shed is prepared beforehand to receive the tobacco.
When cultivating tobacco on a small scale, any shed will do, provided
that it contains a sufficient number of doors and windows to admit of
regulating the circulation of air. A roof made of straw seems to answer
very well. The shed should be high enough to admit of hanging 3 rows of
tobacco in it, one above the other. The bottom tier for the first row
should be about 3–5 feet from the ground, according to the size of the
plants, which should not touch the ground; the second tier should be 3–5
feet higher than the first; the third, 3–5 feet higher than the second;
the whole being 10–17 feet high from the bottom of the shed to the
highest tier. The tiers must be so arranged that the tobacco when hung
on the upper tier should not touch that of the lower one, and that the
rods on which the tobacco has been hung in the field fit exactly. The
windows must face each other, and be placed between the tiers, so that
the bottom part of the window is on the same level as the tier. When
cultivating on a large scale, the same arrangements are made, but the
building is higher, and is provided with a cellar, in which to place the
tobacco for the purpose of stripping, &c.
The drying-shed being ready, the plants immediately on arrival at the
shed are transferred from the conveyance, on the rods, to the lowest
tier. No rule can be given as to the distance the rods should be placed
from each other, as it varies according to the species of the plant, the
degree of ripeness, and especially the state of the weather. The purpose
of hanging the plant here on the lower tier is to cause the leaves to
dry gradually, and assume a good yellow colour, and to create a slight
fermentation in them, while allowing such a circulation of air between
the plants as will facilitate the gradual escape of the moisture from
them, and prevent the injurious development of ammonia and other
combinations that give rise to bad flavour in the tobacco. How to attain
this, exercises the judgment of the cultivator, who, by frequent
examination of the plants, and by careful observation of the changes
going on in the leaves, will soon find out the right way.
The rods should be placed closer together—(_a_) when the plants are much
wilted on reaching the shed; (_b_) when the air is very dry, and the
temperature is high; (_c_) when the leaves of the plant are very thin
and contain little water. Plants which have the leaves closely arranged
on the stems must be hung farther apart. When the air is very dry, and
there is a strong breeze, the windows must be closed. If this is not
sufficient, water may be poured on some heaps of sand, to create a moist
atmosphere in the shed. When the stems of the plant are very thick, and
consequently contain much sap, it is beneficial to open the windows,
especially at morning and evening, for some hours, that the wind may
pass over the butt-ends. As the windows are situated above the lowest
tier, the leaves will not be much affected by it.
The leaves must be examined carefully every day; one plant may progress
very well, whereas another close by may decompose too rapidly, and
another too slowly. Although no change of weather occur, it may yet be
necessary to alter the position of the rods, in order that each plant
and leaf may receive air in such a degree as is most conducive to its
proper decomposition. Any change in the weather necessitates different
arrangements. The plant should remain on the lower tier until the leaves
have turned yellow, which will take place within 6–10 days, according to
circumstances; after this, they are hung on the upper tiers. There they
should be more apart, each plant hanging free. When on the upper tiers,
the tobacco may be said to be in the free-hang; and when on the lowest
tier, in the close-hang. The object in hanging the plants more apart on
the upper tier is to dry them more rapidly there, and for this purpose,
the shutters may be opened, unless there be a strong dry wind. The
light-yellow colour of the leaves should change into a dark
yellow-golden or light-brown colour. After hanging on the upper tier for
about a week, the veins of the leaves will be nearly dry, leaving only
the midribs pliant. The drying of the leaf and the changing of its
colour proceed gradually, commencing from the margin and proceeding to
the midrib. At this time, the plants are hung closer together, the
evaporation from the leaves being little, and the space and sticks being
required. The plants hanging on two or three sticks may be hung on one
stick. All the windows may be kept open from this time; the tobacco may
also be brought into an open shed, or even hung outside exposed to the
sun. In about a week more, the midribs will be entirely dried up, and
the tobacco will be fit for stripping. In some climates, it may be
necessary to facilitate the drying by the aid of artificial heat. For
this purpose, heated air should be conducted into the drying-shed,
without the fire, or the products of combustion, being admitted.
Pursley warns tobacco growers that the plant should not be exposed to
the weather after it is cut, but should "be immediately conveyed to the
barn and hung up. As soon as it gets about half yellowed, a slow fire
should be started under it; if made too hot at first, the tobacco will
turn black. About the second day the ends of the leaves will begin to
curl up; then the fire should be gradually increased, till it heats the
tobacco blood warm; it should be kept up so till the leaf is thoroughly
cured. If this rule be strictly adhered to, the tobacco will be cured
bright. The brighter it is cured the better it sells.
"Our barns are generally built of logs, some have frames. The barn
should be made tight up to the tobacco, which should hang about 8 feet
from the ground; above this leave cracks or air-holes, sufficient for
free ventilation. A barn to hold 2½ acres of tobacco, which is as much
as one man can attend to, should be 24 feet square. It should have 5
tiers of poles, the lowest about 6 feet from the ground; these should
extend across the barn, and be fastened at each end into the walls. The
poles should be 4 feet apart, and the tiers directly one above another.
The sticks which contain the tobacco should be placed within 8 inches of
each other, on all the poles except the bottom ones, which should be
left vacant directly over the fire. When tobacco is nearly cured, it
very readily catches fire. If there be a wet spell of weather before the
stalks are thoroughly dry, build a fire under the tobacco sufficiently
hot to keep it dry. It should not get damp and pliant until the stalks
are dry, then it may be allowed to get damp."
Libhart recommends that the shed "be constructed of timbers strong
enough to resist storms, and boarded 'up and down.' About every 3 feet
one board should be hinged, to readily open and shut. If it is intended
to split and lath the tobacco, the inside of the shed must be divided by
rails into widths to accommodate the lath, and likewise into tiers, one
above the other, far enough apart to allow the stalks to hang from, well
separate. The frame of rails and timbers inside the shed destined to
sustain the weight of the tiers of tobacco (which, when green, is
exceedingly heavy) should be strongly constructed, so as to preclude the
possibility of breaking down, for if this should happen to the upper
tier, in all probability the whole would be tumbled to the ground."
The housing of the crop proceeds, says Dennis, "as fast as it is cured
up on the scaffold, or as the indications of rain make it necessary,
care being taken not to bruise or tear it in hauling. The sticks of
tobacco may be piled upon the waggon or cart, and hauled to the barn and
hung up, commencing in the highest part of the building, and filling up
as you go downwards. If the leaves are pretty well cured, you may hang
it so as to touch, without crowding it; if not, there should be a little
space between. If a cold, rainy spell comes on, you will need to
introduce some means of artificial drying. A trench is sometimes dug,
and a log or two of wood placed in it, and a fire made, taking care to
remove the tobacco immediately over the fire, and avoiding much blaze.
This is dangerous, and a better plan is to make a trench across the
floor of the barn, of mason-work, covered with sheet-iron, and leading
from a furnace outside the house on one side, to a chimney at a safe
distance on the other. The colour and quality of tobacco may be improved
by hanging it closely and curing by artificial heat, watching that it
does not become 'funked,' or moulded, while curing; but the best plan
for a beginner is to dry it safely, and make a sure crop, experimenting
as he goes along, in order to improve the quality, as he may safely do
so. When the stalk becomes dry and entirely cured, which will not
usually be for some weeks, the crop is ready to 'strip.' The hanging
tobacco yields to the influence of a rainy day or a foggy morning, and
'comes in case,' or softens, so it will not crumble. It must never be
handled when dry. When it is just soft, not damp, or when it is barely
so soft that it can be handled (if it is approaching that softened
state), it may be taken down and taken off the sticks, and 'bulked,' by
piling it alongside a partition, or by itself, with the butts of the
stalks outward in every direction, and the tops or leaves in the centre.
Several hundred pounds may be thus bulked down, and can be worked up
while the hanging tobacco has gone out of case, and cannot be touched."
According to Bishop, it usually requires about 12 weeks to cure the
plants thoroughly, that is, so that there is no more juice in the leaves
or leaf-stems; it matters not if the main stalk is not dry, you need not
expect it, and there will be green leaves that will not cure but freeze
while green and are worthless. He calculates that to "hang an acre of
good tobacco requires a building about 30 by 24 feet with 15-feet posts.
Two girths should be framed into the posts on all sides of the building;
one 5 feet above the sill, and the other 10 feet above, to rest the
poles on, also to nail the covering boards to. This gives a space of 5
feet for each tier of plants. Have a beam run across the centre of the
building, with a post in the middle with girths to correspond with those
on the side, extending lengthwise through the middle of the building for
the poles or rails, each 12 feet in length, to be laid upon; or if
sticks are to be used (as hereafter described) lay rails or poles once
in 4 feet for the sticks to rest upon. Place a ventilator upon the
centre of the roof, and have one board in every 4 feet hung on hinges,
to be opened or closed at pleasure. If made with a floor and a cellar
underneath, to let down the tobacco into when ready stripped, it is all
the better. We will now return to the crop, and commence hanging it. A
common way of doing it is by tying with common twine. Tie the end of the
string tightly around the butt of one plant, and by placing it against
the side of the pole nearest you, put another plant on the opposite side
and carry the string over and around it, placing the plants alternately
on each side of the pole until filled, then fasten the string, place the
pole in the right place (it should be nearly right before it is filled),
and commence on the next one in like manner, having some one to hand the
plants as wanted. As to how thick to hang it depends upon the size of
the plants, but in good-sized tobacco about 9 inches on each side is
close enough, that will be from 30–32 on each pole of 12 feet; place the
poles 15–18 inches apart. Another method of hanging, much practised and
approved by many, is to hang on slats or sticks sawed out 4 feet long,
1¼ inches wide, and ⅝ inch thick. Chestnut timber is generally used
here. The common lath answers very well for this purpose. An iron made
something like a chisel is used to slip on to one end of the sticks,
which are sharpened a little at one end to receive it. It is made about
8 inches long, wedge-shaped at the small end, and a socket ½ inch by 1
inch to slip on to the sticks. When ready for use have a place fixed
near where you unload, to hold one of these sticks out at right angles
from a post and about 4 feet from the ground. Let the plants be handed
you from the load and slip them on the stick, piercing the stalk about 6
inches from the butt; put 6 or 7 plants of medium size on each stick,
thicker if smaller; when hung it will appear as in Fig. 9. As each stick
is filled, it may be carried to its place in the barn. In getting them
to the top of the barn, they may be handed up with a pitchfork, lifting
them by the middle of the sticks. These sticks should be about 8 inches
apart. I think a greater amount can be put into a given space by this
method without danger of sweating, as it is more evenly distributed. The
loose leaves that have been broken off while handling, may be cured by
placing 4 or 5 together and securing to a small pole, in the same way as
plants are hung with twine."
Hanging is done in the following manner:—"The 'hanger' stands in an
erect position, having for a foothold the poles on the tier below the
one which he is hanging; he has a ball of tobacco-twine (a twine made of
flax, procurable at any seed-store) which for convenience is carried in
the bosom of the loose blouse generally worn; he stands with the left
side to the pole on which the tobacco is to be hung, left arm over it;
the stalk of tobacco is handed to him by a boy whose duty it is to pass
it to him; the stalk is then taken in the left hand and placed against
the side of the pole, the butt projecting an inch or two, around which
projection the twine is wound from left to right (the twine having
previously been fastened to the pole); the next stalk is placed on the
other side of the pole, just far enough along so that the leaves of the
two stalks will not touch and 'pole-burn,' and so continue, the stalks
being hung alternately on the sides of the pole, as seen in Fig. 9.
After the house is filled, some put fires under the crop to hasten its
drying; but it is found by experience that the practice is not a good
one."
[Illustration: FIG. 9.]
[Illustration: FIG. 10.
TOBACCO-HOUSE.]
Bishop describes the common size of tobacco-house as about 100 feet long
by 24 feet wide, posts 17 feet long, and built upon a wall 18 inches
high; the buildings are framed with girths from bent to bent, for
boarding up and down, the bents being 12 feet apart. The external
appearance is illustrated in Fig. 10. "The boards for closing up the
building should be 1 foot wide, and at intervals of about 5 feet a board
should be hung with light strap hinges, to serve as a ventilator to
admit light and dry air, and to exclude damp. These ventilators or doors
must be closed on frosty nights, but in fair dry weather should remain
open. The tobacco poles, the ends of which rest upon the bents, should
be about 13 feet long, 2 inches thick by 6 inches wide, of some light
timber, such as elm or basswood, and when hung with tobacco should be
8–10 inches apart. A large door should be placed at either end for
ingress and egress. The poles, of which there should be 4 tiers, are
laid from bent to bent, resting the ends of the cross beams in the bent,
tiers 4 feet 4 inches apart." A sectional view of the barn is shown in
Fig. 11.
[Illustration: FIG. 11.]
White suggests that stables, sheds, and barn floors can be arranged "so
as to hang up an acre or two by setting stanchions with holes mortised
in them to hold rests for your poles about 4½ feet apart. Set such ones
on either side with a very stout rail, one end in either post. Set these
as often as you may need them, depending on the length of your poles. No
poles should be so long as to sag very much when filled with plants. But
for another reason I would build a house expressly for hanging and
storing tobacco. Make it of good, liberal dimensions, 30 feet wide, by
40 or more in length; posts, 14 feet, with two tiers of girths for poles
to rest on; one tier can hang on the beams, and another above on the
purlin plates, thus hanging 4 tiers under the same roof. Ventilate by a
ventilator in the roof, also by hanging every other board of the siding
on hinges. For such a building, I would have a tight floor to the whole,
and underneath a good walled cellar lighted with suitable windows, and
chimney in one corner, with a stove, to keep fire in in very cold
weather, to work by when stripping the tobacco. For poles to hang on, I
would get, if possible, straight, slim, white pine staddles about 4–5
inches in diameter; shave the bark off smooth, and we have poles that
will last and remain straight a lifetime, if kept housed.
"Having provided all required, even to the strong cotton or hemp twine
for tying up the tobacco, have a good man to hand it to you. Commence by
tying the end of your twine around the butt of a plant, about 2 inches
from the end, in a slip or loose knot; place this plant at one side of
the pole near the end, your hand carrying the twine over the pole; on
the opposite side of the pole, about 6 inches along, place another
plant, and with a single turn of the twine around it from before, round
back, and by drawing it close, the plant is secure. Proceed thus till
you have filled your pole; then with a knife, cut a notch in the pole
and draw your twine through, and it is fast. You can now cut it off and
commence another pole. Place the poles far enough apart to prevent the
tobacco crowding; about 1 foot will do. In this manner you will have a
row of plants hanging on each side of the pole about 1 foot apart. The
man, in handing up, should take the plant by the butt, carefully from
the pile or load, raise it up and gently shake it sideways, to shake off
dirt and loosen the leaves when stuck together, and also adhering to the
stalk; with the other hand, take hold about midways of the stalk and
pass to the one tying up, enabling him to receive the plant in such a
way as to not need to shift it in his hand, but to place it immediately
into its position beside the pole. All leaves which are accidentally or
otherwise broken from the plants, should be gathered up each day, and
hung three or four in a bunch, the same way as the plants, or string
them on a string; the latter is the best way—with a large needle-thread,
a suitable cord, and on to this string the leaves one at a time, by
running the needle through near the end of the stem. These can be hung
by attaching the two ends to some suitable nail, and having it remain
stretched. In this way they will cure very well.
"Having housed the whole of your crop, give it all the air you can, by
opening doors, shutters, &c. Let them remain open during pleasant
weather, remembering to close them in wet, damp weather, as well as
nights; and also shading the crop so far as may be from the direct rays
of the sun, to prevent blanching. When it has nearly cured, shut it up
and let it remain till perfectly cured. This may be known by the stem of
the leaves being dried up, so that no green sap will show itself. If you
have hung in your stables and other places that you wish to use, it will
be necessary to take it down and strip it at the first favourable
opportunity, which is described farther along. The separate building
elsewhere described is to be preferred, as it does not necessitate any
immediate hurry in getting it down. In such it can be allowed to hang
and freeze and thaw two or three times, which improves the colour and
weight, and will give more leisure in stripping, &c. Watch a favourable
time, when it rains and is damp, to open your buildings, and let in the
damp air till the tobacco is damped, so that it can be handled without
any danger of breaking the leaves. It need not get too damp, as in that
case it is liable to injure in the pile before you can get it stripped.
It will gain dampness from the stalk."
The Cuban tobacco planter, according to Davis, "would force the drying
in wet weather and <DW44> it in dry weather, as either extreme is
injurious; the wet is injurious, as the leaves, when they change from
the natural colour to a pale yellow and light brown, easily mildew; when
dry, as before-named, it is taken down. Damp weather is best, so as not
to break the leaves, which are immediately stripped from the stalks and
sorted into as many grades as the market may require, from one to four
and even more grades, as 'bright yellow, dull, seconds, and
ground-leaves.' But I see no necessity for but three grades, as the
over-ripe, the unripe, and the just ripe at cutting, and when properly
dried they show their grade plain enough to sort. After being stripped
and sorted, they are to be separately piled ('bulked' some say) in
courses of leaves—2, 4, or 6 tiers of leaves, stems end out, and 3–4
feet high. The leaves should be kept straight in all these handlings.
The heap should be made up each day separate, as it begins to make
tobacco in 12 hours or so, by fermenting, which is variously called
'curing,' 'sweating,' 'conditioning,' &c. Soon as the heap begins to get
warm it should be re-piled, putting the inner tier out so as to equalize
the fermentation; some re-pile several times and some none; but the
fermentation should be kept equal, and if covered with old sail-cloth it
can be regulated. This fermenting is allowed to proceed for 4–6 weeks by
careful manufacturers; as it is the process that makes the tobacco to
suit the taste of tobacco-epicures it should be carefully done, yet many
do it in a careless manner, and thus have an article so poor as to not
find many lovers. At the end of the 4–6 weeks the Cuba grower would have
one side of each leaf slightly moistened with the decoction of tobacco,
which is made by letting some leaves rot in clean water, and then he
would tie it up in hanks of 25 or 30 leaves, and hang one day for
drying, then take it down and pack it in tight casks as being best. From
these leaves he would make the best Cuba cigars. The Virginian grower
would not wet his tobacco after it had fermented, but simply tie it in
hanks so that 5 or 6 would weigh a pound, and then pack it in his
hogsheads for market; and this, after it had lain from one to six months
in the 'conditioning bulks.'"
Burton, translating from Mitjen, goes more fully into the Cuban
practice. He advises firstly that the "shoots and the sprouts should be
put apart from the principal tobacco, with which it should never be
mixed, neither in the heaps nor in the packages. The day after the
tobacco has been cut and placed in the curing-houses, the poles should
be pushed together, making thus a compact mass, with the object, that by
means of the warmth, which this contact produces, the fermentation
should commence, called _maduradero_. In this state it should remain 2
or 3 days, according to the consistency of the tobacco and the state of
the atmosphere. By means of this first fermentation it acquires an equal
and a yellowish colour: by the second or third day, at the latest, this
colour should be uniform, and then without loss of time the poles should
be spread apart, and given all the ventilation possible, so that
fermentation may not continue, and the drying of the leaves may be
facilitated—care being taken that they are not exposed to the dew, the
sun, nor to sprinkling of water, should it rain. As the tobacco dries,
the poles should be hung on higher pegs, so as to leave the lower ones
unoccupied for the fresh leaves brought from the fields. This operation
should be performed early in the morning whilst the leaves are flexible
and soft; because later in the day they become crisper, and are more apt
to tear.
"It is not judicious to allow the tobacco to dry too precipitately, by
exposing it to a very strong current of air, because strong wind greatly
injures its quality; many leaves break, and that silkiness of appearance
is destroyed which good leaves should have, and which it is desirable to
preserve. During heavy winds the doors of the drying-house should be
kept closed; they should also be kept closed if there is much dampness
in the atmosphere occasioned by heavy and continuous rain. Dampness
causes mildew, which shows itself first in the points of the leaves, and
is the commencement of the rot. Under these circumstances, and to check
this evil, it is convenient to spread, or part the poles a little; and
if the rains, or the excess of humidity continue, fires should be
kindled and smoke made in the curing-houses, opening at the same time
the doors and the windows, so as to facilitate the circulation of air
whilst the smoking is going on.
"After the tobacco is thoroughly dry, it should be placed on the highest
beams, or pegs, of the framework which support the poles, squeezing them
compactly together. This must be done in the morning whilst the leaves
are soft, and all this should be done with a view of protecting it from
the effects of change in the atmosphere. The house should, after this,
be kept closed, until it is time to make the heaps.
"The object of heaping up the tobacco is to produce a second
fermentation, so as to equalize the colour of the leaf and wear out of
it that excess of gluten or resinous matter which is natural to the
plant; this fermentation makes the leaves more silky and ductile, and
gives them a more agreeable flavour. The place for making the heaps
should be prepared beforehand, in one or more of the rooms of the
tobacco-house, by making a kind of box lined with _yaguas_ (sheets of
palm-tree bark) at the bottom and the sides, the base is a boarding on
which should be placed a sufficient quantity of dry plantain leaves,
which serve as a bed for the heaps.
"In the months of April or May, when the rainy season commences, the
poles which are on the highest pegs of the scaffolding should be taken
down and placed somewhat apart, one from the other, on the lower pegs.
The doors of the house should be left open at night, so that the
humidity from the atmosphere may enter, and when, in the morning, the
tobacco is found to be soft and silky, it is fit to be placed in heaps.
The pairs of leaves should then be collected in armfuls, with all the
bits of stalks placed in one direction; the leaves that may be found
doubled or crooked should be smoothed out, and each armful should be
placed in layers in the heaps, placing the first layer at the bottom
with all the woody pieces of the stalk touching the _yagua_ which forms
the sides of the case; other layers should be placed with the stalk
reversed, and in this manner, crossing the leaves, the pile should be
raised up level. When a pile has a sufficient height, another, and
another, is made until the tobacco is finished or the case is full, so
that each heap may form a compact mass of leaves protected by the pieces
of stalk all round, which should never touch the leaves, but only touch
each other. When the heaps have been thus made, they should be covered
with dry plantain leaves, or palm skins, and, in front, by palm leaves.
"Tobacco should not be packed thus when it is too damp, because a very
strong fermentation would ensue, which, if kept up longer than
necessary, would pass to putrefaction. The tobacco only requires to be
soft, or flexible, before packing, so as to produce a certain degree of
heat, neither is it convenient to pack tobacco when too dry, for then it
would not ferment at all, nor would any beneficial results be produced.
When it has been packed sufficiently soft, it undergoes after the second
or third day a degree of heat of 110° to 120° F. in the centre of the
heap, and if it does not acquire this degree of heat it is because it
has been packed too dry.
"We have already said that reaping or cutting tobacco should be
performed in three distinct sections, preserving always a distinction,
consequently the crown leaves should form one heap, or one set of heaps;
the second and third pairs another, or others; the fourths and the
fifths others; and lastly, the _capaduras_ (second shoots from the same
plants) others. This system, besides having the advantages which we have
in another place described, greatly facilitates the sorting of the
leaves, as the different qualities are from the first kept apart, and
scarcely any other work remains to be done than that of taking out the
broken leaves. Tobacco should be kept for at least 30 days in heaps,
after which, sorting and choosing the leaves may commence, beginning
first with the heaps of the inferior qualities."
_Stripping._—Stripping may be performed at any time, provided the
leaves, after being once properly dried, have again become pliable. For
stripping, such a number of plants as will furnish work for several days
are taken down on a morning, when the plants have absorbed some
moisture, and have become elastic; they are put in a heap, and properly
covered, to check evaporation. If, however, the night air should be so
very dry that the leaves cannot absorb sufficient moisture to become
pliable, a moist atmosphere can be created either by steam, or by
pouring water on the floor, or by keeping vessels with water in the
shed. If this cannot be done, the tobacco must remain hanging until
there is damp weather. Under no condition should the tobacco be stripped
when not pliant, that is if the leaves are so brittle that they would
break when bent or rolled. The best arrangement is to keep the
drying-shed and stripping-room separate, since the latter requires to be
more moist than the former. A cellar under the drying-shed is best
suited for stripping. It should be large enough to admit of the erection
of a scaffold to receive the tobacco.
Pursley looks upon stripping as being labour suited to damp weather. He
says, "the lugs, shipping, and manufacturing, which are worst, medium,
and best qualities, should be separated at stripping. The 'lugs,' or
worst quality, are found at the bottom of the plant; they are chaffy and
light leaves, and should be stripped from the stalk and tied in bundles
by themselves with all of the ragged, black, and injured leaves. The
second quality, or 'shipping tobacco,' is a grade above the lugs; it is
the red or brown tobacco; this should also be tied in separate bundles.
The best, or 'manufacturing,' is the finest and brightest leaves, and
should be put in bundles by itself. In stripping, the stems of the
leaves should be broken off as close as possible to the stalk; this adds
to the weight of the tobacco. In forming a bundle, the butts of the
leaves should be placed evenly, and closely together, and pressed
tightly in the hand; then a leaf should be folded to form a wrapper 2
inches in width; then wrap it tightly and smoothly around the butts of
the leaves, winding it from the end down, about 2½ inches, then open the
bundle in the middle, and tuck the wrapper-leaf through the opening, and
draw it snug, so that when the opening is closed the wrapper-leaf will
remain; this forms a bundle which we call a 'hand of tobacco.' The hands
should be strung on sticks, and hoisted up in the barn on the
tier-poles; 18–20 hands may be put on each stick, at equal distances
apart."
Libhart expresses his opinions on stripping in the following words. "At
the setting in of a warm, drizzling, wet, foggy spell of weather, the
shed must be opened on all sides to allow the damp atmosphere to pervade
the whole interior; after the dry leaves have become damp enough to
allow handling in any degree without breaking, the stalks must be taken
off the lath or pulled down and laid in heaps about 18 inches or 2 feet
high, and any desired length; if it is not intended to strip it
immediately, it should be conveyed to a cellar or other apartment, where
it will remain damp; it should not, however, be suffered to remain
longer than 2 or 3 days in heaps, without examination, as there is
sometimes sufficient moisture remaining in the stalks or frozen leaves
to create heat and rot the good tobacco. If found to be heating, it
should be changed about and aired and be stripped immediately. If found
to be drying out, further evaporation may be checked by covering the
heaps with damp straw or corn-fodder. Tobacco is usually stripped into
two qualities, 'ground-leaf,' or 'fillers,' and 'wrappers'; the leaves
that lie next the ground, generally from 2 to 4, are always more or less
damaged by sand beaten on by the rain and other causes, hence they only
command about half the price of the good tobacco or 'wrappers.' The
ground-leaves are taken off first and tied up separately in bunches.
With a bunch clasped in one hand, take a leaf and wrap it around
(beginning at the end of the bunch), confining the end under the first
turn, continue to wrap smoothly and neatly until about 3 inches of the
leaf remains, then open the bunch in the middle and draw the remaining
part of the leaf through. This forms a neat and compact 'hand,' that
will bear a great deal of handling without coming open. After the
ground-leaves have been removed, the good leaves are stripped off and
tied up the same as the ground-leaves, with this exception: the leaves
of each stalk should be tied in a bunch by themselves, to preserve a
uniformity in colour and size, as tobacco is sold in the market
according to colour and size, therefore if the leaves of a large and a
small plant, or of a dark- and a light one, be tied up together,
it at once diminishes the appearance and value of the crop."
[Illustration: FIG. 12.
HAND OF TOBACCO.]
Dennis describes stripping as being "performed by holding the plant, top
down, with the left hand, while with the right hand the leaves are
pulled off, taking care to have the stems all even in the hand, so that
the ends are together. When 10–15 leaves have thus been grasped by the
right hand, change the handful to the left hand, and with the right,
select a leaf and wrap it around the stems at the end, so as to bind
them altogether and cover up the ends, then split the other leaves apart
with the finger, and pull the end of your wrapping-leaf through, and you
have a 'hand' of tobacco. A small 'hand' of leaves, uniform in size and
colour, will be found the most desirable shape to tie it in, resembling
Fig. 12. The bottom leaves of the plant, and all torn and defective
leaves, should be tied up by themselves, and are known as 'lugs.' These
'hands' should be 'bulked' again, with the wrapped end out, and covered
with straw, or anything that will retain the 'case,' and if subject to
immediate sale, may be boxed up or hauled to market. If boxed, it should
be put in tight boxes—if hauled, it should be kept covered until
unloaded. Care must be taken to avoid 'high case'—extreme dampness or
softness in bulking tobacco after it is stripped—as it may be 'funked'
in bulk, and ruined; and it should not be packed in that condition when
it is liable to remain long. It is a crop that is never off of hands."
According to Perry Hull, stripping, or, as he terms it, "picking,"
should not take place till about December; "at least not until the _fat
stems_ (main stems of the leaves, which are not thoroughly cured at the
butt-end) have mostly or all disappeared, which they will have done by
that time, if the crop reached maturity before harvesting. The
operations of picking and assorting are by many, who make only two
classes or qualities of the tobacco, carried on at the same time. By far
the preferable way is, especially if there is a very large crop to pick,
to take off the leaves during damp or wet weather, tie them into bundles
of 15–20 lb., with twine, and pack it away into cellars, or wherever it
can be kept without drying up. It can then be assorted in any kind of
weather, thus gaining considerable time, as two will pick and tie up in
this way as much during one wet spell as 6 hands would, assorting and
hanking up, at the same time. Another reason why the last practice is
preferable is, that, by the former, the assorting can be but
indifferently done; whereas, by the last, it can be done as carefully as
desired. Tobacco should not be allowed to get too wet before picking; in
fact, should not be allowed to get wet at all, so as to feel wet, only
just damp enough to make the leaves pliable, so as to handle and pack
without breaking or feeling husky. If allowed to get wet, before
picking, it is next to impossible to get it dried to the proper state
again so uniformly but that some of the leaves will still be too wet,
while others will be dry enough to crack and break. So if the rains are
long enough to get it too wet, which they often are, by all means let it
remain upon the poles until the next wet spell."
_Sorting._—Tobacco intended for smoking should be carefully sorted when
stripped. There should be four sorts: 1st, large, equally good ,
untorn leaves; 2nd, leaves of good size and colour, but torn; 3rd,
leaves of inferior colour, and bottom leaves; 4th, refuse, shrivelled-up
leaves, &c., to which may be added the suckers No. 1 leaves, when thin,
elastic, and of good sorts, are mostly valued as wrappers (outside
covers) for cigars, No. 2 may also be used as wrappers, but are less
valued than No. 1; they are adapted for fillers and cut tobacco. The
different sorts are kept separate. The best plan is to let the most
intelligent man strip the leaves from the stem, and at once separate
them according to quality. The leaves should then be made into hands,
i.e. 10–20 leaves should be tied together by twisting a leaf round the
end of the stalks, each sort being attended by a special man, to avoid
mixing. The leaves of the first sort being large, 10–15 will be
sufficient for a hand; more are required of the other sorts. When making
the hands of the two first sorts, each leaf is taken separately,
smoothened on a flat board, and left there while another is treated in
the same way, continuing thus until a sufficient number is ready to make
a hand. When the hand is ready, it is laid aside, and a weight is placed
upon it to keep the leaves smooth.
To sell well, according to Perry Hull, tobacco "should be assorted into
three classes or grades, Wrappers, Seconds, and Fillers. The wrappers
will include the soundest, best- leaves, the colour (a dark
cinnamon) should be as uniform as possible; this quality should include
nothing but what is fit for wrappers. The Seconds, which are used as
binders for cigars, &c., will include the small top leaves, of which, if
the tobacco was topped too high, there will be one or two to each
plant—the bad colours, and those leaves somewhat damaged by worms and
bad handling, but not so much so as to be ragged. The third class, or
Fillers, will include the balance of the crop, bottom leaves, ragged
leaves, &c. The tobacco should be done up into hanks of about ⅓ lb.
each, or about what can be encompassed by the thumb and fingers, winding
at the butt with a pliable leaf, drawing the end through the hank to
secure it."
The Cuban system of sorting is described at considerable length by
Mitjen, whose remarks are interpreted by Burton as follows. The
operation consists in "separating one from the other the different
leaves, according to their strength and quality, and dividing the
produce of the crop into various classes. These are, in practice, styled
_Libra_, 1st quality; _Quebrado_, 2nd quality, broken; _Injuriado de
primera_; _Injuriado de segunda_, _de tercera_, _de cuarta_, _de
quinta_, _de sexta_, _de setima_; _Libra de pie_, and _capadura_.
"Under this classification it is presumed that attention has been
bestowed, not only to the special quality of the leaf, but also to its
size, and its state, whether whole or broken; but it is very seldom that
exactness is found in this classification, because but very few persons
possess the requisite skill which such a complicated mode of sorting
requires. Moreover, by the abuse of mixing in one heap all kinds of
leaves, frequently brought in from the fields all mixed together, the
proper sorting of tobacco becomes a very complicated affair.
"This kind of classification and nomenclature is, moreover, absurd, and
does not positively represent fixed qualities, under the denomination of
which, prices might be arranged which would serve as a guide to the
merchant as well as the grower. In a word, the names, with which the
different qualities of tobacco are to-day distinguished, signify
nothing, and it is ridiculous to be guided in business by them. Until
this kind of classification and nomenclature is changed, it is
impossible to quote the mercantile prices for the different qualities,
because the name does not represent the quality; and this confusion
tends greatly to the prejudice of the planter, and the merchant; and
hinders attaining the perfection after which we should strive.
"We have shown that the practice of making a classification of seven
_Injuriados_ must not be taken as absolute. There are better modes of
sorting in which a separation of 8, and even 9 _Injuriados_ should be
made, and others, and by far the greater proportion, in which only 5
_Injuriados_ should be separated; so that the quality which, in one
sorting, would appear under that of fifths—being the lowest of the
crop—would be equal to eighths, or ninths, if picked more carefully; and
the fifths, in a sorting, whose lowest class may be sevenths, is about
equal in quality to that of thirds of other pickings, whose lowest class
would be fifths, if both crops had produced equal kinds of tobacco.
"There is even more to confirm our opinion. Supposing two crops equal in
all respects, and that each planter makes a separation of 7
_Injuriados_. This would not ensure that the intrinsic value of each
respective quality would be equal; for each _Veguero_ has his own
particular mode of considering the different classes, and some make a
much more careful sorting than others. In the supposed case it may
happen, as it frequently does, that the _Veguero_ A will take from his
crop—which we will suppose to be one hundred packages—2 of the first, 3
of the second, 5 of the third, 8 of the fourth, 12 of the fifth, 30 of
the sixth, and 40 of the seventh; whereas the _Veguero_ B will take from
his, 4 of the first, 6 of the second, 10 of the third, 16 of the fourth,
32 of the fifth, 21 of the sixth, and 11 of the seventh; and it would
result, from the comparison of these two supposed pickings, that each of
these classes of the _Vega_ A would correspond to the immediate superior
one of the _Vega_ B, as will be shown on the following calculation:—
A.
$ $
2 Bales, 1st at 120 = 240
3 " 2nd " 100 = 300
5 " 3rd " 80 = 400
8 " 4th " 60 = 480
12 " 5th " 40 = 480
30 " 6th " 25 = 750
40 " 7th " 20 = 800
—— ———
100 $3450
—— ———
B.
$ $
4 Bales, 1st at 100 = 400
6 " 2nd " 80 = 480
10 " 3rd " 60 = 600
16 " 4th " 40 = 640
32 " 5th " 25 = 800
21 " 6th " 20 = 420
11 " 7th " 10 = 110
——— ————
100 $3450
——— ————
"Here it may be seen that the second of A is worth as much as the first
of B, the third of A as much as the second of B, and so successively in
the other classes; and as it is of importance that names should
represent fixed objects, and that each quality should represent a
relative value, we think that the sortings and the classifications
deserve a reform, which would undoubtedly bring with it advantages to
the planter, to the merchant, the manufacturer, and the consumer.
"The reform in the sortings should take its origin from a reform in the
plantation or field, and principally in the manner of cutting. By
observing a methodical and well-calculated system, each one of the
operations prepares and facilitates the execution of the succeeding one.
In its proper place, we have recommended that the tobacco planter should
not attempt to plant more than 12,000 plants for each labourer employed,
so that all the plants may receive proper cultivation and attention. If
all these plants are equally well taken care of, if the land has been
properly prepared with manure, and all have had the same advantage of
season, it is a necessary consequence that the fruit will be equally
good. If afterwards the cutting or cropping is made in 3 sections,
preserving always the separation we have recommended, we shall have,
naturally, not a capricious assortment of leaves, but one in the order
established by nature.
"None will, we think, question the fact that the pairs of leaves on one
stalk must be equal in quality to those cut from an adjoining stalk,
that is to say, all the crown leaves must be of the same quality, all
the second also, and so successively. This admitted, we have the
separation of qualities made, almost, in the field, and it only remains
to separate the sizes, and the sound leaves from the torn ones, an
operation which any person can make; and thus it will be unnecessary to
employ those workmen who style themselves sorters, who are supposed to
have an exact knowledge of the properties of each leaf. The sortings
ought, therefore, to be made by classes, or by bales, each containing
the separate qualities beginning with the bale of _capaduras_ and
_mamones_, which may be mixed together in the same bale. Of this
quality, however, not more than two classes should be made, which may be
called suckers and sprouts; and in the class called sprouts, the sound
and larger leaves of good consistency should be placed. The result would
be a _tripa_ of good quality, and, after throwing away all those that
are really without substance, the remainder would form the second class,
and would make a useful _tripa_, although inferior to the former.
"When these are made, the next bales should be made of tobacco chosen
from the inferior class of leaves, of which 3 classes ought to be made,
and called _sano_, _quebrado_, and _desecho de tercera_. In the first
class of these, which we will call third quality, should be placed all
the sound leaves which have any consistency; and this would form a weak
_capa_, equal to that which is now called clear fifths, _quinta limpia_,
and this might be called _sano de tercera_. The second class should
contain the torn or broken leaves of good consistency, but not so much
broken or injured as to merit only the name of shavings, as the leaves
which are very much torn, or small pieces of leaves, are called. This
class would be called _quebrado de tercera_, and might be used for
inferior _tripa_. The last class of this quality, after throwing away
all the useless leaves, would be called _desecho_.
"After this, and in the same order as the preceding, three classes
should be made from the sortings for the heaps of bad seconds and
thirds, and called _sano_, _quebrado_, and _tripa_ of the second class.
The first of these should contain all the sound leaves, and should be
called _sano de segunda_, second-class sound. The second should be
composed of the damaged leaves, but good for making _capa_, and should
be called second-class broken; and the third, which will be the most
broken, should be called second-class _tripa_.
"Finally, the picking, or sorting for the pile of pairs of crown leaves
should be made; and of this quality there should also be three classes,
which will be denominated '_sano_,' '_quebrado_,' and '_tripa de
corona_,' observing always the same order as was done for the piles or
heaps of seconds and thirds.
"Sorting carried on in this order is so simplified that we do not doubt
it might be done in one-third the time taken under the present system;
and the labour of the resorters would be dispensed with, which most of
the _vegueros_ have now to employ and pay, as many of them do not
consider themselves sufficiently expert in the matter to classify their
own tobacco. This classification and nomenclature represent exact
qualities to which a relative value can be fixed, and may serve as a
base for mercantile transactions.
"The manufacturer will not have to contend with bales of mixed tobacco
containing all the different classes which the _vega_ may have produced;
and he will find this division very convenient to determine the time
when each class may be used without having any loss from finding in them
leaves that are not seasoned, whilst others of the same bale, and
perhaps of the same _manojo_, may have become deteriorated from having
remained too long in fermentation. The manufacturer will, without any
great trouble, be able to make the assortment for strong and weak
_tripa_ according to the quality of _capa_ which is going to be used, a
most essential point in cigar making, and thus he will be able to make
cigars with all perfection. All these advantages will result from
adopting the reform in the manner of sorting which we propose. And, in
spite of its simplicity, it is much more positive and extensive, as it
will be composed of four qualities subdivided into eleven classes. The
consumer, too, will have the advantage of being able to procure cigars
manufactured completely of the quality which he prefers, and the
contents of each box, or each set of boxes, will be all equal both in
flavour and colour, which, under the present system, it is difficult to
find. The classes will be styled:—
{ 1st class Sound crown.
First quality { 2nd " Broken "
{ 3rd " Stuffing "
{ 1st class Sound seconds.
Second quality { 2nd " Broken "
{ 3rd " Stuffing "
{ 1st class Sound thirds.
Third quality { 2nd " Broken "
{ 3rd " Stuffing "
Fourth quality, 1st and 2nd Suckers and sprouts.
"It is scarcely necessary to add that, according to the preceding system
of sorting, only 3 divisions, cases, or rooms, with _yaguas_, will be
required for depositing the respective qualities which the workmen may
be assorting, until sufficient quantity has been collected in each to
commence the seasoning or painting, _betumeo_, _enmannillado_, or
_engavillado_, _manojo_, and _enterciadura_.
"In all kinds of sortings, the fragments of broken leaves, too small to
use for cigars, should be collected, sponged, and with them packages
made of _picadura_. This should be preserved, and the following year it
will be useful for making _betun_. Wash the tobacco, or rather sponge
it, with a solution made from these pieces of good leaves, and not with
a solution made from stalks and trash of new tobacco, as some do. The
wash (_betun_) has the same effect on tobacco that yeast has on bread.
It is the agent employed to produce a strong and quick fermentation,
from which results that strong and agreeable aroma that may be observed
in old tobacco which has been well _betumeado_ (sponged with tobacco
infusion). This infusion, made with fresh tobacco, is not bad if made
carefully, but we consider that made with old tobacco is the best,
because it instantly imparts an agreeable odour to the leaves on which
it is used; and, instead of the infusion which is generally used, it
would be cleaner and better, if a strong decoction was made from
_picadura_—the small pieces of leaves of good tobacco—and used after it
had become cold, or on the day after the boiling is made.
"If the wash is made by infusion, at least two jugs should be used to
make it in, and it should be only used on the third or fourth day,
renewing it as often as it appears to pass into a state of putrid
fermentation, in which state it is of no use, and on which account two
deposits are necessary, so that one at least may always be in a fit
state to use, whilst the other is acquiring the necessary strength and a
transparent golden colour, in which state it is fit for use.
"Each tobacco leaf should be dyed separately, and not, as some do, after
it has been made up into _gavillas_—small bundles tied at one end of the
leaf. It is very important that all the leaves should equally receive
the benefit, and this is impossible when several are tied together. The
good system of dyeing is used by all practical _vegueros_; to save
labour some do it otherwise, to the great injury of the aroma and
quality, and no small risk of the tobacco becoming spotted, and full of
holes; for tobacco invariably commences to show these spots and small
holes near the heads of the _gavilla_, where the dye has not been able
to penetrate owing to the manner in which the leaves are tied. Each leaf
ought, therefore, to be dyed separately, as the most intelligent people
do. The leaves should be placed separately in rows on a bench, having
all the heads in one line; then the dye should be applied by means of a
sponge, which should be soaked in the dye or infusion, and squeezed, so
that a dampness only will be communicated to the leaf.
"In passing the sponge over the leaf, it should be drawn from the head
or thick part near the stalk, down the large vein to the point, so that
the thick vein down the centre of the leaf may receive the heaviest part
of the infusion, from which the dye pushes along the transversal veins,
and all parts derive benefit from it.
"After dyeing the first layer on the bench, another one is placed above
this, keeping always the leaves in the same direction; and this
operation is repeated, and each layer is sponged, until the pile from
which they are taken is exhausted. As this new pile of dyed leaves
gradually increases in height, it should be gently pressed down with the
hand, and, when finished, should be covered over with green plantain
leaves. This operation should be done in the morning, and by nightfall
the tobacco will have acquired the necessary softness, and soaked up the
infusion, so that the leaves, although very flexible, will have no signs
of excess in moisture. If they have, they should be spread to dry
somewhat, because, when the bundles of leaves are being tied up, they
should not be excessively wet, as the result would probably be so strong
a fermentation that it would degenerate into a putrid one. The leaves
should have a soft silkiness, but should have no positive signs of water
on them after they have been dyed.
"When the tobacco is in a good state of softness, the next operation is
the '_cabeceo_.' This operation consists in uniting the leaves by the
heads—putting them perfectly even, and joining together a uniform number
of each class. The leaves should be collected in the palm of the left
hand, drawing gently the right hand over all the length of each leaf
from the head to the point, and tying them at the heads with a piece of
_yagua_ or vine, or, as most people do, by binding one of the leaves
round the head of the bundle. This operation is generally made in the
evening, and the following morning they should be placed in the bales,
as it injures the tobacco to allow it to dry in _manojos_ before putting
it into bales, for, if too dry, fermentation is retarded, or is
incomplete in the bales.
"We have described the manner of washing or dyeing, in making the
_gavillas_, and tying them in bundles as the most practical _vegueros_
do. In this part we should not, we think, advise any innovation, except
that of using old seasoned tobacco instead of fresh for making the
infusion, and substituting a decoction made by boiling, instead of an
infusion in cold water. But we strongly advise a reform in the sorting
and the classification; and a fixed number of each class of leaves
should be put in each _gavilla_, as a basis from which to start all
calculations for mercantile transactions. We believe, therefore, it
would be convenient to fix, after the following order, the number of
leaves which each head 'gavilla' should contain:-
{ Sound 25 leaves to each _gavilla_
First quality { Broken or torn 30 " "
{ For stuffing 40 " "
{ Sound 30 " "
Second quality { Broken or torn 35 " "
{ Stuffing 43 " "
{ Sound 40 " "
Third quality { Broken or torn 45 " "
{ _Desecho_ } These three classes
} may be added without
Fourth quality { Suckers } counting the
{ Sprouts } number of leaves,
but making the heads (_gavillas_) of a regular uniform size; and the
_manojos_ and bales of about the same size as those of 'sound' and
'broken' of the third quality, the latter weighing 100–125 lb.
"By following strictly this method, and by establishing these quantities
and qualities, as a basis for all contracts, any defects found might
easily be obviated; and very exact calculations might be made of the
number of cigars each bale would yield, after having examined its
special condition; and its real value might be estimated either by bales
or bundles, or by weight."
_Bulking._—Bulking means placing the tobacco-leaves in heaps for the
purpose of heating, in order to develop colour and flavour; this is
carried out in various ways, nearly all involving great labour and risk,
as in most instances tobacco loses more or less in value during the
process called "curing." The more care is taken in raising the crop, the
less attention the tobacco requires in the shed. With a good kind of
tobacco, grown on light, friable soil, treated as described, little care
will be needed, after the leaves are dried and stripped. By the drying
process, the leaves will have undergone a slow fermentation, which makes
it unnecessary to watch or guide a regular fermentation afterwards,
hence bulking and fermenting, as generally understood, are not required.
After being made into hands, the tobacco is put into heaps (bulked)
before it again dries. Every evening, the tobacco that has been stripped
during the day is bulked; but if the weather be very dry, it must be
bulked as soon as a certain number of hands is ready. The heaps should
be made 4–8 feet square and 4–8 feet high; all the stalks are outside,
and the whole is covered by mats, &c., to check evaporation. The drier
the tobacco, the larger must the heaps be made, to encourage a slight
fermentation. The extent of the fermentation can be easily controlled.
If the colour of the leaves is not uniform, or if it is desired to give
them a browner colour, the heaps must be made large, and a somewhat
moist atmosphere is required in the storing-room. This will cause
fermentation to set in after a short time, and the heat to rise after
some days, so much so that rebulking is required, which is done by
putting the top leaves of the old heap at the bottom of the new one.
Under such circumstances, the heap must be frequently examined during
the few first weeks, to prevent overheating. It is advisable to rebulk
the tobacco also, even when not much heated, after the first fourteen
days, and again a month later, to ascertain the exact state in which it
is. Sometimes the tobacco becomes mouldy; this occurs especially with
tobacco which has been manured with chlorides, which cause it to become
more hygroscopic than when manured otherwise. If this occurs, the mould
must be brushed off, and, if necessary, the tobacco be dried. The
tobacco may now remain heaped in the store-room until there is a chance
for sale. It must be remembered, however, that the best time for selling
varies very much. Some tobacco is fit for smoking a few weeks after
drying, whereas others may burn very badly at that time, yet become a
good burning article after being stored for several months.
After assorting, Perry Hull advises that the tobacco "be corded up
awhile, in a dry place, that the butts may be thoroughly cured before
packing in the cases. The pile is made with the butts out, and tips
interlapping in the middle, at every other course, at the ends turning
the butts toward the end. Get upon the pile upon the knees, take hold of
the butt of a hank with one hand, drawing the leaves at the tip together
with the other, and placing it upon the pile in that position,
immediately putting the knee upon it. After the pile is finished, it
should be covered over with boards, to keep it from drying up, and a few
days before packing into the cases, should be well weighted down, which
will save a great deal of pressing at that time. Such a pile should be
made only about 2½–3 feet high, and then closely watched to prevent a
premature sweat, which often, if the weather be mild, will take place in
such a pile, which will not be sufficient to render the tobacco fit for
working, but which, if not intercepted at the commencement, will be
sufficient to prevent a proper sweat afterwards. Check, therefore, the
first symptoms of heat in such a pile, by opening the pile, and
repacking it, shaking out the hanks and giving them time to cool off."
Bowie gives a caution that the tobacco "should not be too moist or
'high,' as it is termed, when put in stalk bulk, or it will get warm,
the leaves stick to the stalk, get a bad smell, and change colour;
besides, if left too long, it will rot. To bulk tobacco requires
judgment and neatness. Two logs should be laid parallel to each other,
about 30 inches apart, and the space between them filled with sticks for
the purpose of keeping the tobacco from the dampness of the ground. The
bundles are then taken one at a time, spread out and smoothed down,
which is most conveniently done by putting it against the breast and
stroking the leaves downward smooth and straight with the right hand. It
is then passed, two bundles at a time, to the man bulking. He takes them
and lays them down and presses them with his hands; they are laid, two
at a time, in a straight line—the broad part of the bundles slightly
projecting over the next two—and two rows of bundles are put in a bulk,
both rows carried on together, the heads being on the outside, and the
tails just lapping one over the other in regular succession. The bulk,
when carried up to a convenient height, should have a few sticks laid
across to keep it in place. It must often be examined, and if getting
warm it ought to be immediately changed and laid down in another bulk of
less height, and not pressed as it is laid down; this is called
'wind-rowing'; being loose and open, it admits the air between the rows
of bundles, hence the term. The next process in this troublesome, but
beautiful crop, is to 'condition' it for 'packing.' The 'bright,'
'yellow,' and 'second' tobacco will condition, but most generally in
such bulks as I have just described, but it is best to hang up the
'dull' as soon almost as stripped. If the bright or second do not dry
thoroughly in the bulks, that should also be hung up in the house to
become well dried. To properly hang up tobacco to condition, small-sized
sticks should be procured, and each one nicely smoothed with the
drawing-knife, and kept for that purpose. After it has once been
perfectly dry, either hanging up or in bulks—so dry that the heads are
easily knocked off, and the shoulders of the bundles crack upon pressure
like pipe-stems—it should be taken down, or if in bulks, removed, the
first soft, moist spell of weather, as soon as it is soft and yielding
enough, as it will become too dry to handle without crumbling or
breaking, and it must be put in 4 or 6 row bulks of any convenient
length and height, the higher the better, laid down close, so that as
little of the leaves or shoulders as possible be exposed on the outside
of the bulk. When completed put sticks and logs of wood, &c., on the top
so as to weigh it down. Here it will keep sweet and in nice order for
packing at any time, no matter what the weather be, if it was
conditioned properly, it will not change a particle while in the
condition-bulk."
_Packing._—Tobacco in America is commonly packed in barrels, the layers
being at right angles to each other alternately, and the butt-ends being
always towards the outside. The usual size is about 4 feet 6 inches
deep, 3 feet 6 inches in diameter at one end, and 3 feet 4 inches at the
other, to enable the contents to be uncovered for examination without
disturbing the mass. The packing is effected under considerable
hydraulic pressure. Elsewhere all kinds of packages are employed, and
their weights are very various.
In Bishop's opinion the best size for boxes is the following:—"3 feet 6
inches long, 2 feet 4 inches wide, 2 feet 6 inches in depth,
manufactured from planed pine boards, 1 inch in thickness, with
standards 2 inches square, inside at each corner to nail to. Having thus
your boxes prepared, and the tobacco in good condition, the first soft,
mild day that comes proceed to packing; the bundles or 'hands' of
tobacco must be taken from the bulk and laid in courses in the box,
laying the butts of the 'hands' to the outside of the box, allowing the
ends to lap over each other, and endeavouring to keep the centre of the
box a little higher than the edges—these courses to be packed as solid
as possible by the hand. If any of the bundles are 'soft' or have an ill
smell, they must be exposed to the fire or sun until sweet and dry
before being packed. When the box is nearly full, a false cover (just
large enough to slip inside the box) must be placed on the tobacco, and
pressed as heavily as possible with the lever or screw power; remove the
pressure and re-fill, pressure finally being applied to the real cover,
which may then be tacked down. A box of the size I have mentioned, when
filled, should contain about 400 lb. of tobacco, and thus packed, will
keep for years."
Another planter considers that parcels of "less than 1500 lb. may be
carried to market almost in any way; but more than that should be
'prized' in hogsheads. Several farmers might combine their crops for
prizing. As to the size, form, and materials of the hogsheads. In
Virginia, the size of the hogsheads is prescribed by law. They must be
made of seasoned pine or poplar. They must be 4 feet 6 inches long; 3
feet 6 inches in diameter, at one end, and 3 feet 4 inches at the other.
This difference of diameter is to allow the tobacco to be inspected.
This may be something new to persons of the North, therefore I will
explain the mode of inspecting tobacco in the hogshead. An inspector is
appointed by law to inspect or examine the tobacco prized in hogsheads.
His first step is, to place the hogshead big end upward. He then removes
the lining, and takes out the head. He next inverts the position of the
hogshead, that is, puts the little end up, and raises it entirely from
the tobacco. The mass of prized tobacco stands before him without a
covering. The outside may be all right, but his sworn duty is to examine
it through and through, as well as round and round. For this purpose he
drives an iron bar to the middle, near the top of the mass, prises up
and takes out a handful of bundles. He repeats that operation on two
other points of the mass. He then inspects or examines the parcels
extracted, and rates the whole hogshead according to their quality. The
hogshead is replaced and made secure. The hogsheads and the samples
taken from them bear corresponding marks, and the former is sold by the
latter. The staves of the hogshead must not be wider than 5, nor
narrower than 3 inches, ⅝ inch thick, and dressed on the inside. The
heading must be seasoned pine or poplar, and 1 inch thick, with 8 hoops.
Such a hogshead will well answer in other States as well as in Virginia.
"Weigh out, say 300 lb. It takes two hands to do this work, one inside
the hogshead and the other out. One is called the 'packer,' the other
the 'waiter.' The packer so arranges the bundles, in placing them, as to
make 4 courses in one layer. Repeat the layers until the 300 lb. are
packed. The weight (lever-power) is then applied. After 6 hours, put in
200 lb. more and apply the weight; 6 hours, and so on, until 1300–1500
lb. have been put in. The softer the tobacco, the more of it can be put
in a hogshead. If the tobacco is of the first quality, 1500 lb. is
enough. But if lower qualities, 1800 lb. can be put in. The finer the
quality the less weight it can bear without injury; and _vice versâ_.
Having prized the crop, it is ready for market."
According to Pursley, a hogshead "4 feet in length, and 3 feet in
diameter, is the medium size; 1000 lb. is considered a full hogshead;
but one of the above dimensions can hold 1500 lb. by hard pressing; but
this blackens the tobacco, and injures the sale of it. Packing in the
hogshead is done by first laying a course or layer of bundles straight
across the bottom, keeping the butts even and close together; then fill
up on each side of the centre course, placing the butts against the
staves; then the butts of the hands that lie against the hogshead should
be covered up with 2 or 3 others, pressed closely down. The next centre
course should be laid across the first, and done in the same manner as
before, and so on, crossing each course in succession, until the
hogshead is two-thirds full; when the press should be applied till the
tobacco is pressed down to within 1½ foot of the bottom of the hogshead.
The press should remain on an hour or more, in order that the tobacco
may settle together; then the press should be raised, and the packing
resumed as before, till the tobacco is within 1½ foot of the top; then
the press should again be applied till the tobacco is pressed half-way
down the hogshead; the same proportion should be observed until the
hogshead is full. Then put the head in, and it is ready for market."
Perry Hull would have packing-cases "made of cheap pine lumber, 3 feet 8
inches long by 2 feet 6 inches wide and high, outside measurement; they
should be made tight and strong; there should be corner-pieces nailed in
1½ inch square, nailing to them well from both ways. The tobacco is
packed in, with the butts towards each end; taking hold of the butt with
one hand, the tip with the other, and giving the hank a slight twist,
lay it in the case in that position. A lever or screw can be used to do
the pressing, whichever is the most convenient. From 360 lb. to 380 is
the proper weight for packing; though if the tobacco is very dry, 400
lb. will probably not sweat too hard; and if quite wet (which it never
should be), 350 may.
"After being packed, the tobacco should never be kept in a damp cellar;
a good tight barn or other outbuilding, where the cases can stand on a
floor, is the best place. The crop usually passes from the hands of
growers, into those of speculators and dealers, before the sweating
season. The first symptoms of sweating appear about as soon as settled
warm weather comes, usually the fore part of May; it then commences to
grow warm, and 'wet' to appearance, which increases for about 3 weeks,
when it reaches its culminating point and commences to cool off. One
unaccustomed to the crop, upon examining it at this period, would be
sure to think it was rotting, but if not too damp when packed, there is
no danger. Sometimes, if a case is known to be too wet, the lids can be
started, to give a little vent to the steam and gases which are
generated, and this is about all that can be done for it; and it is far
safer to see that the proper condition is secured before packing, than
to do even this. The weight will commence to decrease about as soon as
the heat commences, and it has been ascertained by weighing at the
various stages, that more than half of the shrinkage is accomplished by
the time that the sweat has reached its culminating point. About 10 per
cent. is allowed for the shrinkage of a crop, in just the right state
when packed; if wetter, it will shrink as high as 12–13 per cent., and
if very dry, it may shrink less than 10 per cent. The different grades
usually bring about the following prices: Wrappers, 14 cents per lb.;
Seconds, 7–8 cents; Fillers, 3–4 cents. The proportion of the different
grades in a good crop should be, Wrappers, three-fifths, and Seconds and
Fillers, each one-fifth."
Judson Popenoe thinks boxes "should be made 30 inches square by 42
inches in length outside; saw the end-boards 28 inches long, nail them
to two 1¼-inch square slats so that the head will be 28 inches square;
when two heads are made, nail the sides of the box to the heads so as to
come even with the outside of the head, the sides being 28 inches wide;
then nail the bottom on firmly; the top can be nailed slightly until
after the tobacco is packed, when it can be nailed firm. Set your box by
the side of the bulk, and let one hand get in the box and another pass
the tobacco to him, one hand at a time, taking care not to shake it out,
but put in the box as it comes from bulk, with the butt of the hand next
the end of the box. Place close and press with the knee firmly; lay
alternate courses at each end, and if the tobacco is not long enough to
lap sufficiently to fill the centre, put a few hands crosswise in the
centre. When the box is full, place it under a lever; have a follower,
which is a cover made of inch boards, nailed to two pieces of scantling
and made to fit inside of the box; lay this on the tobacco, and build
with blocks of scantling on it of a sufficient height for the lever to
be clear of the box when pressed. Press down firmly with a strong lever,
and, while kneeing in another box full, let the lever remain, so that
the tobacco gets set in the box. When ready, take the lever off and fill
up as before, about 6 inches higher than the box; press it below the top
of the box, take off your lever and nail on the top as quickly as
possible. Some use tobacco-presses for packing, which are perhaps more
convenient; they are of various patterns, but a lever saves the expense
of a press and is in the reach of all. If tobacco is sold at the shed,
it should be sold before packing, being easier examined in bulk than
box."
Mitjen is of opinion that, "except in cases where the extraordinary size
of the leaves will not permit it, all the bales should be made up of 80
'_manojos_'; but in the former case 60 of the first classes of the first
quality will be sufficient. The fixed number of 80 _manojos_ is
convenient for making calculations. We have already said that the day
following that on which the _manojos_ were tied up, they should be
packed in bales, so as not to allow them time to dry too much. Bearing
this in mind, the dyeing and tying up of the _manojos_ should not be
commenced until there is a sufficient quantity of assorted leaves to
make a bale or bales; should there be a surplus of _manojos_ after the
bales are made up, they should be kept protected from the air, until
another set of bales is about to be made up.
"We do not think it is necessary to further explain the manner of
placing the _yaguas_, in order to make the bales, but it is expedient to
state that 8 layers of _manojos_ should not be put in one bale, because
it makes a bad shape, and the tierces or bales appear much smaller than
they really are. The bales should be made of 2 layers, having the heads
of the _manojos_ placed towards the outside. When the first layer of one
of the heads of the bale is placed, the heads of the other layer should
be so arranged that they will be about half-way over the points of the
others; and if the tobacco is very small, to each row of _manojos_ may
be laid crossways, two _manojos_ with their heads touching the _yaguas_,
so that the tobacco placed in the bale may form a compact even mass,
impervious to the air. The same should be done in the other rows, care
being taken that the bale is made somewhat thicker in the middle, and
never have a hollow there,—a sure sign of loose packing,—and into which
the air finds its way, preventing fermentation, proper curing, as well
as aroma—the tobacco becoming dry too soon. After the bales are tied up,
they should be placed in the sun or wind until the humidity of the
_yagua_ is dry. They should then be placed on boards in the storehouse,
putting them two and two, one on the other; and after eight days they
should be moved, placing them below those which had been above, so that
they may ferment and be equally pressed."
[Illustration: FIG. 13.]
For pressing tobacco into the hogshead, Hudson suggests that "a hole be
mortised in a tree, in which the end of the lever can be inserted,
passing over the hogshead, and working by a tree or post, in which
should be pins at intervals of 8–10 inches, by which a small lever may
be used to force the first lever down on the tobacco; 50–100 lb. may be
placed in the hogshead and firmly pressed a few hours, and as much added
again, and so on. Fig. 13 will serve to represent the manner in which
the hands (or ties) may be placed in the hogshead—filling the middle
first, then the outer edges—placing the tops toward the centre, and
observing to keep the centre and edges full."
_Improving._—It is sometimes the custom to subject the tobacco-leaves to
some sort of improvement. There is no doubt that, by proper application
of ingredients, the value of tobacco may be much enhanced. The most
costly tobacco often commands a high price, not so much on account of
its inherent flavour, as from that given to it artificially. In most
instances, the best course to be adopted is to leave the improvement of
the leaves to the manufacturer. Many ingredients are employed to improve
smoking-tobacco. They tend:—1, to make the tobacco more elastic and
flexible; 2, to remove the coarse flavour; 3, to add a particular
flavour; 4, to improve the burning quality; 5, to improve the colour. To
make the tobacco more flexible and pliant, the leaves are macerated in,
or sprinkled with, a solution of sugar. In hot countries, this process
is often necessary, to give tobacco such an elasticity as to fit it for
handling, especially when intended for wrappers. To remove the coarse
flavour, it is often macerated in water, or in very dilute hydrochloric
acid. In Holland, 4–8 oz. of hydrochloric acid, diluted with 25–30
measures of water, is applied to 100 lb. of tobacco. The coarser the
flavour of the tobacco, the stronger is the solution used. The time of
maceration varies between ½ and 1 hour. Sometimes tobacco is steeped in
a mixture of sugar solution and diluted hydrochloric acid. To extract
the fatty matter, it is macerated in alcohol or spirit of wine. To give
a fine flavour, numerous substances are employed, some of which are kept
secret. The following ingredients are mostly in use:—Water, cognac,
vanilla, sugar, rose-wood, cassia, clove, benzoin, citron oil, rose-wood
oil, amber, thyme, lavender, raisins, sassafras-wood, saltpetre, orange,
and many others. The burning quality is improved by macerating in or
sprinkling with solutions of carbonate of potash, acetate of potash,
acetate of lime, or saltpetre, &c. Badly-burning cigars inserted for a
moment in such solutions are much improved. Tobacco treated with acetate
of lime yields a very white ash. The colour is sometimes improved by
fumigating the leaves with sulphur, and by the application of ochre and
saffron.
Although it may be said that fine tobaccos generally do not require any
impregnation with foreign matter for the sake of flavour, yet the
manufacturer frequently endeavours to give the leaf a particular aroma.
An inferior tobacco, however, which often would not find a market, is
sometimes so much improved by artificial means, as to compete
successfully with the genuine fine article. It is said that in Germany
indigenous tobacco is often so much "improved" that the cigars made from
it, after being covered with a fine tobacco leaf, are sold as genuine
Havanas. A special preparation of tobacco for snuff is seldom attempted
by the cultivator. With reference to the preparation of tobacco for
export, the sorting of the leaf is of the utmost importance; only first
and second sorts should be exported. It would be well to remove the
midribs, whereby the cost of transport and customs duty would be greatly
reduced.
The value of a cigar depends, not only on the intrinsic value of the
leaf, but to a great extent on the mode of manufacture. Thus, the raw
material may be of good quality, but if the maker does not classify the
leaves properly, or if he rolls his cigars too hard, which must vary
according to the qualities of the leaves, the cigar will burn badly. The
best-burning leaves must always be used for wrappers. If this should be
neglected, the inside of the cigar burns faster than the covering, the
air has no access to the burning parts, and the empyreumatical
substances are volatilized without being decomposed. Such cigars
therefore make much smoke, and smell badly.
CHAPTER IV.
PRODUCTION AND COMMERCE.
Details concerning the different modes of cultivating and curing, and of
the extent of the production and commerce in tobacco in the various
countries, will best be given in the alphabetical order of the
countries.
_Afghanistan._—The tobacco grown at Kandahar is celebrated in all the
neighbouring states for its mild and agreeable flavour, and is largely
exported to Hindustan and Bokhara. Three kinds are grown,
viz.:—Kandahari, Balkhi, and Mansurabadi. Of these, the last named is
the most esteemed, and fetches the highest price, viz. 6 lb. for
2_s._-4_s._ The Kandahari sells for a little less than half this price,
and the Balkhi for a little more. The Mansurabadi is not much exported,
being mostly consumed in the country. The cultivation is conducted with
great care, and the same plants yield two crops of leaves in the year.
Of these, the first, which is called _sargul_, is the best, the leaves
having a mild and sweet flavour; it is mostly consumed by the wealthy
classes, or exported. The second crop is called _mundhai_: the leaves
have a tough and fibrous texture, and a strong acrid taste; it is
usually smoked by the poor people, and is also made into snuff. The
plants are raised from seed in small beds, prepared for the purpose by
careful manuring with wood-ashes and stable-refuse mixed together. From
these nurseries, the young plants are transplanted into the fields,
previously prepared for their reception, the earth being laid out in
regular ridges and furrows. The plants are fixed into the sides of these
little ridges, and watered by means of the intervening furrows. Often
the young plants, packed in moist clay, and bound up in straw, are
conveyed to distant parts of the country; but the produce of these, it
is said, does not equal that of the plants reared at Kandahar. About six
weeks after transplanting, that is, about May-June, the first crop is
reaped, the whole plant being cut away about 6 inches from the ground,
and only some 5 or 6 of the lowest leaves being left. Each plant, as
cut, is laid on the ridge, and here each side is alternately exposed for
a night and a day to the effects of the dew and sun, by which their
green colour becomes brown. After this, they are collected in large
heaps in a corner of the field, and covered over with mats, or a layer
of straw, &c., and allowed to remain so for 8–10 days, during which the
stems shrivel, and give up their moisture to the leaves. At the end of
this time, the heaps are conveyed away into the villages, where the
stalks are separated from the leaves, the latter are then dried in the
shade and tightly packed in bundles about 14 inches square, and in this
shape are sold by the grower. After the first crop is gathered, the
ground is turned with a spade, well manured, and freely irrigated. In
due course, the old stems shoot up and produce fresh leaves, and in six
weeks or two months, the second crop is cut. Sometimes, though seldom, a
third crop is realized, but the quality of this tobacco is very
inferior, and it is only fit for making snuff.
_Africa._—The tobacco-plant extends throughout Central and East Africa,
wherever the equinoctial rains fall. It is cultivated to some extent in
the Bondei of Usambara, but seems to be the special product of the
Handei district, whence considerable quantities are sent to Pangani for
export. Usambara also exports to Zanzibar stiff, thin, round cakes,
which have been pounded in wooden mortars, and neatly packed in
plaintain-leaves. It is dark and well-flavoured. The Cape of Good Hope,
in 1865, had 933 _morgen_ (of 2·116 acres) under tobacco, yielding
1,632,746 lb.; in 1875, 1243 _morgen_ afforded 3,060,241 lb. Tobacco is
grown considerably in Oudtshorn and other districts of the Cape Colony,
and on the warmer farms in the Transvaal, but to the greatest extent on
the coast. The supply is already sufficient for local demands, and
tobacco promises to become a staple of South African agricultural
industry.
A recent writer on this portion of the British colonies says, "tobacco,
though cultivated as an article of commerce for export, has not met with
much success, as the passion for the weed has become deeply rooted in
the natives of the coast and interior, so that it is cultivated by them
in many parts of the province for their own consumption, and forms a
regular article of sale and barter amongst themselves." The tobacco leaf
is dried very carelessly by the natives, and is made up in a peculiar
way, as follows:—It is first plaited, and when the plait has reached a
length of 3–4 feet, it is wound up in the form of a spiral. Gradually
drying in this shape, it preserves its form without any binding, and it
is unwound and cut off in short pieces when required for use or sale.
This mode of preparation is invariable among the Makua and Yao, between
the Roouma and Zambesi. Consul O'Neill says that "were the natives
instructed in some simple method of drying and pressing the leaf, the
valuable product would be probably brought down by them in considerable
quantities, affording, as it would do, a larger margin for profit than
does the culture of oil seeds, and it might become a regular article of
colonial manufacture and export."
Tobacco-growing is a very important industry in Algeria. The culture and
manufacture are quite free, but the French Government buys all the best
produce, for manufacture and sale by the State factory in Paris. The
cultivation continues to increase, and is highly remunerative where the
land is capable of irrigation. In 1876–7, the 1889 Europeans engaged in
it cultivated 2471 _hectares_ (of 2½ acres), and produced 2,782,500
_kilo._ (of 2·2 lb.); the 8021 natives cultivated 4154 _hectares_, which
yielded 1,889,124 _kilo._ The year 1877–8 was less favourable, and the
area decreased by 425 _hectares_. Still worse results were expected in
1878–9, owing to scarcity of water. The kind most grown is called
_chebli_. The produce per _hectare_ of fine and _chebli_ is estimated at
6–8 _quintals_; the other kinds give 10–12. The exports in 1877 and 1878
respectively were as follows:—Manufactured, 121,090 _kilo._, and 124,117
_kilo._; unmanufactured, 3,445,441 _kilo._ and 1,509,266 _kilo._ In
1879, 1087 Europeans planted 3180 _hectares_, and gathered 1,226,181
_kilo._; 11,079 natives planted 6584 _hectares_, and produced 1,384,802
_kilo._; the exports were 2,481,218 _kilo._ unmanufactured, and 146,345
_kilo._ manufactured.
The figures for 1883 were:—1240 European planters cultivated 2278
_hectares_ and produced 2,250,671 _kilo._, whilst 8735 native planters
cultivated 6416 _hectares_ and produced 2,977,067 _kilo._, the total
product being 5,227,738 _kilo._ This does not differ to any great extent
from the result of the previous year. Tobacco is capable of being
produced in much greater quantity, says the British Consul, but the
market is limited. The colonists themselves and the Government appear to
be the only purchasers.
_Australia._—In the year ending 31st March, 1879, New South Wales had
835 acres under tobacco, and the crop amounted to 7932 cwt. In the same
year, Victoria cultivated 1936 acres, which yielded 15,662 cwt., valued
at 43,853_l._ Queensland grew 36 acres of tobacco in 1879.
_Austro-Hungary._—The manufacture and sale of tobacco is a Government
monopoly in the Austro-Hungarian empire, and the revenue thus derived is
the most lucrative item of the indirect income of the State. The only
tobacco-growing provinces of Austria are Galicia and Bukowina, producing
about 4 million _kilo._ from 2900 _hectares_; and South Tyrol, where 290
_hectares_ yield almost 4 million _kilo._ of green tobacco. The
respective approximate values of the two products are 18⅓ _florin_ (of
1_s._ 11½_d._) and 4⅔ _florin_ per 100 _kilo._ The chief supplies are
furnished by Hungary, which was once so noted for its tobacco, but the
industry is now completely crippled by the fiscal regulations. The area
(in acres) under cultivation fluctuates remarkably; in 1860, it was
679¼; in 1865, 68,141; in 1869, 843¾; in 1875, 26,817; in 1879, 7316.
The total areas (in acres) under cultivation in the whole empire in
1876, 1877, and 1878 respectively were, 144,493, 148,126, 143,447; the
yields in _kilo._, 46,033,163, 44,164,038, 40,978,540; and the yield (in
_kilo._) per _joch_ (of 1·43 acre), 445, 426, 408. Fiume, in 1877,
exported by sea 2862 cwt. of manufactured tobacco; and by land, 31,200
cwt. of leaf, and 53,712 cwt. of manufactured. In 1879, it shipped 9900
_kilo._ of leaf tobacco direct to England. In 1883, the tobacco harvest
was 26,560 metrical centners (about equivalent to cwts.), being 1595 in
advance of 1882. The total exports of raw tobacco were 55,842 metrical
centners in 1883, and 74,475 in 1884. The port of Fiume shipped 613 tons
of tobacco leaf in 1883, of which 189,300 _kilo._ value 75,720 florins,
went to Gibraltar. In 1884, the shipments from Fiume were 1673 tons.
_Borneo._—Tobacco is grown in small quantities by the Dyaks and people
of Bruni; but they are unskilful in its manufacture, though the flavour
of the product of Bruni is much esteemed by Europeans. Under skilful
management, and by introducing a better kind if necessary, it might
become as profitable to this island as it now is to the neighbouring
ones of the Philippines, Java, &c. The Dyaks might be more readily
induced to cultivate this plant, the nature of which they know, than
plants which are strange to them. More recently it is announced that
plantations have been commenced in British North Borneo, and samples of
the leaf sent to Europe have been favourably reported on. The exports
from Sarawak in 1884 were valued at 2020 dollars to foreign ports, and
34,257 dollars in coasting vessels, making a total of 36,277 dollars. In
the same year, British North Borneo shipped 2113 dollars' worth; and
Sandakan, 1537 dollars' worth.
_Bourbon._—Efforts are being made to successfully introduce tobacco into
the rotation of crops on the sugar estates, with the object of supplying
the article to the French _régie_ or Government monopoly, which buys
annually upwards of 40 million francs' worth of tobacco in the islands
of Cuba, Java, and other colonies. The results hitherto obtained are not
unsatisfactory, and this article may shortly acquire importance among
Bourbon products. The exports in 1884 were 10,185 _kilo._, value 61,110
_fr._
_Brazil._—In Brazil, tobacco is chiefly cultivated in the provinces of
Bahia, Minas, Sao Paulo, and Para. The town of Purificaçao, in Bahia, is
the centre of an important district. The cultivation is increasing, and
greater care is being taken in the preparation. The common up-country
method is to pick the leaves from the stalks, dry them under the
hut-roofs, remove the midribs, and spread them in superposed layers,
amounting to 2–8 lb., for rolling together and binding with bark strips.
These rolls are bound very tightly with cord, and left for several days,
when the cord is replaced by strips of _jacitára_, the split stem of a
climbing palm (_Desmoncus sp. <DW37>._), and have a stick-like form 1½ inch
in diameter. They are sold in _masas_ of 4–6 feet in length, but the
tobacco is not considered good till it has fermented for 5–6 months,
when it is hard and black, and shaved off as required for pipes,
cigarettes, and cigars, the last made with wrappers of _tauari_ bark
(_Couratari guianensis_). The Tapajos tobacco is considered the finest
in the Amazon valley. The export of tobacco from Bahia in 1877–8 was
17,272,678 _kilo._, and in 1878–9, 18,149,201 _kilo._, almost the whole
being to Germany. Santos, in 1878–9, shipped 381,310 _kilo._ Bahia sends
away immense numbers of cigars coastwise. Maceio exported 4336_l._ worth
in 1876, but none in 1879.
Some interesting particulars are given in the last report of the United
States Consul-General at Rio de Janeiro, as to the cultivation and
manufacture of tobacco in Brazil. It appears that the cultivation began
about the year 1600, in the province of Bahia, and from thence extended
to all the other districts along the coast. Among the localities
earliest known for their tobacco production was the lake district of
Pernambuco, now the province of Alagoas, where an excellent quality was
produced, which commanded very high prices. During the following century
the cultivation increased so rapidly in Alagoas and Bahia, that at the
commencement of the succeeding century, the average annual export had
reached 2857 tons from the latter, and 285 tons from the former
province. The earliest export statistics available for the whole empire,
are for the year 1839–40, in which the export amounted to 295,966
_arrobas_, the _arroba_ being equivalent to about 32 lb.; and the value
exceeded 65,000_l._ For the next thirteen years, the exports averaged
8,000,000 lb. annually, with a value steadily increasing. During each of
the years 1853–55, the amount exported was 22,000,000 lb., of the total
value each year of 200,000_l._ In 1879–80, the export was 50,000,000
lb., of the value 659,000_l._; in 1880–81, 44,000,000 lb., of the value
of 650,000_l._, and in 1881–82, 52,000,000 lb., of the value of
680,000_l._ Though the principal tobacco-producing province of the
empire is Bahia, tobacco of good quality is grown in every part of
Brazil, from the Amazon to the Rio Grande frontier. Some localities in
the province of Amazonas have long been known for the excellent quality
of their tobacco, while in the Rio market one of the brands most
esteemed comes from the province of Goyaz. The local consumption of
tobacco is very great, and principally in smoking. Bahia tobacco used to
be largely exported in rolls, weighing 8 _arrobas_, or 256 lb. each; of
late years, however, large quantities of the leaves in bales are
exported to Hamburg. Cigar factories are established in all large cities
throughout the tobacco-growing regions, which give employment to a large
number of men, women, and children. The methods employed in the
cultivation and preparation of the plant are very much the same as they
were nearly 200 years ago. The labour employed is that of slaves, to
whom are assigned special descriptions of work. In former times curing
tobacco in rolls required much constant labour, the ropes composing each
roll being unwound, twisted, and re-wound during a period varying from
10 to 15 days. The Brazilian tobacco is generally characterized by its
strength and dark colour, particularly in Bahia. In that province the
practice is to manure heavily, which occasions a very rank growth and
strong flavour. In Minas Geraes the tobacco is somewhat milder, and some
advance has lately been made in a few localities towards improved
processes of curing. This seed may be germinated in any season of the
year, but the months of June, July, and August are generally preferred
for planting, because germination and transplanting are brought into or
near the rainy season. Tobacco plants planted in this season are
considered the best growers, and produce the largest leaves. Those,
however, which are germinated in the dry season, and sustained by
irrigation, grow with greater vigour, and possess a finer aroma. The
land selected for the plants is cleared, and the surface worked with the
hoe, after which it is marked off into parallel rows about 3 feet apart,
according to locality and the size of the mature plants. In
transplanting, the young plants are set from 2 to 3 feet apart, and are
manured heavily in the pits opened for them. Great care is necessary for
a time to protect the shoots from the sun, and to irrigate plentifully
when the transplanting occurs in a dry season. The work of cultivation
and keeping down the weeds is performed entirely with the hoe, and only
two or three times during the season. In gathering in the crops,
planters wait until the plants are fully matured, this being determined
by doubling and breaking one of the top leaves. In Bahia and other
Brazilian provinces the lower leaf is often picked by itself, and in a
few days the next, and so on as long as the plant will develop the lower
leaves into what is classed first quality. These leaves are hung up two
and two, under cover and across poles, 24 hours after picking and
sweating. When it is intended to twist the leaves into ropes, they are
left hanging about 2 days, when they are taken down, carefully freed
from the heavy parts of the midrib, doubled in halves, and laid away for
the rope twister. This operation requires considerable dexterity, and is
generally entrusted to the best slave on the plantation. The operation
requires a rude windlass, which is slowly turned in winding the rope,
which is twisted by hand. A boy is usually employed entirely to hand
leaves to the twister. These ropes are unwound and re-wound once or
twice a day, for a period of 10–15 days, according to the weather, and
are twisted a little harder each time. In curing, the tobacco grows
darker and darker, until it becomes jet black. The juices exuding from
the rolls are carefully caught and preserved until the last winding,
when, mixed with lard, syrup, and various aromatic herbs, they are used
to pass the rope through, previous to the final winding. The last step
is to cut the cured ropes in certain lengths, and to re-wind them upon
light wooden sticks, about 2 feet in length, the winding being very
compact and regular. The rolls are then covered with leather or strong
canvas, when they are ready for market. Formerly, these rolls were made
to weigh 8 _arrobas_, or 256 lb., though rolls of 3 _arrobas_ were made
for the home markets. At the present day the weights vary according to
the locality. The large exportation of tobacco in leaf has considerably
changed the character of tobacco-growing in Bahia, the process of curing
and packing the leaf being simpler than the old process of manufacturing
_rolos_. Tobacco-growing is heavily protected and taxed in Brazil,
nearly all the provinces imposing separate protective taxes, in addition
to those imposed by the Government. Besides these, the municipalities
are permitted to levy taxes on the article. The present export tax on
tobacco, in Brazil, amounts to as much as 18 per cent.
The local market quotations are thus given:—
s. d. s. d.
Patentes 6808–8170 _real_ (=12 2–14 7) per 10 _kilo._ (= 22
lb.)
Santo Amaro, assorted 3 7– 5 8 " "
Alagrinhas 2791–5106 (5 0– 8 2) " "
São Felix 3745–4425 (6 8– 7 10½) " "
The Bahia export in 1883–4 was 15,644,010 _kilo._, value 400,246_l._
_Canary Islands._—With the declining importance of cochineal,
tobacco-growing is gaining ground, and the quality of the article has
been much improved, while factories for drying and preparing the leaf
have been established in various localities. The exports for the year
1883–4 were:—27 lb., value 8_l._, to France; 2268 cwt., value 9809_l._,
to Spain; 1753 lb., value 375_l._, to Germany; and 939 lb., value
189_l._, to West Coast of Africa.
_China._—The chief tobacco-growing provinces of China are Chihli, Hopih,
Hoonan, Szechuen, and Shingking. The use of tobacco is wide-spread and
common, and considerable local trade is carried on in it. The exports
from Amoy were 2573 _piculs_ (of 133⅓ lb.), value 13,561_l._, in 1877;
and 3994½ _piculs_, value 17,936_l._, in 1878. Wenchow exported 27¾
_piculs_ of leaf in 1878, and 321⅓ in 1879. The exports and re-exports
from Hankow in 1878 were 65,070¾ _piculs_ of leaf, and 46,241¾ of
prepared. In 1879, Hankow exported and re-exported 63,180 _piculs_
prepared, value 311,754_l._, and 58,094 of leaf, value 118,534_l._ There
is an immense supply from the provinces, and the leaf is fine in colour,
texture, and fragrance, but though sent to America and England for
cigar-making, the trade has not been remunerative. It is now used in
cigarettes and various cut mixtures as "Turkish," but when better known,
will be smoked on its own merits. Canton exported 1730¾ _piculs_ in
1877, 1742¾ in 1878, and 2397 in 1879. The exports of leaf from Ningpo
were 407 _piculs_ in 1874, 571 in 1875, 211 in 1876, 530 in 1877, 378 in
1878, and 165 in 1879. Kiungchow exported 449¼ _piculs_ of leaf in 1878;
and 85½ _piculs_, value 136_l._, in 1879. Kiukiang exported 28,120½
_piculs_ of leaf, value 35,678_l._, in 1878; and 14,659 of leaf, and 802
of stalk, in 1879.
Chinkiang imported 13,328 _piculs_ of leaf, and 1914 of prepared, in
1879. Macao receives tobacco from the Hokshan district, and prepares it
for exportation to Java, the Straits, and California, the annual export
being about 10,000 _piculs_. The Newchwang imports of prepared native
tobacco were 8052 _piculs_ in 1877, 8354 in 1878, and 6630 in 1879.
Shanghai, in 1879, imported 58,460 _piculs_ of native leaf, 79,081½ of
prepared, and 1187½ of stalk; and exported and re-exported 31,541 of
leaf, and 29,672¼ of prepared. Taiwan imported 3017¼ _piculs_ of
prepared native in 1879. Tientsin exported 1047⅓ _piculs_ native tobacco
in 1878, and 693½ in 1879. Tobacco is grown in the hilly districts near
Wuhu; the leaves are gathered in October, and sun-dried on wicker-work
frames. The exports in 1879 were 597½ _piculs_ of leaf, and 742 of
prepared.
_Cochin-China._—The culture of tobacco is extending in Cochin-China, and
it is even said that a considerable quantity is exported to China, but
it improves little in quality. The area reported to be under tobacco
cultivation in 1878 (including coffee) was 2361 acres.
_Costa Rica._—The free cultivation of tobacco was stopped in January
1884, and its free sale only permitted till December 31, 1885.
_Ecuador._—The tobacco crop of Ecuador for 1879 was not so large as
usual, owing to an unfavourable season. Esmeraldas, the most northerly
port, and whence nearly all the tobacco shipments are made, despatched
about 3000 _quintals_ in 1879. Guayaquil exported 150 _quintals_ in
1877, none in 1878, and 10 in 1879. In 1883, the exports from Guayaquil
were 1374 _quintals_, value 5496_l._; in 1884, only 96 _quintals_,
192_l._
_Fiji._—The Fiji Islands are well adapted to tobacco culture. The
natives produce a good deal, which nearly approaches the American leaf.
With careful curing, it would find a market in England. The native
product is rolled, which prevents its being made into cigars. Samples of
leaf-tobacco in hands, raised from foreign seeds, exhibited very unequal
qualities, and a tendency to revert to American forms, the Havana
returning to the Virginian type. Cut up for smoking, they were deficient
in flavour, but were considered satisfactory as a first experiment.
_France._—The area occupied by tobacco in France in 1873 was 14,858
_hectares_ (of 2½ acres), yielding at the rate of 12 _quintals_ (of 220½
lb.). The amount of land authorized, to grow tobacco in Pas de Calais in
1879 was 2100 acres, and the quantity furnished to the Government was
3,659,636 lb., the prices (per _kilo._) paid by the Government being 1
_fr._ 45_c._ for 1sts, 1 _fr._ 12_c._ for 2nds, 88_c._ for 3rds, and
10–66_c._ for other inferior qualities. The number of plants grown per
acre is about 17,000. The department Nord affords rather more than Pas
de Calais.
By the Imperial decrees of December 29th, 1810, and January 12th, 1811,
it was ordained that the purchase of tobacco in leaf and the fabrication
and sale, whether wholesale or retail, of manufactures of tobacco,
should be exclusively confined to the Administration of Indirect Taxes
(Régie des Droits Unis) in all the departments of France. At present the
Régie has in operation 16 large manufactories, 27 "magasins de culture,"
and 4 "magasins de transit." It employs over 19,000 workpeople, of whom
about 80 per cent. are women and girls. The usual daily earnings are,
for men, from 2_s._ 7_d._ to 3_s._ 11_d._, and for women, from 1_s._
2_d._ to 2_s._ 4_d._ For faithful or exemplary services, the workpeople
receive annually rewards, varying in amounts from 15_s._ to 20_l._ Mr.
Scidmore, the United States Consular Agent in Paris, gives the following
description of the manner in which the operations of the Régie are
carried on. At the beginning of each year the Minister of Finance
designates the number of hectares upon which, and the departments within
which, the cultivation of tobacco may be undertaken during the following
season. The last ministerial decree upon this subject confines the
privilege to the departments of the Alpes Maritimes, Bouches du Rhône,
Dordogne, Gironde, Ille-et-Vilaine, Landes, Lot-et-Garonne,
Meurthe-et-Moselle, Nord, Pas de Calais, Puy de Dôme, Hautes-Pyrénées,
Haute-Saöne, Savoie, Haute Savoie, and Var. In the month of October or
November, an agent of the Régie proceeds to the communes among which the
prefects have apportioned the allotments, and receives the declaration
of every proprietor desiring to profit by the authorization. A
Commission, composed of the prefect, of the director of indirect taxes,
a superior agent of cultivation, a member of the council general, and of
a member of the council of the arrondissement, not being planters, then
examine the declarations, and admit, reduce, or reject them. After a
planter is accorded permission to cultivate, he is subjected to close
official supervision, and to numerous stringent regulations concerning
details as to the prohibition to sow any other seed than that furnished
to him by the administration, the mode of planting, &c.; and, in
addition to the surveillance as to these matters, two official
inventories are taken of the growing crop—the first to ascertain the
extent of land under cultivation and the number of plants, the second to
determine the number of leaves for which the planter will be held
accountable. When the tobacco has been gathered in a manner described by
regulations of minute detail, the planter takes it to the magazine of
the Régie, where it is subjected to the inspection of a commission of
five disinterested experts, who separate the leaves into three portions,
according to quality; the planter is then paid for each portion in
accordance with the tariff of prices promulgated by the Minister of
Finance. Foreign tobacco is obtained through contract with private
parties, after published proposals by the Minister of Finance through
the French Consular Corps abroad, and through a special government
agency established at Havana. At present a little over one-third of the
tobacco purchased by the Régie is of French growth; over one-half
consists of foreign leaf, mostly obtained from the United States, and
the remainder is made up by importations of cigars from Havana and
Manilla, and by cigarettes and miscellaneous productions of various
countries, and by custom-house seizures. The magazines distributed
throughout the country are of two sorts, "magasins de transit" for
foreign tobacco, and "magasins de culture" for indigenous tobacco. In
the "magasins de transit" the foreign leaves have not to submit to any
other manipulation than the sampling of packages, after which they are
forwarded to the factories in such quantities as may be demanded. With
the indigenous tobacco the course is different; this when received from
the hands of the French grower is usually very imperfectly dried, and
has to be subjected to a curing process. After the bundles are
thoroughly thrashed, they are put in heaps according to maturity, and
fermented in a temperature as high as 30° to 40° Centigrade. This
maturation lasts from six to nine months, depending upon the locality,
and the condition of the leaves as received, and is interrupted from
time to time by the operation of shaking and turning in order to prevent
too great fermentation. When this fermentation is concluded, those
leaves containing less than twenty per cent. of water are ready to be
packed. At this point certain of the leaves undergo a stemming process;
they are then packed by hydraulic pressure in bales and hogsheads
weighing from 400 to 500 _kilo._ each, and in this state they remain
stored in the magazine for some months to acquire further ripeness. It
is usually 15–18 months after they are gathered that the leaves are
considered to be in a fit condition to be sent to the manufactory. Upon
arrival at the manufactory, the packages are sorted and emptied; the
leaves are spread out in large bins or receive a preparatory wetting
with water containing 10 per cent. of sea salt, in order to produce
flexibility and prevent powdering. This process occupies 24 hours. Then
follows the sorting according to quality, and the distribution to the
various workrooms for composition.
When intended for the manufacture of snuff, the leaves are put into
machines and chopped into strips of the width of a finger; they are then
moistened with pure water or tobacco juice of various strengths, the
necessary quantity and quality of which is determined by chemical
analysis. These strips are then piled up in masses containing from
35,000 to 40,000 kilogrammes, in rooms where a high and even temperature
is maintained by steam-pipes and ventilators. Here they remain to
ferment during a month or six weeks, when they are dried, ground into
powder, and sifted. This powder then receives a wetting, is packed in
stout wooden bins, in quantities ranging from 25,000 to 30,000 _kilo._,
and so remain to ferment for several months. During the course of the
final fermentation, the powder is tested and moved from one bin to
another from time to time, in order to ensure a successful issue of the
process. When the samples taken from the bins indicate maturity, the
snuff is packed in barrels and casks, and is ready for the market. For
the manufacture of smoking-tobacco, the leaves, after the stemming
process, receive their first moistening, which lasts 24 hours. They are
then neatly arranged, with their edges parallel, and are taken to the
chopping machines; the machines in use at the Régie are capable of
chopping 220 lb. per hour, the knives being renewed twice during that
time. The tobacco, on leaving the choppers, contains about 25 per cent.
of humidity, and is immediately conveyed into one end of a revolving
drying cylinder, heated to a uniform temperature of 203° Fahrenheit,
from the opposite end of which it issues, at the expiration of fifteen
minutes, in a dried state and freed from albumen. It is then put through
a second cylinder, similar in construction to the last, but which
subjects the tobacco to a strong draught of cold air to eliminate all
dust and heat. The tobacco is then packed in well-aired bins, where it
remains from four to six weeks, after which it is carefully overhauled
by hand to remove the pieces of stems and foreign matter that may have
escaped notice in the previous operations. It is then put up in
packages, varying in weight from 40 grammes upwards. These packages are
surrounded with a paper band, upon which are printed the Government tax
stamp, the date of manufacture, the weight, the price, and the letter
"H," followed by figures. The last mark signifies the amount of humidity
contained in the tobacco at the time it was put into the packets. Consul
Scidmore says that in no instance since its inauguration has there been
a year without enormous profits to the tobacco monopoly in France, and
in a table appended to his report, it appears that from the date of its
foundation (1811) to the end of 1878, the net total gain to the French
Government amounted to 287,703,881_l._
The following table from a recent report shows that the consumption of
tobacco in France has been steadily increasing:—
Year. Population. Amount Amount per
consumed. Head.
Kilogrammes. Grammes.
1815 29,250,000 8,981,403 307
1826 31,673,853 11,595,084 366
1831 32,731,256 11,071,088 338
1841 34,018,715 16,461,934 484
1851 35,546,919 19,718,089 555
1864 37,133,424 28,019,803 755
1866 37,807,203 30,627,663 810
1872 35,844,414 27,031,000 754
1876 36,643,087 31,188,846 851
The amount consumed in the different departments varies very much.
Snuff-taking is most practised in Oise, Seine Inférieure, Eure, and
Eure-et-Loir, at the maximum rate of 375 _grm._ per head; and least in
the departments of Doubs, Pyrénées Orientales, Nord, Haut Rhin, and
Haute Savoie, where the average is but 100 _grm._ In smoking, however,
there is rather a reverse order of things, the Nord, Haut-Rhin, and Pas
de Calais consuming at the rate of 2 _kilo._ per head, while the minimum
is found in Haute Savoie, Cantal, Corrèze, Creuse, Aveyron, Dordogne,
Lot, and Lozère. Ten departments only consume tobacco above the average,
while 70 are actually below it. If all France smoked the same quantity
as do the people of Nord, Haut-Rhin, and Pas de Calais, the consumption
for the whole country would be 73,286,174 _kilo._ instead of 31,000,000;
and _vice versâ_ it would be only 6,265,968 _kilo._ if calculated
according to the average of Lozère, which is only at the rate of 171
_grm._ per head.
The department of the Nord, in 1884, had 449 _hectares_ (of 2·47 acres)
under tobacco, the yield of which was 1,168,206 _kilo._
_Germany._—The total area of land engaged in growing tobacco in Germany
in 1878 was about 44,520 acres; nearly two-thirds of this total was
distributed among Rhenish Bavaria, Baden, S. Hesse, and Alsace-Lorraine.
The total consumption of tobacco in the German empire in that year was
2,196,000 cwt. The home production was 596,776 cwt., the remainder being
imported.
The aggregate area of land cultivated with tobacco in the States of the
German Customs Union did not vary considerably during ten years, being
21,509 _hectares_ in 1863, and 20,918 in 1872, to which must be added
the newly annexed provinces of Alsace and Lorraine, which bring up the
total to 24,745 _hectares_. It appears that, with particular regard to
the year 1872, the cultivation was carried on in 4067 different
localities, by 94,916 taxable growers, and by 83,675 smaller growers,
whose production, owing to its limited extent, was exempt from taxation.
By far the larger number were small growers, the area cultivated by each
not exceeding an average of 10 _ares_. In Prussia the aggregate of land
cultivated during the year 1871 amounted to 5925 _hectares_, or 26 per
cent. of the entire territory of the kingdom; the aggregate yield of the
harvest in the same year was 198,890 _centners_. It appears that the
extent of tobacco-growing land has, during the last fifty years, been
gradually diminishing in Prussia, and that accordingly the expectations
entertained in the beginning of that period of a great future
development of this branch of agriculture have not been realized. The
reasons for the gradual decline are considered to be, on the one hand,
the growing competition of the South German growers, and the increase in
the importations of American tobacco; on the other hand, the fact that
the cultivation of beetroot for sugar, and of potatoes for distilling
purposes, has proved to be a more profitable business than tobacco
production. It has, moreover, been found by many years' experience, that
whilst the quality of the tobacco cultivated in most parts of Prussia is
not such as to enable the growers to compete successfully with the
importers of foreign, particularly North American sorts, the labour
attending its cultivation and its preparation for the market, as well as
the uncertainty of only an average crop, are out of proportion, as a
rule, to the average profits arising therefrom. The cultivation of the
plant has consequently gradually "become restricted chiefly to those
districts of the country where either the soil is peculiarly adapted for
the purpose, or where it is carried on for the private use of the
producer.
In Bavaria, as is well known, tobacco is cultivated very extensively,
particularly in the Palatinate and in Franconia, viz. the districts
around Nuremberg and Erlangen. The area of land in 1871 was 4721
_hectares_, which produced 144,153 _centners_. In Saxony but little
tobacco is grown, the total area planted therewith in 1871 not having
exceeded 6 _hectares_, upon which 130 _centners_ were produced. Although
in parts of Wirtemberg the soil and climate are said to be very
favourable to the growth of the plant, the area of land cultivated is,
upon the whole, a very limited one, and did not exceed 178 _hectares_.
The yield of the harvest is given at 5571 _centners_. In the year 1858
the extent of production in Wirtemberg is stated to have been four times
as great as it is at present. The Grand Duchy of Baden has at all times
been the chief tobacco-growing part of Germany, and as far back as the
end of the seventeenth century special laws for regulating the
cultivation, preparation, and warehousing of this article were in force.
The great importance accordingly attaching to this branch of agriculture
and industry for so large a proportion of the inhabitants of Baden,
renders it but natural that any project of increasing the tobacco tax
should meet with very strong opposition amongst most classes of the
Grand Duchy. The most prominent tobacco-growing districts of Baden are
those of Carlsruhe, Mannheim, Heidelberg, Badenburg, Schwitzingen, and
Lahr; the quality of the plant grown in these parts being a very
inferior one. The produce of the districts mentioned is therefore
applied chiefly to the manufacture of "cigar-wrappers," and is exported
in considerable quantities to Bremen, Hamburg, Switzerland, Holland, and
even to America, for the use of the cigar-makers. The prices of the best
kinds of Baden tobacco are consequently also, on an average, much higher
than those realized by other German growers. The area in Hesse was 979
_hectares_, the chief district being around the town of Darmstadt; the
production was 31,311 _centners_. The most prominent amongst the
Thuringian States as regards tobacco production, is the Duchy of
Saxe-Menningen; the land cultivated in 1871 in all of them put together
was 202 _hectares_, the yield of the harvest in that year having been
4806 _centners_. In the two German states of Mecklenburg, 6106
_centners_ were raised from 165 _hectares_ of land. The most important
district is that of Neu-Brandenburg, in Mecklenburgh-Strelitz. Only a
small extent of land, viz. 69 _hectares_, is used for tobacco in the
Duchy of Brunswick, the same being situated near the town of Helmstadt;
the amount raised was 2391 _centners_.
In the recently acquired provinces of Alsace-Lorraine, tobacco
cultivation has been extensively carried on for many years, more
especially in the country around Strasburg, Mülhausen, Schirmeck, and
Münster, and to a smaller extent near Metz and Thionville. The aggregate
area of land cultivated in 1871 in both provinces is given at 3159
_hectares_, upon which 115,518 _centners_ of tobacco were raised.
According to the statistics and information furnished by Consul Ward,
the quantity of tobacco produced in Germany in the year 1871 amounted to
713,845 _centners_, the whole being estimated in value at 60,284,210
dols., or about 9,042,613_l._ sterling.
A Consular report of March 31, 1885, remarks that one of the most
prominent branches of agriculture in Baden is that of tobacco, of which
about 300,000 to 350,000 cwt. annually are grown, whereof large
quantities are exported. Owing to the comparatively high tax on
production of 22½ marks per 50 _kilo._, the grower has been forced to
seek a more rational system of cultivation, and a more careful treatment
of the plant and the curing of the leaf. Government pays particular
attention to this culture. A Commission has been appointed for the
purpose of studying and investigating the treatment of tobacco in
Holland, and the results are to be adopted and propagated, so far as the
climate admits.
It is very doubtful whether the labours of the Commission will greatly
influence the farmers, who are of a very conservative disposition;
moreover, there is a greater obstacle to struggle against, namely, their
desire to increase the quantity of the production, and with it their
income, without regard to the question of deterioration of the quality
of tobacco; the peasantry, like other classes, participates in the
desire to better its material condition.
The surface of land occupied by tobacco plantations represented in 1883
for the whole of the empire the considerable figure of 22,068
_hectares_; this year a reduction is to be noted, as official reports
bring the total to 21,108 _hectares_ only.
The Grand Duchy of Baden participated in the above figures with 7788
_hectares_ for 1883, and 7647 _hectares_ for 1884.
Notwithstanding this difference, the result of the crop will not
essentially be smaller (as regards the weight of the total), the new
produce proving heavier in weight and in substance. While in 1883 the
hectare produced about 1900 _kilo._, it is supposed that for 1884 it
will yield from 1800 to 2000 _kilo._ These figures tend to prove that
the 1884 tobacco is richer in quality, and consequently more durable,
and less capable of treatment than that of the preceding years; although
the quality is somewhat inferior to that of 1882 and 1883 it may fairly
be considered as good.
The subjoined remarks deal with the tobacco trade of Bremen. The number
of casks of Kentucky tobacco sold in 1884 fell considerably below that
disposed of in 1883. This is explainable by the circumstance that lugs
and cuttings were altogether wanting. The prices of leaf on the whole
remained steady, except in October and November, when they soon regained
their firmness through no more supplies from America being expected,
owing to the continued demand for strong tobacco in that country.
Business in Virginia tobacco also suffered from the want of inferior
qualities. Prices, considered high from the beginning, showed even a
rising tendency at the end of the season. Transactions in Maryland and
scrubs exceeded the average of the last five years. Ohio and Bay
suffered, as hitherto, from the protection afforded to home growths.
Operations in stems were, considering the depression in trade, not
unsatisfactory.
A good business was done in almost all descriptions of tobacco in
serons, chests, bales, and baskets, and sales surpassed those of
previous years.
The subjoined table presents a comparison of the transactions in the
various sorts of tobacco during the last two years:—
──────────────────┬──────────────────┬───────────────┬───────────────
Description of │ Description of │ Imports. │ Sales.
Tobacco. │ Packing. │ 1883.│ 1884.│ 1883.│ 1884.
──────────────────┼──────────────────┼───────┼───────┼───────┼───────
Kentucky │Casks │ 20,828│ 12,084│ 20,012│ 12,514
Virginia │ " │ 3,937│ 5,250│ 4,848│ 5,196
Maryland │ " │ 4,929│ 5,615│ 4,579│ 5,811
Scrubs │ " │ 383│ 1,363│ 383│ 1,027
Ohio │ " │ 581│ 1,155│ 566│ 1,174
Bay │ " │ 101│ 136│ 234│ 134
Stems │ " │ 5,013│ 7,332│ 8,163│ 5,403
Havana │Serons │ 16,127│ 15,027│ 13,121│ 11,967
Cuba and Yara │ " │ 22,467│ 22,259│ 29,297│ 17,383
St. Domingo │ " │ 83,836│ 59,665│ 58,121│ 44,065
Seed-leaf │Chests │ 17,070│ 18,723│ 77,000│ 18,203
Porto Rico │Bales │ 1,133│ 300│ 1,137│ 2,210
Esmeralda │ " │ 705│ 549│ 776│ 599
Columbia │Serons and bales │ 11,862│ 21,041│ 14,032│ 22,659
Varnias │Leaves and rolls │ 922│ 2,065│ 3,174│ 2,065
Brazil, in leaves │Bales │131,982│185,061│139,397│189,246
Paraguay │ " │ 2,672│ 2,601│ 2,879│ 2,819
Rio Grande │ " │ 4,571│ ..│ 10,199│ 1,340
Manilla │ " │ 50│ 77│ 21│ 106
Mexican │ " │ ..│ 10│ ..│ 10
Turkish and Greek │ " │ 6,155│ 6,825│ 8,235│ 8,105
Other varieties │ " │ 1,496│ 2,017│ 1,441│ 3,357
──────────────────┴──────────────────┴───────┴───────┴───────┴───────
Good qualities of Havana fetched adequate prices. The demand for Cuba,
Yara, Carmen, and Domingo was brisk; Brazilian and Felix found ready
buyers, owing to the last good crop, the prices rising towards the close
of the year. The stock of Porto Rico was realized at a low figure. In
seed-leaf Pennsylvania plants were chiefly imported, and, being of a
good quality, were for the most part promptly disposed of. Much
inclination was shown for Turkish tobacco, and the same remark applies
to business in Paraguay, of which the supplies might have been greater.
Chinese tobacco, very brisk at first on account of its fine quality,
later on fell off again considerably.
The value of the tobacco consumed in Germany in 1878 is calculated to
have been 353 million marks, or 17,650,000_l._ sterling, the total
return to the revenue being 26,383,966 marks, or 1,319,198_l._ The
quantity consumed in that empire in the year is stated at 2,196,000
cwt., or rather more than 100,000 tons. Of this quantity 582,600 cwt.,
or upwards of 29,000 tons, were consumed in the form of cigars.
Reckoning a hundred cigars to a pound in weight, the number of cigars
consumed in Germany in 1878 would be upwards of seven thousand millions,
which would give two cigars a day all the year round to ten million
smokers. But besides cigars the Germans smoked in the year 1,327,200
cwt., or upwards of 60,000 tons of tobacco more or less manufactured. In
the form of snuff they took 160,600 cwt., or 8000 tons, in the course of
the year, while in the way of chewing-tobacco they limited themselves to
the moderate quantity of 14,200 cwt., or about 700 tons. Rather more
than one-third of the total weight of tobacco consumed was grown within
the limits of Germany, the quantity so produced in 1878 being 596,776
cwt., while the imports amounted to 1,768,855 cwt. of tobacco leaves,
4827 cwt. of roll tobacco, 14,170 cwt. of cigars, 8321 cwt. of stems for
snuffs, 513 cwt. of snuff, and 101 cwt. of chewing-tobacco. The total
area of land engaged in growing the plant in 1878 was 18,016 _hectares_,
or about 44,520 acres. Two-thirds of that quantity was grown in Rhenish
Bavaria, Baden, South Hesse, and Alsace-Lorraine, in which districts
11,623 _hectares_ were employed in the cultivation of the plant.
_Great Britain._—The proposal to re-establish tobacco culture in the
United Kingdom has called for the following sensible article in the
_Planters' Gazette_.
"The question of growing tobacco in the United Kingdom is not so simple
as patriotic Irishmen and enthusiasts of acclimatization might think.
Tobacco has been classed, like tea and coffee, as among those
necessaries of life which could not be grown with any advantage in the
United Kingdom, and might therefore be freely taxed for revenue
purposes. It is, indeed, true that a passable herb may be grown and
called tobacco, in many parts of the United Kingdom, but the fact has
been generally recognized that competition with more tropical countries
is practically fruitless, and therefore to be abandoned. It is easily to
be understood that so aromatic a crop, monopolizing so many of the best
and rarest qualities of the soil, would require high manuring; and that,
just as is the case of any other crop—such as hops, or even wheat—one
could get nothing of the special excellence of the herb required but
what one has previously put into the soil. But, to be profitable, the
plant requires good heat as well as good soil. This, therefore, is the
whole economical question, and upon that the matter mainly hinges. The
claim to grow real tobacco in England or Ireland is based upon the
allegation that the herb can be grown at a profit. The best evidence
furnished to the House of Commons on Monday evening on this point was
that of Lord Harris, who affirmed boldly that Ireland and parts of
England were prepared to enter into a fair competition with the
recognized productive colonies. The Government, and with them, Lord
Iddesleigh, are in favour of an experiment largely granting all that is
asked, and carefully observing the result. Then, when the British
tobacco comes upon the ordinary market, let it be taxed as any other
similar product would be. The Government could not view with anything
but dismay the prospect of a fall in revenue; and there is no question,
therefore, that the home-grown tobacco must pay duty to the full. The
_crux_ of the question is how such duty can be enforced without an army
of revenue officers, whose practical duties would bear no reasonable
proportion to their probable cost. Our own impression is that tobacco
can never be grown in these islands on any large scale to compete with
the growers within the tropics, and that the expense of collecting
revenue would be out of all proportion to the amount collected. At the
same time, it ill becomes us as a Free-trading nation to shut out any
class of our own countrymen, by duties distinctly prohibitive, from
following a branch of agriculture which they think they could make
profitable. It is against our principle to offer a bounty on the forced
cultivation of exotics, such as tobacco undoubtedly is when grown in
these islands, but it would be still worse to maintain, on merely
pedantic grounds, a prohibitive import on a crop which many men think
the smaller tenants could produce to the great advantage of their
holdings. We are by no means sanguine of their success; but that is no
reason why they should not try."
_Greece._—The production of tobacco in Greece is about 4 million _okes_
(of 2¾ lb.) annually. Patras, in 1878, exported 300 tons to Holland,
Austria, and Turkey, at a value of 25–30_l._ a ton. The values of the
exports from Syra, in 1879, were 3503_l._ to Great Britain, 2325_l._ to
Turkey, 88_l._ to the Danubian Principalities, 236_l._ to France,
554_l._ to Austria, 436_l._ to Egypt, 1605_l._ to Russia; and in 1878,
1528_l._ to Turkey, 1875_l._ to Great Britain, 93_l._ to the Danubian
Principalities, 441_l._ to Austria, 334_l._ to France, 266_l._ to
Russia, 39_l._ to Egypt.
In 1884, Nauplia exported 13,000_l._ worth of tobacco; and Calamata,
2400_l._ worth. The value at Patras was 45_s._ per cwt. Syra imported
439_l._ worth of tobacco and 305_l._ worth of tumbeki from Turkey; but
exported 10,459_l._ worth of tobacco to Turkey, 697_l._ worth to Great
Britain, 17,723_l._ worth to Egypt, 200_l._ worth to Russia, 120_l._
worth to Roumania, 2963_l._ worth to Italy, 1176_l._ worth to France,
and 200_l._ worth to Austria.
_Holland._—There were 4117 acres under tobacco in Holland in 1878, which
produced 3,132,875 _kilo._ The imports of tobacco into Holland in 1878
were as follows:—Maryland, 5249, Kentucky, 500, and Virginian, 107
hogsheads; Java, 87,998, seed-leaf, 100, Sumatra, 33,671 packages. In
1876 and 1877, there were 5900 and 3993 packages respectively from Rio
Grande. The exports of leaf from Holland in 1879 were 3,900,000 _kilo._
COMPARATIVE STATEMENT OF THE IMPORTS OF THE VARIOUS KINDS
OF TOBACCO DURING THE FIVE YEARS 1879–83.
───────┬─────────┬─────────┬─────────┬──────────┬─────────┬─────────
│ │Virginia │ │ │ │
│Maryland.│ and │ Java. │Seed-leaf.│ Brazil. │Sumatra.
│ │Kentucky.│ │ │ │
───────┼─────────┼─────────┼─────────┼──────────┼─────────┼─────────
│ Hhds. │ Hhds. │Packages.│ Packages.│Packages.│Packages.
In 1879│ 7,234│ 85│ 102,791│ 192│ 1,548│ 44,477
1880│ 4,775│ 147│ 34,037│ 1,007│ 339│ 52,151
1881│ 2,989│ 151│ 81,225│ 454│ 1,098│ 59,468
1882│ 3,405│ 26│ 103,384│ 905│ Nil.│ 73,444
1883│ 4,240│ 976│ 30,975│ 2,500│ 675│ 10,111
───────┴─────────┴─────────┴─────────┴──────────┴─────────┴─────────
_India._—An immense area is occupied in producing tobacco in India. In
Madras, Dindigul is the great tobacco district, and cheroots are
manufactured at Trichinopoli. The islands in the delta of the Godavari
also yield _lunka_ tobacco, the climate being suitable, and the plants
being raised on rather poor, light soil, highly manured and well
watered. Manilla seeds have been tried on the lower Palnai Hills, but
the Wynaad has proved to be the best locality. In Bombay, the Kaira and
Khandesh tobaccos are superior; altogether over 40,000 acres were under
the crop in this presidency in 1871–2, and the exports were 3 million
lb. Shiraz and Manilla seeds yield good plants in Gujrat and Khandesh.
The total areas under tobacco in 1871–2 were thus returned:—Bengal,
about 300,000 acres; Punjab, over 90,000; Oudh, 69,500; Rungpore, 60.000
(affording the so-called "Burma cheroots"); Central Provinces, 55,000;
Tirhoot, 40,000; Cooch Behar, 24,000; Mysore, 20,000; Dinagepore,
20,000; Purneah, 20,000; Behar, 18,500; Burma, 13,000; Monghyr,
9–10,000; Nuddea, 9–10,000. The best tobacco districts are said to be
Sandoway and the island of Cheduba, in Arracan; Rungpore, in Bengal; and
Bhilsa, in the Central Provinces. The results of many analyses of South
Indian tobaccos show that their ash seldom contains more than 5–6 per
cent. of carbonate of potash, while American range from 20–40 per cent.,
indicating the poverty of the Indian soils in this important ingredient.
It might, however, be supplied at moderate cost in the shape of
saltpetre, which is actually exported largely from the tobacco-growing
districts.
The bulk of the Indian tobacco exported consists of leaf, the kinds
chiefly shipped being the "Bispah" and "Poolah" varieties of the
Rungpore kind; the quantities of cigars and other manufactured tobacco
exported are very small. The exports in lb. for the four years 1875–79
were:—
───────────────┬────────────┬────────────┬────────────┬────────────
│ 1875–76. │ 1876–77. │ 1877–78. │ 1878–79.
───────────────┼────────────┼────────────┼────────────┼────────────
Unmanufactured │ 22,861,711│ 10,508,720│ 10,594,604│ 13,279,158
Manufactured: │ │ │ │
Cigars │ 152,189│ 190,136│ 189,742│ 196,759
Other sorts │ 232,720│ 205,033│ 317,887│ 247,743
│————————————│————————————│————————————│————————————
Total │ 23,246,620│ 10,903,889│ 11,102,233│ 13,723,660
───────────────┴────────────┴────────────┴────────────┴────────────
On the other hand, a considerable quantity of manufactured tobacco,
averaging over 1½ million lb. yearly, is imported, showing that India is
still merely a producer of raw material, and is dependent upon other
countries for the manufactured article in a condition fit for
consumption. Even as regards the raw material, India might do a great
deal more than at present, for there would be a large and constant
demand on the continent of Europe for Indian leaf, if it could be
obtained of somewhat better quality. The French and Italian tobacco
departments are prepared to take Indian tobacco in large quantities, if
it can be supplied of a quality suited to their purposes; and there
would also be an extensive demand from Austria and Germany. Although the
shipments consist mainly of leaf tobacco, and that not of good quality,
tobacco manufacture is now making a promising beginning. In the
enterprise being carried on at Ghazipore, in the North-West Provinces,
and at Poosah, in Bengal, both the cultivation and manufacture are under
the supervision of skilled American growers and curers. Some of this
tobacco sent to the _Administration des Tabacs_ in Paris has been very
favourably reported on. The factory at Ghazipore is now turning out
about 500 lb. a day of all classes, the greater part being black
cavendish and honeydew, for the army. The machinery is capable of
turning out 3500 lb. a day, as soon as sufficient hands have been
trained.
Hitherto no Indian tobacco has realized any valuation approaching that
of American. The average price of the American "shipping tobacco" is
5–6_d._ a lb., higher classes of bright leaf from Virginia realize as
much as 7–13_d._ a lb., while the price of Indian tobacco has generally
been 1–2_d._ a lb. But the 15,000 lb. of Poosah leaf from the 1877 crop
reached England when American shipping leaf was at 4–5_d._ a lb., or 25
per cent. below the normal rate. The consignment was, moreover, packed
in rather damp order, and contained a quantity of moisture which caused
it to be assessed under the highest rate of the new tariff, which
imposes 3_s._ 10_d._ duty when the moisture is over 10 per cent.,
against 3_s._ 6_d._ under 10 per cent. This made a difference in the
value, estimated at 1_d._ a lb. The price obtained was 3¾_d._, which
would have been 4¾_d._ had the tobacco been drier, and the sale has been
followed by orders of large shipments.
The high prices, too, realized for the best samples of the 1876 and 1877
crops, indicate that Indian leaf can be turned out equal to the best
shipping tobacco from America. A tierce of strips from the 1876–77 crop
from Ghazipore sold for 7_d._ a lb., and the greater part of the rest
for 5_d._ or more, while a portion of the Poosah leaf of 1877–78 was
valued at 5_d._ when the market was 25 per cent. below normal rates.
These facts seem to guarantee future success, since the quantity of the
higher classes can be largely increased, and a greater portion of the
crop be brought to the same higher level. The chief point to be
ascertained was whether a sufficiently high level could be attained at
all. It has been attained. The cured leaf of 1878 is very much superior
to any hitherto turned out, especially that from Ghazipore. A new market
is not unlikely to open in France. The French Government have already
asked for a consignment for trial of 1000–1500 lb.
The reason why the manufacture of smoking-tobacco for Indian consumption
has occupied so large a share in the operations is, that the Indian
market, though small, pays far more handsome profits than the English
market.
The price paid for reasonably good American manufactured tobacco in
India ranges from one to three _rupees_ a lb. Ghazipore and Poosah
tobacco is sold at half that price, at a much higher profit than can be
obtained by sending cured leaf to England.
While Indian cured leaf can find a sale in the English market at prices
which will enable it to compete there with American cured leaf, Indian
manufactured leaf is proved to compete successfully with American
manufactured leaf in India itself, with a fair prospect of success in a
similar competition in the colonies. It may be stated in general terms
that 4_d._ a lb. for cured leaf in England, and 6–10 _annas_ for
manufactured leaf in India, will secure sufficient or even handsome
profits. The opening for profits will perhaps be better understood if it
is explained that 1_d._ a lb. represents an asset of about 5_l._ an
acre. The one great advantage which India has over America is cheap
labour. It is now proved that the leaf is, for all practical purposes,
as good as the American leaf, and there is hardly any doubt that America
cannot afford to send home leaf at the price at which India can sell.
The exports of tobacco from British India during the years 1874–5 to
1878–9 have been as follows:—
──────────────────┬──────────┬──────────┬──────────┬──────────┬──────────
│ 1875. │ 1876. │ 1877. │ 1878. │ 1879.
──────────────────┼──────────┼──────────┼──────────┼──────────┼──────────
Unmanufac- } lb.│33,411,504│22,861,711│10,508,720│10,594,604│13,279,158
tured │ │ │ │ │
Manufactured { lb.│ 425,040│ 384,909│ 395,169│ 507,629│ 444,502
No.│ 2,999,940│ ..│ ..│ ..│ ..
──────────────────┴──────────┴──────────┴──────────┴──────────┴──────────
The following letter from the manager of the Poosah tobacco farms,
Tirhoot, describes the system of growing and curing now adopted in
India.
"Preparation of Soil.—Tobacco land should be well-drained upland which
has lain fallow some time or that has had some light crop in it; this
land should be well manured with well-rotted manure. We plough our lands
twice monthly. Just before the time for transplanting the soil is
ploughed up and well pulverized by a henger or beam of wood drawn by
bullocks over the upturned soil so as to bend it and to break up any
lumps of earth. The soil should be sufficiently dry for this purpose so
as not to cake and harden.
"Seed-beds.—These should be made up in a suitable situation, that is,
protected from the afternoon sun, having some building or grove of trees
on the west side. The seed-beds should be raised some six inches off the
ground and have trenches dug all round so as to carry off any
superfluous moisture, the beds should be well worked with a kodalie and
good, rotted manure well worked in. After pulverizing the soil and
levelling it, pick off any stones or other rubbish and it will be ready
for sowing the seed. The size of the bed should be about 4 feet by 15
feet; this is more convenient than square beds, as it enables the plants
to be attended to without risk of destroying them by trampling on them.
"Sowing the Seed.—The seed is sown broadcast with the hand, mixed with
some sand or ashes so as to sow evenly; care should be taken not to sow
too thickly. About one chittak of seed ought to be found sufficient for
one of these beds which would furnish enough plants for one beegah of
land. After having sown and if there is a hot sun, it would be advisable
to cover the beds with light mats. This seed should germinate in seven
or ten days at least. American seed does; Sumatra takes much longer. The
plants may require watering, which should be done with a watering-can
with a rose, when the plants are well up and large. Only water seed-beds
in the evening. As soon as the seedlings have leaves of the size of a
penny, they are capable of bearing transplanting. Before taking up the
seedling to transplant, water the beds well an hour beforehand; this is
done to loosen the earth about the roots so that the plants may be taken
up without injury. To take up the seedlings they should be seized by the
under side of the two largest leaves by the finger and thumb, having one
leaf on each side, not by the stem, then pull up gently, taking care not
to break the leaves. They may then be placed in an open basket. When the
basket is full it should be covered with a cloth if the sun is hot, and
the seedlings slightly sprinkled with water and then carried off to
transplant. The seedlings are planted out in rows 3 feet by 2 feet
apart, for which purpose a knotted cord is used, the knots being 3 feet
apart. This cord is drawn by two men—one at each end. Across the field
or portion of the field at a distance of 2 feet from the outer edge, the
cord is drawn out and then trampled upon by coolies. The knots leave an
impression in the soil where the seedlings have to be planted. The cord
is then raised and put down again at another distance of 2 feet from the
first, and so on till sufficient land has been marked off. This work can
be done during the day, and the transplanting in the evening.
"Transplanting.—Transplanting should be done in the evening if there is
any sun; in cloudy weather it can be done all the day long. Rainy
weather is most suitable as it dispenses with watering and the plants
settle better. A boy takes a basket of seedlings and walks up the row,
dropping a plant here and there where the marks have been made; he is
followed by a man who makes a hole with a _kurpie_, into which he places
a seedling, and then presses the soil around the roots firmly with his
fingers, and then goes on with the rest. As transplanting can hardly be
done here without watering, a boy carrying a can without a rose follows
the man who is transplanting, and waters each plant he comes across;
but, as I mentioned above, if the transplanting could be done in rainy
weather, the watering would be unnecessary. When growing the young
plants require some attention. After the plants have been planted a week
or so, weather permitting, it is advisable to loosen and open the soil
around them with a kurpie, and also to eradicate weeds which may appear.
Later on a kodalie may be used to work the earth between the rows. As
soon as the plants have made growth and begin to throw out flower or
seed-heads, which will take place in about eight weeks or so, they
should be topped, viz. the flower heads should be broken off before they
flower in this way. The stem on which the head was found should be
seized about two to three feet from the ground and snapped clean off by
the hand or fingers. This topping will cause the plant to throw out
heavy leaves. The higher up the stem is broken off, so will the leaves
of the plant become thinner and smaller. We generally leave about ten to
twelve leaves to each plant. After topping, numerous suckers and
offshoots will spring up; these should be promptly broken off as soon as
they appear, as they take a lot of nourishment from the plant. The plant
ripens in about three months. We cut here in January, and none but ripe
plants should be cut.
"How to Cut Ripe Plants.—A tobacco plant is known to be ripe if the leaf
cracks when taken between finger and thumb and pressed, and also when
the leaves present a swollen appearance and have a heavy look. The stem
when cut is full of sap, very thin rind on edge, the leaves are carved
over and look mottled, the ribs of the plant get brittle, and are easily
broken off; when fully ripe, the plant is cut at one stroke close to the
ground. The best instrument to cut the plant with is a kurpie. When cut,
the plant is allowed to hang over on its side and wilt or droop in the
sun. This wilting takes from one to two hours according to the strength
of the sun. When sufficiently wilted (which is known when the plants
look drooping and the ribs can be bent slightly without breaking) the
plants are placed in a cart and taken to the curing-house. Plants should
not be cut in rainy or cloudy weather, as it is obvious the sun would
not be hot enough to wilt were the weather cloudy, and the rain washes
off the gum and thereby decreases the weight of the plant. Plants should
not be cut after the rain unless the gum has returned to the leaves,
which is known by their sticky, gummy feeling."
The results of many analyses of the tobacco of South India show that the
ashes of these tobaccos seldom contain more than 5 or 6 per cent. of
potash carbonate, while the ashes of American tobacco contain from 20 to
40 per cent., proving the poverty of Indian tobacco soils in this
important plant-food—a plant-food, however, easily obtainable in the
shape of saltpetre, and at a moderate cost. But, though saltpetre is
largely exported from the tobacco-growing districts, it is never
employed as a manure for tobacco.
_Italy._—Tobacco is cultivated in Italy in the provinces of Ancona,
Benevento, Terra di Lavoro, Principato Citeriore, Terra d'Otranto,
Umbria, Vicenza, and Sardinia. The area and produce in the following
years were:—in 1870, 9544 acres, 67,192 cwt.; 1872, 12,256 acres, 82,349
cwt.; 1874, 8202 acres, 90,300 cwt. The exports from Naples in 1879 were
2006 _kilo._, value 401_l._
The British Consul at Cagliari reports that the cultivation of tobacco
is only carried on in the district of Sassari, and in the plains of
Sassari, Portotorres, Nurra, Sorso, and Sennori. No positive data on
this branch of industry can be had, it having been exclusively carried
on till 1883 by a private company, called the Regía Cointeressata.
Without fear of being wrong, it may be calculated that the tobacco
cultivators reach the number of 100, who employ during the period of
five months from 600 to 700 labourers; the plantation varies from
4,000,000 to 5,000,000 plants, producing a harvest from 2000 to 2500
_quintals_ of tobacco leaves, at a value of about 125,000 _lire_.
_Japan._—Japanese tobacco is well known in the London market, but it is
often in a soft condition, and then scarcely saleable. More care is
needed in drying it before packing.
_Java._—Tobacco, termed by the natives _tombáku_, or _sáta_, is an
article of very general cultivation in Java, but is only extensively
raised for exportation in the central districts of Kedu and Banyumas. As
it requires a soil of the richest mould, but at the same time not
subject to inundations, these districts hold out peculiar advantages to
the tobacco-planter, not to be found on the low lands. For internal
consumption, small quantities are raised in convenient spots everywhere.
In Kedu, tobacco forms, after rice, by far the most important article of
cultivation, and, in consequence of the fitness of the soil, the plant
grows to the height of 8–10 feet, on lands not previously dressed or
manured, with a luxuriance seldom witnessed in India. Cultivated here
alternately with rice, only one crop of either is obtained within the
year; but after the harvest of the rice, or the gathering of the tobacco
leaves, the land is allowed to remain fallow, till the season again
arrives for preparing it to receive the other. The young plant is not
raised within the district, but procured from the high lands in the
vicinity, principally from the district of Kalibéber, on the <DW72> of
the mountain Diéng or Práhu, where it is raised and sold by the hundred
to the cultivators of the adjoining districts. The transplantation takes
place in June, and the plant is at its full growth in October. The
exports in the year 1877–8 were 212,500 _piculs_ to Holland, and 213 to
Singapore; in 1878–9, they were 248,566 _piculs_ to Holland, and 872 to
Singapore. The value of the export to Holland in 1879 was stated at
1,250,000_l._ The exports in 1884 were 140,351 _piculs_ to Holland, and
2490 to Great Britain.
_New Zealand._—This colony has not yet figured as a tobacco grower, but
the duty on locally produced tobacco is only 1_s._ a lb., and this is
expected to stimulate the home industry.
_Nicaragua._—It appears that the total exports of tobacco were 13,787
lb., value 4830 dollars, in 1883, but only 300 lb., value 240 dollars,
in 1884. At present it is a Government monopoly.
_Paraguay._—Consul Baker, of Buenos Ayres, states that one of the most
valuable crops of Paraguay is tobacco; in 1829, its production amounted
to only 2,675,000 lb., while in 1860, the crop amounted to 15,000,000
lb.; but the war with the allies almost ruined this source of wealth. It
has, however, somewhat recovered its importance, the exports alone last
year amounting to 8,975,000 lb. A large proportion of the crop is
annually worked up into cigars, a branch of industry which is almost
entirely in the hands of the women. The tobacco planted in Paraguay
originally came from Havana, with the exception of a particular kind
which is called in Paraguay, blue tobacco, _peti-hoby_, the origin of
which is unknown. The favourite leaf is a yellow tobacco, _peti-para_,
grown chiefly in Villa Rica, which possesses about 6 per cent. of
nicotine.
_Persia._—The whole of the eastern coast of the Black Sea, i. e.
Mingrelia, Lazistan, Abkhasia, and Circassia, is admirably suited for
tobacco cultivation. The country between Poti and Súkhúm Kalé contains
admirable sites for tobacco plantations, labour for which can be got
from Trebizond. A great demand for tobacco of good quality exists in the
country, and a practical planter should do well. A quantity of coarse,
badly-cured tobacco, of no commercial value, is produced in Imeritia and
Georgia. Great success has attended the culture in Ghilan. The first
seed introduced was from Samsoun; since then Yenija seed has been tried,
and some parcels attained the standard of the best Turkish tobacco. It
can be produced at about 20_s._ a _pood_ (of 36 lb.), giving a profit of
22_s._ a cwt. Hitherto the cultivation has been confined to the plains,
where both soil and atmosphere are damp, but it might be worth trying
the hill-skirts. About 2000 cwt. were produced in 1878. The exports of
tobacco, the produce of Ghilan, from Resht to Russia, were valued at
4615_l._ in 1878, and 6154_l._ in 1879. The values (in rupees) of the
exports in 1879 were 13,000 from Bushire, 73,500 from Lingah, and 35,000
from Bahrein.
At the time when I wrote the article on tobacco in Spons' Encyclopædia,
the true source and history of an article called "tumbeki" was still in
doubt. From researches made at the instigation of my friend E. Morell
Holmes, F.L.S., the Curator of the Pharmaceutical Society's Museum, it
is now clear that it is a Persian tobacco, and as such calls for mention
here. The following paragraph reproduces what I said on the subject in
Spons' Encyclopædia.
"Tumbeki.—This word, under a multitude of forms, is the common name in
several Eastern languages (Bengali, Hindustani, Telugu, Sunda, Javanese,
Malayan, Persian, Guzerati, Deccan) for ordinary tobacco. But in Asia
Minor, it is applied to a narcotic leaf which is spoken of as distinct
from tobacco, and is separately classified in the Consular Returns.
Botanical authorities are at variance as to the plant which affords it,
some attributing it to a _Lobelia_, while others consider it a kind of
tobacco. The latter appears to be the more correct supposition. The
flower resembles the tobacco in being trumpet-shaped; the leaf is
broader, larger, and rounder than that of the tobacco raised in Turkey,
and is also wrinkled like the inner leaf of the cabbage. The plant is
raised from seed in nurseries, and when it has 4 or 5 leaves, is planted
out in April in the prepared field, and watered sparingly. It is 'set'
in a day or two, and is then hoed occasionally to free it from weeds.
After inflorescence, and when the plant is sufficiently 'cooked,' it is
cut down, or pulled up bodily, and re-set in the ground till the leaves
are wilted. These leaves are dried, and, after exposure to the dew, are
pressed heavily, when they undergo a kind of fermentation which develops
the aroma. It is exceedingly narcotic: so much so, that it is usually
steeped in water before use, and placed in the pipe (a _narghilé_ or
water-pipe) while still wet. The exports of this article (the produce of
Persia) from the port of Trebizonde are considerable:—In 1877, they were
13,342 bales (of 1¾ cwt.), value 106,736_l._, to Turkey; in 1878, 11,571
bales, 92,568_l._, to Turkey; in 1879, 9659 bales, 77,272_l._, to
Turkey, and 866 bales, 6928_l._, to Greece. Aleppo, in 1878, sent 4
tons, value 320_l._, to Turkey, and 11 tons, 880_l._, to Egypt. The
exports of the article, the produce of the interior of Persia, from
Resht to Russia, were valued at 5000_l._ in 1877, and 3846_l._ in 1878."
It will be interesting to compare this with Holmes' paper read before
the Pharmaceutical Society on February 10, 1886:—
"Tumbeki is the name under which an article of regular commerce between
Persia and Turkey is mentioned in the consular reports, especially in
that for Trebizonde.
"Two or three years ago an inquiry was made at this institution
concerning the nature and botanical source of umbeki, and the only
information I was then able to give was that in the 'Treasury of Botany'
tumbeky is stated to be the narcotic leaf of a species of lobelia.
"From its frequent occurrence in the Blue Books in the same list with
tobacco, and from the large quantities mentioned as an export from
Trebizonde, my correspondent suggested that it was probably something
used for smoking like tobacco. In the hope that tumbeki might prove to
be some drug possessing important narcotic or possible medicinal
properties, I wrote to Mr. A. Biliotti, Consul at Trebizonde, for
information. In reply, he forwarded samples of tumbeki of different
growths and qualities. This proved on examination to be unquestionably
some kind of tobacco, and being puzzled to know why it figured in the
Blue Books as a distinct article, I asked Mr. Thomas Christy, F.L.S., to
make inquiries for me in Persia. He received the following note through
Mr. Zanni, the well-known chemist at Constantinople, from whom I
received the following information:—
"'There are three qualities of the teymbeki, all derived from the
_Nicotiana persica_.
"'1. Shiraz teymbeki, valued at twenty gold piastres per oke.[A]
"'2. Kechan teymbeki, valued at ten gold piastres.
"'3. Teheran teymbeki, equal in value to No. 2.
"'The Shiraz is the best quality, the leaves are four decimetres long
and half a decimetre wide. The leaves of the two other qualities are not
so large. The quantity of alkaloid in the leaves of teymbeki is more
than in the leaves of _Nicotiana Tabacum_; it is much used in
Constantinople, but more so in Egypt, Syria, and particularly in Persia.
Teymbeki is smoked in a special apparatus known as the narghileh.[B] The
apparatus is found in every coffee-house and even in a great number of
private houses. It resembles somewhat the wash bottle used in
laboratories for washing filters with distilled water, but is often made
of metal. The teymbeki is placed in a small reservoir on the top of the
flask and burns in contact with a piece of incandescent charcoal. The
vapour is drawn through the tube, which passes to the bottom of the
water and collects above it, whence it is inhaled through the longer
tube.[C] It is in fact a water-pipe.'
"Having ascertained then that tumbeki was a species of tobacco, I sought
for further confirmation of the statement that it is the produce of _N.
persica_, and wrote on the subject to Professor Hausknecht, who is well
known as one of the best authorities on the botany of Persia. He kindly
replied as follows:—
"'Tumbeki is the produce of _Nicotiana rustica_, and is almost
exclusively used for the water-pipes called kalian or narghileh. The
plant is cultivated throughout the whole of Persia, especially in
Ispahan and Shiraz, whence the best kind comes.'
"But the statement of M. Zanni that tumbeki contains more alkaloid than
tobacco, and that of Professor Hausknecht that tumbeki is the produce of
_N. rustica_, seemed to conflict with the statements in books that _N.
rustica_ is less active than _N. Tabacum_.
"In the 'Commercial Report,' No. 25, 1883, p. 1056, under 'Smyrna,'
Consul Dennis confirms M. Zanni's statement concerning tumbeki. He
says:—'It is much stronger than ordinary tobacco, and cannot be smoked
in the usual way, therefore it is exclusively used for the narghili.' He
also adds that a large quantity is consumed in the district of Smyrna,
but much is also re-exported to Egypt and other parts of Turkey. It is
imported from Persia, both through Trebizonde and Bushire on the Persian
Gulf.
"Mr. J. B. Fraser, in his work on Persia (1826), remarks, 'The tobacco
smoked in the kalian is called tumbaku in distinction to tootoon, or
that smoked in pipes or cigarettes. It is sold in the leaf, which is
packed dry in layers, and is preserved in bags sewn up in raw hide. It
improves by age, but is quite unsmokable the first year. The best comes
from Jaroum, south of Shiraz.'
"In an interesting article in 'Harper's Magazine' (January 1886, p. 224)
on the 'Domestic and Court Customs of Persia,' the writer remarks
concerning tumbeki:—'The kaliân or water pipe differs from the Turkish
narghileh by having a short straight stem. In it is smoked the tobacco
called tumbakee—a species grown only in Persia. That of Shiraz is very
delicate in flavour and is the best. The tumbakee must be first soaked
in water and squeezed like a sponge or it will cause vertigo. A live
coal, made from the root of the vine, is placed on the tobacco, and the
smoke is drawn through the water with a gentle inhaling, depositing the
oil in its passage through the water.'
"In De Candolle's 'Prodromus,' vol. xii., pt. 1, p. 567, it is stated
under _Nicotiana persica_, that it yields the celebrated tobacco of
Shiraz. This species closely resembles _N. Tabacum_ in the form of its
leaves, which are, however, rather acute than acuminate; but the flowers
are different both in shape and colour. In _N. Tabacum_ the stem leaves
are sessile, and the corolla is funnel-shaped or inflated below the
limb, and is of a pinkish-red colour; in _N. persica_, the tube of the
corolla is club-shaped and the limb more spreading; the colour is white
inside and greenish outside. When in blossom, therefore, the two plants
are easily distinguished. _N. rustica_, on the other hand, has _stalked_
cordate leaves and a short yellowish corolla, with the tube and limb
both short.
"The leaves of tumbeki which I have received from Trebizonde and
Constantinople both correspond with _N. persica_ in character, but not
with _N. rustica_, since they have no trace of a petiole. So far as it
is possible to ascertain therefore, in the absence of flowers, the
weight of evidence is in favour of tumbeki being the produce of _N.
persica_. In order to ascertain the correctness of the statement that
tumbeki is stronger than tobacco, I handed some specimen to Messrs. E.
J. Eastes and W. H. Ince for chemical examination, which they kindly
undertook at my request."
-----
Footnote A:
The oke equals ten kilogrames; a piastre, 2½_d._
Footnote B:
So called from its resemblance in shape to a _narghil_ or coconut.
Footnote C:
A full and interesting account of the forms and uses of the varieties
of the kalian and narghileh is given in the 'Land of the Lion and the
Sun,' p. 29.
-----
Following is the report of these gentlemen on the chemistry of the
subject:—
"Four samples of tumbeki were brought under our notice by Mr. Holmes,
Curator of the Museum of the Pharmaceutical Society, being of interest
on account of their reported greater strength in nicotine as compared
with tobacco. The following are the results of our investigations. We
may state that so far as we have been able to ascertain no previous
researches have been undertaken on the subject.
"Preliminary Examination.—The presence of an alkaloid was demonstrated
on the addition of the usual reagents to the acid infusion.
"Isolation of Alkaloid for Physical Examination.—The powdered tumbeki
was placed in a retort with milk of lime and steam passed through it
till the distillate was no longer alkaline. Alkaloid in abundance was
found in the distillate, which had a distinct odour of nicotine. The
distillate was then extracted with ether, and the ether slowly driven
off. The residue obtained was a light straw oily liquid of
powerful odour, giving off irritating fumes when heated.
"Estimation of Nicotine.—In the estimation of nicotine much difficulty
was experienced, owing to imperfect knowledge of the alkaloid, and to
the imperfect methods recommended in various papers on the subject. The
only method we found reliable was by using a standard solution of
Mayer's reagent, obtained by mixing 13·546 grams of mercuric chloride in
solution with 49·8 grams of potassic iodide, in solution, and adding
water to make 1 litre.[D] One c.c. of this solution represents ·003945
grams of nicotine, the precipitate having the formula C₁₀H₁₆N₂I₂.HgI₂.
"The method we adopted of working with this solution was as follows:—One
or more grams of dried and powdered tumbeki were treated with diluted
sulphuric acid (2·5 per cent.) for several hours on a water-bath,
filtered, and the leaves washed with hot 1 per cent. acid till the
filtrate was colourless.
"The filtrate was then either evaporated to a low bulk and extracted
with alcohol, to get rid of albuminous matters which interfered with the
reaction, or neutralized with sodic hydrate and the alkaloid extracted
with chloroform, the chloroformic solution being shaken with diluted
sulphuric acid as in the ordinary methods of alkaloid extraction.
"The objection to the first method is that the alcohol has to be driven
off before the Mayer's reagent can be added, which is troublesome and
lengthens the process.
"The solution of the alkaloid in excess of sulphuric acid having been
obtained, Mayer's reagent was carefully added till no more precipitation
was observed, the end of the reaction being ascertained when on
filtering some of the nicotine solution into a watch-glass and adding a
drop of the reagent, no precipitate was formed. With careful
manipulation concordant results were obtained.
"Other methods tried were as follows:
"Volumetric method.—Ten or more grams of powdered tumbeki were distilled
with a solution of sodic or potassic hydrate, the distillate being
passed into a known volume of decinormal standard solution of sulphuric
acid, and the amount of acid neutralized by the nicotine was determined
by a standard decinormal solution of soda and the nicotine calculated.
"By this method the results obtained were invariably too high owing to
an appreciable quantity of ammonium salts contained in the leaves. Dr.
Kissling[E] has also noticed the high percentages obtained by this
method of estimating nicotine.
"Kosutány treats the leaves with milk of lime till all the ammonia is
driven off, and then extracts with water; shakes the aqueous solution
with petroleum ether and proceeds as before.
"This method was not found to give good results, for though the ammonium
salts do not interfere with the reaction, yet the petroleum ether does
not extract the whole of the alkaloid, and thus a low percentage is
obtained.
"Extraction by Ammoniacal Ether.—This consists in extracting the
powdered leaves in an upright extractor, by an ethereal solution of
ammonia, and either driving off the ether and weighing the residue as
nicotine; or volumetrically estimating the residue by decinormal
solution of sulphuric acid, or precipitating the alkaloid by platinum
perchloride. In either case, whichever way the residue is estimated, the
results are too high, owing to the difficulty of entirely getting rid of
the ammonia.
"The following are the percentages of nicotine in the tumbeki:—
'_Ispahan._'—I. By Mayer's Reagent.
A. (midrib) 8·156 per cent.
B. (leaf) 5·508 " "
C. (leaf and midrib) 5·589 " "
D. (leaf) 5·3865 " "
——————
5·4945 per cent. average.
II. By Volumetric Method.
By working on 10 grams = 7·2 per cent.
By working on 50 grams = 7·228 " "
'_Hidjaz._'—I. By Mayer's Reagent.
A. (leaf and midrib) 2·025 per cent.
B. (leaf and midrib) 2·268 " "
C. (leaf and midrib) 2·028 " "
D. (leaf and midrib) 1·863 " "
—————
2·046 per cent. average.
II. By Volumetric Process.
A. 2·37 per cent.
III. By Ethereal Solution of Ammonia.
3·6 per cent.
'_Kechan._'—By Mayer's Solution.
A. (leaf and midrib) 2·835 per cent.
B. (leaf and midrib) 3·0375 " "
C. (leaf and midrib) 2·85525 " "
——————
2·90925 per cent. average.
'_Shiraz._'—By Mayer's Solution.
A. (leaf and midrib) 5·8725 per cent.
B. (leaf and midrib) 5·7975 " "
——————
5·835 per cent. average.
-----
Footnote D:
Dragendorff, 'Chemische Werthbestimmung starkwirkender Droguen,' § 63,
p. 52 _et seq._
Footnote E:
The 'Analyst,' January 1886, p. 16; 'Chem. Zeit.,' ix., 1886.
-----
"Estimation of Saccharoid Matter; calculated as cane sugar.—The
fermentation process was the one adopted, not that we consider it by any
means a good one, but because it was the only one practicable. Fehling's
solution was inadmissible, owing to the precipitation of colouring and
other matters, and the polariscope gave no indication. The objections to
the fermentation process are due to the small amount of alcohol produced
in the relatively large bulk of liquid. This renders the solution liable
to acetification, and the ultimate distillate obtained is very weak in
spirit, making it extremely difficult to obtain the correct specific
gravity; the specific gravities obtained were always between ·998 and
unity.
"We worked as follows:—200 grains of dried tumbeki were exhausted by
repeated infusion in boiling water. The filtered liquid when cool was
mixed with 100 grains of German yeast and allowed to stand three days in
a warm place to ferment.
"About one-third was then distilled, the distillate being redistilled
and three successive fractions of 500 fluid grains collected, the
alcohol in each being estimated; the third portion contained little if
any spirit.
"It being stated that basic acetate of lead removes saccharoid matter
from the kindred plant tobacco; we tried its action on the infusion of
tumbeki.
"At the onset it was found impossible to thoroughly wash the bulky
precipitate caused by the lead; so, to ensure a definite result,
sufficient basic acetate of lead was added to the infusion of 200 grains
of tumbeki and the whole made up to 30 fluid ounces with distilled water
and well mixed. An aliquot part (20 fluid ounces) was then filtered off,
excess of lead removed by sulphuretted hydrogen, the sulphide filtered
out, the solution boiled to drive off the sulphuretted hydrogen and the
infusion, when cool, was fermented in the usual way. But acetic acid was
necessarily present from the decomposition of the lead salt by the
sulphuretted hydrogen, and this on distilling would tend to raise the
specific gravity. To remedy this, slaked lime, or preferably potassic
hydrate, was added before redistilling, but considering that from one to
three per cent. of ammoniacal salt is contained in the original tumbeki,
it is probable that some might still remain and by the action of the
fixed alkali furnish a trace of free ammonia which would lower the
specific gravity, and thus apparently raise the percentage of alcohol.
As far as we can judge basic acetate of lead does not seem to remove
fermentable matter from infusion of tumbeki.
────────┬──────────────────┬──────────────────
│ I. │ II.
────────┼─────┬────────────┼─────┬────────────
│ │ Pb │ │ Pb
│ │ treatment. │ │ treatment.
Ispahan │2·64 │ 2·67 │ — │ 2·35
Hidjaz │3·00 │ 2·8 │ 2·7 │ —
Kechan │5·58 │ 5·33 │ — │ —
Shiraz │3·48 │ 3·88 │3·23 │ 3·1
────────┴─────┴────────────┴─────┴────────────
"Ash.—The following bases and acids were uniformly found in the
ashes:—Sodium, potassium, lithium, magnesium, calcium, iron, aluminium,
silica, chlorine, phosphoric acid, sulphuric acid, carbonic acid.
GENERAL TABLE OF RESULTS.
─────────────────────────┬─────────┬─────────┬─────────┬─────────
│Ispahan. │Hidjaz. │Kechan. │Shiraz.
─────────────────────────┼─────────┼─────────┼─────────┼─────────
Nicotine │ 5·4945 │ 2·046 │ 2·909 │ 5·835
Saccharoid matter │ 2·64 │ 2·85 │ 5·58 │ 3·355
Saccharoid matter after │ │ │ │
Pb treatment │ 2·51 │ 2·80 │ 5·33 │ 3·49
Soluble in water │42·0 │42·3 │39·9 │55·6
Insoluble in water │58·0 │57·7 │60·1 │44·4
Ash │22·0 │28·5 │28·5 │26·15
─────────────────────────┴─────────┴─────────┴─────────┴─────────
"The foregoing work has been carried out in the laboratories of the
Pharmaceutical Society."
_Philippines._—The soil and climate of the Philippines are eminently
suited to tobacco culture; but the unjust Spanish monopoly <DW36>s the
industry, and it is declining. Next to the Cuban (Vuelta abajo) and a
few prime Turkish sorts, Manilla tobacco is admitted to be the best.
Most of the Philippines produce it. According to the quality of the
produce, the provinces rank as follows:—(1) Cayagan and Ysabel, (2)
Ygorrotes, (3) Island of Mindanáo, (4) Bisayas, (5) New Ecija. On the
average, over 400 million cigars, and a quantity of tobacco sufficient
to bring up the total weight to 56,000 cwt., are annually exported. The
advantage of the plantations in Cayagan lies in the annual deposit of
alluvial matters by the overflowing of the large streams. The
cultivation in Bisayas promises to become extinct, whereas if the
natives were free to sell in the best market, the industry would
increase immensely. The yield of the Cebu district in 1878 was 8780
_quintals_, the whole of which went to the cigar factories of Cadiz and
Alicante. The exports from Manilla were:—in 1877 17,526,700 lb. tobacco,
value 525,801_l._; 87,007,000 cigars, value 243,619_l._; 1878,
15,630,400 lb. tobacco, value 468,918_l._; 136,835,000 cigars, value
383,136_l._; 1879, 9971 _quintals_ (of 101½ lb.) tobacco leaf to Great
Britain, and 74,490 _quintals_ to Spain; cigars, 10,571,000 to Great
Britain, 6,557,000 to Australia, 44,586,000 to the Straits Settlements
and India, 25,861,000 to China and Japan, 693,000 to the United States,
100,000 to California, 1,521,000 to Spain and the Continent; the total
values amounted to 480,263_l._ The exports of tobacco from Yloilo were
25,454 _piculs_ (of 133⅓ lb.) in 1878, and 20,600 _quintals_ (of 101½
lb.) in 1879, all to Spain.
_Roumania._—Tobacco was extensively cultivated at one time, with
success, near Macin and in other parts; but the monopoly has greatly
affected the condition of the industry.
_Russia._—As regards the production of tobacco, Russia ranks second
among continental countries, but the consumption is less per head than
in other lands. Consul Stanton says that smoking began in the latter
part of the sixteenth century, and the habit steadily increased,
notwithstanding the fact that it was punished by the knout, slitting of
the nostrils, and banishment to Siberia. It is most extensively
cultivated in Tshernigoff, Poltava, Bessarabia, and Samara. In Poland,
the production is not large, and is mainly confined to the vicinity of
Warsaw. It is chiefly cultivated by the peasants and is often their only
occupation.
In 1883, Riga exported 70,722 _pouds_ of leaf tobacco, valued at 194,486
_rubles_. Sevastopol shipped 59 _pouds_, value 1100 _rubles_. Tobacco is
now cultivated largely in all parts of the Crimea, and is likely to
become an export of considerable importance. In Taganrof plantations are
on the increase, and the culture promises well.
_San Salvador._—The exports of tobacco in 1884 were 16,113 dollars'
worth of leaf, 5898 dollars' worth of manufactured, and 826 dollars'
worth of other sorts.
_Servia._—It is estimated that there are 4000 acres under tobacco
culture in Servia.
_Spain._—The port of Cadiz is a great centre of the tobacco industry.
The imports here in 1878 were:—123 _kilo._ from Germany, 304,538 _kilo._
from the United States, and 6,776,900 _kilo._ from Spanish colonies; the
exports were 15,600 _kilo._ to Germany, and 213,846 _kilo._ to France.
Corunna exported 58,280 _kilo._, value 87,420 _pesetas_, in 1884. Cadiz
exported 514,817 _kilo._, value 2,574,085 _pesetas_, in the same year.
_Sumatra._—This great island is assuming a first-rate importance in the
tobacco industry.
The year 1883 was an exceptionally favourable one, as the harvest in
Sumatra was very good, while prices for Java tobacco were higher than of
late years, in consequence of the short harvest of 1882.
Large quantities of Sumatra tobacco found buyers in the United States,
in consequence of the protectionist measure introduced in that country
in favour of the home tobacco producers. The duty was raised from 35 c.
to 75 c. per lb. on and after the 1st July, 1883, and great efforts were
made to import as much as possible at the lower duty before that date.
The principal owners of the plantations are Dutchmen, and the labour
employed is Chinese coolies, brought to the island principally from the
Malaya peninsula. The crop, according to one of these successful
planters, is scarcely ever reared two years in succession on the same
lands. The jungle is first cleared, and then the seed planted. After the
first crop of tobacco is gathered, it is the next season used for rice,
or something else, and tobacco is not planted again until the sixth or
seventh year after the jungle is cleared. By adopting this method, a
better result is obtained.
The drying-house is thus described by a recent visitor to the island:—
"The interior is very much like a rick-yard, with tobacco stalks instead
of hay-ricks, among which a perfect army of half-clad Chinese coolies,
400 strong, are hard at work sorting, ranging and stowing. So
overpoweringly strong is the scent of the half-dried tobacco leaves that
a smoker would have nothing to do but to take in an empty pipe with him
and enjoy a good hard smoke gratis, merely by inhaling the air through
it. But the Chinamen, whether habituated to it by long use, or fortified
against it by the superior power of opium, breathe this perfumed
atmosphere as easily as if it were the purest air of the sea. 'That is
how we measure the heat, you see,' says our host, calling our attention
to the hollow bamboos thrust through the heart of each stack, with a
stick inside it, which, when pulled out, is almost too hot to touch. 'It
must never be above or below a certain point, you know. Instead of
stripping off the leaves at once, we hang up the whole plant to dry, and
do not strip it till it is quite dried. The Sumatra tobacco, however,
will not do for cigars. It is only used for what we call the 'deckblatt'
(cover leaf), which covers the outside of the cigar.'"
Consul Kennedy reports that "the main cause of the prosperity in Deli is
the tobacco, the first crop of which was shipped in 1869.
"The crop for 1884 will turn out about 122,000 bales, valued at
2,080,000_l._
"The accompanying table shows the export during the last 11 years:—
Year. Bales. Value.
──────────────────────────
£
1873 9,238 208,333
1874 12,811 250,000
1875 15,147 291,666
1876 28,947 520,833
1877 36,167 541,666
1878 48,155 750,000
1879 57,544 875,000
1880 64,965 937,500
1881 82,356 1,187,500
1882 102,032 1,750,000
1883 92,000 1,583,333
[Estimated.]
──────────────────────────
NOTE.—One bale equals 176
English lb.
"Prices for Deli tobacco have ruled on the whole fairly high, the
special quality of the leaf lying in the fact of its being light and
elastic in texture, with thin fibres, so that it is admirably adapted to
serve as cover-leaf, and as such is a good substitute for Havana
tobacco. As a smoking-tobacco it lacks flavour. There is a pretty
general concurrence of opinion that the seed of the Deli tobacco was
indigenous, and obtained from Batak tribes in the interior; and although
many experiments have been made with seeds from Java, Manilla, and other
places, the planters have invariably come back to the original seed,
finding that the new kinds develop a coarseness of leaf attributed to
the extraordinary richness of the virgin soil, a soil partly alluvial
and partly volcanic, but covered throughout with dense forests.
"The tobacco estates consist of grants of land taken out by individuals
or companies, and are as a rule of such an extent that every year a new
district can be cleared and used for the coming crop, and this state of
things will continue for many years to come; indeed, hitherto only a
small portion of the ground cultivated (not one-fifth) has borne two
crops, although it is expected that, unless fresh ground is taken up by
the planters, a time will arrive when use must be made of old fallow
lands, and then guano will be required.
"The planters consist of three or four large companies, principally
Dutch—such as the Deli Company, the Amsterdam Deli, and the Batavia
Deli—as well as of individual planters of many nationalities, Germany
and Switzerland being strongly represented, while there are also a good
sprinkling of Englishmen, the principal English firm being the Langkat
Plantations Company, with its headquarters in London.
"The grants of land are taken direct from the chiefs before mentioned,
and are only valid after confirmation at Bengkalis. The term is for 75
years, and for such a grant a sum of money, by way of premium, amounting
to from 1 dol. to 2 dol. per bouw (equal to an acre and two-thirds), is
paid in cash, while an annual rent of 40 c. a bouw, payable at the
expiration of the fifth year, is also reserved. Such at least are the
terms of the last recognised agreements. The whole of the
conveniently-situated land in the three districts before-mentioned has
now been taken up, and it is only in the outlying regions that fresh
ground can be obtained; but as in such outlying regions settled
government is not so well established, the Dutch authorities are now
very chary in confirming grants in places where the tobacco-growing
community would be less under control.
"It is estimated that at least 2,000,000_l._ sterling is now invested in
the tobacco industry in the Deli districts.
"The tobacco when ready for shipment is all sent to Clambia on the L
angle at river, to the Deli river, or the Sirdang river (as the case may
be), and is despatched thence viâ Penang or Singapore to Amsterdam,
which is the tobacco mart for the continent of Europe. The United States
have also bought the Deli tobacco in the Amsterdam market in late years.
Very little of the tobacco goes to England. The leaf remains so moist
that the English import duty would press it heavily in comparison with
other tobaccos, and this circumstance operates as a check on the import
of tobacco from Sumatra into England as compared with tobacco from Java.
The principal purchasers are German manufacturers and Dutch middlemen.
The latter retail the tobacco over the continent, and supply the several
Régies, amongst others the Austrian, Italian, and French. The Americans
confine their purchases to dark-leaved, heavy tobacco, requiring 100
leaves or less to the lb.
"It is worth remarking that the whole of the carrying trade in
connection with the Deli tobaccos is in the hands of Messrs. Holt's
line, the rate of freight from Deli to Amsterdam being about 3_l._ 2_s._
6_d._ per ton. The shipping season may be said to last from January to
June.
"The tobacco crop of 1884 is estimated to yield about 20,000 bales in
excess of that of 1883, but the crop in 1883 was a short one owing to
unfavourable weather. The 1884 crop is the best one ever obtained, both
as regards quantity and quality. Roughly speaking, the Deli tobacco in
the Amsterdam market fetches 1_s._ 4_d._ per lb. English, and the
profits realized may be judged from the dividends given by the most
flourishing companies; the shares of the Deli Company being now quoted
at 500 per cent. premium. Of course there are exceptions where
unsuitable soils have been met with, and losses have been sustained of
no inconsiderable amount. These losses have occurred principally on
Sirdang lands, where the tobacco grown is reputed not equal to that
produced in the other two districts. This comparative defect is
disclosed in the burning, the Sirdang tobacco yielding a brown instead
of a white ash, and being probably therefore lacking in potash.
"The forests when cleared for the tobacco plantations afford splendid
timber, and this is utilized for constructing drying-sheds and coolies'
quarters, but a good deal of the wood which might be exported for
building or fuel is wasted for want of conveyance and burnt on the
ground. As a compensation there can be no doubt that this burnt timber,
or rather the ashes of it, supply an excellent manure.
"The labour employed may be distributed under three classes. There are,
firstly, Malays and Batak tribesmen, who fell heavy timber, do general
clearance, and build sheds; then come the Klings from the Madras
districts, who occupy themselves with drainage and road-making; and
lastly, we have the Chinese for planting, sorting, and preparation of
the weed. The planting is conducted on a co-operative system. Coolies
have their fields allotted to them, and plant at their own risk under
supervision. Their payment depends on the yield. Reckoning from the
estimated out-turn of last year's crop, and that one coolie will raise
seven piculs of tobacco in the season, we arrive at the figure 23,000 as
representing the total number of Chinese engaged at Deli in tobacco
cultivation, to which number 7000 extra hands must be added, employed in
pursuits incidental to the industry. 3000 additional Chinese coolies are
reported to have been engaged for the coming year. The strength of the
Kling community may be taken at about 3000. The Chinamen go into their
clearings and begin work during January and February: those not actually
in service on the tobacco estates earning money as shopkeepers, pedlars,
or gardeners, many of the latter being old hands who, under advances,
have taken to planting patches of tobacco on their own account, for
which they find a ready sale in Penang. The Klings are also to be met
with as drivers of carts and carriages.
"An industrious coolie would, on an average, net in the course of a year
100 to 150 Dutch florins, and on this sum he pays to the Dutch
Government 2 per cent. by way of income tax. The coolie, however,
arrives in the country with a debt of from 100 fl. to 150 fl., and thus
as a rule is not clear and able to leave with a balance in hand till the
end of the second year. The coolie is engaged for a year, but he
generally re-engages, and takes his departure in the beginning of the
third year.
"The Dutch Government regulations with regard to the maintenance of a
medical man by every estate and to the erection of hospitals for sick
coolies are stringent; and, on the whole, the coolie-lines, considering
their temporary nature, are adequate, so that the lot of the coolie in
Deli may be regarded as a favourable one, even when compared with places
where he is under British control.
"The importing of British Indians, as is well known, is not tolerated,
though many have found their way into the country under the stimulus of
high wages, the latter running from 7 dol. to 10 dol. a month, according
to capacity."
The following report by Consul Eckstein on the export of Sumatran
tobacco to the United States, and Dutch dealings in the same in 1882
will be of interest.
Consul Eckstein says "it is not quite three years since a few dealers in
tobacco and manufacturers of cigars in the United States had first their
attention attracted to Sumatra tobacco, with a view of introducing and
using it for cigar-wrappers.
"From this port shipments of the article began to be made during the
latter half of the year 1880, and, considering that this trade has only
so recently taken its rise, and that by this time it has already assumed
rather important proportions, I felt called upon to prepare the present
report, giving some information concerning the same.
"In order to show, as nearly correct as possible, the course this trade
has taken from its commencement to the present time, I made up the
following statement, which exhibits the quantity and value of such
tobacco shipped from Amsterdam to the United States during each quarter
since such shipments first began to be made, viz.:—
──────────────────────────────┬────────────┬────────────
Quarters ending— │Quantities. │ Value.
──────────────────────────────┼────────────┼────────────
│ Bales. │ $
September 30, 1880 │ 311│ 37,694
December 31, 1880 │ 454│ 52,113
│ ————————│ ————————
Total │ 765│ 89,807
│ ————————│ ————————
│ │
March 31, 1881 │ None.│ None.
June 30, 1881 │ 558│ 56,958
September 30, 1881 │ 1,162│ 128,474
December 31, 1881 │ 1,059│ 114,758
│ ————————│ ————————
Total │ 2,779│ 300,190
│ ————————│ ————————
│ │
March 31, 1882 │ 496│ 52,203
June 30, 1882 │ 1,464│ 140,184
September 30, 1882 │ 2,245│ 254,372
December 31, 1882 │ 2,785│ 333,254
│ ————————│ ————————
Total │ 6,990│ 780,013
──────────────────────────────┴────────────┴────────────
"From this statement it will be observed that the export of the article
to the United States is constantly and very largely increasing; and when
it is further taken into account that certain quantities of it were
invoiced and shipped from Rotterdam and Bremen as well, it may safely be
stated that about 9000 bales of Sumatra tobacco entered our markets in
1882.
"What has created, increased, and what sustains this trade appears to
be:
"1st. That certain qualities of Sumatra tobacco in certain dark colours
have been found to be peculiarly and advantageously adaptable for
cigar-wrappers, and are gaining more and more in favour with
manufacturers of cigars in the United States; and
"2nd. The ever-increasing crops of the article, thus also increasing the
supply of the particular sorts especially suitable for the American
market.
"The recent animation in this trade has undoubtedly furthermore been
stimulated by the removal of the 10 per cent. discriminating duty,
formerly payable thereon, being a product of the East Indies, exported
from the west of the Cape of Good Hope.
"This will be clearly evident when I state that many shipments,
aggregating large quantities of this tobacco, purchased or ordered for
months last past, were purposely delayed until late in December, so as
not to arrive until after the law abolishing the discriminating duty had
gone into effect.
"This unlooked-for introduction and now so considerable export of this
staple into the United States has begun to be viewed with great
disfavour by cultivators or growers of 'seed-leaf' tobacco in the United
States.
"They apprehend, as I am informed, that the imports of Sumatra tobacco
into our country will increase still further in the near future, and
seem to consider this would prove greatly detrimental to their
interests.
"I am hardly in position or prepared to express an opinion as to how
well grounded or justified their fears really are, and, moreover, am
inclined to believe that the interested parties are the better judges of
this matter, but so far as I can possibly make myself serviceable by
giving information which may assist them in reaching correct conclusions
on the subject I deem it my duty to do, and do cheerfully.
"Such information may possibly also be of some value to Congress in its
present consideration of our tariff when the article of 'leaf-tobacco'
is reached.
"Thus I would report that up to the present the production of the
article has increased from year to year without any intermission from
the beginning of its cultivation in Sumatra in 1865, when it amounted to
only 189 bales.
"In this connection I would respectfully refer and call attention to my
report on 'The tobacco trade of the Netherlands in 1881,' dated March 7,
1882, and printed in the volume of monthly consular commercial reports
No. 18, of April last, as it contains a statement showing the crops of
Sumatra tobacco each year from 1865 to 1880, inclusive, and the average
prices realized from its sale.
"The crop of 1881 is represented to have footed up 82,356 bales, valued
(approximately) at 5,791,880 dol., being an increase over the crop of
the previous year (1880) of 17,433 bales as to quantity, and of
1,260,000 dol. as to the approximate value thereof.
"From the foregoing it will be seen that about one-ninth of the whole
crop of 1881 has been exported to the United States.
"The entire crop, excepting about 1700 bales remaining in the hands of
the original importers or consignees here, on December 31, 1882, was
disposed of at an advance of about 1 cent, United States currency, in
the average price as compared with that realized in 1881 for the crop of
1880; or, in other words, the total crop of 1880 brought on the average
about 45¾ cents, whereas the crop of 1881 averaged about 46¾ cents,
United States currency, per half-kilogram.
"This refers to the prices originally obtained at the various sales
throughout the year by the importers or consignees, first hands.
"As regards the prices for the particular sorts which during the year
found their way to the United States, and which are usually purchased
from quite a number of firms in the wholesale tobacco trade through the
mediation of brokers, they differed all the way from about 45 cents to
95 cents, United States currency, for the half-kilogram.
"Thinking it might prove interesting, if not important, to parties in
the United States in any way concerned in this matter, to be informed as
to the extent and quality of the crop of 1882, I made inquiries relating
to it, and ascertained as follows, viz.:—'That whilst it is impossible
to state, at this early day, with accuracy the yield of the crop, it is
generally considered and expected to have been again in excess over the
previous one, and that it amounts to about 90,000 bales.'
"Its quality is represented by the planters to be very good, as far as
they are able to judge; but this can, of course, only be determined
later on, after the tobacco has gone through the process of
fermentation.
"The first parcels of this new crop will arrive here about the month of
March next, and will be offered for sale about a month or six months
thereafter.
"In concluding this report, I would remark that the year 1882 has been a
most favourable one for tobacco planters in Sumatra and for those
interested in tobacco plantations there, and so have those connected
with the trade here realized handsomely by the year's operations.
"I am, therefore, induced to state that so long as the present general
demand for the article continues there will be neither lack of capital
nor labour, so long as either can contribute to an increase in its
production, and it would seem to be more a question as to the extent of
acreage in Sumatra adapted for its cultivation, as only once in four or
five years a crop can be raised on the same soil without danger of
producing a very inferior quality of tobacco."
_Turkey._—The Turkish empire has long been known as producing some of
the finest tobaccos in the world. In the sanjac of Drama, which forms
the vice-consular district of Cavalla, tobacco is the staple article of
production and industry, and some 75,000 acres were devoted to its
culture in 1873. The whole crop of 1871 was reckoned at 11,200,000 lb.,
the exports having been 7,600,000 lb., value 37,825_l._ The tobacco of
this district, though derived entirely from one species, is divided into
two classes, known as _Drama_ and _Yenidji_. The former leaf is larger,
stouter, and more potent, and generally of deep reddish-brown colour;
the latter is smaller, slighter, less narcotic, with a peculiarly
delicate aroma, and the best is of a rich yellow colour, whence its name
"golden-leaf." The _Drama_ kind is principally grown in the western
portion of the district, and is the class supplied to European markets.
The differences in the two kinds seem to be due solely to the soil.
The plantations in the Drama district proper occupy both plain and
hill-side. The produce of the former is much the more considerable, and
superior. The best leaves, distinguished by a stronger and more
substantial texture, and a dark-red hue, go to Constantinople; the
inferior and lighter- find a sale in Russia. The mountain
product is much inferior in quality and is sent chiefly to Europe. When
the leaves are petiolate, or furnished with stems, they are made up in
_manoks_ ("hands") of 10–15, and termed _bashi-baghli_ ("head-tied");
when the leaves are sessile, or devoid of stems, they are simply pressed
together in small numbers, and called _bassma_. The whole produce of
this locality varies from 2,100,000 to 2,450,000 lb. yearly. The growth
obtained in the Vale of Pravista is known as _Demirli_. It is inferior,
unsubstantial, and dark-, and usually made up as _bashi-baghli_.
The annual production is about 2 million lb.; the exports to England
were 1,600,000 lb. in 1871. Cavalla affords yearly about 300,000 lb. of
inferior quality, chiefly as _bashi-baghli_, and mostly consumed
locally. The shipping port for all these places is Cavalla.
The district of Sarishaban produces on the average about 2,000,000 lb.
annually, but the crop of 1871 reached 2,800,000 lb. About ⅞ is as
_bashi-baghli_. That grown on the plain and hills is termed _ghynbek_,
and forms the bulk; that from the <DW72>s, about 500,000 lb. a year, is
the best, and is known as _ghubek_. All is packed up in small _boghchas_
(parcels), of 30–50 lb., which are distinguished as _béyaz_, from the
white cotton wrappers used for the best sort, and _kenavir_, from the
canvas coverings of the inferior kinds. The best goes to Constantinople,
secondary to Smyrna and other home markets, and the worst to Europe. The
district of Yenidji, near the Gulf of Lagos, affords some 3,500,000 lb.
per annum, chiefly as _bassma_, and bearing a very general resemblance
to the produce of Sarishaban. The best goes to Constantinople and
Russia. Ghiumirgina (Ghumurdjina, or Komuldsina) grows about 300,000 lb.
yearly of dark- _bassma_, of the _Drama_ class, which is used
locally; and Sultan-Yeri gives 400,000 lb. of still darker
_bashi-baghli_. The produce of these districts is shipped at Lagos
(Karagatch) or Cavalla.
The most delicate and valued of all the tobaccos raised in this portion
of European Turkey is the celebrated "golden leaf" from the caza of
Yenidji, on the Yardar (Nestus) river. After it, in declining order,
come the products of Drama, Persoccian, Sarishaban, Cavalla, and
Pravista. Of the whole Drama and Yenidji produce, it is estimated that
Austro-Hungary takes 40 per cent. Italy buys annually about
150,000–200,000 _kilo._ France, Germany, and Switzerland receive very
little. Russia is a large customer. Before the war, considerable
quantities were sent to the countries on the Lower Danube. England
imports every year some 10,000 bales, or 400,000 _okes_ (of 2·83 lb.) of
Pravista tobacco. The _refusa_, or waste leaves, &c., is sent everywhere
for making into cigarettes, most largely perhaps to Egypt. A kind of
tobacco known as _ayiasoulouk_ is grown in considerable quantities in
the opium districts, almost exclusively for export to Europe, the
natives having a strong prejudice against it.
The necessity for manuring is well understood by the Turks. They dress
the seed-beds with goat- and sheep-dung, and manure the fields during
winter with horse- and cattle-dung. In the spring, sheep and goats are
folded on the land. The soil of tobacco lands will be found quite
impregnated with, ammonia and nitrate of potash, both absorbed by the
plant; the former is thought to influence the aroma, and the latter may
be seen in crystals on the surface of the dried leaf. In order to keep
the leaves small and delicate, the planting is performed very close, the
usual distances being 5 inches apart, and 9 inches between the rows.
The district of Latakia, in the northern part of Syria, has long been
celebrated for its tobacco, which is the chief product of the
mountainous part. There are several kinds:—(1) _Abu Riha_ or _Dgebeli_,
found in its best state among the mountains of the Nesseries (Ansaries),
which possesses a peculiar and much-admired aroma, derived from its
being exposed, from November to April, to the smoke of fires of _ozer_
(_Quercus Ilex_, or _Q. Cerris_); (2) _Dgidar_, including a number of
kinds, of medium strength, and in great favour locally on account of its
low price; (3) _Scheik-el-Bent_, almost equal to _Abu-Riha_, and often
substituted for it.
The plain of Koura is remarkable for its tobaccos, which are rather
strong, but much admired. The villages of Lebail and Serai produce
better tobacco than Koura. The district of Gebail (Gebel) in Kesrasan
(Castravan) affords the best and dearest tobacco in Syria; it is very
brittle, and its ash is quite white. The country south of Lebanon yields
very ordinary qualities, known as _Salili_, _Tanoné_, and _Takibé_, or
generically as _Berraoni_; these are mixed with stronger kinds for use.
The best of the _Abu-Riha_ is yielded by the plant called
_Karn-el-Gazel_; the second quality is termed _Bonati_.
The exports of tobacco from Alexandretta in 1879 were:—To Egypt, 91
tons, value 6380_l._; Turkey, 24 tons, 1920_l._; England, 51 tons,
2550_l._; France, 1 ton, 80_l._ The exports from Aleppo in 1878 were 30
tons, value 1200_l._, to Great Britain. The yield of the crop in
Thessaly was 1,116,000 _okes_ (of 2·83 lb.) in 1877, 210,000 in 1878,
and 890,000 in 1879. The crop of Prevesa in 1878 was 4000 _okes_, value
215_l._ The exports from Dedeagatch were about 260 bales, value
1000_l._, in 1878; and 600 bales, value 2400_l._, in 1879. Considerable
quantities are grown around Sinope. Tobacco is one of the principal
products of the district of Samsoun, and is of good quality. The average
yield is 7,000,000 lb. yearly. It is grown near the sea-shore, and not
eastward of Yomurah, at Matchka and Trebizonde, and especially at
Akché-Abad. But the aggregate crop in these localities is hardly ⅓ of
the quantity produced at Samsoun, and the quality is far inferior. The
Samsoun product is usually purchased largely on account of the French
Government. The exports from Samsoun in 1878 were:—To Turkey, 2,680,000
_kilo._, value 160,800_l._; France, 583,500 _kilo._, 28,008_l._; Russia,
575,000 _kilo._, 57,500_l._; Germany, 400,000 _kilo._, 7200_l._;
Austria, 327,220 _kilo._, 31,266_l._; Great Britain, 87,567 _kilo._,
1576_l._; total, 4,653,287 _kilo._, 286,350_l._ The exports of
Turkey-produced tobacco from Trebizonde in 1879 were:—To Turkey, 14,864
cwt., value 44,592_l._; Russia, 866 cwt., 2598_l._; Great Britain, 490
cwt., 1470_l._; Austria and Germany, 204 cwt., 612_l._; total, 16,424
cwt., 49,272_l._
In 1884, Damascus imported 1313 sacks of tumbeki, value 1674_l._, from
Bagdad. In the same year Erzeroum imported 9000 _okes_, value 1090_l._,
from Persia.
The leaf grown by the Herki Kurds and other cultivators in and around
the district of Shemdina is highly prized in Persia. In 1884, the first
year of their operations, the employés of the tobacco Régie only
succeeded in registering a yield of 25,000 _okes_, but this amount
represents less than a fifth of the estimated produce of the vilayet. It
is believed, however, that 8000–10,000_l._ Turkish worth of Shemdina
tobacco annually crosses the frontier into Persia.
Trebizonde exports in 1884 were 20,167 cwt., value 56,849_l._ Inferior
qualities are sent to Europe, good ones remain in Turkey, and the best
go to Egypt.
The shipments from Samsoun in 1884 were as follows:—
────────────────┬────────┬────────┬────────
│ │ Price. │
│ cwt. │£. s. d.│ £
To Turkey │ 29,210│4 0 0 │ 116,840
Austria │ 8,540│5 0 0 │ 42,700
France │ 5,756│1 4 2 │ 11,512
Egypt │ 4,176│4 0 0 │ 16,704
Germany │ 3,579│1 8 6 │ 5,096
Russia │ 1,730│6 0 0 │ 10,380
Great Britain│ 832│1 4 2 │ 1,002
Holland │ 712│1 12 0 │ 1,140
Greece │ 416│3 0 0 │ 1,248
│ ———————│ │ ———————
│ 54,951│ │ 206,622
────────────────┴────────┴────────┴────────
_United States._—The United States of America occupy the foremost rank
among tobacco-growing countries. The areas and productions have been as
follows:—1875, 559,049 acres, 379,347,000 lb.; 1876, 540,457 acres,
381,002,000 lb.; 1877, 720,344 acres, 489,000,000 lb.; 1878, 542,850
acres, 392,546,700 lb. The crop of 1875 (in millions of lb.) was thus
contributed:—Kentucky, 130; Virginia, 57; Missouri, 40; Tennessee, 35;
Maryland, 22; Pennsylvania, 16; N. Carolina, 14¾; Ohio, 13½; Indiana,
12¾; Connecticut, 10; Massachusetts, 8½; Illinois, 8. The average yields
(in lb. per acre) of the various districts in 1875 were:—Connecticut,
1600; Pennsylvania, 1600; New Hampshire, 1600; Massachusetts, 1350;
Missouri, 850; Arkansas, 822; New York, 800; Florida, 750; Ohio, 700; W.
Virginia, 680; Maryland, 675; Tennessee, 675; Kansas, 670; Texas, 650;
Kentucky, 630; Virginia, 630; Illinois, 550; Georgia, 550; N. Carolina,
500; Indiana, 500; Wisconsin, 500; Alabama, 465; Mississippi, 317. The
exports from New York in 1878 were:—37,484 hogsheads, 2561 bales, and
2,218,200 lb. manufactured, to Great Britain; 15,570 hh., 207 bales, and
14,800 lb. manufactured, to France; 35,700 hh., 78,331 bales, and
147,400 lb. manufactured, to N. Europe ; 23,150 hh., 6058 bales, and
120,000 lb. manufactured, to other Europe; 4628 hh., 14,360 bales, and
4,780,200 lb. manufactured, to S. America, E. and W. Indies, &c.
Baltimore exported 66,039 hh. in 1878. The shipments from New Orleans in
1877–8 were:—1226 hh. to Great Britain, 743 to France, 4552 to N.
Europe, 3222 to S. Europe, Mexico, &c., and 4500 coastwise.
Philadelphia, in 1879, exported 9,564,171 lb. of leaf tobacco, 52,000
cigars, and 515 lb. of snuff. The total American export of
unmanufactured leaf in 1879 was 322,280,000 lb.
The census bulletin on this branch of industry, recently issued, is of a
very interesting nature. The tobacco product in the United States is
divided into classes, types and grades, the basis of a class being its
adaptation to any specific purpose; of a type, to certain qualities or
properties in the leaf, such as colour, strength, elasticity, body or
flavour. It also applies to the method of curing, such as sun, air or
flue curing. Grades represent the different qualities of a type, and
vary much in the several types. The classification of American tobacco
is threefold, viz. domestic cigar tobacco and "smokers,"
chewing-tobacco, export tobacco. The domestic tobacco trade comprises
the various kinds of seed-leaf of Connecticut, New England,
Pennsylvania, Wisconsin, Illinois, New York, Florida and Ohio, as well
as the sorts known as White Burley "lugs," fine-fibred wrappers, Indiana
kite foot, and American-grown Havana. In the chewing class are included
the fine-cut and the plug fillers, principally of the White Burley type
from Kentucky, while under the head of export tobacco are the Virginian
bird's-eye cutting leaf, and the spinning fillers or shag. It is curious
to notice how each market for export tobacco differs in its
requirements. The "closed" markets, or those in which the tobacco trade
is a monopoly of the Government, are France, Italy, Austria and Spain.
The French "Régie" is supplied by wrappers, binders and fillers from
Kentucky, Maryland and Ohio; the Italian Régie from Kentucky and
Virginia; the Austrian Régie by "strips" from the same States, and the
Spanish Régie by common "lugs." The open markets are Germany, to which
are sent the tobaccos known as German saucer and spinners; Ohio and
Maryland, spangled cigar-wrappers and "smokers" fat lugs; Switzerland,
which is supplied with Virginian or Western wrappers and fillers;
Holland, with Dutch saucer (a mottled Virginia, Kentucky or Tennessee
leaf); Belgium, with Belgian cutter (a light, yellowish-brown leaf, well
fired); Norway and Sweden, with heavy types, mainly used for spinning
and "saucing." Kentucky, which stands first of all the States for
production, the annual produce being 171,120,784 lb., gains her chief
profits from the white burley and yellow wrapper; Illinois, from the
production of the seed-leaf; Missouri, from sweet fillers and white
burley; Virginia, from yellow wrappers, bright "smokers," sun, air and
flue-cured fillers. Decidedly the most prosperous tobacco States are
those that grow types suitable for domestic consumption, while those
that grow it mainly for exportation stand low in the scale, the margin
of profit under this head being reduced very low. According to the
researches of Dr. Gideon Moore, the largest amount of nicotine is
contained in the Virginian heavily manured lots (5·81 per cent.), while
the Virginian heavy English shipping has 4·72, the New York domestic
Havana but 2·53, the Connecticut seed-leaf 1·14, while the smallest
amount of all is found in the little Dutch tobacco of the Miami valley,
0·63. Profits in the culture of tobacco have been in direct
proportion—first to its suitableness to domestic consumption; and,
secondly, to the amount of fertilization practised by the growers in its
cultivation. This is true in every case, except the yellow tobacco
districts of North Carolina and Virginia, where poverty in the soil is a
condition of success in the production of quality.
Professor J. T. Rothrock is of the opinion that the early natives of
California smoked the leaves of _Nicotiana clevelandii_—a species only
quite recently described by Professor Asa Gray. It is a small plant with
small flowers, and it was found by Professor Rothrock only in
association with the shell heaps which occur so abundantly on the coasts
of Southern and Central California. He states that perhaps of all the
remains of extinct races so richly furnished by that region, none were
so common as the pipes, usually made of stone resembling serpentine. The
tobacco of _N. clevelandii_ Professor Rothrock found by experience to be
excessively strong.
A recent report of the Commissioner of Agriculture contains a few pages
of sound advice to American planters on the management of this crop,
which is worthy of reproduction here.
"The principal points to be attended to if the best results are to be
attained may be stated in a few paragraphs—paragraphs which, while
referring mainly to shipping, manufacturing, and smoking tobacco as
constituting nine-tenths of the tobacco grown in the United States,
embody principles and prescribe modes of management nearly identical
with those to be considered in the treatment of other tobaccos.
"I. Select good land for the crop; plough and subsoil it _in autumn_ to
get the multiplied benefits of winter's freezes. This cannot be too
strongly urged.
"II. Have early and vigorous plants and _plenty of them_. It were better
to have 100,000 too many than 10,000 too few. They are the corner-stone
of the building. To make sure of them give personal attention to the
selection and preparation of the plant-bed and to the care of the young
plants in the means necessary to hasten their growth, and to protect
them from the dreaded fly.
"III. Collect manure in season and out of season, and from every
available source—from the fence corners, the ditch-banks, the urinal,
the ash-pile. Distribute it with a liberal hand; nothing short of
princely liberality will answer. Plough it under (both the home-made and
the commercial) in _February_, that it may become thoroughly
incorporated in the soil and be ready to answer to the first and every
call of the growing plant. Often (we believe generally) the greater part
of manure applied to tobacco—and this is true of the 'bought' fertilizer
as well as of that made on the farm—is lost to that crop from being
applied too late. Don't wait to apply your dearly-purchased guano in the
hill or the drill from fear that, if applied sooner, it will vanish into
thin air before the plant needs it. This is an exploded fallacy.
Experience, our best teacher, has demonstrated beyond cavil that stable
and commercial manure are most efficacious when used in conjunction. In
no other way can they be so intimately intermixed as by ploughing them
under—the one broadcasted on the other—at an early period of the
preparation of the tobacco lot. This second ploughing should not be so
deep as the first; an average of three to four inches is about the right
depth.
"IV. Early in May (in the main tobacco belt to which this article
chiefly refers, that is to say, between the thirty-fifth and fortieth
parallels of north latitude), re-plough the land to about the depth of
the February ploughing, and drag and cross-drag, and, if need be, drag
it again, until the soil is brought to the finest possible tilth. Thus
you augment many fold the probabilities of a 'stand' on the first
planting, and lessen materially the subsequent labour of cultivation.
Plant on 'lists' (narrow beds made by throwing four furrows together
with the mould-board plough) rather than in hills, if for no other
reason than that having now, if never before, to pay wages in some shape
to labour, whenever and wherever possible horse-power should be
substituted for man-power—the plough for the hoe.
"V. Plant as early as possible after a continuance of pleasant spring
weather is assured. Seek to have a _forward_ crop, as the benefits
claimed for a late one from the fall dews do not compensate for the many
advantages resulting from early maturity. Make it an inflexible rule to
plant no tobacco after the 10th of July—we mean, of course, in the
tobacco belt we have named. Where one good crop is made from later
planting ninety-nine prove utter failures. Far better _rub out and start
afresh the next year_. Take pains in transplanting, that little or no
replanting may be necessary. The cut-worm being a prime cause of most of
the trouble in securing a stand, hunt it assiduously and particularly in
the early morning when it can most readily be found.
"VI. Keep the grass and weeds down, and the soil loose and mellow by
frequent stirring, avoiding as much as possible cutting and tearing the
roots of the plant in all stages of its growth, and more especially
after _topping_. When at all practicable—and, with the great improvement
in cultivators, sweeps, and other farm implements, it is oftener
practicable than generally supposed—substitute for hand-work in
cultivation that of the horse. The difference in cost will tell in the
balance-sheet at the close of the operation.
"VII. Attend closely to 'worming,' for on it hinges in no little degree
the quality and quantity of tobacco you will have for sale. A worm-eaten
crop brings no money. So important is this operation that it may
properly claim more than a passing notice. Not only is it the most
tedious, the most unremitting, and the most expensive operation
connected with the production of tobacco, but the necessity for it
determines more than all other causes the limit of the crop which in
general it has been found possible for a single hand to manage.
Therefore bring to your aid every possible adjunct in diminishing the
number of worms. Use poison for killing the moth in the manner so
frequently described in treatises on tobacco, to wit, by injecting a
solution of cobalt or other deadly drug into the flower of the Jamestown
or 'jimson' weed (_Datura stramonium_), if necessary planting seeds of
the weed for the purpose. Employ at night the flames of lamps, of
torches, or of huge bonfires, in which the moth may find a quick and
certain death.
"In worming, spare those worms found covered with a white film or
net-like substance, this being the cocoon producing the ichneumon-fly,
an enemy to the worm likely to prove a valuable ally to the planter in
his war of extermination.
"Turn your flock of turkeys into the tobacco-field, that they, too, may
prey upon the pest, and themselves grow fat in so doing.
"If these remedies should fail, sprinkle diluted spirits of turpentine
over the plant through the rose of a watering-pot, a herculean task
truly in a large crop, but mere child's play to the hand-picking
process, for the one sprinkling suffices to keep off the worms for all
time, whereas the hand-picking is a continual round of expensive labour
from the appearance of the first worm until the last plant has been
carried to the barn. We have no idea that such sprinkling will at all
affect the odour or flavour of the tobacco when cured.
"If, as stated by a writer in a California paper, the well-known
'yellow-jacket' be useful in destroying tobacco-worms, by all means win
it as an ally. As proving its usefulness, the writer asserts that one of
his neighbours, a Mr. Culp, daring fifteen years growing tobacco, has
never expended a dollar for labour to destroy the worm, trusting all to
this little workman, who, he says, carefully searches the plants for the
worms, and never allows one to escape its vigilance.
"We cannot speak from our own experience as to many of these suggested
means for overcoming the horn-worm, but we have no hesitation in saying
to the farmer, try any, try all of them rather than have your crop eaten
to shreds, and the labour of more than half the year brought to naught
in a few days, it may be, by a single 'glut' of worms.
"VIII. 'Prime high and top low.' While open to objection in particular
cases, even with the character of tobacco chiefly under consideration,
and altogether inadmissible, it may be, in the management of other
varieties of tobacco, this is a safe rule, we think, to follow in
general practice.
"We favour 'priming' by all means; for when no priming is done the lower
leaves (made worthless by constant whipping on the ground) serve only as
a harbour for worms, which are the more difficult to find because of the
increased burden of stooping. Moreover, if the bottom leaves be saved on
the cut stalk, as most likely they will be, there is always the
temptation to put them on the market; and against a _sacrilege_ like
this we are firmly set, let others say and think what they may.
"Yet another advantage to be gained by the removal of these bottom
leaves, which is what the planter terms 'priming,' is the increased
circulation of air and distribution of light thereby afforded, both
essential factors, the merest tyro knows, to the full development of
plant life.
"'Topping' (the pinching off with the finger-nail the bud at the top of
the plant) is an operation requiring considerable skill and judgment.
Let it be performed only by hands having these prerequisites.
"That as many plants as possible may ripen at the same time (a
desideratum not to be undervalued in aiming, as all should, at a
_uniform_ crop) wait until a large number of plants begin to button
before commencing to top. Going about through the crop, topping a plant
here and there because it may chance to have buttoned before its
fellows, is a damaging process not to be tolerated.
"No inflexible rule can be given for the number of leaves that should be
left on a plant. All depends upon the variety of tobacco, the strength
of the soil, the promise of the particular plant, the probable seasons
and time left for ripening, &c.
"One of the most successful growers of heavy dark tobacco we have ever
known, once stated to us his conviction, after years of observation and
practice, that one year with another, taking the seasons as they come,
eight leaves would give a better result than any other number. Our own
experience has tended to confirm this judgment.
"IX. See to it that the suckers are promptly removed. It is work quickly
done, and with worming may constitute a single operation.
"X. We come now to consider the last operation in the field, 'cutting'
the crop. In this, as in topping, a man of judgment, experience, and
fidelity is needed. An inexperienced hand, one without judgment, and
particularly one who is indifferent to the interests of his employer,
will slash away, right and left, not knowing or not caring whether the
tobacco he cuts be ripe or green, doing more damage in a few hours than
his whole year's wages would compensate for, even could they be
garnished.
"Therefore, be on hand to see for yourself, and do not delegate the duty
to any less interested party, that a crop managed well, it may be, so
far, from the initial plant-bed, should not be spoiled in the closing
work by an incompetent or unfaithful cutter.
"Be there, too, to see, in this supreme hour, that injury from sunburn
is warded off by the timely removal, to the shade, of the plants that
have been cut, or by a proper covering, where they lie, against the
scorching rays of the sun. The neglect of this precaution has played
havoc with many a crop when brought under the auctioneer's hammer.
"XI. We should have no space to describe the different methods of
'curing' tobacco, as, for instance, 'sun-curing,' 'air-curing,'
'flue-curing,' 'open-fire-curing,' &c., even though the whole subject
had not been gone over again and again in previous reports of this
Department. We can only say of this operation, as of all others
connected with the production of tobacco, that much depends on its
proper doing, and that, as much as possible, it should have the personal
superintendence of the owner.
"But the crop may have been brought along successfully even to the
completion of this operation and 'lack one thing yet,' if it be not now
properly manipulated.
"Therefore, go yourself, brother planter, into your barns, see with your
own eyes, and not through the medium of others; handle with your own
hands, and _know of a surety_ that the tobacco hanging on the tier-poles
is in proper order for 'striking' and 'bulking,' and act accordingly.
"When, later on, it is being 'stripped,' 'sorted,' and tied into
bundles, or 'hands,' as they are often called, be there again, _propria
persona_, to see that it is properly classed, both as to colour and to
length, the 'lugs' going with lugs, the 'short' with short, the 'long'
with long, &c. Instruct those sorting that when in doubt as to where a
particular leaf should be put, to put it at least one grade lower than
they had thought of doing. Thus any error will be on the safe side.
"Prize in hogsheads to weigh what is usually called for in the market in
which you sell, and, above all, 'let the tobacco in each hogshead be as
near alike as possible, uniform throughout, so that the 'sample,' from
whatever point it may be taken, can be relied on as representing the
whole hogshead,' and that there be left no shadow of suspicion that
'nesting' has been attempted, or any dishonest practice even so much as
winked at.
"We sum up the whole matter by repeating:
"1. That overproduction, the production at all, of _low_ grade tobacco
is the chief cause of the present extremely low price of the entire
commodity.
"2. That the planters of the United States have the remedy in their own
hands; that remedy being the reduction of area, this reduction to
result, from the employment of the means here suggested, in increased
crops; and, paradoxical as it may seem, these increased crops to bring
greatly enhanced values.
"The whole world wants good tobacco, and will pay well for it. Scarcely
a people on earth seeks poor tobacco or will buy it at any price.
"In a word, then, one acre must be made to yield what it has hitherto
taken two or three acres to produce; and this double or treble quantity
must be made (as, indeed, under good management it could not fail to be)
immeasurably superior in quality to that now grown on the greater number
of acres. Either this or the abandonment of the crop altogether—one or
the other."
The exports from Baltimore were 46,239 hogsheads in 1882, 43,620 in
1883, 43,192 in 1884. The State of New York, in 1883, had 5440 acres
under tobacco, producing 9,068,789 lb., value 1,178,943 dollars; and
Connecticut, 8145 acres, 9,576,824 lb., 1,292,871 dollars. The
production of Minnesota was 65,089 lb. in 1879, 48,437 lb. in 1880,
79,631 lb. in 1881, 62,859 lb. in 1882, 14,744 lb. in 1883.
_Venezuela._—The exports from Ciudad Bolivar were, in 1884, 1318
_kilo._, value 1037 _bolivares_, to the British West Indies; 9618
_kilo._, 6691 _bolivares_, to the United States; 275,329 _kilo._,
192,188 _bolivares_, to Germany. The exports of tobacco from this port
in decades have been:—7,650,656 lb. in 1850–59; 2,134,711 in 1860–69;
3,170,812 in 1870–79.
_West Indies._—The Spanish possessions in the West Indies are well known
for their tobacco. The best is produced on the _vuelta abajo_, or
low-lying districts of Cuba, near Havana, which are yearly flooded
during the autumn, just before the tobacco is transplanted. To this
fact, and the peculiar suitability of the seasons, the excellence of
this particular product is attributed. The exports from Havana in 1878
were:—93,603 bales tobacco, 75,212,268 cigars, 203,581 bundles
cigarettes, to the United States; 6169 bales tobacco, 66,795,330 cigars,
5,034,774 bundles cigarettes, to England; 32,582 bales tobacco,
9,541,498 cigars, 133,008 bundles cigarettes, to Spain; 582 bales
tobacco, 3,861,700 cigars, 8206 bundles cigarettes, to N. Europe; 5671
bales tobacco, 18,327,025 cigars, 797,513 bundles cigarettes, to France;
41 bales tobacco, 900,850 cigars, 5,709,442 bundles cigarettes, to other
countries. The totals for 1878 were 7,078,904 _kilo._ of tobacco,
182,356 thousand cigars, and 12,816,903 packets of cigarettes; in 1879,
6,371,014 _kilo._ of tobacco, 145,885 thousand cigars, and 14,098,693
packets of cigarettes. The tobacco exports in 1879 from St. Jago de Cuba
were 9653 bales to Bremen, 4015 to the United States (chiefly for
Bremen), and 1809 coastwise, total 15,477, against 10,249 in 1878. In
the island of Puerto Rico, the tobacco-plant thrives well, and the
quality, especially in the Rio de la Plata district, is very good. In
1878, the island exported 8 _quintals_ (of 101½ lb.) to the United
States, 32,109 to Spain, 4198 to Germany, and 18,123 to other countries.
The British West Indies have only recently appreciated the importance of
tobacco cultivation. Many portions of Jamaica seem as well fitted for it
as the _vuelta abajo_ of Cuba, and already Jamaica tobacco in the
Hamburg market ranks next to the best Havana, and is considered superior
to such Cuban growths as St. Jago, Manzanillo, Yara, &c. Tobacco
cultivation may now be said to have a place in the industries of
Jamaica, a fact mainly due to Cuban refugees. The most extensive
plantations in the island are Potosi in St. Thomas Parish, and Morgan's
Valley in Clarendon. Much of the produce goes to the German market, the
remainder being made into cigars for local consumption, and said to be
quite equal to some of the best Cuban brands. Some experiments made with
Bhilsa tobacco have given great satisfaction, on account of the robust
habit and immense yield of the plant. It is especially adapted for very
wet districts, and its cultivation will be widely extended, if justified
by its market value. Tobacco is, and for very many years has been, grown
by the peasantry in small patches; from this, they manufacture a
smoke-dried leaf, which, twisted together in rope form, sells readily in
the home market. The acreage occupied by the crop was 297 in 1874–5, 442
in 1875–6, 331 in 1876–7, and 380 in 1877–8. The <DW72>s of valleys in
many parts of Dominica, too, are eminently suited to this crop,
particularly the district between Roseau and Grand Bay. The experiment
of tobacco culture in New Providence on a large scale has not proved
satisfactory, owing to the difficulties encountered in curing and
preparing the leaf; the cigars made are fit only for local consumption.
The exports from San Domingo in 1884 were 10,513,940 lb., value 669,500
dollars.
According to a recent Consular Report, it would seem that "Cuban tobacco
has lost its prestige through forcing and artificial manures, and has to
sustain sharp competition from abroad where it formerly commanded the
market; and probably some years must elapse before the soil can recover
from the excessive and indiscriminate use of artificial fertilizers.
"A few years ago the leaf harvested in the Vuelta Abajo was not
sufficient to meet the large demand, and in order to increase the yield,
growers made use of guanos of all sorts, and with such bad results that
they find it now difficult to place on reasonable terms more than half,
and sometimes less, of their crops, at very low prices; in few
localities only the soil has not been spoilt by spurious manures, and
the leaf grown there commands very high prices and is warmly competed
for by local manufacturers and buyers for the United States.
"Notwithstanding the last crop has been of a better quality than
heretofore, growers were compelled to abandon the tobacco cultivation
for a certain time, and devote the ground to other purposes.
"It appears that this change of cultivation is absorbing the
fertilizers, and restoring to the soil its former good qualities, and,
if one can judge from the splendid appearance of the leaf and the ready
sale it now meets with, it would seem that the Vuelta Abajo fields are
regaining their former renown.
"This has been a hard but healthy lesson the Vegueros are not likely to
forget. The soil cannot and should not be taxed beyond a reasonable and
natural yield; any attempt to the contrary would only be a repetition of
the fable of the golden eggs, as the tobacco growers in the Vuelta Abajo
have had occasion to learn to their cost.
"Towards the end of the year buyers, influenced by the pending
negotiations of the Spanish-American Treaty, entered the market and
operated extensively in the expectation of a great reduction of duties
in the United States, paying prices above the established one, and
which, a few weeks later, they were utterly unable to obtain.
"Cuban growers complain much of heavy purchases made in the United
States for account of the Spanish Government for Peninsular consumption;
they say that however low the class of the Cuban leaf may be, it must
necessarily be superior to that of the Virginia and Kentucky tobacco,
and that they might easily cultivate here the quality required, and
place it in the markets at as low a price as any other country.
"Growers are unanimous in denouncing the action of some local merchants
and cigar manufacturers in forwarding at the opening of the last season
samples of leaf tobacco and cigars in condition that by no means gave a
true idea of the quality of the crop, and which necessarily gave a
result contrary to the interests of all parties engaged in the trade;
and they earnestly protest against a repetition of this injudicious
haste.
"The total tobacco production is estimated at between 400,000 and
500,000 _quintals_ (one _quintal_ about 100 lb.), chiefly from the
following districts:—
Tercios.
Vuelta Abajo and semi Vuelta Abajo 150,000 to 200,000
Parlida 30,000 50,000
Remedios 60,000 85,000
Cuba and Java 25,000 35,000
Gibara 20,000 30,000
Total 285,000 400,000
(One tercio about 124 lb.)
"As is well known, that grown in the Vuelta Abajo or district west of
Havana is the best kind, and has given Cuba its well-earned reputation.
About 67,000 acres are cultivated under the denomination.
"I have no reliable statistics to show how much of the raw produce is
manufactured in the island, probably not more than one-fourth. Very
large quantities of the leaf are exported in bales and rolled abroad.
"It is evident, however, that, given the total production and
corresponding result in the manufactured form, but a small portion of
the cigars sold in Europe and elsewhere as Havana cigars have the
slightest claim to a connection with Cuba.
"The chief and only important manufactories of these cigars are in
Havana, and much care and money is expended in producing a
handsome-looking article. As much as 40 dollars gold are paid to skilled
labourers per 1000 for making up first-class goods. About 17,000
operatives are employed in this manufacture in Havana alone. One of the
largest establishments here is that supplying the Henry Clay brands,
which is stated to turn out from 80,000 to 120,000 cigars daily; and
there are many others of considerable importance with a well-earned and
old-established reputation for fine goods.
"The quality of tobacco, like other agricultural produce, depends on
seasons, soil, and many natural causes, which may baffle the most
careful cultivator.
"There are good and bad years; abundant and scanty crops in succession.
"Except in the case of the few rich owners of plantations in the best
districts, brands and names are no guarantees for a permanently good
article. Even these favoured few are exposed to bad seasons, if in a
minor degree than less fortunate holders.
"There has been no really fine-flavoured aromatic leaf harvested since
1881. Much of that since garnered has been simply bad.
"Great hopes are entertained of the coming 1885 crop, and present
indications are in favour of this assumption.
"The manner in which the wholesale trade is carried on in Havana is
incomprehensible to an ordinary outsider, to whom it would appear that
the manufacturers prefer a prospective loss abroad to a present and
certain gain here. They will only execute orders, large or small, for
cash over the counter, giving no, or in some cases the smallest,
discount. No manufactured goods are kept in stock, but are made to order
after sample, and, unless examined in warehouse before delivery, and
that means little, must be paid in full on delivery, and the consequence
but too frequently is that, on arrival at their destination, they do not
correspond with the sample, and the deluded buyer finds that he has made
a bad bargain, and (if an Englishman) discovers that he could have
bought the same article cheaper in the English market with the
additional advantage of examining and testing the goods before purchase.
"I leave the solution of this enigma to the initiated: it probably is
that the makers consign very largely, and London importers are too
experienced and too wary to pay the full invoice price until well
acquainted with the wares, or they get large discounts refused to the
cash purchaser in Havana.
"Complaints are heard of the depressed state of the Cuban tobacco trade
and of the large unsold stocks on hand. I do not think the traders
deserve sympathy, nor have they done anything to earn the confidence of
foreign customers. My experience leads me to advise intending purchasers
to put (I do not advise regular traders) themselves in the hands of
reliable London dealers and avoid all direct purchases.
"Intelligent smokers with sensitive palates will find no cheap tobacco
here fit to smoke; 50_s._ per 100 and upwards is what must be paid at
present for really fine-flavoured aromatic cigars; beyond 80_s._ or
85_s._ prices become fancy ones, and are paid for the smart cases and
envelopes. Even at the rates I quote it is not easy to find what is
wanted. There is abundance of dark powerful tobacco of fine quality at
much lower rates, but not light tobacco with flavour or aroma or without
strength, such as the educated (I allude to taste) Englishman seeks. I
believe that only about 10 per cent. of the tobacco harvested in
ordinary years is of the light colour I refer to, hence the difficulty
in supplying the demand, and the artifices resorted to to supply the
deficiency.
"Cuba's annual tobacco crop may be estimated as between 300,000 and
400,000 _tercios_ of 125 lb. each. About 30,000 persons are employed in
its cultivation, and its value when harvested may be fixed (according to
year's quality) at between 8,000,000 and 12,000,000 dol. of 4_s._
"I cannot estimate the number of persons engaged in working plantation
(Vegueros) and other cigars for home consumption, nor the quantity thus
consumed; but the higher class of operatives employed in cigar-making
for export number about 20,000, and turn out at present probably
200,000,000 cigars annually.
"The export trade has fallen off considerably of late years. In the five
years, 1870 to 1874, about 350,000,000 cigars were annually shipped to
foreign ports, whereas in the period between 1879 and 1884 the annual
average export was only 200,000,000.
"Probably larger quantities have been exported in each period owing to
under valuations to escape export duty; but relative bulk proportions
between the two export periods will hardly be affected by this."
The exports from Havana in 1884 were 11,767,200 lb. to the United
States, 613,000 to Spain, 252,600 to France, 37,500 to Mexico and South
America, 70,000 to Belgium, and 500 to the Mediterranean.
CHAPTER V.
PREPARATION AND USE.
This chapter embraces the manufacture of cut, cake and roll tobacco,
cigars, cigarettes, and snuff. It is impossible to indicate the precise
form in which each kind of tobacco-leaf is manufactured for use; indeed,
no well-defined line marks the qualifications of each sort, and the
great art of the manufacturer is to combine the various growths in a
manner to produce an article suited to the tastes of his customers, at a
price suited to their pockets. But, in a general way, it may be said
that Havana and Manilla are probably exclusively consumed in the form of
cigars; Virginia is a favourite for cavendish, negrohead, and black
twist, and is largely converted into returns, shag, and snuff; Kentucky,
Missouri, and Ohio are used for cavendish, brown twist, bird's-eye,
returns, and shag; Dutch and German make the commonest cigars, k'naster,
moist snuffs, and smoking-mixtures; Java and Japan are selected for
light cigars, mixtures, and light moist shag; Latakia, Turkey, Paraguay,
Brazil, China, and the remainder, are used up in cigarettes, mixtures,
imitations, and substitutes.
_Damping._—The tobacco-leaves are received by the manufacturer in all
kinds of packages, from a hogshead to a seron (raw hide), and of all
weights from 1 to 12 cwt. The first process they undergo is "damping,"
which is necessary to overcome their brittleness, and admit of their
manipulation without breaking. For this purpose, the bunches ("hands")
are separated, and the leaves are scattered loosely upon a portion of
the floor of the factory, recessed to retain the moisture. A quantity of
water, which has been accurately proportioned to the absorbing qualities
of the leaf used, and to the weight present, is applied through a
fine-rosed watering-pot, and the mass is left usually for about 24
hours, that damped on one morning being ready for working on the
following morning. In England, water alone is admissible (by legislative
enactment) for damping, except in special cases to be noted
subsequently; but abroad, many "sauces" are in vogue, their chief
ingredients being salt, sal ammoniac, and sugar.
_Stripping and Sorting._—Quantities of leaf-tobacco are shipped in a
condition deprived of their stem and midrib, and are then known as
"stripts." Those which are not received in this state, after having been
damped, are passed through the hands of workmen, who fold each leaf edge
to edge, and rip out the midrib by a deft twirl of the fingers,
classifying the two halves of each leaf, and ranging the sorts in
separate piles as smooth as possible. The value of the leaf greatly
depends upon the dexterity with which the stripping is done, as the
slightest tear deteriorates it. Stripts require sorting only. The
largest and strongest leaves are selected for cutting and spinning; the
best-shaped are reserved for the wrappers of cigars; broken and
defective pieces form fillers for cigars; and the ribs are ground to
make snuff. For the manufacture of "bird's-eye" smoking-tobacco, the
leaves are used without being previously stripped.
[Illustration: FIG. 14.]
[Illustration: FIG. 15.]
[Illustration: FIG. 16.]
[Illustration: FIG. 17.]
_Cutting._—Cutting is the process by which the damped leaves, whether
stripped or not, are most extensively prepared for smoking in pipes and
cigarettes. The tobacco-cutter which is in general use in this country
is shown in Figs. 14 (side elevation), 15 (sectional elevation), 16
(front elevation), and 17 (plan). The main frames _a_ are united by
stretcher-bolts _b_; _d_ is a wooden-surface feeding-roller, on which
the tobacco is pressed and cut; _c_ are the upper compressing- and
feeding-rollers, mounted in _e_, carriage-plates extended backwards,
forming the sides of the feeding-trough, and hinged to the axle _m_; _f_
are levers; _g_, links by which the weight _w_ presses down the upper
rollers; _h_, a crank, and _i_, a connecting-link for working; _j_, the
cross-head to which the knife _k_ is fixed; _l_, side-levers or
radius-bars for guiding the knife, hinged on the eccentric ends of the
axle; _m_, an axle held in bearings at the back of the machine; on its
middle part, which is concentric with its own bearings, are hinged the
top roll carriage-plates _e_, whilst on its projecting ends, which are
slightly eccentric, the knife-levers _l_ are hinged; _n_ is a worm-wheel
segment; _o_, a worm; _p_, a hand-wheel for turning the eccentric
spindle _m_ through a part of a revolution in its bearings, for
adjusting the contact of the knife with the nose-plate _q_; _r_, a worm;
_s_, a worm-wheel; _t_, a worm-pinion for giving simultaneous movement
to all the rollers; _u_, a spindle, "universal jointed" at both ends,
for driving the upper rollers in positions varying with the thickness of
the feed; _v_, a saw-toothed ratchet-wheel, moved intermittently by a
catch _x_, link _y_, and stud-pin _z_, _v_ being changeable, and the
eccentricity of _z_ variable, for the purpose of regulating the fineness
of the cutting. Both ends of the knife move at the same speed, and its
surface is made to clear the work by describing a slight curve. The
knife is adjusted accurately to the nose-plate, while the machine is in
motion, by varying the direction of eccentricity of the axis of the
knife-levers to that of the roller-levers. The fineness of the cutting
is regulated by varying the eccentricity of a movable stud-pin in a
plate on the crank-shaft which gives motion, through a train of
speed-reducing gear, to the several rollers. The knives are easily
removed and replaced, and require sharpening after every 4–6 hours'
working. Two men attend the machine, one to keep the feed-rollers
supplied, the other to watch that the knife is doing its work, and to
remove the tobacco as fast as it is cut.
_Drying._—The cut tobacco, as removed from the machine, is placed
loosely in a layer several inches deep in a large trough, provided with
a canvas false bottom; steam is introduced between the true and false
bottoms, and finds its way up through the tobacco, which is thus
rendered more easily workable. It is next transferred to a similar
trough having no false bottom, but a steam-jacketed floor instead; here
the tobacco is dry-heated, and at the same time lightened up by hand.
Finally, it is taken to a third trough, where cold air is forced through
the canvas false bottom, by means of a blower or fan. This last
operation dries the tobacco ready for use in the course of some hours;
but it has the disadvantage of dispersing part of the aroma, and is
therefore generally resorted to only when time presses. In other cases,
the drying is conducted on canvas trays. However performed, the drying
operation needs the greatest attention, to prevent the moisture being
extracted to such a degree as to destroy the profit which its presence
confers upon the manufacturer. With drying, the preparation of cut
tobacco for smoking in pipes is completed.
_Cake or Plug._—The manufacture of "cake" or "plug" is little carried on
in this country, as the Excise laws exclude the use of sweetening
matters, except when carried on in bond. The process is sufficiently
simple. Virginian leaf, with or without the addition of flavourings, is
sweated for a day or two, to deepen the colour, worked into a soft mass,
and next placed in moulds, and subjected to sufficient pressure to
ensure the cohesion of the mass. Each cake is then separately wrapped in
perfect leaf, and passes through a series of moulds, each smaller than
the last, and under increasing pressure in steam-jacketed
cupboard-presses, of which there are many forms. The combined effect of
the heat and pressure is to thoroughly impregnate the whole mass with
the natural juices of the leaf and the flavouring (if any has been
used), and to produce a rich dark colour.
A machine for turning out plug-tobacco in ribbons, made by the McGowan
Pump Co., New York, is shown in Fig. 18. The tobacco is first weighed
out in the proper quantities, and spread in a box placed in spaces in a
heavy iron table a. When the latter is filled, it is passed to and fro
under the heavy iron wheels b, which are loose on the shaft, and which
can be adjusted to exert any desired pressure. Twice passing through
suffices. The ribbon is made in lengths of 10 feet, and either 5¾ inches
or 2⅞ inches wide, as desired.
[Illustration: FIG. 18.]
_Roll or Twist._—Roll- or twist-tobacco is made by spinning the leaf
into a rope, and then subjecting it to hot pressure. Until recently, the
spinning was performed by hand, much after the manner of ordinary
rope-making by hand. But this slow process is now superseded by a
machine made by Robinson and Andrew, of Stockport; it is spoken of in
very favourable terms by English manufacturers, and received a diploma
of merit at the Philadelphia Exhibition. The machine consists of a
combination of 3 rollers, whose surfaces are made of segments, to which
lateral to-and-fro motions are given by cams attached to the stands on
which the axles of the rollers rotate. The tobacco occupies the central
space between the 3 rollers, and it is carried through the machine by
the lateral to-and-fro motions given to the segments. The fillers and
wrappers are laid on a table joined to the machine. The filler is placed
in the cover, and they pass together between the rollers, whose action
twists and compresses the tobacco into a roll; this is carried forward
and wound on a bobbin, revolving in an open frame, and provided with a
guide for equalizing the distribution of the tobacco.
[Illustration: FIG. 19.
FIG. 20.]
[Illustration: FIG. 21.]
The machine is shown in Figs. 19 (elevation), 20 (plan), and 21 (end
view). The tobacco is laid on the table _a_, provided with a rib _n_, on
which the sliding rest _b_ is free to move to and fro; _c d_ are the two
lower segmental rollers, the axles of which revolve in stationary
bearings; _e_ is the top roller, the axle of which revolves in sliding
bearings, fitting in the swing-frame _f_, and each acted upon by a
spring _o_, pressing on a pin communicating with the bearing, and
putting an elastic pressure on the tobacco.
[Illustration: FIG. 22.
FIG. 23.]
Each segment-roller consists of an axle with four segments, best shown
in Figs. 22 and 23. The outer shell of the segments is made of hard
wood, fitting an inner shell of malleable cast-iron, the projections on
which suit grooves on the cast-iron axle. The segments of the rollers _c
d_ are moved laterally to and fro by the wedge-shaped cams _p q r s_,
fixed to the bearings of the roller-axles; and the segments of the
roller _e_ are moved in the same manner by cams _t u_, fixed to the
swing-frame _f_. The tobacco occupies the central space between the 3
rollers, and the cams _p r t_ move the segments in the direction of the
arrow where they touch the tobacco, while the cams _q s u_ move them
back. After the tobacco has passed beyond the segment-rollers, it goes
through the hollow trunnion of the open frame _g_, in which the bobbin
_h_ revolves; the other trunnion of the frame _g_ is provided with fast
and loose pulleys, by which the whole machine is driven. To this
trunnion, are also fixed an ordinary friction-break pulley, and a
grooved pulley, around which latter passes a band for driving the pulley
on the axle of the bobbin _h_. To the other end of the axle of the
bobbin, is fixed a pinion, which, by means of a toothed chain, gives
motion to another pinion fixed to the double screw _i_; this double
screw gives a traversing to-and-fro motion to the guide _j_, for
distributing the tobacco evenly on the bobbin, by means of a swivel
=T=-headed stud, connected with the guide, and taking into the thread of
the double screw. The guide is provided with two horizontal grooved
rollers, between which the tobacco passes, and with two other rollers to
guide the tobacco on to the bobbin.
Rotary motion is communicated to the segment-rollers _c d e_ as
follows:—To the hollow trunnion of the open frame _g_, is affixed a
pinion, which drives the wheel _k_, on the same shaft as the
change-pinion that drives the wheel gearing into the pinions on the
axles of the rollers _c_ and _d_, and one of which pinions gears into
the intermediate pinion _l_, which drives the pinions on the axle of the
roller _e_. The driving-strap is held upon the fast pulley by a
drop-catch acting on a weighted lever, one arm of which is connected by
a link to the lower end of a strap fork-lever. When it is requisite to
stop the machine, the attendant kicks the point of a catch off the end
of the lever, which is then raised by the weight, and so moves the
driving-strap from the fast to the loose pulley, the stoppage being
virtually instantaneous. The mode of working is as follows:—The spinner
and assistants stand at opposite sides of the table; the fillers and
wrappers being placed on the table, one assistant spreads out the
wrapper and pushes the end towards the filler, which the spinner
supplies and holds against the sliding-rest _b_; the rotary motion of
the segment-rollers _c d e_ twists the tobacco, and causes the wrapper
to be wound over the filler, and the rest _b_, being movable, enables
the spinner to regulate its position according to the quantity and
quality of the filler and wrapper. The lateral motion of the
segment-rollers passes the roll towards the bobbin, on which it is
wound, as described. The combined rotary and traversing motions of the
rollers consolidate the tobacco, and put the desired face upon the
twist. The roller _e_ is supported in a swing-frame, which is lifted off
the tobacco when starting the machine. When the machine is at work, the
swing-frame is held down by the stud _m_ (Fig. 19). The figures
represent a machine suitable for manufacturing Limerick roll; for
pigtail and other small descriptions, it is necessary to reduce the
diameter of one or more of the segment-rollers.
[Illustration: FIG. 24.]
[Illustration: FIG. 25.]
A more recent improvement in this machine, by J. E. A. Andrew, is shown
in Figs. 24 (side view), 25 (transverse section), and 26 (plan). The
table _a_, rib _n_, and sliding-rest _b_, and two lower segment-rollers
_c d_, are constructed as usual; but the axles of the segment-rollers
revolve in bearings _g h_, bolted to the flanges of swivel-frames _i k_,
hinged upon the fulcrum-shaft _x_; the object of thus supporting the
bottom rollers _c d_ is to be able to vary the distance between them
according to the thickness of the twist of tobacco that is being rolled.
When the distance between the rollers is fixed, the bearings are secured
by bolts passing through segmental slots. The solid top roller _e_
revolves in centres in sliding bearings fitting in the swing-frame _f_.
[Illustration: FIG. 26.]
As the bobbin is filled, it is removed, and replaced by an empty one.
The rope is then unwound, and formed into rolls, by the aid of a spindle
with flanges at the sides, worked by a treadle, under a cushioned weight
which squeezes the coils closely together as they are wound. The
completed rolls are subjected to great pressure in steam-jacketed
presses, in the same way, and with the same object, as the cakes or
plugs.
_Cigars._—Cigars are composed of two parts, a core formed of pieces of
leaf placed longitudinally, known as "fillers," and a covering formed of
perfect leaf, called the "wrapper." Probably all the best cigars are
made by hand, the only tools required being a short-bladed sharp knife,
a receptacle containing an emulsion of gum, and a square wooden disc or
"cutting-board." A portion of perfect leaf is first shaped to form the
wrapper of the cigar; then a bunch of fillers is moulded in the hand,
and rolled up tightly in the wrapper, the taper end being secured by
gumming. Expert workmen make the cigars remarkably uniform in weight and
shape. When made, they are sorted according to colour, deftly trimmed at
the thick end, and placed in their boxes in cupboards heated by
gas-stoves to finally dry or season before being stored for sale.
In America, machinery is introduced wherever possible. Moulds for
shaping the cigars are made of hard wood, sometimes partially lined with
tin, and of every possible size and form. A machine is made by Dubrul
and Co., of Cincinnati, for working 3 sets of moulds at once, 2 being
kept filled up under pressure while the 3rd is being filled, or the
bunches are being rolled up. A handy little machine for rolling the
fillers for cigars is that known as Henneman's, made by Dubrul and Co.
The demand for scrap-made cigars, or those manufactured with short
fillers, has caused the introduction of machines for cutting and sifting
scrap. One made by Dubrul and Co. is shown in Fig. 27. It consists
essentially of a cylinder formed of hook-shaped, double-edged steel
blades, revolving against 3 series of fixed but adjustable steel blades,
thus permitting the size to be regulated at will.
[Illustration: FIG. 27.]
_Cigarettes._—Cigarettes consist of paper tubes filled with cut tobacco,
with or without an external wrapper of leaf tobacco. Preference is
usually given to those made by hand, but machines have been introduced
with some success for making the commoner kinds. A French machine for
making cigarettes is shown in Fig. 28. Its work consists in making the
paper tubes, and filling them with tobacco. The paper, previously
prepared, in a band about 3 inches wide, is unrolled from the coil _a_
by means of the carriage _b_, and cut off in pieces about 1 inch long
for presentation to the mandrel _c_, temporarily introduced into one of
the tubes of the mould-carrier _d_. The mandrel has a clamp which grasps
the paper and rolls it, and, at the moment when the latter escapes from
the carriage, its free end is brought upon a rubber pad covered with
gum, hidden in the illustration. The paper tube is left in the mould,
the mandrel being extracted by means of the cam _e_; the mould-carrier
is then turned ⅟<sub>9</sub> revolution by the cam _f_, a new tube comes
into line, and the operation is repeated. When 6 paper tubes are
completed, the first one is pushed by a small piston, actuated by the
cam _g_, upon the end of the filling-tube; and immediately the rod _h_,
actuated by the cam _e_, drives into this tube a portion of tobacco
already prepared in the compressor _i_. In preparing the tobacco, a
workman, occupying the seat _m_, is necessary to dispose the material in
regular layers on a carrier, by which it is transported into the
compressor. When the cigarette-envelope is filled, the mould-carrier
again makes part of a revolution, and the finished cigarette is pushed
out of the mould by the rod _k_, also actuated by the cam _e_; a device
finally lodges the cigarettes in the box _l_. One workman is said to be
able to turn out 9600 cigarettes in 10 hours by the aid of the machine.
[Illustration: FIG. 28.]
_Snuff._—Snuff is entitled to the last place in the series of tobacco
manufactures, as it is largely made up of the scraps, cuttings, and
rejections of the preceding processes. The materials are chopped very
fine, placed in heaps in warm damp cellars, "doctored" with various
flavourings, left to ferment for several weeks, and then ground to
powder in edge-runner mills, some kinds even undergoing a slight
roasting. When ground, the mass is passed through "mulls," wood-lined,
bottomless bowls, let into a bench, where the snuff is softened and
rendered less powdery by means of pointed pins, resembling domestic
rolling-pins, which slowly travel around the sides of the bowls. Snuff
represents a highly profitable article manufactured from materials that
are otherwise useless, and depending for its flavour chiefly upon the
perfumes and flavourings used. Hence these last are kept profoundly
secret by the manufacturer.
From refuse tobacco which is unfit for any other purpose, is made a
decoction for washing sheep and destroying vermin; often the waste is
ground very fine, and used by gardeners, presumably to keep noxious
insects away.
[Illustration: FIG. 29.]
_Miscellaneous Appliances._—The customary ingenuity of the Americans has
invented a profusion of admirable labour-saving machines for almost all
the operations of the tobacco manufacturer. A few of these only can be
noticed in the present article.
Fig. 29 shows a portable resweating-apparatus, intended for darkening
the colour of tobacco to suit the dealer's market. It measures 4 feet
long, 3 feet wide, and 5 feet high, being just large enough for one case
(400 lb.) of tobacco, including the case; it consists of a water-tank
_a_, a pipe _b_ for conducting the water into the metallic pan _c_, at
the bottom of the apparatus, which is heated by gas-jets _d_. The
tobacco is introduced by the door _e_, which is fitted with a
thermometer. The roof is sloped so as to determine the flow of the water
of condensation. The steaming occupies 3–5 days, and needs occasional
watching. The apparatus is made by C. S. Philips and Co., 188 Pearl
Street, New York.
[Illustration: FIG. 30.]
Fig. 30 illustrates a complicated machine, introduced by C. C. Clawson
and Co., of Raleigh, N. Carolina, for putting up large quantities of
tobacco in parcels of 2 oz. upwards. It consists of a central table
provided with automatic scales for weighing out the portion; four
equidistant guides which determine the form of the package; a plunger
for packing, and a follower for raising the package; a side-table
carrying tongs for holding the empty bags; and another to receive the
packages, and hold them during tying. The hopper being supplied with
tobacco, and the machine put in motion, each form takes a bag from the
tong-table, and the article having been weighed, is carried to the form
by a shute, when it drops into the bag, is packed by the plunger, and
transferred to the tying-table. With 2 girls or boys, it is said to
weigh, pack, and tie 30 bags a minute.
The New York Tobacco Machine Co. make two forms of machines for
granulating tobacco, chiefly for making "Killickinick" and cigarettes,
their working capacity ranging from 200 to 2000 lb. a day. The
cutting-rollers are covered with cross-millings at right angles to each
other, those running lengthwise being deep; the fixed cutters are
adjustable, so that the cutting may be either coarse or fine. When
working, the action is like that of a pair of shears, except that the
cross-millings reduce the strips to a granular state. Both stems and
leaves may be worked up. The great advantage claimed for these machines
is that, though the tobacco should be dry, the percentage of dust
escaping is reduced to a nominal figure.
A cutting-machine made by the same Co. is shown in Fig. 31. It is
adapted to cut leaf, stem, scrap, plug, or any form of tobacco, to any
required degree of fineness, turning out 300–400 lb. a day. The action
is almost precisely that of a chaff-cutter. The Co.'s sifting-machine
consists of an adjustable cylindrical wire sieve, with a rattan-broom
screw-roller revolving inside. The stems are stripped and worked out at
one end, while the remainder is broken up, and passed through the sieve,
falling upon a perforated tray, through which pass the finest particles
for snuff-making. A machine largely used in America is the stem-roller,
for crushing and flattening the stems so that they may be used like
leaves for making cigars. Great benefit is anticipated in the United
States from the adaptation of Ryerson's "attrition mill" to
snuff-grinding, owing to the fact that the pulverization is accomplished
without the particles being heated in the least degree. Of
cigarette-making machines, there are many kinds; the best are those
which deal with the tobacco in a comparatively dry state, thus
preventing shrinkage after packing.
[Illustration: FIG. 31.]
Indebtedness is acknowledged to Hy. Archer and Co., Borough, S.E., and
T. Brankston and Co., Carter Lane, Doctors' Commons, for opportunities
of inspecting their thoroughly representative works, and for much
information readily given concerning the manufacture in this country; to
W. Jollyman, of W. D. and H. O. Wills' London house, for having revised
these sheets before going to press; and to Hy. A. Forrest, 61 Broadway,
agent of the New York Tobacco Machine Co., for valuable material
relating to American machines and processes.
CHAPTER VI.
NATURE AND PROPERTIES.
The active principle of tobacco is a volatile, highly poisonous
alkaloid, called Nicotine (C₁₀H₁₄N₂). Although green tobacco-plants
contain generally more nicotine than the leaves after they have been
prepared for the market, yet the odour is only perceptible after the
fermentation of the leaves has set in. It has been ascertained that
young leaves 2 inches long contained 2·8 per cent., and leaves 10½
inches broad and 16 inches long, as much as 5·6 per cent. of their
weight of nicotine. The amount increases as the plants become ripe, and
decreases on their becoming overripe.
Though the narcotic effects of tobacco experienced by the smoker must
partly be attributed to nicotine, it cannot be said that they are solely
due to it. It is well known that the products of combustion of quite
harmless substances are often stupefying. Good Syrian tobacco contains
no nicotine, yet smokers consider cigars made from this tobacco to be
strong. It is evident that the strength of a cigar, as judged by the
smoker, depends greatly on the circumstance whether the tobacco burns
well or not. If it burns well, a greater amount of nicotine is consumed
and decomposed, and less of the narcotic products of combustion are
created, than when it burns badly. Cigars of the latter description,
containing little nicotine, are more narcotic in their effects when
smoked than well-burning cigars containing much nicotine.
The amount of nicotine in tobacco varies very much, according to the
sort of plant, the climate, the nature of the soil in which the plant
grew, the treatment received during its growth, and the course adopted
to prepare the leaf for the market. Dr. Nessler found that good Syrian
tobacco contained no nicotine, Havana tobaccos between 0·6 and 2·0 per
cent., and German tobaccos between 0·7 and 3·3 per cent. Schlösing found
in French tobacco nearly 8·0 per cent. of nicotine. Fine tobaccos
contain generally little or no nicotine. Broughton found that the amount
of nicotine in Indian tobaccos varies very much. The conditions
favourable to the development of nicotine in the plants are:—Soil in a
bad physical state, strong nitrogenous manure, a dry atmosphere, and
probably a low temperature during the growth.
According to Nessler, green and newly-cut tobacco-plants contain no
ammonia; it is developed during the drying and fermentation of the
leaves, especially when they assume a brown colour. Tobacco-leaves,
which have undergone a strong fermentation, contain more ammonia than
those slightly fermented. Fine tobaccos contain generally less ammonia
than coarser ones. In various smoking-tobaccos, Nessler found:—Havana,
0·2 per cent. of ammonia; Cuba, 0·3; Syrian, 0·6; German, 0·9 per cent.
Schlösing found Havana tobacco to contain 0·8 per cent.
Nitric acid, consisting of nitrogen and oxygen, is formed in animal and
plant substances when decomposed under the influence of atmospheric air
and a sufficiently high temperature; whereas ammonia, consisting of
nitrogen and hydrogen, is formed when those substances decompose in the
absence, or nearly so, of atmospheric air. Organic substances
decomposing under the latter condition emit an objectionable pungent
odour, which must partly be attributed to the formation of ammonia.
Tobacco, soon after harvesting, commences, according to the conditions
under which it is placed, one of these decompositions. The extent of the
decomposition the tobacco has gone through may be partly judged from the
colour the leaves have attained. If leaves be dried so rapidly as to
remain green, the decomposition is probably confined to the formation of
carbonic acid. A yellow colour indicates the formation of nitric acid;
and a dark-brown or black colour, that of ammonia. The conditions under
which nitric acid and ammonia are formed being known, it is possible to
control their development. When the tobacco is hung far apart, so that
the air has free access, the formation of nitric acid will take place;
but if the air be excluded more or less, by hanging the tobacco very
close, or pressing it in heaps or pits, the formation of ammonia is
engendered.
Nitric acid generally promotes the combustion of plant substances, by
supplying a portion of the needed oxygen, and has undoubtedly a similar
effect in tobacco; its occurrence in the tobacco is therefore a
desideratum with the cultivator and manufacturer, and to supply any
deficiency, the manufacturer often resorts to impregnating his tobacco
with a solution of saltpetre. From this, however, it must not be
concluded that every tobacco containing a large amount of nitric acid
will necessarily burn well. Schlösing and Nessler have shown that the
well-burning of a tobacco does not always correspond with a great amount
of nitric acid, thus indicating that other substances or other
conditions also affect the combustibility. The effect of the nitric acid
will most probably vary with the base with which it is in combination.
The nitrogen in the forms of nicotine, ammonia, and nitric acid,
constitutes only a small portion of the total amount present in tobacco;
by far the greater portion (⅔–⅞) exists in the form of albuminoids.
Nessler found that the nitrogen under this form varies from 2 to 4 per
cent., which is equal to 13–26 per cent. of albuminoids. Substances rich
in albuminoids generally burn badly, and emit a pungent noxious odour.
On the condition of these albuminoids, and on the presence of other
substances, as nitric acid, alkalies, &c., in the tobacco, mostly depend
the burning qualities of the leaf, and the flavour of a cigar. The
Eastern habit in smoking, from Malaysia, Japan and China, through India,
Persia and Turkey, even to Hungary, is to inhale the smoke into the
lungs, and natives of these countries maintain that a tobacco should be
of full flavour without burning the throat or catching the breath.
Western nations do not admit the smoke further than the mouth, and
therefore require a strong, rank flavour.
Whilst drying and fermenting, the tobacco undergoes great changes. Some
substances are decomposed, others are newly formed. The highly
complicated compounds, the albuminoids, undergo first decomposition, and
in doing so give rise to more simple combinations. Nitric acid, ammonia,
and other substances less known are chiefly, if not entirely, derived
from the products of the decomposition of albuminoids. The substances
that cause the objectionable pungent smell in tobacco are formed from
the broken-up constituents of these high combinations. The conditions
under which these bad-smelling combinations originate are not properly
known; but it is probable that they are developed with, and under the
same conditions that cause the formation of, ammonia, as the
disagreeable pungent flavour is found generally in tobacco that has
undergone fermentation to a great extent. It is believed that the
conditions that favour the development of nicotine are also conducive to
the formation of albuminous substances in the leaf, viz. fresh
nitrogenous manure, bad physical state of the soil, &c.
According to Nessler, the quality of tobacco depends to a great degree
on the amount of cellulose it contains. He found that a good tobacco
invariably contained more than a bad one, Havana yielding as much as 46
per cent. The fact that tobacco burns better after being stored for a
time may be partly due to an increase of cellulose in it.
Every tobacco contains more or less fat, gum, ethereal oil, &c. It is
not properly known in what way fatty matters affect the quality of
tobacco. Many other organic matters exist in tobacco in combination with
substances from which it is most difficult to separate them; they have
not as yet been quantitatively ascertained, and are therefore little
known. Most of them are only developed during the drying and fermenting
of the leaf; their presence, however, considerably affects the quality
of the tobacco.
The amount of ash constituents in the tobacco is considerable, varying
between 16 and 28 per cent. There cannot be said to exist a definite
relation between the total amount of ash in the tobacco and its quality,
as tobaccos yielding much ash are sometimes of good, and at other times
of bad, quality; a good tobacco may yield much or little ash. The
relative proportion in which the ash constituents exist is, however, of
the greatest importance. It has been ascertained that the presence of
some special mineral elements modify to a great extent the quality of
the tobacco. Of all ash constituents, potash (K₂O), more correctly
speaking potassium carbonate (K₂CO₃), affects the quality of tobacco in
the highest degree. Schlösing has pointed out that the good burning
qualities of a tobacco depend on the presence in it of potash in
combination with a vegetable acid; that a soil deficient in potash is
unfit to produce tobacco of good quality. Numerous analyses have tended
not only to corroborate the assertion made by Schlösing, but to
demonstrate also, that it is not the total amount of potash, but the
potash found as a carbonate, which existed in the plant in combination
with a vegetable acid, that is the constituent chiefly affecting the
combustibility of a tobacco. The complete analyses of Nessler have shown
that, although a tobacco may contain a great amount of potash, it does
not necessarily follow that the tobacco burns well. He found that some
German tobaccos contained more potash than Havana, although the latter
burned much better than the former; and that a great amount of potash
did not always indicate a great amount of carbonate of potash. Although
tobaccos yielding a great amount of carbonate of potash in their ash
generally burn well, there may be conditions which neutralize the good
effect of this combination, as a large proportion of albuminoids. It may
therefore be said that the combustibility of a tobacco is improved in
proportion as its ash yields more carbonate of potash, other conditions
being equal.
Among the minor salts, the chlorides deserve most attention. It has been
found that they generally <DW44> the burning of tobacco, and, that as
they increase, carbonate of potash decreases. Lime is invariably found
more or less in the ash, but it has not been ascertained to what extent
its presence affects the quality of the tobacco; good tobacco may
contain much or little, so that its presence is probably not of great
importance. The same may be said of soda, magnesia, and phosphoric acid.
According to Nessler, their proportions may vary thus:—Potash, 1·95–5
per cent.; lime, 6·5–9·2; soda, 0–1·63; magnesia, 0·12–0·99; phosphoric
acid, 0·57–1·39.
In connection with the chemistry of tobacco, and the rational manuring
of the crop, the name of Prof. S. W. Johnson, Chemist to the Connecticut
State Board of Agriculture, must be placed in the foremost rank.
Indebtedness is acknowledged to Prof. Johnson for a copy of his valuable
report, quoted in the Bibliography at the end of this work.
In November, 1884, a paper was read by Dr. John Clark, on the
composition of tobacco, before the Society of Chemical Industry, which
is sufficiently interesting to be quoted at length.
Dr. Clark remarks that the "tobacco plant is very extensively cultivated
in various parts of the world, and after it has reached its maturity it
is cut and dried on poles. When the plant is in proper condition, the
leaves are stripped from the stalk, sorted and cured, by which means
they are converted into the tobacco of commerce. The good leaves are
called 'wrappers,' and the infirm or defective ones, which are separated
from the others, are called 'mediums and fillers.' The term 'strips' is
applied to tobacco leaves, from which 20 to 25 per cent. of the stem or
midrib has been removed to suit the requirements of manufacturers in
this country more especially. Tobacco is largely imported into the
United Kingdom, partly in the manufactured state, but principally in the
unmanufactured or leaf form.
"Through the kindness of a well-known firm of tobacco manufacturers, I
have been furnished with authentic samples of the principal varieties of
leaf tobacco, imported into this country, and the accompanying table
gives the proportions of mineral matter or ash, alkaline salts, and
sand, which these contain. For the sake of comparison the results are
all stated in the dry tobacco, and in order to ensure greater accuracy,
the analysis was, in each case, made with several leaves, which were
separated into laminæ and stem, and the whole of each incinerated. The
difference in the composition of the laminæ and the stem is very marked,
especially as regards alkaline salts, and is of importance more
especially to the snuff manufacturer.
COMPOSITION OF VARIOUS KINDS OF LEAF TOBACCO.
──────────────────┬─────────────────┬─────────────────┬─────────────────
│ Whole Leaf. │ Laminæ. │ Stem.
│Dried at 212° F.,│Dried at 212° F.,│Dried at 212° F.,
│ per cent. │ per cent. │ per cent.
──────────────────┼─────┬─────┬─────┼─────┬─────┬─────┼─────┬─────┬─────
│ Ash.│ Alk.│Sand.│ Ash.│ Alk.│Sand.│ Ash.│ Alk.│Sand.
│ │Salt.│ │ │Salt.│ │ │Salt.│
──────────────────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────
U. S. Kentucky │19·11│ 6·84│ 2·57│18·93│ 5·43│ 3·06│21·69│13·51│ ·68
do. │18·50│ 6·68│ 1·82│15·50│ 2·77│ 2·39│26·07│16·68│ ·38
do. │25·99│ 9·69│ 3·51│24·88│ 6·70│ 4·17│29·36│20·01│ 1·10
do. Strips │15·73│ 4·31│ 2·61│15·57│ 4·07│ 2·71│16·95│ 6·35│ 1·37
U. S. Missouri │20·96│ 5·07│ 4·63│20·46│ 2·62│ 5·27│22·61│12·72│ 1·90
do. │22·01│ 6·32│ 3·51│21·36│ 4·96│ 3·88│23·62│12·37│ 1·53
do. │18·88│ 4·81│ 2·61│17·18│ 2·88│ 3·21│22·17│10·68│ ·92
do. │18·36│ 4·60│ 3·44│17·05│ 2·50│ 4·07│22·39│11·10│ 1·49
U. S. N. Carolina │14·50│ 5·99│ ·63│12·98│ 3·92│ ·74│18·64│11·72│ ·23
Paraguay │30·80│ 8·15│12·32│31·07│ 6·37│14·41│30·37│14·78│ 4·91
Brazil—Carmen │20·54│ 7·81│ ·42│20·42│ 7·24│ ·46│20·86│ 9·37│ ·31
Holland │21·83│11·37│ ·13│20·16│ 8·99│ ·55│25·15│17·20│ ·12
Turkey—Cavallo │13·79│ 5·05│ 3·06│21·86│ 8·28│ ·72│15·44│ 7·73│ ·24
do. Latakia │19·50│ 7·19│ ·55│21·86│ 8·28│ ·72│15·44│ 7·73│ ·24
do. Samsoun │18·39│ 6·98│ ·49│17·59│ 5·32│ ·44│21·72│13·42│ ·60
Japan │15·67│ 6·86│ ·50│14·60│ 5·59│ ·54│19·84│11·55│ ·35
China │18·58│ 2·40│ 6·30│17·94│ 1·66│ 6·94│20·57│ 5·27│ 3·61
Havana │20·99│ 8·19│ 1·02│20·91│ 7·51│ 1·04│21·02│10·33│ ·92
Manilla │21·80│ 6·54│ ·14│21·25│ 5·49│ ·13│22·50│ 9·09│ ·14
German │22·27│ 3·76│ 1·79│22·12│ 2·78│ 1·87│23·13│ 4·63│ 1·39
Sumatra │18·61│ 7·20│ ·13│18·71│ 6·59│ ·09│18·14│ 9·11│ ·28
──────────────────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────
────────────────┬───────────┬───────────┬───────────
│ Average │ Average │ Average
│ of │ of │ of
│Whole Leaf,│ Laminæ, │ Stem,
│ per cent. │ per cent. │ per cent.
Ash or Inorganic│ 20·32 │ 19·21 │ 21·92
Alk. Salts │ 6·47 │ 4·98 │ 11·41
Sand │ 2·48 │ 2·86 │ 1·15
────────────────┴───────────┴───────────┴───────────
"The unmanufactured tobacco which is imported into this country, is
converted into roll or spun tobacco, cut tobacco and cigars, and the
refuse is used for making snuff. Roll tobacco is the staple manufacture
in Scotland and Ireland, and cut tobacco the staple article in England.
"In the manufacture of roll tobacco, the leaves are moistened with
water, spun into various sizes of twist, made up into rolls, and
pressed. The liquid or juice which exudes under pressure is used as a
sheep dip. Cut tobacco is made by moistening the leaves, cutting them
into the desired size, and drying on plates. Sometimes it is made into
cakes in the first instance, and afterwards cut.
"When we compare the composition of roll and cut tobaccos with that of
the leaf from which they are made, we find that the difference lies
almost entirely in the amount of moisture, and as manufacturers are not
allowed to add anything but water and a little oil to tobacco, you will
not err very much in assuming that as a rule the cheapest qualities of
roll and cut tobaccos contain most water. Thus in 15 samples of the
cheapest roll tobacco I found an average of 41·66 per cent. of water.
"The lowest qualities of cut tobacco, such as are largely manufactured
and consumed in England, contain as much water as the cheapest roll
tobacco, whereas the finer qualities of cut tobacco contain as a rule
from 14 to 22 per cent. Cigars, even the cheapest, are comparatively
dry, and contain, as a rule, only from 10 to 12 per cent. of water.
"The difference in cheap cigars is due chiefly to the weight of the
material, but also to the quality of the tobacco and the labour,
machinery being used in the manufacture of the lower qualities, whereas
the higher qualities are nearly all hand made.
"The large quantity of water contained in the cheapest tobacco, and
which frequently amounts to about 50 per cent., is not, in my opinion,
introduced to please the palate of the working man, but simply on
account of the keen competition between rival manufacturers, and the low
price at which tobacco is sold; and in the interest both of the working
classes and of tobacco manufacturers themselves, I think it is very
desirable that some limit should be placed to the amount of water which
may be sold as tobacco.
"Snuff.—I stated that the refuse tobacco was employed in the manufacture
of snuff. This refuse consists of stems, tobacco smalls, and sweepings.
These are moistened with water, subjected to a process of fermentation,
which lasts from about six weeks to two months, then ground, mixed with
alkaline salts to preserve the snuff, and flavoured when desired.
Nothing is allowed to be added to snuff except the carbonates,
chlorides, and sulphates of potash and soda, and carbonate of ammonia.
It is also provided by Act of Parliament that any snuff found to
contain, after being dried at 212° F., more than 26 per cent. of such
salts, including those naturally in the tobacco, will be liable to
forfeiture and a penalty of 50_l._ From my table of analyses you will
observe that not only does the proportion of alkaline salts vary in
different tobaccos, but the stem contains a much larger proportion than
the leaf. On this account it is necessary that the snuff manufacturer
should know the quantity of alkaline salts in his snuff material, in
order to obtain an article of uniform composition. Some manufacturers go
by rule of thumb, and in attempting to work close to the legal limit,
they run a serious risk of unintentionally incurring the penalty. As a
matter of fact, three samples of snuff, in 1883, were condemned by the
Somerset House authorities because they contained an excessive
proportion of alkaline salts, and the manufacturers were prosecuted. The
more intelligent of the snuff manufacturers, however, analyse their
snuff material, and are thus able to keep within the legal limit.
"The principal alkaline salts which are added to snuff are chloride of
sodium or common salt, carbonate of potash, and carbonate of ammonia,
all of which are allowed by Act of Parliament, and therefore no
exception can be taken to their addition, so long as the total quantity
does not exceed 26 per cent. in the dry snuff. In addition to alkaline
salts, snuffs usually contain from 25 to 45 per cent. of water, with the
exception of a kind of snuff called 'High Toast or Irish Blackguard,'
which is very dry and contains from 5 to 8 per cent. Sometimes they also
contain a considerable quantity of sand. In the several hundred samples
of snuff which I have had occasion to examine for different
manufacturers the average quantity of sand was about 5 per cent. in the
dry snuff, and sometimes fell as low as a half per cent., but in many
samples the quantity exceeded 10 per cent., and in one case I found as
much as 30·94 per cent. of sand in the dry snuff. The greater part of
this sand is probably derived from the sweepings of tobacco, on which
duty has been paid, and I have no doubt the snuff manufacturer considers
himself justified in selling it as snuff. But it appears to me to be
very desirable in the interest of snuffers, that some limit should be
placed on the quantity of sand which may be sold as snuff: more
especially as the particles of sand are frequently very sharp, and have
a tendency to produce inflammation of the mucous membrane of the nose,
and it is to this, probably, that we owe the popular notion that snuff
is sometimes mixed with ground glass to give it additional piquancy.
"When from any cause snuff is spoiled, the manufacturer may export it,
and obtain a drawback of 3_s._ 7_d._ per lb. on the real tobacco which
it contains.
"The Government standard for tobacco is as follows:
Per cent.
Organic matter 70·52
Inorganic 15·48
Water 14·00
——————
100·00
"This is equal to 18 per cent. of ash or inorganic matter in the dry
tobacco. This standard is in my opinion too high, as the average
percentage of inorganic or ash in the dry leaf tobaccos which I have
examined is 20·32, and the stem from which snuff is largely made
contains still more. The result is that the tobacco manufacturer not
only loses the value of the tobacco over and above the duty, but also a
part of the duty which he has paid. This matter concerns the tobacco
manufacturer alone, but I would point out that the authorities in
Somerset House in fixing such a high standard for tobacco are benefiting
the public at the expense of the manufacturer, whereas in the case of
milk the low standard which they employ is a loss to the public and gain
to the dishonest dealer."
CHAPTER VII.
ADULTERATIONS AND SUBSTITUTES.
It is said that in Thuringia, over 1000 tons yearly of dried
beetroot-leaves are passed off as tobacco. These leaves, and those of
chicory and cabbage, are similarly employed in Magdeburg and the
Palatinate. Many of the _Vevey_ cigars of S. Germany are entirely
composed of cabbage- and beetroot-leaves which have been steeped in
tobacco-water for a long time. Other leaves, such as rhubarb, dock,
burdock, and coltsfoot are also used. These are all principally for
cigars. For smoking-tobacco, chamomile flowers, exhausted in water, then
dyed and sweetened with logwood and liquorice, and dried, have been
mixed with tobacco in such proportions as 70–80 per cent. In America, a
specially-prepared brown paper, saturated with the juice expressed from
tobacco-stems and other refuse, is most extensively used, not only for
the "wrappers" of cigars, but also for "filling." Various ground woods,
starches, meals, and pigments are introduced into snuff.
A New York paper mentions that a great quantity of brown straw paper
lately reached Havana, which was to be employed in the manufacture of
Havana cigars. Straw paper impregnated with the juice of tobacco stalks
is wound up with the leaf in such a way that it is often impossible to
detect the adulteration. Dr. Jacobson, writing in the _Industrie
Blätter_, remarks that there is no difficulty in escaping detection, if
the paper be specially prepared for the purpose out of suitable raw
materials. It has long been known that cigar paper soaked in a solution
of soluble glass gives forth no smell of paper on being burnt.
Patent No. 210,538, issued from the United States Patent Office,
December 3, 1878, states the ingredients of a "substitute" to
be—spikenard, red clover, hyssop, hops, slippery-elm bark, tarred rope,
pennyroyal, mullein leaves, kinnikinic, wild cherry bark, and ginseng.
This is an ingenious combination intended to approach in effect,
appearance, and aroma, tobacco; and in so far it might be said to be a
success: as mullein leaves are reputed to be feebly narcotic, hops are
known to possess anodyne properties, clover and hyssop are pectoral in
effect, and slippery-elm febrifuge. Ginseng is aromatic and pungent, and
has a great reputation among the Chinese as a stimulant and restorative.
The tarred rope, we presume, is intended to add to the pyrognostic value
of the mixture. The great point in selecting material for the
fabrication of a mixture of this description is to get leaves containing
a fair percentage of nitrate of potass, as does tobacco; on this depends
its pyrognostic value, and that, next to aroma, is everything.
"Tobacco, like those who smoke it, is credited with many sins of which
it is guiltless. The 'loss of health' so often laid at its door is
probably due in many instances not to tobacco itself, but to some
villainous compound bearing its name. A story told by the principal of
the laboratory of the Inland Revenue Department in his report for the
past year shows how easily this may happen. The supervisor at
Birmingham, observing that an article was being sold at a very cheap
rate in packets, under the name of 'smoking mixture,' sent a sample to
the Inland Revenue laboratory for examination, and it being found to
contain a large proportion of vegetable matter resembling the broken-up
heads of camomile flowers, further inquiry led to the discovery of the
manufactory. The process of manufacture consisted in exhausting the
bitter principle of camomile flower-heads with water, and then dyeing
and sweetening them with a solution of logwood and liquorice, which
brought them, when dried, somewhat to the colour of tobacco. The heads,
when broken up, were then mixed with from 20 to 30 per cent. of cut
tobacco, according to the price at which the mixture was to be sold. The
mixture was supplied to retailers in packets labelled 'The New Smoking
Mixture, Analysed and Approved,' and as agencies had already been
established in several towns, an extensive trade would no doubt soon
have arisen had the manufactory not been suppressed at an early stage of
its existence."
The United States Consul at Smyrna puts the following statement in his
report of January 15, 1883.
Since the establishment of the tobacco monopoly in Turkey, snuff may be
said to be one of the several articles that undergo the most
unscrupulous adulteration. Owing to the high amount of duties imposed on
tobacco by the Turkish Government, and the large profits licensed
manufacturers expect to make on the same, the poorer classes cannot
afford to use the products of doubtful purity coming from the factories,
and so are altogether at the mercy of the clandestine manufacturer and
retailer, who, in order to make the most he can of his vile industry,
adulterates his snuff to such an extent that it can be safely said that
his products contain on an average from 60 to 70 per cent. of inferior
Persian tobacco (tumbeki), fragments of country tobacco leaf, and
tobacco of cigarettes picked up in the streets by beggars, the 30 to 40
per cent. consisting of walnut sawdust, terra umbra, fine sifted sand,
and scum of lead (lead oxide), covered with inferior black writing-ink.
The snuff is manufactured in Smyrna, as follows:
The conscientious manufacturer uses Persian hookah tobacco (tumbeki) and
the fragments of country tobacco-leaf with black ink. These
tobaccos, ground as fine as possible and mixed with grape molasses, are
put in a covered barrel to ferment. Two or three days later the snuff is
taken out and spread in the sun to dry partly, and then rubbed with the
hands and passed through iron wire sieves to be granulated.
The product is afterwards scented with powdered orris root, tonka beans,
and geranium oil; the superior qualities are scented with essences of
roses and jessamine and put up in packages.
The adulterated article is manufactured in the same manner with the
addition of the above-named substances.
The only persons using genuine snuff in this city are the Catholic
priests, who import it directly from France, Italy, Spain and Holland,
and enjoy the privilege of paying no custom-house duties.
CHAPTER VIII.
IMPORTS, DUTIES, VALUES, AND CONSUMPTION.
A comparison of the taxation of the chief nations of the world for the
consumption of tobacco has been published in the _Imperial Statistics of
Germany_. Of the countries where the sale is a Government monopoly,
France last year stood first, the gross duty, with profits, amounting to
7_s._ 1½_d._ per head of the population annually, the net revenue from
the article being 5_s._ 8¼_d._ per head. In Austria the gross was 5_s._
5¾_d._, the net, 3_s._ 5_d._; in Hungary, the gross 3_s._ 3½_d._, the
net 1_s._ 7_d._; in Italy, the gross 3_s._ 11_d._, and the net 2_s._
8¼_d._ In Great Britain, the duty and licenses brought in 4_s._ 10¾_d._
per head of the population for the year, and in the United States 4_s._
4½_d._ In Germany, on the other hand, where the duty was very light, the
average was no more than 7¾_d._ per head of the population.
The duties on unmanufactured tobacco are 3_s._ 6_d._ a lb. when it
contains 10 per cent. or more of moisture; 3_s._ 10_d._ a lb. when it
contains less than 10 per cent. of moisture. Snuff containing no more
than 13 per cent. of moisture, 4_s._ 10_d._ a lb.; 13 per cent. and
upwards, 4_s._ 1_d._ a lb. Cigars pay 5_s._ 6_d._ a lb. Cavendish of
foreign manufacture pays 4_s._ 10_d._ a lb.; that manufactured in bond,
4_s._ 4_d._ Other sorts, including cigarettes, pay 4_s._ 4_d._ a lb.
The approximate relative values in the London market are as
follows:—Maryland, fine yellow, fine, and good , 7–9½_d._ a lb.;
colory, 5–7_d._; light-brown and leafy, 5–(7½)_d._; ordinary and brown,
4–4½_d._ Virginia: Fine Irish and Scotch spinners, 7–10_d._; good and
middling, ordinary light and dry, 6–10_d._; fine black sweet scent, and
middling do., (6½)–(7½)_d._; part blacks, 5–6_d._; ordinary and heated,
3–5_d._; mixed parcels, ordinary and good, middling and fine,
(5½)–(6½)_d._; stripped leaf, 4_d._–1_s._ Kentucky: fine long light
leaf, 7–11_d._; good to middling do., (5½)-(7½)_d._; fine and middling
blacks, 6–8_d._; ordinary and mixed, 2–5_d._; stripped leaf, fine, light
leafy, middling and ordinary, (4½)–11_d._ Negrohead, 11_d._–1_s._ 6_d._
Cavendish, 4½_d._–1_s._ Amersfort and German, 2¾_d._–1_s._ 6_d._ St.
Domingo, 5–(7½)_d._ Havana, Cuba, and Yara, 1_s._ 2_d._–6_s._ Turkish
and Greek, (2½)–9_d._ E. India, Japan, and China, 2–9_d._ Java,
5_d._–2_s._ Colombia (New Granada), 5_d._–2_s._ 6_d._ Manilla,
8_d._–4_s._ Manilla cheroots, 4_s._–7_s._ 6_d._ Havana cigars, 5–40_s._
_Imports, Duties, and Values._—Our imports of tobacco in 1879 were as
follows:—
(_a_) Unmanufactured: From United States, 25,743,880 lb., value
682,253_l._; Holland, 6,215,930 lb., 266,109_l._; China, 1,444,192 lb.,
36,265_l._; Turkey, 1,214,319 lb., 32,627_l._; Japan, 805,928 lb.,
24,003_l._; France, 651,350 lb., 14,585_l._; Belgium, 515,009 lb.,
15,501_l._; Argentine Republic, 470,309 lb., 10,870_l._; Germany,
426,139 lb., 25,602_l._; Straits Settlements, 267,258 lb., 29,718_l._;
British India, 246,305 lb., 3605_l._; New Granada, 241,638 lb.,
9621_l._; Canada, 121,920 lb., 3473_l._; other countries, 497,043 lb.,
14,256_l._; total, 38,861,220 lb., 1,165,488_l._
(_b_) Snuff: From all countries, 7719 lb., value 92_l._
(_c_) Cigars: From Spanish W. Indies, 495,518 lb., value 494,974_l._;
Germany, 150,460 lb., 46,318_l._; Holland, 116,218 lb., 31,348_l._;
Philippines, 80,199 lb., 21,738_l._; France, 73,348 lb., 24,071_l._;
Straits Settlements, 51,191 lb., 13,822_l._; China, 48,762 lb.,
11,240_l._; Belgium, 46,536 lb., 14,211_l._; British India, 33,208 lb.,
10,898_l._; United States, 14,625 lb., 5461_l._; other countries, 43,978
lb., 19,184_l._; total, 1,154,043 lb., 693,265_l._
(_d_) Cavendish or Negrohead: From United States, 2,247,557 lb., value
84,422_l._; other countries, 45,052 lb., 1964_l._; total, 2,292,609 lb.,
86,386_l._
(_e_) Cavendish, manufactured in bond: 33,069 lb., 7126_l._
(_f_) Other sorts, including cigarettes: From United States, 52,206 lb.,
value 7999_l._; Holland, 25,273 lb., 1372_l._; Channel Islands, 15,470
lb., 1279_l._; Germany, 14,474 lb., 4472_l._; France, 9497 lb.,
2368_l._; Belgium, 7939 lb., 2086_l._; other countries, 12,328 lb.,
3845_l._; total, 137,187 lb., 23,421_l._
Following are statistics of the imports of tobacco for the year 1884,
being the latest available.
UNMANUFACTURED TOBACCO.
────────────────────────────────────────────┬──────────┬──────────
│ Quantity.│ Value.
│——————————│——————————
│ lb. │ £
From Germany │ 1,464,350│ 57,435
" Holland │ 5,728,744│ 246,795
" Belgium │ 299,863│ 10,994
" France │ 733,207│ 23,975
" Spain │ 1,265,347│ 24,370
" Malta │ 81,026│ 1,160
" Turkey │ 1,114,143│ 46,545
" Algeria │ 85,580│ 3,081
" British East Indies │ 918,066│ 11,082
" Philippine Islands │ 45,989│ 3,785
│——————————│——————————
Carried forward │11,736,315│ 429,222
│ │
" China and Hong Kong │ 1,813,221│ 63,566
" Japan │ 1,876,787│ 46,081
" British North America │ 150,056│ 5,188
" United States of America │37,186,980│ 1,183,102
" Spanish West India Islands │ 361,095│ 17,972
" United States of Colombia │ 122,570│ 3,589
" Ecuador │ 76,642│ 2,085
" Argentine Republic │ 131,013│ 2,970
" Other Countries │ 75,728│ 2,476
│——————————│——————————
TOTAL │53,530,407│ 1,756,251
────────────────────────────────────────────┴──────────┴──────────
CIGARS.
────────────────────────────────────────────┬──────────┬──────────
│ Quantity.│ Value.
│——————————│——————————
│ lb. │ £
From Denmark │ 2,349│ 1,243
" Germany │ 151,650│ 46,512
" Holland │ 78,471│ 22,231
" Belgium │ 109,388│ 32,789
" Channel Islands │ 2,501│ 1,645
" France │ 49,313│ 24,061
" Gibraltar │ 1,437│ 982
" Malta │ 1,008│ 390
" Greece │ 1,750│ 600
" British Possessions in South Africa │ 1,615│ 687
" British East Indies │ 188,354│ 45,218
" Philippine Islands │ 201,652│ 56,208
" China and Hong Kong │ 25,659│ 6,242
" Australasia │ 3,740│ 883
" United States of America │ 166,740│ 98,510
" British West Indies and Guiana │ 2,313│ 1,198
" Spanish West India Islands │ 467,315│ 453,610
" Danish West India Islands │ 2,448│ 1,519
" Mexico │ 59,727│ 37,249
" United States of Colombia │ 1,004│ 686
" Brazil │ 4,519│ 2,089
" Other Countries │ 4,127│ 2,008
│——————————│———————————
TOTAL │ 1,527,080│ 836,560
────────────────────────────────────────────┴──────────┴──────────
CAVENDISH OR NEGROHEAD.
────────────────────────────────────────────┬──────────┬──────────
│ Quantity.│ Value.
│——————————│——————————
│ lb. │ £
From Channel Islands │ 78,569│ 5,156
" British North America │ 64,910│ 3,244
" United States of America │ 1,243,720│ 59,780
" British West India Islands │ 16,332│ 2,764
" Other Countries │ 32,315│ 1,646
│——————————│——————————
TOTAL │ 1,435,846│ 72,590
────────────────────────────────────────────┴──────────┴──────────
SNUFF.
────────────────────────────────────────────┬──────────┬──────────
│ Quantity.│ Value.
│——————————│——————————
│ lb. │ £
From Brazil │ 4,099│ 830
" Other Countries │ 96│ 24
│——————————│——————————
TOTAL │ 4,195│ 854
────────────────────────────────────────────┴──────────┴──────────
OTHER MANUFACTURED TOBACCO.
────────────────────────────────────────────┬──────────┬──────────
│ Quantity.│ Value.
│——————————│——————————
│ lb. │ £
From Germany │ 9,993│ 2,920
" Holland │ 20,657│ 1,173
" Belgium │ 7,740│ 1,616
" France │ 17,985│ 2,818
" Malta │ 5,968│ 1,592
" Turkey │ 5,444│ 1,674
" Egypt │ 31,662│ 13,306
" Algeria │ 6,410│ 1,580
" United States of America │ 76,472│ 20,039
" Spanish West India Islands │ 6,259│ 865
" Other Countries │ 9,625│ 1,968
│——————————│——————————
TOTAL │ 198,215│ 49,551
────────────────────────────────────────────┴──────────┴──────────
CHAPTER IX.
BIBLIOGRAPHY.
J. Neander.
Tabacologia. Lugduni-Batavorum: 1622.
B. Stella.
Il Tabacco. Rome: 1669.
S. Paulli.
Treatise on Tobacco, &c. London: 1746.
P. Winther.
Tobaks-plantning. Kjoebenhavn: 1773.
J. Carver.
Culture of the Tobacco-plant. London: 1779.
Villeneuve.
Culture, Fabrication et Vente du Tabac. Paris: 1791.
W. Tatham.
Culture and Commerce of Tobacco. London: 1800.
Jens Fr. Becker.
Kort anviisning til tabaks-platning. Viborg: 1809.
J. E. Normann.
Tobaksplantens dyrkning i Norge. Christiania: 1811.
M. de Truchet.
Culture du Tabac en France. Paris: 1816.
M. R. Flor.
Om Tobakavl. Christiania: 1817.
Hermbstädt.
Gründliche Anweisung zur Cultur der Tabakpflanzen.
Berlin: 1822.
T. Brodigan.
Art of Growing and Curing Tobacco in the British Isles.
London: 1830.
J. Jennings.
Practical Treatise on Tobacco. London: 1830.
H. J. Meller.
Nicotiana. London: 1832.
K. C. Antz.
Tabachi historia. Berolini: 1836.
L. A. Demersay.
Du Tabac du Paraguay. Paris: 1851.
Babo und Hofacker.
Der Tabak und sein Anbau. Karlsruhe: 1852.
V. P. G. Demoor.
Culture du Tabac. Luxembourg: 1853.
F. Tiedemann.
Geschichte des Tabaks. Frankfurt: 1854.
C. Fermond.
Monographie du Tabac. Paris: 1857.
A. Steinmetz.
Tobacco. London: 1857.
H. B. Prescott.
Tobacco and its Adulterations. London: 1858.
F. W. Fairholt.
Tobacco; its History. London: 1859, 1876.
M. C. Cooke.
The Seven Sisters of Sleep. London: 1860.
H. Raibaud L'Ange.
Du Tabac en Provence. Paris: 1860.
Nessler.
Der Tabak. Mannheim: 1860.
J. L. P. Fèvre.
Le Tabac. Paris: 1863.
C. E. Guys.
Culture of Latakia Tobacco.
_Technologist_, London: 1863.
Maling.
Tobacco Trade and Cultivation of the District of Cavalla.
_Technologist_, London: 1863.
R. de Coin.
History and Cultivation of Cotton and Tobacco.
London: 1864.
Holzschuher.
Der Tabakbau. Gotha: 1864.
G. A. Henrieck.
Du Tabac. Paris: 1866.
A. Imbert-Courbeyre.
Leçons sur le Tabac. Clermont-Ferrand: 1866.
S. W. Johnson.
Tobacco; Report of Chemist to the
Connecticut State Board of Agriculture. 1873.
A. de Bec.
Culture du Tabac en France. Aix: 1875.
F. A. Allart.
Culture du Tabac. Abbeville: 1876.
B. T. Creighton.
Culture of Tobacco in Ohio.
_Pharmaceutical Journal_, London: 1876.
D. Décobert.
Culture du Tabac. Lille: 1876.
Hofacker und Babo.
Der Tabakbau. Berlin: 1876.
A. Nouvel.
Le Tabac. Brive: 1876.
Notes sur la Culture des Tabacs. Paris: 1876.
R. E. Burton.
Cultivation of Tobacco.
_Sugar Cane_, Manchester: 1877.
G. Cantoni.
L'Industria del Tabacco.
_Annali di Agricoltura_, Rome: 1879.
K. Schiffmayer.
Tobacco and its Culture; Report of Agricultural Department,
Madras Presidency.
Madras: 1879.
F. Alfonso.
Tabacchi della Sicilia. Palermo: 1880.
F. Anderegg.
Tabakbau in der Schweiz. Chur: 1880.
O. Comes.
Tabacco in Italia.
_L'agricolt. meridionale_, Portici: 1881.
K. W. van Gorkom.
De Oost-Indische Cultures. Amsterdam: 1881.
J. H. Zimmermann.
Tabaksbaubüchlein. Aarau: 1881.
J. Clark.
Composition of Tobacco.
_Journal Soc. Chem. Industry_, Manchester: 1884.
A series of Prize Essays on Tobacco Culture in the Southern States of
America, published in pamphlet form by the Orange Judd Co., and
containing much valuable information.
INDEX.
Adulterations of tobacco, 267–70
Afghanistan, tobacco in, 137
African tobacco, 138
After-cultivation, 54–60
Albuminoids in tobacco, 256
American tobacco, 210–22
Amersfort tobacco, 6
Ammonia in tobacco, 255
Analyses of tobacco, 261
—— —— —— plants, 22
Area of nursery, 59
—— to plant, 53
Artificial heat for drying tobacco, 89, 91
Ash of tobacco, 258
Australian tobacco, 141
Austro-Hungarian tobacco, 141
Barrels for tobacco, 125, 127
Best kind of tobacco to grow, 33
Betun, 117
Bibliography of tobacco, 276
Big Frederic tobacco, 34, 37
—— Orinoco tobacco, 35, 37
Black soil for tobacco growing, 10
Blue prior tobacco, 37
—— stalk tobacco, 35
—— vitriol for killing caterpillars, 58
Books on tobacco, 276
Bornean tobacco, 142
Bourbon tobacco, 143
Boxes for tobacco, 126
Brazilian tobacco, 143–7
British tobacco, 164–6
Brittle stem tobacco, 35
Building drying-sheds, 86, 90, 95
Bulking tobacco, 121–5
Bull tongue tobacco, 35
Cake tobacco, 236
Calabrian tobacco, 3
Californian tobacco, 3
Canary Island tobacco, 148
Cases for tobacco, 129
Caterpillars, destroying, 55–60
Cellulose in tobacco, 257
Central Asian tobacco, 3
Chemical ingredients of tobacco soils, 9
Chemistry of tobacco, 259–66
Chinese tobacco, 148
Chlorine compounds to be avoided in tobacco manures, 25
Choosing sort of tobacco, 31–7
Cigarettes, 246–8
Cigars, 244
Classifying tobacco, 109–21
Clay as a tobacco soil, 9, 10
Climate for tobacco growing, 7
Cochin China tobacco, 149
Commerce in tobacco in Afghanistan, 137
—— Africa, 138
—— Australia, 141
—— Austro-Hungary, 141
—— Borneo, 142
—— Bourbon, 143
—— Brazil, 143–7
—— Canary Islands, 148
—— China, 148
—— Cochin China, 149
—— Costa Rica, 149
—— Ecuador, 149
—— Fiji, 150
—— France, 150–6
—— Germany, 156–64
—— Great Britain, 164–6
—— Greece, 166
—— Holland, 166
—— India, 167–76
—— Italy, 176
—— Japan, 176
—— Java, 176
—— New Zealand, 177
—— Nicaragua, 177
—— Paraguay, 178
—— Persia, 178–91
—— Philippines, 191
—— Roumania, 192
—— Russia, 192
—— San Salvador, 192
—— Servia, 192
—— Spain, 192
—— Sumatra, 193–205
—— Turkey, 205–10
—— United States, 210–22
—— Venezuela, 222
—— West Indies, 223–30
Conditions of drying-house, 78
Connecticut seed-leaf, 34
Consumption of tobacco, 271
Corn as a shelter for tobacco, 65
Costa Rica tobacco, 149
Crops adapted for rotation with tobacco, 31
Cuban drying practices, 101
—— harvesting practices, 83–6
—— manuring practices, 15
—— planting practices, 51–4
—— tobacco, 2, 3, 34, 37
Cultivation of tobacco, 7–66
Curing practices in Cuba, 83–6
—— tobacco, 67–136
Cutting machine, 251
—— tobacco for smoking, 233–5
—— —— plants, 68
Damping tobacco, 231
Destroying insects, 55–60
Distance in planting, 50
Doctoring tobacco, 133–6
Drying-house, 75
—— -sheds, building, 86, 90, 95
—— ——, sizes of, 90
—— tobacco, 86–104
—— —— for smoking, 235
Dung for tobacco soils, 14
Dutch tobacco, 166
Duties on tobacco, 271
Dyeing the leaves, 119
Ecuador tobacco, 149
Elements needed by tobacco, 11–29
European tobacco plant, 4
Examining tobacco while drying, 88
Fermenting tobacco, 121–5
Fertilizers, principles of, 18–22
Field, preparing, 48–50
Fiji tobacco, 150
Filling vacancies, 53
Flavouring tobacco, 133–6
Flowers of tobacco plants, 3–6
Foliage of tobacco plants, 3–6
Fowls, protecting seed-beds from, 46
French tobacco, 150–6
Frost at harvest time, 79
——, protecting seed-beds from, 43, 47
German tobacco, 156–64
Grades of tobacco, 109–21
Graham tobacco, 35
Granulating machine, 251
Greek tobacco, 166
Green-soiling for tobacco, 14
Guano for tobacco soil, 16
Gypsum for tobacco soil, 26
Hands of tobacco, 107
Hanging leaves in sheds, 88, 93, 95
—— split leaves, 81
—— tobacco, 72
Harvesting for small planters, 76
—— tobacco, 67–86
Hoeing plants, 54
Hogsheads for tobacco, 126, 128
Hot-bed for seedlings, 44
Hungarian tobacco, 3
Imports of tobacco, 271–5
Improving tobacco, 133–6
Indian tobacco, 167–76
Italian tobacco, 176
Japanese tobacco, 176
Javanese tobacco, 176
Judging condition of leaf, 74
Kainit for tobacco soil, 24
Kentucky leaf, 36
Latakia tobacco, 2, 3
Levant tobacco, 3
Lime for tobacco soil, 28
Literature on tobacco, 276
Little Frederic tobacco, 34, 37
—— Orinoco tobacco, 35, 37
Loading cut tobacco leaves, 70
Loam as a tobacco soil, 10
Magnesia for tobacco soil, 28
Manilla tobacco, 3
Manuring, principles of, 18–22
—— seedlings, 41
—— tobacco, 11–29
Maryland tobacco, 1
—— —— as a crop, 33
Mat for keeping frost off, 47
Mexican tobacco, 3
Moth of tobacco worm, 56
Nature of tobacco, 253–66
New Zealand tobacco, 177
Nicaraguan tobacco, 177
Nicotine, 253
Nitrates for tobacco soils, 13
Nitric acid in tobacco, 255
Nurseries, 38–48
——, shelter for, 38
——, situation for, 38
——, soil for, 38
Nursery, area of, 39
Organic matter in tobacco soils, 8
Packing tobacco, 125–33
Paraguay tobacco, 178
Pegging tobacco, 71
Pennsylvania seed-leaf, 37
Persian tobacco, 3, 178–91
Philippine tobacco, 191
Picking tobacco, 104–9
Planting out, 50–4
Plug tobacco, 236
Pole-burn, 78
Potash for tobacco growing, 11, 27
Preparation of tobacco, 231–52
Preparing field, 48–50
—— seed-beds, 38, 41, 43, 44
Pressing tobacco in casks, 128, 131, 133
Priming, 62, 65
Principles of manuring, 18–22
Production of tobacco in Afghanistan, 137
—— Africa, 138
—— Australia, 141
—— Austro-Hungary, 141
—— Borneo, 142
—— Bourbon, 143
—— Brazil, 143–7
—— Canary Islands, 148
—— China, 148
—— Cochin China, 149
—— Costa Rica, 149
—— Ecuador, 149
—— Fiji, 150
—— France, 150–6
—— Germany, 156–64
—— Great Britain, 164–6
—— Greece, 166
—— Holland, 166
—— India, 167–76
—— Italy, 176
—— Japan, 176
—— Java, 176
—— New Zealand, 177
—— Nicaragua, 177
—— Paraguay, 178
—— Persia, 178–91
—— Philippines, 191
—— Roumania, 192
—— Russia, 192
—— San Salvador, 192
—— Servia, 192
—— Spain, 192
—— Sumatra, 193–205
—— Turkey, 205–10
—— United States, 210–22
—— Venezuela, 222
—— West Indies, 223–30
.sp 1
Properties of tobacco, 253–66
Qualities of tobacco, 109–21
Quantity of manure for tobacco, 12
Quicklime for tobacco soil, 24
Quincunx planting, 52
Rate of growth of tobacco, 24
Removing superfluous leaves, 62, 65
Resweating apparatus, 249
Ridging land, 49
Ripeness, influence on tobacco, 61
—— of tobacco, judging, 70
Ripening, 61
River bottoms for tobacco growing, 10
Roll tobacco, 236–44
Rotation for tobacco soils, 29–31
Roumanian tobacco, 192
Russian tobacco, 192
Saltpetre as tobacco manure, 13
Salts added to snuff, 264
—— in tobacco, 259
Sandy bottoms for tobacco growing, 9
San Salvador tobacco, 192
Saving seed, 37
Scaffolding for tobacco, 75
Seed, 37
—— -beds, 38–48
—— ——, area of, 39
—— ——, preparing, 38, 41, 43, 44
—— ——, protecting from fowls, 46
—— ——, —— —— frost, 43, 47
—— ——, shade frames for, 48
—— ——, shelter for, 38
—— ——, situation for, 38
—— ——, soil for, 38
—— ——, time for sowing, 41
Seedlings, hot-bed for, 44
——, planting out, 50–4
——, thinning out, 40, 48
——, top-dressing, 41
—— , watering, 40
—— , weeding, 40, 46
Seed required for an acre, 39
——, sowing, 39, 45
Servian tobacco, 192
Setting out plants, 50–4
Shade frames for seed-beds, 48
Sheds for holding tobacco as gathered, 68
Shelter for nurseries, 38
Sheltering tobacco from wind, 10
—— —— lands with corn, 65
Shiraz tobacco, 3
Signs of ripening, 72
Situation for nurseries, 38
—— —— plantations, 10
Sizes of tobacco barns, 90
Snuff, 248, 263
—— tobacco, 61
Soil for nurseries, 38
—— —— plantations, 8–10
Sorting tobacco, 109–21
—— —— for use, 232
Sort of tobacco to grow, 31–7
Sorts of tobacco grown in America, 33–7
Sowing seed, 39, 45
Spanish tobacco, 37, 192
Spearing tobacco, 72
Species of tobacco, 1–3
Splitting tobacco, 72
Sponging the leaves, 119
Stacking gathered tobacco leaves, 67
Straw mat for keeping frost off, 47
Stripping tobacco, 104–9
—— —— for use, 232
Substitutes for tobacco, 267–70
Suckering plants, 60–6
Sumatran tobacco, 193–205
Sun-curing shed, 83
—— -drying cut tobacco leaves, 68
Sweating tobacco, 121–5
Tauari wrappers, 143
Temporary hanging for tobacco, 68
Teymbeki, 179–91
Thinning out seedlings, 40, 48
Time for harvesting tobacco, 69
—— —— topping, 60, 61, 63
—— of day for cutting tobacco, 77
—— required for curing, 92
Tobacco horse, 72
—— plant, 1–6
Top-dressing seedlings, 41
Topping plants, 60–6
Tumbeki, 178–91
Turkeys as grub-eaters, 55
Turkish tobacco, 2, 3, 205–10
Twist tobacco, 236–44
Tying tobacco for hanging, 97
United States tobacco, 210–22
Use of tobacco, 231–52
Values of tobaccos, 271
Varieties of tobacco, 1–3
Venezuelan tobacco, 222
Ventilating drying-sheds, 88
Virginian tobacco, 2, 34–7
Washing tobacco when sorting, 117
Watering plants when setting out, 51
—— seedlings, 40
Water in tobacco, 262
Weeding seedlings, 40, 46
Weighing and packing machine, 250
West Indian tobacco, 223–30
Windrowing tobacco, 124
Wind shelter for tobacco, 10
Wood-ashes as tobacco manure, 14
Worms, destroying, 55–60
Yellow prior tobacco, 37
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Transcriber's Note
Errors deemed most likely to be the printer's have been corrected, and
are noted here. The references are to the page and line in the original.
The following issues should be noted, along with the resolutions.
3.1 _N. rustica[.]_ Added.
54.18 they grow rapidly[.] Added.
223.18 900,850 cigars[,] 5,709,442 bundles Added.
cigarettes,
231.25 The first process they undergo is Replaced.
"damping,['/"]
End of the Project Gutenberg EBook of Tobacco: Growing, Curing, &
Manufacturing, by C. G. Warnford Lock
***
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Oriental Orthodoxy in India is a minority that comprises millions within Christianity in India. There is major overlap between this, the Christians in Kerala and the St. Thomas Christians, the latter of whom trace themselves back to Apostle Thomas.
The Oriental Orthodox Churches in India are Malankara Orthodox Syrian Church and Jacobite Syrian Christian Church. Malabar Independent Syrian Church also follows the Oriental Orthodox tradition, but is not in communion with other churches in Oriental Orthodox family.
See also
Roman Catholicism in India
Protestantism in India
References
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On Good Friday Neil Parish, Conservative MEP for Gibraltar and two of his colleagues, Richard Ashworth, who represents the South East, and Philip Bradbourn, from the West Midlands, visited the Rosia Bay area at the invitation of the South District Association to see at first hand the destruction of the Rosia Water Tanks. Neil Parish expressed his dismay that the Government had gone ahead with the demolition of the Water Tanks in spite of all the representations and appeals that had been made.
As part of the intense campaign of lobbying to save the water tanks, he, as well as, Graham Watson, Liberal and Glyn Ford, Labour MEPs for Gibraltar, wrote to the Chief Minister, earlier this year, appealing to him to reconsider his decision to demolish them. Mr Parish confirmed to members of the South District Association, who were showing him around the area, that the CM has still not replied to his letter.
In the hour that Mr Parish spent familiarising himself with the area, he showed great interest in the planning issues that the Nelson's View project had exposed. The SDA members present gave Mr Parish and his colleagues the background and history of the Rosia area and answered their many questions. Mr Parish once again expressed the view that European Funding might be available to fund the restoration and beautification of the area and he offered, once again, to assist in the application process. The SDA thanked Mr Parish for the support that he had given the Save the Rosia Tanks Campaign.
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{"url":"http:\/\/www.mitpress.mit.edu\/books\/conditionals-context","text":"Hardcover | $80.00 Short | \u00a355.95 | ISBN: 9780262072663 | 344 pp. | 5.375 x 8 in | 2 illus.| November 2005 Paperback |$35.00 Short | \u00a324.95 | ISBN: 9780262572316 | 344 pp. | 5.375 x 8 in | 2 illus.| November 2005\n\n# Conditionals in Context\n\n## Overview\n\n\"If you turn left at the next corner, you will see a blue house at the end of the street.\" That sentence\u2014a conditional\u2014might be true even though it is possible that you will not see a blue house at the end of the street when you turn left at the next corner. A moving van may block your view; the house may have been painted pink; a crow might swoop down and peck out your eyes. Still, in some contexts, we might ignore these possibilities and correctly assert the conditional. In this book, Christopher Gauker argues that such context-relativity is the key to understanding the semantics of conditionals. Contexts are defined as objective features of the situation in which a conversation takes place, and the semantic properties of sentences\u2014conditionals included\u2014are defined in terms of assertibility in a context.\n\nOne of the primary goals of a theory of conditionals has to be to distinguish correctly between valid and invalid arguments containing conditionals. According to Gauker, an argument is valid if the conclusion is assertible in every context in which the premises are assertible. This runs counter to what Gauker sees as a systematic misreading of the data by other authors, who judge arguments to be invalid if they can think of a context in which the premises are judged true and some other context in which the conclusion is judged false. Different schools of thought on conditionals reflect fundamentally different approaches to semantics. Gauker offers his theory as a motive and test case for a distinctive kind of semantics that dispenses with reference relations and possible worlds.\n\nChristopher Gauker is Professor of Philosophy at the University of Cincinnati.\n\n## Endorsements\n\n\"Christopher Gauker has produced the most sophisticated and comprehensive theory of the semantics and logic of conditionals yet available, and his book should be read by all philosophers, logicians, and linguists interested in the subject. His theory provides the first completely general and entirely rigorous account of the context-relativity of conditionals. It explains in a novel and extremely plausible fashion the semantic distinction between indicative and subjunctive conditionals and the consequent differences between their logics. Gauker shows convincingly how his theory is superior to its best-known rivals both in accommodating our linguistic intuitions and in avoiding the logical puzzles that beset other approaches.\"\nE. J. Lowe, Department of Philosophy, University of Durham\n\n\"A bold attempt to rethink the analysis of conditionals and the foundations of semantics at one fell swoop. In accounting for the logical validity of arguments involving conditionals, Gauker highlights their context-relativity. The 'context-logical' approach he advocates is comparable in orientation and scope to situation theory, and it deserves a comparably wide audience.\"\nFran\u00e7ois Recanati, Institut Jean-Nicod, Paris\n\n\"Indicative and subjunctive conditionals have presented some of the more difficult problems in the philosophy of language and logic. In this impressive book, Christopher Gauker brings his earlier work on logic and pragmatics to bear on these intriguing problems. His theory is rich in detail, and shows great sensitivity both to issues of how we use language in context and to logical issues such as validity and even decidability. His arguments are carefully crafted and incisive, but also presented in a clear and accessible style.\"\nMichael Glanzberg, Department of Philosophy, University of California, Davis\n\n\"This original account of how semantics might usefully be broadened to include kinds of context-relativity that hitherto have been thought of as belonging to pragmatics weaves together a number of novel lines of thought. The success of an enterprise of this shape and ambition should be judged more by its capacity to stimulate than by its capacity to convince. By that standard, Conditionals in Context is a resounding success.\"\nRobert B. Brandom, Distinguished Service Professor, Department of Philosophy, University of Pittsburgh","date":"2015-01-27 14:31:56","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\": 0.318813681602478, \"perplexity\": 1889.8037041089383}, \"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-2015-06\/segments\/1422121981339.16\/warc\/CC-MAIN-20150124175301-00123-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
| null | null |
William Ampulford (died 1435) was an English politician who was MP for Norwich in 1410. He was also town clerk and tax collector of that place.
References
1435 deaths
Members of the Parliament of England for Norwich
English MPs 1410
Clerks
Tax collectors
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 850
|
{"url":"https:\/\/www.tutorialexample.com\/implement-softmax-cross-entropy-loss-with-masking-in-tensorflow-tensorflow-tutorial\/","text":"# Implement Softmax Cross-entropy Loss with Masking in TensorFlow \u2013 TensorFlow Tutorial\n\nBy | August 24, 2020\n\nWe often need to process variable length sequence in deep learning. In that situation, we will need use mask in our model. In this tutorial, we will introduce how to calculate softmax cross-entropy loss with masking in TensorFlow.\n\n## Softmax cross-entropy loss\n\nIn tensorflow, we can use tf.nn.softmax_cross_entropy_with_logits() to compute cross-entropy. For example:\n\nloss = tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=labels)\n\nHowever, how to calculate softmax cross-entropy loss with masking?\n\nWe will use an tensorflow function to implement it.\n\n## Calculate softmax cross-entropy loss with masking\n\nThis function is:\n\n def masked_softmax_cross_entropy(logits, labels, mask):\n\"\"\"Softmax cross-entropy loss with masking.\"\"\"\nloss = tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=labels)\nreturn tf.reduce_mean(loss)","date":"2021-08-05 09:01:31","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\": 0.7832854986190796, \"perplexity\": 6604.994580232805}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"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-2021-31\/segments\/1627046155458.35\/warc\/CC-MAIN-20210805063730-20210805093730-00106.warc.gz\"}"}
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\section{Introduction}
The key informing principle of general relativity stipulates that matter/energy
and spacetime {\it co-evolve} through a self-consistent loop, whereby, to
say it with Wheeler, \lq\lq spacetime tells matter how to move; matter
tells spacetime how to curve"~\cite{WHEELER}.
Despite its logical simplicity, the mathematical formulation
of the above statement (Einstein equations, EEs for short) faces
with a daunting complexity barrier, mostly on account
of the strong non-linearity of the matter-spacetime interactions
Even for the \lq\lq simple" case of matter at rest, the exact solutions of the
EEs are restricted to very few precious instances, usually characterized
by highly idealized geometries with very special symmetry
properties (often too special), which impair a general
understanding of the problem \cite{REZZOLLA}.
Evaluating the gravitational field generated by matter in motion
clearly adds another layer of mathematical complexity, particularly
in the case where such motion is not regular but {\it turbulent} instead.
For instance, it is not known what kind of spacetime metric results
from a given fluctuating energy-momentum tensor
(energy density, pressure, velocity field).
Likewise, we do not know the fate of turbulent flows in the
presence of gravity: do the associated scales (gravitational and turbulent)
compete or cooperate among them?
Does gravity always dominate in the end, erasing, perhaps beyond some threshold,
all fluid scales, or do the latter leave an appreciable long-standing signature on the
gravitational field despite its dominance?
In other words: can turbulence play an appreciable role on the gravitational collapse
process (or in a cosmological context) of a turbulent fluid?
The relevance of these questions for modern astrophysics and cosmology
cannot be overstated~\cite{Yang:2014tla,Marochnik_I,Marochnik_II,Barreto:2022len,Dahl:2021wyk,Galtier:2021ovg,RoperPol:2021gjc,Waeber:2021xba,Adams:2013vsa}, and this work
represents a preliminary attempt to gain semi-quantitative insights into the above matters.
More specifically, we proceed within a Post-Minkowskian (PM) framework, i.e. starting from a flat space
situation (zero gravity, and Minkowskian fluid dynamics) and adding corrections
to the first order in the gravitational constant $G$, eventually to be
continued with high-order iterative corrections.
This is a standard approach in the study of the
two-body problem in general relativity and seems to offer a
promising avenue also for the case of fluid-driven gravitational field.
In the following, we shall present a \lq\lq warm-up"
investigation along these lines, highlighting on the various difficulties which stand on the
way of a quantitative understanding of the turbulence-gravity coupling.
\section{The turbulent energy cascade and its interference with metric length scales: dimensional estimates}
We begin by considering a gravity-free (flat space) turbulent fluid, whose velocity fluctuations
at statistical steady-state, obey the following generic power-law statistics:
\begin{equation}
u(L) = u(L_0) (L/L_0)^{\alpha}\,,
\end{equation}
where $L_0$ is the typical size of the fluid, related to the typical
velocity size $u(L_0)$, $\alpha$ is a scaling exponent in the range $0 \le \alpha \le 1$, with
$\alpha=0$ corresponding to white uncorrelated noise (total randomness), while
$\alpha=1$ denotes a smooth, differentiable field.
In the following we shall refer to $\alpha$ as to the velocity roughness exponent.
Starting from a mother eddy of size $L_0$, the
nonlinear cascade generates eddies of progressively smaller size, till the smallest
active length is reached, below which nonlinearity is no longer capable of
sustaining coherent motion against dissipation.
This happens at the Kolmogorov (or dissipative) length, which
is given by the following expression \cite{K41,ESS,SREENI}
\begin{equation}
L_d = \frac{L_0}{Re^{1/(1+\alpha)}}\,,
\end{equation}
where $Re=U_0L_0/\nu$ denotes the Reynolds number
of a turbulent fluid with kinematical viscosity $\nu$ \cite{FRISCH}.
Let $L_m=\nu/U_0$ (such that $Re=L_0/L_m$) be a microscale length fixed
by the ratio kinematic of the kinematic viscosity $\nu$ and
the macroscopic velocity $U_0$ of a
fluid of macroscale $L_0$.
A simple rearrangement leads to the following compact expression:
\begin{equation}
\label{LD}
L_d = L_0^{p} L_m^{1-p} \,,
\end{equation}
where the scaling exponent $p$ relates
to the roughness via $p=\alpha/(1+\alpha)$; hence $0 \le p \le 1/2$, assuming $\alpha\in [0,1]$.
Since the kinematic viscosity shows surprisingly
small variations across disparate states of matter \cite{Trachenko:2019ghg},
we keep it within the range $10^{-4}$
(International System units, ten times air in standard conditions) to $10^{-7}$ (quark-gluon plasma, QGP) through
the empirical law $L_m \to L_q (\rho_q/\rho)^{1/6}$. Here $L_q \sim 10^{-15}\, m$
and $\rho_q \sim 10^{18}\, Kg/m^3$ are the QGP mean free path
and density, respectively, while $\rho$ denotes the density of the fluid
introducing another scale, say the density-related gravitational length
\begin{equation}
L_g = \frac{c}{\sqrt{G \rho}}\,.
\end{equation}
This accounts for three orders in magnitude change in viscosity
over eighteen orders of magnitude change in density.
The expression (\ref{LD}) shows that $L_d$ is an intermediate
mesoscale ranging from $L_m$ at $\alpha \to 0$ to $L_0$ as
$\alpha \to \infty$, with $L_d=\sqrt{L_0L_m}$ in the case $\alpha=1$ ($p=\frac12$).
Hence the question is whether, depending on the values of the physical-geometrical
parameters at hand, there exist reasonable values of the roughness exponent such
that the dissipative length can be made comparable or smaller
than the typical curvature length of the spacetime, say the
Schwarzschild scale $L_s = G M_0/c^2$ where $M_0$ is the mass equivalent of $L_s$.
This is a condition for {\it strong-coupling} between the geometrical background curvature scale
and hydrodynamic turbulence of the surrounding matter field \cite{K41}.
By demanding $L_d < L_s$, we obtain:
\begin{equation}
\label{STRONG}
L_s > {L_0}^p L_m^{1-p}\,.
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=0.45]{fig1.eps}
\caption{At sufficiently high Reynolds numbers, the energy cascade
reaches a critical point where dissipative Kolmogorov eddies overlap
with the length scale associated with the metric background, in the case
of the figure, the Schwarzschild radius of a growing black-hole.
}
\end{figure}
Replacing $L_0$ in terms of $M_0$, i.e., $L_0=\left(\frac{M_0}{\rho}\right)^{1/3}$, Eq. \eqref{STRONG} becomes
\begin{equation}
M_0>\left( \frac{c^2}{G\rho^{p/3}} \right)^{\frac{1}{1-\frac{p}{3}}}L_m^{\frac{1-p}{1-\frac{p}{3}}}\,.
\end{equation}
To express this condition in a dimensional form let us divide both sides, for example, by the mass of the Sun, $M_{\rm sun}=2\cdot 10^{30}\, kg$, $G M_{\rm sun}/c^2=L_{\rm sun}\sim 10^3\, m$ and introduce the gravitational length to replace $c^2/G=L_g^2\rho$. We find the relation
\begin{equation}
\label{MSTAR}
m \equiv \frac{M_0}{M_{\rm sun}}>L_g^{a_p} L_m^{1-a_p}L_{\rm sun}^{-1}\equiv m^*\,,
\end{equation}
where
\begin{equation}
a_p=\frac23 \frac{ p}{1-\frac{p}{3}}\,.
\end{equation}
Here $m^*$ represents the critical mass above which the dissipative length
falls below the Schwarzschild scale.
Eq. \eqref{MSTAR} can also we written as
\begin{equation}
\label{MSTAR2}
m^* = \frac{L^*}{L_{\rm sun}}\,,\qquad L^*=L_g^{a_p} L_m^{1-a_p}\,,
\end{equation}
where the numerator defines an effective length, $L^*$, interpolating between
the microscale $L_m$ and the gravitational macroscale $L_g$.
Clearly, $L^*$ is an increasing function of $p$, going from $L_m$
at $p=0$ (random fluid) to $L_g^{2/5} L_m^{3/5}$ for $p=1/2$ (smooth fluid).
The expression \eqref{MSTAR} is the main result of this section.
Summarizing, for any value of $p$, we have the following
length scales related to the density of the fluid
\begin{eqnarray}
&& m_*=10^{-3}L_g^{a_p}L_m^{1-a_p}\,,\qquad L_d=L_0^pL_m^{1-p}\,,\nonumber\\
&& L_g=\frac{c}{\sqrt{G}}\frac{1}{\rho^{1/2}}=\frac{3.7\cdot 10^{13}}{\rho^{1/2}}\,,\nonumber\\
&& L_m=\frac{10^{-12}}{\rho^{1/6}}\,,\nonumber\\
&& L_0=\left(\frac{M_0}{\rho}\right)^{1/3}=\left(\frac{m_*M_{\rm sun}}{\rho}\right)^{1/3}=10\, m_*^{1/3}L_g^{2/3}\,,\nonumber\\
&& L_s=\frac{GM_0}{c^2}=L_{\rm sun}m_*=10^3 m_* \,.
\end{eqnarray}
Despite their simplicity, the above expressions invite a number of
informative remarks.
In the following we analyze three distinguished scenarios of
decreasing roughness, namely:
\begin{enumerate}
\item Fully random fluid ($\alpha=0$, $p=0$, $a_p=0$,);
\item Three-dimensional incompressible turbulence ($\alpha=1/3$, $p=1/4$, $a_p=2/11$);
\item Two-dimensional incompressible ($\alpha=1$, $p=1/2$, $a_p=2/5$).
\end{enumerate}
\subsection{Fully random fluid}
In this case we have $\alpha=0$, hence $p=0$, $a_p=0$, and a $|{\mathbf k}|^{-1}$ spectrum.
The expression \eqref{MSTAR} reduces to
\begin{equation}
m^* = L_m L_{\rm sun}^{-1} \sim 10^{-3} L_m \,,
\end{equation}
and $L_d=L_m$.
In Fig. \ref{fig2} we show $m^*$, $L_0$, and $L_d=L_m$ as a function of $\rho$.
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig2.eps}
\caption{\label{fig2} Random fluid: $m^*$ (black online), $L_0$
(red online), $L_m=L_d=L_s$ (blue online) are plotted as a function of $\rho$.
Note that BH as small as $10^{-16}$ solar
masses, i.e. about $10^{14}$ Kg, can overlap with a turbulent
cascade at Reynolds numbers around $10^{16}$.
}
\end{figure}
These results show that the critical mass ranges from
$10^{-16} \div 10^{-18}$ solar masses, indicating that
pretty small black-holes can potentially interfere with a
turbulent cascade at Reynolds in the order of $10^{16}$.
Since full randomness is less realistic than correlated
turbulence, it is of interest to directly inspect the turbulent cases.
\subsection{Three-dimensional turbulence}
As mentioned above, turbulence is a subtly correlated form of chaos
far from pure randomness. As a result, it shows high
sensitivity to spatial dimensionality.
In $3d$, energy is dissipated even in the (singular) limit of zero viscosity, through the nonlinear
energy cascade from $L_0$ down to $L_d=L_0/Re^{3/4}$, $Re$
being
the Reynolds number. This leads to a roughness exponent
$\alpha=1/3$ and a $|{\mathbf k}|^{-5/3}$ power spectrum.
Hence, we have $\alpha=1/3$, $p=1/4$, $a_p=2/11$.
The expression \eqref{MSTAR} now gives
\begin{equation}
m^* = L_g ^{2/11} L_m^{9/11} L_{\rm sun}^{-1}\,.
\end{equation}
In Fig. \eqref{fig3} we show $m^*$, $L_0$
and $L_m$ as a function of $\rho$.
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig3.eps}
\caption{\label{fig3} $3d$ Turbulence: $m^*$
(black online), $L_0$ (red online),
and $L_m$ (blue online) are plotted as a function of $\rho$.
Overlap between gravity and turbulence occurs
above $10^{-10} \div 10^{-12}$, solar masses, namely
$10^{20} \div 10^{18}$ kg, way more massive than
the full random case. The range of Reynolds number is
comparable to the case of full randomness.
}
\end{figure}
The leading factor $L_g$ is now active, but still largely suppressed
by the small $2/11$ exponent, yet providing a boost of about five orders
of magnitude with respect to the case of a fully random fluid.
The range of Reynolds numbers is more less the same as the full random case.
\subsection{Two-dimensional turbulence}
In two spatial dimensions, energy is conserved while enstrophy (vorticity squared)
is dissipated, which implies a direct (large to small) enstrophy cascade and an
inverse energy cascade (from small to large) \cite{FRISCH}.
The result is a smooth flow field with $\alpha=1$, corresponding to
$p=1/2$, $a_p=2/5$ and a much steeper energy spectrum $|{\mathbf k}|^{-3}$.
The expression (\ref{MSTAR}) now gives
\begin{equation}
m^* = L_g^{2/5} L_m^{3/5} L_{\rm sun}^{-1} \,.
\end{equation}
In Fig. \ref{fig4} we plot $m^*$, $L_0$,
and $L_m$ as a
function of the initial density. From these data it is apparent that
putative black holes (BHs) are another five orders of magnitude more
massive than in the $3d$ case.
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig4.eps}
\caption{\label{fig4} $2d$ turbulent fluid:
$m^*$ (black online), $L_0$ (red online), and
$L_m$ (blue online) are plotted as function of $\rho$.
The corresponding black-holes range from $10^{-5} \div 10^{-9}$
solar masses, namely $10^{25} \div 10^{21}$ kg.
}
\end{figure}
It is worth noting that all turbulent cascades above
involve pretty large values of the Reynolds number
$Re=L_0/L_m$ around $10^{20}$ (as a matter of reference,
the Reynolds number for a standard airline is about $10^8$).
In the above, we have established that turbulent flows are capable of potentially
strong interactions with growing BHs, since they can reach down
to the background curvature scale (i.e., the mass in the case of a Schwarzschild black hole).
The key question, however, is how to describe such strong coupling
in quantitative terms. As discussed in the Introduction, a fully-fledged
answer to this question must necessarily rely upon the non-perturbative
solution of Einstein's equations, driven by a turbulent matter tensor.
However, a few heuristic arguments can be brought up, without
undertaking such a demanding task head-on.
To this regard, let us recall that the standard fate of a dissipative eddy
of size $L_d$ is to de-cohere into a \lq\lq spray" of droplets too small to
sustain collective motion: that's where hydrodynamic
bows away to a microscopic description.
Under strong coupling conditions it is plausible to assume that instead
of being turned into heat, dissipative eddies would rather feed the
black hole (Primordial Black hole, PBH, could be more appropriate in a cosmological context) growth
in a sort of preferential way as compared to eddies of larger size.
This speculation is grounded into the principle of locality of turbulence in reciprocal
(Fourier) space, according to which eddies interact strongly only with eddies of comparable
size. This is known to hold for fluid turbulence, but does by no means imply that the
same principle applies to turbulence-gravity interactions as well.
Indeed, recent results, based on the numerical and analytic
solution of the EEs, show that black hole (BH) horizons can themselves turn
turbulent and in a way which is highly reminiscent of Kolmogorov $3d$ turbulence \cite{K41}.
Such results suggest that the aforementioned principle of locality may indeed
hold true for BH-turbulence interactions as well.
It appears therefore reasonable to speculate that dissipative eddies might
function as \lq\lq catalyzers" of the gravitational collapse, by supplying their
kinetic energy or enstrophy (hence mass) into the newborn BH, long before the entire
mass
is collapsed. The energy contained in the dissipative eddies is only a
tiny fraction of the total fluid energy, but in the presence of near-singular metrics
even small amounts of extra energy supplies may unleash substantial effects on the
incipient primordial BH dynamics.
This putative \lq\lq turbulence-assisted" hierarchical gravitational
collapse should be liable to numerical verification.
\section{Turbulence-driven gravity: first order Post-Minkowskian approach}
So far, we have presented statistical steady-state considerations based on dimensional
analysis, an approach which proves exceedingly insightful in the theory of (gravity-free) turbulence.
In this section, we endeavor to sketch a quantitative analysis of the EEs
under the stochastic drive of a turbulence matter-energy tensor.
To this purpose, we work in a Post-Minkowskian (PM) context which implies
a weak gravitational field (first order in the gravitational constant $G$, treated
as a place-holder in the perturbative expansion) but is not restricted
to small velocities, limiting our considerations to the first-order
(1PM, or $O(G^1)$) approximation level.
Before plunging into the 1PM formalism, we wish to mention that the effects of stochastic
fluctuations of the energy-matter tensors on the gravitational metric have been considered
before in the context of stochastic gravity \cite{Moffat:1996fu}. In this approach, metric fluctuations
are treated by means of a generalized Langevin formalism \cite{Morozov}, whereas in this work we
adopt a strategy inspired by a merger between 1PM and turbulence modeling techniques \cite{TM}.
Let $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (with $\eta_{\mu\nu}={\rm diag}[-1,1,1,1]$) denote a 1PM perturbation of the flat space, with inverse $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ and such that
\begin{equation}
\label{dalemb}
\Box_x h^{\mu\nu}=-16\pi G S^{\mu\nu}+O(\partial \partial hh +h S)\,,
\end{equation}
where $\Box_x=\eta^{\mu\nu}\partial_\mu\partial_\nu$ and
\begin{eqnarray}
S^{\mu\nu}(x)&=&T^{\mu\nu}(x)-\frac12 T(x) g^{\mu\nu}(x)\,,\nonumber\\
T(x)&=&g_{\alpha\beta}(x)T^{\alpha\beta}(x)\,.
\end{eqnarray}
In this case all indices are raised/lowered by using the flat metric $\eta^{\alpha\beta}$, as standard.
Following Ref. \cite{Damour:2016gwp} let us introduce the Green function ${\mathcal G}(x-y)$ of the flat-space D'Alembert operator, such that
\begin{equation}
\Box_x {\mathcal G}(x-y)=-4\pi \delta^{(4)}(x-y) \,.
\end{equation}
This implies
\begin{equation}
\label{h_munu_eq}
h^{\mu\nu}(x)=4G \int d^4y {\mathcal G}(x-y)S^{\mu\nu}(y)+O(G^2)\,,
\end{equation}
and it is fully determined as soon as the source $S^{\mu\nu}(y)$ is specified.
Let us assume $T^{\mu\nu}$ to represent a perfect fluid
\begin{equation}
\label{en_mom}
T^{\mu\nu}=(\rho+p)u^\mu u^\nu+pg^{\mu\nu}\,,\qquad T=-\rho+3p\,,
\end{equation}
so that
\begin{eqnarray}
\label{Smunu}
S^{\mu\nu}&=&(\rho+p)u^\mu u^\nu-\frac12 (p-\rho)g^{\mu\nu}\nonumber\\
&=&\rho \left(u^\mu u^\nu+\frac12 g^{\mu\nu}\right)+p\left(u^\mu u^\nu-\frac12 g^{\mu\nu}\right)\,.
\end{eqnarray}
Furthermore, it is customary to split the fluid components into an
averaged and a fluctuating components $X(x)=X_0(x)+\delta X(x)$, e.g.,
\begin{eqnarray}
\rho(x)&=& \rho_0(x) +\delta \rho (x)\,, \nonumber\\
p(x) &=& p_0(x) +\delta p (x)\,,\nonumber\\
u^\mu(x)&=&u^\mu_0(x) +\delta u^\mu(x)\,,
\end{eqnarray}
where $X_0(x)$), is a slowly varying quantity
whereas $\delta X(x)$ is a rapidly varying component.
In the above "slow (rapid)" implies scales longer (shorter) than the
typical averaging length, namely the heterogeneity scale of the fluid
(infinity in the case of homogeneous turbulence).
Within the 1PM approximation, the source $S^{\mu\nu}$, being prefactored by $G$, can be treated as a
zeroth-order quantity i.e., with $g^{\mu\nu}=\eta^{\mu\nu}$ in Eqs. \eqref{Smunu}.
As a result, we are left with the following source terms in the equation \eqref{h_munu_eq}:
\begin{equation}
S^{\mu\nu}=S_0^{\mu\nu}+S_1^{\mu\nu}+S_2^{\mu\nu}+S_3^{\mu\nu}\,,
\end{equation}
where
\begin{eqnarray}
S_0^{\mu\nu}&=& (\rho_0+p_0)u^\mu_0 u^\nu_0-\frac12 (p_0-\rho_0)\eta^{\mu\nu}\,,\nonumber\\
S_1^{\mu\nu}&=& (\rho_0+p_0)(u_0^\mu \delta u^\nu+u_0^\nu \delta u^\mu)\nonumber\\
&+& (\delta\rho+\delta p) u_0^\mu u_0^\nu -\frac12 (\delta p-\delta \rho) \eta^{\mu\nu}\,,\nonumber\\
S_2^{\mu\nu}&=& (\rho_0+p_0)R^{\mu\nu}_{u,u}+2u_0^{(\mu}R^{\nu)}_{\rho,u}+2u_0^{(\mu}R^{\nu)}_{p,u}\nonumber\\
S_3^{\mu\nu}&=& R^{\mu\nu}_{\rho, u,u}+R^{\mu\nu}_{p, u,u} \,,
\end{eqnarray}
In the above, we have introduced the \lq\lq correlators"
\begin{eqnarray}
R^{\mu\nu}_{u,u} &=& \delta u^\mu \delta u^\nu \,,\quad
R^{\mu}_{\rho,u} = \delta \rho \delta u^\mu \,,\quad
R^{\mu}_{p,u} = \delta p \delta u^\mu \,,\nonumber\\
R^{\mu\nu}_{\rho, u,u}&=& \delta \rho \delta u^\mu \delta u^\nu\,,\qquad
R^{\mu\nu}_{p, u,u}= \delta p \delta u^\mu \delta u^\nu\,,
\end{eqnarray}
where $X_{(ab)}=\frac12(X_{ab}+X_{ba})$ denotes symmetrization.
Notice that $R^{\mu\nu}_{u,u}$ (or more precisely its averaged version) is a
direct analogue of the Reynolds stress tensor in Kolmogorov
turbulence, while $R^{\mu}_{\rho,u}$ and $R^{\mu}_{p,u}$
reflect compressibility effects (in case the pressure is a linear function
of energy they are basically the same).
Next, we move to Fourier space, where the inverse box operator becomes:
\begin{equation}
\hat {\mathcal G}(k)=-\frac{1}{k^2 }\,,\qquad k^2=k\cdot k=\eta_{\alpha\beta}k^\alpha k^\beta\,.
\end{equation}
Consequently,
\begin{eqnarray}
h_{\mu\nu}(x)&=&-16 \pi G \int \frac{d^4k}{(2\pi)^4}\frac{\hat S_{\mu\nu}(k)}{k^2}e^{ik\cdot x}\nonumber\\
&=& -16 \pi G \sum_{r=0}^3 \int \frac{d^4k}{(2\pi)^4}\frac{\hat S_r{}_{\mu\nu}(k)}{k^2}e^{ik\cdot x}\,,\qquad
\end{eqnarray}
where
\begin{equation}
\hat S_{\mu\nu}(k)=\int d^4x e^{-ik\cdot x}S_{\mu\nu}(x)\,.
\end{equation}
We can then consider the parts of $h^{\mu\nu}$ sourced by the various components of $S_{\mu\nu}$,
\begin{equation}
h_r{}_{\mu\nu}(x)= -16 \pi G \int \frac{d^4k}{(2\pi)^4}\frac{\hat S_r{}_{\mu\nu}(k)}{k^2}e^{ik\cdot x}\,,\quad r=0,\ldots 3\,.
\end{equation}
To make further analytical progress, we need to make physically reasonable
assumptions on the source terms $\hat S_r{}_{\mu\nu}(k)$, $r=0,1,2,3$.
For the sake of concreteness, let us start by discussing the
case $\hat S_0{}_{\mu\nu}(k)$ (other source terms can be added later),
under the simplifying hypothesis that $u^\mu_0$ is a constant field,
\begin{equation}
\hat S_0^{\mu\nu}(k)= \hat \rho_0(k) H_+^{\mu\nu}+ \hat p_0 (k) H_-^{\mu\nu}\,,
\end{equation}
where
\begin{equation}
H_{\pm}^{\mu\nu}=u^\mu_0 u^\nu_0\pm \frac12 \eta^{\mu\nu}\,,
\end{equation}
and
\begin{equation}
H_{\pm}^{\mu\nu} u_{0\,\nu}=\lambda_\pm u_0^\mu\,,
\end{equation}
with $\lambda_+=-\frac12$ and $\lambda_-=-\frac32$.
Furthermore, for both energy density and pressure we assume a power-law
scaling, characteristic of turbulent flows
\begin{equation}
\label{hat_rho_p}
\hat \rho_0(k)={\mathcal E}k^n \,,\qquad \hat p_0(k)={\mathcal P}k^m\,,
\end{equation}
with ${\mathcal E}$ and ${\mathcal P}$ constants, and scaling exponents
$n,m$ both negative.
Finally, let us introduce the notation
\begin{eqnarray}
\label{J_q_int}
J_q(x)&=&\int \frac{d^4k}{(2\pi)^4}k^{q}e^{ik\cdot x}\,,
\nonumber\\
&=&-\int \frac{d^3k}{(2\pi)^3}e^{ik_ax^a}\int \frac{d\omega}{2\pi}e^{-i\omega t} (-\omega^2+{\mathbf k}^2)^{q/2}\,.\nonumber\\
\end{eqnarray}
We find then
\begin{equation}
\label{hat_S0_munu}
\hat S_0^{\mu\nu}(k)= H_+^{\mu\nu} {\mathcal E}k^n +H_-^{\mu\nu} {\mathcal P}k^m \,,
\end{equation}
and
\begin{eqnarray}
h_0^{\mu\nu}(x)&=& -16 \pi G H_+^{\mu\nu} {\mathcal E}\int \frac{d^4k}{(2\pi)^4} k^{n-2}e^{ik\cdot x}\nonumber\\
&& -16 \pi G H_-^{\mu\nu} {\mathcal P}\int \frac{d^4k}{(2\pi)^4} k^{m-2}e^{ik\cdot x}\nonumber\\
&=& -16 \pi G [H_+^{\mu\nu} {\mathcal E}J_{n-2}(x) + H_-^{\mu\nu} {\mathcal P}J_{m-2}(x)]\,.\nonumber\\
\end{eqnarray}
A direct computation (see Appendix A) shows that
\begin{equation}
J_q(x)=\frac{C_q}{x^{4+q}}\,,
\end{equation}
where $C_q$ is a constant depending on $q$ (as well as on various convergence conditions). This implies
\begin{eqnarray}
\label{h0_fin_sing}
h_0^{\mu\nu}(x)
&=& -16 \pi G \left(H_+^{\mu\nu} {\mathcal E}\frac{C_{n-2}}{x^{n+2}} + H_-^{\mu\nu} {\mathcal P}\frac{C_{m-2}}{x^{m+2}}\right)\,,\qquad\nonumber\\
\end{eqnarray}
where $x^2=-t^2+{\mathbf x}^2$. Note that the result \eqref{h0_fin_sing} carries a coordinate-dependent information (in particular, it depends on the choice of the origin of the coordinate system, which we implicitly placed at $x=0$).
This simple example shows the onset of a
metric singularity (along the lightcone) for $n>-2$ (and same for $m$).
In particular, $2d$ turbulence, $n=-3$, yields a smooth metric, while
$3d$ turbulence, $n=-5/3$ leads to a mildly singular one, with the same
exponent as turbulent velocity fluctuations, but opposite sign, i.e. $-1/3$.
Next, let us examine the $S_1^{\mu\nu}$ term,
\begin{eqnarray}
\hat S_1^{\mu\nu}(k)&=& u_0^\mu \int d^4x e^{-ik\cdot x} (\rho_0+p_0)\delta u^\nu\nonumber\\
&+&
u_0^\nu \int d^4x e^{-ik\cdot x} (\rho_0+p_0)\delta u^\mu \nonumber\\
&+&H_+^{\mu\nu}\int d^4x e^{-ik\cdot x} \delta \rho(x)\nonumber\\
&+&
H_-^{\mu\nu}\int d^4x e^{-ik\cdot x} \delta p(x)\,.
\end{eqnarray}
The Fourier transform of $\rho_0$, $p_0$, $\delta \rho$ and $\delta p$ leads to the following expression
\begin{eqnarray}
\hat S_1^{\mu\nu}(k)&=& u_0^\mu \int \frac{dk'}{(2\pi)^4}[\hat \rho(k')+\hat p(k')]\hat \delta u^\nu (k-k')\nonumber\\
&+&
u_0^\nu \int \frac{dk'}{(2\pi)^4}[\hat \rho(k')+\hat p(k')]\hat \delta u^\mu (k-k') \nonumber\\
&+&H_+^{\mu\nu}\hat \delta \rho(k)+
H_-^{\mu\nu}\hat\delta p(k)\,.
\end{eqnarray}
Using Eq. \eqref{hat_rho_p}, we find
\begin{eqnarray}
\hat S_1^{\mu\nu}(k)&=& u_0^\mu \int \frac{dk'}{(2\pi)^4}[{\mathcal E}k'{}^n +{\mathcal P}k'{}^m]\hat \delta u^\nu (k-k')\nonumber\\
&+&
u_0^\nu \int \frac{dk'}{(2\pi)^4}[{\mathcal E}k'{}^n +{\mathcal P}k'{}^m]\hat \delta u^\mu (k-k') \nonumber\\
&+&H_+^{\mu\nu}\hat \delta \rho(k)+
H_-^{\mu\nu}\hat\delta p(k)\,,
\end{eqnarray}
Again, to proceed further analytically we need (physically reasonable) assumptions
on $\hat \delta u^\mu(k)$, $\hat \delta \rho(k)$
and $\hat \delta p(k)$.
For instance, the simplest case is:
\begin{equation}
\label{simple_delta u}
\hat \delta u^\nu (k)=(2\pi)^4C_{\delta u}^\nu \delta (k)\,,
\end{equation}
with $C_{\delta u}^\nu $ a constant vector.
This implies
\begin{eqnarray}
\hat S_1^{\mu\nu}(k)&=& (u_0^\mu C_{\delta u}^\nu+u_0^\nu C_{\delta u}^\mu) [{\mathcal E}k^n +{\mathcal P}k^m] \nonumber\\
&+&H_+^{\mu\nu}\hat \delta \rho(k)+
H_-^{\mu\nu}\hat\delta p(k)\,.
\end{eqnarray}
Assuming a power-law for $\hat \delta \rho(k)$ and $\hat\delta p(k)$
(e.g., $\hat \delta \rho(k)\sim C_{\delta \rho} k^p$), we are redirected exactly to
the same mathematical treatment as for the previous case of $S_0^{\mu\nu}$.
Let us now examine the case of $S_2^{\mu\nu}$ and limit our considerations to the first term,
\begin{equation}
S_{2a}^{\mu\nu}(x)=(\rho_0+p_0)R_{u, u}^{\mu\nu},
\end{equation}
since all the others can be treated similarly.
Passing to the Fourier space we find
\begin{eqnarray}
\hat S_{2a}^{\mu\nu}(k)&=& \int \frac{d^4k_1}{(2\pi)^4}\frac{d^4k_2}{(2\pi)^4}(\hat \rho_0(k_1)+\hat p_0(k_1))\hat \delta u^\mu (k_2)\times\nonumber\\
&& \hat \delta u^\nu (k-k_1-k_2)\,.
\end{eqnarray}
Using Eq. \eqref{hat_rho_p} the above expression becomes
\begin{eqnarray}
\hat S_{2a}^{\mu\nu}(k)&=& \int \frac{d^4k_1}{(2\pi)^4}\frac{d^4k_2}{(2\pi)^4}({\mathcal E}k_1^n +{\mathcal P}k_1^m)\hat \delta u^\mu (k_2)\times\nonumber\\
&& \hat \delta u^\nu (k-k_1-k_2)\,,
\end{eqnarray}
and again to proceed further we need a physically reasonable expression for $\hat \delta u^\mu (k)$.
In the simple case \eqref{simple_delta u} we obtain
\begin{eqnarray}
\hat S_{2a}^{\mu\nu}(k)&=& C_{\delta u}^\mu C_{\delta u}^\nu \int d^4k_1 d^4k_2 ({\mathcal E}k_1^n +{\mathcal P}k_1^m)\times \nonumber\\
&& \delta (k_2)\delta (k-k_1-k_2)\nonumber\\
&=&C_{\delta u}^\mu C_{\delta u}^\nu ({\mathcal E}k^n +{\mathcal P}k^m)\,.
\end{eqnarray}
Correspondingly
\begin{eqnarray}
\frac{h_{2a}{}_{\mu\nu}(x)}{16 \pi G}&=& - C_{\delta u}^\mu C_{\delta u}^\nu \int \frac{d^4k}{(2\pi)^4} ({\mathcal E}k^{n-2} +{\mathcal P}k^{m-2}) e^{ik\cdot x}\nonumber\\
&=& - C_{\delta u}^\mu C_{\delta u}^\nu [{\mathcal E}J_{n-2}(x)+{\mathcal P}J_{m-2}(x)]\nonumber\\
&=& - C_{\delta u}^\mu C_{\delta u}^\nu \left[{\mathcal E}\frac{C_{n-2}}{x^{n+2}} +{\mathcal P}\frac{C_{m-2}}{x^{m+2}}\right]
\,.
\end{eqnarray}
Extending these considerations to the other terms entering $S_2^{\mu\nu}$ or to the remaining component of the fluid source
$S_3^{\mu\nu}$ is performed along the same lines as above. Namely, it is conceptually straightforward, albeit
a bit more involved from a mathematical standpoint.
\subsection{Timelike geodesics and particle scattering in fluctuating spacetime}
Armed with the above formalism, we next proceed to study the
timelike geodesics (the orbits of massive particles with mass $m$) of the
turbulence- perturbed metric
$g^{\mu\nu}(x)=\eta^{\mu\nu}-h^{\mu\nu}(x)$, with unit tangent vector
\begin{equation}
p_\alpha=m \frac{dx^\alpha}{d\tau}=\frac{dx^\alpha}{d\sigma}\,,\qquad
\end{equation}
such that
\begin{equation}
\frac{dp_\alpha}{d\sigma}=\frac12 \partial_\alpha h_{ {\mu\nu}}(x) p^\mu p^\nu\,,
\end{equation}
where $\sigma=\frac{\tau}{m}$ and $\tau$ the proper time parameter.
\begin{figure}[hb]
\centering
\includegraphics[scale=0.40]{fig5.eps}
\caption{Geodesic motion of a free particle on a smooth (thick line) and
fluctuating (dashed) metric manifold. Subscripts ``i\rq\rq and ``f\rq\rq denote the
initial and final momenta. The effects of metric fluctuations can be paralleled to
a scattering process (similar to temperature fluctuations in classical fluids)
represented here as the departure of $p'_f$ from $p_f$.
}
\end{figure}
It is also straightforward to derive the variation of the particle's 4-momentum
when scattered by the (spacetime metric generated) fluid, namely
\begin{equation}
\label{variation}
\Delta p_\alpha=\frac12 \int_{-\infty}^\infty d\sigma \partial_\alpha h_{\mu\nu}(x)\bigg|_{x=x(\sigma)} p^\mu p^\nu\,.
\end{equation}
Working at the first order in $G$, since $h_{\mu\nu}$ is already $O(G)$, all the other ingredients entering
the right-hand-side of Eq. \eqref{variation} can be replaced by their zeroth-order (free
motion in a flat spacetime) approximations, i.e., constant momenta
($p^\alpha=p_-^\alpha$=constant) and straight (incoming) world lines:
\begin{equation}
\label{geo_before}
x^\alpha(\sigma)=x^\alpha(0)+p_-^\alpha \sigma\equiv b^\alpha +p_-^\alpha \sigma\,,
\end{equation}
where we have denoted $x(0)=b$.
As a result:
\begin{eqnarray}
\label{variation2}
\Delta p_\alpha&=&\frac12 p^\mu_- p^\nu_- \int_{-\infty}^\infty d\sigma \partial_\alpha h_{{\mu\nu}}(x) \bigg|_{x^\alpha=b^\alpha+p_-^\alpha \sigma}\nonumber\\
&=& -4 G (2\pi)^2 p^\mu_- p^\nu_- i
\int \frac{d^4k}{(2\pi)^4} \frac{\hat S_{\mu\nu}(k)}{k^2}k_\alpha \delta (k\cdot p_-) e^{ik\cdot b} \,,\nonumber\\
\end{eqnarray}
with $\delta (k\cdot p_-)$ arising from the integration over $\sigma$.
Here, we can take $\hat S_{\mu\nu}(k)=\hat S_0^{\mu\nu}(k)$ as given, for
example, by Eq. \eqref{hat_S0_munu},
\begin{equation}
\hat S_{\mu\nu}(k)=H_{+\,\mu\nu}{\mathcal E}k^n +H_{-\,\mu\nu}{\mathcal P}k^m\,.
\end{equation}
Let us introduce the notation
\begin{equation}
H_\pm {}_{\mu\nu} p^\mu_- p^\nu_-=L_\pm={\rm const}\,.
\end{equation}
To perform this integral we choose a coordinate system such that
the unperturbed particle moves along a straight line parallel to the $y$ axis, namely:
\begin{equation}
p_-=m(\gamma \partial_t -\sqrt{\gamma^2-1}\partial_y)\,,
\end{equation}
The parametric equations of the associated orbit then become
\begin{eqnarray}
t(\sigma)&=& m\gamma \sigma\,,\qquad x(\sigma)= b\,, \nonumber\\
y(\sigma)&=& -m \sqrt{\gamma^2-1} \sigma \,,\qquad
z(\sigma)= 0\,.
\end{eqnarray}
As a result, we obtain:
\begin{widetext}
\begin{eqnarray}
\label{variation3}
\Delta p_\alpha&=& -4 G \frac{(2\pi)^2}{m\gamma} i \int \frac{dk_0 d^3k}{(2\pi)^4}[L_+ {\mathcal E}k^{n-2}+L_- {\mathcal P}k^{m-2}] k_\alpha e^{ik_x b} \,
\delta(k_0-\frac{\sqrt{\gamma^2-1}}{\gamma}k_y)\nonumber\\
&=& -4 G \frac{(2\pi)^2}{m\gamma} i \int \frac{d^3k}{(2\pi)^4}[L_+ {\mathcal E}k^{n-2}+L_- {\mathcal P}k^{m-2}] k_\alpha e^{ik_x b}|_{k_0=\frac{\sqrt{\gamma^2-1}}{\gamma}k_y}\,,
\end{eqnarray}
with $k_0=\frac{\sqrt{\gamma^2-1}}{\gamma}k_y$ the on-shell condition and $d^3k=dk_x dk_y dk_z$.
Therefore $k^2=k_x^2+\left(\frac{k_y}{\gamma}\right)^2+k_z^2\equiv k_\perp^2+k_z^2$ on-shell.
Finally
\begin{eqnarray}
\Delta p_\alpha
&=& -4 G \frac{i}{m\gamma(2\pi)^2} \left[L_+ {\mathcal E}\int d^3k k^{n-2}k_\alpha e^{ik_x b}+L_- {\mathcal P}\int d^3k k^{m-2}k_\alpha e^{ik_x b}\right] \,,
\end{eqnarray}
\end{widetext}
On symmetry grounds, we have:
\begin{equation}
\Delta p_z=0\,,
\end{equation}
and, because of the on-shell condition, we obtain
\begin{equation}
\Delta p_0=\frac{\sqrt{\gamma^2-1}}{\gamma}\Delta p_y\,.
\end{equation}
Again for symmetry reasons (the integrand is an odd function of $k_y$), $\Delta p_y=0$ implying
$\Delta p_0=0$,
\begin{eqnarray}
\Delta p_y
&=& - \frac{4iG}{m\gamma(2\pi)^2} \left[L_+ {\mathcal E}\int d^3k k^{n-2}k_y e^{ik_x b}\right. \nonumber\\
&+&\left.L_- {\mathcal P}\int d^3k k^{m-2}k_y e^{ik_x b}\right]=0 \,.
\end{eqnarray}
Finally, $\Delta p_x$ can be expressed as a derivative with respect to $b$,
\begin{eqnarray}
\Delta p_x
&=& - \frac{4 G}{m\gamma(2\pi)^2}\frac{\partial}{\partial b} \Psi \,,
\end{eqnarray}
where
\begin{eqnarray}
\Psi&=& L_+ {\mathcal E}\int d^3k k^{n-2} e^{ik_x b}
\nonumber\\&+&
L_- {\mathcal P}\int d^3k k^{m-2} e^{ik_x b}\,.\qquad
\end{eqnarray}
In this case, the integration over $k_z$ becomes trivial by using the relation
\begin{eqnarray}
\int dk_z k^n&=&\int dk_z (k_\perp^2+k_z^2)^{n/2}\nonumber\\
&=&k_\perp^{n+1}\sqrt{\pi}\frac{\Gamma(-\frac12 -\frac{n}{2})}{\Gamma(-\frac{n}{2})}\nonumber\\
&=&k_\perp^{n+1}B_n
\,.
\end{eqnarray}
Consequently
\begin{eqnarray}
\Psi&=&
L_+ {\mathcal E}B_{n-2} \int dk_xdk_y k_\perp^{n-1} e^{ik_x b}\nonumber\\
&+& L_- {\mathcal P}B_{m-2} \int dk_xdk_y k_\perp^{m-1}e^{ik_x b}\,,
\end{eqnarray}
The basic integral to be computed is then:
\begin{eqnarray}
Y_{n}(b)&=&\int dk_xdk_y k_\perp^{n-1} e^{ik_x b}\nonumber\\
&=&\int dk_xdk_y \left(k_x^2+\left(\frac{k_y}{\gamma}\right)^2\right)^{(n-1)/2} e^{ik_x b}\nonumber\\
&=&\gamma \int dk_xdk_y \left(k_x^2+ k_y^2\right)^{(n-1)/2} e^{ik_x b}\,,
\end{eqnarray}
This yields:
\begin{equation}
\Psi=\left[L_+ {\mathcal E}B_{n-2}Y_n(b) +L_- {\mathcal P}B_{m-2}Y_m(b)\right] \,.
\end{equation}
For $n<0$ ($n$ is assumed to be real)
\begin{eqnarray}
Y_{n}(b)&=& \sqrt{\pi}\gamma \frac{\Gamma(-\frac{n}2)}{\Gamma( \frac{1-n}2)} \int_{-\infty}^\infty dk_x k_x^n e^{ik_x b}\nonumber\\
&=&
\sqrt{\pi}\gamma \frac{\Gamma(-\frac{n}2)}{\Gamma( \frac{1-n}2)}\frac{1}{b^{n+1}}\int_{-\infty}^\infty du u^n e^{iu}
\,,
\end{eqnarray}
For $n=-1$ the above rescaled expression cannot be used.
Working directly with the un-rescaled expression, we have
\begin{equation}
Y_{-1}(b)= \pi \gamma \int dk_x \frac{e^{ik_x b}}{k_x} \,,
\end{equation}
implying the following Dirac-delta result
\begin{equation}
\frac{d}{db}Y_{-1}(b)=2i\pi^2\gamma \delta(b) \,.
\end{equation}
Other values of $n$ are also of interest; for $n=-3$, we compute:
\begin{eqnarray}
Y_{-3}(b)&=& \sqrt{\pi}\gamma \frac{\Gamma(\frac{3}2)}{\Gamma(2)}b^2\int_{-\infty}^\infty du u^{-3} e^{iu} \nonumber\\
&=& \frac12 \pi \gamma b^2 i\int_{-\infty}^\infty du \frac{\sin u}{u^3}\nonumber\\
&=& -\frac{\pi^2}{4} \gamma b^2 i\,,
\end{eqnarray}
where the last integral diverges at $u=0$ and it has been evaluated by using the Partie Finie.
For $n=-5/3$ we find instead:
\begin{eqnarray}
Y_{-5/3}(b)&=& \sqrt{\pi}\gamma \frac{\Gamma( \frac{5}6)}{\Gamma(\frac43)}b^{2/3} \int_{-\infty}^\infty du u^{-5/3} e^{iu}\nonumber\\
&=& \sqrt{\pi}\gamma \frac{\Gamma( \frac{5}6)}{\Gamma(\frac43)}b^{2/3} i 2\int_{0}^\infty du \frac{\sin u}{u^{5/3}} \nonumber\\
&=& 3 \pi^{3/2}\gamma \frac{\Gamma( \frac{5}6)}{\Gamma(\frac43)\Gamma(\frac23)}b^{2/3} i\,,
\end{eqnarray}
where
\begin{equation}
\Gamma\left(\frac43\right)\Gamma\left(\frac23\right)=\frac29 \pi \sqrt{3}\,,\qquad \Gamma\left(\frac56\right)\approx 1.1288\,.
\end{equation}
Therefore
\begin{eqnarray}
Y_{-5/3}(b)
&=& 27 \pi^{1/2}\gamma \frac{\Gamma\left( \frac{5}6\right)}{2 \sqrt{3}}b^{2/3} i\nonumber\\
&\approx& 15.5941 \gamma b^{2/3} i\,.
\end{eqnarray}
Summarizing, in the case $n=-1$ we end up with a $\Delta p_x \propto \delta (b)$, namely a
particle scattered by a fluid with an energy spectrum of the type $\sim k^{-1}$
experiences an instantaneous variation of its (initially constant) linear momentum.
In the case of a energy spectrum of the type $\sim k^{n}$, we find a
corresponding power law for the variation of the linear momentum,
$\Delta p_x \propto b^{-n-1}$. This may permit to discriminate between various
equations of state of the fluid (including a fluid undergoing a turbulent behavior)
by examining the variation of linear momentum for a particle scattered by the fluid itself.
\subsection{Dissipative effects: viscosity and heat conduction}
As discussed in Sec. II, the Kolmogorov cascade is terminated by dissipative
effects; as a result it is of interest to extend our analysis to the case of a viscous fluid with
heat conduction, as discussed in Ex. 22.7 of Ref. \cite{Misner:1973prb}.
To this purpose, we increment Eq. \eqref{en_mom} with additional dissipative components:
\begin{eqnarray}
\label{en_mom}
T_{\rm visc}^{\mu\nu}&=& -\zeta \Theta(u) P(u)^{\mu\nu}-2\eta \sigma(u)^{\mu\nu}\,,\nonumber\\
T_{\rm visc}&=& -3\zeta \Theta(u)\,,\qquad\nonumber\\
T_{\rm heat}^{\mu\nu}&=& 2 u^{(\mu} q^{\nu)}\,,\qquad q\cdot u=0\,,\nonumber\\
T_{\rm heat}&=&0\,,
\end{eqnarray}
where $P(u)_{\alpha\beta}=g_{\alpha\beta}+u_\alpha u_\beta$ projects orthogonally to $u$, $\sigma(u)^{\alpha \beta}={\rm STF}_{\alpha\beta}[P(u)_{\alpha}^\mu P(u)_{\beta}^\nu \nabla_{\mu}u_\nu]$ is the symmetric and tracefree (STF) shear tensor of the fluid and $\Theta(u)=\nabla_\alpha u^\alpha$ is the expansion scalar. Moreover, $\zeta\ge 0$ denotes the coefficient of bulk
viscosity, $\eta\ge 0$ the coefficient of dynamical viscosity and
\begin{equation}
q^\alpha=-\kappa P(u)^{\alpha\beta}[T_\beta+Ta(u)_\beta]\,,
\end{equation}
is the Eckart\cite{Eckart:1940zz} law of conduction heat (later developed by C. Cattaneo \cite{Cattaneo})
with $\kappa$ the (constant) coefficient of thermal conductivity and
$a(u)_\alpha=\nabla_u u_\alpha$ the acceleration of the fluid world lines.
It proves expedient to introduce the dissipative tensors
$S_{\rm visc}^{\mu\nu}=T_{\rm visc}^{\mu\nu}-\frac12 T_{\rm visc}g^{\mu\nu}$ and $S_{\rm heat}^{\mu\nu}=T_{\rm heat}^{\mu\nu}$ and consider their representation in the Fourier space.
Assuming again $u^\mu=u_0^\mu=$constant in a Minkowski (flat) spacetime
referred to Cartesian coordinates $\nabla_\alpha u_\beta=0$,
hence $T_{\rm visc}^{\mu\nu}=0$ (i.e., a viscosity contribution appears
at higher orders of the PM procedure), we obtain:
\begin{eqnarray}
S_{\rm heat}^{\mu\nu}&=& u_0^{\mu} q^{\nu}+u_0^{\nu} q^{\mu}\nonumber\\
&=&-\kappa (u_0^\mu P(u_0)^{\nu\beta}+u_0^\nu P(u_0)^{\mu\beta})T_\beta\nonumber\\
&=&-\kappa (u_0^{ \mu} T^{\nu}+u_0^{ \nu} T^{\mu}+2u_0^{\mu} u_0^\nu u_0^\sigma T_\sigma)\,,
\end{eqnarray}
where $T^\mu=T^{,\mu}$ denotes the temperature gradient.
Moving to Fourier space:
\begin{eqnarray}
\hat S_{\rm heat}^{\mu\nu}(k)&=&-\kappa[ 2 u_0^{( \mu} \tau^{\nu)} (k)
+2 u_0^{\mu} u_0^\nu u_0^\sigma \tau_\sigma (k)] \,,\qquad
\end{eqnarray}
where
\begin{equation}
\tau^\nu (k)=\int d^4x e^{-ik\cdot x} T^{\nu}=ik^\nu \hat T(k)\,,
\end{equation}
and
\begin{equation}
\hat T(k)=\int d^4x e^{-ik\cdot x} T(x)
\end{equation}
is the Fourier transform of the temperature.
Finally, we find
\begin{eqnarray}
\hat S_{\rm heat}^{\mu\nu}(k)&=&-i\kappa \hat T(k)[ 2 u_0^{( \mu} k^{\nu)}
+2 u_0^{\mu} u_0^\nu u_0^\sigma k_\sigma ] \,.\qquad
\end{eqnarray}
To proceed further, consistently with the present analysis,
we consider a power-law spectrum
\begin{equation}
\hat T(k)=T_0 k^\ell\,,
\end{equation}
for some power exponent $\ell$.
This takes us back to the same basic integrals encountered in the non-dissipative treatment,
without adding any further layer of mathematical complexity, at least at
the 1PM level of approximation.
Future work is however needed to investigate the consequences of releasing
the main simplifying assumptions, such as $u^\mu=u_0^\mu=$ constant.
Work along these lines is currently in progress.
\subsection{Nonlocal effects and link to fractional calculus}
Finally, we observe that the integral \eqref{J_q_int} is conducive to fractional
calculus, and particularly to a fractional version of the D'Alembert
operator. As is well known, fractional derivatives
describe nonlocal effects in space an time, which is consistent
with the nature of turbulence \cite{FRACT,Nottale:2009azf,FRAC2}.
Indeed, coherent structures display a finite lifetime, meaning
that by the time their effects are felt in a given spacetime location,
they have already or moved elsewhere in the fluid, or possibly dissipated away,
whence the memory effect in both space and time.
It is very plausible to speculate that such form of dynamic
memory should be enhanced by coupling to the gravitational
degrees of freedom, since the
latter are driven by curvature, itself a source
of nonlocality even in non-relativistic physics
(think of the Poisson equation in electrostatics).
Clearly, the 1PM framework falls short of capturing the interaction of turbulence with
black-holes, but, as mentioned above, it is plausible to speculate that the
emergence of nonlocal effects would only be strengthened in
the presence of strongly curved spacetimes.
\section{Concluding remarks}
We have inspected the perturbative effects of fluid turbulence on
the gravitational metric and viceversa.
Based on purely statistical steady-state scaling arguments, we have first studied the
qualitative viability of gravitational interference on the turbulent energy cascade.
Next, we have performed a detailed dynamic analysis within the simplified
framework of first-order Post-Minkowskian (1PM) gravity.
Despite not being analytically solvable and far from the strong-curvature
conditions characterizing black-hole physics, the 1PM analysis strongly hints
at the onset of turbulence-driven non-local effects on spacetime evolution.
In fact, it permits to pin down the real-spacetime scaling exponents of the
perturbed metric as a function of the spectral exponents of turbulence.
Although firm conclusions are necessarily hinging on a more quantitative
non-perturbative analysis, most likely by numerical means, it is plausible to speculate that such
turbulence-driven nonlocal effects would only be accrued in the presence
of strongly curved spacetimes.
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\section{Introduction}
This paper summarizes our latest explorations in the design space of efficient DNNs. Specifically, we cover two complementary aspects: (a) static architecture design efficiency and (b) dynamic model execution efficiency.
\subsection{Static Architecture Design Efficiency}
Recent AutoML techniques, such as Neural Architecture Search (NAS), aim to automate the design of DNNs~\cite{wen2020neural, cheng2020nasgem, zoph2016neural}. Due to the considerable search cost required to traverse the DNN design space, early approaches can take thousands of GPU hours to select the final model candidates~\cite{gpu_hours}. To this end, in the context of hardware-constrained DNNs, previous work has significantly improved the search efficiency thanks to hardware performance models (e.g., DNN FLOPs, energy consumption, latency, etc.) which allow the AutoML algorithm to efficiently traverse the design space in a hardware-aware manner~\cite{ mnasnet, effnet, stamoulis2020single}, by quickly discarding DNNs that violate the hardware constraints of the target platform.
Hardware-aware AutoML algorithms resort to certain ineffective performance interpretations with respect to the underlying hardware. On the one hand, early AutoML methods~\cite{darts} use FLOPs as a general performance indicator, yet recent works demonstrate a mismatch between FLOPs and hardware metrics~\cite{dong2018dpp, marculescu2018hardware, cai2017neuralpower, yang2017designing}. Nevertheless, this mismatch has been discussed mainly through empirical results and is not comprehensively analyzed. On the other hand, while recent methods replace FLOPs with predictive models (e.g., latency, power consumption), they rely on ``end-to-end'' profiling, which is either limited to discrete design choices (e.g., 50\%, 100\% channels)~\cite{stamoulis2018hyperpower} or follows a look-up table-based manner~\cite{stamoulis2020single}.
To this end, we present a comprehensive ``full-stack'' profiling analysis that dives into individual GPU cores/threads to examine the intrinsic mechanisms of DNN execution~\cite{our_mlsys}. As a key contribution, we shed light into the ``GPU tail'' effect as root cause of FLOPs-latency mismatch and GPU under-utilization. Based on our findings, we revisit the DNN design configuration choices of state-of-the-art AutoML methodologies to eliminate the tail effect, enabling larger, more accurate DNN designs \textbf{at no latency cost}. Hence, our method \textit{concretely improves accuracy-latency trade-offs}, such as 27\% latency and 4\% accuracy improvements on top of SOTA DNN pruning and NAS methods. Moreover, we extend our profiling finding across different GPU configurations.
\iffalse
we also show that previous \textit{multi-path} search space could be unified into a \textit{single-path} search space, thus also greatly reducing the search space size~\cite{single_path}.
For example, the selection of 1x1, 3x3 and 5x5 kernels could be replaced with a single-path super-kernel operation, thus eliminating the multi-path candidate searching cost.
\fi
\textbf{Discussion - Future work}: while our investigation is employed as a fine-tuning (local search) step on top of SOTA designs, our findings can be flexibly incorporated into other AutoML methods. That is, a direction for future work is to revisit the predictive-models of existing single- and multi-path NAS works~\cite{single_path, cai2018proxylessnas} to further improve the accuracy-latency trade-offs by traversing the design space in a ``tail effect''-aware fashion. Moreover, our methodology focuses on eliminating ``tail effects'' at the DNN design level, but improvements from alleviating GPU under-utilization can be realized at other design levels, as shown by novel scheduling- and computational flow-level explorations~\cite{ding2020ios, yu2020dc}.
Next, we hope that our findings could inspire researchers to revisit design-space assumptions, by allowing to identify hardware-optimal DNN candidates while eliminating sub-optimal ones. For example, our channel-level analysis reveals a \textit{discrete} set of DNN channel configurations~\cite{our_mlsys} with optimal GPU utilization, which could potentially reduce the number of candidates by $10\times$ as opposed to traversing a \textit{continuous} channel-number space, e.g., on top of existing channel-pruning methods~\cite{amc, chin2020towards}. Last, an interesting direction would be to investigate the severity of similar under-utilization beyond GPUs, especially in the context of hardware accelerators and co-design NAS methodologies~\cite{zhang2020dna}.
\iffalse
To do so, our work aims to improve the current NAS search efficiency by reducing the search space in two levels: (a) hardware aware optimal configuration pre-identification, and (b) single path efficient search strategy.\newline
\textit{Hardware-aware}: Our ongoing work utilizes the a.\newline
To improve search efficiency, parameter sharing has been widely adapted~\cite{darts, todo}. Existing parameter sharing-based NAS methods usually formulate NAS as a path selection problem by instantiating different candidate operations as distinct trainable variables (paths). Compared to existing multi-path NAS, we instead identify a new perspective, i.e., designing more efficient search space. Specifically, the efficient search space design aims to reduce the size of search configurations, thus serving as an orthogonal dimension to the searching algorithm efficiency improvement.
\textit{Single Path}: Meanwhile, we also show that previous \textit{multi-path} search space could be unified into a \textit{single-path} search space, thus also greatly reducing the search space size~\cite{single_path}. For example, the selection of 1x1, 3x3 and 5x5 kernels could be replaced with a single-path super-kernel operation, thus eliminating the multi-path candidate searching cost.
\fi
\iffalse
For example, the model can contain one set of trained filters (e.g., of shapes 1x1, 3x3, 5x5) but only one (or multiple) of them will be dynamically selected and executed based on the input characteristics (e.g., size of the object). In this way, redundant computations (e.g., 5x5 filters) could be adaptively removed based on the input complexity, improving the
\fi
\subsection{Dynamic Model Execution Efficiency}
Dynamic execution methods aim at selecting between ``switchable'' DNN components at runtime~\cite{xu2020directx, chen2020dynamic, dynamic1, dynamic2, xu2019reform}. The key insight behind these works is to improve the overall model efficiency by adaptively selecting and executing (a subset of) the model based on the input characteristics~\cite{stamoulis2018designing}. In our work, we extend this intuition across a new \textit{model redundancy} dimension, namely dynamic feature map redundancy~\cite{our_date}.
Specifically, we show that feature redundancy exists at the spatial dimensions of DNN convolutions, which allows us to formulate a dynamic pruning methodology
in \textbf{both channel- and spatial-wise dimensions}. Our proposed method can greatly reduce the model computation with up to 54.5\% FLOPs reduction and
negligible accuracy drop on various image-classification DNNs.
\textbf{Discussion - Future work}: Drawing inspiration from our analysis on the FLOPs-latency mismatch, we highlight that when implemented naively, merely pruning convolution weights at the spatial level does not translate to latency savings. To this end, we postulate that advances in sparse DNN operators will be essential to support \textit{dynamic-sparse} execution, as recently shown with CUDA implementations for dynamic convolutions~\cite{verelst2020dynamic}.
\iffalse
Similar to before, such efficient feature map elimination is orthogonal to the structure component switching, thus combining both could potentially lead to further gain.
\fi
\section{Conclusion}
In this paper, we summarize a set of novel efficiency optimization angles for DNN design in both static architecture design and dynamic model execution. New potential advantages can be attained by integrating the proposed new perspectives to current optimization methods.
\vspace{-8pt}
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\section{Introduction}
Analysis of human emotion is the bedrock of affective computing. The majority of research in this field has focussed on predicting emotion from face, voice and text \cite{Pantic2000, Hanjalic2005, ElKaliouby2005, Yang2008, Zeng2009, Schuller2010, Polzehl2011, Schuller2011}. Physiological analysis has garnered comparatively little attention \cite{Kim2008b,Alzoubi2012, Goshvarpour2017}, and explores the neurobiological correlates of emotion within the limbic and autonomic nervous systems.
Physiology-based emotion detection has tremendous potential to compliment existing methods of affective computation. For instance, analysis of face, voice, and text rely heavily on expression, which can vary across individuals and cultures \cite{Ekman1987, Scherer2001}, and can also be easily faked. By comparison, physiological processes are far less volitional. Physiological analyses present a further opportunity for non-invasive continuous monitoring - as physiological signals may be passively measured throughout the day.
For these reasons, physiology-based emotion detection has the capacity to fill critical gaps in domains where it is challenging to continuously collect audiovisual data (e.g. healthcare). Perhaps unsurprisingly, there is a growing physiological resurgence within affective computing. The vast majority of studies rely on a combination of autonomic markers to classify emotional response. These include galvanic skin response (GSR), electroencephalogram (EEG), electromyogram, respiration, skin temperature (ST) and electrocardiogram (ECG). While attractive from a modelling perspective, such multimodal input may not always be available. A select few studies have therefore constructed models capable of predicting emotion from \textit{unimodal} ECG data \cite{harper2019bayesian, Katsigiannis2018, Subramanian2016, Keren2017, Miranda-Correa2017b, Guo2016a, Ferdinando2016, Valenza2014b, Agrafioti2012}. The idea here tends to be that such models could be extended to predict emotion using wearable heart monitoring devices `in the wild'. Indeed, it has been shown that emotional valence can be classified using expensive lab-based wearable ECG recording devices \cite{harper2019bayesian}. However, to be truly relevant for large-scale real-world monitoring today, such classifiers must be compatible with the growing number of affordable consumer fitness trackers. These almost exclusively extract heartbeat from photoplethysmogram (PPG).
Consumer fitness trackers typically extract the peaks of the PPG signal to obtain a heartbeat time series, or `inter-beat intervals' (IBIs). As IBIs can be extracted from ECG and PPG data, we use the notation IBI\textsubscript{ECG} and IBI\textsubscript{PPG} to distinguish between the two. To the best of our knowledge, no previous work has rigorously explored the suitability of IBI\textsubscript{PPG} generated by affordable fitness trackers for predicting human emotion at scale. Such research has the potential for immediate real-world application, as the number of wrist-worn wearable devices continues to rise into the hundreds of millions \cite{idcmedia} and all major brands now incorporate commoditised PPG sensors as standard (e.g. Fitbit, Polar, Samsung Gear, Apple Watch, and Garmin).
In this study, we use a Bayesian deep neural network model that was shown previously to classify emotional valence from IBI\textsubscript{ECG} \cite{harper2019bayesian}. We extend this model for emotion detection using IBI\textsubscript{PPG} collected by a consumer wearable. For this study, we generated a new dataset comprising IBI\textsubscript{PPG} data (collected using a Garmin V\'ivosmart 3 device). This data was recorded during presentation of a number of short emotion-inducing video stimuli. We explore the statistical differences between this IBI\textsubscript{PPG} and previously collected IBI\textsubscript{ECG}. We go on to show that training a neural network classifier on IBI\textsubscript{ECG} confers no performance improvement when tested on IBI\textsubscript{PPG}, demonstrating the necessity for new datasets built around cheap off-the-shelf wearable devices.
\section{Related Work}
This section provides an overview of relevant work, with a focus on (A) unimodal ECG for emotion prediction, and (B) unimodal PPG for emotion prediction.
\subsection{Emotion Prediction from Unimodal ECG Data}
Existing approaches for prediction of emotion using physiological signals typically pool a number of bio-signals to provide multimodal input to a classifier algorithm \cite{Kim2008b, Alzoubi2012, Goshvarpour2017}. Fewer studies narrow their scope to unimodal ECG input in accordance with the heartbeat-centric limitations of affordable wearable devices. Additionally, those studies that have explored unimodal heartbeat models for emotion detection tend to ignore temporal structures of the signal. Instead, they use `static' classification methods that analyse global features of the input time-series, such as Naive Bayes (NB), \cite{Miranda-Correa2017b, Subramanian2016}, linear discrimant analysis (LDA) \cite{Agrafioti2012}, and support vector machine (SVM) \cite{Katsigiannis2018, Guo2016a, Valenza2014b}. A summary of these studies can be found in Table \ref{relevant_work}. Two notable exceptions have implemented temporal neural network models to predict emotional valence from ECG input \cite{Keren2017, harper2019bayesian}. In these examples, convolutional and recurrent network layers were used to perform end-to-end learning, which improved upon computationally expensive manual feature engineering schemes. In \cite{harper2019bayesian}, a Bayesian framework was further used to output probability distributions over valence predictions, making this model particularly suited for applications in domains such as healthcare, where a high premium is placed on predictive certainty.
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Summary of relevant work}
\label{relevant_work}
\centering
\begin{tabularx}{\linewidth}{|Y|Y|c|Y|Y|Y|}
\hline
\textbf{Author} & \textbf{Stimulus} & \textbf{Subjects} & \textbf{Model} & \textbf{Target} & \textbf{Performance}\\
\hline
Harper \& Southern 2018 \cite{harper2019bayesian} & Videos & 40 & LSTM and CNN & High/Low Valence & Acc. 90\% (Chance: 50\%) \\
\hline
Katsigiannis \& Ramzan 2018 \cite{Katsigiannis2018} & Videos & 23 & SVM & High/Low Valence & F1. 0.5305 (Chance: 0.500) \\
\hline
Subramanian et al 2018 \cite{Subramanian2016} & Videos & 58 & NB & High/Low Valence & Acc. 60\% (Chance: 50\%) \\
\hline
Keren et al 2017 \cite{Keren2017} & Naturalistic dyadic interactions & 27 & LSTM and CNN & Continuous Valence (Regression) & Concordance Correlation Coefficient. 0.210 (Baseline: 0.121)\\
\hline
Miranda-Correa et al 2017 \cite{Miranda-Correa2017b} & Videos & 40 & NB & High/Low Valence & F1. 0.545 (Chance: 0.500) \\
\hline
Guo et al 2016 \cite{Guo2016a} & Videos & 25 & SVM & High/Low Valence & Acc. 71.40\% (Chance: 50\%) \\
\hline
Ferdinando et al 2016 \cite{Ferdinando2016} & Videos \& Images & 27 & KNN & High/Medium/Low Valence & Acc. 59.2\% (Chance: 33.3\%) \\
\hline
Valenza et al 2014 \cite{Valenza2014b} & Images & 30 & SVM & High/Low Valence & Acc. 79.15\% (Chance: 50\%) \\
\hline
Agrafioti et al 2012 \cite{Agrafioti2012} & Images & 32 & LDA & Gore, Erotica & Acc. 46.56\% (Chance: 50\%) \\
\hline
\end{tabularx}
\end{table*}
\subsection{Emotion Prediction from Unimodal PPG Data}
Very few studies have explored emotion detection with a focus on PPG data. One study combined GSR and PPG, collected by a Shimmer3 sensor \cite{shimmer:xxx}, to classify High/Low valence and arousal \cite{Udovicic:2017:WER:3132635.3132641}. In another study, unimodal PPG data collected by an expensive wrist-worn wearable device (Empatica E4 \cite{7015904}) was compared with data collected by a laboratory sensor (Biopac MP150 \cite{biopac:xxx}) \cite{10.1007/978-3-319-60639-2_2}. Although these studies represent an important step towards real-world applicability, we are not aware of any studies that have explored emotion recognition using IBI\textsubscript{PPG} data of the type collected by affordable consumer fitness trackers.
\section{Experimental Setup}
In this section, we describe the experimental procedure for collecting IBI\textsubscript{PPG} from a consumer fitness tracker (Garmin V\'ivosmart 3).
\subsection{Experimental Protocol}
We used an emotion-inducing stimulus setup combined with participant self-reporting, as is conventional within the field of affective computing. The experiment involved 17 study participants (5 female; 12 male). Each participant received an initial tutorial on how to self-report their emotional state using the widely-used Self-Assessment Manikin (SAM) framework for measuring emotion \cite{Morris1995}. Next, the Garmin V\'ivosmart 3 was secured to the left wrist of the participant, and IBIs extracted by the embedded PPG sensor were collected. The participants were seated directly in front of a computer screen (at a distance of 60cm) and were asked to wear headphones in order to reduce external distractions. The experimenter then left the room and the recording session began.
Emotion-inducing video stimuli were presented on the computer screen in randomised order. At the end of each video stimulus, the participant was asked to complete an emotional valence self-report using the SAM framework \cite{Morris1995}. After completing the emotion self-report, the participant experienced one minute of a neutral scene and was asked to clear their mind as much as possible prior to the next video stimulus. This was done to reduce carry-over of emotions between video stimuli. A schematic overview of the experimental setup can be found in Fig.~\ref{fig:Experimental_Setup}A, with photograph shown in Fig.~\ref{fig:Experimental_Setup}B.
\begin{figure}\centering
\includegraphics[width=\linewidth]{Experimental_Setup_4.pdf}
\caption{Experimental setup. The participant was seated in front of a single computer monitor. A video stimulus from Table \ref{tab:Videos} was selected randomly and played. After the stimulus had ended, the participant was asked to report their emotional valence using the SAM framework. The next randomly selected video stimulus was presented after a one minute break. This process repeated until all 24 stimuli had been presented.}
\label{fig:Experimental_Setup}
\end{figure}
\subsection{Stimuli}
\label{section:Video_Stimuli}
The participants each viewed 24 short video stimuli presented in random order. 96 potential videos were initially chosen, and independently annotated for emotional valence by 30 volunteers using the SAM framework. The variance of these annotations was then calculated for each video, and the 96 potential videos ranked from lowest to highest variance (lowest variance at the top, representing highest agreement amongst the 30 annotators). The top 25\% of videos were selected (24 stimuli). Of these, 8 videos had been independently scored as inducing pleasure (high valence); 16 videos were independently scored as inducing displeasure (low valence). The average stimulus length was 02:29 (See Table \ref{tab:Videos}).
To confirm that the 24 test videos induced the expected emotional valence in study participants, we show in Fig.~\ref{fig:Reported_Emotional_State} the density of valence scores obtained from study participants during the experiment. We see that these self-reports broadly match those of the 30 volunteer annotators (Table \ref{tab:Videos}). All video stimuli were presented, and the SAM administered, using a custom-made web app.
\begin{figure*}[ht]\centering
\includegraphics[width=\linewidth]{Reported_Emotional_State.png}
\caption{Self-reported emotional valence induced by each video clip. Study participants rated their emotional state after each video clip on a five-point scale from `Strong Displeasure' to `Strong Pleasure' in accordance with Self-Assessment Manikin (SAM) framework for measuring emotion \cite{Morris1995}. The image shows the density of reports for each emotional state: blue to yellow with greater density of ratings (see colour bar). Note the videos 1-8 elicited more pleasurable emotions compared to 9-24, which elicited more displeasurable emotions. This is in agreement with Table \ref{tab:Videos}.}
\label{fig:Reported_Emotional_State}
\end{figure*}
\subsection{Measuring PPG} \label{section:Measuring_PPG}
IBI data extracted from the PPG signal was collected using the Garmin V\'ivosmart 3, which retails at around £70 (\$90). For this, we developed a custom Android Wear app using the Android Wear SDK. This app collected the IBIs locally on a mobile device, synchronised these with the timing of video stimuli presentation, and sent the resulting data files to cloud servers upon experiment completion.
\subsection{Facial Video Recordings}
Frontal face video was also recorded during the experiment using a web-cam positioned centrally on the computer screen. Although our present study does not incorporate visual data for affect recognition, this data can be used for future work comparing facial and physiological signals for prediction of emotion.
\begin{table}[h]
\caption{Selected Video Stimuli}
\label{tab:Videos}
\begin{tabularx}{\linewidth}{|c|Y|Y|}
\hline
\textbf{Video} & \textbf{Elicitation} & \textbf{Duration} \\
\hline
\#1 & Pleasure & 0:53 \\
\#2 & Pleasure & 3:52 \\
\#3 & Pleasure & 1:14 \\
\#4 & Pleasure & 2:15 \\
\#5 & Pleasure & 2:39 \\
\#6 & Pleasure & 4:28 \\
\#7 & Pleasure & 0:51 \\
\#8 & Pleasure & 1:11 \\
\hline
\#9 & Displeasure & 2:11 \\
\#10 & Displeasure & 6:08 \\
\#11 & Displeasure & 1:22 \\
\#12 & Displeasure & 2:06 \\
\#13 & Displeasure & 1:51 \\
\#14 & Displeasure & 1:47 \\
\#15 & Displeasure & 2:53 \\
\#16 & Displeasure & 5:09 \\
\#17 & Displeasure & 0:56 \\
\#18 & Displeasure & 2:13 \\
\#19 & Displeasure & 2:47 \\
\#20 & Displeasure & 4:52 \\
\#21 & Displeasure & 0:40 \\
\#22 & Displeasure & 2:19 \\
\#23 & Displeasure & 2:51 \\
\#24 & Displeasure & 2:18 \\ \hline
\end{tabularx}
\end{table}
\section{Model} \label{section:Model}
\subsection{Neural Network Architecture}
Deep neural networks have obtained promising results for end-to-end classification of valence from unimodal ECG data \cite{Keren2017, harper2019bayesian}. In this study, we use the neural network architecture described in \cite{harper2019bayesian}, which incorporates a Bayesian framework to model probability distributions over model output. (For details of the model hyperparameters and training protocol, please see the original text).
An overview of our model is shown in Fig.~\ref{fig:ModelArchitecture}. In brief, the IBI time series passes through two concurrent streams. The first stream comprises four stacked convolutional layers with filter size set to 128, and window size decreasing from 8 to 2 time steps with network depth. This extracts features from larger receptive fields as the data passes through each successive layer. Monte Carlo dropout is applied after each convolutional layer, as well as a ReLU activation function (for details, see \cite{harper2019bayesian}).
The second stream comprises a bidirectional LSTM (each with 32 hidden units), also followed by Monte Carlo dropout. This recurrent structure permits temporal modelling of the heartbeat time series, which is non-linear and non-stationary \cite{Weber1992, Sunagawa1998}. The output of these two streams is finally concatenated into a 192-length vector (128 from the convolutional stream; 64 from the LSTM stream) before passing through a dense layer to output a regression estimate for valence.
Uncertainty is a key component of decision-making in many real-world domains, especially healthcare \cite{Ghahramani2015}. It therefore follows that applications of physiology-based emotion detection in this area must incorporate probabilistic considerations. We therefore use Monte Carlo dropout to recast our neural network as a Bayesian model, performing $N$ stochastic forward passes through the network to approximate a posterior distribution over model predictions \cite{Gal2016}.
\subsection{Binary Classification Framework}
In order to translate from a regression to a classification scheme, we introduce decision boundaries in continuous space. For a binary (high/low) classification, this can be done by including a decision boundary at the central point of the valence axis. We next introduce a confidence threshold parameter, $\alpha$, to tune predictions to a specified level of model uncertainty. For example, when $\alpha = 0.95$, at least $95\%$ of the output distribution must lie above or below the valence scale midpoint in order for the input sample to be classified as belonging to the high or low valence class respectively. If this is not the case, no prediction is made (the model respectfully makes no comment). As our model may not classify all instances, we adopt the term 'coverage' to denote the set of cases for which it is confident enough to make a prediction. For an in-depth discussion, see \cite{harper2019bayesian}.
Note that for a binary classification problem, and $N$ is an odd integer, there will always be at least $50\%$ of the output distribution above or below the valence midpoint. Thus, when $\alpha = 0.5$, classification is determined by the median of the output distribution, and the coverage is $100\%$. As $\alpha$ increases, model behaviour moves from risky to cautious $-$ lower coverage, but more confidence in the classification. This aligns with our goal of providing real-world relevance to physiology-based emotion prediction in domains such as healthcare.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Limbic_Model.pdf}
\caption{End-to-end model architecture (adapted from \cite{harper2019bayesian}). Data flows through two temporal processing streams: 1D convolutions (green) and a bi-directional LSTM (blue). The output from both streams is then concatenated before passing through a dense layer to output a regression estimate for valence, $\hat{y}$. }
\label{fig:ModelArchitecture}
\end{figure}
\section{External Data}
We applied the Bayesian deep learning framework described above (and in \cite{harper2019bayesian}) to achieve end-to-end prediction of emotion using IBI\textsubscript{PPG} collected by the Garmin V\'ivosmart 3. However, we further wished to explore the differences between these IBI\textsubscript{PPG} and IBI\textsubscript{ECG} extracted from a laboratory-grade monitor. For this comparison, we used the established AMIGOS dataset \cite{Miranda-Correa2017b}.
The AMIGOS dataset consists of 40 healthy participants (13 female; 27 male) aged between 21 and 40 years old (mean: 28.3). The ECG was recorded using a Shimmer\textsuperscript{TM} ECG wireless monitoring device (256 Hz, 12 bit resolution) \cite{shimmer:xxx}. The participants watched 18 film clips (duration $< 395$ seconds), which had been selected for their ability to elicit strong emotional responses \cite{Miranda-Correa2017b}. The videos were presented to the subjects in a random order with a 5-second baseline recording of a fixation cross being shown before each video. Each film clip was followed by self-assessment of valence on a scale of 1 to 9 using SAM \cite{Morris1995}.
\section{Methods}
\subsection{Pre-processing}
The IBI\textsubscript{PPG} extracted from the PPG sensor in the Garmin V\'ivosmart 3 were z-score normalized and zero padded to the length of the longest training sample. For the AMIGOS data, IBIs were extracted manually from the ECG time-series using a combined adaptive threshold method \cite{Christov2004}. The resulting IBI\textsubscript{ECG} was then also z-normalized and zero padded or cut to the length of the longest IBI\textsubscript{PPG} training sample.
\subsection{Training and Hyperparameters} \label{section:Training_and_Hyperparams}
The hyper-parameters of the model were set to those specified previously \cite{harper2019bayesian}. The convolutional kernels were initialized as He normal \cite{He2015} with a filter size set to 128, and a window size decreasing from 8 to 2 time steps with network depth. A dropout of $50\%$ was applied after each convolutional block, and $80\%$ dropout followed the bi-directional LSTM, which comprised 32 hidden units. The training phase was run for 1000 epochs using Adam optimization \cite{Kingma2014} and the learning rate decreased from $e^{-3}$ to $e^{-4}$, halving with a patience of 100 epochs. The model was implemented using Tensorflow \cite{GoogleResearch2015}.
\subsection{Evaluation}
Model performance was assessed using 10 iterations of leave-one-subject-out cross-validation to show the ability of the model to generalize to new people. For each iteration, one subject was randomly selected and their data held out as a test set. Dropout was applied at test time with $N = 1000$ forward propagations made through the network to generate an empirical distribution over model output. As outlined in section \ref{section:Model}, a given test input sample was classified into a binary high/low valence class provided a proportion of at least $\alpha$ posterior distribution mass fell above or below the valence midpoint respectively. If this was not the case, then no prediction was made. The model's F1 score was then calculated based on those classifications that the model attempted. We chose to evaluate our model using the F1 score, rather than accuracy, due to the unbalanced high/low valence videos in the dataset (selected as described in Section \ref{section:Video_Stimuli}).
\section{Results}
\subsection{Comparison of IBIs Extracted from ECG and PPG}
In order to gain an understanding of the differences between IBIs extracted from ECG data (collected by the commonly-used laboratory-grade Shimmer\textsuperscript{TM}), and IBIs extracted from a consumer PPG sensor (Garmin V\'ivosmart 3), we calculated a number of features for all IBI samples across both datasets.
Frequency domain features included (1) spectral power in the frequency range [0.15, 0.4] Hz (HF power), (2) spectral power in the frequency range [0.04, 0.15] Hz (LF power), spectral power in the frequency range [0.003, 0.04] Hz (VLF power), and (4) ratio of low frequency to high frequency signal (LF/HF). Time domain features included (1) mean, (2) median, (3) standard deviation (SDSD), (4) number of instances where the change between successive IBIs is greater than 0.02 (NN20), (5) normalised NN20 (pNN20), (6) the root mean square of the successive differences (rMSSD), and (7) the multiscale entropy.
Non-parametric Mann-Whitney test was performed for each feature to identify statistically significant differences between IBIs extracted from ECG and PPG. Statistically significant differences were observed for VLF power, SDSD, NN20, pNN20, rMSSD and the multiscale entropy (See Fig.~\ref{fig:StatisticalDifferences} and Table \ref{tab:pvalues}).
To further probe these statistical differences, a simple SVM classifier was used to differentiate IBIs extracted from ECG and PPG using the previously calculated features as input. The sklearn library in Python \cite{Pedregosa:2011:SML:1953048.2078195} was used to build a C-Support Vector Classification with `rbf' kernel and penalty parameter, C, set to 1. 10-fold cross-validation was implemented and accuracy of the classifier was found to be $70\%$. This supports the conclusion that there are structural differences in the statistical properties between IBI\textsubscript{PPG} and IBI\textsubscript{ECG}.
\boldmath
\begin{table}[h]
\caption{Mann-Whitney P-values between IBI\textsubscript{ECG} and IBI\textsubscript{PPG} features}
\label{tab:pvalues}
\begin{tabularx}{\linewidth}{|c|Y|Y|}
\hline
& \textbf{Feature} & \textbf{P-value} \\ \hline
1 & HF power & 0.22 \\ \hline
2 & LF power & 0.22 \\ \hline
3 & \textbf{VLF power} & $< 0.001$ \\ \hline
4 & LF/HF & 0.22 \\ \hline
6 & Mean & 0.27 \\ \hline
7 & Median & 0.26 \\ \hline
8 & \textbf{SDSD} & $< 0.001$ \\ \hline
9 & \textbf{NN20} & $< 0.001$ \\ \hline
10 & \textbf{pNN20} & $< 0.001$ \\ \hline
11 & \textbf{rMSSD} & $< 0.001$ \\ \hline
12 & \textbf{Multiscale Entropy} & $< 0.001$ \\ \hline
\end{tabularx}
\end{table}
\unboldmath
\begin{figure}
\centering
\includegraphics[width=\linewidth]{ppg_ecg_histograms.pdf}
\caption{Histograms showing the values of different features calculated from the time-series of both the IBI\textsubscript{PPG} (yellow) and IBI\textsubscript{ECG} data (purple).}
\label{fig:StatisticalDifferences}
\end{figure}
\subsection{Predicting Emotion Using IBIs Extracted from PPG}
We implemented the Bayesian neural network described in Section \ref{section:Model} using IBI\textsubscript{PPG} data collected by the Garmin V\'ivosmart 3 (see Section \ref{section:Measuring_PPG}). As $\alpha$ increased, so too did the F1 score, demonstrating a clear relationship between model confidence and propensity to make accurate predictions (Fig.~\ref{fig:ModelResults}A). As expected, model coverage decreased as $\alpha$ increased, due to the fact that fewer output distributions met the necessary threshold for a prediction to be made (Fig.~\ref{fig:ModelResults}B). When $\alpha = 0.95$, our model achieved a peak F1 score of 0.7 (Fig.~\ref{fig:ModelResults}A).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Results.pdf}
\caption{Classification performance of high/low valence using IBI\textsubscript{PPG} input from a consumer wearable. (A) Model F1 score as a function of $\alpha$. (B) Model coverage as a function of $\alpha$. Results are shown for model trained on the IBI\textsubscript{PPG} data alone (yellow, triangles), IBI\textsubscript{ECG} and IBI\textsubscript{PPG} data together (blue, circles), and IBI\textsubscript{ECG} data alone (green, crosses). Grey dashed line in (A) shows F1 score of a chance model for comparison.}
\label{fig:ModelResults}
\end{figure}
\subsection{Further Training with IBIs Extracted from ECG}
We next investigated whether IBI\textsubscript{ECG} collected by the commonly-used laboratory-grade Shimmer\textsuperscript{TM} conferred any advantage to the task of predicting emotion from IBI\textsubscript{PPG} collected by the consumer fitness tracker. The IBI\textsubscript{ECG} data from the AMIGOS dataset was added to the IBI\textsubscript{PPG} training set, and the model was evaluated, as before, on the IBI\textsubscript{PPG} test set (see Section \ref{section:Training_and_Hyperparams} for train-test subdivision). No significant difference was observed in model performance when trained on IBI\textsubscript{PPG} data alone versus IBI\textsubscript{PPG} combined with IBI\textsubscript{ECG} (p = 0.16, computed using Mann-Whitney test between the 10 F1 scores, $\alpha = 0.5$).
For completeness, we further trained the model using the IBI\textsubscript{ECG} data alone, and then evaluated on the IBI\textsubscript{PPG} test data. In this setting, the model performed no better than chance. (Here, the chance F1 score of 0.57 is the F1 score obtained when a video is naively classified as either high or low valence with equal probability). The performance of the model with these different combinations of training data is shown in Fig.~\ref{fig:ModelResults}.
\section{Discussion}
The growing prevalence of affordable consumer wearable monitoring devices has created an opportunity for emotion detection at scale. Recent work has tried to bridge the gap from laboratory to real-world through the analysis of unimodal heartbeat data (in accordance with the availability of heartbeat sensors). However, no study has explored affect recognition on heartbeat data collected by a cheap off-the-shelf consumer wearable device. This is important if physiology-based emotion detection is to have immediate relevance today.
In this study, we have shown that the IBI data collected by a popular fitness tracker is statistically different to that which is collected by a widely-used laboratory-grade ECG monitor. Of particular note is that significant differences were found for more time domain features of the heartbeat signal, as compared to frequency domain features. Additionally, the IBI\textsubscript{ECG} data did not confer any performance advantage when used to train our neural network model for the task of predicting valence from IBI\textsubscript{PPG} samples generated by the consumer fitness tracker. This supports the conclusion that real-world applications of physiology-based emotion detection would benefit from new datasets built around cheap off-the-shelf wearable devices. This study represents a good first attempt, which, using a Bayesian neural network classifier, achieved a promising peak F1 score of 0.70 from our new dataset comprising of 17 participants.
Our probabilistic classification framework includes a confidence parameter, $\alpha$, which allowed the F1 score and coverage of our model to be tuned according to varying demands on prediction certainty. The use of a regression output further allows the experimenter to switch easily between regression and classification tasks, and indeed allows her to specify bespoke decision boundaries appropriate for binary- or multi-class tasks. We chose to incorporate these Bayesian considerations to align with our overarching goal of making physiology-based emotion detection relevant to real-world applications. For instance, emotion detection for mental health monitoring might reasonably require high levels of certainty to predict the onset of major depressive disorder. Additionally, clinical triaging is possible, where uncertain model predictions are sent to a human expert for review (or perhaps a more computationally expensive model). Similar levels of certainty may not, however, be absolutely necessary in many consumer products.
\bibliographystyle{ieeetr}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
The brightest of five main spiral galaxies that form the Sculptor group, NGC 300 is a fairly typical late-type galaxy \citep{1988ngc..book.....T} at a distance of $\sim 2.1$ Mpc \citep{1992ApJ...396...80F}.
Most of the measurements of the distance to this galaxy are based on the luminosity of its Cepheid variables population. Based on near-infrared H-band observations of two long-period Cepheid variables, \citet{1987ApJ...320...26M} reported a distance modulus $(m-M)_0=26.35 \pm 0.25$. The distance was slightly revised by \citet{1988PASP..100..949W} who derived a distance modulus $(m-M)_0=26.4 \pm 0.2$. Additional photometry of the same sample of variables by \citet{1992ApJ...396...80F} resulted in the already quoted distance $(m-M)_0=26.66 \pm 0.10$, subsequently revised to $(m-M)_0=26.63 \pm 0.06$ in \citet{2004ApJ...608...42S}.
More recently, NGC 300 has been selected as a key target for the Araucaria Project\footnote{http://ifa.hawaii.edu/$\sim$bresolin/Araucaria}. \citet{2002AJ....123..789P} presented an extensive characterization of 117 Cepheid variables, most of which were new discoveries, observed with the 2.2m ESO/MPI telescope at La Silla, Chile. Additional V and I data were obtained by \citet{2004AJ....128.1167G} at Las Campanas and Cerro Tololo. Deep, near-infrared J and K band observations were obtained with ESO VLT using the ISAAC camera, resulting in a final distance modulus $(m-M)_0=26.37 \pm 0.05 \,{\rm (random)} \pm 0.03 \,{\rm (systematic)}$ \citep{2005ApJ...628..695G}.
The superb angular resolution offered by the Hubble Space Telescope has recently open the possibility of determining the distance to NGC 300 using the Tip of the Red Giant Branch (TRGB). A set of HST WFPC2 fields were analyzed by \citet{2004ApJ...608...42S} and \citet{2004AJ....127.1472B}, and more recently by \citet{2005A&A...431..127T}. The derived distance moduli are $(m-M)_0=26.65 \pm 0.09$,
$(m-M)_0=26.56 \pm 0.07 \pm 0.13$, and $(m-M)_0=26.50 \pm 0.15$, respectively.
In this paper, we present the first TRGB distance based on deep ACS observations of NGC 300. These data are the deepest ever obtained for this galaxy, and they sample both the inner bulge and the outer disk. The paper is organized as follows: Section \ref{data} presents the data, the reduction techniques we adopted, and the resulting color-magnidute diagrams (CMD). We describe the TRGB method and its application to NGC 300 in Section \ref{distance}. We discuss our results in Section \ref{discussion} and a brief summary is presented in Section \ref{conclusions}.
\section{Observations, Data reduction, and Color-Magnitude Diagrams}
\label{data}
The ACS observations used to derive a new TRGB distance to NGC 300 were obtained during HST Cycle 11, as part of program GO-9492 (PI: Bresolin), from July 2002 to December 2002. The main purpose of these observations was to complement the extensive ground-based CCD photometry of Cepheid variable stars and blue supergiant stars collected in the framework of the Araucaria project.
Two-orbit HST visits allowed to obtain deep photometry in the F435W, F555W (1080 seconds), and F814W (1440 seconds) filters. A total of six fields were observed.
Stellar photometry was performed with the DOLPHOT (version 1.0) package, an adaptation of HSTphot
\citep{2000PASP..112.1383D} to ACS images. Pre-computer Point Spread Functions were adopted, and the final calibrated photometry was then transformed to the standard BVI system using the equations provided by \citet{2005astro.ph..7614S}. The transformation from one photometric system to another inevitably introduces additional uncertainties but it seems necessary given that most of the calibrations of the absolute magnitude of the TRGB are in the I band. For a more extended discussion of the issues related to calibration see \citet{Bresolin:rb}.
As an example of the quality of the results, the final calibrated CMDs are shown in Figures \ref{cmd_1.ps} and \ref{cmd_3.ps} for Fields 1 and 3, respectively. Field 1 is situated close to the eastern outer edge, while Field 3 is centered on the nucleus of the galaxy \citep[see][for a map of the observed Fields]{Bresolin:rb}. All the CMDs show a very well pronounced sequence of blue young stars, reaching down to the lower age limit of isochrone sets \citep[$\sim 60$ Myr,][]{2000A&AS..141..371G}. Blue-loop stars occupy the central region of the diagrams, and a well defined red giant branch (RGB) extends from I $\sim 22$ down to the photometric detection limit, I $\sim 26$.
A full discussion of the CMD features, along with a reconstruction of the star formation history, will be presented in a forthcoming paper.
\begin{figure}
\plotone{f1.eps}
\caption{(V-I,I) color-magnitude diagram for Field 1 of NGC 300. The Field is situated close to the eastern edge of the galaxy.}
\label{cmd_1.ps}
\end{figure}
\begin{figure}
\plotone{f2.eps}
\caption{(V-I,I) color-magnitude diagram for Field 3 of NGC 300. The Field is situated on top of the nucleus of the galaxy.}
\label{cmd_3.ps}
\end{figure}
\section{The Distance to NGC300}
\label{distance}
\subsection{Detection of the tip}
The distance estimates based on the RGB tip rest on a solid physical basis: low-mass stars reach the end of their ascent along the RGB with a degenerate helium core and they ignite helium burning within a very narrow range of luminosity \citep[][and references therein]{2002PASP..114..375S}.
The potential of the method was revealed in a seminal paper by Lee and collaborators \citep{1993ApJ...417..553L}, along with a first attempt at objectively estimate the position of the tip on the CMD based on a digital edge-detection (ED) filter in the form [-2,0,2], applied to the I band luminosity function. This filter effectively responds to changes in the slope of the luminosity function and displays a peak corresponding to the TRGB. A refined version of this method was presented in \citet{1996ApJ...461..713S}. More recently, a different approach was suggested by \citet{2002AJ....124..213M}. To avoid problems related to binning, this method uses a maximum-likelihood (ML) analysis to get the best fit of a parametric RGB luminosity function to the observed one. Each of these methods has advantages and disadvantages. ED methods are quite sensitive to binning, but they don't require any {\it a priori} assumption on the shape of the RGB luminosity function. ML methods use much more information, because every star of the sample contributes to the probability distribution, but they use a theoretical luminosity function as an input parameter. In this work, we use both approaches and we discuss the different results.
Whenever color information is available, it is advisable to restrict the analysis of the luminosity function to a suitable region carefully chosen to represent the RGB.
To perform this selection, we took into account the available calibrations of the absolute magnitude of the RGB tip. As discussed in Section \ref{calib}, one of the most reliable calibrations available to date is based on the absolute magnitude of the RGB tip measured on a large sample of stars of the globular cluster $\omega$ Centauri \citep{2001ApJ...556..635B}, at a metallicity of $\rm{[Fe/H]} \sim -1.7$. To be able to apply this calibration to our data, we decided to select our RGB stars using the ridge line of $\omega$ Centauri and selecting stars in a narrow ($\sim$0.1 mag) range on both sides of it. Section \ref{calib} will present a discussion of the implications of this choice.
The upper panels of Figures \ref{trgb_1.ps} - \ref{trgb_6.ps} show the detection of the RGB tip using the ML approach presented by \citet{2002AJ....124..213M}. The continuous line shows the observed RGB luminosity function, while the best fit is shown by a dashed line. The results of the detection are presented in columns 4 and 5 of Table \ref{tab1}. The lower panels of the same set of Figures show the detection of the RGB tip using the ED filter in a version similar to the one presented in \citet{1996ApJ...461..713S}. The continuous line shows the response of the ED filter, while the vertical line indicates the position of the center of the highest peak. The results of the measurements are reported in columns 2 and 3 of Table \ref{tab1}.
The discontinuity in the luminosity function due to the RGB tip is conspicuous in most cases, although a significant amount of contamination from AGB stars is affecting Fields 2 and 3, producing a rather smooth slope at the level of the RGB tip. The effect of an AGB contamination has been investigated in many studies, \citep[e.g., see][]{makarov:co,2004ApJ...606..869B}. The conclusion is that in most cases the RGB tip detection is quite insensitive to the effect of this contamination. This result is further confirmed by looking at the results presented here. Indeed, the RGB tip positions measured in Fields 2 and 3 do not significantly differ from the positions measured in any other field.
To estimate the errors connected with the detection of the RGB tip, we adopted a bootstrap resampling strategy similar to the one presented in \citet{2002AJ....124..213M}. The sample of stars chosen to represent the RGB was resampled 500 times, and the RGB tip measured for each realization. The r.m.s. of the results is then quoted in columns 3 and 5 of Table \ref{tab1}, for ED and ML methods, respectively.
\begin{table}
\begin{tabular}{c|cc|cc}
\tableline
\tableline
& \multicolumn{2}{c|}{Edge detector} & \multicolumn{2}{c}{Maximum likelihood} \\
Field & I$_{RGBT}$ & $\sigma$ & I$_{RGBT}$ & $\sigma$ \\
\tableline
1 & 22.48 & 0.09 & 22.50 & 0.03 \\
2 & 22.40 & 0.03 & 22.48 & 0.02\\
3 & 22.48 & 0.06 & 22.50 & 0.02\\
4 & 22.42 & 0.16 & 22.48 & 0.06\\
5 & 22.50 & 0.10 & 22.50 & 0.02\\
6 & 22.39 & 0.12 & 22.45 & 0.08\\
\tableline
\end{tabular}
\caption{Results of the measurements of the magnitude of the RGB tip.\label{tab1}}
\end{table}
\begin{figure}
\plotone{f3.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 1. Lower panel: Detection of the TRGB using ED method applied to Field 1.}
\label{trgb_1.ps}
\end{figure}
\begin{figure}
\plotone{f4.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 2. Lower panel: Detection of the TRGB using ED method applied to Field 2.}
\label{trgb_2.ps}
\end{figure}
\begin{figure}
\plotone{f5.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 3. Lower panel: Detection of the TRGB using ED method applied to Field 3.}
\label{trgb_3.ps}
\end{figure}
\begin{figure}
\plotone{f6.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 4. Lower panel: Detection of the TRGB using ED method applied to Field 4.}
\label{trgb_4.ps}
\end{figure}
\begin{figure}
\plotone{f7.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 5. Lower panel: Detection of the TRGB using ED method applied to Field 5.}
\label{trgb_5.ps}
\end{figure}
\begin{figure}
\plotone{f8.eps}
\caption{Upper panel: Detection of the TRGB using ML method applied to Field 6. Lower panel: Detection of the TRGB using ED method applied to Field 6.}
\label{trgb_6.ps}
\end{figure}
\subsection{Distance modulus}
\label{calib}
The first calibration of the absolute magnitude of the RGB tip dates back to the early 1990's.
\citet{1993ApJ...417..553L} defined the distance modulus based on the RGB tip as $$(m-M)_I=I_{TRGB}+BC_I-M_{bol,TRGB}$$ where $BC_I$ is the bolometric correction to the I magnitude, and $M_{bol,TRGB}$ is the bolometric magnitude of the TRGB. $BC_I$ and $M_{bol,TRGB}$ are given in \citet{1990AJ....100..162D} as $\rm{BC_I}=0.881-0.243(V-I)_0$
and $M_{bol}=-0.19\rm{[Fe/H]}-3.81$. These calibrations are based on the distance scale of \cite{1990ApJ...350..155L} where the magnitude of RR Lyrae stars is $M_V(RR)=0.82 + 0.17 \rm{[Fe/H]}$.
All these relations are based on a small sample of RGB stars observed in a few template globular clusters, and they only cover the range $-2.17 < \rm{[Fe/H]} < -0.71$.
An extensive set of computer simulations was performed by \citet{1995AJ....109.1645M} to test for possible systematic effects on the detection of the RGB tip. The authors found that a reasonable lower limit to the number of stars within 1 magnitude from the tip is 50. Below this level, strong biases can affect the determined magnitude of the tip. Note that in the sample of \citet{1990AJ....100..162D} the number of stars within 1 magnitude from the tip is never larger than 20, and can be as low as 2.
A significant improvement on this situation was presented by \citet{2001ApJ...556..635B}. In their work, the authors derive a new calibration of the magnitude of the tip in the form $M_I^{TRGB}=0.14\rm{[Fe/H]}^2 + 0.48\rm{[Fe/H]} -3.66$.
The result is based on an extensive sample of stars observed in different bands including the near-IR and presented in
\citet{1999AJ....118.1738F,2000AJ....119.1282F}. Although based on a larger sample of stars than the one presented in \citet{1993ApJ...417..553L}, this calibration still does not meet the completeness criteria established by \citet{1995AJ....109.1645M}. In addition, both this calibration and the one by \citet{1993ApJ...417..553L} require a knowledge of the metallicity of the underlying population, either measured independently or deduced from the color of the RGB, iterating through measurements of the distance and the metallicity.
The only calibration based on a sufficient number of stars is derived for $\omega$ Centauri by
\citet{2001ApJ...556..635B}. According to this calibration, the absolute magnitude of the RGB tip is
$M_I^{TRGB}=-4.04 \pm 0.12$ at a metallicity of ${\rm [Fe/H]} \sim -1.7$. This value is tied to the distance of the eclipsing binary OGLEGC 17 in $\omega$ Centauri \citep{2001AJ....121.3089T}, and it's completely independent from any other optical RR Lyrae distances. A possible source of uncertainty associated with this calibration is the wide and complex color/metallicity distribution observed in $\omega$ Centauri, but several studies have shown that the dominant population is rather metal-poor, and that the peak of the metallicity distribution is at ${\rm [Fe/H]} \sim -1.7$ \citep{2000ApJ...534L..83P,1996AJ....111.1913S}. In this work, we will adopt the value $M_I^{TRGB}=-4.04 \pm 0.12$. We note that this assumption is the reason behind our choice of the selection criteria we have adopted to define the RGB sample, as can be verified in Figure \ref{omegacen.ps}. The left panel of Figure \ref{omegacen.ps} shows the CMD of NGC 300, Field 2. Only 20 \% of the stars are plotted, for easier reading. The right panel shows the CMD of $\omega$ Centauri from \citet{2000A&AS..145..451R,2000A&AS..144....5R}. Horizontal and vertical lines show the position of the RGB tip as measured in NGC 300. It is evident that it is possible to define in NGC 300 a sample of RGB stars that perfectly overlaps with the RGB of $\omega$ Centauri.
Assuming $E(B-V)=0.096 \pm 0.008$ \citep{2005ApJ...628..695G}, we derived distance moduli both with the ED and ML methods, and for the 6 ACS Fields. The results are presented in Table \ref{tab3}.
To estimate the errors attached to these measurements, we separate the errors connected to the detection of the tip and the photometric calibration ({\em internal} error) and the errors due to the extinction correction and the calibration of the absolute magnitude of the tip ({\em external} error). The errors due to the detection of the tip have already been discussed earlier in this Section. The errors connected with the conversion from ACS photometric system and BVI system can be quantified in 0.02 mag
\citep{2005astro.ph..7614S}. The error attached to the $E(B-V)$ measurement provided by \citet{2005ApJ...628..695G} is 0.006 mag, which accounts for a total of 0.01 mag attached to $A_I$. Finally, the error in the absolute calibration is 0.12 mag \citep{2001ApJ...556..635B}, and it's basically determined by the uncertainty in the distance to $\omega$ Centauri \citep{2001AJ....121.3089T}. The total amount of {\em internal} errors attached to the different distance moduli computed for the six Fields are reported in columns 3 and 5 of Table \ref{tab3}, for ED and ML methods, respectively.
\begin{table}
\begin{tabular}{c|cc|cc}
\tableline
\tableline
& \multicolumn{2}{c|}{Edge detector} & \multicolumn{2}{c}{Maximum likelihood} \\
Field & $(m-M)_0$ & $\sigma$ & $(m-M)_0$ & $\sigma$ \\
\tableline
1 & 26.35 & 0.09 & 26.37 & 0.03 \\
2 & 26.26 & 0.04 & 26.35 & 0.03\\
3 & 26.35 & 0.06 & 26.37 & 0.03\\
4 & 26.28 & 0.16 & 26.35 & 0.06\\
5 & 26.37 & 0.10 & 26.37 & 0.03\\
6 & 26.26 & 0.13 & 26.32 & 0.08\\
\tableline
\end{tabular}
\caption{Results of the measurements of the distance modulus. \label{tab3}}
\end{table}
To derive our final distance moduli, we computed a weighted mean of the measurements in the six Fields. The results are:
$$(m-M)_0=26.30 \pm 0.03 \pm 0.12 (ED)$$
and
$$(m-M)_0=26.36 \pm 0.02 \pm 0.12 (ML).$$
\begin{figure}
\plotone{f9.eps}
\caption{ Left panel shows the CMD of NGC 300, right panel shows the CMD of $\omega$ Centauri. Vertical and horizontal lines indicate the color and the magnitude of the TRGB as measured in NGC 300.}
\label{omegacen.ps}
\end{figure}
\section{Discussion}
\label{discussion}
Our selection of the sample of the stars representing the RGB is entirely motivated by our choice of the absolute calibration of the RGB tip. This approach actually limits the analysis to about 20 \% of the total number of available RGB stars.
As an alternative approach, one could choose to adopt a much larger sample of RGB stars, reaching the high-metallicity edge of the RGB. We argue that this approach would provide consistent results, but with a lower precision. This is shown in Figure \ref{ferraro2.ps}. In this Figure we plot the CMD of NGC 300, Field 2, in the absolute plane, using the distance and the reddening provided by \citet{2005ApJ...628..695G}. The continuous line shows the color dependence of the RGB tip according to \citet{2001ApJ...556..635B}. It is evident that the slope of the function $M_I^{TRGB} vs. (V-I)_0$ reproduces very closely the observed data. On the other hand, using the high-metallicity part of the CMD would introduce additional errors due the still uncertain slope of the high-metallicity extension of the calibration.
\begin{figure}
\plotone{f10.eps}
\caption{CMD of NGC 300 in the absolute plane. The continuous line shows the color dependence of the TRGB according to \citet{2001ApJ...556..635B}}
\label{ferraro2.ps}
\end{figure}
Another issue that should be given attention to is the age of the underlying population used to define the RGB sample. Whenever the RGB tip technique is applied to a composite stellar population, the possibility of biases arises, due to the fact that the presence of a well-developed and populated RGB does not necessarily imply the presence of a globular cluster-like population, while the calibration of the absolute magnitude of the RGB tip relies completely on a sample of globular clusters.
\citet{2004ApJ...606..869B} reported that the RGB distances are rather insensitive to the stellar populations provided most of the stars are more metal poor than $\rm{[Fe/H]}=-0.3$ and that there is not a strong star formation burst between 1 and 2 Gyr. \citet{2005MNRAS.357..669S} extended this analysis to real cases, and showed that applying the standard technique for RGB tip distances to the LMC and to the SMC could result in significant deviations from the real value, due to the underestimation of the correct metallicity. We argue that the TRGB method can be safely applied to NGC 300, without introducing age- or metallicity-related biases. Indeed, \citet{2004AJ....127.1472B} have shown that the star formation history of this galaxy has been rather uniform throughout all its life, and they found no indication for an increased star formation rate at young ages, except for a possible final burst at 200-100 Myr. Besides, both \citet{2004AJ....127.1472B} and the results of the Araucaria project \citep{2002ApJ...567..277B, 2003ApJ...584L..73U} show that the metallicity of NGC 300 has probably been lower than ${\rm [Fe/H]}=-0.5$ for the whole life of the galaxy.
The result presented in this paper is fully consistent with the results recently derived by \citet{2005ApJ...628..695G}, based on the luminosity of Cepheids variable stars. Our distance modulus is also consistent with the one derived by \citet{2004AJ....127.1472B}, provided the difference in the adopted reddening correction is taken into account. Indeed, the observed magnitude of the RGB tip that we derived is consistent within the errors with the value $I_{TRGB}=22.52 \pm 0.02$ measured by \citet{2004AJ....127.1472B}, but the authors then apply a reddening correction $E(B-V)=0.013$ \citep{1998ApJ...500..525S}, which is much lower than the value adopted in this paper, resulting in a distance modulus $(m-M)_0=26.56 \pm 0.07 \pm 0.13$. Similar considerations apply to the results published by \citet{2005A&A...431..127T}, although in this case we do not know what is the adopted calibration of the absolute magnitude of the RGB tip, and the reddening correction applied.
On the other hand, the results presented here show a significant discrepancy with the measurements of
\citet{2004ApJ...608...42S}, who published a distance modulus $(m-M)_0=26.65 \pm 0.09$. The total difference between this value and our value is $\sim 0.3$ magnitudes. Half of this difference can be explained by the different assumption of the reddening, as in the case of the distance presented by
\citet{2004AJ....127.1472B}, but a further difference of $\sim 0.16$ remains to be explained. It appears that this difference can be accounted for by the difference in the estimated level of the RGB tip, measured at $I_{TRGB}=22.49 \pm 0.01$ in this paper, and at $I_{TRGB}=22.62 \pm 0.07$ by \citet{2004ApJ...608...42S}. It is difficult to provide an explanation for this difference, but a value of $I_{TRGB}=22.60$ is not compatible with our data. Besides, it is interesting to notice that the data analyzed by \citet{2004ApJ...608...42S} were also analyzed by \citet{2004AJ....127.1472B}, indicated as field F3. Both groups determined the RGB tip around 22.6, but they also warned the reader that the field analyzed was poorly populated, and that the determination could be uncertain. Indeed, \citet{2004AJ....127.1472B} rejected the result derived from this WFPC2 field as non reliable. \citet{2004AJ....127.1472B} also analyzed an additional field, indicated as field F1, and for that field they derived the already quoted value of $I_{TRGB}=22.52 \pm 0.02$, in agreement with our determination. Our conclusion is that WFPC2 and ACS measurements agree within the errors when sufficient number of stars are used, as is the case of field F3 of \citet{2004AJ....127.1472B}.
Finally, \citet{2004ApJ...608...42S} also reported that the Cepheids distance to NGC300, based on the measurements of \citet{1992ApJ...396...80F}, is $(m-M)_0=26.63 \pm 0.06$, but using the calibration of
\citet{1999AcA....49..201U} the distance would be $(m-M)_0=26.53 \pm 0.05$, which would be in agreement with our determination if our value for the reddening would be used.
\section{Conclusions}
\label{conclusions}
We have presented a new measurement of the distance to NGC 300 based on the deepest available photometry catalog, obtained with the Advanced Camera for Survey on board the Hubble Space Telescope. We have used both edge-detection and maximum likelihood methods, and we have applied the methods independently to six different ACS Fields. All the Fields give consistent results. We have also discussed the possibility of biases in our results related to the application of the TRGB method to a composite stellar population, and we have concluded that NGC 300 is likely to be a case in which this distance estimator can be safely applied. Our result is fully consistent with the recent distance determination from near-infrared photometry of Cepheids variables \citep{2005ApJ...628..695G}. Since their result is tied to an assumed LMC distance modulus of 18.50, our independent TRGB distance determination of NGC 300 supports a distance of LMC of, or very close to, 18.50. The distance modulus we derive is also consistent with other recent determinations based on the TRGB \citep{2005A&A...431..127T,2004AJ....127.1472B} if our reddening value is used in these studies; however, our present determination has succeeded in reducing the internal errors of the result by a factor $\sim 3$.
\acknowledgements{WP and GP gratefully acknowledge support for this work from the Chilean FONDAP Center for Astrophysics 15010003 and the Polish KBN grant No 2P03D02123. Support for program \# GO-9492 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We would like to thank the referee for useful suggestions and comments that helped improve this paper.}
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Ellie Dehn (born Elizabeth Dehn, 1980) is an American soprano.
Biography
The daughter of Catherine Dehn and Douglas Dehn, Dehn was born and raised in Anoka, Minnesota. She is a 1998 graduate of Anoka High School, where she played flute and sang in the school choir. She received voice training from Judy Bender and sang in her high school's choir. She attended a vocal summer camp at Oberlin Conservatory of Music, where she later went to college. For graduate school, she attended the Academy of Vocal Arts in Philadelphia.
After seeing Dehn in a singing competition, Eve Queler of the Opera Orchestra of New York cast her as an understudy in Il Corsaro in 2005. Since then, Dehn has performed as Elvira in Don Giovanni at the Metropolitan Opera and the Countess in Le Nozze di Figaro at the Royal Opera House. She has been a frequent performer at San Francisco Opera, where she has performed as Musetta in La Boheme, Manon in Manon, Fiordiligi in Così fan tutte, and Arabella in Arabella. She has portrayed Donna Anna in Don Giovanni for San Francisco Opera, Opera Colorado, Michigan Opera Theatre, and Opera Memphis. Dehn has also performed on fellow Anoka native Garrison Keillor's radio show, A Prairie Home Companion.
Dehn has a daughter, Arabella, born in 2018.
References
External links
Official website of Ellie Dehn
IMG Artists agency page on Ellie Dehn
American operatic sopranos
21st-century American women opera singers
Living people
People from Anoka, Minnesota
Academy of Vocal Arts alumni
Oberlin College alumni
1980 births
Anoka High School alumni
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\section{Intoduction}
Let us consider a Cauchy problem of a dispersive equation in $\mathbb R^{n+1}$
\begin{equation} \label{gen_form}
\left\{
\begin{aligned}
i \partial_t u + \Phi(D)u &= 0,\\
u(0) &= f,
\end{aligned}
\right.
\end{equation}
where $\Phi(D)$ is the corresponding Fourier multiplier to the function $\Phi$. We assume that $\Phi \in C^\infty(\mathbb R^n \setminus \{0\})$ is a real-valued function satisfying the following conditions:
\begin{condition} \label{homog_cond}
{$ $}
\begin{itemize}
\item
$|\nabla \Phi(\xi)| \neq 0$ for all $\xi \neq 0$.
\item
There is a constant $\mu \ge 1$ such that $\mu^{-1} \le |\Phi(\xi)| \le \mu$ for any $\xi$ with $|\xi|=1$.
\item
There is a constant $m \ge 1$ such that $\Phi(\lambda\xi)=\lambda^m\Phi(\xi)$ for all $\lambda>0$ and all $\xi \neq 0$.
\item
The Hessian $H_{\Phi}(\xi)$ of $\Phi$ has rank at least $1$ for all $\xi \neq 0$.
\end{itemize}
\end{condition}
The solution $u$ to \eqref{gen_form} becomes the Schr\"odinger operator $e^{-it\Delta}f$ if $\Phi(\xi)=|\xi|^2$ and the wave operator $e^{it\sqrt{-\Delta}}f$ if $\Phi(\xi)=|\xi|$. When $\Phi(\xi)=|\xi|^m$ for $m>1$, the solution is called the fractional Schr\"odinger operator $e^{it(\sqrt{-\Delta})^{m/2}}f$.
Let $e^{it\Phi(D)}f$ denote the solution to \eqref{gen_form}.
Our interest is to find suitable pairs $(q,r)$ which satisfy the global space-time estimate
\begin{equation}\label{main_goal}
\|e^{it\Phi(D)} f\|_{L_x^q(\mathbb R^n; L_t^r(\mathbb R))}
\le C \|f\|_{\dot H^s(\mathbb R^n)},
\end{equation}
where $\dot H^s(\mathbb R^n)$ denotes the homogeneous $L^2$ Sobolev space of order $s$.
By scaling invariance the regularity $s=s(r,q)$ should be defined as
\begin{equation} \label{regularity}
s = n(\frac{1}{2} - \frac{1}{q}) -\frac{m}{r}.
\end{equation}
This problem for $\mu =1$ has been studied by many researchers. For the Schr\"odinger operator,
Planchon \cite{Pl} conjectured that the estimate \eqref{main_goal} is valid if and only if $r \ge 2$ and $\frac{n+1}{q} + \frac{1}{r} \le \frac{n}{2}$.
Kenig--Ponce--Vega \cite{KPV} showed the conjecture is true for $n=1$. In higher dimensions $n \ge 2$ it was proven by Vega \cite{V1} that \eqref{main_goal} holds for $q \ge \frac{2(n+2)}{n}$ and $\frac{n+1}{q} + \frac{1}{r} \le \frac{n}{2}$. When $n=2$ Rogers \cite{R} showed it for $2\le r < \infty$, $q > \frac{16}5$ and $\frac{3}{q} + \frac{1}{r} < 1$, and later the excluded endline $\frac{3}{q} + \frac{1}{r} = 1$ was obtained by Lee--Rogers--Vargas \cite{LRV}. When $n \ge 3$, Lee--Rogers--Vargas \cite{LRV} improved the previous known result to $r \ge 2$, $q > \frac{2(n+3)}{n+1}$ and $\frac{n+1}{q} + \frac{1}{r} = \frac{n}{2}$.
Recently it is shown by Du--Kim--Wang--Zhang \cite{DKWZ} that the estimate \eqref{main_goal} with $r=\infty$, that is, the maximal estimate fails for $n \ge 3$.
For a case of the wave operator it is known that \eqref{main_goal} holds for $(r,q)$ pairs such $2 \le r \le q$, $q \neq \infty$ and $\frac{1}{r} + \frac{n-1}{2q} \le \frac{n-1}{4}$ (see \cites{GV, KT, Pe, S}). Particularly, when $r=\infty$, Rogers--Villarroya \cite{RV} showed that \eqref{main_goal} with regularity $s> n(\frac{1}{2} - \frac{1}{q})-\frac{1}{r}$ is valid for $q \ge \frac{2(n+1)}{n-1}$.
For the fractional Schr\"odinger operator the known range of $(r,q)$ for which the estimates hold is that $2 \le r \le q$, $q \neq \infty$ and $\frac{n}{2q} + \frac{1}{r} \le \frac{n}{4}$ (see \cites{C,T2,Pa,CHKL,CO}).
The case of $\mu > 1$ has an interesting in its own right. The solution $u$ is formally written as
\begin{equation*}
u(t,x) = e^{it\Phi(D)}f(x) := \frac{1}{(2\pi)^n}\int_{\mathbb R^n} e^{i(x \cdot \xi + t \Phi(\xi))} \hat f(\xi) d\xi.
\end{equation*}
From this form we see that the space-time Fourier transform of $u$ is supported in the surface $S=\{(\xi, \Phi(\xi))\}$. It is known that the operator $u$ is related to the curvature of $S$ such as the sign of Gaussian curvature and the number of nonvanishing principle curvature. The Schr\"odinger operator corresponds to a paraboloid which has a positive Gaussian curvature, and the wave operator corresponds to a cone whose Gaussian curvature is zero. We are also interested in operators corresponding to a surface with negative Gaussian curvature. When $\mu >1$ there is a surface with negative Gaussian curvature. For instance, the surface $\{(\xi_1,\xi_2,\xi_1^4+2\xi_1^3\xi_2 - 2\xi_1\xi_2^3 +\xi_2^4) \}$ has negative Gaussian curvature on a neighborhood of the point $(1,0,1)$.
In this paper we will establish a local-to-global approach as follows.
\begin{thm}\label{main_thm2}
Let $\mathbb I = (0,1)$ be a unit interval and $\mathbb B = B(0,1)$ a unit ball in $\mathbb R^n$.
Let $q_0, r_0 \in [2,\infty)$, $s(r,q)$ defined as \eqref{regularity} and $\Phi$ satisfy Condition \ref{homog_cond}.
Suppose that the local estimate
\begin{equation}\label{thm_assum}
\|e^{it\Phi(D)} f \|_{L_x^{q_0}(\mathbb{B};L_t^{r_0}(\mathbb{I}))}
\leq C_{\epsilon} \|f\|_{H^{s(r_0,q_0)+\epsilon}(\mathbb{R}^n)}
\end{equation}
holds for all $\epsilon>0$.
Then for any $q> q_0$ and $r> r_0$, the global estimate
\begin{equation}\label{thm_goal}
\|e^{it\Phi(D)} f \|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
\le C \|f\|_{\dot H^{s(r,q)}(\mathbb R^n)}
\end{equation}
holds, where $H^s(\mathbb R^n)$ denotes the inhomogeneous $L^2$-Sobolev space of order $s$ and $\dot H^s(\mathbb R^n)$ denotes homogeneous one.
\end{thm}
The maximal estimate, which is \eqref{thm_assum} with $r_0=\infty$, is related to pointwise convergence problems.
When $n=2$ it was proven that the maximal estimates with $m>1$ and $\mu=1$ are valid for $q_0=3$ and $s>\frac{1}{3}$ (see \cites{CK, DGL}). By interpolating with a Strichartz estimate
\begin{equation*}
\|e^{it\Phi(D)} f \|_{L_x^6(\mathbb{R}^2;L_t^2(\mathbb{R}))}
\le \|e^{it\Phi(D)} f \|_{L_t^2(\mathbb{R};L_x^6(\mathbb{R}^2))}
\leq C\|f\|_{\dot{H}^{(2-m)/4}(\mathbb{R}^2)},
\end{equation*}
we have \eqref{thm_goal} for the line $\frac{3}{q} + \frac{1}{r} = 1$ with $r \ge 2$.
By Theorem \ref{main_thm2}, we can obtain the following global space-time estimates which is the Planchon conjecture for $n=2$ except the endline.
\begin{cor} \label{main_thm}
Let $m>1$ and $\mu=1$. For $2 \leq r < \infty$ and $\frac{3}{q} + \frac{1}{r} < 1$, the global estimate
\begin{equation*}
\|e^{it\Phi(D)} f \|_{L_x^q(\mathbb R^2;L_t^r(\mathbb R))}
\le C \|f\|_{\dot H^{1 - \frac{2}{q} -\frac{m}{r}}(\mathbb R^2)}.
\end{equation*}
\end{cor}
\textit{Notation.} Throughout this paper let $C>0$ denote various constants that vary from line to line, which possibly depend on $n$, $q$, $r$, $m$ and $\mu$. We use $A \lesssim B$ to denote $A \le CB$, and if $A \lesssim B$ and $B \lesssim A$ we denote by $A \sim B$.
\section{Proof of Theorem \ref{main_thm2}}
In this section we prove Theorem \ref{main_thm2} by using two propositions. In subsection 2.1 we consider a Littlewood--Paley type inequality by which the initial data $f$ can be assumed to be Fourier supported in $\{1/2 \le |\xi| \le 2\}$. In subsection 2.2 we prove a mixed norm version of Tao's $\varepsilon$-removable lemma by which the global estimates with a compact Fourier support are reduced to a local ones. In subsection 2.3 we show the two propositions imply Theorem \ref{main_thm2}.
\subsection{A Littlewood-Paley type inequality}
We discuss a Littlewood-Paley type inequality for the operator $e^{it\Phi(D)}$ in a time variable.
Let a cut-off function $\phi \in C_0^{\infty} \big( [\frac{1}{2}, 2] \big)$ satisfy
$ \sum_{k\in\mathbb{Z}} \phi( {2^{-k}} x) =1 $.
We define Littlewood-Paley projection operators $P_{k}$ and $\widetilde{P_k}$ by
\[
\widehat{P_k f}(\xi) = \phi(2^{-k}|\xi|) \hat{f}(\xi)
\quad\mbox{and}\quad
\widehat{\widetilde{P_k} f}(\tau) = \phi (2^{-mk}|\tau| ) \hat{f}(\tau)
\] for $\xi \in \mathbb R^n$ and $\tau \in \mathbb R$, respectively.
\begin{lem}\label{LP_lemma}
Suppose that $\Phi$ satisfies Condition \ref{homog_cond}. Then for $1<r<\infty$,
\begin{equation*
\big\|e^{it\Phi(D)} f(x) \big\|_{L_t^r(\mathbb R)}
\leq C_{m,\mu} \Big\| \Big(\sum_{\substack{j,k \in \mathbb Z: \\|k-j| \leq \frac{\log_2 \mu}{m}+2}} | \widetilde{P_j} e^{it\Phi(D)}P_kf(x) |^2\Big)^{1/2} \Big\|_{L_t^r(\mathbb R)}
\end{equation*}
for all functions $f$ and all $x \in \mathbb R^n$.
\end{lem}
\begin{proof}
For simplicity,
\[
F (t) := e^{it\Phi(D)} f(x) \quad\mbox{and}\quad F_k (t) := e^{it\Phi(D)}P_kf(x).
\]
Since the projection operators are linear, we have an identity
\[
F(t) = \sum_{j\in \mathbb{Z}} \widetilde{P_j}F(t) = \sum_{j\in \mathbb{Z}} \sum_{k\in \mathbb{Z}} \widetilde{P_j}F_k(t).
\]
For any test function $\psi \in C_0^{\infty} \big( [-2, 2] \big)$ with $\psi = 1$ in $[-1,1]$,
the Fourier transform $\widehat{f}$ of $f$ is defined by
\[
\widehat{f}(\tau) = \lim_{R\rightarrow\infty} \frac{1}{2\pi} \int_{\mathbb{R}} e^{it\tau} \psi\Big(\frac{t}{R}\Big) f(t) dt
\]
in the distributional sense.
We claim that $\widetilde{P_j}F_k(t) =0$
if
\begin{equation}\label{freq_cond_null}
|k -j| > \frac{\log_2 \mu}{m}+2.
\end{equation}
Indeed, using the above definition of the Fourier transform we can write
\[
\begin{aligned}
\widehat{\widetilde{P_j}F_k}(\tau)
&= \frac{1}{(2\pi)^{n+1}} {\phi}\Big(\frac{|\tau|}{2^{mj}} \Big) \lim_{R\rightarrow\infty} \int_{\mathbb{R}^n} e^{ix\cdot\xi} \bigg( \int_{\mathbb{R}} e^{it\tau} e^{it\Phi(\xi)} \psi\Big(\frac{t}{R}\Big) dt \bigg) \phi\Big(\frac{|\xi|}{2^k}\Big) \hat{f}(\xi) d\xi .
\end{aligned}
\]
In the right side of the above equation, we see that the range of $(\tau, \xi)$ is contained in
\[
2^{m(j-1)}\leq |\tau|\leq 2^{m(j+1)} \quad\mbox{and}\quad 2^{(k-1)}\leq |\xi| \leq 2^{(k+1)}.
\]
From Condition \ref{homog_cond}
we have a bound
\[
{\mu^{-1}} 2^{m(k-1)}\leq |\Phi(\xi)| \leq \mu 2^{m(k+1)}.
\]
Then it follows that for $k$ and $j$ satisfying \eqref{freq_cond_null},
\[
|\tau + \Phi(\xi)| >0.
\]
By the integration by parts it implies that there exists a constant $C_0>0$ such that
\[
\Big| \int_{\mathbb{R}} e^{it\tau} e^{it\Phi(\xi)} \psi\Big(\frac{t}{R}\Big) dt \Big| \le \frac{1}{C_0 R}.
\]
From this estimate and the Lebesgue dominated convergence theorem we obtain
$\widehat{\widetilde{P_j}F_k}=0,$
which implies the claim.
By the claim, the Littlewood-Paley theory and the Cauchy-Schwarz inequality,
\begin{equation*}
\begin{aligned}
\big\|e^{it\Phi(D)} f(x) \big\|_{L_t^r(\mathbb R)}
&= \Big\| \sum_{j \in \mathbb{Z}} \widetilde{P_j} \Big( \sum_{k \in \mathbb Z} F_k(\cdot,x) \Big) \Big\|_{L_t^r(\mathbb R)} \\
&\leq C \Big\| \Big(\sum_{j \in \mathbb{Z}} \Big| \sum_{k \in \mathbb Z: |k -j| \le \frac{\log_2 \mu}{m}+2} \widetilde{P_j} F_k(\cdot,x) \Big|^2\Big)^{1/2} \Big\|_{L_t^r(\mathbb R)} \\
&\leq C_{m,\mu} \Big\| \Big( \sum_{j \in \mathbb{Z}} \sum_{k \in \mathbb Z: |k -j| \le \frac{\log_2 \mu}{m}+2} | \widetilde{P_j} F_k(\cdot,x)|^2\Big)^{1/2} \Big\|_{L_t^r(\mathbb R)}.
\end{aligned}
\end{equation*}
This is the desired inequality.
\end{proof}
Using the above lemma we can have the following proposition.
\begin{prop}\label{LP_prop}
Let $2 \leq q,r < \infty$. Suppose that $\Phi$ satisfies Condition \ref{homog_cond}.
If the estimate
\begin{equation}\label{LP_prop_supp}
\|e^{it\Phi(D)} f \|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
\leq C \| f\|_{L^2(\mathbb R^n)}
\end{equation}
holds for all $f$ with $\supp \hat f \subset \{1/2 \le |\xi| \le 2 \}$, then the estimate
$$
\|e^{it\Phi(D)} f \|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
\leq C_{m,\mu} \| f\|_{\dot{H}^{\frac{n}{2}-\frac{n}{q}-\frac{m}{r}}(\mathbb R^n)}
$$
holds for all $f$.
\end{prop}
\begin{proof}
The Minkowski inequality and Lemma \ref{LP_lemma} allow that
\[
\big\|e^{it\Phi(D)} f \big\|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
\leq C_{m,\mu} \bigg\|\bigg( \sum_{|k -j| \le \frac{\log_2 \mu}{m}+2} \Big\| \widetilde{P_j} \big( e^{it\Phi(D)} P_kf \big) \Big\|_{L_t^r(\mathbb R)}^2 \bigg)^{1/2} \bigg\|_{L^q_x(\mathbb R^n)}.
\]
Since $\widetilde{P_j}$ is bounded in $L^p$, it is bounded by
\[
C_{m,\mu} \bigg\|\bigg( \sum_{k \in \mathbb Z} \big\| e^{it\Phi(D)} P_kf \big\|_{L_t^r(\mathbb R)}^2 \bigg)^{1/2} \bigg\|_{L^q_x(\mathbb R^n)}.
\]
By the Minkowski inequality we thus have
\[
\big\|e^{it\Phi(D)} f \big\|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
\leq C_{m,\mu} \bigg( \sum_{k\in \mathbb{Z}} \big\| e^{it\Phi(D)} P_kf \big\|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}^2 \bigg)^{1/2}.
\]
Apply \eqref{LP_prop_supp} to the right side of the above estimate after parabolic rescaling. Then we obtain
\begin{align*}
\|e^{it\Phi(D)} f \|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}
&\leq C_{m,\mu} \bigg(\sum_{k\in\mathbb{Z}} 2^{2k(\frac{n}{2}-\frac{n}{q}-\frac{m}{r} )}\| P_kf \|_2^2 \bigg)^{1/2} \\
&= C_{m,\mu} \| f\|_{\dot{H}^{\frac{n}{2}-\frac{n}{q}-\frac{m}{r}}(\mathbb R^n)}.
\end{align*}
\end{proof}
\subsection{Local-to-global arguments}
We will show that the global estimate
\eqref{LP_prop_supp} is obtained from its local estimate. Adopting the arguments in \cite{T1}, we consider the dual estimate of \eqref{LP_prop_supp}.
Let $S=\{(\xi, \Phi(\xi)) \in \mathbb R^n \times \mathbb R: 1/2 \le |x| \le 2 \}$ be a compact hypersurface with the induced (singular) Lebesgue measure $d\sigma$.
We define the Fourier restriction operator $\mathfrak R$ for a compact surface $S$ by the restriction of $\hat f$ to $S$, i.e.,
\[
\mathfrak Rf = \hat f \big|_S.
\]
Its adjoint operator $\mathfrak R^*f = \widehat{fd\sigma}$ can be viewed as $e^{it\Phi(D)}\hat{g}$, where the Fourier transform $\hat g(\xi)$ of $g$ corresponds to $f(\xi, \Phi(\xi))$.
Let $\rho>0$ be the decay of $\widehat{d\sigma}$, i.e.,
\begin{equation} \label{surdecay}
|\widehat{d\sigma}(x)| \lesssim (1+|x|)^{-\rho}, \qquad x \in \mathbb R^{n+1}.
\end{equation}
It is known that $\rho$ is determined by the number of nonzero principal curvatures of the surface $S$, which is equal to the rank of the Hessian $H_{\Phi}$. Specifically, if $H_{\Phi}$ has rank at least $k$ then
\[
\rho = k/2,
\]
see \cite{St}*{subsection 5.8, VIII}. From Condition \ref{homog_cond} we have $k \ge 1$.
When a function $f$ has a compact Fourier support, the $\widehat{fd\sigma}$ decays away from the support of $\hat f$ because of the decay of $\widehat{d\sigma}$. Thus if $f$ and $g$ are compactly Fourier supported and their supports are far away from each other then the interaction between $\widehat{fd\sigma}$ and $\widehat{gd\sigma}$ is negligible.
\begin{defn}
A finite collection $\{Q(z_i,R)\}_{i=1}^{N}$ of balls in $\mathbb{R}^{n+1}$ with radius $R>0$ is called $(N,R)$-\textit{sparse} if the centers $\{z_i\}$ are $(NR)^\gamma$-separated where $\gamma := n/\rho~ (\ge 2)$.
\end{defn}
From the definition of $(N,R)$-sparse we have a kind of orthogonality as follows. Let $\phi$ be a radial Schwartz function which is positive on the ball $B(0,3/2)$ and $\phi = 1$ on the unit ball $B(0,1)$ and whose Fourier transform is supported on the ball $B(0,2/3)$.
\begin{lem}[\cite{T1}*{in the proof of Lemma 3.2}] \label{lem:spase_decp}
Let $\{Q(z_i,R)\}_{i=1}^{N}$ be a $(N,R)$-sparse collection
and $\phi_i(z)=\phi(R^{-1}(z-z_i))$ for $i=1,\cdots, N$. Then there is a constnat $C$ independent of $N$ such that
\begin{equation} \label{eqn:dep}
\Big\| \sum_{i=1}^{N} f_i \ast \hat \phi_i \big|_S \Big\|_2 \le CR^{1/2} \Big( \sum_{i=1}^{N} \|f_i\|_2^2 \Big)^{1/2}
\end{equation}
for all $f_i \in L^2(\mathbb R^{n+1})$.
\end{lem}
A proof of the above lemma is given in Appendix.
Let $\mathbb I_R=(0,R)$ denote an $R$-interval and $\mathbb B_R$ the ball of radius $R$ centered at the origin in $\mathbb R^n$. Using Lemma \ref{lem:spase_decp} we have an intermediate result.
\begin{prop} \label{lem:sparseEst}
Let $R>0$ and $1 < q,r \leq 2$.
Suppose that there is a constant $A(R)$ such that
\begin{equation}\label{eqn:loc_rest}
\|\mathfrak R(\chi_{\mathbb{I}_R \times \mathbb B_R} f) \|_{L^2(d\sigma)}
\leq A(R) \|f\|_{L_{x}^{q}(\mathbb R^n;L_{t}^{ r}(\mathbb R))}
\end{equation}
for all $f \in L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))$.
Then for any $(N,R)$-sparse collection $\{Q(z_i,R)\}_{i=1}^{N}$ there is a constant $C$ independent of $N$ such that
\begin{equation}\label{eqn:loc_rest_result}
\|\mathfrak Rf\|_{L^2(d\sigma)}
\leq C A(R) \|f\|_{L_{x}^{q}(\mathbb R^n;L_{t}^{ r}(\mathbb R))}
\end{equation}
for all $f$ supported in $\cup_{i=1}^{N} Q(z_i,R)$.
\end{prop}
\begin{proof}
Let $f_i = f \chi_{Q(z_i,R)}$. Then,
\[\mathfrak R f_i = \hat f_i \big|_S =\widehat{ f_i \phi_i}\big|_S = (\hat f_i \ast \hat \phi_i) \big|_S,\] where $\phi_i(z)$ is defined as in Lemma \ref{lem:spase_decp}. Since $\hat \phi_i$ is supported on the ball $B(0,\frac{2}{3R})$, we may restrict the support of $\hat f_i$ to a $O(1/R)$-neighborhood of the surface $S$ and write
\[
\mathfrak R f_i = (\hat f_i \big|_{\mathcal N_{1/R} (S)} \ast \hat \phi_i) \big|_S
\]
where $\mathcal N_{1/R} (S)$ is a $O(1/R)$-neighborhood of the surface $S$. Let $\tilde{\mathfrak R}$ be another restriction operator defined by $\tilde{\mathfrak R} f = \hat f \big|_{\mathcal N_{1/R}(S)}$. If $f$ is suppported in $\cup_{i=1}^{N} Q(z_i,R)$, we write
\[
\mathfrak R f = \sum_{i=1}^{N} (\tilde{\mathfrak R}f_i \ast \hat \phi_i) \big|_{S}.
\]
By Lemma \ref{lem:spase_decp},
\[
\|\mathfrak R f \|_{L^2(d\sigma)} \le C R^{1/2} \Big( \sum_{i=1}^{N} \| \tilde{\mathfrak R} f_i \|_{L^2(\mathcal N_{1/R} (S))}^2 \Big)^{1/2}.
\]
Since the estimate \eqref{eqn:loc_rest} is translation invariant, by a slice argument we have
\[
\| \tilde{\mathfrak R} f_i \|_{L^2(\mathcal N_{1/R} (S))}
\le CR^{-1/2} A(R) \|f_i\|_{L_x^{q}(\mathbb R^n;L_t^r(\mathbb R))}.
\]
By combining the previous two estimates,
\[
\|\mathfrak R f \|_{L^2(d\sigma)} \le C A(R) \Big( \sum_{i=1}^{N} \|f_i \|^2_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))} \Big)^{1/2}.
\]
If $1 \le r \le q \le 2$ then by $\ell^r \subset \ell^{q} \subset \ell^{2}$,
\begin{align*}
\Big( \sum_{i=1}^{N} \|f_i \|^2_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))} \Big)^{1/2}
&\le \Big( \sum_{i=1}^{N} \|f_i \|^q_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))} \Big)^{1/q} \\
&=\Big(\int_{\mathbb R^n}\sum_{i=1}^{N}\|f_i\|_{L_t^{r}(\mathbb R)}^{q} dx\Big)^{1/q} \\
&\le \Big(\int_{\mathbb R^n} \Big(\sum_{i=1}^{N}\|f_i\|_{L_t^{r}(\mathbb R)}^{r} \Big)^{q/r} dx \Big)^{1/q} \\
&=\|f\|_{L_{x}^{q}(\mathbb R^n;L_{t}^{ r}(\mathbb R))}.
\end{align*}
If $1 \le q \le r \le 2$ one can use the embedding $\ell^r \subset \ell^2$ and the Minkowski inequality to get
\begin{align*}
\Big( \sum_{i=1}^{N} \|f_i \|^2_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))} \Big)^{1/2}
&\le \Big( \sum_{i=1}^{N} \|f_i \|^r_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))} \Big)^{1/r} \\
&\le \Big(\int_{\mathbb R^n} \Big(\sum_{i=1}^{N}\|f_i\|_{L_t^{r}(\mathbb R)}^{r} \Big)^{q/r} dx\Big)^{1/q} \\
&=\|f\|_{L_{x}^{q}(\mathbb R^n;L_{t}^{ r}(\mathbb R))}.
\end{align*}
Therefore we have \eqref{eqn:loc_rest_result}.
\end{proof}
We now extend the $(N,R)$-sparse sets to the whole space. For this we need the following decomposition lemma.
\begin{lem}[\cite{T1}]\label{lem:DecE}
Let $E$ be a subset in $\mathbb R^n$ with $|E| >1$. Suppose that $E$ is a finite union of finitely overlapping cubes of side-length $c \sim 1$.
Then for each $K \in \mathbb{N}$, there are subsets $E_1, E_2, \cdots, E_K$ of $E$ with
\[
E = \bigcup_{k=1}^{K} E_k
\]
such that each $E_k$ has $O(|E|^{1/K})$ number of $(O(|E|), |E|^{O(\gamma^{k-1})})$-sparse collections
\[
\mathbf S_1, \mathbf S_2, \cdots, \mathbf S_{O(|E|^{1/K})}
\] of which the union $\mathbf S_1 \cup \mathbf S_2 \cup \cdots \cup \mathbf S_{O(|E|^{1/K})}$ is a covering of $E_k$.
\end{lem}
This lemma is a precise version of Lemma 3.3 in \cite{T1}. A detailed proof can be found in Appendix.
Using the above lemma we have the following proposition.
\begin{prop}\label{prop:glob_d}
Let $1 < q_0,r_0 < \infty$.
Suppose that for any $\epsilon>0$ and any $(N,R)$-sparse collection $\{Q(z_i,R)\}_{i=1}^{N}$ in $\mathbb R^{n+1}$,
the estimate
\begin{equation} \label{eqn:sparse_rest}
\| \mathfrak R f \|_ {L^{2}(d\sigma)}\leq C_{\epsilon} R^{\epsilon} \|f\|_{L_x^{q_0}(\mathbb R^n; L_t^{r_0}(\mathbb R) )}
\end{equation}
holds for all $f$ supported in $\cup_{i=1}^{N} Q(z_i,R)$.
Then for any $1\leq q < q_0$ and $1\leq r < r_0$, the estimate
\begin{equation*}
\| \mathfrak R f \|_ {L^{2}(d\sigma)}\le C\|f\|_{L_x^{q}(\mathbb R^n; L_t^{r}(\mathbb R) )}
\end{equation*}
holds for all $f \in L_x^{q}(\mathbb R^n; L_t^{r}(\mathbb R) )$.
\end{prop}
\begin{proof}
By interpolation (see \cite{F}),
it suffices to show that for $1 \leq q < q_0$ and $1 \leq r < r_0$,
the restricted type estimate
\begin{equation*}
\|\mathfrak R \chi_E \|_{L^2(d\sigma)}
\leq C \|\chi_E\|_{L^{q}(\mathbb R^n;L^{r}(\mathbb R))}
\end{equation*}
for all subset $E$ in $\mathbb R^{n+1}$. We may assume $|E| > 1$, otherwise the estimate is trivial.
Since the set $S$ is compact,
$\chi_E$ can be replaced with $\chi_E \ast \varphi$, where $\varphi$ is a bump function supported on a cube of sidelength $c \sim 1$ such that $\hat \varphi$ is positive on $S$. Thus, we may further assume that $E$ is the union of $c$-cubes.
We denote by $\mathrm{proj} (E)$ the projection of $E$ onto the $x$-plane. For each grid point $x \in c\, \mathbb Z^n \cap \mathrm{proj} (E)$, we define $E_x$ to be the union of $c$-cubes in $E$ that intersect $\mathbb R \times \{x\}$. Let $E^j$ be the union of $E_x$ which satisfies
\[2^{j-1} < \text{ the number of}\,\, c\,\text{- cubes contained in}\,\, E_x \le 2^{j+1} \]
for $j \in \mathbb N$, (see Figure \ref{fig1}).
Then,
$$
E= \bigcup_{j \ge 1} E^j.
$$
\begin{figure}
\begin{center}
{\includegraphics[width=0.9\textwidth]{fig1.eps}}
\end{center}
\caption{The sets $E$, proj$E$, $E_x$ and $E^j$ in the proof of Proposition \ref{prop:glob_d}.}
\label{fig1}
\end{figure}
By using Lemma \ref{lem:DecE} with
\[
K := \frac{\log(1/\epsilon)}{2\log \gamma} + 1,
\]
the $E^j$ is decomposed into $E^j_k$'s which are covered by $O(|E^j|^{1/K})$ number of $(O(|E^j|),|E^j|^{C\gamma^{k-1}}))$-sparse collections. We apply \eqref{eqn:sparse_rest} to these sparse collections and obtain
$$
\|\mathfrak R \chi_{E^j_k} \|_{L^2(d\sigma)}
\leq C_{\epsilon} |E^j|^{1/K} (|E^j|^{C\gamma^{k-1}})^{\epsilon} \|\chi_{E^j_k} \|_{L_x^{q_0}(\mathbb R^n ; L_t^{r_0}(\mathbb R))}.
$$
Summing over k, we have
\begin{align*}
\|\mathfrak R \chi_{E^j} \|_{L^2(d\sigma)}
&\leq \sum_{k=1}^K \|\mathfrak R \chi_{E^j_k} \|_{L^2(d\sigma)} \\
&\leq C_{\epsilon} |E^j|^{1/K} (|E^j|^{C\gamma^{K-1}})^{\epsilon} \|\chi_{E^j} \|_{L_x^{q_0}(\mathbb R^n ; L_t^{r_0}(\mathbb R))}
\end{align*}
where $K$ is absorbed into $C_\epsilon$.
Since $|E^j| \leq 2^{j+1} |\mathrm{proj\,} (E^j)|$, we have
\[
\|\mathfrak R \chi_{E^j} \|_{L^2(d\sigma)}
\le C_{\epsilon} 2^{j(\frac{1}{r_0} + \delta(\epsilon) )} |\mathrm{proj} (E^j)|^{\frac{1}{q_0} + \delta(\epsilon)},
\]
where
\[
\delta(\epsilon) :=\frac{1}{K} +C\gamma^{K-1}\epsilon.
\]
Since $\lim_{\epsilon \to 0}\delta(\epsilon) = 0$,
we can take $\epsilon >0$ such that
\[
0< \delta(\epsilon)+\epsilon \le \min \bigg(\frac{1}{q}-\frac{1}{q_0}, \frac{1}{r} - \frac{1}{r_0} \bigg).
\]
Thus,
\begin{align*}
\|\mathfrak R \chi_E \|_{L^2(d\sigma)}
&\leq \sum_{j\geq1} \|\mathfrak R \chi_{E^j} \|_{L^2(d\sigma)} \\
&\leq C_{\epsilon} \sum_{j\geq1} 2^{j(\frac{1}{r_0} + \delta(\epsilon))} |\mathrm{proj} (E^j)|^{\frac{1}{q_0} + \delta(\epsilon)}\\
&\leq C \sum_{j\geq1} 2^{-\epsilon j} 2^{\frac{1}{r} j} |\mathrm{proj} (E^j)|^{\frac{1}{q}} \\
&\leq C \sum_{j\geq1} 2^{-\epsilon j} \|\chi_{E}\|_{L_x^{q}(\mathbb R^n;L_t^{r}(\mathbb R))} \\
&\leq C \|\chi_{E}\|_{L_x^{q}(\mathbb R^n;L_t^{r}(\mathbb R))}.
\end{align*}
\end{proof}
Combining Proposition \ref{lem:sparseEst} and Proposition \ref{prop:glob_d} we obtain an extension of Tao's epsilon removal lemma as follows.
\begin{prop}\label{eps_remv}
Let $1 < q_0,r_0 \leq 2$.
Suppose that
$$
\|\mathfrak R(\chi_{\mathbb{I}_R \times \mathbb B_R} f) \|_{L^2(d\sigma)}
\leq C_{\epsilon} R^{\epsilon} \|f\|_{L_{x}^{q_0}(\mathbb R^n;L_{t}^{r_0}(\mathbb R))}
$$
for all $\epsilon>0$, $R>1$ and all $f \in L_{x}^{q}(\mathbb R^n;L_{t}^{ r}(\mathbb R))$.
Then for any $1\leq q < q_0$ and $1\leq r < r_0$,
$$
\| \mathfrak R f \|_ {L^{2}(d\sigma)}\le C\|f\|_{L_x^{q}(\mathbb R^n; L_t^{r}(\mathbb R) )}
$$
for all $f \in L_x^{q}(\mathbb R^n; L_t^{r}(\mathbb R) )$.
\end{prop}
Now we are ready to prove Theorem \ref{main_thm2}. The theorem follows from Proposition \ref{LP_prop} and Proposition \ref{eps_remv} as follows.
\subsection{Proof of Theorem \ref{main_thm2}}
Let $P_0$ be the Littlewood-Paley projection operator as in subsection 2.1.
By rescaling $x \mapsto 2^{-k}x$ and $t \mapsto 2^{-mk}t$, the estimate \eqref{thm_assum} implies
$$
\| e^{it\Phi(D)} P_0 f\|_{L_x^{q_0}(\mathbb{B}_{R}; L_t^{r_0}(\mathbb{I}_{R^{m}}) )}
\leq C_{\epsilon} 2^{k\epsilon} \|P_0 f\|_{L^2(\mathbb R^n)}
$$
for all $k\geq1$ and $\epsilon>0$.
Since $m\geq 1$, we have
$$
\| e^{it\Phi(D)} P_0 f\|_{L_x^{q_0}(\mathbb{B}_{R}; L_t^{r_0}(\mathbb{I}_{R}) )}
\leq C_{\epsilon} 2^{k\epsilon} \|P_0 f\|_{L^2(\mathbb R^n)} .
$$
By Proposition \ref{eps_remv} and duality,
$$
\| e^{it\Phi(D)} P_0 f\|_{L_x^{q}(\mathbb{R}^n; L_t^{r}(\mathbb{R}) )}
\leq C\|P_0 f\|_{L^2(\mathbb R^n)}.
$$
By Proposition \ref{LP_prop}, we obtain the desired estimate. \qed
\section{Appendix} \label{sec:append}
\subsection{Proof of Lemma \ref{lem:spase_decp}}
We divide the left side of \eqref{eqn:dep} into two parts
\[
\| \sum_{i=1}^{N} f_i \ast \hat \varphi_i |_{S} \|_2^2
= \sum_{i} \|f_i \ast \hat \varphi_i |_{S} \|_2^2
+ \sum_{i \neq j} \int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j} d\sigma.
\]
By a basic restriction estimate we have $\| f_i \ast \hat \varphi_i |_{S}\|_2 \lesssim R^{1/2} \|f_i\|_2$. Thus,
\begin{equation} \label{Bres}
\sum_{i=1}^{N} \| f_i \ast \hat \varphi_i |_{S}\|_2^2 \lesssim R \sum_{i=1}^{N} \|f_i\|_2^2.
\end{equation}
By Parseval's identity,
\[
\int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j} d\sigma
= \int \overline{\check f_j \varphi_j} ((\check f_i \varphi_i) \ast \widehat{d\sigma}),
\] where the $\check{ }$ denotes the inverse Fourier transform.
It is bounded by
\[
\big(\sup_{z,w} | \varphi_j^{1/2}(z) \varphi^{1/2}_i(w) \widehat{d\sigma}(z-w) | \big) \|\check f_i \varphi^{1/2}_i\|_1 \|\check f_j \varphi^{1/2}_j\|_1.
\]
By the Cauchy-Schwarz inequality and Plancherel's theorem,
\[
\|\check f_i \varphi^{1/2}_i \|_{1} \lesssim R^{(n+1)/2} \| f_i\|_{2}.
\]
By \eqref{surdecay},
\[
\sup_{z,w} | \varphi^{1/2}_j(z) \varphi^{1/2}_i(w) \widehat{d\sigma}(z-w) | \lesssim |z_i-z_j-2R|^{-\rho}.
\]
Since $|z_i-z_j| \ge (NR)^{\gamma}$ and $\gamma \ge 2$, we have that $|z_i-z_j-2R|$ is comparable to $|z_i-z_j|$. Thus,
\[
\sup_{z,w} | \varphi^{1/2}_j(z) \varphi^{1/2}_i(w) \widehat{d\sigma}(z-w) | \lesssim |z_i-z_j|^{-\rho}.
\]
Combining these estimates we have
\begin{align*}
\sum_{i \neq j} \int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j} d\sigma &\lesssim R^{n+1} \sum_{i=1}^{N}\sum_{j=1}^{N} |z_i-z_j|^{-\rho} \| f_i\|_{2} \| f_j\|_{2} \\
&\lesssim R^{n+1} N \max_{i,j} |z_i-z_j|^{-\rho} \sum_{i=1}^{N} \| f_i\|_{2}^2.
\end{align*}
Since $|z_i-z_j| \ge (NR)^{\gamma} \ge N^{\frac{1}{\rho}}R^{\frac{n}{\rho}}$, it follows that
\[
\sum_{i \neq j} \int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j} d\sigma \lesssim R \sum_{i=1}^{N} \| f_i\|_{2}^2.
\]
From the above estimate and \eqref{Bres} we obtain \eqref{eqn:dep}.
\qed
\subsection{Proof of Lemma \ref{lem:DecE}}
Fix $K \in \mathbb N$. We define $R_0=1$ and $R_k$ for $k=1,2,\cdots, K$ recursively by
\begin{equation} \label{sepB}
R_{k} = |E|^{\gamma} R_{k-1}^{\gamma}.
\end{equation}
From this definition we have $R_k = |E|^{\frac{\gamma^{k+1}-\gamma}{\gamma-1}}$. Let $E_0 = \emptyset$. We define $E_k$ for $k=1,2,\cdots, K$ to be the set of all $x \in E \setminus \cup_{j=0,1,2,\cdots,k-1} E_j$ such that
\begin{equation} \label{Econ1}
|E \cap B(x, R_k)| \le |E|^{k/K}.
\end{equation}
Then, $E = \bigcup_{k=1}^{K} E_k$. From this construction it follows that that for $x \in E_k$, $k=2,3, \cdots, K$,
\begin{equation} \label{Econ2}
|E \cap B(x, R_{k-1})| > |E|^{(k-1)/K}.
\end{equation}
We cover $E_k$ with finitely overlapping $R_k$-balls $\mathbf C_{E_k} := \{B_i=B(x_i,R_k): x_i \in E_k\}$.
Since $E$ is a finite union of cubes of side-length $c \sim 1$, it is obvious that $\#\mathbf C_{E_k} \lesssim |E|$. For each $B_i \in \mathbf C_{E_k}$ we cover $E_k \cap B_i$ with finitely overlapping $R_{k-1}$-balls $\mathbf C_{E_k \cap B_i} :=\{B'_{ij}=B'(y_j, R_{k-1}):y_j \in E_k \cap B_i \}$, that is,
\[
E_k \cap B_i = \bigcup_{B'_{ij} \in \mathbf C_{E_k \cap B_i}} E_k \cap B_{ij}'.
\]
Since $((E \setminus E_k) \cap B_{ij}') \subset ((E \setminus E_k) \cap B_{i})$ for all $j$, we have
\[
(E_k \cap B_i) \cup ((E \setminus E_k) \cap B_{i}) \supset \bigcup_{B'_{ij} \in \mathbf C_{E_k \cap B_i}} (E_k \cap B_{ij}') \cup ((E \setminus E_k) \cap B_{ij}'),
\]
thus
\[
E \cap B_i \supset \bigcup_{B'_{ij} \in \mathbf C_{E_k \cap B_i}} E \cap B_{ij}'.
\]
By finitely overlapping,
\[
\# \mathbf C_{E_k \cap B_i} \lesssim \max_{B'_{ij} \in \mathbf C_{E_k \cap B_i}} \frac{|E \cap B_i|}{|E \cap B_{ij}'|}.
\]
By \eqref{Econ1} and \eqref{Econ2} the above is bounded by $C|E|^{1/K}$, and we have $\#\mathbf C_{E_k \cap B_i} \le C|E|^{1/K}$ for all $i$.
Thus,
\[
E_k \subset \bigcup_{i=1}^{O(|E|)} \bigcup_{j=1}^{O(|E|^{1/K})} B_{ij}'.
\]
We choose $O(R_k)$-separated balls $\{B'_{ij(i)}\}_{i=1}^{O(|E|)}$. Then it becomes a $(O(|E|), R_{k-1})$-sparse collection because of \eqref{sepB}. Since $R_{k-1} = |E|^{O(\gamma^{k-1})}$ and every $B_i \in \mathbf C_{E_k}$ has the covering $\mathbf C_{E_k \cap B_i}$ of cardinality $O(|E|^{1/K})$, there are $O(|E|^{1/K})$ number of $(O(|E|), |E|^{O(\gamma^{k-1})})$-sparse collections $\mathbf S_1, \mathbf S_2, \cdots, \mathbf S_{O(|E|^{1/K})}$ such that
\[
E_k \subset \bigcup_{j=1}^{O(|E|^{1/K})} \bigcup_{B' \in \mathbf S_j} B'.
\]
\qed
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|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 416
|
\section{Bandit linear optimization}\label{sec:blo}
Our second general application is to bandit linear optimization, in which at each round $i$ of task $t$ we play a vector $\*x_{t,i}\in\mathcal K$ for some convex set $\mathcal K$ and observe loss $\langle\ell_{t,i},\*x_{t,i}\rangle\in[-1,1]$.
We will again use a variant of mirror descent on top of estimated losses, this time setting $\phi$ to be a self-concordant barrier function with specialized loss estimators as in \citet{abernethy2008competing}.
This class of algorithms is picked because of its general applicability to any convex domain $\mathcal K$ via the construction of such barriers and the optimal dependence of its regret on the number of rounds $m$.
Note that our ability to handle non-smooth regularizers via the structural result in Theorem~\ref{thm:meta} is even more important here, as the barrier functions are infinite at the boundaries.
Indeed, in this section we will no longer learn a $\beta$ parameterizing the regularizer and instead focus on learning an offset $\varepsilon>0$ away from the boundary.
For each such offset define $\mathcal K_\varepsilon=\{\*x\in\mathbb R^d:\pi_{\*x_{1,1}}(\*x)\le1/(1+\varepsilon)\}\subset\mathcal K$, where $\*x_{1,1}=\argmin_{\*x\in\mathcal K}\phi(\*x)$ and $\pi_{\*x_{1,1}}(\*x)=\inf_{\lambda\ge0,\*x_{1,1}+(\*x-\*x_{1,1})/\lambda\in\mathcal K}\lambda$ is the Minkowski function.
As before we obtain the $\varepsilon$-restricted optima-in-hindsight via the primitive $\operatorname{OPT}_\varepsilon(\hat\ell_t)=\argmin_{\*x\in\mathcal K_\varepsilon}\langle\hat\ell_t,\*x\rangle$.
With this specified, we can again adapt our meta-approach of Algorithm~\ref{alg:meta}, roughly summarized for BLO as doing the following at each task $t>1$:
\begin{samepage}
\begin{enumerate}[noitemsep]
\item sample $\theta_t=(\eta_t,\varepsilon_t)$ from a distribution $\*p_t$ over the discretization $\Theta$
\item run $\texttt{OMD}_{\eta_t}$ using the initialization $\*x_{t,1}
=\frac1{t-1}\sum_{s<t}\hat{\*x}_t^{(\theta_t)}
=\frac1{t-1}\sum_{s<t}\operatorname{OPT}_{\varepsilon_t}(\hat\ell_t)$
\item update $\*p_{t+1}$ using multiplicative weights with losses $\frac1{\eta_t}B_\phi(\hat{\*x}_t^{(\varepsilon_t)}||\*x_{t,1})+(32d^2\eta_t+\varepsilon_t)m$
\end{enumerate}
\end{samepage}
Note that this algorithm is very similar to that for MAB, with both being special cases of Algorithm~\ref{alg:meta}, with the main difference being the different upper bound passed to multiplicative weights.
The procedure has the following guarantee
\begin{Thm}\label{thm:blo}
Suppose $\texttt{OMD}_{\eta,\beta}$ is online mirror descent with a self-concordant barrier $\phi$ as a regularizer and loss estimators specified as in \citet{abernethy2008competing}.
Then for every $\overline\varepsilon\in(0,1/\sqrt m]$ and $\underline\varepsilon\in(0,\overline\varepsilon]$ there exists an integer $k=\mathcal O(D_{\underline\varepsilon}^2d\lceil\sqrt{mT}\rceil)$, where $D_{\underline\varepsilon}^2$ is a bound on $B_\phi$ over $\mathcal K_{\underline\varepsilon}$, and $\alpha,\underline\eta,\overline\eta\in(0,\infty)$ such that running Algorithm~\ref{alg:meta} with $\Theta$ the product of uniform grids of size $k$ over each dimension of $[\underline\eta,\overline\eta]\times[\underline\epsilon,\overline\epsilon]$ and $\alpha$ the meta-step-size yields the expected task-averaged regret
\begin{align}
\begin{split}
\mathbb E\frac1T\sum_{t=1}^T\sum_{i=1}^m\langle\ell_{t,i},\*x_{t,i}-\*x_t^\ast\rangle
&\le72d\sqrt m\sqrt[4]T\left(D_{\underline\varepsilon}\sqrt{\frac mT\log k}+\frac{S_{\underline\varepsilon}K^2}{D_{\underline\varepsilon}T}(1+\log T)\right)\\
&\qquad+\min_{\*x\in\mathcal K,\eta>0,\varepsilon\in[\underline\varepsilon,\overline\varepsilon]}\mathbb E\frac1T\sum_{t=1}^T\frac{B_\phi(\operatorname{OPT}_\varepsilon(\hat\ell_t)||\*x)}\eta+(32\eta d^2+\varepsilon)m\\
&=\tilde\mathcal O\left(\frac{D_{\underline\varepsilon}dm}{\sqrt[4]T}+\frac{S_{\underline\varepsilon}K^2d\sqrt m}{D_{\underline\varepsilon}T^\frac34}\right)+\min_{\*x\in\mathcal K,\varepsilon\in[\underline\varepsilon,\overline\varepsilon]}4d\hat V_\varepsilon\sqrt{2m}+\varepsilon m
\end{split}
\end{align}
where $S_{\underline\varepsilon}=\max_{\*x\in\mathcal K_{\underline\varepsilon}}\|\nabla^2\phi(\*x)\|_2$, $K$ is the Euclidean diameter of $\mathcal K$, and $\hat V_\varepsilon$ is what we call the {\bf barrier-divergence at level $\varepsilon$} defined by $\hat V_\varepsilon^2=\min_{\*x\in\mathcal K}\mathbb E\frac1T\sum_{t=1}^TB_\phi(\operatorname{OPT}_\varepsilon(\hat\ell_t)||\*x)$.
\end{Thm}
For self-concordant barriers we generally have $D_{\underline\varepsilon}=\mathcal O(1/\underline\varepsilon)$ and $S_{\underline\varepsilon}=\mathcal O(1/\underline\varepsilon^2)$~\citep{abernethy2008competing}, so setting $\underline\varepsilon=1/m$ and yields
\begin{equation}
\mathbb E\frac1T\sum_{t=1}^T\sum_{i=1}^m\langle\ell_{t,i},\*x_{t,i}-\*x_t^\ast\rangle
\le\tilde\mathcal O\left(\frac{dm^2}{\sqrt[4]T}\right)+\min_{\frac1m\le\varepsilon\le\frac1{\sqrt m}}4d\hat V_\varepsilon\sqrt{2m}+\varepsilon m
\end{equation}
As before, this shows that as the number of tasks $T\to\infty$ the average regret improves with a notion of task-similarity $\hat V_\varepsilon$ that decreases if the estimated task-optima are close together.
Roughly speaking, if tasks have barrier-divergence $\hat V_\varepsilon$ then the average regret will be $\mathcal O(\hat V_\varepsilon\sqrt m+\varepsilon m)$, which can be a significant improvement over the single-task case, e.g. if $\hat V_\frac1m$ is small.
In-particular, our analysis removes explicit dependence on the square root of the self-concordance constant of $\phi$ in the single-task case \citep{abernethy2008competing};
as an example, this constant is equal to the number of constraints if $\mathcal K$ is defined by linear inequalities, as in the bandit shortest-path application below.
Note that the use of $\varepsilon$-constrained optima is necessary for this problem due to the regularizers being infinite at the boundaries, where all true optima lie.\looseness-1
To make the above result and task-similarity notion more concrete, consider the following corollary for BLO over the unit sphere $\mathcal K=\{\*x\in\mathbb R^d:\|\*x\|_2\le1\}$:
\begin{Cor}\label{cor:sphere}
Let $\mathcal K$ be the unit sphere with the self-concordant barrier $\phi(\*x)=-\log(1-\|\*x\|_2^2)$.
Then Algorithm~\ref{alg:meta} attains expected task-averaged regret bounded by
\begin{equation}
\tilde\mathcal O\left(\frac{dm^2}{\sqrt[4]T}\right)+\min_{\frac1m\le\varepsilon\le\frac1{\sqrt m}}4d\mathbb E\sqrt{2m\log\left(\frac{1-\|\hat{\bar\ell}^{(\varepsilon)}\|_2^2}{2\varepsilon-\varepsilon^2}\right)}+\varepsilon m
\end{equation}
for $\hat{\bar\ell}^{(\varepsilon)}=\frac1T\sum_{t=1}^T\operatorname{OPT}_\varepsilon(\hat\ell_t)=\frac{\varepsilon-1}T\sum_{t=1}^T\frac{\hat\ell_t}{\|\hat\ell_t\|_2}$ the average over normalized estimated task-optima.\looseness-1
\end{Cor}
Thus in this setting if all tasks have similar estimated losses then $\hat{\bar\ell}^{(\varepsilon)}$ will be an average over similar vectors and thus have large Euclidean norm close to $1-\varepsilon$, making the term in the logarithm above close to 1.
In this case $\hat V_\varepsilon$ is close to zero and so the average regret is $\varepsilon m$ as $T\to\infty$;
setting $\varepsilon=1/m$ yields constant asymptotic averaged regret.
This demonstrates the usefulness of the barrier-divergence as a measure of task-similarity.
As a final application, we apply our meta-BLO result to the shortest-path problem in online optimization \citep{takimoto2003path,kalai2005efficient}.
In its bandit variant \citep{awerbuch2004adaptive,dani2008price}, at each time step $i=1,\dots,m$ the player must choose a path $p_i$ from a fixed source $u\in V$ to a fixed sink $v\in V$ in a directed graph $G(V,E)$.
At the same time the adversary chooses edge weights $\ell_i\in\mathbb R^{|E|}$ and the player suffers the sum $\sum_{e\in p_t}\ell_i(e)$ of the weights in their chosen path $p_t$.
This can be transformed into BLO over vectors $\*x$ in a convex set $\mathcal K\subset[0,1]^{|E|}$ defined by a set $\mathcal C$ of $\mathcal O(|E|)$ linear constraints $(\*a,b)$ s.t. $\langle\*a,\*x\rangle\le b$ enforcing flows from $u$ to $v$;
paths from $u$ to $v$ can then be sampled from any $\*x\in\mathcal K$ in an unbiased manner \citep[Proposition~1]{abernethy2008competing}.
In the single-task case the BLO method of \citet{abernethy2008competing} yields an $\mathcal O(|E|^\frac32\sqrt m)$-regret algorithm for this problem.
In the multi-task case consider a sequence of $t=1,\dots,T$ shortest path instances, each consisting of $m$ edge loss vectors $\ell_{t,i}$ selected by an adversary.
The goal is to minimize average regret across instances.
Note that our setup may be viewed as learning a prediction of the optimal path in a manner similar to the algorithms with predictions paradigm in beyond-worst-case-analysis \citep{mitzenmacher2021awp};
in-particular, we have incorporated predictions into the algorithm of \citet{abernethy2008competing} via the meta-initialization approach and now present the learning-theoretic result for an end-to-end guarantee \citep{khodak2022awp}.
\begin{Cor}\label{cor:path}
Let $\mathcal K=\{\*x\in[0,1]^{|E|}:\langle\*a,\*x\rangle\le b~\forall~(\*a,b)\in\mathcal C\}$ be the set of flows from $u$ to $v$ on a graph $G(V,E)$, where $\mathcal C\subset\mathbb R^{|E|}\times\mathbb R$ is a set of $\mathcal O(|E|)$ linear constraints.
Suppose we see $T$ instances of the bandit online shortest path problem with $m$ timesteps each.
Then sampling from probability distributions over paths from $u$ to $v$ returned by running Algorithm~\ref{alg:meta} with regularizer $\phi(\*x)=-\sum_{\*a,b\in\mathcal C}\log(b-\langle\*a,\*x\rangle)$ attains the following expected average regret across instances:
\begin{equation}
\tilde\mathcal O\left(\frac{|E|m^2}{\sqrt[4]T}\right)+\min_{\frac1m\le\varepsilon\le\frac1{\sqrt m}}4|E|\mathbb E\sqrt{2m\sum_{\*a,b\in\mathcal C}\log\left(\frac{\frac1T\sum_{t=1}^Tb-\langle\*a,\hat{\*x}_t^{(\varepsilon)}\rangle}{\sqrt[T]{\prod_{t=1}^Tb-\langle\*a,\hat{\*x}_t^{(\varepsilon)}\rangle}}\right)}+\varepsilon m
\end{equation}
Here $\hat{\*x}_t^{(\varepsilon)}=\operatorname{OPT}_\varepsilon(\hat\ell_t)$ is the $\varepsilon$-constrained estimated optimal flow for instance $t$.
\end{Cor}
Corollary~\ref{cor:path} shows that the average regret on the $T$ bandit shortest-path problems scales with the sum across all constraints $\*a,b\in\mathcal C$ of the log of the ratio between the arithmetic and geometric mean of the distances $b-\langle\*a,\hat{\*x}_t^{(\varepsilon)}\rangle$ from the estimated optimum flow $\hat{\*x}_t^{(\varepsilon)}$ to the constraint boundary.
Since the arithmetic and geometric mean are equal exactly when all entries are equal---and otherwise the former is larger---this means that the regret is small when the estimated optimal flows $\hat{\*x}_t^{(\varepsilon)}$ for each task are at similar distances from the constraints.
\section*{Checklist}
\begin{enumerate}
\item For all authors...
\begin{enumerate}
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\answerYes{}
\item Did you describe the limitations of your work?
\answerYes{}
\item Did you discuss any potential negative societal impacts of your work?
\answerNA{}
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\answerYes{}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\answerYes{Assumptions are stated in Sections~\ref{sec:setup}, \ref{sec:mab}, and~\ref{sec:blo}.}
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\answerYes{Proofs are given in the Appendix.}
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\section{Conclusion}
In this work, we develop a meta-algorithm for learning to initialize and tune OMD for regularizers used in adversarial bandit tasks. We apply our meta-algorithm to obtain task-averaged regret guarantees for both the multi-armed and the linear bandit settings that depend on natural, setting-specific notions of task similarity. For MAB, we use OMD with the Tsallis regularizer as our base-learner and meta-learn the initialization, step-size, and entropy parameter. For BLO, we again use a variant of mirror descent with self-concordant barrier regularizers as our base-learner, and meta-learn the initialization, step-size, and boundary-offset.
\newpage
There are several exciting directions for future work.
A limitation of our current results is the dependence of the task-similarity on optima estimated by the within-task algorithm;
while they have the benefit of being computable, one may wish to obtain task-averaged regret bounds which do not depend on algorithmic quantities.
In particular, defining task-similarity via the true optima or losses may be more natural.
Achieving this may be possible by making further assumptions on the structure of the problem, e.g. gap conditions or best-arm identifiability.
Another direction is to extend these results to other adversarial bandits settings such as contextual bandits and Lipschitz bandits.
\section{Introduction}\label{sec:intro}
Many real-world problems involve decision-making under partial feedback. For example, an administrator of a news website will not get to observe user engagement for undisplayed content. Likewise, the administrator does not know what their commute time would have been had they taken a different route to work that day. Such feedback, where a \emph{learner} only observes the outcome of the action taken, is often referred to as \emph{bandit feedback}. This is in contrast to the \emph{full feedback} setting, in which the learner gets to observe what would have happened under all possible actions they could have taken.
While there are many methods with performance guarantees for bandit learning, most do not take into consideration the information the learner has gained from previous experience completing similar tasks. When selecting content for a website targeting a new demographic, the administrator will likely consider which types of content generated high levels of engagement with similar subpopulations of users. Likewise, they will most likely use knowledge about traffic patterns gained from their daily commute to work to inform routes to other locations.
Can our learning algorithms do the same?
\emph{Meta-learning}, also known as \emph{learning-to-learn} \cite{thrun1998ltl}, is a popular approach to studying such problems in the context of multi-task learning, changing environments, and beyond.
The goal of such meta-learning algorithms is to leverage information from previously-seen tasks in order to achieve better performance on the current task at hand. While most meta-learning algorithms are designed for the full feedback setting, there is a small but growing amount of recent work which aims to design meta-learning algorithms capable of operating under bandit feedback.
The two most commonly studied types of feedback in the bandit literature are \emph{stochastic bandit feedback}, where feedback is sampled i.i.d. from some distribution, and a more general notion called \emph{adversarial bandit feedback}, where it is chosen by an adversary possibly trying to harm the learner. To the best of our knowledge, we are the first to study the problem of meta-learning under adversarial bandit feedback.
We consider a setting in which a meta-learner interacts with a sequence of bandit tasks.
In the single-task setting, the goal is to minimize \emph{regret} with respect to the best fixed action in hindsight.
Lifting this to the multi-task setting, our goal will be to design algorithms which achieve low regret \emph{on average} across tasks. Ideally, an algorithm's task-averaged regret should be no worse than that of algorithms for the single-task setting, e.g. if the tasks are not very similar, and should be much better on tasks that are closely related, e.g. if the same small set of arms do well on all of them.
We design a meta-algorithm based on learning the initialization and tuning parameters of online mirror descent (OMD) when it uses regularizers employed by bandit algorithms such as Exp3 \citep{auer2002exp3}.
Theoretically, our work differs from past approaches to parameterizing OMD in the full information setting because the regularizers used by bandit methods are non-Lipschitz near the boundary of the action set;
thus the results of past work~\citep{khodak2019provable,khodak2019adaptive,denevi2019meta} do not apply.
To overcome this issue, we only initialize away from the boundary and adapt our algorithms to handle the resulting error.
We apply our meta-algorithm for adversarial bandit feedback to both multi-armed bandits (MAB) and bandit linear optimization (BLO), obtaining in both settings new meta-learning algorithms with provable guarantees. For MAB, the average $m$-round regret across $T$ tasks of our algorithm is
\begin{equation}\label{eq:result}
o_T(1)+2\min_{\beta\in(0,1]}\sqrt{\hat H_\beta d^\beta m/\beta}
\end{equation}
where $d$ is the number of actions and $\hat H_\beta$ is the Tsallis entropy~\citep{tsallis1988possible,abernethy2015fighting} of the empirical distribution over the estimated optimal actions across tasks.
At $\beta=1$ the Tsallis entropy reduces to the Shannon entropy;
both are small if most tasks are estimated to be solved by the same few arms and large if all are used roughly the same amount, making it a natural task-similarity notion.
The bound of $\log d\ge\hat H_1$ means that the bound~\eqref{eq:result} recovers Exp3's guarantee in the worst-case of dissimilar tasks.
In the important case of $s\ll d$ arms always being estimated to be optimal we have $\hat H_\beta=\mathcal O(s)$, so using $\beta=\frac1{\log d}$ in bound~\eqref{eq:result} yields a task-averaged regret of $\mathcal O(\sqrt{sm\log d})$ as $T\to\infty$.
For $s=\mathcal O_d(1)$ this beats the single-task lower bound of $\Omega(\sqrt{dm})$~\citep{audibert2011minimax}.
We also obtain natural task-averaged regret bounds for BLO, albeit with different setting-specific notions of task similarity.
Our main technical contributions are as follows:
\begin{enumerate}[noitemsep]
\item We design a unified meta-learning algorithm to set the initialization and tuning parameters of OMD when using regularizers used by different bandit algorithms (Algorithm~\ref{alg:meta}).
Apart from strong guarantees and generality, our approach is notable for its adaptivity:
we do not need to know anything about the task-similarity---e.g. the size of the subset of optimal arms---to adapt to similar tasks.\looseness-1
\item We apply our meta-approach to obtain a meta-learning algorithm for the adversarial MAB problem. In particular, we use the method of \citet{abernethy2015fighting}---OMD with the Tsallis regularizer---as our within-task algorithm to achieve bounds on task-averaged regret that depend on a natural notion of task similarity:
the Tsallis entropy of the estimated optima-in-hindsight.
\item We adapt Algorithm \ref{alg:meta} to the adversarial BLO problem by setting the regularizer to be a self-concordant barrier function, as in~\citet{abernethy2008competing}. As in MAB, we obtain task-averaged regret bounds which depend on a natural notion of task similarity based on the constraints defining the convex action space.
We instantiate the BLO result in two settings:
linear bandits over the sphere and an application to the bandit shortest-path problem~\citep{takimoto2003path,kalai2005efficient}.
\end{enumerate}
\section{Multi-armed bandits}\label{sec:mab}
We now turn to our first application:
the multi-armed bandits problem.
In this setting at each round $i$ of task $t$ we take action $a_{t,i}\in[d]$ and observe loss $\ell_{t,i}(a_{t,i})\in[0,1]$.
As algorithms for MAB are probabilistic, we often sample methods from distributions $\*x\in\overline\mathcal K=\triangle_d$ in the $k$-simplex, thus making the inner product $\langle\ell_{t,i},\*x_{t,i}\rangle$ the expectation.
In this paper we use as a base-learner a generalization of the popular Exp3 method of \citet{auer2002exp3}, which runs multiplicative weights over unbiased estimators of the losses.
The first generalization is of the OMD regularizer, which for Exp3 is the negative Shannon entropy;
we employ the negative Tsallis entropy $\phi_\beta(\*p)=\frac{1-\sum_{a=1}^d\*p^\beta(a)}{1-\beta}$ for $\beta\in[0,1]$, which was used by \citet{abernethy2015fighting} to improve the dependence of the regret on the dimension from $\mathcal O(\sqrt{dm\log d})$ to the optimal $\mathcal O(\sqrt{dm})$.
Note that $\phi_\beta$ recovers the Shannon entropy in the limit $\beta\to1$, and also that $B_{\phi_\beta}(\*x||\cdot)$ is non-convex in the second argument, making ours the first known application of the online learnability of non-convex Bregman divergences.
The second generalization is in the loss estimators;
for $\gamma>0$ we employ $\hat\ell_{t,i}(a)=\frac{\ell_{t,i}(a)1_{a_{t,i}=a}}{\*x_{t,i}(a)+\gamma}$, where $\*x_{t,i}(a)$ is the probability of sampling $a$ on round $i$ of task $t$.
While this is an under-estimate of $\ell_{t,i}(a)$, its lower variance compared to the unbiased estimator---recovered by setting $\gamma=0$---allows \citet{neu2015explore} to obtain high probability bounds.
As the Tsallis entropy is non-smooth at the simplex boundary, learning Tsallis divergences will require the tools developed previously for initializing $\texttt{OMD}$ in the interior of $\overline\mathcal K$.
We set $\mathcal K_\varepsilon=\{\*x\in\triangle_d:\min_a\*x(a)\ge\varepsilon/d\}$, so that the offset optimum $\hat{\*x}_t^{(\theta)}$ then has the very simple form $\operatorname{OPT}_\varepsilon(\hat\ell_t)=(1-\varepsilon)\hat{\*x}_t+\varepsilon\*1_d/d$, i.e. it is the mixture of the estimated optimum $\hat{\*x}_t$ over the entire simplex with the uniform distribution.
Note that for MAB we will {\em not} need to the capability of Algorithm~\ref{alg:meta} to learn $\varepsilon$ using multiplicative weights and can just set it assuming knowledge of the number of tasks.
Thus the method in this setting can be roughly summarized as doing the following at each task $t>1$:
\begin{enumerate}[noitemsep]
\item sample $\theta_t=(\eta_t,\beta_t)$ from a distribution $\*p_t$ over the discretization $\Theta$
\item run $\texttt{OMD}_{\beta_t,\eta_t}$ using the initialization $\*x_{t,1}
=\frac1{t-1}\sum_{s<t}\hat{\*x}_t^{(\theta_t)}
=\frac\varepsilon d\*1_d+\frac{1-\varepsilon}{t-1}\sum_{s<t}\hat{\*x}_t$
\item update $\*p_{t+1}$ using multiplicative weights with the expert losses $\frac1{\eta_t}B_{\phi_{\beta_t}}(\hat{\*x}_t^{(\varepsilon)}||\*x_{t,1})+\frac{\eta_td^\beta_tm}{\beta_t}$
\end{enumerate}
The latter regret-upper-bound is derived from the within-task regret of OMD with the Tsallis regularizer.
This simple procedure achieves the following guarantee on the task-averaged regret:
\begin{Thm}\label{thm:mab}
Suppose $\texttt{OMD}_{\eta,\beta}$ is online mirror descent with the Tsallis entropy regularizer $\phi_\beta$ over $\gamma$-offset loss estimators.
Then for every $\varepsilon>0$ and $\underline\beta\in(0,1]$ there exists integer $k=\tilde\mathcal O(\lceil d^4\sqrt{mT}\log\frac1\varepsilon\rceil)$ and $\alpha,\underline\eta,\overline\eta\in(0,\infty)$ such that running Algorithm~\ref{alg:meta} with $\Theta$ the product of uniform grids of size $k$ over each non-singleton dimension of $[\underline\eta,\overline\eta]\times[\max\{\underline\beta,1/\log d\},1]\times\{\varepsilon\}$ and $\alpha$ the meta-step-size yields w.p. at least $1-\delta$ the task-averaged regret
\begin{align}
\begin{split}
\frac1T&\sum_{t=1}^T\sum_{i=1}^m\ell_{t,i}(a_{t,i})-\ell_{t,i}(a_t^\ast)\\
&\le\tilde\mathcal O\left(
\frac{\sqrt d}{\gamma T}\log\frac4\delta
+\left(\frac\varepsilon{\gamma d}+\gamma d\right)m
+\frac{d^{2-\underline\beta}\sqrt m}{\rho\varepsilon^{2-\underline\beta}T}
+\left(\frac{\sqrt d}\rho+d\right)\sqrt{\frac{md}T\log\frac4\delta}
+\rho d\sqrt m
\right)\\
&+\qquad\min_{\eta>0,\beta\in[\underline\beta,1]}\frac{H_\beta(\hat{\bar{\*x}})}\eta+\frac{\eta d^\beta m}\beta+\frac{\varepsilon^\beta d^{1-\beta}1_{\beta<1}}{(1-\beta)\eta}
\end{split}
\end{align}
where $H_\beta=-\phi_\beta$ is the Tsallis entropy and $\hat{\bar{\*x}}$ is the mean of the estimated optima $\hat{\*x}_1,\dots,\hat{\*x}_T$.
\end{Thm}
We see that the regret-upper-bound is highly dependent on the loss estimator offset $\gamma$, the boundary offset $\varepsilon$, the step-size offset $\rho$, and the lower bound $\underline\beta$ on the parameter of the Tsallis entropy.
Thus to clarify the guarantee we consider three regimes of $\underline\beta$:
$\underline\beta=1$, i.e. always using Exp3;
$\underline\beta=1/2$, which corresponds to the standard setting when using the Tsallis entropy \citep{abernethy2015fighting};
and $\underline\beta=1/\log d$, below which the OMD regret-upper-bound always worsens and so it does not make sense to try $\beta<1/\log d$.
\begin{Cor}\label{cor:mab}
Suppose we run Algorithm~\ref{alg:meta} as in Theorem~\ref{thm:mab}.
For $\underline\beta=1$, if we set $\varepsilon=\frac1{\sqrt T}$, $\gamma=\frac{\sqrt{\log\frac4\delta}}{d\sqrt[4]T}$, and $\rho=\frac{\sqrt d}{\sqrt[4]T}$ then w.p. $1-\delta$ the task-averaged regret satisfies
\begin{equation}\label{eq:exp3}
\tilde\mathcal O\left(\frac{d^\frac32+\sqrt m}{\sqrt[4]T}\sqrt{m\log\frac4\delta}\right)+2\sqrt{H_1(\hat{\bar{\*x}})dm}
\end{equation}
For $\underline\beta=\frac12$, if $\varepsilon=\sqrt{\frac dT}$, $\gamma=\frac{\sqrt{\log\frac4\delta}}{d\sqrt[4]T}$, and $\rho=\frac1{\sqrt{md}}$ then w.p. $1-\delta$ the task-averaged regret is
\begin{equation}\label{eq:half}
\tilde\mathcal O\left(\frac{dm\sqrt{d\log\frac4\delta}}{\sqrt[4]T}\right)+2\sqrt d+2\min_{\beta\in\left[\frac12,\frac{\log d-1}{\log d}\right]}\sqrt{H_\beta(\hat{\bar{\*x}})d^\beta m/\beta}
\end{equation}
For $\underline\beta=\frac1{\log d}$, if $\varepsilon=\frac1{\sqrt[3]T}$, $\gamma=\frac{\sqrt{\log\frac4\delta}}{d\sqrt[6]T}$, and $\rho=\frac{\sqrt d}{\sqrt[6]T}$ then w.p. $1-\delta$ the task-averaged regret is
\begin{equation}\label{eq:full}
\tilde\mathcal O\left(\frac{d^\frac32+\sqrt m}{\sqrt[6]T}\sqrt{m\log\frac4\delta}\right)+2\min_{\beta\in(0,1]}\sqrt{H_\beta(\hat{\bar{\*x}})d^\beta m/\beta}+\sqrt{\frac{d1_{\beta<1}}{\beta(1-\beta)mT^\frac\beta 3}}
\end{equation}
\end{Cor}
These results show that for all three settings of $\underline\beta$, as the meta-learner sees more tasks the average regret depends directly on the entropy of the estimated optima-in-hindsight, a natural notion of task-similarity since it is small if most tasks are estimated to be solved by the same arms and large if all arms are used roughly the same amount.
It also demonstrates how our algorithm's automatic tuning of the step-size $\eta$ allows us to set the asymptotic rate optimally depending on the entropy.
The algorithm's tuning of the entropy itself via $\beta$ also enables adaptation to similar tasks;
specifically, a smaller $\beta$ weights the $H_\beta(\hat{\bar{\*x}})/\eta$ term higher and is thus beneficial if tasks are similar.
As a natural example, suppose a constant $s\ll d$ actions are always minimizers, i.e. $\hat{\bar{\*x}}$ is $s$-sparse.
Then the last bound~\eqref{eq:full} implies that Algorithm~\ref{alg:meta} can achieve task-averaged regret $o_T(1)+\mathcal O(\sqrt{sm\log d})$, albeit at the cost of slow convergence.
In-general, for the case of tuning over all $\beta\ge1/\log d$ the speed of the convergence depends on the optimal $\beta$;
the algorithm will converge very slowly at rate $\tilde\mathcal O(1/\sqrt[6\log d]T)$ if the optimal $\beta$ is around $1/\log d$, but for $\beta$ near 1 the rate will be $\tilde\mathcal O(1/\sqrt[4]T)$.
Note that we show in the intermediate case of tuning only as low as $\beta=1/2$ that we can still achieve $\tilde\mathcal O(1/\sqrt[4]T)$ at the cost of a fast $2\sqrt d$ term per-task.
Finally, note that because the entropy is bounded by $d^{1-\beta}$ we do asymptotically recover worst-case guarantees in all three cases if the tasks are dissimilar.
To put these results in some theoretical context, we can compare them to those of \citet{azizi2022non}, who achieve task-averaged regret bounds of the form $\tilde\mathcal O(1/\sqrt T+\sqrt{sm})$ in the {\em stochastic} MAB setting, where $s$ is an unknown subset of optimal actions.
Unlike their result, we study the harder adversarial setting and do {\em not} place restrictions on how the tasks are related;
despite this greater generality, our bounds are asymptotically comparable if the estimated and true optima-in-hindsight are roughly equivalent, as we also have $\tilde\mathcal O(\sqrt{sm})$ average regret as $T\to\infty$.
On the other hand, the rate in the number of tasks of \citet{azizi2022non} is much better, albeit at a cost of runtime exponential in $s$.
Apart from generality, we believe a great strength of our result is its adaptiveness;
unlike this work, we do not need to know how many optimal arms there are or their entropy in order to improve task-averaged regret with task-similarity.
\section*{Acknowledgments}
This material is based on work supported by the National Science Foundation under grants CCF-1910321, FAI-1939606, IIS-1901403, SCC-1952085, and SES-1919453; the Defense Advanced Research Projects Agency under cooperative agreement HR00112020003; a Simons Investigator Award; an AWS Machine Learning Research Award; an Amazon Research Award; a Bloomberg Research Grant; a Microsoft Research Faculty Fellowship; a Google Faculty Research Award; a J.P. Morgan Faculty Award; a Facebook Research Award; a Mozilla Research Grant; and a Facebook PhD Fellowship.
\medskip
\bibliographystyle{plainnat}
\section{Proof of Theorem~\ref{thm:meta}}
\begin{proof}
We define $\underline\eta=\frac{\rho D}{G\sqrt m}$, $\overline\eta=\frac{2DM}{G\sqrt m}$, number of grid points $k=\Omega(\lceil(4D^2M^2LG+C)\sqrt{mT}\rceil)$, $\Theta=\left\{\underline\eta+\frac jk(\overline\eta-\underline\eta)\right\}_{j=0}^k\times\{\underline\beta+\frac jk(\overline\beta-\underline\beta)\}_{j=0}^{k1_{\overline\beta>\underline\beta}}\times\{\underline\varepsilon+\frac jk(\overline\varepsilon-\underline\varepsilon)\}_{j=0}^{k1_{\overline\varepsilon>\underline\varepsilon}}$, and meta-step-size $\alpha=\frac1{DG/\rho+2DMG+C\sqrt m}\sqrt{\frac{3\log k}{2Tm}}$.
Note that
\begin{equation}
\frac{\rho D}{G\sqrt m}
\le\argmin_{\eta>0}\min_{\*x\in\mathcal K_{\underline\varepsilon},\beta,\varepsilon}\sum_{t=1}^T\tilde U_t(\*x,(\eta,\beta,\varepsilon))
\le\frac{DM}G\sqrt{\frac{1+\rho^2}m}
\le\frac{2DM}{G\sqrt m}
\end{equation}
so
\begin{equation}
\max_{t\in[T]}U_t^{(\rho)}(\*x_t^{(\theta_t)},\theta_t)
\le\frac{DG\sqrt m}\rho+2DMG\sqrt{m}+Cm
\end{equation}
Therefore applying the regret guarantee for exponentiated gradient \citep[Corollary~2.14]{shalev-shwartz2011oco} followed by the regret of follow-the-leader on a sequent of Bregman divergences (Lemma~\ref{lem:bregman}) yields
\begin{align}
\begin{split}
\mathbb E&\sum_{t=1}^TU_t(\*x_t^{(\theta_t)},\theta_t)\\
&\le\mathbb E\sum_{t=1}^TU_t^{(\rho)}(\*x_t^{(\theta_t)},\theta_t)\\
&\le\left(C\sqrt m+DG\left(\frac1\rho+2M\right)\right)\sqrt{2mT\log|\Theta|}+\min_{\theta\in\Theta}\mathbb E\sum_{t=1}^TU_t^{(\rho)}(\*x_t^{(\theta)},\theta)\\
&\le\left(C\sqrt m+DG\left(\frac1\rho+2M\right)\right)\sqrt{2mT\log|\Theta|}\\
&\qquad+\min_{(\eta,\beta,\varepsilon)\in\Theta}\frac{8SK^2}\eta(1+\log T)+\min_{\*x\in\mathcal K_\varepsilon}\mathbb E\sum_{t=1}^T\frac{B_{\phi_\beta}(\hat{\*x}_t^{(\varepsilon)}||\*x)+\rho^2D^2}\eta+(\eta G_\beta^2+C\varepsilon)m\\
&\le\left(C\sqrt m+DG\left(\frac1\rho+2M\right)\right)\sqrt{2mT\log|\Theta|}+\frac{8SK^2\overline G\sqrt m}{\rho D}(1+\log T)\\
&\qquad+\min_{(\eta,\beta,\varepsilon)\in\Theta}\eta G_\beta^2mT+C\varepsilon mT+\frac{\rho^2D^2T}\eta+\mathbb E\sum_{t=1}^T\frac{\phi_\beta(\hat{\*x}_t^{(\varepsilon)})-\phi_\beta(\hat{\bar{\*x}}^{(\varepsilon)})}\eta\\
&\le\left(C\sqrt m+DG\left(\frac1\rho+2M\right)\right)\sqrt{2mT\log|\Theta|}+\frac{8SK^2G\sqrt m}{\rho D}(1+\log T)+\rho DGT\sqrt m\\
&\qquad+\left(4D^2M^2+\left(C\sqrt m+\frac{2LG}{\rho D}\right)(\overline\varepsilon-\underline\varepsilon)\sqrt m+2\left(\frac{2M}G+\frac G\rho\right)DL\sqrt m(\overline\beta-\underline\beta)\right)\frac Tk\\
&\qquad+\min_{(\eta,\beta,\varepsilon)\in\overline\Theta}\eta G_\beta^2mT+C\varepsilon mT
+\mathbb E\sum_{t=1}^T\frac{\phi_\beta(\hat{\*x}_t^{(\varepsilon)})-\phi_\beta(\hat{\bar{\*x}}^{(\varepsilon)})}\eta\\
&\le\left(C\sqrt m+DG\left(\frac1\rho+2M\right)\right)\sqrt{6mT\log k}+(4D^2M^2LG+C)\frac{Tm}{\rho k}\\
&\qquad+\frac{8SK^2G\sqrt m}{\rho D}(1+\log T)+\rho DGT\sqrt m+\min_{\*x\in\mathcal K,\theta\in\Theta^\ast}\mathbb E\sum_{t=1}^TU_t(\*x,\theta)
\end{split}
\end{align}
where the fourth inequality follows by Claim~\ref{clm:bregman}, the fifth by Lipschitzness of $1/\eta$ on $\eta\ge\frac{\rho D}{G\sqrt m}$ and of $\phi_\beta(\hat{\*x}_t^{(\varepsilon)}||\cdot)$ on $\mathcal K_{\underline\varepsilon}$, and the sixth by simplifying and substituting the lower bound for $k$.
The w.h.p. version of the bound follows by applying \citet[Lemma~4.1]{cesa-bianchi2006prediction} when obtaining the second inequality.
\end{proof}
\begin{Lem}\label{lem:bregman}
Let $\phi:\mathcal K\mapsto\mathbb R_{\ge0}$ be a strictly-convex function with $\max_{\*x\in\mathcal K}\|\nabla^2\phi(\*x)\|_2\le S$ over a convex set $\mathcal K\subset\mathbb R^d$ with $\max_{\*x\in\mathcal K}\|\*x\|_2\le K$.
Then for any points $\*x_1,\dots,\*x_T\in\mathcal K$ the actions $\*y_1=\argmin_{\*x\in\mathcal K}\phi(\*x)$ and $\*y_t=\frac1{t-1}\sum_{s<t}\*x_s$ have regret
\begin{equation}
\sum_{t=1}^TB_\phi(\*x_t||\*y_t)-B_\phi(\*x_t||\*y_{T+1})
\le\sum_{t=1}^T\frac{8SK^2}{2t-1}
\le8SK^2(1+\log T)
\end{equation}
\end{Lem}
\begin{proof}
Note that
\begin{equation}
\nabla_{\*y}B_\phi(\*x||\*y)
=-\nabla\phi(\*y)-\nabla_{\*y}\langle\nabla\phi(\*y),\*x\rangle+\nabla_{\*y}\langle\nabla\phi(\*y),\*y\rangle
=\operatorname{diag}(\nabla^2\phi(\*y))(\*y-\*x)
\end{equation}
so $B_\phi(\*x_t||\*y)$ is $2SK$-Lipschitz w.r.t. the Euclidean norm.
Applying \citet[Proposition~B.1]{khodak2019adaptive} yields the result.
\end{proof}
\begin{Clm}\label{clm:bregman}
Let $\phi:\mathcal K\mapsto\mathbb R$ be a strictly-convex function over a convex set $\mathcal K\subset\mathbb R^d$ containing points $\*x_1,\dots,\*x_T$.
Then their mean $\bar{\*x}=\frac1T\sum_{t=1}^T\*x_t$ satisfies
\begin{equation}
\sum_{t=1}^TB_\phi(\*x_t||\bar{\*x})
=\sum_{t=1}^T\phi(\*x_t)-\phi(\bar{\*x})
\end{equation}
\end{Clm}
\begin{proof}
\begin{align}
\begin{split}
\sum_{t=1}^TB_\phi(\*x_t||\bar{\*x})
&=\sum_{t=1}^T\phi(\*x_t)-\phi(\bar{\*x})-\langle\nabla\phi(\bar{\*x}),\*x_t-\bar{\*x}\rangle\\
&=\sum_{t=1}^T\phi(\*x_t)-\phi(\bar{\*x})-\langle\nabla\phi(\bar{\*x}),\sum_{t=1}^T\*x_t-\bar{\*x}\rangle\\
&=\sum_{t=1}^T\phi(\*x_t)-\phi(\bar{\*x})
\end{split}
\end{align}
\end{proof}
\newpage
\section{Proof of Theorem~\ref{thm:mab}}
\begin{proof}
Since $\varepsilon$ is constant we use the shorthand $\hat{\*x}_t^{(\varepsilon)}=\hat{\*x}_t^{(\theta)}~\forall~\theta\in\Theta$.
Note that we use search space $\Theta=\left[\frac\rho{\sqrt m},2\sqrt{\frac{d\log d}{em}}\right]\times\left[\max\left\{\frac1{\log d},\underline\beta\right\},1\right]\times\{\varepsilon\}$.
We have the constants $D\le\sqrt d$, $G\le\sqrt d$, $M\le\sqrt{\frac{d\log d}e}$, $S\le\left(\frac d \varepsilon\right)^{2-\underline\beta}$, and $K=1$.
Note that the second term $d^{1-\beta}m/\beta$ is decreasing on $\beta<1/\log d$, so since $\phi_\beta$ is always increasing in $\beta$ we know that the optimal $\beta$ is in $[1/\log d,1]$.
Note that by Lemma~\ref{lem:tsallis} we have that $L=d\log\frac d\varepsilon$.
We thus have
\begin{align}
\begin{split}
\sum_{t=1}^T&\sum_{i=1}^m\ell_{t,i}(a_{t,i})-\ell_{t,i}(a_t^\ast)\\
&\le\sum_{t=1}^T\sum_{i=1}^m\langle\hat\ell_{t,i},\*x_{t,i}-\ell_{t,i}(a_t^\ast)\rangle+\gamma\sum_{a=1}^d\hat\ell_{t,i}(a)\\
&\le\sum_{t=1}^T\frac{B_{\phi_{\beta_t}}(\hat{\*x}_t^{(\varepsilon)}||\*x_{t,1})}{\eta_t}+\sum_{i=1}^m\langle\hat\ell_{t,i},\hat{\*x}_{t,i}^{(\varepsilon)}\rangle-\ell_{t,i}(a_t^\ast)+\frac{\eta_t}{\beta_t}\sum_{a=1}^d\*x_{t,i}^{2-\beta_t}(a)\hat\ell_{t,i}^2(a)+\gamma\sum_{a=1}^d\hat\ell_{t,i}(a)\\
&\le\frac{\varepsilon mT}{\gamma d}+\sum_{t=1}^T\frac{B_{\phi_{\beta_t}}(\hat{\*x}_t^{(\varepsilon)}||\*x_{t,1})}{\eta_t}+\sum_{i=1}^m\hat\ell_{t,i}(a_t^\ast)-\ell_{t,i}(a_t^\ast)\\
&\qquad+\sum_{t=1}^T\frac{\eta_t}{\beta_t}\sum_{i=1}^m\sum_{a=1}^d\*x_{t,i}^{1-\beta_t}(a)\hat\ell_{t,i}(a)+\gamma\sum_{a=1}^d\hat\ell_{t,i}(a)\\
&\le\frac{\varepsilon mT}{\gamma d}+\frac{1+\frac{\overline\eta}{\underline\beta}+\gamma}{2\gamma}\log\frac4\delta+\sum_{t=1}^T\frac{B_{\phi_{\beta_t}}(\hat{\*x}_t^{(\varepsilon)}||\*x_{t,1})}{\eta_t}\\
&\qquad+\sum_{t=1}^T\frac{\eta_t}{\beta_t}\sum_{i=1}^m\sum_{a=1}^d\*x_{t,i}^{1-\beta_t}(a)\ell_{t,i}(a)+\gamma\sum_{a=1}^d\ell_{t,i}(a)\\
&\le\frac{\varepsilon mT}{\gamma d}+\frac{1+\sqrt{\frac{d\log^3d}{em}}}\gamma\log\frac4\delta+\gamma dmT+\sum_{t=1}^T\frac{B_{\phi_{\beta_t}}(\hat{\*x}_t^{(\varepsilon)}||\*x_{t,1})}{\eta_t}+\frac{\eta_td^{\beta_t}m}{\beta_t}\\
&\le\frac{\varepsilon mT}{\gamma d}+\frac{1+\sqrt{\frac{d\log^3d}{em}}}\gamma\log\frac4\delta+\gamma dmT+\min_{\*x\in\triangle_d,\eta>0,\beta\in[\underline\beta,1]}\sum_{t=1}^T\frac{B_{\phi_\beta}(\hat{\*x}_t^{(\varepsilon)}||\*x)}\eta+\frac{\eta d^\beta m}\beta\\
&\qquad+2d\left(\frac1\rho+\sqrt{\frac{d\log d}e}\right)\sqrt{6mT\log\frac{4k}\delta}+\frac{8d^{2-\underline\beta}\sqrt{m}}{\rho\varepsilon^{2-\underline\beta}}(1+\log T)+\rho dT\sqrt m\\
&\le\left(\frac\varepsilon{\gamma d}+\gamma d\right)mT+\frac{1+\sqrt{\frac{d\log^3d}{em}}}\gamma\log\frac4\delta+T\min_{\eta>0,\beta\in[\underline\beta,1]}\frac{H_\beta(\hat{\bar{\*x}})}\eta+\frac{\eta d^\beta m}\beta+\frac{\varepsilon^\beta d^{1-\beta}1_{\beta<1}}{(1-\beta)\eta}\\
&\qquad+2d\left(\frac1\rho+\sqrt{\frac{d\log d}e}\right)\sqrt{6mT\log\frac{4k}\delta}+\frac{8d^{2-\underline\beta}\sqrt{m}}{\rho\varepsilon^{2-\underline\beta}}(1+\log T)+\rho dT\sqrt m
\end{split}
\end{align}
where the second inequality follows by Lemma~\ref{lem:mirror}, the third by H\"older's inequality and the definitions $\hat\ell_{t,i}$ and $\hat{\*x}_{t,i}^{(\varepsilon)}$, the fourth by \citet[Lemma~1]{neu2015explore}, the fifth by the definition of $\ell_{t,i}$, the sixth by Theorem~\ref{thm:meta}, and the last by the derivation below for $\beta<1$ (otherwise it holds by joint convexity of the KL-divergence) followed by Claim~\ref{clm:bregman} combined with the fact that the entropy of optima-in-hindsight is zero.
\begin{align}
\begin{split}
-\phi_\beta((1-\varepsilon)\*x+\varepsilon\*1_d/d)
&=\frac{\sum_{a=1}^d((1-\varepsilon)\*x(a)+\varepsilon/d)^\beta-1}{1-\beta}\\
&\le\frac{\varepsilon^\beta d^{1-\beta}+(1-\varepsilon)^\beta\sum_{a=1}^d\*x^\beta(a)-1}{1-\beta}
\le\frac{\varepsilon^\beta d^{1-\beta}}{1-\beta}
\end{split}
\end{align}
\end{proof}
\begin{Lem}\label{lem:mirror}
Suppose we play $\texttt{OMD}_{\beta,\eta}$ with regularizer $\phi_\beta$ the negative Tsallis entropy and initialization $\*x_1\in\triangle_d$ on the sequence of linear loss functions $\ell_1,\dots,\ell_T\in[0,1]^d$.
Then for any $\*x^\ast\in\triangle_d$ we have
\begin{equation}
\sum_{t=1}^T\langle\ell_t,\*x_t-\*x^\ast\rangle
\le\frac{B_{\phi_\beta}(\*x^\ast||\*x_1)}\eta+\frac\eta\beta\sum_{a=1}^d\*x_t^{2-\beta}(a)\ell_t^2(a)
\end{equation}
\end{Lem}
\begin{proof}
Note that the following proof follows parts of the course notes by \citet{luo2017tsallis}, which we reproduce for completeness.
The OMD update at each step $t$ involves the following two steps: set $\*y_{t+1}\in\triangle_d$ s.t. $\nabla\phi_\beta(\*y_{t+1})=\nabla\phi_\beta(\*x_t)-\eta\ell_t$ and then set $\*x_{t+1}=\argmin_{\*x\in\triangle_d}B_{\phi_\beta}(\*x,\*y_{t+1})$ \citep[Algorithm~14]{hazan2015oco}.
Note that by \citet[Equation~5.3]{hazan2015oco} and nonnegativity of the Bregman divergence we have
\begin{equation}
\sum_{t=1}^T\langle\ell_t,\*x_t-\*x^\ast\rangle
\le\frac{B_{\phi_\beta}(\*x^\ast||\*x_1)}\eta+\frac1\eta\sum_{t=1}^TB_{\phi_\beta}(\*x_t||\*y_{t+1})
\end{equation}
To bound the second term, note that when $\phi_\beta$ is the negative Tsallis entropy we have
\begin{align}
\begin{split}
B_{\phi_\beta}&(\*x_t||\*y_{t+1})\\
&=\frac1{1-\beta}\sum_{a=1}^d\left(\*y_{t+1}^\beta(a)-\*x_t^\beta(a)+\frac\beta{\*y_{t+1}^{1-\beta}(a)}(\*x_t(a)-\*y_{t+1}(a)\right)\\
&=\frac1{1-\beta}\sum_{a=1}^d\left((1-\beta)\*y_{t+1}^\beta(a)-\*x_t^\beta(a)+\beta\left(\frac1{\*x_t^{1-\beta}(a)}+\frac{1-\beta}\beta\eta\ell_t(a)\right)\*x_t(a)\right)\\
&=\sum_{a=1}^d\left(\*y_{t+1}^\beta(a)-\*x_t^\beta(a)+\eta\*x_t(a)\ell_t(a)\right)
\end{split}
\end{align}
Plugging the following result, which follows from $(1+x)^\alpha\le1+\alpha x+\alpha(\alpha-1)x^2~\forall~x\ge0,\alpha<0$, into the above yields the desired bound.
\begin{align}
\begin{split}
\*y_{t+1}^\beta(a)
=\*x_t^\beta(a)\left(\frac{\*y_{t+1}^{\beta-1}(a)}{\*x_t^{\beta-1}(a)}\right)^\frac\beta{\beta-1}
&=\*x_t^\beta(a)\left(1+\frac{1-\beta}\beta\eta\*x_t^{1-\beta}(a)\ell_t(a)\right)^\frac\beta{\beta-1}\\
&\le\*x_t^\beta(a)\left(1-\eta\*x_t^{1-\beta}(a)\ell_t(a)+\frac{\eta^2}\beta\*x_t^{2-2\beta}(a)\ell_t(a)^2\right)\\
&=\*x_t^\beta(a)-\eta\*x_t(a)\ell_t(a)+\frac{\eta^2}\beta\*x_t^{2-\beta}(a)\ell_t(a)^2
\end{split}
\end{align}
\end{proof}
\newpage
\begin{Lem}\label{lem:tsallis}
For any $\rho\in(0,1/d]$ and $\*x\in\triangle_d$ s.t. $\*x(a)\ge\rho~\forall~a\in[d]$ the $\beta$-Tsallis entropy $H_\beta(\*x)=-\frac{1-\sum_{a=1}^d\*x^\beta(a)}{1-\beta}$ is $d\log\frac1\rho$-Lipschitz w.r.t. $\beta\in[0,1]$.
\end{Lem}
\begin{proof}
Let $\log_\beta x=\frac{x^{1-\beta}-1}{1-\beta}$ be the $\beta$-logarithm function and note that by \citet[Equation~6]{yamano2002tsallis} we have
$\log_\beta x-\log x=(1-\beta)(\partial_b\log_\beta x+\log_\beta x\log x)\ge0~\forall~\beta\in[0,1]$.
Then we have for $\beta\in[0,1)$ that
\begin{align}
\begin{split}
|\partial_\beta H_\beta(\*x)|
&=\left|\frac{-H_\beta(\*x)-\sum_{a=1}^d\*x^\beta(a)\log\*x(a)}{1-\beta}\right|\\
&=\frac1{1-\beta}\left|\sum_{a=1}^d\*x^\beta(a)(\log_\beta\*x(a)-\log\*x(a))\right|\\
&=\frac1{1-\beta}\sum_{a=1}^d\*x^\beta(a)(\log_\beta\*x(a)-\log\*x(a))\\
&\le\frac1{1-\beta}\left(\sum_{a=1}^d\*x(a)\right)^\beta\left(\sum_{a=1}^d(\log_\beta\*x(a)-\log\*x(a))^{\frac1{1-\beta}}\right)^{1-\beta}\\
&\le\frac1{1-\beta}\sum_{a=1}^d\log_\beta\*x(a)-\log\*x(a)\\
&\le\frac d{1-\beta}(\log_\beta\rho-\log\rho)\\
&\le-d\log\rho
\end{split}
\end{align}
where the fourth line follows by H\"older's inequality, the fifth by subadditivity of $x^a$ for $a\in(0,1]$, the sixth by the fact that $\partial_x(\log_\beta x-\log x)=x^{-\beta}-1/x\le0~\forall~\beta,x\in[0,1)$, and the last line by substituting $\beta=0$ since $\partial_\beta\left(\frac{\log_\beta\rho-\log\rho}{1-\beta}\right)
=\frac{2(\rho-\rho^\beta)-(1-\beta)(\rho^\beta+\rho)\log\rho}{\rho^\beta(1-\beta)^3}\le0~\forall~\beta\in[0,1),\rho\in(0,1/d]$.
For $\beta=1$, applying L'H\^opital's rule yields
\begin{equation}
\lim_{\beta\to1}\partial_\beta H_\beta(\*x)
=-\frac12\lim_{\beta\to1}\sum_{a=1}^d\*x^\beta(a)\log^2\*x(a)(1-(1-\beta)\log\*x(a))
=-\frac12\sum_{a=1}^d\*x(a)\log^2\*x(a)
\end{equation}
which is bounded on $[-2d/e^2,0]$.
\end{proof}
\newpage
\section{Proof of Theorem~\ref{thm:blo}}
\begin{proof}
Applying Theorem~\ref{thm:meta} with constants $D=D_{\underline\varepsilon}$, $G=4d\sqrt 2$, $M=1$, $S=S_{\underline\varepsilon}$, and $K=K$ yields
\begin{align}
\begin{split}
\mathbb E&\sum_{t=1}^T\sum_{i=1}^m\langle\ell_{t,i},\*x_{t,i}-\*x_t^\ast\rangle\\
&\le\mathbb E\sum_{t=1}^T\varepsilon_tm+\sum_{i=1}^m\langle\ell_{t,i},\*x_{t,i}-\operatorname{OPT}_{\varepsilon_t}(\ell_t)\rangle\\
&=\mathbb E\sum_{t=1}^T\varepsilon_tm+\sum_{i=1}^m\langle\hat\ell_{t,i},\*x_{t,i}-\operatorname{OPT}_{\varepsilon_t}(\ell_t)\rangle\\
&\le\mathbb E\sum_{t=1}^T\varepsilon_tm+\sum_{i=1}^m\langle\hat\ell_{t,i},\*x_{t,i}-\hat{\*x}_t^{(\theta_t)}\rangle\\
&\le\mathbb E\sum_{t=1}^T\frac{B_\phi(\hat{\*x}_t^{(\theta_t)}||\*x_{t,1}^{(\theta_t)})}{\eta_t}+(\eta_tG^2+\varepsilon_t)m\\
&\le\left(\sqrt m+\frac{4D_{\underline\varepsilon}G}\rho\right)\sqrt{6mT\log k}+\frac{8S_{\underline\varepsilon}K^2G\sqrt m}{\rho D_{\underline\varepsilon}}(1+\log T)+\rho D_{\underline\varepsilon}GT\sqrt m\\
&\qquad+\min_{\*x\in\mathcal K,\eta>0,\varepsilon\in[\underline\varepsilon,\overline\varepsilon]}\mathbb E\sum_{t=1}^T\frac{B_\phi(\operatorname{OPT}_\varepsilon(\hat\ell_t)||\*x)}\eta+(\eta G^2+\varepsilon)m\\
&\le72d\sqrt m\sqrt[4]T\left(D_{\underline\varepsilon}\sqrt{mT\log k}+\frac{S_{\underline\varepsilon}K^2}{D_{\underline\varepsilon}}(1+\log T)\right)\\
&\qquad+\min_{\*x\in\mathcal K,\eta>0,\varepsilon\in[\underline\varepsilon,\overline\varepsilon]}\mathbb E\sum_{t=1}^T\frac{B_\phi(\operatorname{OPT}_\varepsilon(\hat\ell_t)||\*x)}\eta+(32\eta d^2+\varepsilon)m\\
\end{split}
\end{align}
where the third inequality follows from Lemma~\ref{lem:concordant}.
\end{proof}
\begin{Lem}\label{lem:concordant}
Let $\overline\mathcal K\subset\mathbb R^d$ be a convex set and $\phi$ be a self-concordant barrier.
Suppose $\ell_1,\dots,\ell_T$ are a sequence of loss functions satisfying $|\langle\ell_t,\*x\rangle|\le1~\forall~\*x\in\mathcal K$.
Then if we run OMD with step-size $\eta>0$ as in \citet[Algorithm~1]{abernethy2008competing} on the sequence of estimators $\hat\ell_t$ our estimated regret w.r.t. any $\*x^\ast\in\mathcal K_\varepsilon$ for $\varepsilon>0$ will satisfy
\begin{equation}
\sum_{t=1}^T\langle\hat\ell_t,\*x_t-\*x^\ast\rangle\le\frac{B_\phi(\*x^\ast||\*x_1)}\eta+32d^2\eta T
\end{equation}
\end{Lem}
\begin{proof}
The result follows from \citet{abernethy2008competing} by stopping the derivation on the second inequality below Equation~10.
\end{proof}
\section{Related work}
While we are the first to consider meta-learning under adversarial bandit feedback, many have studied meta-learning in various {\em stochastic} bandit settings~\citep{sharaf2021meta, simchowitz2021bayesian, kveton2020meta, cella2020meta, kveton2021meta, basu2021no, azizi2022non, lazaric2013sequential}. \citet{kveton2021meta}, \citet{basu2021no}, and \citet{simchowitz2021bayesian} study meta-learning algorithms for the Bayesian bandit setting. \citet{kveton2020meta} and \citet{sharaf2021meta} consider meta-learning for contextual bandits, although they allow their algorithms to have offline access to a set of training tasks for which full feedback is available. \citet{cella2020meta} and \citet{moradipari2022multi} provide algorithms based on OFUL~\citep{abbasi2011improved} for meta-learning in stochastic linear bandits under various assumptions on how the bandit learning tasks are generated. \citet{azizi2022non} study a setting in which a meta-learner faces a sequence of stochastic multi-armed bandit tasks. While the sequence of tasks may be adversarially designed, the adversary is constrained to choose the optimal arm for each task from a smaller but unknown subset of arms. In contrast to \cite{cella2020meta, moradipari2022multi, azizi2022non}, we make no assumptions about how the sequence of tasks is generated and our guarantees adapt to a natural measure of similarity between tasks.
Theoretically our analysis draws on the average regret-upper-bound analysis (ARUBA) framework of \citet{khodak2019adaptive}, which was designed for meta-learning under full information.
While the general approach is not restricted by convexity~\citep{balcan2021ltl} and has been combined with bandit algorithms on the meta-level~\citep{khodak2021fedex}, the existing results cannot be applied to OMD methods for within-task learning under bandit feedback because the associated regularizers are non-Lipschitz or sometimes even unbounded near the boundaries of the action space.
We thus require a specialized analysis for the bandit setting.
\citet{denevi2019meta} also study an OMD-based algorithm for meta-learning in the online setting, but their results are also only applicable in the full information setting.\looseness-1
\section{Learning the regularizers of bandit algorithms}\label{sec:setup}
We consider the problem of meta-learning across bandit tasks $t=1,\dots,T$ over some fixed set $\mathcal K\subset\mathbb R^d$.
On each round $i=1,\dots,m$ of task $t$ we play action $\*x_{t,i}\in\mathcal K$ and receive feedback $\ell_{t,i}(\*x_{t,i})$ for some function $\ell_{t,i}:\mathcal K\mapsto[-1,1]$.
Note that all functions we consider will be linear and so we will also write $\ell_{t,i}(\*x)=\langle\ell_{t,i},\*x\rangle$. Additionally, we allow each $\ell_{t,i}$ to be chosen by an \emph{oblivious adversary}, i.e. an adversary with knowledge of the algorithm that must select $\ell_{t,i}$ independent of $\*x_{t,i}$.
We will also denote $\*x(a)$ to be the $a$th element of the vector $\*x\in\mathbb R^d$, $\overline\mathcal K$ to be the convex hull of $\mathcal K$, and $\triangle_n$ to be the simplex on $n$ elements.
Finally, note that all proofs can be found in the Appendix.\looseness-1
In online learning, the goal on a single task $t$ is to play actions $\*x_{t,1},\dots\*x_{t,m}$ that minimize the regret $\sum_{i=1}^m\ell_{t,i}(\*x_{t,i})-\ell_{t,i}(\*x_t^\ast)$, where $\*x_t^\ast\in\argmin_{\*x\in\mathcal K}\sum_{i=1}^m\ell_{t,i}(\*x)$.
Lifting this to the meta-learning setting, our goal as in past work \citep{khodak2019adaptive,balcan2021ltl} will be to minimize the {\bf task-averaged regret}
\begin{equation}\label{eq:tar}
\frac1T\sum_{t=1}^T\sum_{i=1}^m\ell_{t,i}(\*x_{i,t})-\ell_{t,i}(\*x_t^\ast)
\end{equation}
In-particular, we hope to use multi-task data in order to improve average performance as the number of tasks $T\to\infty$, e.g. by attaining a task-averaged regret of $o_T(1)+\tilde\mathcal O(V\sqrt m)$, where $V\in\mathbb R_{\ge0}$ is a measure of task-similarity that is small if the tasks are similar but still yields the worst-case single-task performance if they are not.
\subsection{Online mirror descent as a base-learner}\label{ssec:mirror}
In meta-learning we are commonly interested in learning a within-task algorithm or {\bf base-learner}, a parameterized method that we run on each task $t$.
A popular approach, both empirically~\citep{finn2017maml,nichol2018reptile} and theoretically~\citep{khodak2019adaptive,denevi2019ltlsgd}, is to learn the initialization and sometimes other parameters of a gradient-based method such as stochastic gradient descent.
The hope is that optimal parameters for each task are close to each other and thus a meta-learned initialization will result in a strong model after only a few steps.
In this paper we take a similar approach applied to online mirror descent, a generalization of gradient descent to non-Euclidean geometries~\citep{beck2003mirror}.
Given a strictly convex {\bf regularizer} $\phi:\overline\mathcal K\mapsto\mathbb R$ and step-size $\eta>0$, this method performs the update\looseness-1
\begin{equation}\label{eq:mirror}
\*x_{t,i+1}=\argmin_{\*x\in\overline\mathcal K}B_\phi(\*x||\*x_{t,1})+\eta\sum_{j<i}\langle\nabla\ell_{t,j}(\*x_{t,j}),\*x\rangle
\end{equation}
where $B_\phi(\*x||\*y)=\phi(\*x)-\phi(\*y)-\langle\nabla\phi(\*y),\*x-\*y\rangle$ is the {\bf Bregman divergence} of $\phi$.
OMD recovers online gradient descent when $\phi(\*x)=\frac12\|\*x\|_2^2$, in which case $B_\phi(\*x||\*y)=\frac12\|\*x-\*y\|_2^2$;
another important example is {\bf exponentiated gradient}, for which $\phi(\*p)=\langle\*p,\log\*p\rangle$ is the negative Shannon entropy on probability vectors $\*p\in\triangle_n$ and $B_\phi$ is the KL-divergence~\citep{shalev-shwartz2011oco}.
An important property of $B_\phi$ is that the sum over functions $B_\phi(\*x_t||\cdot)$ is minimized at the mean $\bar{\*x}$ of the points $\*x_1,\dots,\*x_T$.
While originally developed for online convex optimization, mirror descent using {\bf loss estimators} $\hat\ell_{t,i}$ constructed using bandit feedback $\ell_{t,i}(\*x_{t,i})$ forms an important class of methods for bandit settings~\citep{abernethy2008competing,neu2015explore,abernethy2015fighting}, including the famous Exp3 method \citep{auer2002exp3}.
Learning the initialization for online mirror descent has been considered for full-information meta-learning \citep{khodak2019adaptive,denevi2019meta}, but these papers do not apply to the types of regularizers $\phi$ required for bandits, which are often non-Lipschitz and sometimes even unbounded on the boundary of $\overline\mathcal K$ \citep{abernethy2008competing,abernethy2015fighting}.
For example, \citet{khodak2019adaptive} also take advantage of the mean-as-minimizer property of $B_\phi$ and learn both the initialization and step-size $\eta$, but they assume the gradient of $B_\phi$ is bounded on the domain, which does not hold if $\phi$ is non-Lipschitz, e.g. if $\phi$ is the negative Shannon entropy as in Exp3.
In this paper we resolve these issues by meta-learning to initialize and tune mirror descent when it employs a regularizer used by bandit methods.
Following the average regret-upper-bound analysis (ARUBA) framework of \citet{khodak2019adaptive}, we do this by online learning a sequence of losses $U_t(\*x,\theta)$, each of which is a hyperparameter $\theta$-dependent affine function of a Bregman divergence from an initialization $\*x\in\overline\mathcal K$ to some known fixed point in $\overline\mathcal K$.
We are interested in learning such functions because the regret after $m$ rounds of OMD initialized at $\*x$ with step-size $\eta$ is usually upper-bounded by $\frac1\etaB_\phi(\*x_t^\ast||\*x)+\mathcal O(\eta m)$ for $\*x_t^\ast$, the optimum-in-hindsight on task~$t$~\citep{shalev-shwartz2011oco,hazan2015oco}.\looseness-1
Unlike past work, we use a parameter $\varepsilon>0$ to constrain this optimum to lie in a convex subset $\mathcal K_\varepsilon\subset\overline\mathcal K$ whose boundary is $\varepsilon$-away from that of $\overline\mathcal K$ and which satisfies $\mathcal K_\varepsilon\subset\mathcal K_{\varepsilon'}$ whenever $\varepsilon\le\varepsilon'$;
for example, we use $\mathcal K_\varepsilon=\{\*x\in\triangle_d:\min_a\*x(a)\ge\varepsilon/d\}$ for the simplex.
Thus, unlike with full-information, the feedback we receive from the within-task algorithm will be the minimizer $\operatorname{OPT}_\varepsilon(\hat\ell_t)=\argmin_{\*x\in\mathcal K_\varepsilon}\langle\hat\ell_t,\*x\rangle$ of the estimated loss $\hat\ell_t=\sum_{i=1}^m\hat\ell_{t,i}$ over the $\varepsilon$-constrained subset, where we can pick $\varepsilon\in(0,1)$.
This allows us to handle regularizers that diverge near the boundary, but also introduces $\varepsilon$-dependent error terms to handle.
In the BLO case it also forces us to automatically tune $\varepsilon$ itself, as initializing too close to the boundary leads to unbounded regret while initializing too far away does not take advantage of the task-similarity.
Thus in full generality the upper bounds of interest are functions of the initialization $\*x$ and three parameters:
the step-size $\eta>0$, a parameter $\beta$ of the regularizer $\phi_\beta$, and the boundary offset $\varepsilon>0$.
\begin{equation}\label{eq:rub}
U_t(\*x,(\eta,\beta,\varepsilon))
=\frac{B_{\phi_\beta}(\operatorname{OPT}_\varepsilon(\hat\ell_t)||\*x)}\eta+(\eta G_\beta^2+C\varepsilon)m
\end{equation}
Here $G_\beta\ge1,C\ge0$ are constants and $\beta$ parameterizes the regularizer $\phi_\beta$, e.g. the negative Tsallis entropy used to attain optimal dependence on dimension for MAB~\citep{abernethy2015fighting}.
The reason to optimize this sequence of upper bounds is because the resulting average regret directly bounds the task-averaged regret, apart from some $o_T(1)$ terms.
Furthermore, an affine sum over Bregman divergences is minimized at the average optimum in hindsight, which leads to natural and problem specific task-similarity measures $V$ \citep{khodak2019adaptive};
specifically, $V$ is the square root of the average divergence between optima in hindsight and their mean, which is small if the tasks are optimized by similar parameters.\looseness-1
\subsection{A meta-algorithm for tuning bandit algorithms}\label{ssec:algo}
Having specified our meta-goal---learning to initialize and tune OMD for regularizers $\phi_\beta$ use in bandit tasks---we now detail our meta-algorithm for doing so, pseudo-code for which is in Algorithm~\ref{alg:meta}.
At a high level, the method simultaneously learns the initialization by taking the mean of $\mathcal K_\varepsilon$-constrained estimated optima-in-hindsight---i.e. follow-the-leader over the Bregman divergences in \eqref{eq:rub}---while simultaneously tuning OMD via multiplicative weights over a discrete grid $\Theta$ over $\theta=(\eta,\beta,\varepsilon)$.
In more detail, the algorithm assumes two primitives discussed above:
(1) the base-learner $\texttt{OMD}_{\eta,\beta}$ that outputs an estimated cumulative loss $\hat\ell_t\in\mathbb R^d$ after running online mirror descent over the $m$ losses $\ell_{t,1},\dots,\ell_{t,m}$ of task $t$, and (2) an optimizer $\operatorname{OPT}_\varepsilon$ that, given a vector $\*c\in\mathbb R^d$, finds the minimum of $\langle\*c,\cdot\rangle$ over $\mathcal K_\varepsilon$.
Algorithm~\ref{alg:meta} maintains a categorical distribution $\*p_t$ over a finite set $\Theta\subset\mathbb R^3$ containing triples $\theta=(\eta,\beta,\varepsilon)$, each with its own associated initialization $\*x_t^{(\theta)}$;
at each task $t$ it samples $\theta_t=(\eta_t,\beta_t,\varepsilon_t)$ from $\Theta$ using $\*p_t$ and runs $\texttt{OMD}_{\eta_t,\beta_t}$ from initialization $\*x_t^{(\theta_t)}$, obtaining a loss estimate $\hat\ell_t$.
Then for each $\theta=(\eta,\beta,\varepsilon)$ in $\Theta$ the method updates the corresponding initialization $\*x_t^{(\theta)}$ by taking the average of the $\varepsilon$-constrained optima-in-hindsight $\operatorname{OPT}_\varepsilon(\hat\ell_1),\dots,\operatorname{OPT}_\varepsilon(\hat\ell_t)$ seen so far.
Finally, the algorithm updates the distribution $\*p_t$ using multiplicative weights over the following modification of the regret-upper-bound \eqref{eq:rub} above for some $\rho>0$:
\begin{align}\label{eq:modrub}
U_t^{(\rho)}(\*x,\theta)=\frac{B_{\phi_\beta}(\hat{\*x}_t^{(\theta)}||\*x)+\rho^2D^2}\eta+(\eta G_\beta^2+C\varepsilon)m
\end{align}
Note that given $\rho>0$ this function is fully defined after running $\texttt{OMD}_{\eta_t,\beta_t}$ on task $t$ to obtain loss estimates $\hat\ell_t$ and then computing the $\varepsilon$-constrained optimum-in-hindsight $\hat{\*x}_t^{(\theta)}=\operatorname{OPT}_\varepsilon(\hat\ell_t)$ for each $\theta=(\eta,\beta,\varepsilon)$.
This allows us to use full-information multiplicative weights for $\theta$.
$\rho>0$ is necessary for learning $\eta$, as if its optimum is near zero then $U_t$ will not be Lipschitz near the optimum.
Theorem~\ref{thm:meta} shows a sublinear regret guarantee for Algorithm~\ref{alg:meta} over the unmodified regret-upper-bounds \eqref{eq:modrub} w.r.t. all elements in $\overline\mathcal K$ and in a continous set of hyperparameters $\Theta^\ast\subset\mathbb R^3$.
\begin{algorithm}[!t]
\DontPrintSemicolon
\KwIn{compact $\overline\mathcal K\subset\mathbb R^d$,
meta-hyperparameters $\alpha,\rho>0$,
finite $\Theta\subset\mathbb R^3$ over $(\eta,\beta,\varepsilon)$,
base-learner $\texttt{OMD}_{\eta,\beta}:\overline\mathcal K\mapsto\mathbb R^d$,
constrained linear minimizer $\operatorname{OPT}_\varepsilon:\mathbb R^d\mapsto\mathcal K_\varepsilon$}
\For{$\theta=(\eta,\beta,\varepsilon)\in\Theta$}{
$\*x_1^{(\theta)}\gets\argmin_{\*x\in\overline\mathcal K}\phi(\*x)$\tcp*{maintain an initialization for each $\theta\in\Theta$}
}
$\*p_1\gets\*1_{|\Theta|}/|\Theta|$\tcp*{multiplicative weights (MW) initialization}
\For{task $t=1,\dots,T$}{
sample $\theta_t=(\eta_t,\beta_t,\varepsilon_t)\sim\*p_t$ from $\Theta$\\
$\hat\ell_t\gets\texttt{OMD}_{\eta_t,\beta_t}(\*x^{(\theta_t)})$\tcp*{run bandit OMD within-task}
\For{$\theta=(\eta,\beta,\varepsilon)\in\Theta$}{
$\*x_{t+1}^{(\theta)}\gets\frac1t\sum_{s=1}^t\operatorname{OPT}_\varepsilon(\hat\ell_s)$\tcp*{update all initializations}
$\*p_{t+1}(\theta)\gets\*p_{t+1}(\theta)\exp\left(-\alpha U_t^{(\rho)}(\*x_t^{(\theta)},\theta)\right)$\tcp*{MW update using loss in \eqref{eq:modrub}}
}
$\*p_{t+1}\gets\*p_{t+1}/\|\*p_{t+1}\|_1$
}
\caption{\label{alg:meta}
Algorithm for tuning an online mirror descent (OMD) base-learner $\texttt{OMD}_{\eta,\beta}$ with parameterized regularizer $\phi_\beta:\overline\mathcal K\mapsto\mathbb R$ and step-size $\eta>0$ that runs OMD on loss estimators $\hat\ell_{t,1},\dots,\hat\ell_{t,m}$ from an initialization $\*x\in\overline\mathcal K$ and returns estimated loss $\hat\ell_t=\sum_{i=1}^m\hat\ell_{t,i}\in\mathbb R^d$.
Then for every $\varepsilon>0$ the constrained optimizer $\operatorname{OPT}_\varepsilon(\hat\ell)=\argmin_{\*x\in\mathcal K_\varepsilon}\langle\hat\ell,\*x\rangle$ returns the minimizer of the estimated loss over the constrained subset $\mathcal K_\varepsilon\subset\overline\mathcal K$ (set $\operatorname{OPT}_\varepsilon(\*0_d)=\argmin_{\*x\in\overline\mathcal K}\phi(\*x)$).
}
\end{algorithm}
\begin{Thm}\label{thm:meta}
Let $\Theta^\ast=(0,\infty)\times[\underline\beta,\overline\beta]\times[\underline\varepsilon,\overline\varepsilon]$ for $0\le\underline\beta\le\overline\beta\le1$ and $0\le\underline\varepsilon\le\overline\varepsilon\le1$ be the set of hyperparameters $(\eta,\beta,\varepsilon)$ of interest.
Then there exists integer $k=\mathcal O(\lceil\sqrt{mT}\rceil)$ and $\alpha,\underline\eta,\overline\eta\in(0,\infty)$ such that running Algorithm~\ref{alg:meta} with $\Theta$ the product of uniform grids of size $k$ over each non-singleton dimension of $[\underline\eta,\overline\eta]\times[\underline\beta,\overline\beta]\times[\underline\varepsilon,\overline\varepsilon]$ and $\alpha$ the meta-step-size yields regret
\begin{align}\label{eq:metareg}
\begin{split}
\mathbb E&\sum_{t=1}^TU_t(\*x_t^{(\theta_t)},\theta_t)-\min_{\*x\in\overline\mathcal K,\theta\in\Theta^\ast}\sum_{t=1}^TU_t(\*x,\theta)
\\
&\le\left(C\sqrt m+2DG\left(\frac1\rho+M\right)\right)\sqrt{6mT\log k}+\frac{8SK^2G\sqrt m}{\rho D}(1+\log T)+\rho DGT\sqrt m
\end{split}
\end{align}
for $G=\max_\beta G_\beta\ge1$,
$M=\frac G{\min_\beta G_\beta}$, $D^2=\max_{\beta,\varepsilon,\*x,\*y\in\mathcal K_\varepsilon}B_{\phi_\beta}(\*x||\*y)\ge1$,
$L$ the maximum Lipschitz constant of $\phi_\beta(\operatorname{OPT}_\varepsilon(\ell))$ w.r.t. $(\beta,\varepsilon)$ over $\ell\in\mathbb R^d$, $S=\max_{\beta,\varepsilon,\*x\in\mathcal K_\varepsilon}\|\nabla^2\phi_\beta(\*x)\|_2$,
$K=\max_{\*x,\*y\in\mathcal K}\|\*x-\*y\|_2$,
and the expectation is over sampling $\theta_t\sim\*p_t$.
The result without the expectation holds w.p. $1-\delta$ at the cost of an additional $\left(C\sqrt m+2DG\left(\frac1\rho+M\right)\right)\sqrt{\frac T2\log\frac1\delta}$ term.\looseness-1
\end{Thm}
\begin{proof}[Proof sketch]
At a high-level, we use the $\mathcal O(\log T)$ regret of follow-the-leader over Bregman divergences~\citep{khodak2019adaptive} for the initialization, the $\mathcal O(\sqrt{T\log k})$ regret of multiplicative weights over $k$ experts~\citep{shalev-shwartz2011oco} to tune over a large grid of hyperparameters, the fact that $U_t$ is an affine function of the Bregman divergence to combine the two methods, and the identity $\sum_{t=1}^TB_\phi(\*x_t||\bar{\*x})=\sum_{t=1}^T\phi(\*x_t)-\phi(\bar{\*x})$ to bound the discretization error.
The w.h.p. result follows by \citet[Lemma~4.1]{cesa-bianchi2006prediction}.
\end{proof}
Note that we keep details of the dependence on values like Lipschitz constants because they are important in applying this result;
however, in general setting $\rho=1/\sqrt[4]T$ in \eqref{eq:metareg} yields $\tilde\mathcal O(T^\frac34)$-regret.
While a slow rate, note that Algorithm~\ref{alg:meta} is learning a sequence of affine functions of Bregman divergences that are non-smooth and non-convex in-general.
Theorem~\ref{thm:meta} is an important structural result;
our main contributions to multi-armed and linear bandits follow by applying its instantiations for specific regularizers $\phi$ and hyperparameter sets $\Theta^\ast$.
We also believe Theorem~\ref{thm:meta} may be of independent interest as it holds for any choice of Bregman divergence beyond those we consider, and unlike past work \citep{khodak2019adaptive} allows for explicit control of non-smooth regularizers near the boundaries.
The theorem allows tuning the hyperparameters over user-specified intervals for $\beta$ and $\varepsilon$ and over an infinite interval for the step-size $\eta>0$.
Note that a similar result is straightforward to show for $\beta$ outside $[0,1]$ or for discrete rather than continuous set of hyperparameters.
|
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{"url":"https:\/\/codereview.stackexchange.com\/questions\/86787\/transforming-an-array-in-ruby","text":"# Transforming an array in Ruby\n\nI watched a presentation by Dave Thomas on Elixir where he gave an example problem that he solved using functional programming. He mentioned using Ruby to solve the same problem, but did not show an example. I decided to give it a try. The problem goes something like this:\n\nFor a list (Ruby doesn't have lists, but for our purposes an Array is close enough) of n numbers, create a new list where unique numbers are represented once, and repeated numbers are represented by a tuple (again, Ruby doesn't have tuples, but for our purposes a Hash or an Array would do) where the first element is the number itself, and the second is its count.\n\nFor Example, the following list:\n\n[ 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 6, 6 ]\n\n\nWould become:\n\n[ 1, {2, 3}, 3, {4, 2}, 5, {6, 3} ]\n\n\nI came up with the following solution.\n\nlist = [ 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 6, 6 ]\nlist.each_with_object(Hash.new(0)) { |n, hash| hash[n] += 1 }.map { |k, v| v > 1 ? { k => v } : k }\n#=> [ 1, {2 => 3}, 3, {4 => 2}, 5, {6 => 3} ]\n\n\nHow would you go about solving this problem?\n\nSome notes:\n\n\u2022 From the moment you write hash[n] += 1, your solution is not functional anymore.\n\u2022 each_with_object is also imperative, for FP you should use reduce.\n\u2022 reduce works well with linked lists, not so well for generating new arrays. In any case, reduce is a generic abstraction, there is a more specific one to the problem at hand: Enumerable#chunk.\n\nThat's how I'd write it:\n\nxs.chunk(&:itself).map { |y, ys| ys.size == 1 ? y : {y => ys.size} }\n\n\u2022 Thanks great. I am wondering about &:itself, there's no mention of it in the docs. Is that a shortcut or a keyword? \u2013\u00a0Mohamad Apr 13 '15 at 20:10\n\u2022 ruby-doc.org\/core-2.2.1\/Object.html#method-i-itself \u2013\u00a0tokland Apr 13 '15 at 20:11\n\u2022 Ahhh, ok, I was looking at Enumerable. So I suppose &:itself translates to chunk { |n| n.itself }, also the same as chunk { |n| n } \u2013\u00a0Mohamad Apr 13 '15 at 20:12\n\u2022 @Mohamad that is correct, and by extension &:any_method is translated the same way \u2013\u00a0Devon Parsons Apr 14 '15 at 17:24\n\u2022 Immutability is a key concept of functional programming. You can't change anything in-place (add, update, remove, and so on). goo.gl\/4OyjP \u2013\u00a0tokland Apr 15 '15 at 18:46\n\nI would write a slower but easier to read version:\n\ndef same_or_tuple(list)\nlist.map {|x| if list.count(x) == 1 then x\nelse [x, list.count(x)] end}\n.uniq\nend","date":"2019-12-10 17:33:29","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\": 0.320523202419281, \"perplexity\": 1642.5076458376182}, \"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-2019-51\/segments\/1575540528457.66\/warc\/CC-MAIN-20191210152154-20191210180154-00546.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.r-bloggers.com\/2013\/05\/strategic-zombie-simulation-animation\/","text":"Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\n# Escape Zombie Land!\n\n# This is a simulation an escape from a hot zombie zone. It freezes and gives an error if you get get killed so you had best not. You attempt to navigate the zone by constructing waypoints.\n\n# This is not a very clean set up and I would like to clean it up. However, I have spent way more time on it than I intended. So I might come back to it another day.\n\n# Zombies are distributed on a 10 x 10 grid.\ngridxy = c(10,10)\n\n# The number of zombies on the map\nnzombies = 40\n\n# How close a zombie needs to be to take out a human is defined here\nsame.space = .05\n\n# This is how close a human needs to be to consider that the human has reached the waypoint.\nwaypoint.hit = .2\n\n# I set up the zombie distribution randomly initially.\nset.seed(1)\nzombiexy = cbind(runif(nzombies)*gridxy[1], runif(nzombies)*gridxy[2])\nplot(zombiexy, main=\"Zombies!\", xlab=\"X\", ylab=\"Y\", col=grey(.2), xlim=c(0,gridxy[1]), ylim=c(0,gridxy[2]))\n\n# Humans\nstartpoint = c(.5,.5)\nhumans = data.frame(x=c(0,-.25, .25), y=c(0,.25, -.25), name=c(\"You\",\"Pete\", \"Jimmy\"))\n\nhumansxy = humans[,1:2]\n\n# Count humans\nnhumans = nrow(humansxy)\n\n(humansxy = humansxy+rep(startpoint, each=nhumans))\n\n# Plot humans\npoints(humansxy, pch=8)\n\n# Safety\nsafety = c(9.5,9.5)\n\n# Waypoints, specify the waypoints the humans are to take to get to the destination.\nwaypoints = rbind(c(2.5,2), c(5,6), c(9.75, 7))\n\n# Route\nroute = rbind(startpoint, waypoints, safety, safety, safety)\nlines(route)\n\n# A vector that will be shortenned as the simulation progresses\nroute.unreached = route\n\npoints(safety[1], safety[2], pch=7)\n\n# Now let's imagine that each zombie has a sensory distance in which the zombie can detect humans.\ndetection.dist = 3\n\n# How fast the zombies can move. Zombies have no inertia.\nzombie.acceleration = .075\n\n# How fast humans can move\nhuman.acceleration = .075\n\n# Humans can outrun zombies by building up inertia\nhuman.inertia = .6\n\n# Initially everybody is at rest.\nhmovement = zmovement = 0\n\n# ---------------------------------------------------\n#### Set up a single loop to check programming.\n\n# First the zombies move\n\n# First let's check how close each zombie is to each human.\n# We will accomplish this by going through each zombie and checking how far away each zombie is from each human.\ndistances = matrix(NA, nrow=nzombies, ncol=nhumans)\nfor (i in 1:nzombies) for (ii in 1:nhumans) distances[i,ii] = (sum((zombiexy[i,]-humansxy[ii,])^2))^.5\n\ntarget = matrix(1:nrow(humansxy), ncol=nzombies, nrow=nrow(humansxy))[apply(distances, 1, order)[1,]]\n# The apply command will apply the order command to each row while the [1,] selects only the critter that is closes.\n\nplot(zombiexy, xlab = \"X\", ylab = \"Y\", main=\"If zombies did not have perception limitations\")\nfor (i in 1:nzombies) arrows(x0=zombiexy[i,1], y0=zombiexy[i,2],\nx1=humansxy[target,][i,1],\ny1=humansxy[target,][i,2],\nlength=.1, col=\"red\")\n\n points(humansxy, pch=8)\n\n# Safety\npoints(9.5,9.5, pch=7)\n\n# However, if the target is outside of detection range then zombies cannot target that human.\ntarget[distances[cbind(1:nzombies,target)]>detection.dist]=NA\n\n# Plot the relationship between zombies and humans\nplot(zombiexy, xlab = \"X\", ylab = \"Y\", main=\"Escape Zombie Land\")\nfor (i in 1:nzombies) arrows(x0=zombiexy[i,1], y0=zombiexy[i,2],\nx1=humansxy[target,][i,1],\ny1=humansxy[target,][i,2],\nlength=.1, col=\"red\")\n# Plot humans\npoints(humansxy, pch=8)\n\n# Safety\npoints(9.5,9.5, pch=7)\n\n# This calculates the difference between the current position of each zombie and that of the closest human.\nab = zombiexy-humansxy[target,]\n\nab=ab[!is.na(target),]\n\n# Now calculate the difference in the horizontal and vertical axes that the zombies will move as a projection into the direction of the closest zombie outside of the perceptive zone.\na.prime = zombie.acceleration\/(1 + (ab[,2]^2)\/(ab[,1]^2))^.5\nb.prime = (zombie.acceleration^2-a.prime^2)^.5\n\n# This corrects the movement to ensure that the zombies are moving at the humans rather than away from them.\nzmovement = cbind(a.prime * sign(ab[,2]), b.prime * sign(ab[,1]))\nbetween = function(xy1,xy2,point) ((point>xy1&pointxy2&point1) warntxt = paste(humans[humans.down,3], \"are down!\")\n\n# Remove any \"captured\" humans\nif (length(humans.down)>0) {\nhumansxy = humansxy[-humans.down,]\nnhumans = nrow(humansxy)\n}\n\n# Now the surving humans get to move.\n\n# However, we only calculate the movement for the leader (you) since all of the other humans move in parrellel to you.\n\n# Movement is also much simpler since humans just run from one waypoint to the next.\n\n# First we check if we have reached any waypoints (which we have since we start on one).\nway.distance =\n(sum((humansxy[1,]-route.unreached[1,])^2))^.5\n\nif (way.distancexy1&pointxy2&point2) {\n\n# First let's check how close each zombie is to each human.\n# We will accomplish this by going through each zombie and checking how far away each zombie is from each human.\ndistances = matrix(NA, nrow=nzombies, ncol=nhumans)\nfor (i in 1:nzombies) for (ii in 1:nhumans) distances[i,ii] = (sum((zombiexy[i,]-humansxy[ii,])^2))^.5\n\nif (nrow(humansxy)>1) target = matrix(1:nrow(humansxy), ncol=nzombies, nrow=nrow(humansxy))[apply(distances, 1, order)[1,]]\nif (nrow(humansxy)==1) matrix(1, ncol=nzombies, nrow=1)\n# The apply command will apply the order command to each row while the [1,] selects only the critter that is closes.\n\ntarget[distances[cbind(1:nzombies,target)]>detection.dist]=NA\n\n# Plot the relationship between zombies and humans\nplot(0,0, type=\"n\", xlab = \"X\", ylab = \"Y\", main=\"Escape Zombie Land\", xlim=c(0,gridxy[1]), ylim=c(0,gridxy[2]))\n\n# Safety\npoints(9.5,9.5, pch=7)\n\ntext(5,.25,warntxt)\n\n# This calculates the difference between the current position of each zombie and that of the closest human.\nab = zombiexy-humansxy[target,]\n\nab=ab[!is.na(target),]\n\n# Now calculate the difference in the horizontal and vertical axes that the zombies will move as a projection into the direction of the closest zombie outside of the perceptive zone.\na.prime = zombie.acceleration\/(1 + (ab[,2]^2)\/(ab[,1]^2))^.5\nb.prime = (zombie.acceleration^2-a.prime^2)^.5\n\n# This corrects the movement to ensure that the zombies are moving at the humans rather than away from them.\nzmovement = cbind(a.prime * sign(ab[,2]), b.prime * sign(ab[,1]))\nbetween = function(xy1,xy2,point) ((point>xy1&pointxy2&point1) warntxt = paste(humans[humans.down,3], \"are down!\")\n\n# Remove any \"captured\" humans\nif (length(humans.down)>0) {\nhumansxy = humansxy[-humans.down,]\nnhumans = nrow(humansxy)\n}\n\n# Now the surving humans get to move.\n\n# However, we only calculate the movement for the leader (you) since all of the other humans move in parrellel to you.\n\n# Movement is also much simpler since humans just run from one waypoint to the next.\n\n# First we check if we have reached any waypoints (which we have since we start on one).\nway.distance = (sum((humansxy[1,]-route.unreached[1,])^2))^.5\n\nif (way.distancexy1&pointxy2&point\n# Let's see how we do at escaping zombie land\nani.options(ani.width=600, ani.height=600, interval=.25)\nsaveGIF(flocking( ani.pause=T), movie.name = \"Zombies.gif\", replace=T)\n\nvar vglnk = {key: '949efb41171ac6ec1bf7f206d57e90b8'};\n(function(d, t) {\nvar s = d.createElement(t);\ns.type = 'text\/javascript';\ns.async = true;\n\/\/ s.defer = true;\nvar r = d.getElementsByTagName(t)[0];\nr.parentNode.insertBefore(s, r);\n}(document, 'script'));\n\nRelated\nShareTweet","date":"2021-05-15 10:52:27","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\": 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\": 0.45125752687454224, \"perplexity\": 7754.985085240259}, \"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\/1620243991801.49\/warc\/CC-MAIN-20210515100825-20210515130825-00459.warc.gz\"}"}
| null | null |
Ethiopian Refugees Worry About COVID-19 Outbreak in Sudanese Camps
QADARIF, UM RAKOUBA REFUGEES CAMP - Thousands of Ethiopians who fled fighting in Tigray for camps in Sudan fear a new threat - a COVID-19 outbreak. While there are no confirmed cases, concern is running high.
The more than 40,000 Ethiopians who left Tigray for eastern Sudan over the past month have been placed in three camps in the cities of Qadarif and Kassala.
Tesfai Alley, 32, and his pregnant wife fled the fighting, but are now worried about the spread of COVID-19 in the camp due to the low living standards, crowded rooms and a shortage of water.
Social distancing is difficult in the camp, where at least six refugees stay in each small tent or room, and people gather in large groups to get food, water or blankets.
To help reduce the risk, health workers are teaching refugees methods to prevent the spread of the virus, and isolation centers are being built.
Isaac Yousif, a doctor at Um Rakouba refugee camp, said the densely populated camps pose a risk, should a coronavirus outbreak occur. In addition, he said, there is a possibility some suspected cases fled isolation centers in the war zone and brought the virus to the camps.
A second wave of COVID-19 is sweeping across Sudan, which reports at least 17,000 registered cases and a high mortality rate.
The government has shut down universities and schools, and is considering whether to impose more restrictions.
The U.N. refugee agency is calling for more international aid to deal with the crisis.
Refugees are trying to adjust by constructing small markets inside the camps and close to hosting villages.
Meanwhile, the Sudanese government is setting up a new refugee camp 250 kilometers from Khartoum to help ease overcrowding in the existing camps.
Get a daily dose of Baltimore Star news through our daily email, its complimentary and keeps you fully up to date with world and business news as well.
Publish news of your business, community or sports group, personnel appointments, major event and more by submitting a news release to Baltimore Star.
Fair in Baltimore
© Copyright 1999-2021 Baltimore Star. All rights reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,484
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Q: Show Icons instead of Text for Stock Availability in WooCommerce I am trying to show icons instead of text as the availability in WooCommerce. The code I have shows text, but I like it to be icons instead.
I have three icons: Red, Orange and Green for Out of stock, Half stock left and fully stocked.
Here's the code I need help changing:
add_filter( 'woocommerce_get_availability', 'dispay_custom_icons_for_availability', 1, 2);
function dispay_custom_icons_for_availability( $availability, $product ) {
global $product;
// available
if ( $product->is_in_stock() ) {
$availability['availability'] = __('GREEN ICON HERE', 'woocommerce');
}
// middle stock
if ( $product->is_in_stock() && $product->get_stock_quantity() <= 20 ) {
$availability['availability'] = sprintf( __('ORANGE ICON HERE', 'woocommerce'), $product->get_stock_quantity());
}
// out of stock
if ( ! $product->is_in_stock() ) {
$availability['availability'] = __('RED ICON HERE', 'woocommerce');
}
return $availability;
}
All help is appreciated.
A: Try the following, based on Fontawesome icons that are embedded in WooCommerce:
add_filter( 'woocommerce_get_availability', 'dispay_custom_icons_for_availability', 1, 2);
function dispay_custom_icons_for_availability( $availability, $product ) {
global $product;
// available
if ( $product->is_in_stock() ) {
$availability['availability'] = '<i class="fa fa-lg fa-smile" style="color:green;"></i>';
$availability['class'] = 'in_stock';
}
// middle stock
if ( $product->is_in_stock() && $product->get_stock_quantity() <= 20 ) {
$availability['availability'] = '<i class="fa fa-lg fa-meh" style="color:orange;"></i>';
$availability['class'] = 'low_stock';
}
// out of stock
if ( ! $product->is_in_stock() ) {
$availability['availability'] = '<i class="fa fa-lg fa-frown" style="color:red;"></i>';
$availability['class'] = 'out_of_stock';
}
return $availability;
}
Code goes in functions.php file of your active child theme (or active theme). Tested and work.
You will get one of the following icons
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,414
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%%
%% KDE4xHb - Bindings libraries for Harbour/xHarbour and KDE 4
%%
%% Copyright (C) 2020 Marcos Antonio Gambeta <marcosgambeta AT outlook DOT com>
%%
$project=KDE4xHb
$module=KDEUI
$header
$includes
$beginSlotsClass
$slot=|ratingChanged( int rating )
$endSlotsClass
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3
|
\section{A Motivating Example}
\label{sec:motex}
Consider the problem of measuring racial discrimination in prosecutorial
charging decisions. After an individual has been arrested, prosecutors in the
district attorney's office read the arresting officer's incident report and then
decide whether or not to press charges. For simplicity, suppose prosecutors
only have access to the incident report---and to no other information---when
making their decisions. We allow for the possibility that the arrest decision
that preceded the charging decision may have suffered from racial discrimination
in complex ways that cannot be inferred from the incident reports themselves.
Finally, suppose that a researcher has access to these incident reports for
arrested individuals, but, importantly, not to any data on individuals that
officers considered but ultimately decided against arresting. What, if
anything, might one hope to discover about racial discrimination in charging
decisions in light of the fact that the people about whom the prosecutor makes
charging decisions have been selected---that is, arrested---not randomly, but
rather in ways that likely depended on their race?
The first challenge is to rigorously define causal estimands of interest. The
inherent difficulty is captured by the statistical refrain ``no causation
without manipulation''~\citep{holland1986statistics}, since it is often unclear
what it means to alter attributes like race and gender~\citep{sekhonneyman}.
One common maneuver is to instead consider the causal effect of \emph{perceived}
attributes (e.g., perceived race or perceived gender), which ostensibly can be
manipulated---for example, by changing the name listed on an employment
application~\citep{bertrand2004emily}, or by masking an individual's
appearance~\citep{goldin2000orchestrating, grogger2006testing, OPP}. In our
case, one might imagine a hypothetical experiment in which explicit mentions of
race in the incident report are altered (e.g., replacing ``white'' with
``Black''). The causal effect is then, by definition, the difference in
charging rates between those cases in which arrested individuals were randomly
described (and hence may be perceived) as ``Black'' and those in which they were
randomly described as ``white.'' This conceptualization of discrimination
conforms to one common causal understanding of discrimination used, for example,
in audit studies. This framing also maps closely to the legal notion of
disparate treatment, a form of discrimination in which actions are motivated by
animus or otherwise discriminatory intent~\citep{goel2017combatting}.
While researchers have carried out such audit studies---including in the case of
prosecutorial charging decisions \citep{chohlaswood2020blind}---it is often
infeasible to study important policy questions through randomized experiments.
In the absence of a controlled experiment, one can in theory identify this type
of causal estimand from purely observational data by comparing charging rates
across pairs of cases that are identical in all aspects other than the stated
race of the arrested individual.\footnote{%
In fact, it suffices to compare groups of cases that have the same
distribution of potential outcomes---even if the cases themselves are not
identical---a property we formalize in Definition~\ref{def:si} below.
} That strategy, which mimics the key features of the randomized experiment
described above, is formally justified when treatment assignment (i.e.,
description of race on the incident report, and subsequent perception by the
prosecutor) is \emph{ignorable} given the observed covariates (i.e., features of
the incident report)~\citep{imbens2015causal}. In practice, though, this
approach may suffer from omitted-variable bias when the full incident report is
not available to researchers, and may suffer from lack of overlap when suitable
matches cannot be found for each case---limitations common to many observational
studies of causal effects. To address these issues, one can restrict attention
to the overlap region and gauge the robustness of estimates to varying forms and
degrees of unmeasured confounding~\citep{cornfield1959smoking, rr,
cinelli2018making}, an approach we demonstrate below.
Finally, there is the issue of post-treatment bias, especially due to sample
selection. \citet{knox-2019}\ argue that researchers often inadvertently
introduce post-treatment bias in observational studies of discrimination by
subsetting on apparently intermediate outcomes---such as, in our charging
example, being arrested---that themselves may be the product of discrimination.
As a result, the authors caution that causal quantities of interest cannot be
identified by the data in the absence of implausible assumptions, such as lack
of discrimination in the initial arrest decision. In making their argument,
\citeauthor{knox-2019}\ focus on the use of force by police officers in civilian
encounters, but they suggest their formal critique applies more broadly, casting
doubt on a wide range of observational studies of discrimination.
Here we show that such customary subsetting does not pose an insurmountable
threat to discrimination research. To understand why, one must precisely define
the causal estimand, and carefully consider the timing of events. For instance,
in our charging example, there are two relevant treatments, the officer's
perception of race, affecting the officer's arrest decision, and the
prosecutor's perception of race, affecting the prosecutor's charging decision.
The arrest decision is post-treatment relative to the officer's perception of
race but, importantly, it is pre-treatment relative to the prosecutor's
perception of race. Similarly, the features of the incident report---which we
must adjust for in this type of benchmark analysis---are post-treatment relative
to the officer's perception of race but pre-treatment relative to the
prosecutor's perception of race. In such a two-decider situation, as
\citet{greiner2011causal} suggest, it is possible to recover estimates of
discrimination by the second decider (e.g., in the charging decision) even if
there is discrimination by the first decider (e.g., in the arrest decision).
\section{A Measure of Discrimination}
\label{sec:post_treat_bias}
We present a simple two-stage model to characterize discriminatory decision
making in a variety of real-world situations and define our main causal quantity
of interest---the second-stage sample average treatment effect, or
\(\cdes\)---within this general framework. In the context of our motivating
example, the \(\cdes\) corresponds to the quantity that would be measured in the
audit study of prosecutorial decisions described in Section~\ref{sec:motex}. A
central aim of this paper is to formalize technical assumptions that allow one
to statistically identify discrimination---more precisely, disparate
treatment---in the second stage (e.g., in prosecutorial charging decisions) when
data are only available for individuals who made it past the first stage (e.g.,
those who were arrested). Importantly, our formalization accommodates scenarios
in which first-stage decisions may themselves be discriminatory.
In the first stage, we assume each individual in some population is subject to a
binary decision \(M\), such as an offer of employment, admission to college, or
law enforcement action. Those who receive a ``positive'' first-stage decision
(e.g., those who are arrested) proceed to a second stage, where another binary
decision \(Y\) is made. In our running example, the case of each arrested
individual is reviewed in the second stage by a prosecutor who may or may not
choose to press charges. Those who are not arrested do not have a case that
requires review by a prosecutor and, indeed, there may be no administrative
record of those individuals.
When considering racial discrimination in decisions involving Black and white
individuals, our primary quantity of interest is the second-stage sample average
treatment effect, \(\EE[Y(b) - Y(w)]\), where \(Y(z)\) indicates the potential
second-stage decision and the expectation is taken over individuals reaching the
second stage. Here, we imagine that the perception of race is counterfactually
determined after the first-stage decision but before the second-stage decision
(e.g., after arrest but before charging, perhaps by altering the description of
race on the incident report viewed by a prosecutor). The second-stage sample
average treatment effect thus captures discrimination in the second-stage
decision among those who made it past the first stage (e.g., discrimination in
charging decisions among those who were arrested). This estimand maps onto a
common understanding of disparate treatment in second-stage decisions, including
in our charging example.
\subsection{A formal model of discrimination}
\label{ssec:discrimination_model}
We now formalize the above discussion to explicitly include decisions made at
both the first and second stages. For ease of interpretation, we follow
\citet{greiner2011causal} and motivate our statistical model by considering
settings where there are two deciders (e.g., an officer and a prosecutor) whose
perceptions of race---or gender, or another trait---can in theory be
independently altered prior to their decisions. There are, however, examples
in which one can plausibly intervene twice even when a single decider makes both
decisions. For instance, an officer may decide to stop a motorist based in part
on a brief impression of the motorist's skin tone as they drive
past~\citep{grogger2006testing, OPP}. This visual impression of race could
subsequently be altered if the motorist presents a driver's license bearing a
name characteristic of another race group, or speaks a dialect of English at
odds with the officer's expectation. It thus may be possible to apply our
framework whether one imagines there are two deciders or a single one.
We begin by denoting the race of an individual as perceived by the first decider
at the first stage by \(D \in \{w,b\}\), where, for simplicity, we consider a
population consisting of only white and Black individuals. We focus on racial
discrimination for concreteness, but similar considerations apply to
discrimination based on other attributes, such as gender. Assuming that there
is no interference between units~\citep{imbens2015causal}, we let the binary
variables \(M(w)\) and \(M(b)\) denote the potential first-stage decisions for
an individual (e.g., whether they were arrested), and write \(M = M(D)\) for the
observed first-stage decision.
Next, we let \(Z \in \{w,b\}\) denote the race of an individual as perceived by
the second decider, at the second stage. In our running example, \(Z\) denotes
race as perceived by the prosecutor reviewing that person's file, while \(D\)
denotes race as perceived by the police officer during the encounter. Finally,
we define the second-stage potential outcomes as a function of both the
first-stage outcome \(M\) (e.g., the arrest decision) and the second decider's
perception of race \(Z\). Thus, assuming once again that there is no
interference, the observed second-stage outcome for an individual can be denoted
\(Y = Y(Z, M)\), where we consider four potential second-stage outcomes for each
individual: \(Y(z, m)\), where \(z\in \{w,b\}\) and \(m \in \{0,1\}\). In our
example, only those who were arrested can be charged, and so \(Y(b, 0) = Y(w, 0)
= 0\) for all individuals.
We further allow each individual to have an associated vector of (non-race)
covariates \(X\), representing, for example, their behavior during a police
encounter, their recorded criminal history, or both. We imagine these
covariates are fixed prior to the second-stage treatment (e.g., prior to the
prosecutor's perception of race), since otherwise the key ignorability
assumption in Definition~\ref{def:si} below is unlikely to hold. In practice,
\(X\) is only observed for a subset of the population (e.g., those who were
arrested and hence in the dataset), but we nonetheless define the covariate
vector for all individuals in our population of interest. These covariates are
not necessary to define our causal estimands of interest, but they play an
important role in constructing our statistical estimators.
In this model of discrimination, we have taken care to distinguish between the
(realized) first- and second-stage perceptions of race, \(D\) and \(Z\), because
this helps to clarify the timing of events and the meaning of causal quantities.
Importantly, this makes it clear that we can conceive of \(D\) and \(Z\) as
separately manipulable. At the same time, our focus is observational settings,
in which disagreement between \(Z\) and \(D\) may be realized only rarely, if at
all, in the data we observe. For instance, barring manipulation of the incident
report, it seems unlikely that an arresting officer's perception of race will
frequently differ from a prosecutor's perception. Our simulation in
Section~\ref{sec:ex} thus imposes the further constraint that perceived race is
the same at each stage, though this restriction is not necessary in general.
With this framing, we now formally describe the primary causal estimand we
consider. This quantity, which we call the second-stage sample average treatment
effect (\(\cdes\)) reflects discrimination in the second stage of the
decision-making process outlined above, such as discrimination in the
prosecutor's charging decision.\footnote{%
The \(\cdes\) is notationally equivalent to the \(\cde_{M = 1}\) defined in
\citet{knox-2019}. In our case, however, we have taken care to specify that
the first parameter in the quantity \(Y(z, m)\) denotes intervening on the
\emph{second-stage} perception of race. Moreover, the \(\cdes\) is distinct
from what \citeauthor{knox-2019}\ call the \(\ate_{M = 1}\).
}
\begin{defn}[\(\cdes\)]
\label{def:cdes}
The \emph{second-stage sample average treatment effect}, denoted \(\cdes\),
is:
\begin{equation}
\label{eq:cdes}
\cdes = \EE[Y(b, 1) - Y(w, 1) \mid M = 1].
\end{equation}
\end{defn}
The estimand in Eq.~\eqref{eq:cdes} compares the potential second-stage
decisions under two race perception scenarios. For example, it compares the
potential charging decisions when the prosecutor perceives the individual to be
either Black or white; importantly, though, the estimand does not explicitly
consider the arresting officer's perception of race. Moreover, this estimand
restricts to the subset of individuals who had a ``positive'' first-stage
decision (e.g., those who were in reality arrested).
Because we condition on \(M = 1\) in the definition of the \(\cdes\), we may
equivalently write Eq.~\eqref{eq:cdes} as
\begin{equation}
\label{eq:cdes_alt}
\cdes = \EE[Y(b, M) - Y(w, M) \mid M = 1].
\end{equation}
We can further write
\begin{equation}
\label{eq:cdes_second_alt}
\cdes = \EE[Y(b) - Y(w) \mid M = 1],
\end{equation}
where we define \(Y(z) = Y(z, M)\). Among those who reach the second stage
(i.e., individuals with \(M=1\)), \(Y(z) = Y(z, 1)\) denotes the outcome of
intervening \emph{only} on the second decider's perception of race. Among those
who do not reach the second stage (i.e., individuals with \(M=0\)), \(Y(z) =
Y(z, 0) = 0\).
Eqs.~\eqref{eq:cdes},~\eqref{eq:cdes_alt},~and~\eqref{eq:cdes_second_alt}, as
well as the informal estimand introduced at the beginning of
Section~\ref{sec:post_treat_bias}, are equivalent ways of capturing the same
underlying quantity, varying only in the degree to which they are explicit about
the staged nature of the process.
\subsection{Estimating the \texorpdfstring{\(\cdes\)}{CDE-Ob}}
Having defined the \(\cdes\), our goal is now to estimate it using only
second-stage data. That is, we aim to estimate the \(\cdes\) only using
observations for those individuals who received a ``positive''---and potentially
discriminatory---decision in the first stage. For example, we seek to estimate
discrimination in charging decisions based only on data describing those who
were arrested. As we show now, an ignorability assumption, together with an
overlap condition, is sufficient to guarantee the \(\cdes\) can be
nonparametrically identified by data on the second-stage decisions.
\begin{defn}[Subset ignorability]
\label{def:si}
We say that \(Y(z, 1)\), \(Z\), \(M\), and \(X\) satisfy \emph{subset
ignorability} if
\begin{equation}
\label{eq:ci}
Y(z, 1) \indep Z \mid X, M=1
\end{equation}
for \(z\in\{w,b\}\).
\end{defn}
In our recurring example, subset ignorability means that among arrested
individuals, after conditioning on available covariates, race (as perceived by
the prosecutor) is independent of the potential outcomes for the charging
decision. As above, we can equivalently write Eq.~\eqref{eq:ci} as
\begin{equation}
\label{eq:ci_alt}
Y(z) \indep Z \mid X, M=1.
\end{equation}
This latter expression makes clear that subset ignorability is closely related
to the traditional ignorability assumption in causal inference, but where we
have explicitly referenced the first-stage outcomes to accommodate a staged
model of decision making.
Almost all causal analyses implicitly rely on a version of subset ignorability,
since researchers rarely make inferences about the full population of interest.
For example, analyses are typically limited to the individuals who agreed to
participate in the study. Even ``gold standard'' randomized experiments, while
ideal for internal validity, frequently lack external validity because the study
participants do not resemble a larger population of interest. Whenever
ascribing causal interpretations to non-experimental data, one should never
merely assume that ignorability or other assumptions hold. As we discuss in
detail in Sections~\ref{sec:ex}~and~\ref{sec:empirical} below, it is critical to
evaluate the credibility of these assumptions in any given application---but we
note that the assumptions we rely on are similar to those invoked in nearly
every observational study of causal effects.
In the traditional, single-stage setting, ignorability is sufficient to obtain
consistent estimates of the average treatment effect. Likewise, we now show
that in our two-stage model of discrimination, subset ignorability is sufficient
to guarantee consistent estimates of the \(\cdes\). In practice, if one can
adjust for (nearly) all relevant factors affecting second-stage decisions, one
can (approximately) satisfy subset ignorability, and in particular, one can
estimate the \(\cdes\) only using data available at the second stage. In the
Appendix, we compare subset ignorability to several alternatives, and show that
those variants tend either to be too weak to guarantee identifiability, or
unnecessarily demanding for real-world applications. We emphasize that since
the first-stage decision, \(M\), and the covariates, \(X\), can be viewed as
pre-treatment relative to the second-stage intervention, concerns about
post-treatment bias corrupting estimates of the \(\cdes\) are more naturally
thought of as familiar concerns about omitted-variable bias.
In the following, we assume that \(X\) is discrete for simplicity of exposition.
\begin{thm}
\label{thm:main}
Suppose \(Y(z,1)\), \(Z\), \(M\), and \(X\) satisfy subset ignorability, and
that overlap holds, meaning that \(\Pr(Z = z \mid X = x, M = 1) > 0\) for all
\(x\) and \(z\). Then, the \(\cdes\) equals
\begin{align*}
\begin{split}
& \sum_x \EE[Y \mid Z = b, X=x, M=1] \cdot \Pr(X=x \mid M = 1) \\
& \hspace{1cm} - \sum_x \EE[Y \mid Z = w, X=x, M=1]
\cdot \Pr(X=x \mid M = 1).
\end{split}
\end{align*}
\end{thm}
\begin{proof}
Conditioning on \(X\) in Eq.~\eqref{eq:cdes}, we have
\begin{align}
\begin{split}
\label{eq:decomp}
\cdes
& = \sum_x \EE[Y(b, 1) \mid X = x, M = 1] \cdot \Pr(X = x\mid M = 1)\\
& \hspace{1cm} - \sum_x \EE[Y(w, 1) \mid X = x, M = 1] \cdot
\Pr(X = x\mid M = 1).
\end{split}
\end{align}
By the subset ignorability assumption, and our assumption
of overlap, we can condition the summands in Eq.~\eqref{eq:decomp} on
\(Z = b\) and \(Z = w\), respectively, without changing their values, yielding
\begin{align}
\begin{split}
\cdes
& = \sum_x \B E[Y(b, 1) \mid Z = b, X = x, M = 1] \cdot
\Pr(X = x\mid M = 1) \\
& \hspace{1cm} - \sum_x \EE[Y(w, 1) \mid Z = w, X = x, M = 1] \cdot
\Pr(X = x\mid M = 1).
\end{split}
\end{align}
Finally, the statement of the proposition follows by replacing the potential
outcomes with their realized values.
\end{proof}
\begin{cor}
\label{cor:main}
Suppose subset ignorability and overlap hold, and that we have \(n\) i.i.d.\
observations \((X_i, Z_i, Y_i)_{i=1}^n\) with \(M_i = 1\). Let \(S_{zx} = \{i
: Z_i = z \land X_i = x\}\) represent the set of observations with \(Z = z\)
and \(X = x\). Finally, let \(n_{zx}\) denote the number of observation in
\(S_{zx}\), and let \(n_x\) denote the total number of observations with \(X =
x\). Then the stratified difference-in-means estimator,
\begin{equation*}
\label{eq:estimator}
\Delta_n =
\sum_x \left[ \frac {1} {n_{bx}} \sum_{i \in S_{bx}} Y_i \right]
\frac {n_x} {n} - \sum_x \left[\frac {1} {n_{wx}} \sum_{i \in
S_{wx}} Y_i \right]\frac {n_x} {n},
\end{equation*}
yields a consistent estimate of the \(\cdes\).
\end{cor}
\begin{proof}
Note that
\begin{align*}
\lim_{n \rightarrow \infty} \frac {1} {n_{zx}} \sum_{i \in S_{zx}} Y_i
& \stackrel{\text{a.s.}}{=} \EE[Y \mid Z = z, X=x, M=1], \ \text{and} \\
\lim_{n \rightarrow \infty} \frac{n_x}{n} & \stackrel{\text{a.s.}}{=}
\Pr(X = x \mid M = 1).
\end{align*}
Consequently, each of the terms in \(\Delta_n\) converges to the corresponding
terms in the definition of the \(\cdes\).
\end{proof}
A straightforward calculation further shows that the following expression yields
a consistent estimate of the standard error of \(\Delta_n\):
\begin{equation}
\label{eq:sedim}
\wh{\se}(\Delta_n) = \sqrt{\sum_x \left( \frac{n_x}{n} \right)^2 \left [
\frac{c_{bx}(1-c_{bx})}{n_{bx}} + \frac{c_{wx}(1-c_{wx})}{n_{wx}}
\right] },
\end{equation}
where
\begin{align*}
c_{zx} = \frac {1} {n_{zx}} \sum_{i \in S_{zx}} Y_i.
\end{align*}
Eq.~\eqref{eq:sedim} accordingly allows us to form confidence intervals for
\(\Delta_n\).
The nonparametric stratified difference-in-means estimator \(\Delta_n\) is the
basis for nearly all applications of benchmark analysis in discrimination
studies. In practice, as we discuss further in Section~\ref{sec:ex}, it is
common to approximate \(\Delta_n\) via a parametric regression model---but the
two estimators share the same theoretical underpinnings. As such, our analysis
above simply grounds traditional benchmark analysis within a specific causal
framework, and demonstrates that a particular ignorability assumption, together
with overlap, is sufficient to yield valid estimates.
\subsection{An alternative measure of discrimination}
\label{ssec:causal_comparison}
To better understand the \(\cdes\), we now contrast it with the total effect
(\(\te\))~\citep{imai-2010-general}, a second estimand considered by
discrimination researchers~\citep{knox-2019, heckman2020comment, zhao2020note}.
The total effect and the \(\cdes\) differ in our setting in two ways: (1) the
population of individuals about which we make inferences; and (2) the potential
outcomes being contrasted. The total effect is not restricted to individuals who
had a ``positive'' first-stage decision (e.g., it is not restricted to those who
were arrested). Additionally, we imagine a causal variable that reflects a
situation where the perception of race is counterfactually determined
\emph{before} the first-stage decision (instead of \emph{after} the first-stage
decision, as with the \(\cdes\)), and is the same at both stages.
We note that, in general---as discussed in Section~\ref{sec:motex} and
below---there is no coherent notion of a ``total effect'' of race, since one
cannot intervene on race, \emph{per se}. In our running example, the two
treatments (i.e., the officer's perception of race and the prosecutor's
perception of race) represent distinct, situation-specific notions of
intervening on race. In this restricted context, then, there is a natural
estimand that captures the spirit of a ``total effect'': comparing an
individual's potential outcomes had they been perceived as white or Black when
\emph{both} the first- and second-stage decisions were made. We formalize this
as follows:
\begin{defn}[\(\te\)]
\label{def:te}
The \emph{total effect}, denoted \(\te\), is given by:
\begin{equation}
\label{eq:te}
\te = \EE[Y(b, M(b)) - Y(w, M(w))].
\end{equation}
\end{defn}
Unlike the \(\cdes\), which only measures discrimination in the second decision,
the total effect measures cumulative discrimination across \emph{both}
decisions. In our recurring example, the total effect captures the effect of
race at the time of arrest on the subsequent charging decision. In particular,
if a charged Black individual had instead been perceived as white by an officer,
they might never have been arrested, and hence never been at risk of being
charged, a possibility encompassed by the definition of the total effect, but
not by the \(\cdes\).
However, in studies of discrimination---particularly racial
discrimination---there is often no clear intervention point, and the difference
between the \(\te\) and the \(\cdes\) is largely an artifact of how one defines
both the population of interest and the start of the decision-making process.
What is the \(\te\) in one description of events may be the \(\cdes\) in
another, equally valid description of the same events, as we describe next.
\begin{figure*}[t]
\begin{center}
\begin{tikzpicture}[
minimum size=0.25cm,
inner sep=0pt,
yscale=0.35
]
\draw[fill=blue1] (0, 10) node[anchor=north west, inner sep=5pt] {Spotted}
rectangle (3, 0);
\draw[fill=blue2] (3, 9) node[anchor=north west, inner sep=5pt] {Stopped}
rectangle (6, 1);
\draw[fill=blue3] (6, 8) node[anchor=north west, inner sep=5pt] {Arrested}
rectangle (9, 2);
\draw[fill=blue4] (9, 7) node[anchor=north west, inner sep=5pt] {Charged}
rectangle (12, 3);
\node[shape=circle, fill=white, draw] at (1.5, 2) (A1) {};
\node[shape=circle, fill=white, draw] at (1.5, 3) (B1) {};
\node[shape=circle, fill=white, draw] at (1.5, 4) (C1) {};
\node[shape=circle, fill=black, draw] at (1.5, 5) (D1) {};
\node[shape=circle, fill=black, draw] at (1.5, 6) (E1) {};
\node[shape=circle, fill=black, draw] at (1.5, 7) (F1) {};
\node[shape=circle, fill=white, draw] at (4.5, 3) (B2) {};
\node[shape=circle, fill=white, draw] at (4.5, 4) (C2) {};
\node[shape=circle, fill=black, draw] at (4.5, 5) (D2) {};
\node[shape=circle, fill=black, draw] at (4.5, 6) (E2) {};
\node[shape=circle, fill=white, draw] at (7.5, 4) (C3) {};
\node[shape=circle, fill=black, draw] at (7.5, 5) (D3) {};
\node[shape=circle, fill=black, draw] at (7.5, 6) (E3) {};
\node[shape=circle, fill=black, draw] at (10.5, 5) (D4) {};
\draw[->] (B1) -- (B2);
\draw[->] (C1) -- (C2);
\draw[->] (D1) -- (D2);
\draw[->] (E1) -- (E2);
\draw[->] (C2) -- (C3);
\draw[->] (D2) -- (D3);
\draw[->] (E2) -- (E3);
\draw[->] (D3) -- (D4);
\draw[thin, dotted] (0, 0) -- (0, -7);
\draw[thin, dotted] (3, 0) -- (3, -7);
\draw[thin, dotted] (6, 1) -- (6, -7);
\draw[thin, dotted] (9, 2) -- (9, -7);
\draw[thin, dotted] (12, 3) -- (12, -7);
\draw[draw=blue2, fill=blue2] (3, -1.1) rectangle (6, -1.9);
\draw[draw=blue3, fill=blue3] (6, -1.1) rectangle (9, -1.9);
\draw[draw=blue4, fill=blue4] (9, -1.1) rectangle (12, -1.9);
\draw[fill opacity=0] (3, -1.1) rectangle (12, -1.9);
\draw[
decorate, decoration={brace, amplitude=0.8em, aspect=7/12, mirror},
yshift=-4pt
] (3, -2) -- (12, -2);
\node at (7.5, -3.5) {\(\cdes = \te\)};
\draw[
decorate, decoration={brace, amplitude=1em, aspect=37/64}, yshift=4pt
] (0, -5) -- (12, -5);
\draw[draw=blue2, pattern={crosshatch dots}, pattern color=blue1]
(0, -5.1) rectangle (3, -5.9);
\draw[draw=blue2, fill=blue2] (3, -5.1) rectangle (6, -5.9);
\draw[draw=blue3, fill=blue3] (6, -5.1) rectangle (9, -5.9);
\draw[draw=blue4, fill=blue4] (9, -5.1) rectangle (12, -5.9);
\draw[fill opacity=0] (0, -5.1) rectangle (12, -5.9);
\draw[thick, <->] (-1.5, -7) node[anchor=east] {\(\cdots\)} -- (13.5, -7)
node[anchor=west] {\(\cdots\)};
\draw (1.5, -7cm + 4pt) -- (1.5, -7cm - 4pt) node[anchor=north]
{\(t_{k-1}\)};
\draw (4.5, -7cm + 4pt) -- (4.5, -7cm - 4pt) node[anchor=north]
{\(t_{k}\)};
\draw (7.5, -7cm + 4pt) -- (7.5, -7cm - 4pt) node[anchor=north]
{\(t_{k+1}\)};
\draw (10.5, -7cm + 4pt) -- (10.5, -7cm - 4pt) node[anchor=north]
{\(t_{k+2}\)};
\end{tikzpicture}
\end{center}
\caption{%
\emph{%
One can measure combined discrimination in arrest and charging decisions
either via the \(\te\) or the \(\cdes\). In studies of discrimination,
there is no clear point at which race is ``assigned'' and so both the
\(\te\) and the \(\cdes\) can be used interchangeably to express the same
underlying causal effect. The diagram illustrates a multistage process,
where one seeks to measure discrimination culminating at stage
\(t_{k + 2}\) (e.g., charging decisions) among those who make it to stage
\(t_k\) (e.g., those who were stopped by the police). This quantity can
be viewed as the \(\te\), where one imagines the process starting at time
\(t_k\). Alternatively, it can be viewed as the \(\cdes\), where one
views the process as starting earlier (at, say, \(t_{k-1}\), indicating
that an officer spotted an individual), and then conditioning on those who
made it to stage \(t_k\).
}
}
\label{fig:cde-te}
\end{figure*}
In our running example, the implicit population of interest consists of those
individuals stopped by the police, and the \(\te\) reflects a description of
events in which the decision-making process starts---and perception of race is
counterfactually determined---after the stop decision has occurred but before
the arrest decision has been made. We can, however, imagine moving back the
clock and starting the process after an officer first spots an individual but
before a stop decision has been made, with the population of interest now
comprising those individuals spotted by an officer. In this case, the original
\(\te\) is equivalent to the \(\cdes\) on this newly defined population, where
the first-stage decision indicates whether an individual was \emph{stopped}.
Both the original \(\te\) and the new \(\cdes\) capture combined discrimination
in the arrest and charging decisions, among the subset of individuals who were
stopped.\footnote{%
To be explicit, our point is that the old \(\te\) and the new \(\cdes\) are
the same quantities, and hence are estimable using the same data. However,
the new \(\cdes\) (which subsets on individuals who are \emph{stopped} among
those who are \emph{spotted}) and the old \(\cdes\) (which subsets on
individuals who are \emph{arrested} among those who are \emph{stopped}),
are, in contrast, not equal in general, and not necessarily estimable using
the same data.
}
But the moment when an individual is spotted is no more privileged as a starting
point than the moment when an officer makes a stop decision. One could similarly
measure cumulative discrimination that includes the stop decision itself, either
in terms of the \(\te\) or the \(\cdes\). For the \(\te\), as above, we imagine
time starting immediately after a potential police encounter, with the
first-stage decision indicating whether an individual was stopped (among a
population of individuals spotted by the officer). For the \(\cdes\), we back up
the clock once again and imagine the first-stage decision indicating whether an
individual was spotted by an officer, among an even larger population of people
walking through the neighborhood where the officer patrols.
Figure~\ref{fig:cde-te} provides a graphical depiction of this
interchangeability.
Although the \(\te\) may appear to avoid conditioning on intermediate outcomes,
it simply masks a complex chain of events that came before the nominal start of
the process, a chain that itself was likely influenced by discriminatory
decisions. For instance, the officer spotting and stopping motorists in our
running example could be patrolling the neighborhood in question because of its
racial composition. The very idea of ``intermediate outcomes''---a concept
central to concerns about post-treatment bias---is a slippery notion in the
context of discrimination studies, where there is no clear point in time where
one can imagine that race is ``assigned.'' Even birth cannot be considered the
ultimate starting point since, in theory, one might include, at the least, the
race of a child's parents, determined at an earlier stage, when assessing
discrimination.\footnote{%
In the case of biological sex, one might consider assignment to occur at
conception, though that is typically not the primary moment of interest in
studies of sex discrimination.
} Indeed, such generational counterfactuals may be critical for understanding
systemic, institutional discrimination.
\section{Assessing Discrimination in a Stylized Scenario}
\label{sec:ex}
Subset ignorability, in theory, is sufficient to ensure nonparametrically
identified estimates of the \(\cdes\), even when the first-stage decisions are
discriminatory. We illustrate that idea by investigating in detail a
hypothetical scenario involving discriminatory arrest decisions in the first
stage and discriminatory charging decisions in the second stage. We explore the
properties of simple estimators in this setting through a simulation study. We
demonstrate that failing to adjust for a factor that directly influences
charging decisions can result in biased estimates of discrimination in those
decisions, but by accounting for all factors that directly influence charging
decisions---and hence satisfying subset ignorability---one can accurately
estimate the \(\cdes\), even when there is unmeasured confounding in arrest
decisions. This example further clarifies the conceptual importance of
distinguishing between an officer's perception of race and a prosecutor's
perception of race when defining and estimating our quantities of interest.
We consider a hypothetical jurisdiction in which police officers observe the
behavior and race of individuals who are potentially engaged in specific
criminal activity (e.g., a drug transaction) and then decide whether or not to
make an arrest. Subsequently, the case files of arrested
individuals---consisting of a written copy of the officer's description of the
encounter and the arrested individual's criminal history---are brought to a
prosecutor who decides whether or not to press charges. We assume the
prosecutor only observes the documented race and criminal record of the
arrestee, and the arresting officer's written description of the encounter;
accordingly, by construction, the charging decision depends only on these three
factors. For example, the prosecutor may choose only to charge individuals who
have several previous drug convictions and who were reported to be engaging in a
drug transaction. Importantly, while the prosecutor has access to an officer's
written report, the prosecutor does not directly observe the individual's
behavior leading up to the arrest.
Our goal is to estimate discrimination in charging decisions, formalized in
terms of the \(\cdes\). Intuitively, if we can adjust for every arrested
individual's criminal history, race, and officer report, then subset
ignorability would hold because the prosecutor's charging decision depends only
on these factors. Thus, with these three covariates, we could generate valid
estimates of discrimination in prosecutorial decisions, even without knowing all
of the factors that led to an arrest, a decision that may itself have been
discriminatory. However, if any of these three covariates---criminal history,
race, or officer report---are unobserved, we will, in general, be unable to
accurately assess discrimination in prosecutorial decisions. In both scenarios,
with and without unmeasured confounding, our analysis is based on the
subpopulation of arrested individuals, where we note that the subsetting (i.e.,
arrest) is not influenced by the prosecutor's perception of race. In this
setting, the primary concern is thus omitted-variable bias, not post-treatment
bias.
We emphasize that we seek only to estimate discrimination in the second-stage
charging decision, not cumulative discrimination stemming from both the arrest
and charging decisions. In particular, while officer reports may represent an
inaccurate---and discriminatory---account of events, such discrimination is
distinct from that in the charging decision itself. Similarly, criminal
histories reflect a form of complex, long-term discrimination that we do not aim
to measure here. Alternative, and more expansive, notions of discrimination are
important to understand, but here we focus on assessing the prosecutor's narrow
contribution to inequities at a specific point in the process, a common
statistical objective closely tied to policy decisions and legal theories of
disparate treatment~\citep{jung2018omitted}.
\subsection{The data-generating process}
\label{sec:simulation}
\begin{figure}[t]
\centering
\begin{center}
\begin{tikzpicture}[xscale = 3.75, yscale = 2, align = center]
\node (race) at (0, 0) {\(T\)\\{\footnotesize``True'' Race}};
\node (behavior) at (4/3, 1.8) {\(A\)\\{\footnotesize Behavior}};
\node (race_o) at (4/3, 0) {\(D\)\\{\footnotesize Officer-Perceived Race}};
\node (arrest) at (8/3, 0) {\(M\)\\{\footnotesize Arrest}};
\node (history) at (2, -2) {\(X\)\\{\footnotesize Criminal History}};
\node (race_p) at (8/3, -1) {\(Z\)\\{\footnotesize Prosecutor-Perceived Race}};
\node (report) at (8/3, 1.8) {\(R\)\\{\footnotesize Officer Report}};
\node (charge) at (4, 0) {\(Y\)\\{\footnotesize Charge}};
\draw[->, bend left = 25] (race) to (behavior);
\draw[->] (race) to (race_o);
\draw[->, bend right = 30] (race) to (history);
\draw[->] (behavior) to (arrest);
\draw[->, bend left = 15] (behavior) to (report);
\draw[->] (race_o) to (arrest);
\draw[line width = 5pt, white] (race_o) to (report);
\draw[line width = 5pt, white] (race_o) to (race_p);
\draw[->] (race_o) to (report);
\draw[->] (race_o) to (race_p);
\draw[->, bend right = 30] (history) to (charge);
\draw[->] (race_p) to (charge);
\draw[->] (arrest) to (charge);
\draw[->, bend left = 25] (report) to (charge);
\end{tikzpicture}
\end{center}
\caption{\emph{%
A causal DAG depicting our stylized example of arrest and charging
decisions, where \(D\) represents the officer's perception of race, and
\(Z\) represents the prosecutor's perception of race. Officer arrest
decisions (\(M\)) are directly influenced by observed criminal behavior
(\(A\)) and officer-perceived race (\(D\)); the officer reports of the
encounters (\(R\)) are directly influenced by \(A\) and \(D\).
Prosecutorial charging decisions are made for all arrested individuals,
and are directly influenced by officer reports (\(R\)), criminal history
(\(X\)), and prosecutor-perceived race (\(Z\)). Finally, ``true'' race
(\(T\)) influences the officer's perception of race (\(D\)), as well as
criminal history (\(X\)) and behavior (\(A\)).
}}
\label{fig:sim-dag}
\end{figure}
We now formally describe the data-generating process for our stylized example.
Under the structural causal model we consider, we can both compute the true
\(\cdes\) and compute estimates based only on select information available to
the prosecutor. In defining the generative process, we closely follow the
terminology and conventions of \citet{pearl2016causal}.\footnote{%
We do deviate from \citeauthor{pearl2009causality} in one aspect of our
notation: we write counterfactuals as \(Y(z, m)\) instead of \(Y_{z, m}(u)\),
suppressing the notational dependence on \(u\). The former notation aligns
with the popular Rubin-Neyman potential outcome notation that we use when
defining the \(\cdes\).
}
Our model is defined in terms of the causal directed acyclic graph (DAG)
depicted in Figure~\ref{fig:sim-dag}. In this model, \(T \in \{w,b\}\) indicates
one's ``true'' race, and \(D\) and \(Z\) indicate, respectively, an officer's
and a prosecutor's perception of race. Further, \(M \in \{0,1\}\) indicates the
arrest decision, and \(Y \in \{0,1\}\) indicates the charging decision. Finally,
\(A\) corresponds to an individual's behavior, as observed by an officer, and
\(X\) and \(R\) correspond, respectively, to criminal history and an officer's
description of an encounter. For simplicity, in our example these latter three
variables are operationalized as being binary---for example, one can imagine
that \(X\) indicates whether an individual had at least one previous drug
conviction, \(A\) indicates whether they were seen actively engaging in a drug
transaction, and \(R\) indicates whether they were reported by the officer to be
actively engaging in a drug transaction. Officers observe \(D\), \(A\), and
\(R\) for all individuals; prosecutors observe \(Z\), \(X\), and \(R\) only for
the subset of arrested individuals.
Structural causal models are defined by a set of exogenous random variables and
deterministic structural equations specifying the values of all other variables
in the DAG. In our example, the independent exogenous variables are:
\begin{align*}
U_T & \sim \bern(\mu_{T}), \\
U_A,\ U_M,\ U_X,\ U_R,\ U_Y & \sim \unif(0,1),
\end{align*}
where \(\mu_T\) is an appropriately defined constant and, for ease of
exposition, we assume the Bernoulli variable \(U_T\) takes values in \(\{w,b\}\)
rather than \(\{0,1\}\).
In line with our discussion in Section~\ref{ssec:discrimination_model}, we set
the prosecutor's perception of race (\(Z\)) equal to the officer's perception of
race (\(D\)), and, for simplicity, we set both equal to one's ``true'' race
(\(T\)). This choice yields the following structural equations:
\begin{align*}
f_T(u_T) & = u_T, \\
f_D(t) & = t, \\
f_Z(d) & = d.
\end{align*}
Now, for constants \(\mu_A\), \(\gamma\), \(\mu_X\), and \(\delta\), the
structural equations for behavior (\(A\)) and criminal history (\(X\)) are
given by:
\begin{align*}
f_A(t, u_A) & = \B 1(u_A \leq \mu_A + \gamma \cdot \B 1(t = b)), \\
f_X(t, u_X) & = \B 1(u_X \leq \mu_X + \delta \cdot \B 1(t = b)).
\end{align*}
This specification allows for the distributions of criminal history and behavior
to vary by race. For example, stopped Black individuals may be less likely to
be engaged in criminal activity than stopped white individuals, corresponding to
\(\gamma < 0\).
Finally, for constants \(\alpha_0\), \(\alpha_A\), \(\alpha_{\text{black}}\),
\(\lambda_0\), \(\lambda_A\), \(\lambda_{\text{black}}\), \(\beta_0\),
\(\beta_X\), \(\beta_R\), and \(\beta_{\text{black}}\), the structural equations
for stop decisions (\(M\)), police reports (\(R\)), and charging decisions
(\(Y\)) are given by:
\begin{align*}
f_M(d,a,u_M) &= \B 1(u_M \leq \alpha_0 + \alpha_A \cdot a +
\alpha_{\text{black}} \cdot \B 1(d = b)), \\
f_R(d,a,u_R) &= \B 1(u_R \leq \lambda_0 + \lambda_A \cdot a +
\lambda_{\text{black}} \cdot \B 1(d = b)), \\
f_Y(z,m,r,x,u_Y) &= m \cdot \B 1(u_Y \leq \beta_0 + \beta_X \cdot x +
\beta_R \cdot r + \beta_{\text{black}} \cdot \B 1(z = b)).
\end{align*}
In particular, stop decisions and police reports depend on an officer's
perception of race, whereas charging decisions depend on a prosecutor's
perception of race.
The above structural equations, together with the distributions on the exogenous
variables, fully define the joint distribution of realized and potential
outcomes. In particular,
\begin{align*}
T &= f_T(U_T), & D &= f_D(T),\\
Z &= f_Z(D), & A &= f_A(T, U_A),\\
X &= f_X(T, U_X), & M &= f_M(D, A, U_M),\\
R &= f_R(D, A, U_R), & Y &= f_Y(Z, M, R, X, U_Y).
\end{align*}
The primary causal quantity we seek to estimate---the \(\cdes\)---is defined in
terms of counterfactuals \(Y(z,m)\). As discussed in \cite{pearl2009causality}
and \cite{pearl2016causal}, such counterfactuals require some care to define, as
one must appropriately account for the exogenous variables \(U\). In particular,
for the causal DAG in Figure~\ref{fig:sim-dag}, the bivariate charge potential
outcomes, for counterfactual versions of prosecutor-perceived race, are given by
\(Y(z, m) = f_Y(z, m, R, X, U_Y)\). Further, the arrest potential
outcomes---where we consider counterfactual versions of officer-perceived
race---are given by \(M(d) = f_M(d, A, U_M)\). In general, counterfactuals
defined in this way obey the consistency rule, meaning that \(M = M(D)\) and \(Y
= Y(Z, M)\).
When \(\alpha_\text{black} \geq 0\), anyone who would be arrested if white would
also be arrested if Black (i.e., \(M(b) \geq M(w)\)). When
\(\alpha_\text{black} > 0\), we say arrest decisions are discriminatory since,
all else being equal, an individual is more likely to be arrested if they were
Black than if they were white. Likewise, \(Y(b, 1) \geq Y(w, 1)\) when
\(\beta_\text{black} \geq 0\), meaning that an individual who would be charged
if arrested and white would also be charged if arrested and Black. We say the
charging decision is discriminatory when \(\beta_\text{black} > 0\).
\begin{table}[t]
\begin{center}
\caption{\emph{%
A sample of potential and realized outcomes for individuals in our
hypothetical example. The data-generating process produces the full set
of entries, but the prosecutor only observes the realized outcomes for
those who were arrested, indicated by the shaded cells.
}}
\label{tb:o}
\begin{tabular}{@{\extracolsep{5pt}} cccccccccccc}
\\[-1.8ex]
\hline \hline \\[-1.8ex]
\(T\) & \(D\) &
\(Z\) & \(A\) &\(R\) & \(X\) & \(M(b)\) & \(M(w)\) & \(Y(b, 1)\) &
\(Y(w, 1)\) & \(M\) & \(Y\) \\
\hline \\[-1.8ex]
\(b\) & \(b\) & \(b\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\(b\) & \(b\) & \(b\) & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
\(b\) & \(b\) & \cellcolor{gray!25}\(b\) & 1 & \cellcolor{gray!25}0 & \cellcolor{gray!25}1 & 1 & 0 & 1 & 1 & \cellcolor{gray!25}1 & \cellcolor{gray!25}1 \\
\(w\) & \(w\) & \cellcolor{gray!25}\(w\) & 0 & \cellcolor{gray!25}1 & \cellcolor{gray!25}0 & 1 & 1 & 0 & 0 & \cellcolor{gray!25}1 & \cellcolor{gray!25}0 \\
\(w\) & \(w\) & \(w\) & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline \\[-1.8ex]
\end{tabular}
\end{center}
\end{table}
\paragraph{Features of our data-generating process.}
Table~\ref{tb:o} displays a sample of five rows of data generated from our
model. From the full set of potential outcomes, we can compute the true
\(\cdes\) by directly applying Definition~1 to the generated data, taking the
average difference between \(Y(b, 1)\) and \(Y(w, 1)\) among arrested
individuals.\footnote{%
Because \(Z\) and \(D\) are separately manipulable in our framing, this
quantity---obtained by first subsetting on arrested individuals, and then
computing the average difference between potential outcomes---can also be
expressed in the \emph{do}-calculus: \(\EE[Y \mid \Do(Z = b), M = 1] - \EE[Y
\mid \Do(Z = w), M = 1]\). However, as is common in causal mediation
analysis, if there were only one indecomposable treatment (e.g., if one
instead imagined directly manipulating \(T\)) then the \(\cdes\) could no
longer be expressed using \emph{do}-operations alone. This limitation
motivates the definition of the counterfactuals above, which are expressive
enough to capture the \(\cdes\)~\citep{pearl2009causality,
pearl2015conditioning}. \label{fn:pearl}
}
However, given the simple linear form of our structural equations, a
straightforward calculation also shows that the \(\cdes\) is exactly equal to
\(\beta_{\text{black}}\).
Our hypothetical example captures three key features of real-world
discrimination studies. First, prosecutorial records do not contain all
information that influenced officers' first-stage arrest decisions (i.e.,
prosecutors only observe \(R\), not \(A\)). Second, our set-up allows for
situations where the arrest decisions are themselves discriminatory---those
where \(\alpha_\text{black} > 0\). Third, the prosecutor's records include the
full set of information on which charging decisions are based (i.e., \(Z\),
\(X\), and \(R\)). Among those who were arrested, the charging potential
outcomes depend only on one's criminal history (\(X\)) and the arrest report
(\(R\)). In particular, they do not depend on one's realized,
prosecutor-perceived race (\(Z\)). Consequently, \(Y(z, 1) \indep Z \mid X, R,
M = 1\), meaning that the model satisfies subset ignorability relative to \(X\)
and \(R\). As a result, access to \(X\) and \(R\) guarantees the stratified
difference-in-means is a consistent estimator of the \(\cdes\), even if one does
not have access to \(A\). However, in general, \(Y(z, 1) \not \indep Z \mid X, M
= 1\) (and, likewise, \(Y(z, 1) \not \indep Z \mid R, M = 1\)), and so if one
only has partial information on charging decisions, there is no guarantee the
\(\cdes\) can be consistently estimated. Indeed, when there is such unmeasured
confounding in the prosecutor's decisions, one should expect biased estimates of
the \(\cdes\).
\subsection{Estimating the \texorpdfstring{\(\cdes\)}{CDE-Ob}}
Although the data-generating procedure produces the full set of potential
outcomes for each individual, the prosecutor only observes a subset of the
cells---realized outcomes for arrested individuals, highlighted in gray in
Table~\ref{tb:o}. We explore the performance of two statistical methods for
estimating the \(\cdes\) based on data observed by the prosecutor: the
stratified difference-in-means estimator described in Eq.~\eqref{eq:estimator},
and a regression-based estimator. We apply each of these methods to two types of
data: the full set of information available to prosecutors (i.e., \(Y\), \(Z\),
\(X\) and \(R\)), and an incomplete dataset comprised only of \(Y\), \(Z\), and
\(X\)---in which case we view \(R\) as an unmeasured confounder.
One can compute the stratified difference-in-means estimate in three steps.
First, partition arrested individuals into subsets that have the same value of
the available control variables (i.e., \(X\) and \(R\) in the complete data
setting, and \(X\) alone in the partial data setting). Second, on each resulting
subset, compute the average difference in charging rates between Black and white
individuals. Third, take a weighted average of these differences, where the
weights reflect the proportion of arrested individuals in each subset. In
addition, one can apply Eq.~\eqref{eq:sedim} to estimate the standard error of
this point estimate to generate confidence intervals.
The stratified difference-in-means estimator is theoretically appealing in that
it is guaranteed to yield consistent estimates of the \(\cdes\) when subset
ignorability and overlap hold. But the estimator can have high variance when the
dimension of the covariate space is high and the sample size is small. Thus, in
practice, it is common to model potential outcomes as a function of observed
covariates---also known as response surface modeling~\citep{hill2011bayesian}.
In particular, on the subset of arrested individuals, one can estimate the
\(\cdes\) via a parametric model that estimates observed charging decisions as a
function of the available information.
To demonstrate this latter approach, we use a linear probability model. In the
complete data setting, we have:
\begin{align}
\label{eq:linear-model}
\EE[Y \mid Z, X, R] = \beta_0 +
\beta_1 Z + \beta_2 X + \beta_3 R,
\end{align}
where the model is fit on the full set of arrests seen by the prosecutor. Under
this model, the \(\cdes\) is approximated by the fitted coefficient
\(\hat{\beta}_1\), since that term captures the difference in charging potential
outcomes after adjusting for the observed covariates. For our specific stylized
example, the linear regression model in Eq.~\eqref{eq:linear-model} is in fact
perfectly specified---exactly mirroring the prosecutor's charging
decisions---and so we are guaranteed to obtain statistically consistent
estimates. In the partial data setting, where an analyst only has access to
\(X\), one must fit a reduced model that excludes \(R\):
\begin{align}
\label{eq:linear-model-incomplete}
\EE[Y \mid Z, X] = \beta_0 +
\beta_1 Z + \beta_2 X.
\end{align}
In this case, \(\hat{\beta}_1\) in general yields a biased estimate of the
\(\cdes\), because of the omitted variable \(R\). The stratified
difference-in-means estimator will in general similarly yield a biased estimate
of the \(\cdes\) in this omitted-variable setting.
\subsection{Simulation results}
We perform a simulation study to understand the properties of the above
estimators, varying our assumptions about discrimination and confounding. We
simulate 10,000 datasets of size 100,000 for each of 25 different parameter
settings. Each setting is defined as a combination of our two key
discrimination parameters, \(\alpha_\text{black}\) and \(\beta_\text{black}\),
where each parameter is allowed to take one of five values: 0.20, 0.25, 0.30,
0.35, and 0.40. Across all simulation settings, we assume the population of
individuals encountered by police is \(30\%\) Black (i.e., \(\mu_T = 0.3\));
that \(30\%\) of white individuals and \(40\%\) of Black individuals have a past
drug conviction, indicated by \(X\); and that \(30\%\) of white individuals and
\(20\%\) of Black individuals are seen engaging in a drug transaction, indicated
by \(A\).\footnote{%
More specifically, the full set of parameters in our simulation was set as
follows: \(\mu_T = 0.3, \mu_X = 0.3\), \(\mu_A = 0.3\), \(\delta = 0.1, \gamma
= -0.1\), \(\alpha_0 = 0.1\), \(\alpha_A = 0.3\), \(\alpha_{\text{black}} \in
\{0.2, 0.25, 0.3, 0.35, 0.4\}\), \(\lambda_0 = 0.2\), \(\lambda_A = 0.6\),
\(\lambda_{\text{black}} = 0.1\), \(\beta_0 = 0.2\), \(\beta_X = 0.4\),
\(\beta_R = 0.2\), and \(\beta_{\text{black}} \in \{0.2, 0.25, 0.3, 0.35,
0.4\}\).
} These settings allow for a substantial amount of overlap across race groups
with regard to the key covariates.
\begin{figure}[t!]
\begin{center}
\includegraphics[height=3in]{figures/millisim.pdf}
\caption{\emph{%
In our hypothetical example of officer and prosecutor behavior,
estimates of discrimination in charging decisions are biased when
information directly influencing those decisions---in this case, an
officer's report---is omitted (left). However, one can obtain accurate
estimates of discrimination when accounting for all information directly
influencing charging decisions (right). Each plot shows the results of
10,000 simulations for each of 25 different combinations of
discrimination in officer and prosecutor decisions, given by
\(\alpha_\text{black}\) and \(\beta_\text{black}\), respectively. The
true value of the \(\cdes\), indicated by the horizontal colored lines,
is computed based on the full set of potential outcomes for each
individual, and does not depend on the degree of discrimination in the
first stage, as seen by the constant value of the \(\cdes\) across
different values of \(\alpha_\text{black}\). For each parameter choice,
we display the mean of the sampling distribution for the stratified
difference-in-means estimator (solid circle) and the regression-based
estimator (hollow circle), along with the interval spanned by the 2.5th
and 97.5th percentiles of the sampling distribution. In the right plot
(``unconfounded''), estimates are based on all three factors that
directly influence charging decisions: race, criminal history, and
officer report; in the left plot (``confounded''), we omit the report.
When all variables directly influencing charging decisions are
available, both estimators recover the true value of the \(\cdes\), even
when there is an unknown degree of discrimination in arrest decisions.
}}
\label{fig:cdes_sampling}
\end{center}
\end{figure}
On each synthetic dataset, we estimate the \(\cdes\) using both the stratified
difference-in-means estimator and the regression-based estimator, and compare
the results to the true population-level \(\cdes\) in two scenarios. In the
first scenario, we assume the officer's report \(R\) is unavailable---meaning
there is unmeasured confounding---and therefore only stratify based on \(X\) in
the difference-in-means estimator, and fit the model in
Eq.~\eqref{eq:linear-model-incomplete} for the regression-based estimator. In
the second scenario, we assume that \(R\) is available, and stratify on both
\(X\) and \(R\) in the difference-in-means estimator, and fit the model in
Eq.~\eqref{eq:linear-model} for the regression-based estimator. For each
combination of \(\alpha_\text{black}\) and \(\beta_\text{black}\), the estimates
on the 10,000 synthetic datasets yield the approximate sampling distributions
for the difference-in-means and regression-based estimators. In
Figure~\ref{fig:cdes_sampling}, we summarize each sampling distribution by its
mean, 2.5th percentile, and 97.5th percentile. The solid points correspond to
the difference-in-means estimator, and the hollow points to the regression-based
estimator. The horizontal lines indicate the true population-level \(\cdes\).
In the left panel (``confounded'') of Figure~\ref{fig:cdes_sampling}, the points
lie below the horizontal lines in all cases, meaning we underestimate
discrimination in charging decisions. In this setting, estimates do not account
for the officer reports \(R\), and so there is unmeasured confounding in the
charging decisions. We set \(\gamma < 0\) in our simulations, and thus stopped
and arrested Black individuals are less likely to be engaging in criminal
activity, a pattern (noisily) reflected in the officer reports. Because we
assume these arrest reports are not available for analysis, we cannot fully
adjust for their direct influence on prosecutor decisions. As a result, by
adjusting for \(X\) alone, we miss an important, unmeasured difference between
arrested white and Black individuals, leading us to underestimate discrimination
in prosecutorial decisions.
In the right panel (``unconfounded'') of Figure~\ref{fig:cdes_sampling}, the
points lie on the horizontal lines in all cases, meaning the estimators are
unbiased, and the range between the 2.5th and 97.5th percentiles is relatively
narrow, indicating estimates are typically close to the true value. These
results hold even when one is unable to assess the degree of discrimination
\(\alpha_{\text{black}}\) in the arrest decisions. As implied by
Theorem~\ref{thm:main}, to accurately estimate the \(\cdes\), it is sufficient
to measure all covariates that directly influence the prosecutor's decisions. In
practice, it is nearly always impossible to do so perfectly, and thus it is
important to gauge the sensitivity of estimates to unmeasured confounding in
those decisions, as we demonstrate with real-world data in
Section~\ref{sec:empirical} below. The key point is that it is sufficient to
adjust for unmeasured confounding in the charging decisions alone; to estimate
discrimination in these charging decisions---formalized by the \(\cdes\)---one
need not account for unmeasured confounding in the arrest decisions themselves.
Finally, in addition to examining the sampling distributions, we assessed the
coverage of our 95\% confidence intervals. For the difference-in-means
estimator, confidence intervals were constructed via the estimated standard
error given by Eq.~\eqref{eq:sedim}; and for the regression-based estimator, we
used the conventional OLS estimate of standard error. For each parameter
setting, we computed the proportion of confidence intervals for the 10,000
datasets that contained the true value of the \(\cdes\). In the no-confounding
scenario, we found the true coverage was in line with the nominal coverage,
ranging from 94\% to 96\% across parameter specifications. In the confounding
scenario, the intervals rarely covered the true values, as expected, with
coverage ranging from 1\% to 30\% across parameters.
\section{An Empirical Analysis of Prosecutorial Charging Decisions}
\label{sec:empirical}
We now apply the statistical framework developed above to assess possible race
and gender discrimination in real-world prosecutorial charging decisions. We
start with the set of individuals in a major U.S.\ county who were arrested for
a felony offense between 2013 and 2019. For our race-based analysis, we then
limit to the 25,918 instances in which the race of the arrested individual was
identified as either Black (14,686) or non-Hispanic white (11,232), and for our
gender-based analysis we limit to the 34,871 instances in which the gender of
the arrested individual was recorded as either male (29,283) or female
(5,588).\footnote{%
Both Hispanic and non-Hispanic white individuals in our dataset appear to have
been recorded simply as ``white''. To disentangle these two categories, we
followed past work and imputed Hispanic ethnicity from
surnames~\citep{word2008demographic, word1996building, OPP}.
}
Our dataset includes a variety of information about each case, including the
criminal history of the arrested individual; the alleged offenses (e.g.,
burglary); the location, date, and time of the incident; whether there is
body-worn camera footage; whether a weapon was involved; whether an elderly
victim was involved; and whether there was gang involvement. We also know the
ultimate charging decision for each case. Disaggregating by gender, 51\% of
cases involving a male arrestee were charged, compared to 45\% of cases
involving a female arrestee; and disaggregating by race, 51\% of cases involving
a Black arrestee were charged, compared to 50\% of cases involving a white
arrestee.
To gauge the extent to which charging decisions may suffer from disparate
treatment by race or gender, we estimate the \(\cdes\). We start by checking
that overlap is satisfied for both our race-based and our gender-based analyses.
Recall that overlap means \(\Pr(Z = z \mid X= x, M=1) > 0\), where \(Z=1\)
indicates an individual's ``treatment'' status (i.e., whether an individual is
male in our analysis of gender discrimination, or Black in our analysis of
racial discrimination), \(X\) is a vector of observed case features, and \(M =
1\) means we restrict to those individuals who were arrested. In contrast to
ignorability, overlap can be assessed directly by examining the data. To do so,
we estimate propensity scores~\citep{rosenbaum1983central}, \(\Pr(Z = z \mid X=
x, M=1)\), via an \(L^1\)-regularized (lasso) logistic regression model. In
Figure~\ref{fig:overlap}, we plot the distribution of the estimated propensity
scores. In the left panel we disaggregate by gender, and in the right panel we
disaggregate by race (Black and white). In situations where overlap does not
hold, it is common to restrict one's analysis to a region of the covariate space
where it does hold. In our case, however, the vast majority of the data are
already far from the endpoints of the unit interval, so we work with the dataset
in its entirety.
\begin{figure}[t]
\begin{center}
\begin{subfigure}{.48\textwidth}
\begin{center}
\includegraphics[height=2in]{figures/overlap_gender.pdf}
\caption{Gender-based analysis}
\label{fig:gender_overlap}
\end{center}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
\begin{center}
\includegraphics[height=2in]{figures/overlap_race.pdf}
\caption{Race-based analysis}
\label{fig:race_overlap}
\end{center}
\end{subfigure}
\caption{\emph{%
We plot, for both our gender-based (left) and race-based (right)
analyses, the distribution of propensity scores, disaggregated by
observed treatment status. We find that the propensity scores are
concentrated away from the interval endpoints, satisfying overlap.
}}
\label{fig:overlap}
\end{center}
\end{figure}
As discussed in Section~\ref{sec:ex}, regression-based estimators can be viewed
as a parametric variant of the stratified difference-in-means estimator
\(\Delta_n\). Thus, to help account for the high dimensionality of our feature
set, we now estimate the \(\cdes\) via linear regression. In particular, for
ease of interpretation, we use a linear probability model:
\begin{equation}
\label{eq:empirical_linear_model}
\EE[Y \mid Z, X] = \beta_0 + \beta_1 Z + \beta_2^T X,
\end{equation}
where \(Y\) indicates whether an arrested individual was charged, and \(X\)
denotes the vector of covariates.
In the gender model, we find that the \(\cdeshat\)---as given by
\(\hat{\beta}_1\)---is 0.025 (95\% CI: [0.014, 0.037]); and in the race model,
we have \(\cdeshat\) is \(-\)0.008 (95\% CI: [\(-\)0.018, 0.002]). These results
indicate that the charging rate for men is slightly higher than the rate for
similar women, and that the charging rate for Black individuals is on par with
that of similar white individuals, mirroring the patterns we saw with the raw,
unadjusted charging rates. If there are no unmeasured confounders (i.e., if
subset ignorability holds) and our parametric model is appropriate, these
results suggest race and gender have a relatively modest impact on charging
decisions in the jurisdiction we consider.
To help contextualize these results, we note that past studies have found mixed
evidence of discrimination in prosecutorial charging decisions, likely due in
part to differences in the jurisdictions and time periods analyzed, and the
methods employed. In one of the most comprehensive investigations to date,
\citet{rehavi2014racial} examined nearly 40,000 individuals in the federal
criminal justice system from initial arrest to final sentencing. The authors
found that disparate treatment in prosecutorial charging
decisions---specifically for charges with statutory mandatory minimum
sentences---was a primary driver for sentencing disparities between Black and
white individuals. In contrast, in a recent experimental study,
\citet{robertson_race_2019} found no evidence of racial bias in charging
decisions when they presented prosecutors with vignettes in which the race of
the suspect was randomly varied. Similarly, in an observational analysis of
prosecutors at the San Francisco District Attorney's Office,
\citet{macdonald2017analysis} found little evidence of discrimination in
charging decisions---in fact, the authors found that white individuals were
charged slightly more often than similarly situated Black individuals. Finally,
in a recent quasi-random study of charging decisions at a large metropolitan
district attorney's office, \citet{chohlaswood2020blind} similarly found little
evidence of disparate treatment.
The AUC of our outcome model in Eq.~\eqref{eq:empirical_linear_model}
above---fit with all available covariates, including race and gender---is 86\%,
indicating that it can predict charging decisions well. Our model, however,
cannot capture all aspects of prosecutorial decision making, as at least some
information used by prosecutors (e.g., forensic evidence) is not recorded in our
dataset, meaning that subset ignorability is likely violated. To check the
robustness of our causal estimates to such unmeasured confounding, one may use a
variety of statistical methods for sensitivity analysis~\citep{%
rr, imbens2003sensitivity, carnegie2016assessing, dorie2016flexible,
mccandless2007bayesian, mccandless2017comparison, jung2020bayesian,
franks2019flexible%
}. At a high level, these methods posit relationships between
the unmeasured confounder and both the treatment variable (e.g., race or gender)
and the outcome (e.g., the charging decision), and then examine the sensitivity
of estimates under the model of confounding.
We apply a technique for sensitivity analysis recently introduced by
\citet{cinelli2018making}. In brief, their approach bounds the extent to which
a coefficient estimate in a linear model---like \(\hat{\beta}_1\) in
Eq.~\eqref{eq:empirical_linear_model}---might change if one were to refit the
model including an unmeasured confounder \(U\). More specifically, under the
extended model
\begin{equation*}
\EE[Y \mid Z, X, U] = \beta_0 + \beta_1 Z + \beta_2^T X + \gamma U,
\end{equation*}
\citeauthor{cinelli2018making} bound the change in \(\hat{\beta}_1\) in terms of
two partial \(R^2\) values: \(\rsqy\) and \(\rsqz\). These two values
respectively quantify how much residual variance in the outcome \(Y\) and
treatment \(Z\) is explained by \(U\). Formally, \(\rsqy\) is defined in terms
of the \(R^2\) of two linear regressions: one using all the covariates \(X\),
\(Z\), and \(U\) to estimate \(Y\) (\(R^2_{\text{full}}\)), and one excluding
\(U\) (\(R^2_{\text{red}}\)). Then, \(\rsqy = (R^2_{\text{full}} -
R^2_{\text{red}}) / (1 - R^2_{\text{red}})\). The quantity \(\rsqz\) is defined
analogously. As these partial \(R^2\) values increase, so does the amount by
which \(\hat{\beta}_2\) could change.
\begin{figure*}[t]
\begin{center}
\begin{subfigure}{.48\textwidth}
\begin{center}
\includegraphics[height=2.75in]{figures/sensitivity.pdf}
\caption{Gender-based analysis}
\end{center}
\label{fig:sens_gender}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
\begin{center}
\includegraphics[height=2.75in]{figures/race_sensitivity.pdf}
\caption{Race-based analysis}
\end{center}
\label{fig:sens_race}
\end{subfigure}
\caption{\emph{%
Contour plots describing the sensitivity of the \(\cdeshat\) to
unmeasured confounding, for our analysis of gender (left) and race
(right). The plots indicate the maximum amount the \(\cdeshat\) may
change under the \citet{cinelli2018making} model of confounding,
parameterized by two partial \(R^2\) values. The red curves correspond
to a change equalling the magnitude of the \(\cdeshat\) estimated from
the available data. Thus, an unobserved confounder corresponding to a
point above the red curve would be capable of changing the sign of our
estimate. To aid interpretation, both plots display the partial \(R^2\)
values associated with several observed subsets of covariates.
}}
\label{fig:sens}
\end{center}
\end{figure*}
The contour plots in Figure~\ref{fig:sens} show the maximum amount by which the
\(\cdeshat\) may change as a function of \(\rsqy\) and \(\rsqz\) for our
analysis of gender and race---with that change potentially increasing or
decreasing the estimate. The red lines trace out values for which the maximum
change equals our empirical point estimates of the \(\cdeshat\). In particular,
an unmeasured confounder lying above the red line could be sufficient to change
the sign of our estimate.
A key hurdle in sensitivity analysis is positing a reasonable range for the
strength of a possible unmeasured confounder. To aid interpretation, we compute
the partial \(R^2\) values for various subsets of observed covariates, as
recommended by \citeauthor{cinelli2018making}. For each such subset, we fit the
regression model in Eq.~\eqref{eq:empirical_linear_model} both with and without
that subset, which in turn yields a pair of partial \(R^2\) values for that
subset of covariates.
The contour plots in Figure~\ref{fig:sens} contain these reference points for
five different subsets of covariates: (1) the subset describing criminal history
(e.g., number of prior convictions and number of prior arrests); (2) the alleged
offenses (e.g., burglary); (3) the subset of all covariates except for the
alleged offenses; (4) the district in which the alleged incident took place; and
(5) whether a weapon was alleged to have been used. We find that the partial
\(R^2\) values associated with criminal history and whether a weapon was used
are far below the red curves for both our analysis of gender and race,
indicating that a confounder with comparable marginal explanatory power to these
covariates would not be sufficient to change the sign of our estimates.
However, the partial \(R^2\) values corresponding to the alleged offenses and
the district in which the charges were filed are near the red curve for our
gender-based analysis and far above the curve for our race-based analysis,
meaning that omitting a covariate with similar explanatory power could
qualitatively change our conclusions. Furthermore, the partial \(R^2\) values
corresponding to everything except the alleged offenses are far above the red
curve in both cases, suggesting that an unobserved confounder of similar
strength could again substantially alter our results. For instance, in this
extreme scenario, inclusion of a currently omitted confounder with similar
characteristics in the race-based analysis could yield an estimated treatment
effect of more than 13\%.
One cannot know the exact nature and impact of unmeasured confounding. Thus, as
in many applied statistical problems, we must rely in large part on domain
expertise and intuition to form reasonable conclusions. In this case, we
interpret our results as providing moderately robust evidence that perceived
gender and race have limited effects on prosecutorial charging decisions in the
jurisdiction we consider.
\section{Discussion}
\label{sec:discussion}
We have outlined a formal causal framework to ground observational studies of
discrimination. We specifically showed that subset ignorability, together with
overlap, is sufficient to guarantee that one important causal measure of
discrimination (the \(\cdes\)) is nonparametrically identified in a canonical
two-stage decision-making setting, and that potential issues of post-treatment
bias in this context are more appropriately thought of as concerns about omitted
variables. In particular, we demonstrated that a traditional regression-based
analysis can be used to assess discrimination in real-world prosecutorial
charging decisions, even though the underlying arrests may have been
discriminatory in unknown ways. In that example---as in many applied
settings---subset ignorability may only hold approximately, and our empirical
analysis illustrates the importance of sensitivity analysis for robust
inference.
Stepping back, there are at least two broad notions of discrimination, which
approximately map to the legal concepts of disparate treatment and disparate
impact. Both involve causal interpretations, though with key differences in the
definition of the estimand. Disparate treatment concerns the causal effect of
race on outcomes, with behavior often driven by animus or explicit racial
categorization. Disparate impact, on the other hand, concerns the causal effect
of policies or practices on unjustified racial disparities, regardless of
intent. Disparate treatment and disparate impact both play important roles in
legal and policy discussions, but the perspective one adopts in any given
situation affects the choice of statistical estimation strategy and the
interpretation of results~\citep{jung2018omitted}.
We have throughout focused on the statistical foundations and measurement of
disparate treatment. In our primary example, we estimate---assuming subset
ignorability holds---that perceived race and gender have relatively small
effects on prosecutorial charging decisions in the jurisdiction we examine. We
further demonstrate that these estimates are moderately robust to potential
omitted-variable bias. However, that finding, in and of itself, does not mean
charging decisions are equitable in a broader sense. Consider, for example, the
1,637 cases in our data involving alleged possession of controlled substances by
Black or non-Hispanic white individuals. Of these, 748 cases (46\%) were
ultimately charged, and charging rates by race were nearly identical across race
groups, suggesting little disparate treatment. However, among the 748 charged
cases, 464 (62\%) involved a Black individual---far exceeding the proportion of
Black residents in the county we study. Charging decisions for these cases thus
impose a heavy burden on Black individuals, even if those decisions were not
tainted by animus. To the extent that prosecution of drug crimes is misaligned
with community goals, these decisions create an unjustified, and discriminatory,
disparate impact.
Rigorously estimating discrimination is a daunting task that requires careful
consideration. At an empirical level, it is often difficult to obtain detailed
data on individual decisions, in which case benchmark analysis may be
inadequate---even if coupled with sensitivity analysis. At a theoretical level,
we have a limited statistical language to make precise concepts such as animus
and implicit bias that are central to discrimination research. Further, as we
note above, past work has often framed discrimination as the causal effect of
race on behavior, but other conceptions of discrimination, such as disparate
impact, are equally important for assessing and reforming practices. Finally,
the conclusions of discrimination studies are generally limited to specific
decisions that happen within a long chain of potentially discriminatory actions.
Quantifying discrimination at any one point (e.g., charging decisions) does not
yield estimates of specific or cumulative discrimination at other points---for
example, arrests. Despite these important considerations, we hope our work
helps place discrimination research on more solid statistical footing, and
provokes further interest in the subtle conceptual and methodological issues at
the heart of discrimination studies.
\bibliographystyle{abbrvnat}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,147
|
{"url":"https:\/\/mathoverflow.net\/questions\/linked\/133050","text":"5k views\n\n### What's a noncommutative set?\n\nThis issue is for logicians and operator algebraists (but also for anyone who is interested). Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]): Let $H$ ...\nLet $\\mathcal{M}$ be a finite dimensional von Neumann algebra, then : $$\\mathcal{M} \\simeq \\bigoplus_i M_{n_i}(\\mathbb{C})$$ Question : Is it singly generated (as von Neumann algebra)? how ? ...","date":"2020-02-29 06:52:37","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\": 0.9225941300392151, \"perplexity\": 1207.1296783394844}, \"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\/1581875148671.99\/warc\/CC-MAIN-20200229053151-20200229083151-00520.warc.gz\"}"}
| null | null |
Q: Azure Linux App Service : Installing packages after deploy from Devops pipeline I'm currently setuping a CI/CD pipeline in Azure Devops to deploy a NodeJS app on a linux hosted app service (not a VM).
My build and deploy both go smoothly, BUT I need to make sure some packages are installed in the environment after the app has been deployed.
The issue is: whatever apt-get script I create after the deploy, I have to run then manually for them to actually take effect. In the Pipeline log they seem to have been executed, though.
Here is the part of my yaml code responsible for the deploy, did I miss something?
- stage: Deploy
displayName: Deploy stage
dependsOn: Build
condition: succeeded()
jobs:
- deployment: Deploy
displayName: Deploy
environment: $(environmentName)
pool:
vmImage: $(vmImageName)
workspace:
clean: all
strategy:
runOnce:
deploy:
steps:
- task: AzureWebApp@1
displayName: 'Azure Web App Deploy:'
inputs:
azureSubscription: $(azureSubscription)
appType: webAppLinux
appName: $(webAppName)
runtimeStack: 'NODE|16-lts'
package: $(Pipeline.Workspace)/drop/drop$(Build.BuildNumber).zip
startUpCommand: 'pm2 start index.js --no-daemon'
on:
success:
steps:
- script: sudo apt-get update
displayName: apt update
- script: sudo apt-get -y [SOME LIBS]
displayName: try install dependencies
Thanks !
A: For now, went with a "startup.sh" file which I run manually after each deploy. Gonna go through docker later though
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,922
|
Bahamian Orphanage Facing Devastation
Asia International
Hong Kong Protesters are Now Raging Against a System, Not Just a Bill
Celebrating Our AAPI Activists: An Interview With @asian.actiivist
How Jeunesse Global is Changing the World Through Health and Philanthropy
Sudan: The Counterrevolution No One's Heard of Until Now
Gay Sex Decriminalised in Botswana
Sophie Sawyer
When one thinks of the Bahamas, they most likely think of paradise, beaches, and expensive vacations. In Nassau, there is a special area hidden from many called the Elizabeth Estates Children's Home. The Elizabeth Estates Children's Home provides education and shelter for children growing up without parents. Established in 1992, the main mission is to give children a life worth living. Because of the work done by the children's home, more Bahamian adults are thriving in society instead of living on the streets.
I had the privilege of visiting the orphanage with a small mission group in March 2016. It is an experience I will never forget. My group brought hundreds of dollars worth of toys, school supplies, and other necessities in suit cases. With the children, we made bracelets and decorated T-shirts. It was obvious how essential the home and programs provided were to the children. As one of the home director's said, "These children are relying on us for survival, education, and more. They deserve only the best, and every gift given to us goes directly into giving these children a better life." The people that work at the shelter are unbelievably grateful for every donation and helping hand given.
The Elizabeth Estates Children's Home is now in trouble. In October 2016, Hurricane Matthew hit a large portion of the western hemisphere. Unfortunately, the Bahamas were one of the areas hit the hardest. According to Florence Pratt, the home's director, the orphanage was left "devastated." The roof was reduced to shambles. The classrooms, including the few computers, were demolished. Only around half of their materials and supplies could be salvaged. The only sense of a home that the children had is now ruined. They desperately need our help. You can go to their GoFundMe, where you can donate directly to the orphanage and save the children.
In this article:bahamas, help, mission, nassau, orphanage, save
Why Canada Isn't The Perfect Country You Think It Is
Refugees Are Humans Too
Written By Sophie Sawyer
Sophie is a sixteen year old Floridian that is passionate and ready to write. She is a proud Christian and lover of theatre.
To Parents of Depressed Children
Stop Telling Suicidal People They Are Selfish
AP and IB Classes: Behind the Scenes for a Good Choice
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,859
|
Thackerville, Okla. (Thur, July 13, 2017)– Bellator MMA is returning to WinStar World Casino and Resort in Thackerville, Okla. The main event will be headlined by Derek Campos as he takes on Brandon Girtz for the 3rd time on Friday, July 14. Bellator 181 will be broadcast live and free on SPIKE at 9 p.m. ET/8 p.m. CT, while preliminary action will stream on FightBookMA.com, Bellator.com and the Bellator Mobile App.
FightBookMMA sat down with both fighters headlining this event. Click here for the link to our interview with Derek Campos and click here for the link to our interview with Brandon Girtz.
Bellator 181 Weigh-ins will take place today Thursday at 5 p.m. CST and you can watch the Live Stream below.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,642
|
I came to a saving knowledge in Jesus Christ when I was 34 years old. I was a cultural Christian growing up and then oscillated between agnosticism and atheism in my early adulthood. My testimony is not one of hardship and struggle leading me to a dependence on God. I had so much in life and yet wasn't satisfied. I made the Olympic trials in swimming, swam and played water polo for UC Berkeley, sang the lead role in South Pacific and The Pirates of Penzance and toured Europe and recorded a CD with an a cappella singing group. I've been sky diving, hang gliding, scuba diving, hiking in the Swiss Alps and swimming in the Andaman sea off the coast of Thailand. My life has been filled with great friends, fun times, and a loving, devoted family. Yet it always seemed incomplete.
I gave up a career in economic consulting to become a teacher, thinking that perhaps if I just chose the right profession I could achieve self-fulfillment and satisfaction…but it didn't satisfy my search for deeper meaning in my life. I decided that if I just met the right woman it would make me whole and happy, so I started attending one of Menlo Park Presbyterian Church's fellowship groups to find a wife. Thankfully, God had so much more in store for me before being married. While getting to know true Christian believers and developing relationships with committed followers of Jesus Christ, I was more comfortable with investigating Christianity.
It dawned on me in the year 2000 that I had never really investigated who Jesus was or the main tenets of the Christian faith. Though I had spent much of my life openly ridiculing those who called themselves Christian and completely rejecting the truth of the Bible, I had never read an entire gospel with an open mind. I had rejected Christianity not because I had investigated it and found it lacking, but because I simply didn't like what I perceived it to be. I would encourage anyone who doesn't believe that Jesus is the only way to heaven, to honestly and openly ask God to reveal Himself to them and then read the Gospel of John. If Christianity is true, it has eternal significance. Isn't it worth dedicating some time to truly investigate who Jesus claimed to be and with an open mind, learn about the Christian faith? Click on this link to locate an Alpha class near you and take an 11-week course on the foundational truths of the Christian faith.
Lee Strobel's books, the Case for Christ and the Case for Faith, were also instrumental in breaking down some walls that I had put up to the truth. Through taking the Alpha class and reading various books, I was finally given enough faith to commit my life to Jesus and ask for Him to reveal Himself to me in a real way.
God is so good to us and faithful in answering prayer when we seek His will for our lives. On April 16th, 2001 the Lord answered that baby step of faith that I took in a huge and abundant way. I woke up at 4:30am and felt an overwhelming desire to read the Bible. Although I had read bits and pieces of Scripture, I had not yet read a gospel in its entirety. During my reading of John's gospel, I couldn't put the book down. This amazing depth of understanding and knowledge of the truth of those words literally came to life inside me. It was not just a head knowledge of the truth either, but an experiential knowledge of the truth. I experienced the very presence of God Himself in my heart through His Holy Spirit in a life altering, transforming, way that will forever solidify my trust, confidence and belief in the Lord.
If given a choice, I would throw away all of my life's accomplishments–activities, fun times, great adventures, achievements, the sum total of every worldly joy I've ever experienced in life–for that one day after Easter where God revealed Himself to me and opened my eyes to the truth. Everything else is relatively meaningless compared to a personal relationship with God by His Holy Spirit living in us through belief in Jesus Christ. And this was just the beginning of that relationship. Since coming to Christ and receiving the Holy Spirit, my life has been totally transformed with the fruits of the Spirit. I was finally shown the meaning of life.
All praise and glory to God for coming to earth in the person of Jesus Christ who died to set us free from the bonds of sin in our lives through the transforming power of the indwelling Holy Spirit. Praise God!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,846
|
Wzmacniacz optyczny – urządzenie wzmacniające sygnał optyczny (promieniowanie świetlne) bezpośrednio, bez konwersji na sygnał elektryczny. Podobnie jak laser, wykorzystuje zjawisko emisji wymuszonej w ośrodku czynnym. Używa się go, między innymi, w światłowodach.
Rodzaje wzmacniaczy
Rodzaje wzmacniaczy optycznych:
wzmacniacz półprzewodnikowy (SOA)
wzmacniacze światłowodowe – domieszkowane jonami metali ziem rzadkich:
erbem (EDFA) – pracujący w III oknie telekomunikacyjnym (1550 nm)
prazeodymem (PDFA) – pracujący w II oknie telekomunikacyjnym (1300 nm)
iterbem (YDFA) – pracujący w zakresie 1 μm
neodymem (NDFA) – pracujący w zakresie 1 μm
tulem (TDFA) – pracujący w zakresie 2 μm
holmem – pracujący w zakresie 3 μm
wzmacniacz Ramana
wzmacniacz Brillouina
Wzmacniacze półprzewodnikowe
Wykonuje się je z półprzewodników podobnie jak lasery półprzewodnikowe Fabry'ego-Perota, ale nie mają one luster odbijających światło, a – wręcz przeciwnie – powierzchnie wejścia i wyjścia sygnału formuje się tak, by nie odbijały światła. Inwersję obsadzeń uzyskuje się poprzez elektryczne wzbudzanie ośrodka.
Wzmacniacze światłowodowe
Podstawowym elementem tego wzmacniacza jest odcinek światłowodu domieszkowanego jonami o energii stanów metastabilnych nieco większych od energii fali przenoszonej przez światłowód. Światło lasera wzbudza jony do stanu o wysokiej energii (pompowanie optyczne). W wyniku emisji wymuszonej wzbudzony światłowód generuje światło, gdy przechodzi przez niego światło wzmacniane. Ośrodek światłowodu wzmacnia tylko określony zakres długości fali (tak zwane okno wzmocnienia), zależnej od właściwości spektroskopowych jonów domieszki, struktury szkła światłowodu oraz długość fali i mocy lasera pompującego.
Powszechnie stosowaną domieszką we wzmacniaczach światłowodowych są jony metali ziem rzadkich, przeważnie erbu. Jony erbu wzbudzone światłem o długości fali około 980 nm umożliwiają efektywne wzmacnianie sygnałów o długości fali około 1500 nm. W zakresie tym pracują często światłowody telekomunikacyjne.
Przypisy
Optoelektronika
Technika światłowodowa
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,859
|
{"url":"https:\/\/www.physicsforums.com\/threads\/help-with-recurrence-formula.605548\/","text":"# Help with Recurrence Formula\n\n1. May 12, 2012\n\n### QCM~\n\nHi guys. I'm having a bit of trouble with what I thought was a simple math question.\n\n1. The problem statement, all variables and given\/known data\n$x_{n}$ = $\\int_0^1 \\frac{t^n}{t+7}dt$\n\nShow that $x_0$=ln(8\/7) and $x_n = n^{-1} - 7x_{n-1}$\n\n2. The attempt at a solution\n\nShowing x0 = ln(8\/7) is a vanilla textbook log question. I'm having trouble with the second part. I am using integration by parts on the form:\n$\\int^1_0 t \\frac{t^{n-1}}{t+7}dt$\nand letting u=t and dv=$\\frac{t^{n-1}}{t+7}dt$\nThis give:\n$tx_{n-1}|^1_0 - \\int_0^1 x_{n-1} dt\\\\ = x_{n-1} - \\int_0^1 x_{n-1} dt$\n\nat which point I'm stuck. I'm not sure if I've used the right IBP substitution or if I'm just almost there and it's just a case of simplifying what I have into a more general case (but can't see it).\n\nThanks\n\n2. May 12, 2012\n\n### cjc0117\n\nI'm still trying to figure it out using IBP, but try using long division on the integrand instead to arrive at a recursive relationship for the quotient. Then integrate.\n\nLast edited: May 13, 2012\n3. May 12, 2012\n\n### Dick\n\nVery astute! I don't think integration by parts leads anywhere. You might change your post to just the hint, \"try long division\" instead of giving the whole solution. It's more subtle.\n\n4. May 12, 2012\n\n### cjc0117\n\nThanks Dick. And yes, I suppose you're right. I just edited it.","date":"2017-12-16 13:23:37","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\": 0.8339523673057556, \"perplexity\": 856.6669392234845}, \"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-2017-51\/segments\/1512948588072.75\/warc\/CC-MAIN-20171216123525-20171216145525-00663.warc.gz\"}"}
| null | null |
Tutore – nel diritto, rappresentante legale di una persona
Tutore – in ambito scolastico e universitario, docente che svolge didattica aggiuntiva
Tutore – in ortopedia, struttura di sostegno che mantiene una parte del corpo nella giusta posizione
Tutore – in botanica, struttura di sostegno per piante rampicanti
Giulio Tutore – ufficiale dell'esercito romano coinvolto nella rivolta batava del 69 d.C.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,273
|
Texas law, as well as statutes established by many local communities in our state, has placed certain restrictions on the types of activities that can occur within the perimeter of a determined school zone. Speed limits are greatly reduced, at least during the days on which school is in session. Anyone caught with drugs while on school property can expect a greatly heightened fine or increased jail time. Now, the House has tentatively approved another way in which behavior must be modified around our school children. This time, those indispensable cell phones are involved.
Yesterday, the Texas House voted for a ban on the use of cell phones by drivers in an active school zone. Exceptions would be made for hands-free devices, on-duty emergency services, and certain licensed radio operators. Emergencies experienced by private citizens while moving through a school zone also may be reason for forgiveness when a fine is issued.
The ban brings with it a maximum fine of $25 for a first offense and up to $50 for any subsequent offenses. It still requires a final vote in the House and then approval by the Senate before being submitted to Governor Rick Perry for his needed signature. Similar measures have already been approved in Texas cities such as El Paso as well as in towns across our country. With police officers regularly patrolling school zones looking for speeders or other threats to the safety of our children, the belief is that enforcing this new law would be easy to accomplish.
Whether or not this new law is put into place, we know that talking or texting with a cell phone while driving can be dangerous to others on the road. If you experience an injury due to a driver's distracted operation of his vehicle, the personal injury attorneys at Bertolino LLP can help. If you are in need of legal assistnace, please contact our Austin, Houston, or San Antonio office today.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 291
|
import {remote} from 'electron';
import settings from '../../common/settings';
class AppConfig {
constructor(file) {
this.fileName = file;
try {
this.data = settings.readFileSync(file);
} catch (e) {
this.data = {
teams: [],
};
}
}
set(key, value) {
this.data[key] = value;
settings.writeFileSync(this.fileName, this.data);
}
}
export default new AppConfig(remote.app.getPath('userData') + '/config.json');
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 169
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{"url":"https:\/\/tex.stackexchange.com\/questions\/474117\/writing-limits-with-additional-conditions","text":"# Writing limits with additional conditions\n\nI gave a quick search, but didn't find something similarly, atleast for limits.\n\nI am trying to write something like:\n\n $\\lim_{\\binom{x\\to 0}{x>0}} f(x)$\n\n\nMy intention is to put the limit with the two conditions under it. For now I came up with the idea to use the \\binom, but that has some braces near it. I would appreciate a better idea.\n\n\u2022 Welcome to TeX.SE! Something like \\documentclass{article} \\usepackage{amsmath} \\begin{document} $\\lim_{\\substack{x\\to 0\\\\ x>0}} f(x)$ or $\\lim\\limits_{\\substack{x\\to 0\\\\ x>0}} f(x)$ \\end{document}? \u2013\u00a0user121799 Feb 9 at 22:25\n\n## 1 Answer\n\nThis is an alternative for your question using the option smallmatrix.\n\n\\documentclass[12pt]{article}\n\\usepackage{mathtools}\n\\begin{document}\n$\\lim_{\\begin{smallmatrix} x \\to 0 & \\\\ x>0 \\end{smallmatrix}} f(x)$\n\\end{document}\n\n\u2022 Thanks! This still has some braces near it. Writing 'psmallmatrix' as 'smallmatrix' works though. \u2013\u00a0Zacky Feb 9 at 22:34\n\u2022 @Zacky You're absolutely right. I knew it right away, then, but I'm slow as a snail. \u2013\u00a0Sebastiano Feb 9 at 22:37\n\u2022 Much appreciated. Can you also give me a quick hint on how to write tags? For example with mathjax I just have to use \\tag1 and a: (1) appears at the end of the line. With latex do I have to include a package or something?Or this would better be put in another question? \u2013\u00a0Zacky Feb 9 at 22:41\n\u2022 @Zacky Excuse me for my bad English. I not use often Mathjax but with LaTeX the \\tag{1} into an ambient in math-mode you have (1) at the end of the line correctly. Use for example \\usepackage{amsmath} package. \u2013\u00a0Sebastiano Feb 9 at 22:46\n\u2022 Sebastiano's solution has not the text under \"lim\" centered, and more space between \"lim\" and \"f(x)\" than Marmot solution under the OP question. \\substack is specifically designed for the task asked, as explained here. \u2013\u00a0quark67 Feb 9 at 23:14","date":"2019-08-25 15:22:49","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\": 2, \"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\": 0.858373761177063, \"perplexity\": 1563.194124055753}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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\/1566027330750.45\/warc\/CC-MAIN-20190825151521-20190825173521-00268.warc.gz\"}"}
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\section{Introduction}
Quantum entanglement, primarily a source of quantum information,
has developed into one of the most studied subfields of many-body physics.
In the last decade, quantum entanglement has mainly been used to study
phase structure in condensed matter physics \cite{Amico2008}.
The entanglement spectrum of a bipartite system of subsystems A and B
is defined in terms of the Schmidt decomposition of its ground state
$ \vert \psi \rangle $ as
\begin{eqnarray}
\vert \psi \rangle = \sum_{n} e^{-\frac{\xi_{n}}{2}}
\vert \psi_{n}^{A} \rangle \vert \psi_{n}^{B} \rangle
\end{eqnarray}
where the states $ \vert \psi_{n}^{A} \rangle $ ($ \vert \psi_{n}^{B} \rangle $)
are orthonormal states of the subsystem
A (B), respectively, and the non-negative quantities $ \xi_{n} $ represent
the levels of the entanglement spectrum.
Further, Haldane and Li in Ref.\cite{Li08} have reported a remarkable relationship
between the excitation spectrum and the edges separating the subsystems,
considering the entanglement spectrum of the fractional quantum Hall system
obtained using a spatial cut. This connection between
the edge spectrum and entanglement spectrum is observed in many condensed matter systems including ladders systems
\cite{Qi12, Chen13, Lundgren13}
In many previous studies, the proportionality between the energetic Hamiltonian
of the subsystem A $ \mathcal{H}_{A} $ and the entanglement Hamiltonian
$ \mathcal{H}_{ent} $ in the strong coupling regime \cite{POI10, CIR11,
SCH11, PES11, LAE12, SCH12, SCH13} has been observed.
However, this does not hold in general, even in the strong coupling limit
which is illustrated by counterexamples in Ref. \cite{LUN12}where four spin terms of the Kugel-Khomskii model
are considered in Ref. \cite{SCH12}, in which anisotropic spin ladders of arbitrary
spin length were considered, where even the unperturbed nondegenerate ground state
has a nontrivial entanglement spectrum.
Here we study the entanglement spectrum of the Heisenberg spin
ladders in a time-dependent magnetic field via the degenerate perturbation theory,
where couplings between legs are considered as a perturbation.
The entanglement Hamiltonian is, within the first-order perturbation theory,
proportional to the energy Hamiltonian of a chain in the magnetic field
when the ground state is degenerate.
This holds, although the entanglement spectrum of the unperturbed
ground state has a nontrivial entanglement spectrum.
\section{Motivation}
\label{motivacija}
\begin{figure}[t]
\includegraphics[width=1\columnwidth]{ladders1.png}
\caption{Illustration of the two leg spin ladder considered in this paper. The entanglement spectrum is performed by tracing out subsystem A.
}
\label{fig1}
\end{figure}
We consider a bipartite system consisting of two subsystems described
by Hamiltonians $ \mathcal{H}_{A} $ of subsystem A and $ \mathcal{H}_{B} $ of subsystem B,
which are coupled by the Hamiltonian $ \mathcal{H}_{0} $. We assume that the Hamiltonians
$ \mathcal{H}_{A} $ and $ \mathcal{H}_{B} $ are small compared to $ \mathcal{H}_{0} $,
and it will be treated as a small perturbation.
This problem can be illustrated by two leg spin ladders, where interaction between rungs is considered as a small perturbation (see Fig. \ref{fig1}).
The projector onto subsystem orthogonal on the
nondegenerative ground state $ \vert \psi_{0} \rangle $ is defined as
\begin{equation}
Q_{l} = 1 - \vert \psi_0 \rangle \langle \psi_0 \vert = \sum_{l\neq 0} \vert \psi_l
\rangle \langle \psi_l \vert.
\end{equation}
Then, the first correlation $ \vert \psi_{l}^{(1)} \rangle $ of the nondegenerative
ground state $ \vert \psi_{0} \rangle $ reads
\begin{small}
\begin{align}
\vert \psi_{l}^{(1)} \rangle = & \vert \psi_{0} \rangle \nonumber \\
+ &
\frac{1}{E_{0}-\mathcal{H}_{0}}Q_{l}\left(\left(\mathcal{H}_{A}+\mathcal{H}_{B}\right)-
\langle \psi_{0}\vert \left(\mathcal{H}_{A}+\mathcal{H}_{B}\right) \vert \psi_{0} \rangle \right)
\vert \psi_{l} \rangle
\label{def_corr2}
\end{align}
\end{small}
where $ E_{0} = \langle \psi_{0} \vert \mathcal{H}_{0} \vert \psi_{0} \rangle $.
We also use that $ \frac{1}{E_{0}-\mathcal{H}_{0}} Q_{l}=
\sum_{l\neq 0}\frac{\vert\psi_{l}\rangle \langle \psi_{l}\vert}{E_{0}-E_{l}} $,
where $ E_{l} = \langle \psi_{l} \vert \mathcal{H}_{0} \vert \psi_{l} \rangle $
and the fact that $ \frac{1}{E_{0}-\mathcal{H}_{0}} Q_{l} \vert \psi_{0} \rangle = 0 $
by definition.
In the following, we will assume that $ \frac{1}{E_{0}-E_{l}} $ is equal for every $ l $.
This allows us to rewrite Eq.(\ref{def_corr2}) as
\begin{eqnarray}
\vert \psi_{l}^{(1)} \rangle = \vert \psi_{0} \rangle + \frac{1}{E_{0}-E_{l}}
(\mathcal{H}_{A}+\mathcal{H}_{B}) \vert \psi_{0} \rangle.
\end{eqnarray}
The density matrix within the first order of the perturbation theory
has the following form
\begin{small}
\begin{align}
\rho =& \vert \psi_{l}^{(1)} \rangle \langle \psi_{l}^{(1)} \vert \nonumber \\
\rho =& \vert \psi_{0} \rangle \langle \psi_{0} \vert \nonumber
\\ &+
\frac{1}{E_{0}-E_{l}}
( (\mathcal{H}_{A}+\mathcal{H}_{B})\vert \psi_{0} \rangle
\langle \psi_{0} \vert + \vert \psi_{0} \rangle \langle \psi_{0} \vert
(\mathcal{H}_{A}+\mathcal{H}_{B})).
\end{align}
\end{small}
Owing to the fact that, here, the Hamiltonian $ \mathcal{H}_{A} $ acts only on the subsystem A,
the reduced density matrix can be calculated by tracing out the subsystem B
\begin{eqnarray}
\rho_{red}=\rho_{1} + \frac{1}{E_{0}-E_{l}}
\left(\mathcal{H}_{A} \rho_{1} + \rho_{1} \mathcal{H}_{A} \right)
\end{eqnarray}
where $ \rho_{1} = {\textrm tr}_{2} \vert \psi_{0} \rangle \langle \psi_{0} \vert $
is the reduced density matrix within the zeroth order of the perturbation theory.
When the two subsystems are maximally entangled, $ \rho_{1} $ is proportional to the unit matrix.
In this case, we obtain
\begin{eqnarray}
\rho_{red} = \rho_{1}\left(1-\frac{2}{E_{0}-E_{l}}\mathcal{H}_{A}\right).
\end{eqnarray}
The reduced density matrix can be reformulated as
\begin{eqnarray}
\rho_{red}=\frac{1}{Z}\exp(-\mathcal{H}_{ent}^{(1)})
\end{eqnarray}
where the entanglement Hamiltonian $ \mathcal{H}_{ent} $ is the entanglement
Hamiltonian, and the partition function $Z$ is $ Z=tr\left(exp(-\mathcal{H}_{ent}^{(1)})\right) $.
The entanglement Hamiltonian
\begin{eqnarray}
\mathcal{H}_{ent}=\frac{2}{E_{0}-E_{l}}\mathcal{H}_{A}
\end{eqnarray}
is proportional to the Hamiltonian of subsystem A, with the proportionality
factor $ \beta = \frac{2}{E_{0}-E_{l}} $ interpreted as an inverse temperature.
To conclude, we assume that
\begin{enumerate}[label=\textnormal{(\arabic*)}]
\item $ \frac{1}{E_{0}-E_{l}} $ is equal for every $ l $, and\label{itm:1}
\item $ \rho_{1} $ is proportional to the unit matrix\label{itm:2}.
\end{enumerate}
These two assumptions directly lead to the proportionality between
the entanglement and subsystem Hamiltonians in the strong coupling limit
within the first order of the perturbation theory, when the ground state is
nondegenerate.
In Ref. \cite{POI10}, Poilblanc stressed a remarkable similarity between the chain--
chain entanglement spectrum in the two-leg spin-1/2 ladders and the energy spectrum of a
single spin-1/2 Heisenberg chain.
L\"auchli and Schliemann \cite{LAE12} analytically showed that the entanglement
Hamiltonian of the two coupled anisotropic XXZ chains is proportional to the energy
Hamiltonian of the single chain with renormalized anisotropy in the first order of the
perturbation theory in the strong coupling limit. There, the first assumption \ref{itm:1}
is not valid. In the case of the
isotropic Heisenberg ladders, both assumptions \ref{itm:1} and \ref{itm:2} are valid, and for that reason,
they found that the entanglement spectrum is directly
proportional to the energy of the single chain.
The authors in Ref. \cite{SCH12} generalized this observation
for the isotropic Heisenberg ladders to the case of the arbitrary spin length
$ \textit{S} $. They found that for arbitrary spin, the entanglement spectrum
of the isotropic Heisenberg ladders is proportional to the energy of the single chain
within the first-order perturbation theory. This is also a consequence of the fact that both
assumptions \ref{itm:1} and \ref{itm:2} hold.
However, they found that there is no proportionality
between the entanglement Hamiltonian of anisotropic spin ladders of arbitrary
spin length. Since here, the reduced density matrix in zeroth order of the perturbation
theory is not proportional to the unit matrix, there is no mention of proportionality.
\section{Model}
\label{model}
We investigate the Hamiltonian of the Heisenberg spin-1/2 ladder in a time-dependent
circularly polarized magnetic field $ B $ described by the Hamiltonian
\begin{align}
\tilde{H}&=J_{rung}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+1}
+B\sum_{i}\left(S_{i}^{x}\cos\omega t-S_{i}^{y}\sin\omega t\right) \nonumber \\
&+J_{leg}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+2}
+J_{leg}\sum_{i}\vec{S}_{2i+1}\vec{S}_{2i+3}. \label{Ham_cir}
\end{align}
where $ \omega $ is the angular velocity of the rotation of the magnetic field.
The sites on the first (second) leg are denoted by even (odd) labels, such
that the $ i $th rung consists of sites $ 2i $ and $ 2i+1 $.
All spin-1/2 operators are taken to be dimensionless, such that the couplings along
the rungs $ J_{rung} $ and the legs $ J_{leg} $ have the dimensions of energy.
We will consider antiferromagnetic coupling when $ J_{rung}>0 $.
This time-dependent Hamiltonian can be factorized to a time-independent Hamiltonian
by unitary transformations that represent a rotation around the z-axis
$ R(t)=\exp(-iS_{z}\omega t/ \hbar) $ \cite{QT}.
Since
\begin{align}
&R(t)S_{x}R^{-1}(t)=S_{x}\cos \omega t+S_{y}\sin \omega t \nonumber \\
&R(t)S_{y}R^{-1}(t)=-S_{x}\sin \omega t+S_{y}\cos \omega t \nonumber \\
&R(t)S_{z}R^{-1}(t)=S_{z} \label{transformations}
\end{align}
Hamiltonian Eq. (\ref{Ham_cir}) can be transformed into a time-independent Hamiltonian
\begin{align}
\widehat{H}&=R(t)\tilde{H}(t)R(t)^{-1}\\
\widehat{H}&=J_{rung}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+1}+B\sum_{i}S_{i}^{x}
+J_{leg}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+2} \nonumber \\
&+J_{leg}\sum_{i}\vec{S}_{2i+1}\vec{S}_{2i+3}. \label{Ham_tid}
\end{align}
Defining the propagator that confirms
\begin{align}
&\frac{\partial}{\partial t}K(t,t_{0})=-\frac{i}{\hbar}\tilde{H}(t)K(t,t_{0}) \\
&\frac{\partial}{\partial t}K(t,t_{0})=-\frac{i}{\hbar}R_{-1}(t)\widehat{H}(t)R(t)K(t,t_{0})
\end{align}
we find
\begin{eqnarray}
\frac{\partial}{\partial t}\left(R(t)K(t,t_{0})R^{-1}(t_{0})\right)=-\frac{i}{\hbar}H\left(R(t)K(t,t_{0})R^{-1}(t_{0})\right). \nonumber
\end{eqnarray}
Then, the Hamiltonian becomes
\begin{align}
H=&J_{rung}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+1}
+B\sum_{i}S_{i}^{x}+\omega\sum_{i}S_{i}^{z} \nonumber \\
+&J_{leg}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+2}
+J_{leg}\sum_{i}\vec{S}_{2i+1}\vec{S}_{2i+3}\label{Hami}
\end{align}
where the propagator is
\begin{footnotesize}
\begin{eqnarray}
K(t,t_{0})=\exp\left(\frac{i}{\hbar}S_{z}\omega t\right)
\exp\left(-\frac{i}{\hbar}H(t-t_{0})\right)\exp
\left(-\frac{i}{\hbar}S_{z}\omega t_{0}\right).
\label{propagator}
\end{eqnarray}
\end{footnotesize}
In order to use the perturbation theory, we will rewrite the Hamiltonian
Eq.(\ref{Hami}) as $ H=H_{0}+H_{1} $ where
\begin{eqnarray}
H_{0}=J_{rung}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+1}+B\sum_{i}S_{i}^{x}+\omega\sum_{i}S_{i}^{z}
\label{Hami0}
\end{eqnarray}
and
\begin{eqnarray}
H_{1}=J_{leg}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+2}
+J_{leg}\sum_{i}\vec{S}_{2i+1}\vec{S}_{2i+3} \label{Hami1}
\end{eqnarray}
and consider $ H_{1} $ as a small perturbation.
The Hamiltonian Eq.(\ref{Hami0}) is independent of the
direction of the magnetic field and
it can be considered as the isotropic Heisenberg chain in the magnetic field
$ \sqrt{B^{2}+\omega^{2}} $
\begin{eqnarray}
H_{0}&=J_{rung}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+1}
+\sqrt{B^{2}+\omega^{2}}\sum_{i}S_{i}^{z} \label{Hami10}
\end{eqnarray}
and
\begin{eqnarray}
H_{1}&=J_{leg}\sum_{i}\vec{S}_{2i}\vec{S}_{2i+2}
+J_{leg}\sum_{i}\vec{S}_{2i+1}\vec{S}_{2i+3}.
\label{Hami11}
\end{eqnarray}
The energies of a rung of the singlet and triplet states are
\begin{align}
&E_{s_{i}}=-\frac{3}{4} J_{rung}\\
&E_{t_{i}^{+}}=\frac{1}{4} J_{rung}+\sqrt{B^{2}+\omega^{2}} \\
&E_{t_{i}^{0}}=\frac{1}{4} J_{rung} \\
&E_{t_{i}^{-}}=\frac{1}{4} J_{rung}-\sqrt{B^{2}+\omega^{2}},
\end{align}
The ground state changes from the spin singlet $ \vert s_{i} \rangle $,
to the triplet state $ \vert t^{-}_{i} \rangle $
by increasing the value of $ \sqrt{B^{2}+\omega^{2}} $ .
When $ J_{rung}= \sqrt{\omega^{2}+B^{2}} $, the ground state
is two-fold degenerate, since the singlet states $ \vert s_{i}\rangle $
and triplet states $ \vert t_{i}^{-}\rangle $ have the same eigenenergy.
The situation when ground state is two-fold degenerate is quite interesting
and it will be considered in the following section.
\section{Entanglement spectrum}
When $ J_{rung}= \sqrt{\omega^{2}+B^{2}} $,
it is necessary to use degenerate perturbation theory, while
any combination of eigenstates $ \vert s_{i}\rangle $
and $ \vert t_{i}^{-}\rangle $ can be taken as the ground state
$ \vert \psi_{0} \rangle $.
In order to achieve an analytically manageable situation, we will assume
a finite number of rungs $ i = 4 $.
Let us suppose that the unperturbed ground state $ \vert psi_0\rangle $
of the Hamiltonian $ \mathcal{H}_{0} $ is an unknown
combination of eigenvectors $ \vert s_{i}\rangle $
and $ \vert t_{i}^{-}\rangle $. In the following, we will note
eigenvectors of the ground state $ \vert \psi_0\rangle $ as
$ \lbrace\vert \overline{n}\lambda \rangle \vert, \lambda=1,...,4\rbrace $,
where
\begin{align}
&\vert\overline{n}1\rangle=\vert s_{1}\rangle \vert s_{2}\rangle
\vert s_{3}\rangle \vert s_{4}\rangle ,
\vert\overline{n}2\rangle=\vert s_{1}\rangle \vert s_{2}\rangle
\vert s_{3}\rangle \vert t_{4}^{-}\rangle, \nonumber \\
&\vert\overline{n}3\rangle=\vert s_{1}\rangle \vert s_{2}\rangle
\vert t_{3}^{-}\rangle \vert s_{4}\rangle,
\vert\overline{n}4\rangle=\vert s_{1}\rangle \vert t_{2}^{-}\rangle
\vert s_{3}\rangle \vert s_{4}\rangle, \nonumber \\
&\vert\overline{n}5\rangle=\vert t_{1}^{-}\rangle \vert s_{2}\rangle
\vert s_{3}\rangle \vert s_{4}\rangle,
\vert\overline{n}6\rangle=\vert t_{1}^{-}\rangle \vert t_{2}^{-1}\rangle
\vert s_{3}\rangle \vert s_{4}\rangle, \nonumber \\
&\vert\overline{n}7\rangle=\vert t_{1}^{-1}\rangle \vert s_{2}\rangle
\vert t_{3}^{-1}\rangle \vert s_{4}\rangle,
\vert\overline{n}8\rangle=\vert t_{1}^{-1}\rangle \vert s_{2}\rangle
\vert s_{3}\rangle \vert t_{4}^{-1}\rangle, \nonumber \\
&\vert\overline{n}9\rangle=\vert s_{1}\rangle \vert t_{2}^{-1}\rangle
\vert t_{3}^{-1}\rangle \vert s_{4}\rangle,
\vert\overline{n}10\rangle=\vert s_{1}\rangle \vert t_{2}^{-1}\rangle
\vert s_{3}\rangle \vert t_{4}^{-1}\rangle, \nonumber \\
&\vert\overline{n}11\rangle=\vert s_{1}\rangle \vert s_{2}\rangle
\vert t_{3}^{-1}\rangle \vert t_{4}^{-1}\rangle,
\vert\overline{n}12\rangle=\vert t_{1}^{-1}\rangle \vert t_{2}^{-1}\rangle
\vert t_{3}^{-1}\rangle \vert s_{4}\rangle, \nonumber \\
&\vert\overline{n}13\rangle=\vert t_{1}^{-1}\rangle \vert t_{2}^{-1}\rangle
\vert s_{3}\rangle \vert t_{4}^{-1}\rangle,
\vert\overline{n}14\rangle=\vert t_{1}^{-1}\rangle \vert s_{2}\rangle
\vert t_{3}^{-1}\rangle \vert t_{4}^{-1}\rangle, \nonumber \\
&\vert\overline{n}15\rangle=\vert s_{1} \rangle \vert t_{2}^{-1}\rangle
\vert t_{3}^{-1}\rangle \vert t_{4}^{-1}\rangle,
\vert\overline{n}16\rangle=\vert t_{1}^{-1}\rangle \vert t_{2}^{-1}\rangle
\vert t_{3}^{-1}\rangle \vert t_{4}^{-1}\rangle
\end{align}
The projector $ P_{n}^{(0)} $ of $ \mathcal{H}_{0} $ projects
on the subspace, and is defined by the eigenvalue $ E_{\overline{n}}^{(0)}=-\frac{3}{4} J_{rung} $
of the Hamiltonian $ \mathcal{H}_{0} $.
Furthermore, the projector $ P_{n}^{(0)} $ satisfies
\begin{eqnarray}
P_{n}^{(0)} \mathcal{H}^{'} P_{n}^{(0)} \vert \psi_{0} \rangle =
E^{(1)} \vert \psi_{0} \rangle
\end{eqnarray}
where $ E^{(1)} $ is the eigenvalue of $ P_{n}^{(0)} \mathcal{H}^{'} P_{n}^{(0)} $
for eigenvector $ \vert \psi_{0} \rangle $. In order to find the eigenvalue
$ E^{(1)} $ of the perturbation $ \mathcal{H^{'}} $ and the ground state
$ \vert \psi_0 \rangle $, it is sufficient to diagonalize a $16\times16$ matrix
\begin{equation}
\left[
\begin{array}{ccc}
\langle \overline{n} 1 \vert \mathcal{H}^{'} \vert \overline{n} 1 \rangle &
\cdot \cdot \cdot &
\langle \overline{n} 1 \vert \mathcal{H}^{'} \vert \overline{n} 16 \rangle \\
\cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\
\langle \overline{n} 16 \vert \mathcal{H}^{'} \vert \overline{n} 1 \rangle &
\cdot \cdot \cdot & \langle \overline{n} 16 \vert \mathcal{H}^{'} \vert \overline{n} 16 \rangle
\end{array}
\right].
\end{equation}
By elementary calculations, one finds the uniquely determined ground state $
\vert \psi_0 \rangle $ and the first correction of the energy $ E^{(1)} $.
The unperturbed density matrix is constructed from this ground state and is given,
after simplification, by
\begin{eqnarray}
\rho^{(0)}=& \sum_{\lambda}^{16} \vert \overline{n} \lambda \rangle \langle
\overline{n} \lambda \vert.
\end{eqnarray}
By again tracing out one leg, we obtain the reduced unperturbed density matrix
\begin{eqnarray}
\rho_{red}^{(0)}=\frac{1}{2^{4}}\bigotimes_{i=1}^{4}\left(1-S_{2i+1}^{z}\right)
\end{eqnarray}
It is obvious that this reduced density matrix is not proportional to the unitary matrix
and possesses a nontrivial entanglement spectrum.
The first corrections to the ground state in the degenerate perturbation theory
are defined by
\begin{align}
\vert 1\rangle=&\sum_{n\neq \overline{n}} \vert n \lambda \rangle
\frac{\langle n \lambda \vert H_{1} \vert \psi_{0}\rangle}
{E_{\overline{n}}^{(0)}-E_{n}^{(0)}}, \\
\vert 1^{'}\rangle=&\sum_{n\neq \overline{n},n'} \sum_{\lambda'}
\vert \overline{n} \lambda \rangle
\frac{\langle \overline{n} \lambda \vert H_{1} \vert n' \lambda\rangle}
{E^{(1)}-E_{\lambda}^{(1)}}
\frac{\langle n' \lambda \vert H_{1} \vert \psi_{0}\rangle}
{E_{\overline{n}}^{(0)}-E_{n'}^{(0)}}.
\end{align}
One finds the reduced density matrix to the first order
\begin{align}
\rho_{red}^{(1)}=&\frac{1}{2^{4}}\bigotimes_{i=1}^{4}\left(1-S_{2i+1}^{z}\right)
-\frac{J_{leg}}{8J_{rung}}\left((1-S_{1}^{z})(1-S_{3}^{z})\vec{S}_{5}\vec{S}_{7}\right.
\nonumber \\
+&\left.(1-S_{1}^{z})\vec{S}_{3}\vec{S}_{5}(1-S_{7}^{z})
+\vec{S}_{1}\vec{S}_{3}(1-S_{5}^{z})(1-S_{7}^{z})
\right)
\label{rdm}
\end{align}
The reduced density matrix can be rewritten as
\begin{eqnarray}
\rho_{red}=\frac{1}{Z}\exp(-\mathcal{H}_{ent}^{(1)})
\end{eqnarray}
where the partition function $Z$ is $ Z=\textrm{tr} \exp(-\mathcal{H}_{ent}^{(1)}) $
and the entanglement Hamiltonian within the first order of the perturbation theory
has the following form
\begin{eqnarray}
\mathcal{H}_{ent}^{(1)}=\frac{1}{J_{rung}}\sum_{i=0}^{3}\left(2J_{leg}\vec{S}_{2i+1}\vec{S}_{2i+3}+
J_{rung} S_{2i+1}\right).
\label{ent_Ham1}
\end{eqnarray}
The entanglement Hamiltonian is simply proportional to the
Hamiltonian of a chain in the magnetic field with the proportional factor
$ \beta=\frac{1}{J_{rung}} $ defined as an inverse temperature.
The system of the Heisenberg chain in the longitudinal magnetic field is
exactly solvable by the Bethe ansatz. The ground state becomes the spin--liquid
one and gapless up to when $ \frac{\sqrt{B^{2}+\omega^{2}}}{J_{leg}} = 2 $,
where the phase transition of the Pokrovsky--Talapov type takes place and the
ground state becomes a completely ordered gapped ferromagnetic state.
One of the most important features of the energy spectra of spin chains is the absence
of an excitation gap over the ground state for the integer spin length.
We restrict ourselves to the case when the two chains are strongly
coupled; therefore, $ \frac{\sqrt{B^{2}+\omega^{2}}}{J_{leg}} >> 1 $ and the Hamiltonian
of a subsystem stays gapless in this region. The entanglement spectrum
Eq.(\ref{ent_Ham1}) remains gapless owing to the proportionality to the energy spectrum.
\section{Summary}
\label{conclusion2}
Here we investigated the entanglement spectrum of Heisenberg ladders in a
time-dependent magnetic field using the degenerate perturbation theory, where couplings
between legs are taken as a small perturbation. When the ground state is not
degenerate, the existence of the trivial entanglement Hamiltonian in the zeroth order of the perturbation
theory is identified as an important condition that guarantees the proportionality
between the entanglement and subsystem Hamiltonians.
We find that although the entanglement spectrum of the unperturbed
ground state has a nontrivial entanglement spectrum,
the entanglement Hamiltonian of Heisenberg ladders in a
time-dependent magnetic field, within the first-order perturbation theory,
is proportional to the energy Hamiltonian of a chain in the magnetic field
when the ground state is degenerate.
\section{ACKNOWLEDGMENTS}
The author kindly acknowledges John Schliemann.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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<?php
namespace App\Http;
use Illuminate\Foundation\Http\Kernel as HttpKernel;
class Kernel extends HttpKernel
{
/**
* The application's global HTTP middleware stack.
*
* @var array
*/
protected $middleware = [
\Illuminate\Foundation\Http\Middleware\CheckForMaintenanceMode::class,
\App\Http\Middleware\EncryptCookies::class,
\Illuminate\Cookie\Middleware\AddQueuedCookiesToResponse::class,
\Illuminate\Session\Middleware\StartSession::class,
\Illuminate\View\Middleware\ShareErrorsFromSession::class,
// \App\Http\Middleware\VerifyCsrfToken::class,
];
/**
* The application's route middleware.
*
* @var array
*/
protected $routeMiddleware = [
'auth' => \App\Http\Middleware\Authenticate::class,
'auth.basic' => \Illuminate\Auth\Middleware\AuthenticateWithBasicAuth::class,
'guest' => \App\Http\Middleware\RedirectIfAuthenticated::class,
'role' => \Zizaco\Entrust\Middleware\EntrustRole::class,
'permission' => \Zizaco\Entrust\Middleware\EntrustPermission::class,
'ability' => \Zizaco\Entrust\Middleware\EntrustAbility::class,
'jwt.auth' => 'Tymon\JWTAuth\Middleware\GetUserFromToken',
'jwt.refresh' => 'Tymon\JWTAuth\Middleware\RefreshToken',
'cors' => \App\Http\Middleware\Cors::class,
'log' => \App\Http\Middleware\LogMiddleware::class,
];
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 621
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\section{Introduction}
In 1971, Zakharov and Shabat (ZS) showed that the nonlinear Schr\"odinger equation (NLSE)
\begin{equation}\label{nlse}
i\frac{\partial q}{\partial z}+\frac{\sigma}{2}\frac{\partial^2 q}{\partial t^2}+|q|^2q=0,\quad \sigma=\pm 1
\end{equation}
can be integrated by the inverse problem method (or so-called nonlinear Fourier transform -- NFT) previously applied to the Korteweg de Vries equation~\cite{ZakharovShabat1972}. After that, interest in the NLSE arose in all areas of physics connected with wave systems, because the NLSE describes the envelope for narrow wave beams.
In 1973, Hasegawa and Tappert numerically investigated the NLSE in respect to the propagation of light pulses in optical fibers~\cite{Hasegawa1973a}. They proposed using solitons as an information carrier for fiber lines with anomalous dispersion at $\sigma=1$. For normal dispersion at $\sigma =-1$, solitons do not exist, as is well known.
Since that time, the study of the NLSE and its generalizations to describe the propagation of light pulses in optical fibers has begun. Analytical and numerical studies were carried out, as well as work on the development of numerical methods for integrating NLSE. Currently, the most popular and effective method is splitting into physical processes, the so-called split-step Fourier method (SSFM)~\cite{Hardin_SIAM_1973}.
On the other hand, attempts to create fast numerical algorithms for solving the inverse scattering problem for the NLSE have not stopped. Such methods are combined under the general name Fast Nonlinear Fourier Transform (FNFT)~\cite{Wahls2013, Wahls2015a, Wahls2016, Turitsyn2017Optica}.
In this paper, we propose a special fourth-order numerical method for solving the direct ZS problem (ZSP) and a fast algorithm for its numerical implementation.
The main advantage of the presented scheme is the conservation of the quadratic invariant for real spectral parameters, even in the fast version. This is the first proposed fast scheme with such property for the best of our knowledge.
The quadratic invariant conservation by numerical scheme allows calculating precisely the reflection coefficient, what is valuable for various telecommunication problems connected with NFT-based coding schemes (for example, NFDM~\cite{Yousefi2014III} and $b$-modulation~\cite{Wahls2017}).
\section{Direct spectral ZS problem}\label{sec:headings}
Direct spectral ZS problem for the NLSE~(\ref{nlse}) with the complex spectral parameter $\zeta$ can be rewritten as an evolutionary system
\begin{equation}\label{psit}
\frac{d{\Psi}(t)}{dt}=Q(t){\Psi}(t),
\end{equation}
where $q=q(t,z_0)$ is the initial field for the NLSE at the point $z_0$, which is the potential in the ZS problem, and
$$
{\Psi}(t)=\left(\begin{array}{c}\psi_1(t)\\\psi_2(t)\end{array}\right),\quad
Q(t)=\left(\begin{array}{cc}-i\zeta&q\\-\sigma q^*&i\zeta\end{array}\right).
$$
Here $z_0$ plays the role of the parameter and we will omit it. For details, we refer to the numerous literature, in particular,~\cite{medvedev2019exponential}.
Moreover, the system~(\ref{psit}) can be written in the gradient form
\begin{equation}
\left(\begin{array}{c}\psi_1\\\psi_2\end{array}\right)_t=KD
\left(\begin{array}{c}\psi_1\\\psi_2\end{array}\right)=K
\left(\begin{array}{c}\frac{\partial H}{\partial\psi_1^*}\\\frac{\partial H}{\partial\psi_2^*}\end{array}\right),
\end{equation}
where $H=({\Psi}^*,D{\Psi})$,
\begin{equation}
K=\left(\begin{array}{cc}-i\zeta&\sigma q\\-\sigma q^*&i\sigma\zeta\end{array}\right),\quad
D=\left(\begin{array}{cc}1&0\\0&\sigma\end{array}\right).
\end{equation}
For real~$\zeta=\xi$ the matrix~$K$ before the gradient becomes anti-Hermitian $K=-K^\dagger$ for any $\sigma=\pm 1$ and, therefore, the system~(\ref{psit}) will conserve the quantity~$H$.
Assuming that $q(t)$ decays rapidly when $t\to \pm \infty $, the specific solutions (Jost functions) for ZSP~(\ref{psit}) can be derived as
\begin{equation}\label{psi0}
\Psi =\left(
\begin{array}{c}
\psi_{1}\\\psi_{2}
\end{array}
\right) = \left(
\begin{array}{c}
e^{-i\zeta t}\\0
\end{array}
\right)[1+o(1)],\quad t\to-\infty,
\end{equation}
and
\begin{equation}\label{psi0right}
\Phi =\left(
\begin{array}{c}
\phi_{1}\\\phi_{2}
\end{array}
\right) = \left(
\begin{array}{c}
0\\e^{i\zeta t}
\end{array}
\right)[1+o(1)],\quad t\to \infty,
\end{equation}
Then we obtain the Jost scattering coefficients $a(\xi )$ and $b(\xi )$ as
\begin{equation}\label{ab}
a(\xi)=\lim_{t\to\infty}\,\psi_1(t,\xi)\,e^{i\xi t},\quad b(\xi)=\lim_{t\to\infty}\,\psi_2(t,\xi)\,e^{-i\xi t}.
\end{equation}
The functions $a(\xi)$ and $b(\xi)$ can be extended to the upper half-plane $\xi \to \zeta $, where $\zeta $ is a complex number with the positive imaginary part $\eta =Im\zeta >0$. The spectral data of ZSP~(\ref{psit}) are determined by $a(\zeta )$ and $b(\zeta )$ in the following way:\\
(1) zeros of $a(\zeta)=0$ define the discrete spectrum $\{\zeta_k\}$, $k=1,...,K$ of ZSP~(\ref{psit}) and phase coefficients
$$r_k=\left.\frac{b(\zeta)}{a'(\zeta)}\right|_{\zeta=\zeta_k},\quad\mbox{where}\quad a'(\zeta)=\frac{da(\zeta)}{d\zeta};$$
(2) the continuous spectrum is determined by the reflection coefficient
$$r(\xi)={b(\xi)}/{a(\xi)},\quad \xi\in\mathbb{R}. $$
These spectral data were defined using the "left"\, boundary condition (\ref{psi0}). Both conditions (\ref{psi0}) and (\ref{psi0right}) can be used to calculate the coefficient~$b(\zeta _{k} )$ of the discrete spectrum with the relation $\Psi(t,\zeta_k)=\Phi(t,\zeta_k)b(\zeta_k)$.
For real values of the spectral parameter $\zeta=\xi$, we have the quadratic invariant~$H=|\psi_1|^2+\sigma|\psi_2|^2$.
Taking into account the boundary conditions~(\ref{psi0}), we get the same condition $H=1$ for $\sigma=\pm 1$.
\section{Computational features of ZS system}
We solve a linear system of the form~(\ref{psit}) with the matrix~$Q(t)$ linearly dependent on the complex function~$q(t)$. The numerical implementation of the continuous function~$ q(t)$ is a discrete function~$q_n=q(t_n)$, which is defined at the integer nodes $t_n$ of the uniform grid with the step $\tau$. Since we are considering a finite time interval, we will solve the problem on the interval $[-L,L]$ with the total number of points equal to $M+1$, the grid step in this case is $\tau=2L/M$ and $t_n=-L+\tau n$, where $n=0,...,M$.
The main features of the discrete problem are the following:
\begin{enumerate}
\item The matrix of the system $Q$ is given on a uniform grid with a step $\tau$, therefore the unknown function $\Psi$ must also be calculated on a uniform grid with a step $\tau$. One cannot use Runge-Kutta methods on such grid. If, for example, an explicit 4th-order RC scheme is used, then it is necessary to take the computational grid with a double step. In this case, the values of $Q_n$ will enter unequally.
\item For small values of the potential $|q(t)|<<|\zeta|$ and $\mbox{Im}\,\zeta>0$, the ZS system contains exponentially increasing and decreasing solutions therefore A-stability of difference methods is required~\cite{dahlquist1963special}. {\it Dalquist's Second Barrier} restricts multistep methods.This barrier says that there are no explicit A-stable multistep methods. 2nd order of convergence is maximum for implicit multi-step methods~\cite{hairer1991solving}.
\item The ZS system has a second-order matrix; therefore, inverse matrices and the exponent of the matrix are easily calculated. This allows us to include almost any functions of the matrix~$Q$ in difference schemes.
\item To calculate the spectral data, it is required to solve the ZS system for a large number of spectral parameter values~$\zeta$ with the fixed potential $q(t)$. If possible, this should be taken into account when implementing the algorithms.
\end{enumerate}
\section{Scheme}
Previously, we found the necessary conditions for the existence of a one-step fourth-order scheme $\Psi(t_n+\tau/2)=T_n\Psi(t_n-\tau/2)$ in the form of the Taylor series for the transition matrix~$T_n$. We obtained several fourth-order schemes that exactly conserve the quadratic invariant~$H$. However, only one scheme in the form of three exponentials is convenient for the fast computation~\cite{medvedev2019exponential}:
\begin{equation}\label{tripleexp}
T_n=e^{\left\{\frac{\tau^2}{12}Q^{(1)}_n+\frac{\tau^3}{48}Q^{(2)}_n\right\}}e^{\tau Q_n}e^{\left\{-\frac{\tau^2}{12}Q^{(1)}_n+\frac{\tau^3}{48}Q^{(2)}_n\right\}},
\end{equation}
here $Q_n=Q(t_n)$ and $Q^{(k)}_n$, $k =1,2,$ are the $k$-th derivative of the matrix $Q$ approximated by central finite differences of the second order.
To construct the fast algorithm, we must express $\exp(\tau Q)$, where $Q=A+B$, and
\begin{equation}
A=\left(\begin{array}{cc}-i\zeta&0\\0&i\zeta\end{array}\right),\quad
B(t)=\left(\begin{array}{cc}0&q\\-\sigma q^*&0\end{array}\right),
\end{equation}
in the form of a polynomial in exponentials of $A$ and $B$ with rational weights. For example, one can use the expansions for $\exp(\tau Q)$ suggested in \cite{Prins2018a}. However, representing the sum of the product of exponentials does not guarantee the exact conservation of the invariant~$H$. In order for the scheme to be suitable for the fast algorithm and conserve the invariant~$H$, it suffices to represent the matrix $\exp(\tau Q)$ as the product of exponentials of $A$ and $B$ with real rational coefficients.
Since for $\sigma=1$ the matrices $A$ and $B$ are Hermitian, then, in this case, each exponent will be unitary and the resulting scheme will conserve the quadratic invariant~$H$. Conserving~$H$ also takes place for $\sigma=-1$. Details can be found in \cite{medvedev2019exponential}. The rationality condition for weight coefficients provides an opportunity to represent the transition matrix in the form of the ratio of two polynomials in~$\exp(A)$.
\section{Suzuki factorization}
Since the scheme~(\ref{tripleexp}) has a fourth order of accuracy in $\tau$, it is necessary to have factorization of the same order. In addition, factorization should be suitable for a fast algorithm, i.e. have rational ratios. An example of such factorization is given in
\cite{suzuki1992general}:
\begin{equation}\label{factor4}
e^{\tau(A+B)}=
e^{\frac{7}{48}\tau B}e^{\frac{1}{3}\tau A}e^{\frac{3}{8}\tau B}e^{-\frac{1}{3}\tau A}e^{-\frac{1}{48}\tau B} e^{\tau A}e^{-\frac{1}{48}\tau B}e^{-\frac{1}{3}\tau A}
e^{\frac{3}{8}\tau B}e^{\frac{1}{3}\tau A}e^{\frac{7}{48}\tau B}.\nonumber
\end{equation}
We introduce the notation $Z = \exp(-\frac{i}{3}\tau\zeta)$, then the three exponents participating in this expansion take the form
\begin{equation}
e^{\frac{1}{3}\tau A}=\left(\begin{array}{cc}Z&0\\0&Z^{-1}\end{array}\right)=Z^{-1}\left(\begin{array}{cc}Z^2&0\\0&1\end{array}\right),
\end{equation}
\begin{equation}
e^{-\frac{1}{3}\tau A}=\left(\begin{array}{cc}Z^{-1}&0\\0&Z\end{array}\right)=Z^{-1}\left(\begin{array}{cc}1&0\\0&Z^2\end{array}\right),
\end{equation}
\begin{equation}
e^{\tau A}=\left(\begin{array}{cc}Z^3&0\\0&Z^{-3}\end{array}\right)=Z^{-3}\left(\begin{array}{cc}Z^6&0\\0&1\end{array}\right).
\end{equation}
Thus, the right-hand side of Eq.~(\ref{factor4}) is a rational function
\begin{equation}\label{SZ}
\frac{S(Z)}{Z^7},
\end{equation}
where $S(Z)$ is a polynomial not higher than 14 degrees in $Z$.
Since $Z$ is included only in the square, it is possible to introduce the variable $W=Z^2$, then (\ref{SZ}) takes the form
\begin{equation}\label{SW}
\frac{\hat{S}(W)}{W^{\frac{7}{2}}},
\end{equation}
where $\hat{S}(W)$ is a polynomial of degree 7 or less in $W$. The denominator is taken out and calculated independently.
It should be noted, that except of factorization~(\ref{factor4}) the symmetrical representation can be applied, when the matrices $A$ and $B$ are interchanged.
Such factorization leads to the polynomial~$\hat{S}(W)$ of degree 52, which is more computationally difficult and is less accurate, thus it is not considered here.
\section{Numerical examples}
The presented scheme implementation was based on FNFT software library~\cite{Wahls2018}. It was compared with
Boffetta-Osborne scheme (BO)~\cite{Boffetta1992a} and
the triple-exponential scheme with not factorized exponential~(\ref{tripleexp}) (TES4).
All these algorithms conserve the quadratic invariant~$H$ for real spectral parameters~$\xi$.
We do not compare our scheme with other fast algorithms (see, for example, \cite{Prins2018a, Wahls2018, Vaibhav2018}), due to the theoretical lack of such property among them.
We have considered both variants of the scheme with Suzuki factorization: conventional (TES4SB) and fast (FTES4SB). The last letter in the scheme name denotes the decomposition type: TES4SA denotes the scheme with the exponential with matrix~$A$ at the edges of Suzuki decomposition, while TES4SB is referred to the scheme~(\ref{factor4}).
A model signal was considered in the form of a chirped hyperbolic secant
\begin{equation}\label{Chirped}
q(t) = A[\mbox{sech}(t)]^{1+iC}.
\end{equation}
with the following parameters: $A = 5.2$, $C = 4$ for both anomalous and normal dispersion.
The detailed analytical expressions of the spectral data for this type of potentials can be found in~\cite{medvedev2019exponential}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{figure1.pdf}
\caption{The continuous spectrum errors for the chirped hyperbolic secant~(\ref{Chirped}) in the case of anomalous dispersion~$\sigma=1$.}
\label{fig:errorS1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{ContinuousTest_OverSoliton_sigma=-1_chirp=4_5,2_Errors.pdf}
\caption{The continuous spectrum errors for the chirped hyperbolic secant~(\ref{Chirped}) in the case of normal dispersion~$\sigma=-1$.}
\label{fig:errorSm1}
\end{figure}
We present the numerical errors of calculating the spectral data for continuous spectrum only, because of focusing on the conservation of the invariant~$H$ for real spectral parameters~$\xi$. To find the calculation errors of the continuous spectrum energy~$E_c$ the following formula was used
\begin{equation}\label{error}
\mbox{error}[E_c]=\frac{|E_c^{comp} - E_c^{exact}|}{\phi_0}, \quad
\phi_0 =
\begin{cases}
E_c^{exact}, \mbox{ if } |E_c^{exact}| > 1\\
1, \mbox{ otherwise},
\end{cases}
\end{equation}
For the continuous spectrum we calculated the root mean squared error
\begin{equation}\label{MSE}
RMSE[\phi]=\sqrt{\frac{1}{N}\sum_{j=1}^{N}\frac{|\phi^{comp}(\xi_j) - \phi^{exact}(\xi_j)|^2}{|\phi_0(\xi_j)|^2}}, \quad
\phi_0 =
\begin{cases}
\phi^{exact}(\xi_j), \mbox{ if } |\phi^{exact}(\xi_j)| > 1\\
1, \mbox{ otherwise},
\end{cases}
\end{equation}
where $\phi$ can represent $a(\xi)$, $b(\xi)$, $r(\xi)$ or $|H^{comp}(\xi) - H^{exact}(\xi)|$. Here we assume the spectral parameter $\xi \in [-20, 20]$ with the total number of points $N = 1025$.
Figures~\ref{fig:errorS1}, \ref{fig:errorSm1} present the errors calculated using the schemes under consideration. All schemes with triple-exponential representation demonstrated 4th order for the model signal and similar values of error in the case of anomalous dispersion.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{ContinuousTest_OverSoliton_sigma=1_chirp=4_5,2_ErrorVsTime.pdf}
\caption{The continuous spectrum errors depending on the execution time trade-off for the chirped hyperbolic secant~(\ref{Chirped}) in the case of anomalous dispersion~$\sigma=1$.}
\label{fig:errorTimeS1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{ContinuousTest_OverSoliton_sigma=-1_chirp=4_5,2_ErrorVsTime.pdf}
\caption{The continuous spectrum errors depending on the execution time trade-off for the chirped hyperbolic secant~(\ref{Chirped}) in the case of normal dispersion~$\sigma=-1$.}
\label{fig:errorTimeSm1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{ContinuousTest_OverSoliton_sigma=1_chirp=4_5,2_Times.pdf}
\caption{Execution times for different algorithms in the case of anomalous dispersion~$\sigma=1$.}
\label{fig:timeS1}
\end{figure}
The efficiency of the schemes is compared in figures~\ref{fig:errorTimeS1} and \ref{fig:errorTimeSm1}. The fast variant of the proposed algorithm FTES4SB demonstrated the best speed when getting the desired error value across all considered schemes for both signs of dispersion. Of course, due to an asymptotic complexity of fast methods~\cite{Wahls2013}, one can determine the temporal grid size~$M$ for a fixed number of spectral parameter values~$N$ when the speed and efficiency of the fast scheme FTES4SB become comparable with conventional algorithms, which is demonstrated by Figure~\ref{fig:timeS1}. We should note here that the execution times of all methods don't depend on the signs of dispersion.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{ContinuousTest_OverSoliton_chirp=4_5,2_invariant}
\caption{Maximum value of the quadratic invariant~$H$ conservation error for anomalous dispersion~$\sigma=1$~(a) and normal dispersion~$\sigma=-1$~(b).}
\label{fig:invariantS1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\linewidth]{Invariant_of_xi}
\caption{Quadratic invariant conservation error~$|1 - |\psi_1|^2-\sigma|\psi_2|^2|$ depending on the spectral parameter~$\xi$: (a)~$\sigma=1$, (b)~$\sigma=-1$.}
\label{fig:invariant}
\end{figure}
The conservation properties of the schemes are considered in Figures~\ref{fig:invariantS1} and \ref{fig:invariant}. All algorithms demonstrated good conservation of the quadratic invariant~$H$ for the anomalous dispersion, but in case of normal dispersion, an error sufficiently increases. This is caused by the subtraction of large modulo quantities. All conventional schemes are comparable in the magnitude of the error. The accuracy of the proposed scheme in a fast variant (FTES4SB) reaches close value, though the fast computational technique caused an increase in error by two orders of magnitude for the normal dispersion.
\section{Conclusion}
In conclusion, we have developed a new multi-exponential scheme based on our three-exponential scheme and Suzuki decomposition, which allows fast computation and conserves the quadratic invariant for the real spectral parameter. The scheme consists of 13 matrix exponentials and has the 4th order of approximation. Also, it works for uniform grids, what together with the quadratic invariant conservation makes the proposed scheme attractive for telecommunication problems.
\section*{Funding}
Russian Science Foundation (RSF) (17-72-30006).
\bibliographystyle{unsrt}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,038
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"Castles Made of Sand" is a song written by Jimi Hendrix and recorded by the Jimi Hendrix Experience for their 1967 second album, Axis: Bold as Love. Produced by manager Chas Chandler, the song is a biographical story about Hendrix's childhood, and was recorded towards the end of the production cycle for Axis: Bold as Love.
Recording and production
The Jimi Hendrix Experience began and finished work on the recording for "Castles Made of Sand" at London's Olympic Sound Studios on October 29, 1967, the penultimate day of recording for Axis: Bold as Love on which the songs "Up from the Skies", "Bold as Love", "One Rainy Wish" and "EXP" were also completed. As with the rest of the album, "Castles Made of Sand" was produced by Chas Chandler and engineered by Eddie Kramer, and was mixed at Olympic on October 31.
Composition and lyrics
Writing the Hendrix biography Jimi Hendrix: Electric Gypsy, commentators Harry Shapiro and Caesar Glebbeek summarise "Castles Made of Sand" as "a sharply observed reflection on life's bitter ironies". Addressing the lyrics of the song, Shapiro and Glebbeek go on to discuss the significance of sand within the track as a metaphor "for the temporary nature of existence, of time slipping away, how nothing can be taken for grantedlove, loyalty, family bonds, [and] friendship". It is claimed that "Castles Made of Sand" is one of Hendrix's more obviously biographical songs, said to be written about his uncertain and transitional childhood involving "different homes, different schools, different careers and a mother who was here one minute and gone the next". Hendrix's brother, Leon Hendrix, has commented that the lyrics allude to their father's alcoholism, Leon being taken away so suddenly by Child Protective Services without announcement, and the abusive relationship between their parents (or from stories told by their grandmother).
Writer Tom Maginn for AllMusic outlines the lyrical delivery of the song:
Musically, "The track begins with overdubbed backwards guitar creeping in, as Hendrix lays down his signature clean guitar sound", drawing comparison with fellow Axis: Bold as Love track "Little Wing". According to Maginnis, "The backwards-recorded guitar... [creates] a dreamy atmosphere and [lends] the song its distinctive character", with fellow band members Mitch Mitchell and Noel Redding providing a "laid-back groove" with their "mid-tempo drum shuffle" and "concise bass line", respectively. Similarly, Chris Jones of the BBC also notes the "signature backwards guitar" in the song.
Reception
Writing a five-star review of the album for Allmusic, Cub Koda has cited "Castles Made of Sand", along with "Little Wing", "One Rainy Wish" and "Bold as Love", as evidence of Jimi Hendrix's "remarkable growth and depth as a tunesmith, harnessing Curtis Mayfield soul guitar to Dylanesque lyrical imagery and Fuzz Face hyperactivity to produce yet another side to his grand psychedelic musical vision", describing it as a "beautiful, wistful ballad". Speaking about the track specifically, fellow Allmusic commentator Tom Maginnis cites "Castles Made of Sand" as evidence that, in writing material for The Experience's second album, "Hendrix [was] becoming a songwriter of depth, while unafraid to make use of the latest studio technology available to him". Far Out and American Songwriter both named "Castles Made of Sand" as Hendrix's seventh-greatest song.
Legacy
A live recording by funk rock band Red Hot Chili Peppers is included as the B-side to 1989 single "Taste the Pain" and later on 1994 compilation Out in L.A.. Tom Maginnis of AllMusic calls it a "largely faithful tribute where guitarist John Frusciante displays a considerable Hendrix influence". The band also recorded another version, which is included as a bonus track on Blood Sugar Sex Magik.
Personnel
The Jimi Hendrix Experience
Jimi Hendrix – guitars, vocals
Noel Redding – bass guitar
Mitch Mitchell – drums
Additional personnel
Chas Chandler – production
Eddie Kramer – engineering
Notes
References
1967 songs
The Jimi Hendrix Experience songs
Songs written by Jimi Hendrix
Songs about death
Songs about Native Americans
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,991
|
\section{Introduction}
\label{sec:introduction}
Powerful radio galaxies at redshifts z$\ge$1-2 (HzRGs) are
particularly massive \citep[several $10^{11}$ M$_{\odot}$,
e.g.,][]{seymour07, debreuck10}, intensely star-forming
\citep[on the order of 1000 M$_{\odot}$ yr$^{-1}$][]{archibald01, reuland04,
seymour12, ivison12, barthel12, drouart14}, relatively evolved
\citep[][]{nesvadba11a} galaxies.
Many are surrounded by overdensities
of satellite galaxies, making them interesting tracers of overdense
regions in the early Universe that may ultimately evolve into
massive, rich galaxy clusters \citep[e.g.,][]{chambers96,lefevre96,
Kurk2004a, venemans07, galametz12, wylezalek13}. Their co-moving number
density corresponds to that of massive clusters, when corrected for
duty-cycle effects \citep[][]{venemans07,
Nesvadba2008}. Particularly detailed studies of individual HzRGs
suggest that at least one subset of this population could be the
progenitors of the central galaxies in massive clusters in the nearby
Universe \citep[e.g.,][]{hatch09}. They follow the low-redshift
Kormendy relation of powerful radio and brightest cluster galaxies
when allowing for passive luminosity evolution
\citep[][]{targett11}. Many HzRGs are themselves gas rich
\citep[e.g.,][]{vanOjik1997, nesvadba08, ogle12, emonts14}, and while
they have no well established, virialized halos of hot gas
\citep[][]{overzier05}, HzRGs do have extended (few 100 kpc)
reservoirs of diffuse warm ionized gas \citep[][]{villar03} with
embedded clouds or filaments of neutral \citep[][]{vanOjik1997, adams09}
and molecular gas \citep[][]{nesvadba09, debreuck03, emonts14}. As
suggested by, e.g., \citet[][]{villar03} these could be the vestiges
of the gaseous reservoirs from which the brightest cluster and other
galaxies formed.
The central, high surface-brightness regions of HzRGs are well
fitted with de Vaucouleurs profiles with relatively large radii,
R$\sim$8~kpc \citep{targett11}, but extended, often irregular line
and continuum emission in HzRGs is found preferentially along
the radio jet axis \citep[e.g.,][]{cimatti93}. The reasons for this
alignment effect are still not fully understood. Polarimetry suggests
that light scattering on dust grains could be the origin of at least
some of the extended continuum light; other authors favor
jet-triggered star formation \citep[][]{chambers88, klamer05} as
the cause of the extended continuum emission.
The kinematics in the extended gas around HzRGs are often strongly
perturbed \citep[e.g.,][]{baum00, villar03}. SINFONI imaging
spectroscopy showed this is the case in
individual regions of these galaxies, and throughout their extended
emission-line gas \citep[][]{nesvadba06, nesvadba08}. Warm ionized
gas seems to be a major component of the ISM in these galaxies, with
ionized gas masses of up to $\gtrsim 10^{10}$ M$_{\odot}$ in some
cases \citep[][]{nesvadba08, nesvadba06}, comparable to the molecular
gas masses found in some HzRGs
\citep[e.g.,][]{emonts14,nesvadba09,debreuck05}. Broad line widths and
abrupt velocity offsets of up to 2000 km s$^{-1}$ found in this gas
are not reconcilable with gravitational motion. Instead they suggest that a
small part of the kinetic energy release from the AGN, through powerful
radio jets, accelerates large fractions of the ISM in these galaxies to
velocities well above the escape velocity from the underlying
dark-matter halo, thereby driving winds and removing the fuel for
subsequent star formation. This scenario is broadly supported by
hydrodynamic models showing that jets may indeed accelerate relatively
dense gas to the observed velocities of a few 100 km s$^{-1}$
\citep[][]{Wagner2012}.
Using VLT near-infrared imaging spectroscopy we have carried
out a large survey of the rest-frame optical line emission of the warm
ionized gas in a set of 50 radio-selected HzRGs at
z$\ga$2 spanning three orders of magnitude in radio power from
a few times $10^{26}$ to a few times $10^{29}$ W Hz$^{-1}$ at 1.4 GHz in the
rest frame, and more than two orders of magnitude in radio size,
from 1~kpc to about 300~kpc (Collet et al. 2015, submitted, Nesvadba
et al. 2015, in prep.). Emission-line regions are well aligned with
the radio jet axis in all but two galaxies with extended line
emission, which in addition to kinematic, energy, and timing arguments
suggests that this gas represents outflows of warm ionized gas driven
by the overpressurized cocoon of hot plasma inflated by the radio
source. These galaxies are described in greater detail in \citet{nesvadba06,
nesvadba07, nesvadba08, nesvadba11a} and Collet et al. (2015).
Here we discuss the two radio galaxies, TXS~2353$-$003 and
NVSS~J210626$-$314003, that do not follow these global trends. Their
extended regions of warm ionized gas with similar surface brightness
to the other sources show strong offsets in position angle to the
radio jet axis, contrary to what is expected from the alignment effect and
observed in the other galaxies within the SINFONI sample. Gas
associated with the radio jet axis, if present at all, is not a
distinctive component compared to gas along other directions in our
SINFONI data. Gas that is seen in projection to be associated
with the jet axis may or may not imply a physical association; however, it is
clear that gas that is globally and significantly offset from the jet
axis as projected on the sky is not reconcilable with a scenario where
the gas is embedded in an overpressured cocoon inflated by the
on-going jet activity. In addition, the line widths are low compared
to the rest of our sample. Line widths near the nucleus are higher,
and may be affected by the central radio source.
TXS~2353-003 and NVSS~J210626-314003 are among the sources with the
largest radio sizes in our sample, with largest angular sizes
LAS$=$38.8\arcsec\ and 24.2\arcsec\ at 4.86 GHz \citep[][]{debreuck01}
and 2.368~GHz, respectively (corresponding to 310~kpc and 190~kpc at
their redshifts of z=1.49 and z=2.10, respectively). Both have very
regular centimeter radio morphologies (Collet et al. 2015, Nesvadba et
al. 2015 in prep.), and do not have bright radio cores. Both are
associated with overdensities seen in projection along nearby lines of
sight of distant (z$\ge$1.3) IRAC sources in the CARLA survey of
\citet{wylezalek13}. TXS~2353-003 is within the densest region of
their overall sample of 200 HzRGs. We highlight the similarities
between our galaxies and brightest cluster galaxies at low redshift,
including the extended emission-line gas, which resembles the warm
ionized gas in and around the central galaxies of cool-core clusters
in several respects.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{2355-003_radioAndSINFONI.eps}
\caption{Results of our observations of TXS~2353-003.
{\it Top left:} 5.5~GHz contours from ATCA.
{\it Clockwise from top right to bottom left:} SINFONI maps of
[OIII]$\lambda$5007 surface brightness, FWHM line widths, and relative
velocities. The square in the top left panel shows the size of the SINFONI
maps compared to radio size. Black contours show the 5.5~GHz radio
continuum, and the dotted square in the top right panel illustrates
the region from which the bottom spectrum in
Fig.~\ref{fig:TXS2355spectra} was extracted. The arrow in the
lower left panel shows the direction of the radio jet, and the
number gives the jet size in arcsec. Ellipses in the lower left
corner of each panel show the FWHM size of the point spread
function.}
\label{fig:SINFONIobsTXS2355}
\end{center}
\end{figure}
We describe our SINFONI and ISAAC observations and data reduction in
\S\ref{sec:observations}, and discuss the results of these
observations in \S\ref{ssec:results:linemission} for the line emission
and in \S\ref{sec:contmorphologies} for the continuum, where we also
fit surface brightness profiles. In \S\ref{sec:environment} we present
our narrowband search for line emitters around TXS~2353-003 and argue
that this galaxy is surrounded by an overdensity of line emitters,
like many other HzRGs at somewhat greater redshifts. We compare these results with
the IRAC results obtained as part of the CARLA survey
\citep[][]{wylezalek13} in \S\ref{sec:bcgs}. In \S\ref{sec:nature} we
discuss several hypotheses regarding the nature of the emission-line
regions in these two galaxies, which, given their strong misalignment
relative to the jet axis, are unlikely to be directly associated with
an expanding cocoon driving an outflow. We summarize our results in
\S\ref{sec:summary}. Throughout our analysis we adopt a flat
cosmology with H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$,
$\Omega_{\Lambda}$=0.7, $\Omega_{M}=0.3$. For NVSS~J210626-314003 at
z=2.1, this corresponds to D$_L=$ 16.6~Gpc, and for TXS~2353$-$003 at
z=1.49 to D$_L$=10.8~Gpc, respectively; 1\arcsec\ corresponds to
8.4~kpc for both galaxies.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{NVSS2106.eps}
\caption{Results of our observations of NVSS~J210626$-$314003.
{\it Top left:} 5.5~GHz contours from ATCA.
{\it Clockwise from top right to bottom left:} SINFONI maps of
[OIII]$\lambda$5007 surface brightness, FWHM line widths, and relative
velocities. The square in the top left shows the size of the SINFONI
maps compared to radio size. The black contour marks the continuum
position in our SINFONI cube. Ellipses in the lower left corner
of each panel show the FWHM size of the point spread function.}
\label{fig:SINFONIobsNVSS2106}
\end{center}
\end{figure}
\section{Observations and data reduction}
\label{sec:observations}
\subsection{VLT/SINFONI imaging spectroscopy}
\label{ssec:observations:sinfoni}
We observed both galaxies with the near-infrared imaging spectrograph
SINFONI on UT~4 of the Very Large Telescope (VLT) of ESO. The
NVSS~J210626-314003 data were taken on 2009 October 12, and those
of TXS~2353-003 on 2010 August 13. The data were obtained as part
of a comprehensive observational program to study the emission-line
gas kinematics of 50 powerful radio galaxies at z$\ge$2. We used the
H$+$K band covering a spectral range of 1.45-2.45$\mu$m at a
spectral resolving power R$=$1500. The data were taken during good to
moderate observing conditions, with a typical point spread function of
1\arcsec$\times$1\arcsec\ along right ascension and declination, dominated by the seeing. We used the 250~mas pixel
scales with a field of view of 8\arcsec$\times$8\arcsec, covering a
field of 67~kpc$\times$67~kpc at z$=$2.
Total integration times were 145~min for TXS~2353-003 and 120~min for
NVSS~J210626-314003, and were obtained in one-hour
sequences of individual 300~s exposures.
Our data reduction relied on a combination of IRAF and IDL
routines \citep[for details see, e.g., ][]{nesvadba11a, nesvadba11b}.
The absolute flux scale was measured by comparing our data with observations of
standard stars with known magnitudes from the 2MASS survey that
were observed at the end of each observing session. We also determined
the size of the seeing disk from these stars.
\subsection{VLT/ISAAC imaging}
\label{ssec:observations:isaac}
Both galaxies were also observed with ISAAC on UT~3 of the VLT between
2012 September 30 and 2012 October 2 through the
service-mode program 090.A-0614. TXS~2353$-$003 was observed in the
H band and through the narrowband FeII filter (NB~1.64) with
FWHM$=0.025\mu$m (corresponding to a velocity range of $\Delta v=$4570
km s$^{-1}$), so we cover H$\alpha$ and [NII] at
z$=$1.49. NVSS~J210626-314003 was only observed in the Ks band. Both
broadband images cover roughly the rest-frame R band.
With the NB~1.64 filter we obtained a total of 248~minutes of
on-source observing time of TXS~2353$-$003 split into 99 exposures
with individual observing times of 150~seconds, and grouped into
observing sequences of one hour each. After
visual inspection we discarded 17 of these frames because of residuals
from a strongly saturated star. The on-source observing time was thus
205~minutes.
In the H band we obtained 29~exposures for TXS~2353-003 with
individual observing times of 72~seconds, corresponding to six detector
readouts after 12~seconds. We discarded one frame which contained
the trail of a satellite. In the K band for NVSS~J210626$-$314003, we
obtained 25~exposures of 90~seconds, corresponding to six detector readouts after 15~seconds. Total observing times in the H and K bands were
therefore 35~minutes and 38~minutes for TXS~2353$-$003 and
NVSS~J210626$-$314003, respectively.
Data were dark-subtracted and flat-fielded, where we constructed the
sky flats directly from the science data. All frames were then
combined with the IRAF task {\verb xdimsum }. Flux calibration relied
on standard stars observed during the same night and at similar air
mass to the science data. We used the Starlink GAIA software
\citep[][]{draper14} to calculate the astrometry relative to 2MASS
reaching rms$=$0.07\arcsec\ for TXS~2353$-$003, and
rms$=$0.3\arcsec\ for NVSS~J210626$-$314003, respectively. We also
used 2MASS to verify our flux calibration, finding a good agreement
within $\la$10\%. The sizes of stars in the image suggest the seeing
was FWHM$=$0.5\arcsec\ in the H-band and NB~1.64 image of
TXS~2353$-$003, and of FWHM$=$0.4\arcsec\ in the Ks-band image of
NVSS~J210626-314003. The limiting rms surface brightness in these images
is 26.2 mag arcsec$^{-2}$ and 23.6~mag~arcsec$^{-2}$ in the narrow
and broadband image of TXS~2353$-$003, respectively. For the Ks-band
image of NVSS~J210626$-$314003, we find 23.7~mag arcsec$^{-2}$.
\subsection{Narrowband observations of TXS~2353$-$003}
\label{ssec:narrowband}
To construct a continuum-subtracted line image from the narrowband
image of TXS~2353-003, we approximate the underlying continuum in the
narrowband filter by a scaled version of the flux measured through
the broadband filter, taking into account the relative filter
bandwidths of the narrow and broadband filters, and their
transmission. Here we assume that the continuum is flat and uniform
across the H-band filter.
We measured the zero points in the narrowband filter from
standard stars. From the zero points in each filter, filter
bandwidths, and transmissions of the narrow and broadband filters,
we determine that the continuum flux density observed through the
narrowband image corresponds to about 7.2\% of the flux density
measured through the broadband image. We would therefore need to
scale the broadband image by this value before subtracting it from
the narrowband image to obtain the flux density from the emission
lines. When directly comparing the flux density of 12 stars in both
images, we empirically find that a very similar scaling factor of
7.6\% is best to minimize the residuals of the stellar continuum
after the subtraction, and this is the factor that we used.
The redshifted wavelength of H$\alpha$ in the radio galaxy is slightly
blueshifted relative to the wavelength of peak throughput of the
narrowband filter. Comparing this with the flux measured in the SINFONI
data cube, and assuming that the relative flux calibration between
both images is perfect, we find that we are missing between 20 and
25\% of the line flux in the continuum-subtracted line image. For
other H$\alpha$ emitters (\S\ref{sec:environment}), for which we have
no precise redshift estimates, we would first need to obtain
spectroscopy before we can make similar corrections.
To evaluate the completeness limit of our data, we added faint
artificial sources with the size of the seeing disk at sky positions
in each frame of the narrow- and broadband images
before re-reducing these data and extracting sources in the same way
as for the final scientific analysis. We used fluxes between 2 and
11 times the rms in the central regions of the image. Adding the
artificial sources to the individual frames rather than the final
image helps to take misalignments between frames into account. We find
that we recover 90\% of sources with surface brightness above $1.5
\times 10^{-19}$~erg s$^{-1}$ cm$^{-2}$~\AA$^{-1}$ arcsec$^{-2}$ (or
above 23.6~mag$_{AB}$~arcsec$^{-2}$).
\section{Imaging spectroscopy of the warm ionized gas}
\label{ssec:results:linemission}
The SINFONI maps of NVSS~J210626-314003 and TXS~2353-003 are shown in
Figs.~\ref{fig:SINFONIobsTXS2355} and \ref{fig:SINFONIobsNVSS2106},
respectively. In NVSS~J210626$-$314003 at z$=$2.104,
[OIII]$\lambda\lambda$4959,5007 and H$\beta$ fall into the H band, and
H$\alpha$, [NII]$\lambda\lambda6548,6583$, and
[SII]$\lambda\lambda6716,6731$ fall into the K band, both of which we cover
with our data cubes. In TXS~2353$-$003, we identify H$\alpha$ and
[NII]$\lambda\lambda6548,6583$ at wavelengths corresponding to a
redshift of z$=$1.49. At this redshift,
[OIII]$\lambda\lambda4959,5007$ and H$\beta$ fall at redshifts below
the lower wavelength cutoff of our grating, $\lambda_{\rm
min}=1.45\ \mu$m.
Our redshift measurement for TXS~2353-003 does not agree with the
previous redshift of z$=$2.59 measured with longslit spectroscopy,
which was thought to have detected Ly$\alpha$ in the observed optical wavelength
range \citep{debreuck01}, but was in hindsight probably affected by a
Cosmic ray. We did not identify any possible emission line consistent
with the previous redshift z$=$2.59, although
[OIII]$\lambda\lambda$4959,5007 and H$\beta$ should fall into the
H band, and [NII]$\lambda\lambda$6548,6583, and H$\alpha$ should
fall into the K band at that redshift. The rest-frame UV spectrum
shows no other line candidate consistent with z$=$2.59; the value for Ly$\alpha$ at
the redshift $z=1.49$ we measure here falls well below the UV
atmospheric cutoff.
Both galaxies have bright, spatially well-resolved line emission. We
fitted the extended line emission in both galaxies with single
Gaussian profiles. Spectra were extracted from small apertures of
0.4\arcsec$\times$0.4\arcsec, which helps to reduce the strongest
pixel-to-pixel noise while safely oversampling the seeing disk so that
no spatial information is lost. The data were also smoothed in the
spectral direction by three pixels, without loss of spectral
resolution. Our fitting routine interpolates over wavelength ranges
dominated by bright night-sky line residuals, and we only map spatial
pixels where a line was detected at a S/N $>$3.
Line emission extends beyond the bright continuum seen in the SINFONI
cubes (but not in the ISAAC broadband image which we will discuss
later in more detail and show in Figs.~\ref{fig:txs2353cont}
and~\ref{fig:nvss2106cont}, which is deeper in the continuum). The
sizes of the extended emission-line regions of NVSS J210626-314003 and
TXS~2353$-$003 are 5.3\arcsec$\times$2.0\arcsec\
(44~kpc$\times$17~kpc) and 2.5\arcsec$\times$2.0\arcsec
(21~kpc$\times$17~kpc) along the major and minor axis, respectively,
and down to surface-brightness limits of $4.3\times 10^{-17}$ erg
s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ and $6.2\times 10^{-17}$ erg s$^{-1}$
cm$^{-2}$ arcsec$^{-2}$.
In NVSS~J210626$-$314003 the emission-line region is roughly centered
on the continuum peak. In TXS~2353$-$003, the gas appears more
lopsided. This is also seen in the continuum-subtracted ISAAC
narrowband image (source $a$ in Fig.~\ref{fig:thumbnailsHAE}). The
morphology of the ionized gas in both observations is fairly
consistent: a bright emission-line region associated with the radio
galaxy, more extended toward the southeast and northwest. The integrated
spectrum extracted from this fainter region, shown as a dashed region in
Fig.~\ref{fig:SINFONIobsTXS2355}, clearly shows H$\alpha$ and
[NII]$\lambda$6583 (Fig.~\ref{fig:TXS2355spectra}). Fluxes and other
line properties are listed for both galaxies in
Table~\ref{tab:propertiesUnalignedGalaxies}. In TXS~2353$-$003, we
note a faint continuum emitter along the direction of extended gas,
but at a larger distance. In NVSS~J210626$-$314003 no further
continuum source is observed along the major axis of the emission-line
region.
In the SINFONI cube, NVSS~210626-3134003 shows a smooth velocity
gradient of about 600~km s$^{-1}$ over 25~kpc. In
TXS~2353$-$003 we observe a more irregular velocity pattern, which
could be at least in part due to the lower signal-to-noise ratio and
irregular line profiles. In NVSS~J210626-3134003, where we observe
[OIII]$\lambda$5007 and H$\alpha$ at good signal-to-noise ratios, the
gas kinematics and morphologies are similar in both lines.
Both galaxies have broad line widths with FWHM$\sim$800~km~s$^{-1}$
around the continuum peak which we associate with the central regions
of the galaxies. These lines are broader than those of star-forming
galaxies at similar redshifts, including the most massive, intensely
star-forming dusty galaxies such as submillimeter galaxies
\citep[e.g.,][]{swinbank06}, and are consistent with those found in other
powerful radio galaxies at similar redshifts
\citep[e.g.,][]{nesvadba08, collet14}. Following the arguments given
in these studies, we consider the broad line widths as evidence that
the gas in the central regions of our sources is stirred up by the
transfer of kinetic energy from the radio source.
The gas kinematics at larger radii are more quiescent in both
sources. Line widths in the extended gas of NVSS~J210626-314003 and
TXS~2353-003 are FWHM$=$300-400 km s$^{-1}$
(Figs.~ \ref{fig:SINFONIobsTXS2355} and \ref{fig:SINFONIobsNVSS2106}),
more akin to those observed in a quasar illumination cone at z$\sim$2
\citep[][]{lehnert09} and Ly$\alpha$ blobs without a powerful central
radio source \citep[e.g.,][]{wilman05,overzier13}, than to powerful
radio galaxies, including the central regions of our two sources here.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{2355_Halpha_NII_SII_IntegratedSpectrum.eps}
\includegraphics[width=0.48\textwidth]{2355_Halpha_NII_SII_IntegratedSpectrumNorthEast.eps}
\end{center}
\caption{{\it Top}:
Spectrum of TXS~2353$-$003, integrated over all spatial pixels
where the H$\alpha$ emission from the galaxy is detected at $\ge
3\sigma$, and corrected for velocity shifts.
{\it Bottom}: Integrated
spectrum of the northeastern part of the emission-line region of
TXS~2353$-$003, defined by the dotted square in
Fig.~\ref{fig:SINFONIobsTXS2355}.}
\label{fig:TXS2355spectra}
\end{figure}
A marked difference to other HzRGs with similar data is that most of
the emitting gas is at large position angles from the radio jet
axis. Typically, the jet axis and the semi-major axis of the gas are
aligned within about 20$^\circ$-30$^\circ$
(Fig.~\ref{fig:illustrationParticularityMisaligned}, see
also Collet et al. 2015, Nesvadba et al. 2015 in prep.). Given
the size of the minor axis of the extended line emission of up to
$\sim 15-20$ kpc and beam smearing effects in the radio and
near-infrared, this range is on the order of what would be expected if gas
and jet were associated with each other, for example, if the jet
inflated a hot bubble of overpressurized gas like in the cocoon model
\citep[][]{begelman89}. In NVSS~J210626$-$314003 and TXS~2353$-$003;
however, the relative position angle between extended gas and radio
jet axis is much larger, 60$^\circ$ and 90$^\circ$, respectively, and
the jet is not embedded within the emission line gas, as expected from
a cocoon inflated by the
jet. Figure~\ref{fig:illustrationParticularityMisaligned} shows that
both galaxies clearly stand out from the overall sample of HzRGs with
SINFONI observations.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{PA.eps}
\caption{Position angle (north through east) of the gas (ordinate) as
a function of the position angle of the radio jet for our two
sources. The outer boundaries of the red, blue, and green stripes
indicate offsets of 20$^\circ$, 45$^\circ$, and 90$^\circ$,
respectively. Red stars indicate our two sources discussed here,
small black dots show the remaining HzRGs with SINFONI data and well-determined position angles (Nesvadba et al. 2015, in prep.).}
\label{fig:illustrationParticularityMisaligned}
\end{center}
\end{figure}
\subsection{Line diagnostics}
The ratios of bright optical line emission provide important
constraints on the ionizing source and other gas properties in the
extended emission-line regions of galaxies
\citep[][]{baldwin81,VO87,kewley06}. \citet{collet14} discussed the
emission-line ratios of NVSS~J2106-314003 in the context of their
overall sample of moderately powerful HzRGs, finding [NII]/H$\alpha$
and [OIII]/H$\beta$ ratios that correspond to neither the classical
starburst nor the AGN branch. They investigate several potential reasons for
this, including low metallicites, relatively diffuse gas clouds, high
gas pressures, or a mixture of heating from shocks and
photoionization. In TXS~2353-003, we only have [NII] and H$\alpha$
measured with a fairly high ratio of [NII]/H$\alpha$=0.7 in the
extended gas (Fig.~\ref{fig:TXS2355spectra}), which alone
is sufficient to safely attribute the line emission to either shocks or
photoionization from the AGN (Fig.~\ref{fig:bpt}). Off-axis gas not
aligned with the radio jet axis and out to a galactocentric radius of
65~kpc in the radio galaxy 3C265 at z=0.8 may indeed be powered by
shocks, perhaps with additional photoionization from soft X-ray
emission from the precursor of the shock \citep{solorzano02}. However,
as argued by \citet{leTiran2011b} and \citet{Nesvadba2006}, it is
difficult to reach high H$\alpha$ surface brightnesses as observed
with shocks. The same is the case for shock-heated, likely infalling
clouds in the cluster central galaxy NGC~4696 \citep[][]{Farage2010}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{coolflow_bpt.ps}
\caption{Diagnostic diagram showing the ratio of [OIII]/H$\beta$
as a function of [NII]/H$\alpha$ line ratio \citep[][]{baldwin81,
VO87}. The red dot shows the position of
NVSS~J210626-314003. The red hatched area shows the region spanned
by the [NII]/H$\alpha$ ratio in the off-nucleus gas in
TXS~2353-003. Light blue dots indicate the distribution of line
ratios for galaxies in the local Universe \citep[from SDSS;
see][]{kauffmann03}, and the dashed blue line shows the
diagnostic of \citet{kewley06} to separate HII regions and
star-forming galaxies (below) from AGN (above). The solid blue
lines indicate how the distribution of line ratios for both SF
galaxies and AGN are expected to evolve out to high redshift
\citep[][their model 4 with metal-poor narrow-line
regions]{kewley13}: the two HzRGs are clearly consistent with
the predictions for high redshift AGN.}
\label{fig:bpt}
\end{figure}
Like TXS~2353-003, NVSS~J210626-314003 also falls outside
the star formation branch; however, it also falls outside the sequence
formed by low-redshift AGN in the SDSS. The line ratios are consistent
with low-metallicity gas photoionized by an AGN, as expected in
the models of \citet{groves06} and \citet{kewley13}, and as shown with
the solid blue lines in Fig.~\ref{fig:bpt}. These lines show the range
of line ratios expected for galaxies at z=1.5 with intense
star formation typical for galaxies at these redshifts, and AGN
with metal-poor narrow line regions and a metal enrichment history predicted
by cosmological models of galaxy evolution \citep[][]{dave11}. These
conditions are referred to as ``model 4'' by
\citet[][]{kewley13}. J-band spectroscopy would be required to
investigate whether TXS~2355-003 falls onto the low-redshift AGN
branch or above.
\section{Continuum morphologies}
\label{sec:contmorphologies}
\subsection{Line-free continuum images from SINFONI}
We also constructed line-free continuum images from our SINFONI cubes
(shown as contours in
Figs.~\ref{fig:SINFONIobsTXS2355}~and~\ref{fig:SINFONIobsNVSS2106}) for
both galaxies by collapsing the data cubes along the spectral
direction over all wavelengths that are free from emission lines and
prominent night-sky line residuals. We consider the continuum peak in
the cube to be the location of the galaxy nucleus, which may be
dominated either by stellar light or the AGN. At a projected distance
of about 3\arcsec\ (24~kpc) northeast of TXS~2353-003, we find
another, fainter H-band source. We do not detect any line emission
from this galaxy, so we cannot determine whether it is physically
associated with the radio galaxy, or an interloper along the line of
sight. NVSS~J210626$-$314003 appears to be an isolated source,
except for two much fainter sources at 1.7\arcsec\ and
3.9\arcsec\ toward the southeast (corresponding to projected
separations of 14.3~kpc and 32.8~kpc at z$=$2.1,
respectively), which however are not aligned with the extended
emission-line regions. We do not detect any line emission from the
nearer source, and the more distant source falls outside the field
of view of SINFONI. It is therefore not clear if they are associated
with the radio galaxy. For TXS~2353-003, for which we do not have
broadband K-band imaging, we measure a K-band magnitude of
18.1$\pm$0.2~mag from our SINFONI data cube within a
3\arcsec\ aperture.
\subsection{ISAAC broadband imaging}
\label{ssec:results:continuum}
In Figs.~\ref{fig:txs2353cont} and ~\ref{fig:nvss2106cont},
we have already shown the
rest-frame R-band morphologies of TXS~2353$-$003 and NVSS~J210626-314003
observed in the H and Ks band, respectively. Both galaxies are clearly
spatially resolved, even with ground-based, seeing-limited
images. This is best seen in
Fig.~\ref{fig:luminosityProfileChi2map_TXSB2353-003} which shows the
azimuthally averaged surface-brightness profiles of both sources
compared to the profiles of nearby stars taken from the same images,
showing a clear excess of continuum emission out to large radii
in both galaxies.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{Sersic_N2106_2.ps}
\includegraphics[width=0.48\textwidth]{Sersic_T2353.ps}
\caption{Surface-brightness profiles of NVSS~J210626-314003 ({\it
top}) and TXS~2353$-$003 ({\it bottom}) extracted from ISAAC
H-band and Ks-band images, respectively. Red dots show the measured
azimuthally averaged surface brightnesses as a function of
R$^{1/4}$. Red lines show a Sersic fit with n$=$4, corresponding
to the de Vaucouleurs law for classical elliptical galaxies, and
convolved with the point spread function in each image. The dashed
black lines show the size of the seeing disk. Dotted horizonal lines
show the 3$\sigma$ limiting surface brightness of our images. Excess
flux at large radii compared to the R$^{1/4}$ law is characteristic for
brightest cluster galaxies in evolved galaxy clusters
\citep[e.g.,][]{Schombert1987}.}
\label{fig:luminosityProfileChi2map_TXSB2353-003}
\end{center}
\end{figure}
Both galaxies have a bright central region (which is more regular in
NVSS~J210626-314003 than in TXS~2353$-$003) surrounded by extended,
low-surface-brightness halos. This is remarkable, given that the
general population of passively evolving early-type galaxies at high
redshift tends to be more compact than their local analogs
\citep[e.g.,][]{Daddi2005, vanDokkum2008, Schawinski2011,
Whitaker2012}. Furthermore, although about 1/3 of HzRGs are still
actively star-forming \citep[][]{drouart14}, their molecular gas
reservoirs of a few $10^{10}$ M$_{\odot}$ \citep[e.g.,][]{emonts14}
compared to few $10^{11}$ M$_\odot$ of stellar mass
\citep[][]{seymour07, debreuck10} suggest that most of their stellar
mass growth has already been completed. \citet{targett11} found
extended broadband emission when stacking 13 HzRGs at redshifts
between z=1.5 and 2.0 \citep[as did][at slightly lower
redshifts]{best98}. However, only \citet{hatch09} have so far reported
a qualitatively similar surface-brightness profile in an individual
radio galaxy, MRC~1138$-$262 at z=2.16. Previous studies, e.g.,
\citet{pentericci01}, have drawn attention to the irregularity of
broadband morphologies in many cases, which they attributed to the
presence of on-going mergers.
\subsection{Surface brightness profiles}
\label{ssec:surfbrightprofile}
Both galaxies are spatially resolved into four or five resolution
elements in the ISAAC images, which enables us to do a basic analysis
of their surface-brightness profiles shown in
Fig.~\ref{fig:luminosityProfileChi2map_TXSB2353-003}. The goal of this
analysis is to demonstrate that the extended low-surface brightness
regions seen in the broadband images are not just the faint
extensions of the surface-brightness profiles of the central regions,
but separate components of the light profiles. We are
mainly interested in whether (and over what radii) the profiles are consistent with the
$r^{1/4}$ or de Vaucouleurs profile typical of early-type galaxies,
corresponding to a Sersic profile with index n$=$4
\citep[][]{Sersic1963, Sersic1968}.
The profiles shown in
Fig.~\ref{fig:luminosityProfileChi2map_TXSB2353-003} were obtained
with the IRAF task \verb+Ellipse+ \citep[][]{freudling93}, measuring
the surface brightness in circumnuclear elliptical annuli with radii
increasing in steps of 0.1\arcsec. \verb+Ellipse+ fits the ellipticity
of these annuli, finding between 0.05 and 0.4 in our case. Solid red
lines in Fig.~\ref{fig:luminosityProfileChi2map_TXSB2353-003} show the
expected curves of n$=$4 Sersic profiles that match the central
1\arcsec\ of the light profile, with effective radii R$=$261~kpc
and R$=$15~kpc for TXS~2353-003 and NVSS~J21062-314003,
respectively. These effective radii are consistent with those we
found with two-dimensional fits using \verb+Galfit+ \citep[][]{Peng2002,
Peng2010}, which we ran on images that were slightly smoothed to
minimize pixel-to-pixel noise (by convolution with a 3$\times$3 pixel
Gaussian) and where we left all parameters free, except imposing that
n$=$4.
The comparison between the expected profiles for n$=$4 and the
observations shown in
Fig.~\ref{fig:luminosityProfileChi2map_TXSB2353-003} illustrates that
only the inner regions of NVSS~J210626$-$314003 can be
fitted with a de Vaucouleurs profile, with radius $R_e=15$~kpc. However, the
extended halo produces residuals greater than $3
\sigma$. For TXS~2353$-$003, the surface-brightness contrast between
the central regions and the extended outer halo is lower, resulting in
a very large size of $R_e = 261$~kpc for a de Vaucouleurs profile,
which matches the central and outermost isophotes, but also leads to
significant residuals at intermediate radii. This is in agreement with
the visual impression that only NVSS~J210626$-$314003 has a
distinctive, high-surface brightness core, whereas TXS~2353$-$003 is
more diffuse. High-resolution imaging with the Hubble Space
Telescope, for example, would be needed for a full analysis of the
surface-brightness profiles in the inner regions of our galaxies.
Nonetheless, these results already show that both galaxies show important
departures from simple de Vaucouleurs profiles at intermediate and large radii.
Such extended wings are characteristic of
central cluster galaxies \citep[e.g.,][]{Schombert1987}.
\subsection{Line contamination}
Line emission in high-z radio galaxies can reach high equivalent
widths and might therefore be a significant contaminant in morphology
measurements through broadband filters \citep[][]{pentericci01,
nesvadba08, targett11}. We will now investigate the significance of
this contamination in our analysis.
\begin{figure}
\includegraphics[width=0.24\textwidth]{TXS2355_cont_map.eps}
\includegraphics[width=0.24\textwidth]{TXS2355_Ha_map.eps}
\caption{{\it left:} ISAAC H-band broadband image of TXS~2353$-$003
convolved with a two-dimensional Gaussian to match the spatial
resolution of the H$\alpha+[NII]$ map shown in the right panel. {\it
right:} H$\alpha$ morphology of TXS~2353$-$003, convolved with a
two-dimensional Gaussian to emphasize the morphology of the faint
northeastern extension at a spatial resolution of 1.2\arcsec.
The color bar shows the contamination to the flux density measured in
the broadband filter from H$\alpha$ line emission in units of
$10^{-19}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ arcsec$^{-1}$. Contours show the
broadband K-band morphology and are identical to those shown in the
left panel. Starting from the outermost contour, the broadband
fluxes represented by each contour are 1.1, 1.8, 2.5, and $5.4 \times
10^{-19}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ arcsec$^{-1}$ (3, 5, 7,
and 15$\sigma$ in the convolved image, where $\sigma=3.9\times
10^{-20}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ arcsec$^{-1}$). The
circles in the lower left of each panel show the size of the convolved
PSF. \label{fig:txs2353cont}}
\end{figure}
>From our SINFONI imaging spectroscopy we know the line fluxes of the
most prominent optical emission lines in both galaxies, H$\alpha$,
[NII]$\lambda\lambda$6548,6583, and
[OIII]$\lambda\lambda$4959,5007. In TXS~2353-003, the combined flux of
all emission lines in the H band reaches an average surface brightness
of $1.6\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ in a
2.15\arcsec\ aperture around the center of the galaxy, and $3\times
10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ along the northeastern
extension. Outside this region, line emission is not detected and must
therefore be below the 3$\sigma$ upper surface-brightness limit of
$1\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$.
The broadband morphology of TXS~2353$-$003 is shown in the left panel
of Fig.~\ref{fig:txs2353cont} as a gray scale image and contours. The
image was convolved with a two-dimensional Gaussian to a resolution of
1.2\arcsec, to match the size of the
seeing disk of the H$\alpha$ and [NII]$\lambda$6583 line image shown
in the right panel of Fig.~\ref{fig:txs2353cont}, and to enhance the
contrast of the faint extended emission in the broadband image
against the background noise.
The right panel of Fig.~\ref{fig:txs2353cont} shows the
H$\alpha$+[NII] line image of TXS~2353$-$003, obtained by collapsing
the data cube along the spectral direction over wavelengths where
H$\alpha$+[NII] is detected. This makes the line morphology
measurement more robust than the maps obtained from Gaussian fits to
each individual pixel, in particular in the faint outer parts of the
emission-line regions, and allows the measurement of the upper limits
from the same image that is also used to estimate the emission-line
surface brightness. Dividing by the 2700~\AA\ width of the H-band
filter of
ISAAC\footnote{http://www.eso.org/sci/facilities/paranal/decommissioned/
isaac/inst/isaac\_img.html}, we find that line emission contributes on
average 2.75$\times$ 10$^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$
\AA$^{-1}$ to the measured flux density in the H band near the center
of TXS~2353$-$003, and of $1.1\times 10^{-19}$ erg s$^{-1}$ cm$^{-2}$
arcsec$^{-2}$ \AA$^{-1}$ in the northeastern periphery. Outside this
extended emission-line region, the flux densities from emission lines
drop to below $3.7\times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$
arcsec$^{-2}$ \AA$^{-1}$ (our 3$\sigma$ limit). H-band surface
brightnesses at the location of the extended H$\alpha$ emission-line
region are above $2.5\times 10^{-19}$ erg s$^{-1}$ cm$^{-2}$
\AA$^{-1}$ arcsec$^{-2}$, which suggests that line contamination does
not dominate the overall broadband morphology of TXS~2353$-$003.
\begin{figure}
\includegraphics[width=0.24\textwidth]{NVSS2106_cont_map.eps}
\includegraphics[width=0.24\textwidth]{NVSS2106_Ha_map.eps}
\caption{{\it left} ISAAC K-band broadband image of
NVSS~J210626-314003 smoothed to the same spatial resolution as the
SINFONI imaging spectroscopy, 1\arcsec. {\it right} H$\alpha$
morphology of NVSS~J210626-314003. The color bar shows the
contamination to the flux density measured in the broadband filter
from H$\alpha$ line emission in units of $10^{-19}$ erg s$^{-1}$
cm$^{-2}$ \AA$^{-1}$ arcsec$^{-1}$. Contours show the broadband
K-band morphology and are the same as in the left panel. The
contour levels correspond to 3, 5, 7, 15, and 30$\times$ the root-mean
square of the broadband image smoothed to the SINFONI resolution of
1\arcsec, which has r.m.s.$=3\times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$
\AA$^{-1}$ arcsec$^{-1}$. The circles in the lower left of each panel
show the size of the seeing disk.\label{fig:nvss2106cont}}
\end{figure}
The broadband image of NVSS~J210626$-$314004 is shown in
Fig.~\ref{fig:nvss2106cont} as a gray scale image with contours (left panel),
and as contours on top of the H$\alpha$ and [NII] line image (right
panel), which was constructed in the same way as described above for
TXS~2353$-$003. The line emission is closely approximated by
Gaussian line profiles, and we therefore use the emission-line map
derived from those fits and shown in
Fig.~\ref{fig:SINFONIobsNVSS2106}. In a 2.15\arcsec\ aperture around
the nucleus, we find an average surface brightness of H$\alpha$ and
[NII] of $6.9\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arsec$^{-2}$ and
$2.2\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ in the
periphery, respectively. This corresponds to a surface flux density of
$2.3\times 10^{-19}$ and $9.4\times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$
\AA$^{-1}$ arcsec$^{-2}$, respectively, after dividing by the filter
width of the ISAAC K-band filter, 3000~\AA$^{1}$. Outside the ridge of
extended line emission, the K-band flux densities from the lines drop
to below $3\times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ arcsec$^{-2}$
(our 3$\sigma$ limit on the emission-line surface brightness).
Line emission in NVSS~J210626$-$314003 extends over regions of
broadband isophotes of $1.5\times 10^{-19}$ erg s$^{-1}$ cm$^{-2}$
and brighter (Fig.~\ref{fig:nvss2106cont}). This comparison suggests
that, outside the brightest areas seen in the H$\alpha$ line image
near the galaxy center, line emission is not a major contaminant of
the broadband morphology in NVSS~J210626$-$314003.
The diffuse Ly$\alpha$ halos discovered by \citet{villar02} and
\citet{villar03} would also produce fainter emission-line surface
brightnesses than we observe here. For a typical $SB_{Ly\alpha}=2-3
\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$, we would expect
H$\alpha$ surface brightnesses between $2-5\times 10^{-18}$ erg
s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$, assuming typical
Ly$\alpha$/H$\alpha$ ratios of 8$-$13 \citep[][]{villar03}. Dividing
by the 2700\AA\ filter width of the ISAAC H-band filter, this would
correspond to $0.7-1.5\times 10^{-21}$ erg s$^{-1}$ cm$^{-2}$
\AA$^{-1}$ arcsec$^{-2}$, much lower than the surface brightness seen
in the broadband filter.
\section{H$\alpha$ candidates surrounding TXS~2353-003}
\label{sec:environment}
Narrowband imaging is a standard way of identifying emission-line
galaxies at high redshifts, and has been successfully used to identify
over-densities of actively star-forming galaxies around high-redshift
galaxies for almost two decades \citep[e.g.,][]{chambers96,lefevre96,
Kurk2004a, venemans07, hatch11, koyama13, hayashi12, cooke14}. Most of
these observations were carried out for HzRGs at z$\gtrsim$2, where
H$\alpha$ falls in the K band, which is relatively free from telluric
night-sky lines.
We also obtained H$\alpha$ narrow-band imaging of TXS 2353-003, where
H$\alpha$ falls fortuitously into the NB 1.64 filter (although not
perfectly into the center). Our main goal was to search for
possible bright emission-line gas outside the small SINFONI field of
view of 8"x8", that would be associated with the environment rather
than the radio galaxy itself. This also enables us to identify emission-line galaxies that
are within the same dark-matter environment, and within a
velocity range of $-$1580~km~s$^{-1}$ to $+$3000~km~s$^{-1}$ from
the HzRG. Typical galaxy overdensities around HzRGs have velocity
dispersions of $<$900 km s$^{-1}$ \citep[][]{venemans07}, which
suggests that the asymmetry of the velocity range in our case does
not hinder our detection of a potential overdensity of line emitters
around the radio galaxy out to several times the velocity dispersion
of the cluster. We present the results of both parts of this
narrow-line imaging project in this section.
\subsection{Identification of candidate H$\alpha$ emitters}
\label{ssec:hacandidates}
We identified candidate emission-line galaxies associated with the
dark-matter environment of TXS~2353-003 candidate HAEs in two different ways. The first is
a method introduced by \citet{Bunker1995} that searches for a
flux excess in the narrow compared to the broadband image, and the second is the
direct inspection of the continuum-subtracted line image
described in \S\ref{ssec:narrowband}.
For the flux excess method of \citet{Bunker1995}, we used SExtractor
v.2.5.0 \citep[][]{BertinArnouts1996} to construct catalogs from the
narrow- and broadband images, setting the parameters
\verb+DETECT_THRESH+ = 2.0 and \verb+ANALYSIS_THRESH+ = 2.0, while
leaving all others at their default values. We then take the positions
of the galaxies in the catalog extracted from the broadband image to
perform aperture photometry within 2\arcsec\ apertures in both
images. Using fixed apertures is appropriate for marginally or
unresolved galaxies like ours, and has the advantage of providing
errors that are independent of galaxy size \citep[see
also][]{Kurk2004a}. Selecting our candidates from the broadband image, rather
than the narrowband, is different from many other narrowband
searches, but makes our selection particularly robust, since it
requires independent detections in two images. The disadvantage is
that we might miss galaxies with particularly high emission-line
equivalent widths. However, comparison of the catalogs extracted from
the two images shows that this was not the case in our analysis.
We use the ``significance of excess'', $\Sigma$,
as defined by \citet{Bunker1995}, i.e., ``the number of standard
deviations between the counts measured in the broadband and the
number expected on the basis of the narrowband counts (assuming a
flat spectrum)'', to select the best H$\alpha$ candidates. In
addition, and again following \citet{Bunker1995} we require that the
sources fall above a given equivalent width threshold. Following
\citet{Kurk2004a}, we choose EW$_0\ge$50\AA\ and 25\AA. These thresholds
correspond to broad- and narrowband colors of 0.38 and 0.17, respectively. Three
sources with $\Sigma\ge3.0$ even have EW$_0\ge 100$\AA, including the
radio galaxy, which corresponds to a broad- to narrowband color of
0.69.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{2355_colorMagnitude_goodVersion_4.eps}
\caption{Broad- to narrowband color-magnitude diagram of our candidate
H$\alpha$ emitters. H$\alpha$ candidates are labeled with
colored symbols, depending on the significance of their
detection. Letters refer
to the sources shown in Fig.~\ref{fig:thumbnailsHAE}. Source ``a'' is
the HzRG.}
\label{fig:colorMagnitudeDiagram}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{positionsOfHAE_2.eps}
\caption{Positions of the candidate H$\alpha$ emitters surrounding TXS~2353-003 on the sky. The HzRG is at the field center and
labeled with the symbol ``a''. Offsets are given relative to the
position of TXS~2353-003. North is up and east to the left.}
\label{fig:halphapos}
\end{center}
\end{figure}
The resulting color-magnitude diagram is shown in
Fig.~\ref{fig:colorMagnitudeDiagram}. We
find seven sources with $\Sigma\ge2.0$ and $EW_0\gtrsim 50$\AA.
We also show another seven H$\alpha$ candidates
that have somewhat lower excess significances $\Sigma=$1.5-2.0. A
fraction of these are probably part of the overdensity of
TXS~2353-003. However, the fraction of misidentifications due to larger
spectral slopes, for example, is likely to be larger among these sources,
which is why we restrict our analysis to the most robust candidates
above $\Sigma=$2.0.
\subsection{An overdensity of H$\alpha$ candidates around TXS~2353$-$003}
\label{ss:overdensityHaEmittersAroundTXSB2353-003}
Given the relatively good seeing of our narrowband data of 0.4\arcsec, most
H$\alpha$ candidates are spatially resolved
(Fig.~\ref{fig:thumbnailsHAE}) and we estimate H$\alpha$ equivalent
widths of 50$-$250~\AA\ in the rest frame, corresponding to H$\alpha$
fluxes between about 1 and a few $\times 10^{-16}$ erg s$^{-1}$
cm$^{-2}$. Their properties are listed in
Table~\ref{table:HalphaEmitters}. We did not correct the H$\alpha$
flux and equivalent widths for [NII] emission, which should fall into
the same filter. Observations of UV/optically
selected galaxies at high redshift typically show very low
[NII]/H$\alpha$ line ratios on the order of 10\%
\citep[e.g.,][]{nmfs09,lehnert09,Queyrel2012}. Using
the conversion of \citet{Kennicutt1998} with a $1-100$~M$_{\odot}$
Salpeter initial mass function, we find star formation rates of 16-35
M$_{\odot}$ yr$^{-1}$ per galaxy, and a moderate total
star formation rate of 138 M$_{\odot}$ yr$^{-1}$ in these galaxies
combined. These values were not corrected for extinction, so
intrinsic star formation rates could be higher. For example,
\citet[][]{garn10} and \citet{stott13} typically find ~1 mag
extinction in H-alpha meaning that intrinsic star formation rates
are probably a factor of $\sim$2.5 times higher. The expected fraction
of low-redshift interlopers is much lower in our case than for
narrowband searches using Ly$\alpha$. The brightest emission lines
longward of H$\alpha$ are the Paschen and Brackett hydrogen lines
in the near-infrared, with typically much lower equivalent widths than
H$\alpha$.
To quantify whether the observed number of H$\alpha$ candidates around
TXS~2353-003 may herald an overdensity of galaxies, we have to compare our data
with samples of H$\alpha$ emitters in the field at the same epoch. A
blind redshift search for HAEs at z~=~1.48 has recently
been performed by \citet{Sobral2013} through the NB~1.617 filter at
UKIRT, which has a very similar width of 210~\AA\ (compared to
250~\AA\ for our ISAAC filter), that we use as reference for
the expected source density in the field. The HiZEL survey
\citep[][]{Sobral2009b, Sobral2010, Sobral2012, Sobral2013} gives an
estimate of the H$\alpha$ emitters that we can expect in a blank
field: \citet{Sobral2012} concentrated on H$\alpha$ emitters at
z~$\simeq$~1.47, very close to the z=1.49 of TXS~2353$-$003. They
found 295 NB emitters over the 0.67~deg$^2$ field of their main
analysis, and 411 over the entire 0.79~deg$^2$ field of their NB
observations. Among them, 190 objects also show an excess in deep NB
imaging centered on the [OII]$\lambda3727$ emission line and have
colors compatible with a photometric redshift z~$\simeq$~1.47. This
leads to densities of candidate HAEs in the
field of $\Sigma_{HAE} \simeq$~0.14~HAE~arcmin$^{-2}$ (with all NB
emitters) ; $\Sigma_{HAE} \simeq$~0.12~HAE~arcmin$^{-2}$ (with NB
emitters in their main field) and $\Sigma_{HAE}
\simeq$~0.08~HAE~arcmin$^{-2}$ (with NB emitters having an adequate
photometric redshift). These values include galaxies down to a
3$\sigma$ flux level of $7\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$,
compared to $1\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ in our case.
Around TXS~2353$-$003, with an effective field of view of
5.4~arcmin$^2$ (neglecting the trimmed edges where our data did not
reach the full depth) we find six H$\alpha$ emitting candidates with
fluxes $> 7\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ and significances
$\Sigma \ge$~2.0 in addition to the radio galaxy. This
corresponds to a source density of 1.1~arcmin$^{-2}$, and an
overdensity by a factor $\sim$8 relative to the field.
\citet{Kurk2004b} detected 28 HAEs in two ISAAC pointings around
MRC~1138-262 at z~=~2.16 down to an H$\alpha$ flux limit of
$2.5\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$, which corresponds to
the same luminosity limit as the current analysis. They found a
source density of 2.2~arcmin$^{-2}$.
Although our source density at z$=$1.5 is lower than that in the
z$=$2.16 overdensity around MRC~1138-262 in absolute terms, the
significance of the overdensity relative to the field at the same
epoch is high, because the star formation rate density at z~=~1.5 in
the field is already strongly declining for the highest H$\alpha$
luminosities \citep[e.g.,][]{Sobral2013}.
Finding high surface-brightness gas out to the periphery of our
SINFONI data cube raises the question whether there is similarly
bright line emission outside the small,
8\arcsec$\times$8\arcsec\ field of view of SINFONI
(67~kpc$\times$67~kpc at z=1.49). We do not find any such emission
down to a 3$\sigma$ detection limit of $6\times 10^{-20}$ erg s$^{-1}$
cm$^{-2}$ \AA$^{-1}$ in the continuum subtracted narrowband
image. With the 250\AA\ width of the NB1.64 filter, and cosmological
surface-brightness dimming by a factor $(1+z)^4=39$, we are sensitive
to the equivalent of $\ge 60\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$
arcsec$^{-2}$ in nearby clusters. This is more than an order of
magnitude fainter than the surface brightness observed, e.g., in parts
of the filaments in the Perseus cluster \citep[][]{Hatch2007}.
\section{Comparison of TXS~2353-003 and NVSS~J210620-214003 with cluster central galaxies}
\label{sec:bcgs}
High-redshift radio galaxies do not often reside in solitude, but are surrounded by
overdensities of line or continuum emitters, suggesting that a
significant fraction of the HzRG population may be the progenitors of
the central galaxies of massive galaxy clusters
\citep[e.g.,][]{chambers96,lefevre96, Kurk2004a, venemans07,
galametz12}. \citet{wylezalek13} obtained Spitzer warm-mission IRAC
photometry at 3.6$\mu$m and 4.5$\mu$m of 200 HzRGs, including our two
sources. They find that, compared to the Spitzer UKDSS Ultra Deep
Survey \citep[][]{kim11}, 55\% of HzRGs are surrounded by $\ge2\sigma$
overdensities of galaxies with colors between the 3.6$\mu$m and
4.5$\mu$m channels that are consistent with redshifts z$\ge$1.3. Like
any method identifying galaxy overdensities, however, the IRAC color
selection only samples parts of the population of putative satellite
galaxies around HzRGs, and can therefore only provide lower limits to
the actual density of such galaxies around a particular HzRG. It can
therefore only demonstrate the presence, but not the absence of such
an overdensity of galaxies around a given radio galaxy.
\citet{wylezalek13} show that TXS~2353-003 is surrounded by one of the
highest overdensities of IRAC-selected sources, with an excess of
$>5\sigma$, and consistent with the results of our narrowband search
presented in \S\ref{sec:environment}. NVSS~J210626-314003, however,
does not stand out as a source with a particularly dense environment
of IRAC-selected sources in the \citet{wylezalek13} sample, a 1.2
$\sigma$ excess. Unfortunately, the observing program committee of ESO
only granted us narrowband imaging of TXS~2353-003, not of
NVSS~J210626-314003, so that we were unable to do a narrowband search
around this galaxy. We note, however, that selecting a galaxy based on
its bright radio emission already leads to significant biases toward
galaxies in dense environments \citep[e.g.,][]{Best2000, ramos13,
hatch14}, and we argue in the following that the stellar component
of NVSS~J210626-314003 itself shows some of the characteristic
signatures of cluster central galaxies.
\subsection{K$-$z relationship}
\label{ssec:kmag}
\citet{kauffmann98} argued that the observed K-band magnitude can
be used to approximate the stellar mass of all but the most rapidly
growing galaxies out to redshifts z$\ge$2, with a scatter of about a
factor of~2. This is consistent with the empirical tight K-z
relationship between observed K-band magnitudes of powerful radio
galaxies and their redshifts \citep{lilly84,DeBreuck2002, Willott2003a,
bryant09b} and the small mass range of these galaxies
in the Spitzer photometric survey of HzRGs by
\citet{seymour07} and \citet{debreuck10}.
In Fig.~\ref{fig:K-redshift} we show where TXS~2353$-$003 and
NVSS~J210626$-$314003 fall relative to the K-z relationship of
\citet{seymour07} and \citet{debreuck10}, and that of
\citet{Lidman2012}, who found a similar relationship for brightest cluster galaxies (BCGs). This
figure shows the good agreement of our targets with both samples, as
well as the self-consistency of both samples with each other (as
expected if HzRGs are the progenitors of BCGs).
Encouraged by the results of \citet{kauffmann98, seymour07}, and
\citet{debreuck10}, which, when considered together, suggest that the
observed K-band magnitude of HzRGs should scale with the stellar mass,
we used the K-band magnitudes of $z\sim2$ HzRGs in the
\citet{seymour07} sample and the stellar mass estimates given in the
same paper, to obtain a simple empirical relationship between stellar
mass, $M_{stellar}$, and observed K-band magnitude, $m_K$,
$\log{(M_{stellar}/M_{\odot})}= 6.18-0.189\ m_K$. We find a 1 $\sigma$
scatter of about 0.3~dex in mass, consistent with that previously
found by \citet{kauffmann98}. We note that we are using observed
magnitudes from galaxies in a similar redshift range, and therefore
do not need a k-correction.
This allows us to derive a rough mass
estimate for our two sources. For TXS~2353$-$003 and
NVSS~J210626$-$314003, K-band magnitudes of 18.1$\pm$0.3 mag and
18.7$\pm$0.2 mag suggest stellar masses in the range
$\log{(M^{2353}/M_{\odot})}=11.1-11.4$ and
$\log{(M^{2106}/M_{\odot})}=11.2-11.5$, respectively. While such an
approach should not replace more detailed analyses of the continuum
emission where additional photometric constraints are available, it
does highlight that our two galaxies fall into the typical range of
stellar masses of powerful radio galaxies at these redshifts.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{K-z_relationUnaligned_2.eps}
\caption{Observed $K-$band magnitude of brightest-cluster galaxies
(shown as `+') and radio galaxies (shown as `x') as a function of
redshift \citep[plot reproduced from][and slightly extrapolated to
z=2.5]{Lidman2012}. Some of the data points were taken from
\citet{stott08} and \citet{stott10}. The large red stars illustrate
that TXS~2353$-$003 and NVSS~J210626$-$314003 fall well within the
expected range of $K-$band magnitude, and also have typical $K-$band
magnitudes for HzRGs at their redshifts \citep[black crosses][see also
\citealt{DeBreuck2002, Willott2003a, bryant09b}]{seymour07}. At
z$\ge2$, H$\alpha$ and [NII]$\lambda\lambda$6548,6583 fall into the
K-band filter and can affect the continuum magnitudes by up to about
0.3~dex in extreme cases \citep[][]{nesvadba06}.}
\label{fig:K-redshift}
\end{center}
\end{figure}
\subsection{Surface-brightness profiles}
In Section~\ref{ssec:results:continuum} we showed that both galaxies
are spatially resolved in our ground-based imaging, and that both have
surface-brightness profiles that are not consistent with the pure
Sersic profiles of early-type galaxies (n$=$4). The central arcsec of
the profile of NVSS~J210626$-$314003 can be fitted with a de
Vaucouleurs profile with R=15~kpc, but fainter extended emission
persists at larger radii. TXS~2353-003 can only be fitted with a de
Vaucouleurs profile when assuming an unphysically large radius of
261~kpc.
\citet{targett11} analyzed the K-band morphologies of 13 radio
galaxies at z$=$1.5$-$2.5 observed with UKIRT during very good seeing,
and found that most sources are well fit with Sersic indices n$=3-5$
and effective radii between 5 and 10~kpc. This is in contrast to the
findings of \citet{pentericci01} with HST/NICMOS through the F160W
filter that only 5 out of 19 sources have well-established de Vaucouleurs
profiles \citep[although the][sample does extend to higher redshifts
than that of \citealt{targett11} and, with H-band observations,
samples much shorter wavelengths]{pentericci01}. Only one source of
the \citet{targett11} sample, 0128$-$264, has an effective radius
R=15~kpc, as does NVSS~J210626$-$314003. This galaxy is
also included in our parent sample with SINFONI data. It has a small
emission-line region, although elongated along the radio jet axis, and
a very extended (294~kpc) radio size, resembling the two sources
discussed here (Nesvadba et al. 2015, in prep.).
Massive early-type galaxies at high redshift appear to be generally
characterized by their greater compactness compared to similar, more nearby
galaxies \citep[][who found $R_e \le 1$~kpc and 0.9~kpc,
respectively ]{Daddi2005, vanDokkum2008}. Typically, these galaxies
are only resolved with deep, high-resolution HST imaging, showing no
deviations from S\'ersic laws \citep[e.g.,][]{Szomoru2010,
Cassata2010}. The HzRGs of \citet{targett11}, which have relatively
large radii (average 8~kpc) compared to submillimeter galaxies
(average 3~kpc) at similar redshifts, show an average decrease in size
by a factor 1.5 compared to equally massive galaxies at low
redshift. A size decrease by a factor of $\sim2$ has also been found by
\citet{pentericci01} for their HzRGs that are well modeled with a de
Vaucouleurs profile.
Extended stellar envelopes are a primary characteristic of central
cluster galaxies in the low-redshift Universe \citep[cD galaxies;
e.g.,][]{Schombert1987}, and likely formed from the debris of repeated
accretion of several satellite galaxies. This interpretation also
agrees with the finding of \citet{targett11} that HzRGs fall onto the
same Kormendy relationship between central surface brightness and
effective radius as massive low-redshift galaxies, which, as stated by
the authors, would imply that the size increase from z$\sim$2 to today
must be accommodated by an increase in stellar mass at large
galactocentric radii, as could be produced by repeated accretion of
satellites. \citet{best98} identified similar structures in a stack of
12 radio galaxies from the 3CRR sample at z$=0.6-1.4$, but to our
knowledge, the only example of a HzRG with extended stellar halo at
greater redshift currently known in the literature is MRC~1138$-$262
at z=2.2 \citep[the Spiderweb Galaxy,][]{miley06}. \citet{Hatch2008,
hatch09} discussed the envelope around this source in detail, finding
that the rest-frame UV colors of the faint continuum emission are
consistent with a stellar envelope (rather than scattering from a
dusty gaseous halo) with a formation history that is
coeval with that of the high-surface brightness central regions. This
is in contrast to semi-analytical simulations which postulate that
such envelopes assembled gradually through dry mergers over
cosmological timescales \citep[e.g.,][]{delucia07}.
\section{The nature of the extended ionized gas}
\label{sec:nature}
A good alignment between jet and gas is typical of distant radio
galaxies \citep[the alignment effect; e.g.,][]{cimatti93}, and this
alignment was also among the prime arguments of \citet{collet14} and
of \citet{Nesvadba2006,nesvadba08} that jet and gas are physically
related in their SINFONI samples of HzRG. They also found very large,
super-gravitational velocity gradients reaching to more than~1000~km
s$^{-1}$ associated with the most powerful radio sources, and broad
line widths in the extended gas with FWHM $\gtrsim$ 800 km
s$^{-1}$. They considered this as evidence of fast, AGN-driven
outflows in HzRGs.
None of this applies here. Velocity gradients are well below 1000 km
s$^{-1}$, line widths outside of the central regions are only on the order of
200-300~km s$^{-1}$, and the mismatch in position angle between jet
and gas is striking. This mismatch makes it difficult to postulate
that jet cocoons are driving the gas out as they expand through the
ambient gas, because it appears unavoidable in this model that
cocoon (and hence, jet) and gas are cospatial, both in three
dimensions and in projection.
The presence of this off-axis component of line emission
is worth discussing, as is the absence of bright, extended emission-line
regions along the jet axis. TXS~2353$-$003 may have a faint, not well
resolved emission-line component along the jet axis and within the
galaxy itself. However, this does not stand out compared to the gas in
other regions of the galaxy, and is therefore difficult to associate
uniquely with the radio jet. We find no similar emission at larger
radii, including the cluster scales that we cover with ISAAC around
TXS~2353-003 (\S\ref{sec:environment}).
The absence of bright gas on scales much larger than the host galaxy
cannot be a mere observational effect, since the sensitivities of these
data are comparable to the overall sample, and the surface brightness
of the line emission in our targets is not significantly lower than in
other HzRGs with SINFONI data. However, for the largest sources with
'regular' line emission, in particular MRC~2104-242 at z=2.5, the
extended line emission can become very faint and is concentrated along
long, thin, very filamentary structures, suggesting that the warm gas
is either being heated to temperatures T$\gg 10^4$ K, or becoming
strongly dispersed as the cocoon continues to expand (Nesvadba et
al. 2015, in prep.; see also \citealt{pentericci01} who observed the
same gas as faint, extended filaments in their HST/NICMOS broadband
imaging).
We stress that finding extended gas around HzRGs that is unrelated to
the radio jets is by itself not uncommon. Many HzRGs are surrounded by
diffuse halos of warm ionized gas \citep{villar03}, however,
dense, high-surface-brightness Ly$\alpha$ emission consistent with the
H$\alpha$ surface brightnesses that we observe here is only found
within the jet cocoon in the galaxies of \citet{villar03}
H$\alpha$ associated with the diffuse ionized gas would be about 2
orders of magnitude fainter than what we observe.
We will now discuss several hypotheses, from AGN orientation to galaxy
interactions to extended gas disks or filaments within the radio
galaxy itself or its immediate surroundings, to explain the nature of
this gas.
\subsection{Partial AGN illumination of ambient halo clouds}
The line ratios of both sources suggest that they are photoionized by the
central AGN (\S\ref{ssec:results:linemission}). This could imply that
they do not represent a distinct structure, but are merely part of a
population of ambient clouds that are distributed over large solid
angles, and of which a subset is being lit up as it intercepts
the quasar ionization cone. Filaments of neutral gas have also been
observed around some HzRGs \citep[][]{vanOjik1997, jarvis03, wilman04,
debreuck05, nesvadba09, emonts14}.
However, not all HzRGs show signatures of neutral gas clouds in their
halos. \citet{vanOjik1997} found Ly$\alpha$ absorption only
in galaxies with radio sizes $<$50~kpc, in stark contrast to the large
radio sizes, $\ga 200$~kpc, of TXS~2353-003 and NVSS~210626-314003.
\citet{binette00} showed from an analysis of the ionization properties
that these absobers must also lie at larger radii than the radio
source. In the more compact radio galaxies, Ly$\alpha$ absorbers are
found not only against the nucleus, but also against more extended
Ly$\alpha$ emission around the galaxy \citep[][]{Kurk2004a}, which
suggests that this result holds for large solid angles around the
radio galaxies, and not just along the radio jet axis.
This scenario would also require a misalignment between radio jet axis
and quasar illumination cone, as is possible because of jet
precession, or a misalignment between jet axis and the normal of the
torus, for example. \citet{drouart14} recently argued, based on Spitzer
mid-infrared imaging and the core-to-lobe fraction of the radio
sources, that the unified model must hold for the global population of
HzRGs, but the presence of nuclear broad H$\alpha$ emission in a few
sources does also suggest that this might not be the case for each
individual HzRG \citep[][]{nesvadba11a}.
Alternatively, we may be seeing gas in an old cocoon from a previous
radio-loud activity phase of the super-massive black hole. Jet
precession has been invoked to explain the X-shaped radio
sources at low redshift, where about 5-10\% of FRII radio sources have
two pairs of radio lobes \citep[][]{leahy02}. The second bar of the
``X'', typically secondary pairs of fairly diffuse, low
surface-brightness wings in centimeter radio continuum imaging, may be the
relic of a previous burst of nuclear activity along a different jet
axis. Detecting such low surface-brightness emission directly would be
very challenging at high redshift.
This scenario, however, also has its
difficulties. \citet[][]{kaiser02} argued that clouds in relic cocoons
will be destroyed by shocks within a few $10^6$ yrs, which is of the
same order as the ages of bright radio sources in high-redshift
galaxies \citep[e.g.,][]{blundell99}. In the cocoon model, dense
clouds are confined by the high pressure of the cocoon material, and
should disperse within a sound-crossing time once the jet has stopped
maintaining the cocoon pressure high \citep[e.g.,][]{fabian87}. From
the size of the two lobes in our sources, 310~kpc and 190~kpc, and
assuming a jet advance speed on the order of $0.1c$, we can roughly
constrain that the feeding of a putative older jet component must have
ended at least about $1\times 10^7$ yrs ago in our two sources, so
that most of the emission-line clouds should have already been
destroyed or evaporated. The relatively old age of our two sources
does not make them good candidates for seeing relic cocoons, in
particular since many younger radio sources, which should in principle
have brighter relic cocoons, do not show evidence of two sets of radio
lobes or extended emission-line regions.
We conclude that neither partial AGN illumination of a general
population of ambient clouds nor repeated cycles of radio-loud AGN
activity with different jet orientations appears to be a good
explanation for the nature of our sources.
\subsection{External gas supply from a satellite galaxy?}
Gas transfer from a satellite galaxy undergoing accretion onto the
HzRG, as previously observed in the z=3.8 HzRG 4C60.07
\citep[][]{Ivison2008}, is another interesting hypothesis for the
origin of this misaligned gas in our two sources. In this case, the
extended line emission could trace a gas tail produced by ram-pressure
stripping or tidal forces. However, only TXS~2353$-$003 has a
companion along the direction of the extended gas, but at a larger
distance from the radio galaxy. NVSS~J210626-314003 has no obvious
companion within several tens of kpc that can be associated with the
extended emission-line region.
It is uncertain whether the galaxy near TXS~2353-003 (Fig.~1) is
physically associated with the radio galaxy, although its infrared
colors as measured with IRAC are consistent with a redshift z$\ga$1.3
using the criteria of \citet{papovich08}
(\S\ref{sec:environment}). However, we detect no H$\alpha$ line
emission down to $3.3 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. Tidal
forces during galaxy interactions produce extended tails, and
also funnel gas toward the nuclei of the galaxies
\citep[e.g.,][]{Barnes1996}, fueling intense nuclear
starbursts. Likewise ram pressure should not only strip parts of the
ISM, but also compress the gas along the head of the infalling galaxy,
enhancing the gas surface brightness within the galaxy, and
potentially star formation \citep{Kapferer2008}.
It appears therefore difficult for a galaxy to produce a tail of
$\sim 10^9$ M$_{\odot}$ of ionized gas without sustaining significant
star formation. Our upper limit of H$\alpha$ flux corresponds to
star formation rates on the order of 10 M$_{\odot}$ yr$^{-1}$ \citep[using
the][calibration and neglecting extinction]{Kennicutt1998}. Since
the source is bright in the continuum at similar wavelengths, the
H$\alpha$ equivalent width must also be low. Although gas, dust, and
young stars are not necessarily co-spatial
\citep[][]{calzetti97}, this would imply here that the galaxy, in
spite of losing considerable fractions of its ISM in the putative
tail, is also maintaining a highly efficient obscuring dust screen
around its star-forming regions. This appears contradictory, in
particular for high-z galaxies, which typically have extended star
formation in multiple knots across the galaxy
\citep[e.g.,][]{nmfs09}. Other HzRGs, e.g., MRC~1138-262 at z~=~2.16,
are already known to be surrounded by companions within a few tens of
kpc that are already on the red sequence \citep[][]{Kodama2007} and
lack line emission \citep[][]{Nesvadba2006,Kuiper2011}. It
thus appears unlikely that external gas supply from a satellite galaxy
is a good explanation of the extended emission-line regions in our
galaxies.
\subsection{Extended gas disks within the radio galaxy}
\citet{Best2000} pointed out that radio galaxies with very extended
radio sources are particularly good candidates for being cluster
central galaxies because the pressure from the Mpc-scale intracluster
medium boosts the luminosity of the radio source even for
comparably extended jets \citep[see also][]{athreya98, klamer06}. Many
authors have previously suggested that the progenitors of brightest
cluster galaxies are likely to be found among HzRGs
\citep[e.g.,][]{pentericci01,miley06,hatch09}, and \citet{wylezalek13}
find from IRAC imaging that TXS~2353-003 is even associated
with the most strongly pronounced overdensity within their
sample.
The gas that we see here may actually have much in common with the
extended gas disks or filaments that are found in and near
cluster central galaxies in the more nearby Universe.
Mismatches between the position angles of jet and gas are
fairly common in about 30\% \citep{McDonald2010} of cool-core
galaxy clusters with extended H$\alpha$ filaments. A particularly
clear example, where the warm ionized gas appears to avoid the
cavities that the radio jet has inflated in the intracluster medium is
Abell 1795, in particular at the highest emission-line surface
brightnesses \citep{vanBreugel1984}. In total, such structures have up
to a few $10^{10}$~M$_{\odot}$ \citep[e.g.,][]{Salome2006} of warm and
cold gas with T$\le 10^4$~K, typically dominated by cold molecular
gas. We have already argued in \S\ref{sec:bcgs} that our sources
follow the overall trends found in BCGs between H$\alpha$ luminosity and surface
luminosity, size, and ratios of low-ionization emission lines.
Imaging spectroscopy of cool-core clusters with bright optical line
emission \citep[e.g.,][]{Hatch2006,Wilman2006,Hatch2007,Wilman2009,
Farage2010, Farage2012} shows that the gas has typically FWHM $\sim
100-300$~km s$^{-1}$ outside the near-nuclear regions, with a
characteristic broadening toward the center, with typical FWHM $\ge$
500~km s$^{-1}$ and up to the values we find near the center of
NVSS~J210626$-$314003 and TXS~2353$-$003
(\S\ref{ssec:results:linemission} and
Figs. \ref{fig:SINFONIobsTXS2355} and \ref{fig:SINFONIobsNVSS2106}). However,
regular velocity gradients of a few 100~km s$^{-1}$, similar to what we
observe in our sources, are found only in some BCGs, e.g., Abell~262
\citep{Hatch2007} or 2A~0335$+$096 \citep{Farage2012}. Alternatively,
disks with well-ordered velocity fields, perhaps more like in
NVSS~J210626-3140003 than in TXS~2353$-$003, have also been found, for
example in Hydra~A \citep{Hamer2014}. Such disks are also typically
not aligned with the radio jet axis, but can be illuminated by the
central AGN, in particular if the opening angle of the AGN
illumination cone is large.
The observed velocity gradients in our two sources are in fact
consistent with rotational motion in a gravitational potential as
suggested by the stellar mass estimates of $1.5-3\times 10^{11}$
M$_{\odot}$ (\S\ref{ssec:kmag}). A dynamical mass estimate, $v$, can
be derived by setting $v = \sqrt{M \sin{i}^2 G / R}$, where $M$ is the
stellar mass, $R$ the radius of the putative disk, $i$ the inclination
angle, and $G$ the gravitational constant. With R$=$10.5~kpc, measured
for the very regular velocity field of NVSS~J210626-314003, we find
v$=250-350\ \sin{i}^{-1}$ km s$^{-1}$, compared to 220 km s$^{-1}$
observed. Even if the gradients
were dominated by radial outflow motion, the small velocity range implies that most of this gas is
unlikely to escape.
This is consistent with the finding that the gas extends over similar
radii to the faint continuum wings in the ISAAC image of TXS~2353-003
on both sides, and in NVSS~J210626-314003 on the southwestern side. If the
wings in the continuum surface brightness profiles of our two sources
represent extended stellar halos \citep[][]{hatch09}, perhaps
originating from a phase of rapid accretion of many satellite galaxies
in the very early evolution of the HzRGs \citep[][]{burkert08}, then
we may be seeing leftover gas that is settling down after this
phase. \citet{McDonald2010} suggested that the condensation of such
gas into fairly dense clouds could be accelerated by weak shocks
caused by the passage of a companion galaxy, as may be the case for
the putative companion in TXS2353-003.
Weak shocks could also be produced by the vestiges of the relic cocoon
that initially formed when the jet was passing through the inner
regions of the halo. This scenario has some resemblance to the
cyclical AGN feedback scenarios invoked, e.g., by
\citet{PizzolatoSoker2005} or \citet{AntonuccioDeloguSilk2010} to
explain extended gas disks and filaments in nearby cluster central
galaxies. \citet{nesvadba10,nesvadba11b} argued that isolated
radio galaxies with extended gas disks may also require repeated,
episodic AGN activity, broadly akin to the \citet{PizzolatoSoker2005}
model, in order to understand the observed properties of the gas.
Winds associated with the large radio jets we observe may have
dispersed the ISM of these galaxies just a few $10^7$~yrs ago, and
parts of this gas may now be raining back onto the galaxy, possibly
feeding a new feedback cycle. The age of the Universe at z=1.5-2
is already $>3$~Gyr, so that repeated activity cycles are possible
for duty cycles of a few times $10^8$ yrs, as seems appropriate at low
redshift \citep[][]{PizzolatoSoker2005}. The gas cooling times
could be lowered by compression of the outflow and diffuse ambient gas
that is being swept up as the two bubbles of the AGN cocoon continue
to inflate in the form of a momentum-driven wind after the feeding from
the AGN has ceased \citep[][]{kaiser02}. This cyclical feedback may
contribute to forming the hot intracluster X-ray gas in massive galaxy
clusters today, which presumably took place around redshift z$\ge$2
\citep[][]{nath02, mccarthy08}. Energy injection through repeated
episodic AGN activity has also been invoked as an explanation of how
the X-ray halos surrounding individual massive early-type galaxies can
be maintained over a Hubble time \citep[e.g.,][]{mathews03,best06}.
\section{Summary}
\label{sec:summary}
We have presented an analysis of rest-frame optical imaging spectroscopy
and deep broadband near-infrared imaging of two radio galaxies at
z$\sim$2, which have extended emission-line regions that are strongly
misaligned relative to the axis of the radio jets. This is in stark
contrast to the majority of powerful radio galaxies at similar
redshifts, where the gas is aligned within 20-30$^\circ$ of the jet
axis, as expected from a cocoon of turbulent gas that is being
inflated by the expanding radio jets. For one source, TXS~2353-003 at
z$=$1.5, we also present H$\alpha$ narrowband imaging through the
[FeII]1.64 filter of ISAAC, finding an overdensity of H$\alpha$
candidates which resembles those of radio galaxies at higher redshift.
The gas in both galaxies is less perturbed than in radio galaxies with
more typical gas properties. In particular, line widths in the extended
gas are lower and, with FWHM $\sim$200-300 km s$^{-1}$ comparable to
those in the extended gas disks and filaments surrounding brightest
cluster galaxies at lower redshifts. Overall, we find remarkable
similarities to the BCGs in low-redshift cool-core clusters, ranging
from the extended, faint continuum halos in both galaxies that are
atypical for massive high-redshift galaxies except for the particularly
well-studied Spiderweb Galaxy MRC~1138-262, where such structures
are interpreted as extended stellar halos broadly akin to cD galaxies
at lower redshifts. Both galaxies fall onto the relationship between
K-band magnitude and redshift for brightest cluster galaxies and
HzRGs, suggesting they are among the most massive galaxies at their
epoch. We note that, given that HzRGs are now generally
considered to be the progenitors of BCGs, the distinction between
both classes is not mutually exclusive, and our two sources may
simply be further evolved than many other HzRGs, but still following the same
overall evolutionary path.
We discuss several scenarios for the nature of this gas, finding that
simple illumination effects in otherwise typical HzRG halos is not a
good explanation, and neither are galaxy interactions. Generally
speaking, our results support the hypothesis that the extended
line emission arises from extended gas disks or filaments within or
near the radio galaxy, by analogy with broadly similar structures in
nearby cluster central galaxies. This could be evidence for cyclical
AGN feedback, which has been discussed on several occasions for nearby
clusters.
\section*{Acknowledgments}
We are very grateful to the staff at Paranal for having carried out
the observations on which our analysis is based. We also thank the
anonymous referee for the detailed comments which helped improve the
paper. CC wishes to acknowledge support from the Ecole Doctorale
Astronomie \& Astrophysique de l'Ile de France. Parts of this research
were conducted by the Australian Research Council Centre of Excellence
for All-sky Astrophysics (CAASTRO), through project number
CE110001020.
\bibliographystyle{aa}
|
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@private
// paramValues_ contains the parameters used in requests and responses
NSMutableDictionary *paramValues_;
NSString *realm_;
NSString *privateKey_;
NSString *timestamp_; // set for testing only
NSString *nonce_; // set for testing only
// flag indicating if the token in paramValues is a request token or an
// access token
BOOL hasAccessToken_;
// flag indicating if authorizeRequest: adds a header or parameters
BOOL shouldUseParamsToAuthorize_;
id userData_;
}
// OAuth protocol parameters
//
// timestamp (seconds since 1970) and nonce (random number) are generated
// uniquely for each request, except during testing, when they may be set
// explicitly
//
// Note: we're using "assign" for these since they're stored inside
// the dictionary of param values rather than retained by ivars.
@property (nonatomic, copy) NSString *scope;
@property (nonatomic, copy) NSString *displayName;
@property (nonatomic, copy) NSString *hostedDomain;
@property (nonatomic, copy) NSString *domain;
@property (nonatomic, copy) NSString *iconURLString;
@property (nonatomic, copy) NSString *language;
@property (nonatomic, copy) NSString *mobile;
@property (nonatomic, copy) NSString *consumerKey;
@property (nonatomic, copy) NSString *signatureMethod;
@property (nonatomic, copy) NSString *version;
@property (nonatomic, copy) NSString *token;
@property (nonatomic, copy) NSString *callback;
@property (nonatomic, copy) NSString *verifier;
@property (nonatomic, copy) NSString *tokenSecret;
@property (nonatomic, copy) NSString *callbackConfirmed;
@property (nonatomic, copy) NSString *timestamp;
@property (nonatomic, copy) NSString *nonce;
// other standard non-parameter OAuth protocol properties
@property (nonatomic, copy) NSString *realm;
@property (nonatomic, copy) NSString *privateKey;
// service identifier, like "Twitter"; not used for authentication or signing
@property (nonatomic, copy) NSString *serviceProvider;
// user email and verified status; not used for authentication or signing
//
// The verified string can be checked with -boolValue. If the result is false,
// then the email address is listed with the account on the server, but the
// address has not been confirmed as belonging to the owner of the account.
@property (nonatomic, copy) NSString *userEmail;
@property (nonatomic, copy) NSString *userEmailIsVerified;
// property for using a previously-authorized access token
@property (nonatomic, copy) NSString *accessToken;
// property indicating if authorization is done with parameters rather than a
// header
@property (nonatomic, assign) BOOL shouldUseParamsToAuthorize;
// property indicating if this auth has an access token so is suitable for
// authorizing a request. This does not guarantee that the token is valid.
@property (nonatomic, readonly) BOOL canAuthorize;
// userData is retained for the convenience of the caller
@property (nonatomic, retain) id userData;
// Create an authentication object, with hardcoded values for installed apps
// with HMAC-SHA1 as signature method, and "anonymous" as the consumer key and
// consumer secret (private key).
+ (GTMOAuthAuthentication *)authForInstalledApp;
// Create an authentication object, specifying the consumer key and
// private key (both anonymous for installed apps) and the signature method
// ("HMAC-SHA1" for installed apps).
//
// For signature method "RSA-SHA1", a proper consumer key and private key
// may be supplied (and the GTL_OAUTH_SUPPORTS_RSASHA1_SIGNING compiler
// conditional must be set.)
- (id)initWithSignatureMethod:(NSString *)signatureMethod
consumerKey:(NSString *)consumerKey
privateKey:(NSString *)privateKey;
// clear out any authentication values, prepare for a new request fetch
- (void)reset;
// authorization entry point for GTL library
- (BOOL)authorizeRequest:(NSMutableURLRequest *)request;
// add OAuth headers
//
// any non-OAuth parameters (such as scope) will be included in the signature
// but added as a URL parameter, not in the Auth header
- (void)addRequestTokenHeaderToRequest:(NSMutableURLRequest *)request;
- (void)addAuthorizeTokenHeaderToRequest:(NSMutableURLRequest *)request;
- (void)addAccessTokenHeaderToRequest:(NSMutableURLRequest *)request;
- (void)addResourceTokenHeaderToRequest:(NSMutableURLRequest *)request;
// add OAuth URL params, as an alternative to adding headers
- (void)addRequestTokenParamsToRequest:(NSMutableURLRequest *)request;
- (void)addAuthorizeTokenParamsToRequest:(NSMutableURLRequest *)request;
- (void)addAccessTokenParamsToRequest:(NSMutableURLRequest *)request;
- (void)addResourceTokenParamsToRequest:(NSMutableURLRequest *)request;
// parse and set token and token secret from response data
- (void)setKeysForResponseData:(NSData *)data;
- (void)setKeysForResponseString:(NSString *)str;
- (void)setKeysForResponseDictionary:(NSDictionary *)dict;
// persistent token string for keychain storage
//
// we'll use the format "oauth_token=foo&oauth_token_secret=bar" so we can
// easily alter what portions of the auth data are stored
- (NSString *)persistenceResponseString;
- (void)setKeysForPersistenceResponseString:(NSString *)str;
// method for distinguishing between the OAuth token being a request token and
// an access token; use the canAuthorize property to determine if the
// auth object has an access token
- (BOOL)hasAccessToken;
- (void)setHasAccessToken:(BOOL)flag;
// methods for unit testing
+ (NSString *)normalizeQueryString:(NSString *)str;
//
// utilities
//
+ (NSString *)encodedOAuthParameterForString:(NSString *)str;
+ (NSString *)unencodedOAuthParameterForString:(NSString *)str;
+ (NSDictionary *)dictionaryWithResponseData:(NSData *)data;
+ (NSDictionary *)dictionaryWithResponseString:(NSString *)responseStr;
+ (NSString *)scopeWithStrings:(NSString *)str, ...;
+ (NSString *)stringWithBase64ForData:(NSData *)data;
+ (NSString *)HMACSHA1HashForConsumerSecret:(NSString *)consumerSecret
tokenSecret:(NSString *)tokenSecret
body:(NSString *)body;
#if GTL_OAUTH_SUPPORTS_RSASHA1_SIGNING
+ (NSString *)RSASHA1HashForString:(NSString *)source
privateKeyPEMString:(NSString *)key;
#endif
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,039
|
\section{Introduction}
\subsection{} \label{section.intro1}
Let
\[ F = \sum\limits_{i=0}^d \, \binom{d}{i} \, a_i \, x_1^{d-i} \,
x_2^i, \qquad (a_i \in \complex) \]
denote a binary form of order\footnote{Usually $d$ would be called
the degree of $F$, but `order' is the common usage in classical
invariant theory.} $d$ in the variables $\ux =
\{x_1,x_2\}$. Its Hessian is defined to be
\[ \He(F) = \frac{\partial^2 F}{\partial x_1^2} \, \frac{\partial^2 F}{\partial x_2^2}
- \left(\frac{\partial^2 \, F}{\partial x_1 \, \partial x_2} \right)^2. \]
It is well-known that
\[ \He(F) = 0 \iff F = (p \, x_1 + q \, x_2)^d \quad \text{for some $p,q \in
\complex$}. \]
(The implication $\Leftarrow$ is obvious, and $\Rightarrow$ easily
follows by a simple integration -- see~\cite[Proposition~2.23]{Olver}.)
The Hessian is a covariant of binary $d$-ics, in the sense
that its construction commutes with a linear change of variables in
the $\ux$. More precisely, let $g = \left( \begin{array}{cc} \alpha & \gamma
\\ \beta & \delta \end{array} \right)$ denote a complex matrix such
that $\det g = 1$. Given a binary form $A(x_1,x_2)$, write
\[ A^g = A(\alpha \, x_1 + \beta \, x_2, \gamma \, x_1 + \delta \,
x_2). \] Then we have an identity
\[ \He(F^g) = [ \, \He(F) \, ]^g. \]
By definition, $\He(F)$ is of order $2d-4$ (in the $\ux$), and its coefficients are
quadratic in the $a_i$; hence it is said to be a covariant of degree $2$ and order $2d-4$.
\subsection{} \label{power.problem}
Now suppose that $r$ is a divisor of $d$ (say $d = r \, \mu$), and we are looking for a
similar covariant which vanishes exactly whether $F$ is the perfect $\mu$-th
power of an order $r$ form. About a decade ago, the second author
(JC) had constructed such a covariant using Wronskians. It will be
described below in \S\ref{definition.alphaF}; but tentatively let us denote it by $\Gott_{r,d}(F)$. Subsequently, he learnt
from the report of a colloquium lecture by Gian-Carlo
Rota~\cite{RotaLecture} that Hilbert~\cite{Hilbert} had already solved this
problem. Hilbert's construction (see~\S\ref{Hilb.construction}) is
based upon an entirely different idea; it will be denoted by $\Hilb_{r,d}(F)$.
In fact, either of the constructions makes sense even if
$r$ does not divide $d$. If we let $e = \gcd (r,d)$ and $d = e \, \mu$, then we have the property
\begin{equation}
\Gott_{r,d}(F) = 0 \iff F = G^\mu \; \text{for some order $e$ form
$G$} \iff \Hilb_{r,d}(F) =0.
\label{HG2} \end{equation}
Both covariants turn out to be of degree $r+1$ and order $N = (r+1)(d-2)$. This, of
course, creates a strong \emph{presumption} that they might indeed be the
same (but see~\S\ref{misconception} below). This is our first result.
\begin{Theorem} \sl There exists a nonzero rational scalar
$\kappa_{r,d}$ such that
$\Gott_{r,d} = \kappa_{r,d} \, \Hilb_{r,d}$. \label{Theorem.HGeq} \end{Theorem}
The proof will be given in \S\ref{section.Hilb.construction}. When
$r=1$, either covariant reduces to the Hessian.
\subsection{} For $p \ge 0$, let $S_p$ denote the $(p+1)$-dimensional
space of order $p$ forms in $\ux$. We have an embedding
\[ \P S_e \lra \P S_d, \qquad [G] \lra [G^\mu] \]
whose image $X = X_{e,d}$ is the variety of binary $d$-ics which are perfect
$\mu$-th powers of order $e$ forms. (In particular, $X_{1,d}$ is the rational
normal $d$-ic curve.) Let $R = \complex[a_0, \dots, a_d]$ denote the co-ordinate ring of $\P S_d
\simeq \P^d$. Write
\[ \Hilb_{r,d}(F) = \sum\limits_{i=0}^{N} \,
\binom{N}{i} \, h_i \, x_1^{N-i} \, x_2^i, \]
and let $J = (h_0,h_1, \dots, h_N) \subseteq R$ denote the ideal generated
by the coefficients of $\Hilb_{r,d}$ (or what is the same, $\Gott_{r,d}$). By construction,
the zero locus of $J$ is precisely $X$. We show that, when $r$ divides
$d$, the ideal $J$ defines $X$ as a scheme.
\begin{Theorem} \sl
Assume that $r$ divides $d$. Then the saturation of $J$ coincides with the defining ideal $I_X \subseteq
R$.
\label{Theorem.Jsaturation} \end{Theorem}
The proof will be given in \S\ref{proof.Jsaturation}. For the
case $r=1$, this theorem appears in~\cite{AluffiFaber}.
\subsection{} In \S\ref{section.Gott.cov}, we use the plethysm
decomposition of $SL_2$-representations to exhibit $\Gott_{r,d}$ as a
special case of a family of covariants which vanish on $X$. We
baptise them as G{\"o}ttingen covariants, to commemorate the
G{\"o}ttingen school of which Hilbert was a distinguished member for
nearly five decades. In
\S\ref{GottPsi.formula}, we give an algorithm for the
symbolic computation of these covariants. The examples in
\S\ref{section.rnc.ideal}-\S\ref{section.g26} suggest the conjecture
that the G{\"o}ttingen covariants generate all of $I_X$ when $r$
divides $d$.
The $J$-ideals seem to obey complicated containment relations for
varying values of $r$, and there is much here that we do not understand. We give a preliminary result in this direction in
Proposition~\ref{prop.containmentJ}. The example in
\S\ref{section.twistedcubic} shows that when $r$ does not divide $d$,
the Hilbert covariants can create interesting nonreduced scheme structures on $X$.
\subsection{}
The problem discussed in \S\ref{power.problem} makes sense in any number of
variables. There is a classical construction due to Clebsch called the
`transfer principle', which allows us to lift the binary solution to
$n$-ary forms. We explain this in \S\ref{section.Clebsch}, and
construct a concomitant $\tGott_{r,d}$ of $n$-ary $d$-ics which
has exactly the same vanishing property that $\Hilb_{r,d}$ does for
binary forms (see Theorem~\ref{theorem.clebsch}). For instance, let $F$
denote a quartic form in three variables
$x_1,x_2,x_3$, which we write symbolically as
\[ F = a_\ux^4 = b_\ux^4 = c_\ux^4. \]
Then $F$ is the square of a quadratic form, if and only if the
concomitant
\[ \tGott_{2,4} = (a \, b \, u) \, (a \, c \, u)^2 \, a_\ux \, b_\ux^3 \, c_\ux^2, \]
vanishes on $F$.
\subsection{} Although the Hilbert covariants were defined over a
century ago, they do not seem to have been studied much in the
subsequent years.\footnote{There is a later short note by Brioschi
\cite{Brioschi}, but it is mostly a report on Hilbert's original
paper and contains little that is new.} This may be partly due to Hilbert
himself, whose papers around 1890 in the \emph{Mathematische Annalen}
changed the texture of modern algebra, and to some extent caused the earlier
themes to be seen as \emph{pass{\' e}} (cf. \cite[\S II]{Fisher}). We are convinced, however, that
these covariants (and their generalisation, namely the G{\"o}ttingen
covariants) encapsulate a large amount of hitherto unexplored algebraic
geometry.
\section{Preliminaries}
In this section we establish notation, and explain the necessary
preliminaries in the invariant theory of binary forms. Since the
latter are less widely known now than they were a century ago, we have
included rather more background material. Some of the classical
sources for this subject are~\cite{Glenn, GY, Hilbert2, Salmon}, whereas more modern treatments
may be found in~\cite{Dolgachev1, KungRota, Olver, Processi,
Sturmfels}. In particular, for explanations pertaining to the symbolic
calculus the reader is also referred to \cite[\S 2]{Abd1}.
\subsection{$SL_2$-representations}
The base field will be $\complex$. Let $V$ denote a two-dimensional
complex vector space with basis $\ux = \{x_1,x_2\}$, and a natural action of
the group $SL(V) \simeq SL_2$. For $p \ge 0$, let $S_p = \text{Sym}^p \, V$ denote the
$(p+1)$-dimensional space of binary $p$-ics in $\ux$. Recall that $\{S_p:
p \ge 0\}$ is a complete set of finite-dimensional irreducible $SL_2$-representations,
and each finite-dimensional representation is a direct sum of
irreducibles. The reader is referred to \cite[\S 6]{FH} and
\cite[\S I.9]{Knapp} for the elementary theory of
$SL_2$-representations.
For brevity, we will write $S_p(S_q)$ for $\text{Sym}^p (S_q)$ etc.
\subsection{Transvectants} \label{section.trans}
Given integers $p,q \ge 0$, we have a decomposition of representations
\begin{equation}
S_p \otimes S_q \simeq \bigoplus\limits_{k=0}^{\min(p,q)} \,
S_{p+q-2k}. \label{Clebsch-Gordan} \end{equation}
Let $A,B$ denote binary forms in $\ux$ of respective orders $p,q$. The $k$-th transvectant
of $A$ with $B$, written $(A,B)_k$, is defined to be the image of
$A \otimes B$ via the projection map
\[ \pi_k: S_p \otimes S_q \lra S_{p+q-2k} \, . \]
It is given by the formula
\begin{equation} (A,B)_k = \frac{(p-k)! \, (q-k)!}{p! \, q!} \,
\sum\limits_{i=0}^k \; (-1)^i \binom{k}{i} \,
\frac{\partial^k A}{\partial x_1^{k-i} \, \partial x_2^i} \,
\frac{\partial^k B}{\partial x_1^i \, \partial x_2^{k-i}} \ .
\label{trans.formula} \end{equation}
Usually $k$ is called the index of transvection. By convention,
$(A,B)_k = 0$, if $k > \min \, (p,q)$.
If we symbolically write $A = a_\ux^p, B = b_\ux^q$ as in \cite[Ch.~I]{GY}, then
$(A,B)_k = (a \, b)^k \, a_\ux^{p-k} \, b_\ux^{q-k}$. A useful method for calculating transvectants of
symbolic expressions is given in \cite[\S 3.2.5]{Glenn}.
There is a canonical isomorphism of representations
\begin{equation}
S_p \stackrel{\sim}{\lra} S_p^* \, ( \, = \text{Hom}_{SL(V)}(S_p,S_0))
\label{self-duality} \end{equation}
which sends $A \in S_p$ to the functional $B \lra (A,B)_p$. It is
convenient to identify each $S_p$ with its dual via this isomorphism, unless it is necessary to maintain a
distinction between them.
\subsection{The Omega Operator} \label{section.omegaop}
If $\ux = \{x_1,x_2\}$ and $\uy = \{y_1,y_2\}$ are two sets of binary variables, then the corresponding Omega
operator is defined to be
\[ \Omega_{\ux \, \uy} = \frac{\partial^2}{\partial x_1 \, \partial y_2}
- \frac{\partial^2}{\partial x_2 \, \partial y_1}. \]
Given forms $A, B$ as above,
\[ (A,B)_k = \frac{(p-k)! \, (q-k)!}{p! \, q!} \, \left\{ \Omega_{\ux \,
\uy}^k \left[ A(\ux) \, B(\uy) \right] \right\}_{\uy:=\ux}. \]
That is to say, change the $\ux$ to $\uy$ in $B$, operate $k$-times by
$\Omega$, and then revert back to the $\ux$.
\subsection{Covariants} \label{section.covariants}
We will revive an old notation due to Cayley, and
write $(\alpha_0,\dots,\alpha_n \cb u,v)^n$ for the
expression
\[\sum\limits_{i=0}^n \; \binom{n}{i} \, \alpha_i \, u^{n-i} v^i. \]
In particular
\begin{equation} \F = (a_0,\dots,a_d \cb x_1,x_2)^d
\label{F.gen} \end{equation}
denotes the {\sl generic} $d$-ic, which we identify with the natural trace form in $S_d^* \, \otimes \, S_d$.
Using the duality in~(\ref{self-duality}), this amounts to the identification
of $a_i \in S_d^*$ with $\frac{1}{d!} \, x_2^{d-i} \, (-x_1)^i$; but
it is convenient to think of the $\ua = \{a_0, \dots, a_d\}$ as independent variables. Let $R$ denote the symmetric algebra
\[ \bigoplus\limits_{m \ge 0} \, S_m(S_d^*) = \bigoplus\limits_{m \ge 0} \, R_m =
\complex \, [a_0,\dots,a_d], \]
so that $\Proj R = \P S_d \simeq \P^d$.
Consider an $SL(V)$-equivariant embedding
\[ S_0 \hookrightarrow R_m \otimes S_q. \]
Let $\Phi$ denote the image of $1$ via this map, then we may write
\begin{equation} \Phi = (\varphi_0,\dots,\varphi_q \cb x_1,x_2)^q,
\label{Phi.cov} \end{equation}
where each $\varphi_i$ is a homogeneous degree $m$ form in the
$\ua$. One says that $\Phi$ is a covariant of degree $m$ and order
$q$ (of the generic $d$-ic $\F$). In other words, the space
\[ \text{Span} \, \{\varphi_0, \dots, \varphi_q\} \subseteq R_m \]
is an irreducible subrepresentation isomorphic to $S_q$.
The {\sl weight} of $\Phi$ is defined to be $\frac{1}{2}(d \, m-q)$
(which is always a nonnegative integer).
In particular, $\F$ itself is a
covariant of degree $1$ and order $d$. A covariant of order $0$ is called an invariant.
Any transvectant of two covariants is also one, hence expressions such as
\[ (\F,\F)_4, \quad (\F,(\F,\F)_2)_3, \quad ((\F,\F)_2, (\F,\F)_4)_5, \dots \]
are all covariants. The Hessian coincides with $(\F,\F)_2$ up to a scalar.
A fundamental result due to Gordan says that each covariant is a $\complex$-linear combination of such compound
transvectants (see~\cite[\S 86]{GY}). The weight of a compound
transvectant is the sum of transvection indices occurring in it; for
instance, $((\F,\F)_2, (\F,\F)_4)_5$ is of weight $2+4+5=11$.
\subsection{} \label{ex.isobaric}
Recall that a homogeneous form in $R$ is called isobaric
of weight $w$, if for each monomial $\prod\limits a_k^{n_k}$ appearing
in it, we have $\sum\limits_k k \, n_k = w$. If $\Phi$ is a covariant of degree-order $(m,q)$, then its coefficient
$\varphi_k$ is isobaric of weight $\frac{1}{2}(d \, m-q)+k$. For instance,
let $d=6$, and $\Phi = (\F,(\F,\F)_2)_1$, which is a covariant of degree
$3$, order $3d-6$, and hence weight $3$. Its expression begins as
\[ \begin{aligned}
\Phi = \; & (a_0^2 \, a_3+2 \, a_1^3-3 \, a_0 \, a_1 \, a_2) \,
x_1^{12} \, + \\
& (12 \, a_1^2 \, a_2-15 \, a_0 \, a_2^2+3 \, a_0^2 \, a_4) \, x_1^{11} \,
x_2 \, + \\ &
(15 \, a_1 \, a_2^2+3 \, a_0^2 \, a_5+18 \, a_0 \, a_1 \, a_4+24 \,
a_1^2 \, a_3-60 \, a_0 \, a_2 \, a_3) \, x_1^{10} \, x_2^2 \, + \\
& ( 25 \, a_2^3+60 \, a_1^2 \, a_4-80 \, a_0 \, a_3^2+a_0^2 \, a_6-30 \,
a_4 \, a_0 \, a_2+24 \, a_1 \, a_0 \, a_5) \, x_1^9 \, x_2^3 \, + \dots,
\end{aligned} \]
and one sees that the successive coefficients are isobaric of weights
$3,4,5$ etc.
\subsection{The Cayley-Sylvester formula} Let $C(d, m,q)$ denote the
vector space of covariants of degree-order $(m,q)$ for binary $d$-ics;
its dimension is the same
as the multiplicity of $S_q$ in the irreducible decomposition of $R_m
\simeq S_m(S_d)$. This number is given by the Cayley-Sylvester formula (see~\cite[Corollary
4.2.8]{Sturmfels}). For integers $n,k,l$, let $\pi(n, k, l)$ denote
the number of partitions of $n$ into $k$ parts such that no part
exceeds $l$. Then
\[ \zeta(d,m,q) = \dim C(d,m,q) = \pi \left(\frac{d \, m-q}{2},d,m \right) -
\pi \left(\frac{d \, m-q-2}{2},d,m \right). \]
For instance, $\zeta(6,3,6)=\pi(6,6,3)-\pi(5,6,3)=7-5=2$, and it is easy to check (e.g., by
specialising $\F$) that
\[ \F \, (\F,\F)_6, \quad (\F, (\F,\F)_4)_2, \] is a basis of $C(6,3,6)$.
\subsection{} \label{misconception}
This is perhaps the correct place to forestall one possible misconception about
Theorem~\ref{Theorem.HGeq}. Recall that $\Gott_{r,d}$ and $\Hilb_{r,d}$ both
have degree $r+1$ and order $N = (r+1) \, (d-2)$. If it were the case that
\begin{equation}
\zeta(d,r+1,N)=1,
\label{zetaeq1} \end{equation}
then one could immediately conclude that the two
must be equal up to a scalar. But such may not be the case. For instance, if
$r=5,d=15$, then $\zeta(15,6,78) = 4$.
Hence, Theorem~\ref{Theorem.HGeq} does not follow from general
multiplicity considerations, but instead requires an explicit hard
calculation. However, (\ref{zetaeq1}) is true for $r=1,2$. (This
can be seen from the plethysm formulae in~\cite[\S I.8]{MacDonald}.)
\subsection{} An alternate equivalent definition of a covariant is as follows. Let
$g = \left( \begin{array}{cc} \alpha & \gamma \\ \beta &
\delta \end{array} \right)$, where $\alpha, \dots, \delta$ are
regarded as independent indeterminates. Write
\[ x_1 = \alpha \, x_1' + \beta \, x_2', \quad x_2 = \gamma \, x_1' +
\delta \, x_2', \]
and substitute into (\ref{F.gen}). Determine expressions $a_0',\dots,
a_d'$ such that we have an equality
\[ (a_0',\dots,a_d' \cb x_1',x_2')^d = (a_0,\dots,a_d \cb
x_1,x_2)^d; \]
then each $a_i'$ is a polynomial expression in the $\ua$ and $\alpha,
\dots, \delta$. Now let $\Phi \in \complex[a_0, \dots, a_d; x_1,x_2]$ be a bihomogeneous form
of degrees $m,q$ respectively in $\ua, \ux$. Then $\Phi$ is a
covariant, if and only if the following identity holds:
\begin{equation}
\Phi(a_0', \dots, a_d'; x_1', x_2') = (\alpha \, \delta - \beta \,
\gamma)^{\frac{dm-q}{2}} \, \Phi(a_0, \dots, a_d; x_1,
x_2). \label{cov.identity} \end{equation}
\subsection{Covariants and Differential
Operators} \label{section.cov.diff}
Consider the following differential operators:
\begin{equation}
E_{+} = \sum\limits_{i=0}^{d-1} \, (d-i) \, a_{i+1} \,
\frac{\partial}{\partial a_i}, \qquad
E_{-} = \sum\limits_{i=1}^d i \, a_{i-1} \, \frac{\partial}{\partial
a_i}, \qquad
E_0 = \sum\limits_{i=0}^d \, (2i-d) \, a_i \,
\frac{\partial}{\partial a_i},
\label{cayley.equations2} \end{equation}
and
\[
\Gamma_{+} = E_{+} - x_1 \, \frac{\partial}{\partial x_2}, \qquad
\Gamma_{-} = E_{-} - x_2 \, \frac{\partial}{\partial x_1}, \qquad
\Gamma_0 = E_0 + (x_1 \, \frac{\partial }{\partial x_1} -
x_2 \, \frac{\partial }{\partial x_2}). \]
\begin{Proposition} \sl
A bihomogeneous form $\Phi$ is a covariant, if and only if
\begin{equation}
\Gamma_{+} \, \Phi = \Gamma_{-} \, \Phi = \Gamma_{0} \, \Phi =
0. \label{GammaPhi} \end{equation}
\end{Proposition}
A proof is given in~\cite[\S 149]{Salmon} (also see~\cite[\S
4.5]{Sturmfels}), but the central idea is the following: $\Phi$ is a covariant exactly when it
remains unchanged by an $SL_2$-action, i.e., when it is annihilated
by the Lie algebra ${\mathfrak{sl}}_2$. Let
\[ J_{+} = \left( \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right), \quad
J_{-} = \left( \begin{array}{rr} 0 & 0 \\ 1 & 0 \end{array} \right), \quad
J_0 = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right), \]
denote the standard generators of ${\mathfrak{sl}}_2$. Choose a path
$t \lra g_t$ starting from the identity element in $SL_2$,
and apply condition (\ref{cov.identity}) to $g_t$. For the three
cases $J_\star = \left[\frac{d g_t}{d t}\right]_{t=0}$ where $\star \in
\{+,-,0\}$, we respectively get the identities in (\ref{GammaPhi}). \qed
\smallskip
The first coefficient $\varphi_0$ is called the source (or
seminvariant) of $\Phi$. From~(\ref{GammaPhi}), we get equations
\begin{equation}
E_{-}(\varphi_0) =0, \quad \text{and} \quad
\varphi_k = \frac{(q-k)!}{q !} \, E_{+}^k(\varphi_0) \quad \text{for $0 \le k \le
q$.}
\label{source.eqns} \end{equation}
Thus one can recover the entire covariant from the source alone.
Moreover, a homogeneous isobaric form $\psi$ in the $\ua$ can be a
source (of some covariant), if and only if it satisfies the condition $E_{-}(\psi)=0$.
The commutation relations between the $E_\star$ are parallel to the ones between the standard generators of
${\mathfrak{sl}}_2$, i.e.,
\[ [E_{+}, E_{-}] = E_0, \quad [E_0, E_{+}] = 2 \, E_{+}, \quad [E_0,
E_{-}] = - 2 \, E_{-}. \]
The following lemma will be needed in \S\ref{Hilb.construction}.
\begin{Lemma} \sl
For $n \ge 0$, we have an identity
\[ E_{-} \, E_{+}^{n+1} = E_{+}^{n+1} \, E_{-} -
(n+1) \, E_{+}^n \, E_0 - n \, (n+1) \, E_{+}^n. \]
\label{lemma.E} \end{Lemma}
\demo This follows by a straightforward induction on $n$. \qed
\subsection{Wronskians}
Let $m,n \ge 0$ be integers such that $m \le n+1$.
Consider the following composite morphism of representations
\[ w: \wedge^m S_n \stackrel{\sim}{\lra} S_m(S_{n-m+1}) \lra S_{m(n-m+1)}, \]
where the first map is an isomorphism (described in~\cite[\S2.5]{AC2}) and the second is the natural surjection.
Given a sequence of binary $n$-ics $A_1,\dots,A_m$, define
their Wronskian $W(A_1,\dots,A_m)$ to be the image $w(A_1 \wedge \dots
\wedge A_m)$. It is given by the determinant
\[ (i,j) \lra \frac{\partial^{m-1} \, A_i}{\partial x_1^{m-j} \, \partial \, x_2^{j-1}}, \quad
(1 \le i,j \le m). \]
The $\{A_i\}$ are linearly dependent over $\complex$, if and only if
$W(A_1, \dots, A_m)=0$. (The `only if' part is obvious. For
the converse, see~\cite[\S 1.1]{Meulien}.)
\section{The G{\"o}ttingen covariants} \label{section.Gott.cov}
\subsection{} \label{section.defn.rde}
Henceforth assume that $r,d$ are positive integers, and let $e =
\gcd(r,d)$. Write $d = e \, \mu$ and $r = e \, \mu'$. Consider the embedding
\[ \P S_e \stackrel{\imath}{\lra} \P S_d, \quad [G] \lra [G^\mu]. \]
Let $X_{e,d}$ denote the image variety. We have a factorisation
\[ \diagram
{} & \; \P S_\mu(S_e) \ar@{.>}[d]^\pi \\
\P S_e \ar[ur]^{v_\mu} \ar[r]^\imath & \P S_d
\enddiagram
\]
where $v_\mu$ is the $\mu$-fold Veronese embedding, and $\pi$ is the
projection coming from the surjective map $S_\mu(S_e) \lra S_{e \, \mu} =
S_d$. Thus $\imath$ corresponds to the incomplete linear series $S_d
\subseteq H^0({\mathcal O}_{\P S_e}(\mu))$.
\subsection{} \label{definition.alphaF}
In this section we will define the covariants
$\Gott_{r,d}$. For $F \in S_d$, we have a morphism
\[ \alpha_F : S_r \lra S_{r+d-2}, \quad A \lra (A,F)_1 = \frac{1}{rd}
\, \left| \begin{array}{cc} A_{x_1} & A_{x_2} \\ F_{x_1} &
F_{x_2} \end{array} \right|, \]
where $A_{x_i}$ stands for $\frac{\partial A}{\partial x_i}$ etc.
\begin{Proposition} \sl With notation as above,
\[ \ker \alpha_F \neq 0 \iff [F] \in X_{e,d}. \]
\end{Proposition}
\demo Assume $F = G^\mu$ for some $G$, then
$\alpha_F(G^{\mu'}) = (G^{\mu'}, G^\mu)_1 =0$.
Alternately, assume that $(A,F)_1=0$ for some nonzero $A$. We will
construct a form $G$ such that (up to scalars) $A = G^{\mu'}, F =
G^\mu$. Let $\ell \in S_1$ be a linear form which divides
either $A$ or $F$; after a change of variables we may assume $\ell =
x_1$. Suppose that $a,f$ are the highest powers of $x_1$ which
divide $A,F$ respectively, and write $A = x_1^a \, \tA, F = x_1^f
\, \tF$. Starting from the relation
$A_{x_1} \, F_{x_2} = A_{x_2} \, F_{x_1}$, after expanding and rearranging the terms, we get
\[ x_1^{a+f} \, (\tA_{x_1} \, \tF_{x_2} - \tA_{x_2} \, \tF_{x_1}) =
x_1^{a+f-1} \, (-a \, \tA \, \tF_{x_2} + f \, \tA_{x_2} \, \tF), \]
hence $x_1$ must divide $(-a \, \tA \, \tF_{x_2} + f \,
\tA_{x_2} \, \tF)$. Thus, either $\tA_{x_2} = \tF_{x_2} =0$ (and so
$a=r,f=d$), or the terms with highest powers of $x_2$ in $a \,
\tA \, \tF_{x_2}$ and $f \, \tA_{x_2} \, \tF$ cancel against each
other. In the latter case,
\[ a \, (d-f) = f \, (r-a) \implies a \, d = f \, r \implies a \, \mu
= f \, \mu'. \]
In either case, $\mu' \, | \, a$ and $\mu \, | \, f$. Define $G$
such that $x_1$ appears in it exactly to the power $\frac{f}{\mu}$, and similarly
for all $\ell$. \qed
\medskip
Now consider the composite morphism
\[
S_0 \simeq \wedge^{r+1} S_r \stackrel{\wedge^{r+1} \alpha_F}{\lra}
\wedge^{r+1} S_{r+d-2} \simeq S_{r+1}(S_{d-2}) \lra S_{(r+1)(d-2)}.
\]
The image of $1 \in S_0$ is the Wronskian $W(\alpha_F(x_1^r),
\alpha_F(x_1^{r-1} x_2), \dots, \alpha_F(x_2^r))$,
which we define to be $\Gott_{r,d}(F)$. To recapitulate,
$\Gott_{r,d}(F)$ is the determinant of the $(r+1) \times (r+1)$ matrix
\begin{equation}
(i,j) \lra \frac{\partial^r \, C_i}{\partial x_1^{r-j} \, \partial
\, x_2^j}, \quad
(0 \le i,j \le r). \label{formula.Grd} \end{equation}
where $C_i = (x_1^{r-i} \, x_2^i,F)_1$. Then
\[ \Gott_{r,d}(F) = 0 \iff \ker \alpha_F \neq 0 \iff [F] \in X_{e,d}. \]
Each matrix entry is linear in the coefficients of $F$,
and of order $d-2$ in $\ux$, hence $\Gott_{r,d}$ has degree $r+1$ and order
$N = (r+1)(d-2)$.
In the next section, we will generalise this construction to obtain a
family of covariants vanishing on $X_{e,d}$. The reader who is
more interested in Hilbert's solution may proceed directly to
\S\ref{Hilb.construction}.
\subsection{}
Let
\[ \B = (b_0, b_1, \dots, b_{d-2} \cb x_1,x_2)^{d-2}, \]
denote a generic form of order $d-2$, with a new set of
indeterminates $\ub$. As in \S\ref{section.covariants}, the
$\ub$ can be seen as forming a basis of $S_{d-2}^* \simeq
S_{d-2}$. Let $\Psi (\ub, \ux)$ denote a covariant of degree $r+1$
and order $q$ of $\B$. Then $\Psi$ corresponds to an embedding $S_q \lra
S_{r+1}(S_{d-2})$, which can be described as follows: if we realise
$S_{r+1}(S_{d-2})$ as the space of degree $r+1$ forms in the $\ub$, then $A(\ux) \in S_q$ gets sent to
$(A,\Psi)_q$. After dualising, we get a morphism
\[ f_\Psi: S_{r+1}(S_{d-2}) \lra S_q. \]
Now consider the composite morphism
\[ S_0 \simeq \wedge^{r+1} S_r \stackrel{\wedge^{r+1} \alpha_F}{\lra}
\wedge^{r+1} S_{r+d-2} \simeq S_{r+1}(S_{d-2})
\stackrel{f_\Psi}{\lra} S_q, \]
and let $\Gott_\Psi(\F)$ denote the image of $1 \in S_0$, which will
be called the G{\"o}ttingen covariant of $\F$ associated to $\Psi$. It is of
the same degree and order as $\Psi$, and hence its weight is $(r+1)$ more than that of $\Psi$.
In particular, $\Gott_{r,d}(\F)$ is the same as $\Gott_{\B^{r+1}}(\F)$. As before,
\begin{equation} [F] \in X_{e,d} \implies \Gott_\Psi(F) = 0. \label{Psi.implications} \end{equation}
\subsection{The calculation of $\Gott_\Psi$} \label{GottPsi.formula}
One can calculate $\Gott_\Psi(\F)$ explicitly by following the sequence of
maps above, which amounts to the following recipe:
\begin{itemize}
\item Introduce $2 \, (r+1)$ sets of binary variables
\[ \uy_{(i)} = \{y_{i1}, y_{i2} \}, \quad \uz_{(i)} =
\{z_{i1},z_{i2}\} \quad \text{for \; $0 \le i
\le r$; } \]
and let $\Omega_{\uy_{(i)} \, \uz_{(i)}}$ be the corresponding Omega
operators.
\item
Let $\cW$ denote the determinant
\[ (i,j) \lra \left. \frac{\partial^r (x_1^{r-i} \, x_2^i,\F)_1}{\partial
x_1^{r-j} \, \partial x_2^j} \right|_{\ux \lra \uy_{(i)}}, \quad (0 \le i, j \le r). \]
This is similar to~(\ref{formula.Grd}), except that the $\uy_{(i)}$ variables are
used throughout the $i$-th row. Let $\cW^\sharp$ denote the
symmetrisation of $\cW$ with respect to the sets $\uy_{(i)}$, i.e.,
\[ \cW^\sharp = \sum\limits_\sigma \; \cW(\uy_{\sigma(0)}, \dots,
\uy_{\sigma(r)}), \]
the sum quantified over all permutations $\sigma$ of $\{0,
\dots, r\}$. Then $\cW^\sharp$ is of degree
$r+1$ in $\ua$, and of order $d-2$ in each $\uy_{(i)}$.
\item Write
\[ \Psi = (\psi_0,\dots, \psi_q \cb x_1,x_2)^q, \]
where each $\psi_i$ is a degree $r+1$ form in $\ub = \{b_0, \dots,
b_{d-2} \}$.
Introduce $r+1$ sets of variables $\ub_{(0)}, \dots, \ub_{(r)}$, where
\[ \ub_{(i)} = \{b_{i \, 0}, \dots, b_{i \, d-2} \}, \]
and let $\widetilde \Psi$ be the total polarisation of $\Psi$ with respect to
the new variables (see~\cite[\S 1.1]{Dolgachev2}). Then $\widetilde \Psi$ is linear in each set
$\ub_{(i)}$.
\item Let $\widehat \Psi$ denote the form obtained from $\widetilde \Psi$ by replacing
$b_{ik}$ with $\frac{1}{(d-2)!} \, z_{i2}^{d-2-k}(-z_{i1})^k$ for $0 \le i \le r$ and
$0 \le k \le d-2$. (This is similar to the identification of $a_i$
as in \S\ref{section.covariants}.) Thus $\widehat \Psi$ is of order $q$ in $\ux$,
and of order $d-2$ in each $\uz_{(i)}$.
\item
Finally,
\[ \Gott_\Psi(\F) = [ \, \Omega_{\uy_{(0)} \, \uz_{(0)}}^{d-2} \circ \dots \circ
\Omega_{\uy_{(r)} \, \uz_{(r)}}^{d-2} \, ] \, \widehat\Psi \, \cW^\sharp \,. \]
This removes all the $\uy_{(i)}$ and $\uz_{(i)}$ variables, which leaves a
form of degree $r+1$ in the $\underline{a}$ and order $q$ in $\ux$.
\end{itemize}
\subsection{} It may happen that $\Gott_\Psi$ is identically zero,
even if $\Psi$ is nontrivial. (Hence the implication
in~(\ref{Psi.implications}) is not reversible in general.) For
instance, recall that a generic binary $d$-ic has a cubic
invariant exactly when $d$ is a multiple of $4$. Now let
$r=2$, and assume $d \equiv 2 \; (\text{mod 4})$. Then $\Psi
=(\B,(\B,\B)_{\frac{d-2}{2}})_{d-2}$ is a nontrivial cubic invariant of
$\B$, but $\Gott_\Psi$ must vanish identically.
If $d$ is a divisor of $r$, then $X_{e,d} = \P S_d$, and
in that case all $\Gott_\Psi$ are identically zero.
\medskip
\noindent {\bf N.B.} Henceforth, if $A,B$ are two quantities, we will write $A \doteq B$ to
mean that $A = c \, B$ for some unspecified nonzero rational scalar
$c$. This will be convenient in symbolic calculations, where more and
more unwieldy scalars tend to accumulate at each stage.
\subsection{} \label{Gott.quadratic}
As an example, we will follow this recipe when
$r=1$ and $\Psi$ is any quadratic covariant. Write symbolically
\[ \F = \alpha_\ux^d = \beta_\ux^d, \qquad \B = p_\ux^{d-2} = q_\ux^{d-2}. \]
Every quadratic covariant of $\B$ must be of the form
\[ \Psi = (\B,\B)_{2n} = (p \, q)^{2n} \, p_\ux^{d-2-2n} \,
q_\ux^{d-2-2n}, \]
for some $n$ in the range $0 \le n \le \frac{d-2}{2}$.
Using $\alpha, \beta$ for the two rows of $\cW$, we get
\[ \cW \doteq \left| \begin{array}{rr}
\alpha_1 \, \alpha_2 \, \alpha_{\uy_{(0)}}^{d-2} & \alpha_2^2 \, \alpha_{\uy_{(0)}}^{d-2}
\\
\beta_1^2 \, \beta_{\uy_{(1)}}^{d-2} & \beta_1 \, \beta_2 \,
\beta_{\uy_{(1)}}^{d-2} \end{array} \right| = \alpha_2 \, \beta_1 \, (\alpha
\, \beta) \, \alpha_{\uy_{(0)}}^{d-2} \, \beta_{\uy_{(1)}}^{d-2}, \]
and hence
\[ \cW^\sharp \doteq (\alpha \, \beta)^2 \, \alpha_{\uy_{(0)}}^{d-2} \,
\beta_{\uy_{(1)}}^{d-2}. \]
Now the symbolic expression for $\widetilde \Psi$ is the same as the
one for $\Psi$, once we make the convention that $p,q$ respectively refer
to the $\ub_{(0)}, \ub_{(1)}$ variables. Then
\[ \widehat \Psi \doteq (\uz_{(0)} \, \uz_{(1)})^{2n} \, (\ux \, \uz_{(0)})^{d-2-2n} \, (\ux
\, \uz_{(1)})^{d-2-2n}; \]
and finally,
\[ \Gott_\Psi \doteq (\alpha \, \beta)^{2n+2} \, \alpha_\ux^{d-2-2n} \, \beta_\ux^{d-2-2n}. \]
We have proved the following:
\begin{Proposition} \sl
If $\Psi = (\B,\B)_{2n}$, then $\Gott_{\Psi} \doteq (\F, \F)_{2n+2}$.
\label{prop.quadG} \end{Proposition}
In particular, $\Gott_{1,d} \doteq \Gott_{\B^2} =
\Gott_{(\B,\B)_0} \doteq (\F, \F)_2$ is the
Hessian of $\F$. Similar calculations show that
\begin{equation}
\begin{array}{l}
\Gott_{2,d} \doteq (\F, (\F,\F)_2)_1, \\
\Gott_{3,d} \doteq 3 \, (2 \, d-3) \, (\F,\F)_2^2 - 2 \, (d-2) \, \F^2 \,
(\F,\F)_4, \\
\Gott_{4,d} \doteq 2 \, (3 \, d-4) \, (\F,\F)_2 \, (\F, (\F,\F)_2)_1 - (d-3) \,
\F^2 \, (\F,(\F,\F)_4)_1. \end{array}
\label{Gott.lowr} \end{equation}
(Such formulae are derived for the $\Hilb_{r,d}$ in~\cite{Brioschi}
and~\cite{Hilbert}, but this makes no difference in view of
Theorem~\ref{Theorem.HGeq}.) However, as $r$ grows, it quickly
begins to get more and more tedious to execute this recipe.
\section{Hilbert's construction}
\label{section.Hilb.construction}
In this section we will describe Hilbert's construction of his
covariants $\Hilb_{r,d}$, and later prove that the outcome
coincides with $\Gott_{r,d}$ up to a scalar.
The underlying idea is as follows. Suppose, for instance, that
$F$ is an order $10$ form such that $F = G^5$ for some quadratic $G$. Then substituting $x_1=1, x_2=z$, we have
\[ \frac{d^3}{d z^3} \; \sqrt[5]{F(1,z)} = 0. \]
One should like to convert the left-hand side into a covariant
condition on $F$; but this requires some technical modifications. We begin
by constructing the source of Hilbert's covariant.
\subsection{} \label{Hilb.construction}
Define
\[ h_0 = a_0^{r+1-\frac{r}{d}} \, E_+^{\, r+1} \,
(a_0^{\, \frac{r}{d}}). \]
This is easily seen to be an isobaric homogeneous form of degree and weight
$r+1$ in the $\ua$. For instance,
\[ h_0 =
\begin{cases} (d-1) \, (a_0 \, a_2 - a_1^2) & \text{if $r=1$,} \\
(2 d^2 -6d+4) \, a_0^2 \, a_3 -(6d^2-18d+12) \, a_0 \, a_1 \, a_2 +
(4 \, d^2- 12 \, d + 8) \, a_1^3 & \text{if $r=2$.}
\end{cases} \]
\begin{Lemma} \sl
The form $h_0$ is a source.
\end{Lemma}
\demo We want to show that $E_{-} \, h_0 = a_0^{r+1-\frac{r}{d}} \,
E_{-} \, E_{+}^{\, r+1}(a_0^{\, \frac{r}{d}})$ vanishes. Apply Lemma~\ref{lemma.E}, and
note that
\[ E_0 \, (a_0^{\frac{r}{d}})= - r \, a_0^{\frac{r}{d}}, \quad E_{-} (a_0^{\frac{r}{d}})=0, \]
which implies the result. \qed
\smallskip
Since $h_0$ has weight $r+1$, the covariant corresponding to $h_0$
must have order $N = (r+1)(d-2)$. The Hilbert covariant is defined to be
\begin{equation}
\Hilb_{r,d}(\F) = (h_0, \dots, h_N \cb x_1,
x_2)^N, \label{Hilb.cov.definition} \end{equation}
where
\begin{equation}
h_k = \frac{(N-k)!}{N !} \, E_{+}^k(h_0)
\qquad \text{for $0 \le k \le N$.}
\label{formula.hk} \end{equation}
\subsection{} In order to prove Theorem~\ref{Theorem.HGeq}, it will
suffice to show that $\Hilb_{r,d}$ and $\Gott_{r,d}$ have the same
source up to a scalar. We will avoid writing such scalars explicitly in the course of the
calculation, but see formula~(\ref{formula.krd}) below.
Let $\ua = (a_0, \dots, a_d)$ denote a $(d+1)$-tuple of
complex variables. For $t \in \complex$ define
\[ \gamma_t: \complex^{d+1} \lra \complex^{d+1}, \quad
(a_0,a_1, \dots, a_d) \lra (a_0(t), a_1(t), \dots, a_d(t)), \]
by the formula
\[ (a_0, \dots, a_d \cb 1,z+t)^d = (a_0(t), \dots, a_d(t) \cb
1,z)^d. \]
It is easy to see that
\[ a_i(t) = a_i + (d-i) \, a_{i+1} \, t + O(t^2) \quad \text{for $0
\le i \le d-1$}. \]
Hence, given an analytic function $\phi: \complex^{d+1} \lra
\complex$, we have an equality
\[ E_+ \, \phi = \left[\frac{d}{d \, t} (\phi(\gamma_t))
\right]_{t=0}. \]
Iterating this formula,
\begin{equation}
E_+^n \, \phi =\left[\frac{\partial^n \phi(\gamma_{t_1+\dots +t_n})}{\partial t_1\cdots\partial t_n}\right]_{t_1=\cdots=t_n=0}
= \left[\frac{d^n \, \phi(\gamma_t)}{d \, t^n}\right]_{t=0}.
\label{formula.Ediff} \end{equation}
Now write $f(z) = (a_0, \dots, a_d \cb 1,z)^d$, and apply this to the function
\[ \phi(\ua) = a_0^{\, \frac{r}{d}} = f(0)^{\frac{r}{d}} , \]
which gives the expression
\begin{equation}
h_0=f(0)^{r+1-\frac{r}{d}} \, \left[ \frac{d^{\, r+1}}{d \, t^{r+1}}
\, f(t)^{\frac{r}{d}} \right]_{t=0}, \label{h0.formula} \end{equation}
for the source of $\Hilb_{r,d}$.
\subsection{}
We make a small digression to prove that $\Hilb_{r,d}$ has the
required vanishing property.
\begin{Proposition} \sl
Let $F$ be a $d$-ic. Then
\[ \Hilb_{r,d}(F) =0 \iff [F] \in X_{e,d}. \]
\end{Proposition}
\demo
This will of course follow from Theorem~\ref{Theorem.HGeq}, but even so,
we include an independent proof. After a change of variables, we may
assume $a_0 \neq 0$. Write
\begin{equation} f(t)^{\frac{r}{d}} = a_0^{\frac{r}{d}} \; (1 + \sum\limits_{m \ge 1}
\; \frac{\theta^m}{m!} \, t^m).
\label{powseries.f} \end{equation}
Using the reformulation of $E_+$ above, we have
\[ a_0^{\frac{r}{d}} \, \theta_{m+1} = E_+(a_0^{\frac{r}{d}} \,
\theta_{m}). \]
Now a simple induction shows that there are identities
\[ a_0^{r+1} \, \theta_{r+1} = h_0, \quad
a_0^{r+2} \, \theta_{r+2} \doteq a_0 \, h_1 + \Box_1 \, h_0, \]
and in general
\[ a_0^{r+1+k} \, \theta_{r+1+k} \doteq a_0^k \, h_k + \sum\limits_{i=1}^k \; \Box_i \, h_{k-i} , \]
for some homogeneous polynomials $\Box_i(\ua)$ of degree $k$ and
weight $i$. (Here we have set $h_i = 0$ for $i > N$.) If $h_0, h_1,
\dots $ etc.~all vanish, then so do $\theta_k$ for $k
\ge r+1$, and the power series in~(\ref{powseries.f}) becomes a polynomial of degree
$\le r$. Thus $f(t)$ reduces to a perfect $\mu$-th power. Conversely, if
$f(t) = g(t)^\mu$, then $f^{\frac{r}{d}} = g^{\mu'}$ is of degree $\le
r$, and hence $h_0 = h_1 = \dots =0$. \qed
\subsection{}
We should like to calculate the source of $\Gott_{r,d}$ as defined by
the determinant in~(\ref{formula.Grd}). However, the dehomogenisation
in the previous section is with respect to the other variable, so a
preparatory step is needed. The Wronskian construction is equivariant,
hence in the notation of \S\ref{section.intro1}, we have an identity
\[ W(C_0,\dots, C_r)(x_1,x_2) = W(C_0^g, \dots, C_r^g)(x_2,-x_1), \]
for $g = \left( \begin{array}{rr} 0 & -1
\\ 1 & 0 \end{array} \right)$. Let $z = -x_2/x_1$, then (up to a
scalar) the right-hand side becomes
\[(-x_1)^N \times \left|
\begin{array}{ccc}
v_0^{(r)} & \cdots & v_0\\
\vdots & & \vdots\\
v_r^{(r)} & \cdots & v_r \end{array} \right|, \]
where
\[ v_i(z)=C_i^g(z,1)=C_i(-1, z), \quad \text{and} \quad v_i^{(k)} =
\frac{d^k}{dz^k} v_i.\]
Substituting $x_1=1,x_2=0$,
\[ g_0 = \text{source of $\Gott_{r,d}$} \doteq \left|
\begin{array}{ccc}
v_0^{(r)}(0) & \cdots & v_0(0)\\
\vdots & & \vdots\\
v_r^{(r)}(0) & \cdots & v_r(0) \end{array} \right|. \]
By definition,
\[ C_i \doteq \frac{\partial \, [x_1^{r-i} \, x_2^i]}{\partial x_1}
\frac{\partial F}{\partial x_2} -
\frac{\partial \, [x_1^{r-i} \, x_2^i]}{\partial x_2} \, \frac{\partial
F}{\partial x_1}. \]
Using $F(x_1,x_2)=x_1^d \, f(\frac{x_2}{x_1})$, this can be rewritten as
\[ C_i \doteq \left(
i \, d \, x_1^{d+r-i-1} \, x_2^{i-1} \, f\left(\frac{x_2}{x_1}\right)
-r \, x_1^{d+r-i-2} \, x_2^i \, f'\left(\frac{x_2}{x_1}\right)
\right), \]
and therefore
\[ v_i(z) \doteq \left( i \, d \, (-1)^{d+r-i-1} \, z^{i-1} \, f(-z)
-r \, (-1)^{d+r-i-2} \, z^i \, f'(-z) \right). \]
Let
\begin{equation} b_i(z)= i \, d \, z^{i-1} \, f(z)-r \, z^i \, f'(z), \label{definition.biz} \end{equation}
so that $v_i(-z)=(-1)^{d+r} \, b_i(z)$. After reordering the columns,
\[
g_0 \doteq \left|
\begin{array}{ccc}
b_0(0) & \cdots & b_0^{(r)}(0)\\
\vdots & & \vdots\\
b_r(0) & \cdots & b_r^{(r)}(0)
\end{array} \right|. \]
Now the key step is to write
\[ b_i \doteq f(z)^{1+\frac{r}{d}} \, \underbrace{\frac{d}{dz} \, \left(z^i
f(z)^{-\frac{r}{d}}\right)}_{c_i}, \]
which is similar to the idea of an integrating factor in the theory of
ordinary differential equations. Then
\begin{equation} g_0 \doteq (a_0^{1+\frac{r}{d}})^{r+1}\times \left|
\begin{array}{ccc}
c_0(0) & \cdots & c_0^{(r)}(0)\\
\vdots & & \vdots\\
c_r(0) & \cdots & c_r^{(r)}(0)
\end{array}
\right|\ .
\label{det.ci} \end{equation}
Now let $\mu_i =z^i$ and $\omega=f(z)^{-\frac{r}{d}}$, so that
$c_i=(\mu_i \, \omega)'$. By the Leibniz rule,
\[ c_i^{(j)} = (\mu_i \, \omega)^{(j+1)}=\sum_{k=0}^{r+1} \binom{j+1}{k} \,
\mu_i^{(k)} \omega^{(j+1-k)} \quad \text{for $0\le j\le r$,} \]
with the convention that $\binom{j+1}{k}=0$ if $k > j+1$. Since
$\mu_i^{(r+1)}=0$, we can stop the summation at $k=r$, and factor the determinant
in~(\ref{det.ci}) as
\begin{equation} \left|
\begin{array}{ccc}
\mu_0(0) & \cdots & \mu_0^{(r)}(0)\\
\vdots & & \vdots\\
\mu_r(0) & \cdots & \mu_r^{(r)}(0)
\end{array}
\right|\times \Delta, \label{det.mu.delta} \end{equation}
where
\[ \Delta=
\left| \begin{array}{ccccc}
\left(\begin{array}{c} 1\\ 0\end{array}\right)\omega' &
\left(\begin{array}{c} 2\\ 0\end{array}\right)\omega'' &
\cdots & \cdots &
\left(\begin{array}{c} r+1\\ 0\end{array}\right)\omega^{(r+1)}\\
\left(\begin{array}{c} 1\\ 1\end{array}\right)\omega &
\left(\begin{array}{c} 2\\ 1\end{array}\right)\omega' &
\cdots & \cdots &
\left(\begin{array}{c} r+1\\ 1\end{array}\right)\omega^{(r)}\\
0 &
\left(\begin{array}{c} 2\\ 2\end{array}\right)\om &
\cdots & \cdots &
\left(\begin{array}{c} r+1\\ 2\end{array}\right)\om^{(r-1)}\\
\vdots & \ddots & \ddots & & \vdots\\
0 & \cdots & 0 &
\left(\begin{array}{c} r\\ r\end{array}\right)\om &
\left(\begin{array}{c} r+1\\ r\end{array}\right)\om'
\end{array}
\right|
\]
evaluated at $z=0$. The first determinant in~(\ref{det.mu.delta}) is a
pure rational constant, so it only remains to calculate $\Delta$.
\subsection{}
Let $\nu=\omega^{-1}=f(z)^{\frac{r}{d}}$. For $f(0)\neq 0$,
these are holomorphic functions of $z$ near the origin. From
\[ \omega \, \nu = (\omega_0+\omega_1 \, z+\omega_2 \, z^2+\cdots) \,
(\nu_0+\nu_1 \, z+\nu_2 \, z^2+\cdots) =1, \]
we get the linear system
\[
\omega_0 \, \nu_0=1, \quad \text{and} \quad
\sum\limits_{i=0}^k \, \omega_i \, \nu_{k-i} =0, \quad
\text{for $1 \le k \le r+1$.} \]
Solving for $\nu_{r+1}$ by Cramer's rule,
\begin{equation} \nu_{r+1}=\frac{1}{\omega_0^{r+2}}\times
\left|
\begin{array}{ccccc}
\omega_0 & 0 & \cdots & \cdots & 1\\
\omega_1 & \om_0 & 0 & \cdots & 0\\
\vdots & & \ddots & & \vdots\\
\omega_r & & & \om_0 & 0\\
\omega_{r+1} & \cdots & \cdots & \omega_1 & 0
\end{array}
\right| =
\frac{(-1)^{r+1}}{\omega_0^{r+2}} \times
\left|
\begin{array}{ccccc}
\omega_1 & \om_2 & \cdots & \cdots & \omega_{r+1}\\
\omega_0 & \omega_1 & \cdots & \cdots & \omega_{r}\\
0 & \omega_0 & \ddots & & \vdots\\
\vdots & \ddots & \ddots & \ddots & \vdots\\
0 & \cdots & 0 & \omega_0 & \omega_1
\end{array}
\right|, \label{two.det} \end{equation}
where the second determinant is obtained by expanding the first by its last column and
transposing.
\subsection{}
On the other hand,
\[
\Delta = \left|
\binom{j+1}{k} \, \omega^{(j+1-k)} \right|_{0\le k,j\le r}
= \left| \frac{(j+1)!}{k!} \, \bbone_{\{k\le j+1\}} \, \omega_{j+1-k}
\right|_{0\le k,j\le r}, \]
where the characteristic function $\bbone_{\{k\le j+1\}}$ assumes the
value $1$ if $k \le j+1$, and $0$ otherwise. This is the same as the
rightmost determinant in~(\ref{two.det}), hence
\begin{equation} g_0 \doteq (a_0^{1+\frac{r}{d}})^{r+1} \, \Delta \doteq
(a_0^{1+\frac{r}{d}})^{r+1} \, \omega_0^{r+2} \,
\nu_{r+1}. \label{g0.final.formula} \end{equation}
Now recall that
\[ \omega_0=\om(0)=a_0^{-\frac{r}{d}}, \]
and
\[
\nu_{r+1}=\frac{1}{(r+1)!} \, \nu^{(r+1)}(0)
=\frac{1}{(r+1)!} \left[\frac{d^{\, r+1}}{d \, t^{\, r+1}} \, f(t)^{\frac{r}{d}} \right]_{t=0}. \]
The exponent of $a_0$ reduces to
\[
\left(1+\frac{r}{d}\right)(r+1)-\frac{r}{d}(r+2)=r+1-\frac{r}{d}. \]
Hence, by comparing~(\ref{h0.formula}) with~(\ref{g0.final.formula}),
we get
\[ g_0 \doteq h_0, \]
which completes the proof of Theorem~\ref{Theorem.HGeq}. \qed
\bigskip
\noindent If we keep track of the unwritten scalars in the intermediate stages,
then the connecting relation is
\begin{equation} g_0 = \left\{\frac{\prod\limits_{i=0}^r \, i! \,
(d+i-2)!}{\left[ \, r \times (d-2)! \, \right]^{r+1}}
\, \right\} h_0. \label{formula.krd} \end{equation}
This of course implies a parallel relation between $\Gott_{r,d}$ and
$\Hilb_{r,d}$.
\subsection{} One can give a formula for the Hilbert covariant directly,
without constructing its source first. It is merely the homogenised version of the
formula (\ref{formula.hk}) combined with (\ref{formula.Ediff}), so we
omit the proof. Introduce binary variables
$\uy = \{y_1, y_2\}$.
\begin{Proposition} \sl
We have an identity
\[ \Hilb_{r,d}(\F) \doteq \frac{\F(x_1,x_2)^{r+1-\frac{r}{d}}}{(x_1 \, y_2 - x_2 \, y_1)^{r+1}} \,
\left[ \, \left( y_1 \, \frac{\partial}{\partial x_1} + y_2 \,
\frac{\partial}{\partial x_2} \right)^{r+1} \F(x_1,x_2)^{\frac{r}{d}} \right]. \]
\label{proposition.Hpolar} \end{Proposition}
\section{The three ideals}
Let $X = X_{e,d}$ be as in \S\ref{section.defn.rde}, with
$I_X \subseteq R$ its homogeneous defining ideal. Let $J$ (respectively $\fg$) denote the ideal in $R$
generated by the coefficients of $\Gott_{r,d}$ (respectively all
possible $\Gott_\Psi$). In other words, $\fg$ is the ideal generated
by the maximal minors of a matrix representing the morphism
$\alpha_\F: S_r \lra S_{r+d-2}$ from \S\ref{definition.alphaF}. There are inclusions
\[ J \subseteq \fg \subseteq I_X. \]
The zero locus of each of these ideals is $X$, but depending on the values
of $r$ and $d$, either of these inclusions may be proper.
Since $I_X$ has nonzero elements in degree $e+1$ (arising from the coefficients of
$\Gott_{e,d}$), we must have a proper containment $\fg \subsetneq
I_X$, if $r$ does not divide $d$.
\subsection{} \label{section.rnc.ideal}
Suppose $r=1$, so that $X$ is the rational normal $d$-ic
curve. We have a decomposition
\[ R_2 \simeq S_2(S_d) \simeq \bigoplus\limits_{n=0}^{\lfloor\frac{d}{2}\rfloor}
\, S_{2d-4n}, \]
where the summand $S_{2d-4n}$ is spanned by the coefficients of $(\F,\F)_{2n}$. It is classically known
that $I_X$ is minimally generated in degree $2$, and $(I_X)_2 \simeq \bigoplus\limits_{n \ge 1} \, S_{2d-4n}
\subseteq R_2$ (see~\cite{Ch2}). By Proposition \ref{prop.quadG}, we have $\fg =
I_X$. Moreover, $J$ and $\fg$ coincide for $d \le 3$ and differ
afterwards.
\subsection{} \label{section.g36}
Assume $r=3, d=6$. One can explicitly calculate the ideal of $X = X_{3,6}$
using the following elimination-theoretic technique. Let
$Q = (q_0,q_1,q_2,q_3 \cb x_1,x_2)^3$, where the $q_i$ are independent
indeterminates. Write
\[ (a_0, \dots, a_6 \cb x_1,x_2)^6 = Q^2 \]
and equate the corresponding coefficients on both sides. This
gives expressions $a_i = f_i (q_0, \dots, q_3)$, defining a ring homomorphism
\[ {\mathfrak f}: R \lra \complex[q_0, \dots, q_3], \qquad
a_i \lra f_i(q_0, \dots, q_3). \]
Then $I_X$ is the kernel of $\mathfrak f$. We carried out this computation in the
computer algebra system {\sc Macaulay-2} (henceforth {\sc M2}); it
shows that $I_X$ is minimally generated by a $45$-dimensional subspace
of $R_4$.
In order to determine the piece $(\fg)_4$, we need to list the
degree $4$ covariants of $\B$. By the Cayley-Sylvester formula,
\[ S_4(S_4) = S_{16} \oplus S_{12} \oplus S_{10}
\oplus (S_8 \otimes \complex^2)
\oplus (S_4 \otimes \complex^2) \oplus S_0. \]
It is classically known (see~\cite[\S 89]{GY}) that each covariant of a generic binary quartic $\B$ is a polynomial in these
fundamental covariants:
\[ C_{1,4} = \B, \quad C_{2,4} = (\B,\B)_2, \quad C_{2,0} = (\B,\B)_4, \quad
C_{3,6} = (\B, (\B,\B)_2)_1, \quad C_{3,0} = (\B,(\B,\B)_2)_4, \]
where $C_{m,q}$ is of degree-order $(m,q)$. Hence, the space of degree $4$ covariants of $\B$ is spanned by
\[ \begin{array}{lllll}
\Psi_{4,16} = C_{1,4}^4, & \Psi_{4,12} = C_{1,4}^2 \, C_{2,4}, & \Psi_{4,10} = C_{1,4}
\, C_{3,6}, & \Psi_{4,8}^{(1)} = C_{2,4}^2, & \Psi_{4,8}^{(2)} = C_{1,4}^2 \, C_{2,0}, \\
\Psi_{4,4}^{(1)} = C_{1,4} \, C_{3,0}, & \Psi_{4,4}^{(2)} = C_{2,4} \,
C_{2,0}, & \Psi_{4,0} = C_{2,0}^2.
\end{array} \]
We have calculated $\Gott_\Psi$ in each case using the recipe of
\S\ref{GottPsi.formula}. It turns out that the ones coming from $\Psi_{4,16}, \Psi_{4,12},
\Psi_{4,0}$ are nonzero, whereas $\Gott_{\Psi_{4,10}}$ vanishes
identically. Moreover, we have identities
\[ 6 \, \Gott_{\Psi_{4,8}^{(1)}} = \Gott_{\Psi_{4,8}^{(2)}}, \qquad
29 \, \Gott_{\Psi_{4,4}^{(1)}} = 36 \, \Gott_{\Psi_{4,4}^{(2)}}, \]
and thus both $\Psi_{(4,8)}^{(i)}$ lead to the same G{\"o}ttingen covariant (up to a
scalar), and similarly for $\Psi_{(4,4)}^{(i)}$. Hence
\[ (\fg)_4 \simeq S_{16} \oplus S_{12} \oplus S_8 \oplus S_4 \oplus
S_0, \]
which is exactly $45$-dimensional; this forces $\fg = I_X$.
\subsection{} \label{section.g26}
Assume $r=2$, and $d$ even. Now \cite[Theorem 7.2]{AC1} says that
$I_X$ is minimally generated by cubic forms, and its generators are explicitly
described there. If $d=4$, then $(I_X)_3 \simeq S_6$, with the only
piece coming from $\Gott_{2,4}$. If $d=6$, then
\[ (I_X)_3 \simeq S_{12} \oplus S_8 \oplus S_6. \]
The three summands are respectively generated by the coefficients
of:
\[ \Phi_{3,12} = (\F^2,\F)_3, \quad \Phi_{3,8} = (\F^2,\F)_5, \quad
\Phi_{3,6} = 33 \, (\F^2,\F)_6 - 250 \, (\F,(\F,\F)_2)_4. \]
Now, following the recipe of \S \ref{GottPsi.formula}, one finds that
\[ \Gott_{\B^3} \doteq \Phi_{3,12}, \quad
\Gott_{\B \, (\B,\B)_2} \doteq \Phi_{3,8}, \quad
\Gott_{(\B, (\B,\B)_2)_1} \doteq \Phi_{3,6}; \]
and hence $\fg = I_X$ once again.
\medskip
We have calculated several such examples, which suggest the
following pair of conjectures:
\begin{Conjecture} \sl
Assume that $r$ divides $d$. Then
\begin{itemize}
\item[(c1)]
the ideal $I_X$ is always minimally generated in degree $r+1$, and
\item[(c2)] $\fg = I_X$.
\end{itemize}
\end{Conjecture}
At least for $r=1, 2$, something much stronger than (c1) is true;
namely $I_X$ has Castelnuovo regularity $r+1$, and its graded minimal
resolution is linear (see~\cite[Theorem 1.4]{AC1}). We do not know of
a counterexample to this when $r> 2$.
Referring to the diagram at the beginning of \S\ref{section.Gott.cov}, note that the ideal of the Veronese
embedding is generated by quadrics; but the projection $\pi$ (which
implicitly involves elimination theory) will tend to increase the degrees of the
defining equations of its image.
\subsection{The saturation of $J$} \label{proof.Jsaturation}
In this section we will prove Theorem~\ref{Theorem.Jsaturation}. We
want to show that the ideal $J$ defines $X$ scheme-theoretically when $r$ divides $d$, that is to say,
\[ \Proj R/I_X \lra \Proj R/J \]
is an isomorphism of schemes. The following example
should convey the essential idea behind the proof.
Assume $r=2, d=6$. Write $t_i = a_i/a_0$ for $1 \le i \le 6$. Let
$A = \complex[t_1, \dots, t_6]$, and consider the corresponding degree
zero localisation $\fra = (J_{a_0})_0 \subseteq A$. The zero
locus of $\fra$ is $X \setminus \{a_0=0\} \simeq \Aff^2$. Since the
question is local on $X$, it would suffice to show that $A/\fra$ is isomorphic to a polynomial algebra
$\complex[v_1,v_2]$.
Now $\Hilb_{2,d} \doteq (\F, (\F, \F)_2)_1$, and we have explicitly
written down its first few terms in \S\ref{ex.isobaric}. Note that the
monomial $a_0^2 \, a_3$ occurs in its source, and similarly $a_0^2 \, a_4, a_0^2 \, a_5, a_0^2
\, a_6$ occur in the successive coefficients. Hence, modulo $\fra$, we have identities of
the form
\[ t_k = \text{a polynomial expression in $t_1, t_2, \dots, t_{k-1}$,}
\quad \text{for $3\le k \le 6$}. \]
Thus we have a surjective ring morphism
\[ \complex[v_1,v_2] \lra A/\fra, \quad v_i \lra t_i. \]
Since $\text{Krull-dim} \, A/\fra=2$, this must be an isomorphism.
\medskip
For the general case, write $\Hilb_{r,d}$ as in~(\ref{Hilb.cov.definition}),
and recall that $h_{k-(r+1)}$ is isobaric of weight $k$.
\begin{Lemma} \sl The coefficient of $a_0^r \, a_k$ in $h_{k-(r+1)}$ is
nonzero for $r+1 \le k \le d$.
\end{Lemma}
\demo
The monomial $a_0^r \, a_{r+1}$ can appear in $h_0$ only by one route,
namely by applying the sequence
\[ \left[ (d-r) \, a_{r+1} \, \frac{\partial}{\partial a_r} \right]
\circ \dots \left[(d-1) \, a_2 \, \frac{\partial}{\partial a_1}
\right] \circ \left[ d \, a_1 \,\frac{\partial}{\partial a_0} \right] \]
to $a_0^{\frac{r}{d}}$, and then multiplying by
$a_0^{r+1-\frac{r}{d}}$. Hence its coefficient is nonzero.
Now $a_0^r \, a_k$ can appear in $h_{k-(r+1)} \doteq E_+ \, h_{k-1-(r+1)}$ only by
applying $(d-k+1) \, a_k \, \frac{\partial}{\partial a_{k-1}}$ to
$a_0^r \, a_{k-1}$, so we are done by induction. \qed
\medskip
We can always change co-ordinates such that $a_0 \neq 0$ at any
given point of $X$. Write $t_i = a_i/a_0$ and $\fra < A = \complex[t_1,
\dots, t_d]$ as above. By the lemma, each of $t_{r+1}, \dots,
t_d$ is a polynomial in $t_1, \dots, t_r$ modulo $\fra$. This gives a bijection
\[ \complex[v_1,\dots, v_r] \lra A/\fra, \quad v_i \lra t_i, \]
which shows that the scheme $\Proj R/J$ is locally isomorphic
to the affine space $\Aff^r$, and hence $J_\text{sat} = I_X$.
This completes the proof of Theorem~\ref{Theorem.Jsaturation}. \qed
\subsection{} It follows that $J$ and $I_X$ coincide in sufficiently
large degrees. Let $\SI(r,d)$ denote the saturation index of $J$,
namely it is the smallest integer $m_0$ such that
\[ J_m = (I_X)_m \qquad \text{for all \, $m \ge m_0$}. \]
It would be of interest to have a bound on this quantity in either
direction. It is proved in~\cite{Ch2} that
\[ \frac{1}{d-2} \sqrt{\frac{(d-1)(d^2-2)}{2}} \le \SI(1,d) \le
d+2; \]
but those techniques do not seem to generalise readily to the case
$r>1$. We have obtained the following few values by explicit
calculations in {\sc M2:}
\[ \begin{array}{lllll}
\SI(2,4) = 3, & \SI(2,6) = 7, & \SI(2,8) = 9, & \SI(2,10) = 9, & \SI(2,12) = 10, \\
\SI(3,6) = 9, & \SI(3,9) = 11, & {\mathfrak S}(4,8) = 13.
\end{array} \]
A similar (but larger) table for $r=1$ is given in \cite{Ch2}, where the value of
$\SI$ is related to transvectant identities involving the Hessian.
\subsection{} Suppose $e_i = \gcd (r_i,d)$ for $i=1,2$. Then $X_{e_1,d}
\subseteq X_{e_2,d}$ exactly when $e_1 \, | \, e_2$.
However, the containment relations between the ideals $J_{r_i,d}$ are
not altogether obvious. For $J_{r_1,d} \supseteq J_{r_2,d}$
to be true, it is necessary that $r_1 \le r_2$ and $e_1 \, |
\, e_2$, but these conditions are not sufficient. For
instance, we have obtained the following miscellaneous data by calculating these ideals in {\sc M2}:
\[ \begin{array}{llll}
J_{2,5} \not\supseteq J_{3,5}, & J_{3,5} \not\supseteq J_{4,5}, &
J_{2,5} \supseteq J_{4,5}, & J_{4,5} \not\supseteq J_{6,5}, \\
J_{2,4} \supseteq J_{6,4}, & J_{6,4} \supseteq
J_{10,4}; \end{array} \]
which at least shows that the general pattern is not so easily
guessed. Nevertheless, we have the following modest result:
\begin{Proposition} \sl
There are inclusions $J_{1,d} \supseteq J_{r,d}$ for arbitrary $d$, and $r=2,3,4$.
\label{prop.containmentJ} \end{Proposition}
\demo It is clear from the formula for a transvectant (see
\S\ref{section.trans}), that if the coefficients of $A$
belong to an ideal, then all the coefficients of $(A,B)_k$
also belong to this ideal. Hence, given any covariants $\Phi_1, \dots,
\Phi_n$ of $\F$, all the coefficients of any transvectant of the form
\[ (\dots ((\Gott_{1,d}, \Phi_1)_{k_1}, \Phi_2)_{k_2}, \dots , \Phi_n)_{k_n} \]
are in $J_{1,d}$. Thus the result would follow if we could obtain $\Gott_{r,d}$ as a linear
combination of such expressions.
Observe the formulae in (\ref{Gott.lowr}). It is clear that
$\Gott_{2,d}$ is itself such an expression. Let $r=4$, then this is also true
of the first term in $\Gott_{4,d}$. Now the so-called Gordan syzygies give relations between cubic covariants
of $\F$. In particular, the syzygy which is written as
$\left( \begin{array}{ccc} \F & \F & \F \\ d & d & d \\ 0 & 1 &
4 \end{array} \right)$ in the notation of \cite[Ch.~IV]{GY}, gives an
identity
\[ (\F, (\F, \F)_4)_1 = \frac{2 \, (2d-5)}{d-4} \, (\F, (\F, \F)_2)_3, \]
for any $d \ge 5$. Hence the same follows for the second
term. (If $d \le 4$, then the second term is identically zero.) This proves the result for $r=4$.
The argument for $r=3$ is similar. The first term in $\Gott_{3,d}$
is already of the required form. Moreover, we have an identity
\[ \F^2 \, (\F,\F)_4 = \frac{d \, (2d-5)}{(d-3) \, (2d-1)} \, (\F,\F)_2^2
+ \frac{2 \, (2d-5)}{d-3} \, (\F^2,(\F,\F)_2)_2, \]
for $d \ge 4$. (This can be shown by a routine but tedious symbolic calculation as in
\cite[Ch.~IV-V]{GY}.) Hence the same is true of the second term, which completes the proof. \qed
\smallskip
Since the argument depends on specific features of these formulae,
it seems unlikely that this technique will generalise
substantially. Even so, we suspect that the proposition may well be true of all $r$.
\subsection{The twisted cubic curve} \label{section.twistedcubic}
Assume $d=3$, and $r$ arbitrary (but
not divisible by $3$). Then $X \subseteq \P^3$ is the twisted cubic curve. Since $\B$ is a linear form, the only
possibility for $\Psi$ is $\B^{r+1}$, hence $J = \fg$. It follows that
the Hilbert-Burch complex (see~\cite[\S 20]{Eisenbud1}) of $\alpha_\F$
gives a resolution
\[ 0 \la R/J \la R \la R(-r-1) \otimes S_{r+1} \la R(-r-2) \otimes S_r
\la 0. \]
Its first syzygy shows that we have an identity $(\Gott_{r,3},
\F)_2=0$. (The correspondence between syzygies and transvectant
identities is discussed in \cite[\S 4]{Ch1}.) The scheme $\Proj R/J$
has degree $\binom{r+2}{2}$, that is to say, it is a nonreduced
$\frac{(r+1) \, (r+2)}{6}$--fold structure on $X$ for $r>1$.
We have $\sqrt{J_{r,3}} = I_X$ for any $r$. Some experimental calculations in {\sc M2} suggest the
following narrow but interesting conjecture:
\begin{Conjecture} \sl There is an inclusion
$(I_X)^r \subseteq J_{r,3}$, and moreover $r$ is the smallest such power.
\end{Conjecture}
This problem is related to identities between the covariants of a
generic cubic form. For instance, we have an identity
\[ \Gott_{1,3}^2 = - \frac{1}{2} \, (\F,\Gott_{2,3})_1, \]
which can be verified by a direct symbolic computation. This
immediately shows that $(I_X)^2 \subseteq J_{2,3}$. (Compare the
argument of Proposition~\ref{prop.containmentJ} above.)
In general, if $r$ and $d$ are coprime, then $\fg$ is a perfect ideal of height $d-1$,
which is resolved by the Eagon-Northcott complex (see \cite[Appendix 2]{Eisenbud1}) of $\alpha_\F$.
By the Porteous formula (see~\cite[Ch.~II, \S 4]{ACGH}), the scheme $\Proj R/\fg$ supported on the
rational normal $d$-ic curve has degree $\binom{r+d-1}{d-1}$.
\section{The Clebsch transfer principle} \label{section.Clebsch}
In this section we generalise the G{\"o}ttingen covariants to $n$-ary
forms.
\subsection{}
Let $W$ be an $n$-dimensional complex vector space with basis $\ux = \{x_1,
\dots, x_n\}$, and a natural action of the group $SL(W)$.
Given an $n$-tuple of nonnegative integers $I = (i_1, \dots, i_n)$ adding up to
$d$, let
\[ \binom{d}{I} = \frac{d!}{\prod\limits_k \, i_k!}, \qquad x^I =
\prod\limits x_k^{i_k}. \]
We write a generic form of order $d$ in the $\ux$ as
\[ \Gamma = \sum\limits_I \binom{d}{I} \, a_I \, x^I, \]
where the $a_I$ are independent indeterminates. As in the binary case,
the $\{a_I\}$ can be seen as forming a basis of $S_d \, W^*$. Define the symmetric algebra
\[ \cA = \bigoplus\limits_{m \ge 0} \, S_m(S_d \, W^*) = \complex[\{a_I\}] \]
so that $\Proj \cA = \P S_d \simeq \P^{\binom{d+n-1}{d}-1}$ is the space of
$n$-ary $d$-ics.
\subsection{}
Each irreducible representation of $SL(W)$ is a
Schur module of the form $S_\lambda = S_\lambda \, W$, where $\lambda$ is a
partition with at most $n-1$ parts (see~\cite[\S 15]{FH}). Moreover,
we have an isomorphism
\[ S_{(\lambda_1, \lambda_2, \dots, \lambda_{n-1},0)} \, W \simeq
S_{(\lambda_1, \lambda_1-\lambda_{n-1}, \dots, \lambda_1 -
\lambda_2,0)} \, W^*. \]
An inclusion $S_\lambda \, W^* \subseteq \cA_m$ corresponds to a morphism
\[ S_0 \hookrightarrow \cA_m \otimes S_\lambda \, W, \]
then the image of $1 \in S_0$ will be called a concomitant of $\Gamma$ of degree
$m$, and type $\lambda$.
\subsection{} \label{section.clebsch33}
In the case of ternary forms, for $\lambda = (\lambda_1, \lambda_2)$,
we have an embedding (see~\cite[\S 15]{FH})
\[ S_\lambda \subseteq S_{\lambda_2}(\wedge^2 \, W) \otimes S_{\lambda_1
- \lambda_2}. \]
Using the basis $u_1 = x_1 \wedge x_2, \, u_2 = x_2 \wedge x_3,
\, u_3 = x_3 \wedge x_1$ for $\wedge^2 \, W \simeq W^*$, we can write the concomitant as a form of degree
$m$ in the $a_I$, degree $\lambda_1 - \lambda_2$ in $\ux$, and
degree $\lambda_2$ in $\uu$.
For instance, assume $m=2,d=3$. We have a plethysm decomposition
$S_2(S_3^*) \simeq S_6^* \oplus S_{4,2}^*$, and hence (up to a scalar) a unique morphism
\[ S_0 \hookrightarrow \cA_2 \otimes S_{4,2}. \]
If we symbolically write $\Gamma = a_\ux^3 = b_\ux^3$, then this
concomitant is $(a \, b \, u)^2 \, a_\ux \, b_\ux$. We refer the reader
to~\cite[Ch.~XII]{GY} or~\cite{Littlewood} for
the symbolic calculus of $n$-ary forms and their concomitants.
\subsection{} \label{section.clebsch34}
The `Clebsch transfer principle' is a type of construction used to lift a binary covariant to a concomitant of $n$-ary
forms in a geometrically natural way. As such, it comes in many
flavours depending on the specifics of the geometric situation in
play. (See~\cite[\S 4]{Briand}, \cite[\S 3.4.2]{Dolgachev2} or \cite[\S 215]{GY} for variant descriptions
of this principle.) Clebsch's own statement of this technique may be found
in~\cite[p.~28]{Clebsch}, but Cayley and Salmon seem to have been
aware of it earlier (see~\cite[p.~28]{SalmonCD}).
The following example should convey an idea of how the transfer principle is used.
Let $n=3$ and $d=4$, so that $\P S_4 \simeq \P^{14}$ is the space of
quartic plane curves. Let $Z \subset \P^{14}$ be the $5$-dimensional
subvariety of double conics, i.e.,
\[ Z = \left\{ [\Gamma] \in \P S_4: \Gamma = Q^2 \; \, \text{for some
ternary quadratic $Q$} \right\}. \]
A line $L$ in the plane $\P W^* \simeq \P^2$ will intersect a general
quartic curve $\Gamma(x_1,x_2,x_3)=0$ in four points, which become
two double points when $\Gamma \in Z$. With the identification $L
\simeq \P^1$, let $\Gamma|_L$ denote the `restriction' of $\Gamma$ to
$L$, regarded as a binary quartic form. Hence the `function'
\[ L \lra \Gott_{2,4}(\Gamma|_L) \]
should vanish identically when $\Gamma \in Z$.
In order to make this precise, write $p = [p_1, p_2, p_3], q =
[q_1,q_2,q_3]$, where $p_i,q_i$ are indeterminates. We think of a generic $L$ as spanned by the points $p,q \in
\P^2$, and thus $L$ has line co-ordinates
\[ u_1 = \left| \begin{array}{cc} p_1 & q_1 \\ p_2 & q_2 \end{array}
\right|, \quad
u_2 = \left| \begin{array}{cc} p_2 & q_2 \\ p_3 & q_3 \end{array} \right|,
\quad
u_3 = \left| \begin{array}{cc} p_3 & q_3 \\ p_1 & q_1 \end{array}
\right|. \]
Introduce binary variables $\lambda = \{\lambda_1, \lambda_2 \}$,
and substitute $x_i = \lambda_1 \, p_i + \lambda_2 \, q_i$ in
$\Gamma$ to get a form $\Theta$ (which represents the restriction).
Now evaluate $\Gott_{2,4}$ on $\Theta$ by regarding the
latter as a binary form in the $\lambda$; then the final result is the
required lift $\tGott_{2,4}$. The actual symbolic calculation proceeds as follows.
Let
\[ \Gamma = a_\ux^4 = b_\ux^4 = c_\ux^4, \]
where $a_\ux = a_1 \, x_1 + a_2 \, x_2 + a_3 \, x_3$ etc. After
substitution, $a_\ux$ becomes $\lambda_1 \, a_p + \lambda_2 \, a_q$, which we rewrite as
\[ \alpha_\lambda = \alpha_1 \, \lambda_1 + \alpha_2 \, \lambda_2 \quad
\text{where $\alpha_1 = a_p, \, \alpha_2 = a_q$,} \]
and similarly $b_\ux = \beta_\lambda, c_\ux = \gamma_\lambda$.
Thus $\Theta = \alpha_\lambda^4 = \beta_\lambda^4 = \gamma_\lambda^4$.
Recall from \S\ref{Gott.quadratic} that
\begin{equation}
\Gott_{2,d}(\Theta) \doteq (\Theta, (\Theta,\Theta)_2)_1 \doteq
(\alpha \, \beta)^2 \, (\alpha \, \gamma) \, \alpha_\lambda \, \beta_\lambda^2
\, \gamma_\lambda^3. \label{Gott.24.Theta} \end{equation}
Now
\[ (\alpha \, \beta) = \left| \begin{array}{cc} \alpha_1 & \alpha_2 \\
\beta_1 & \beta_2 \end{array} \right| = a_p \, b_q - b_p \, a_q =
\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ u_1 &
u_2 & u_3 \end{array} \right| = (a \, b \, u), \]
and similarly for the bracket factor $(\alpha \, \gamma)$. Hence we
arrive at the expression
\begin{equation} \tGott_{2,4}(\Gamma) = (a \, b \, u)^2 \, (a \, c \, u) \, a_\ux \,
b_\ux^2 \, c_\ux^3, \label{tGott.24} \end{equation}
which is a concomitant of degree $3$ and type $(9,3)$. We have the property
\[ [\Gamma] \in Z \iff \tGott_{2,4}(\Gamma) \; \text{vanishes identically as a
polynomial in $\ux, \uu$}. \]
The implication $\Rightarrow$ follows by construction. The converse says
that if $\Gamma=0$ were not a double conic, then a line could be found which does not intersect it in two
double points. This is clear on geometric grounds.
\subsection{} The case of a general G{\"o}ttingen covariant is
similar. Assume that $\Gott_\Psi$ is of degree $r+1$, order $q$, and
weight $w=\frac{(r+1) \, d-q}{2}$. Let
$p=[ p_1, \dots, p_n], \, q=[ q_1, \dots, q_n]$, substitute
\begin{equation} x_i = \lambda_1 \, p_i + \lambda_2 \, q_i, \quad (1 \le i \le n),
\label{subs.xi} \end{equation}
into $\Gamma$, and evaluate $\Gott_\Psi$ on the new binary form in the $\lambda$
variables. The resulting concomitant $\tGott_\Psi$ is of degree $r+1$
and type $(q+w,w)$. If $\Gamma = G^\mu$, then $\tGott_\Psi(\Gamma)$ vanishes identically for the same
reason as above.
If $\Gott_\Psi$ is written as a symbolic expression in $r+1$ binary letters
$a,b,\dots$ and their brackets $(a \, b)$ etc., then $\tGott_\Psi$ is
obtained by simply treating them as $n$-ary letters and replacing the corresponding brackets by $(a \, b \, u)$
etc. This follows immediately by tracing the passage from~(\ref{Gott.24.Theta}) to~(\ref{tGott.24}).
In particular, the concomitant in \S\ref{section.clebsch33} is the
Clebsch transfer of the Hessian of a binary cubic. The formal symbolic
expression for $\tGott_\Psi$ does not depend on $n$, although of
course, its interpretation does.
\begin{Theorem} \sl
Let $\Gamma$ be an $n$-ary $d$-ic. Then $\tGott_{r,d}(\Gamma)$ is
identically zero, if and only if $\Gamma = G^\mu$ for some $n$-ic $G$ of
order $e$.
\label{theorem.clebsch} \end{Theorem}
\demo The `if' part follows from the discussion above. Let $\Gamma =
\prod\limits H_i^{\nu_i}$ be the prime decomposition, where $H_i$ is
an irreducible form of degree $c_i$. If $\Gamma$ cannot be written as $G^\mu$, then at
least one $\nu_i$ is not divisible by $\mu$. A general line $L$ will
intersect each hypersurface $H_i=0$ in $c_i$ distinct
points. Altogether $L$ intersects $\Gamma=0$ in $c_1+ c_2 + \dots$
points, at least one of which occurs with multiplicity not divisible by
$\mu$. Thus $\tGott_{r,d}(\Gamma)$ will not vanish if the $\uu$ variables in
it are specialised to the Pl{\"u}cker co-ordinates of a general $L$. \qed
\subsection{} This is a continuation of \S\ref{section.clebsch34}. We have calculated the ideal of $Z$
using a procedure similar to the one in \S\ref{section.g36}, and it turns
out that $I_Z$ is minimally generated by a $218$-dimensional space of forms in
degree $3$. We have a plethysm decomposition
\[ \cA_3 = S_3(S_4^*) \simeq S_{12}^* \oplus S_{10,2}^* \oplus S_{9,3}^*
\oplus S_{8,4}^* \oplus S_6^* \oplus S_{6,3}^* \oplus S_{6,6}^* \oplus
S_{4,2}^* \oplus S_0^*, \]
where the summands are of respective dimensions
\[ 91, \, 162, \, 154, \, 125, \, 28, \, 64, \, 28, \, 27, \, 1. \]
Now $(I_Z)_3$ is a subdirect sum of the above, and we
already know that $S_{9,3}^*$ is one of its pieces. This forces
$(I_Z)_3 \simeq S_{9,3}^* \oplus S_{6,3}^*$ on dimensional
grounds. Hence there is a concomitant of type $(6,3)$ vanishing on
$Z$. We have checked by a direct calculation that it can be
written as
\[ (a \, b \, c) \, (a \, b \, u)^2 \, (a \, c \, u) \, b_\ux \,
c_\ux^2. \]
In fact, all that needs to be checked is that this symbolic expression
is not identically zero, which can be done by specialising
$\Gamma$. This suffices, since we have up to scalar only one concomitant of this type in degree $3$.
Recall from \S\ref{section.g26} that for $r=2, d=4$, that there are
no G{\"o}ttingen covariants other than $\Gott_{2,4}$. Hence we have
found a concomitant vanishing on $Z$ which is not the Clebsch transfer
of any binary covariant.
Let $J \subseteq \cA$ denote the ideal generated by the coefficients of
$\tGott_{2,4}$. We have checked using {\sc M2} that
the saturation of $J$ is $I_Z$, and moreover the two ideals coincide
in degrees $\ge 7$. But in general, we do not know whether there is an
analogue of Theorem \ref{Theorem.Jsaturation} in the $n$-ary case.
\subsection{} We end with an example which is at least a pleasing
curiosity. Assume that $\Gamma=0$ is a \emph{nonsingular} plane quartic
curve. A line $L \subset \P^2$ with co-ordinates $[u_1,u_2,u_3]$
passes through the points $p = [u_3,0,-u_1], q = [u_2,-u_1,0]$,
and moreover these points are distinct (and well-defined) when $u_1
\neq 0$. Now make substitutions into $\tGott_{2,4}(\Gamma)$ as in~(\ref{subs.xi}) to get a binary sextic
$\cE_1(\lambda)$; it represents the binary form $\Gott_{2,4}$ as
living on $L \simeq \P^1$. (This is no longer correct if $u_1=0$, hence
in order to avoid spurious solutions we also
need to consider the forms $\cE_2(\lambda), \cE_3(\lambda)$ similarly obtained from
\[ p = [0,u_3,-u_2], \; q=[u_2,-u_1,0], \qquad p = [0,u_3,-u_2], \;
q=[u_3,0,-u_1].) \]
Now all the $\cE_i(\lambda)$ are identically
zero exactly when $\{\Gamma=0\} \cap L$ represents two double points, i.e., when $L$ is a
bitangent to the curve defined by $\Gamma$. Let $B = \complex[u_1,u_2,u_3]$ denote the
co-ordinate ring of the dual plane, and ${\mathfrak b}_\Gamma
\subseteq B$ the ideal generated by the coefficients of all the monomials
in $\lambda$ for $\cE_i(\lambda), i=1,2,3$. Then the zero locus of ${\mathfrak
b}_\Gamma$ is the set of $28$ points (see~\cite[Ch.~6]{Dolgachev2})
corresponding to the bitangents of the curve.
We have verified in {\sc M2} that ${\mathfrak b}_\Gamma$ is not
saturated, but its saturation has resolution
\[ 0 \la B/({\mathfrak b}_\Gamma)_\text{sat} \la B \la B(-7)^8 \la B(-8)^7 \la 0, \]
which is characteristic of $28$ general points in the plane
(see~\cite[Ch.~3]{Eisenbud2}). In much the same way, the concomitant
in \S\ref{section.clebsch33} can be used to give equations for the
$9$ inflexional tangents of a nonsingular plane cubic curve.
\bigskip
{\small {\sc Acknowledgements:} The second author was funded by NSERC, Canada. We are thankful to Daniel Grayson and
Michael Stillman (the authors of {\sc Macaulay-2}). We have used John Stembridge's `SF'
package for {\sc Maple} for calculating plethysm decompositions.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,682
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{"url":"https:\/\/asmedigitalcollection.asme.org\/turbomachinery\/article\/140\/5\/051001\/378348\/A-New-Approach-for-Centrifugal-Impeller","text":"This paper introduces a new approach for the preliminary design and aerothermal analysis of centrifugal impellers using a relative diffusion effectiveness parameter. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller. It represents the quality of the relative diffusion process in an impeller. This parameter is used to evaluate impeller performance by correlating the relative diffusion effectiveness with the impeller isentropic efficiency using the experimental data acquired on a single-stage centrifugal compressor (SSCC). By including slip, which is appropriate considering it is an inviscid effect that should be included in the determination of maximum available diffusion in the impeller, a linear correlation between impeller efficiency and relative diffusion effectiveness resulted for all operating conditions. Additionally, a new method for impeller preliminary design was introduced using the relative diffusion effectiveness parameter, in which the optimal design is selected to maximize relative diffusion effectiveness. While traditional preliminary design methods are based on empirical loss models or empirical knowledge for selection of diffusion factor (DF) in the impeller, the new method does not require any such models, and it also provides an analytical approach for the selection of DF that gives optimal impeller performance. Validation of the method was performed using three classic impeller designs available in the open literature, and very good agreement was achieved. Furthermore, a sensitivity study shows that the method is robust in that the resulting flow angles at the impeller inlet and exit are insensitive to a wide range of blockage factors and various slip models.\n\n## Introduction\n\nHigh pressure ratio centrifugal compressors have been widely used in turbochargers and turboshaft engines because of their compact size, high efficiency, and wide operating range. Design considerations for high pressure ratio centrifugal compressor have been systematically studied by many researchers for decades. Empirical equations were established for preliminary design of impellers and diffusers, including Stodola [1], Cordier [2], Herbert [3], Rodgers [4], and Wiesner [5]. Rodgers and Sapiro [6,7] performed a detailed parametric study on compressor performance and successfully correlated the efficiency of a single-stage centrifugal compressor (SSCC) with four major parameters: inlet specific speed, impeller tip diameter, inducer tip relative Mach number, and exit discharge Mach number. With the help of these empirical loss correlations and slip estimation, the most essential aerodynamic parameters and geometric dimensions (including stage loading, efficiency, surge margin, inlet shroud radius, impeller exit radius, and blade number) could be determined in the preliminary design process.\n\nSince the impeller is essentially a rotating diffusion system, the diffusion ratio (or reciprocal of the de Haller number) is of great importance and has been investigated by a variety of researchers including Rodgers [8], Young [9], and Benvenuti [10,11]. Rodgers [8] showed diffusion ratios between 1.9 and 2.0 at surge flow rates, while Young [9] provided guidance for the maximum attainable relative diffusion in the discussion of Rodgers of results, as it applies to three-dimensional impellers. Furthermore, Benvenuti [10,11] analyzed the data from industrial centrifugal compressors and provided the guidance for industrial two-dimensional impellers. In addition to the diffusion ratio, reduced static pressure and reduced static pressure coefficient are other parameters that have been used in investigating the secondary flows in rotating diffusion systems [12], and they are particularly useful for low-speed machines and pumps.\n\nOne challenge in modeling the impeller flow using one-dimensional (1D) tools is that the actual discharge flow pattern of impellers is different from the ideal pattern predicted by a potential flow solver. The existence of a jet-wake flow pattern at the impeller discharge was introduced by Dean and Senoo [13] and further confirmed in the studies of Eckardt [14], Krain [15], and Skoch et al. [16]. The results from Eckardt [14] on a conventional centrifugal impeller showed a well-conditioned flow in the inducer, but the development of a wake near the suction surface starting at the beginning of the radial turn to the impeller exit. The study performed by Krain [15] and Skoch et al. [16] in modern backswept impellers also showed similar measurements.\n\nBased on the experimental observation of the jet-wake flow pattern, Dean and Senoo [13] developed a model for the impeller discharge flow where the impeller discharge flow was categorized into jet flow and wake flow (a two-zone model). The jet flow is modeled as isentropic and follows the impeller blade, while the wake flow contains all the losses. Furthermore, Japikse [17] assessed the single-zone and jet-wake models in evaluating the component performance in centrifugal compressors. The results showed improved accuracy and advantages for designs optimized using two-zone modeling.\n\nThe inlet conditions for impellers studied by Eckardt [14], Krain [15], and Skoch et al. [16] are subsonic and free of shocks. However, modern turbochargers and turboshaft engines are continuously pushing the boundary of pressure ratio and flow capacity. Size limitations on the outer diameter lead to larger rotational speeds and result in transonic flow conditions at the compressor inlet, where additional losses due to the interaction of shock waves and blade surface boundary layers and tip clearance flow are possible. Investigations by Rodgers [18,19] showed a drop in impeller peak efficiency (PE) with the increase of impeller inlet shroud relative Mach number, and this has prompted several investigations on transonic impellers.\n\nThe flow inside a centrifugal impeller with transonic inlet conditions has been studied by researchers including Senoo et al. [20,21], Krain et al. [22,23], Ibaraki et al. [24,25], and Hagashimori et al. [26] with conflicting results. The results from Senoo et al. [20,21] showed the existence of two shock waves, a detached wave at the impeller leading edge and a passage shock on the pressure surface, at supersonic flow conditions but with no deterioration in impeller performance. However, the results from Krain et al. [22,23] and Ibaraki et al. [24,25] showed increased loss and size of the wake region from tip leakage flow at the impeller exit due to the interaction between the shock wave and the tip leakage flow in the inducer. The results from Hagashimori et al. [26] showed the presence of an oblique shock at the inducer leading edge together with a passage shock at the inducer throat. Additionally, reversed flow near the shroud in the inducer was present due to the interaction between the shock wave and the tip leakage flow.\n\nIn general, compared to the impeller flow with subsonic inlet conditions, the flow in transonic impellers is more complex due to the presence of shock waves, and flow instabilities start further upstream at the throat of inducer due to the interaction of shock waves and tip leakage flow. The tip leakage flow plays a more important role for these machines, both within the impeller and also at the exit of the impeller. Reversed flow near the shroud in the inducer may occur due to the presence of shock waves.\n\n## Scope of the Paper\n\nDespite the complexity of flow inside centrifugal impellers, this paper demonstrates use of a meanline, 1D approach for impeller preliminary design and aerothermal analysis. In light of this goal, a relative diffusion effectiveness parameter is introduced to evaluate the impeller performance at both design and off-design operating conditions and also to optimize the impeller exit geometry and velocity triangles during the preliminary design phase. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller, and it can be calculated knowing the static pressure and geometry. The application of this parameter in the preliminary design phase and aerothermal analysis of centrifugal impeller performance are discussed.\n\n## Methodology\n\nThe purpose of a centrifugal compressor is to raise the static pressure of the working fluid. Rearranging the equation for conservation of rothalpy, the enthalpy rise across the impeller for an adiabatic process is\n$h2\u2212h1=U22\u2212U122+W12\u2212W222$\n(1)\n\nwhere $h$ is the static enthalpy, $U$ is the wheel speed, $W$ is the relative velocity, subscript $1$ represents impeller inlet, and subscript $2$ indicates impeller exit.\n\nThe thermodynamic relationship for enthalpy gives\n$dh=Tds+dP\/\u03c1$\n(2)\n\nwhere $T$ stands for static temperature, $s$ is entropy, $P$ is static pressure, and $\u03c1$ is density.\n\nIntegrating Eq. (2) gives\n$h2\u2212h1=\u222b12Tds+\u222b12dP\/\u03c1$\n(3)\nCombining Eqs. (1) and (3), the relationship between static pressure rise and impeller velocity triangles is\n$\u222b12dP\/\u03c1+\u222b12Tds=U22\u2212U122+W12\u2212W222$\n(4)\n\nThe first term on the left-hand side represents the static pressure rise achieved in impeller. The second term on the left-hand side represents loss in terms of entropy generation. The first term on the right-hand side is related to the static pressure rise associated with the centrifugal effect. The second term on the right-hand side represents the static enthalpy rise associated with the relative-frame diffusion that occurs in the impeller.\n\nOne advantage of an impeller is that there is always a contribution to the static pressure rise from the centrifugal effect regardless of the flow quality inside the impeller. Even though the centrifugal effect contributes to the static pressure rise and impeller efficiency, it also undermines the capability of using impeller efficiency or the traditional diffusion effectiveness as the parameters in evaluating the quality of flow in impeller. As a result, efficiency is not the best indicator for the designer aiming to improve the aerodynamic design of an impeller.\n\nThus, relative diffusion effectiveness ($\u03b5$) is introduced as a parameter used to describe the quality of the flow inside the impeller. Instead of using reduced static pressure, the relative diffusion effectiveness is defined in terms of diffusion ratio, which is directly related to the velocity triangles and, thus, reduces the complexity of its application within the design process. It is described as\n$\u03b5=RMR\/RMRi$\n(5)\nwhere subscript $i$ stands for the ideal case, and $RMR$ represents the diffusion in terms of relative Mach number ratio to account for compressibility and is described as\n$RMR=Mrel_1\/Mrel_2$\n(6)\n\nwhere $Mrel$ represents the relative Mach number.\n\nCombining Eqs. (5) and (6), the relative diffusion effectiveness becomes\n$\u03b5=Mrel_2i\/Mrel_2$\n(7)\n\nIt is worth noting that the relative diffusion effectiveness is linear with respect to the relative Mach number ratios at the impeller exit, which is different from the traditional diffusion effectiveness obtained in terms of static pressure. Since centrifugal effects contribute to the static pressure rise in radial impellers, the relative diffusion effectiveness isolates the impeller aerodynamic performance from the centrifugal effect and provides a more direct metric for the quality of the impeller aerodynamic design.\n\n### Evaluation of Impeller Exit Relative Mach Number, Mrel_2.\n\nThe flow inside the impeller can be categorized as a primary flow zone and secondary flow zone. The primary flow represents the well-diffused isentropic flow, and the secondary flow generates all the entropy and does not contribute to the static pressure recovery. Thus, the primary flow relative Mach number represents the diffusion in the impeller with high fidelity.\n\nThe primary flow relative Mach number could be obtained using the temperature and pressure information measured at the impeller inlet with the static pressure measured at impeller exit by\n$s2p=s1=s(T1,P1)$and\n(8)\n$T2p=T(P2,s2p)$\n(9)\n\nwhere subscript $p$ represents the primary flow.\n\nThe impeller inlet rothalpy is\n$I1=ht1\u2212U1V\u03b81$\n(10)\nin which $I$ is rothalpy, $V$ is the absolute velocity, subscript $t$ stands for the stagnation condition, and $\u03b8$ represents the tangential component. Furthermore, for cases with zero prewhirl, the rothalpy at the impeller inlet equals the stagnation enthalpy at the impeller inlet.\nThe equation for conservation of rothalpy for an adiabatic process gives\n$I2=I1=h2\u2212U222+W222$\n(11)\nImpeller relative velocity can then be obtained with\n$W2=2I1\u2212h2+U222$\n(12)\n\nIn this and the following procedures, all the fluid properties (including enthalpy, entropy, speed of sound, etc.) are obtained from the National Institute of Standards and Technology reference fluid thermodynamic and transport properties database, REFPROP [27].\n\n### Evaluation of Mrel_2i.\n\nThe ideal relative Mach number could be calculated in two different ways depending on the assumptions used. This section discusses the method based on the assumption of isentropic flow, zero slip, and no blockage. This relative Mach number represents the maximum diffusion available for a given geometry. The procedure is iterative and starts with an initial guess of impeller exit static pressure, $P2,j$\n$T2,j=T(P2,j,s1)$\n(13)\n$h2,j=h(T2j,P2,j)$\n(14)\n$\u03c12,j=\u03c1(T2,j,P2,j)$\n(15)\n\nwhere subscript $j$ stands for the $jth$ iteration.\n\nRearranging the equation for conservation of rothalpy gives\n$W2i_cor,j=2I1\u2212h2,j+U222and$\n(16)\nrearranging the equation for conservation of mass gives\n$W2i_com,j=m\u02d9\u03c12,jA2cos\u03b22b$\n(17)\n\nwhere $A$ is the effective area at the impeller exit (considering blade thickness), $\u03b22b$ is the blade angle at the impeller exit, subscript $cor$ represents the value obtained from the equation for conservation of rothalpy, and subscript $com$ represents the value calculated from the equation for conservation of mass.\n\nIn each iteration, two values of relative velocity, $W2i$, were obtained from equations for conservation of rothalpy and conservation of mass. The iteration of impeller exit static pressure occurs until both the equation for conservation of rothalpy and the equation for conservation of mass are satisfied. Thus, the ideal relative Mach number at the impeller exit is calculated using the converged relative velocity and speed of sound at the impeller exit:\n$Mrel_2i=W2i\/a2i$\n(18)\n\n### Evaluation of Mrel_2i_slip.\n\nThe parameter $Mrel_2i$ represents the maximum diffusion available for a given geometry. However, due to the inviscid nature of slip, the maximum diffusion obtained based on the no-slip assumption provides an overestimation to maximum diffusion. Additionally, the work input by the impeller is overestimated using the assumption of zero slip. To mitigate the inconsistency in the work input calculation, a corrected ideal relative Mach number, $Mrel_2i_slip$ is introduced. This method removes the no-slip assumption. The corrected relative Mach number represents the diffusion available for a given geometry, together with a known work input based on the assumptions of isentropic flow and zero blockage. The procedure is iterative and includes an outer iteration and an inner iteration. It starts with initial guess of impeller exit static pressure, $P2,j$ (inner loop) and slip angle, $\u03b2slip,k$ (outer loop)\n$\u03b22f,k=\u03b22b+\u03b2slip,k$\n(19)\n\nwhere $\u03b22f$ is the relative flow angle at the impeller exit, and $\u03b2slip$ is the slip angle.\n\nThe relative velocity at the impeller exit considering slip can be found both by using conservation of rothalpy (as shown in Eq. (20)) and conservation of mass (as shown in Eq. (21))\n$W2i_slip_cor,j=2I1\u2212h2,j+U222$\n(20)\n$W2i_slip_com,j=m\u02d9\u03c12,jA2cos\u03b22f,j$\n(21)\nThe static pressure is iteratively adjusted until the relative velocity is matched from both equations. Then, the total energy is used to iterate on slip angle. This is done by calculating the absolute velocity at impeller exit\n$V2,k=W2i_slipcos\u03b22f,k2+U2\u2212W2i_slipsin\u03b22f,k2$\n(22)\nThe impeller exit stagnation enthalpy is calculated from static enthalpy and kinetic energy\n$ht2,k=h2+V2,k2\/2$\n(23)\nThe impeller exit stagnation enthalpy obtained from the total pressure and total temperature is\n$ht2=h(Tt2,Pt2)$\n(24)\nThe slip angle is adjusted until the calculated impeller stagnation enthalpy matches the measured stagnation enthalpy. Thus, the process essentially adjusts the impeller exit static pressure and slip angle until all the equations for conservation of rothalpy, conservation of mass, and conservation of energy are satisfied. The corrected ideal relative Mach number at impeller exit is calculated as\n$Mrel_2i_slip=W2i_slip\/a2i_slip$\n(25)\nThere will be two different values for the relative diffusion effectiveness depending on the slip assumptions used in calculating the minimum relative Mach number at impeller exit. The relative diffusion effectiveness calculated with $Mrel_2i$ is defined as the relative diffusion effectiveness without slip\n$\u03b5no_slip=Mrel_2i\/Mrel_2$\n(26)\nThe relative diffusion effectiveness calculated with $Mrel_2i_slip$ is defined as the relative diffusion effectiveness with slip\n$\u03b5slip=Mrel_2i_slip\/Mrel_2$\n(27)\n\nSince the maximum diffusion derived from the assumption of zero slip is an overestimate of what can actually be achieved, $\u03b5slip$ is greater than $\u03b5no_slip$.\n\n## Performance Evaluation\n\nThe relative diffusion effectiveness is correlated with impeller isentropic efficiency using experimental data acquired on a single-stage centrifugal compressor. The compressor features a transonic impeller with backswept blades. The impeller has 17 main blades and 17 splitter blades. The design operating speed for the compressor is about 45,000\u2009rpm, and the total pressure ratio for the entire stage is on the order of 6.5. The details of SSCC facility were documented by Lou et al. [28].\n\nThe steady performance is characterized by the total pressure ratio, total temperature ratio, and efficiency. The performance of the entire compressor stage is calculated from the area-averaged measurements acquired at the compressor inlet and exit. The impeller-only performance is evaluated by using the area-averaged measurements at the compressor inlet and the impeller exit condition determined using static pressure measurements at that location. The total temperature at the impeller exit is assumed to be the same as that measured at the deswirl exit (stage exit) based on the adiabatic assumption. The impeller exit total pressure is derived from the measured total temperature at the deswirl exit, the inlet mass flow rate, and the area-averaged static pressure measured at the impeller trailing edge using the continuity equation and the turbomachinery Euler equation [29]. The compressor corrected conditions (speed and mass flow rate) [30] and efficiency [31] are calculated using properties for humid air retrieved from REPROP. The results presented in this section are normalized using the operating condition at design point. The compressor inlet pressure is measured using highly accurate 2.5 psid modules with an uncertainty less than 0.12%. Compressor exit pressure is measured using 100 psid modules with an uncertainty less than 0.05% full scale. This renders the uncertainty in the total pressure ratio less than 0.2% from 80% to 100% corrected speed. The mass flow rate is measured using a calibrated bellmouth with uncertainty less than 0.5%.\n\nFigure 1 shows the normalized total pressure ratio versus the normalized corrected mass flow rate. The results for both the impeller and the entire stage from 80% to 100% corrected speed are presented. The PE conditions are represented by the solid green symbols. The low loading conditions are represented by the solid blue symbols. The choke conditions are shown as solid red symbols. Those color schemes are consistent with the rest of figures presented in this section. Regardless of the variation in the total pressure ratio for the entire compressor stage relative to the changes in the loading conditions, the total pressure ratio for the impeller stays very consistent along the choke line due to the choked flow in the diffuser.\n\nFig. 1\nFig. 1\nClose modal\n\nFigure 2 shows the performance of the impeller and entire compressor stage in terms of isentropic efficiency. Compared to the entire stage, the impeller operates more efficiently over the entire operating range, from choke to near surge. There is no obvious deterioration in impeller efficiency along the choke line as loading decreases. In fact, at subsonic inlet conditions from 80% to 95% corrected speed, the impeller efficiency increases as loading decreases. At design speed with supersonic inlet conditions, the trend in impeller efficiency lines up with the trend of the entire compressor stage in that they both increase from choke to PE condition.\n\nFig. 2\nFig. 2\nClose modal\n\nThe effect of inlet tip relative Mach number on impeller efficiency was investigated, and the results are shown in Fig. 3. The inlet tip relative Mach number is mainly determined by the inlet mass flow rate and prewhirl angle. At supersonic inlet conditions, the impeller peak efficiency drops with the increase of inlet tip relative Mach number, and there is about a 0.7 point drop in the peak efficiency from 95% to 100% corrected speed. This agrees with the observation from Rodgers [18,19]. However, at subsonic inlet conditions, the impeller peak efficiency increases as the inlet tip relative Mach number increases, and there is a 3.2 point improvement in the impeller peak efficiency from 80% to 90% corrected speed. Additionally, the impeller efficiency is closely related to loading condition. At design speed, there is about a 1.0 point change in the impeller efficiency from choke to near surge. At 80% corrected speed, the variation in the impeller efficiency with loading is about 1.5 points.\n\nFig. 3\nFig. 3\nClose modal\n\nThe relationship between impeller peak efficiency and inlet tip relative Mach number is very helpful in optimizing the inducer size during preliminary design. However, its utility is limited in correlating the impeller performance to the entire operating range. Figure 4 shows the relationship between the impeller isentropic efficiency and the relative diffusion effectiveness without slip. The impeller efficiency is proportional to the relative diffusion effectiveness, and this applies to all the operating points from 80% corrected speed to 100% corrected speed. The variation in impeller efficiency with respect to the changes in loading condition for each speed line observed in Fig. 3 is associated with the relative diffusion in the impeller. Despite the differences in the peak efficiency at various corrected speeds, the slope between the impeller efficiency and the relative diffusion effectiveness is similar from 80% to 100% corrected speed. Additionally, for each 0.2-point change in the relative diffusion effectiveness, there is about a 1.0-point change in the efficiency, which indicates that the relative diffusion effectiveness without slip is a reliable parameter in comparing the performance of various impellers.\n\nFig. 4\nFig. 4\nClose modal\n\nThe correlation between the impeller isentropic efficiency and the relative diffusion effectiveness with slip is shown in Fig. 5. There is also a similar trend between impeller efficiency and the relative diffusion effectiveness with slip. However, the advantage of using the relative diffusion effectiveness that incorporates slip is that all the operating points from 80% to 100% corrected speed are correlated by a single linear fit. This greatly reduces the complexity of using this parameter in analyzing impeller performance.\n\nFig. 5\nFig. 5\nClose modal\n\n## Preliminary Design\n\nIn addition to the aerothermal analysis that has been presented, the relative diffusion effectiveness parameter also offers a new approach for optimizing the geometry and velocity triangles at impeller exit during the preliminary design phase. The objective of the impeller preliminary design is to determine the principle aerodynamic and geometrical parameters for a required design duty. Traditional preliminary design methods rely on empirical loss models, and the selection of diffusion factor (DF) in the impeller is usually subject to the designer's experience. The new method of using relative diffusion effectiveness has zero dependence on empirical loss models, and thus, it reduces the empirical input parameters needed to simplify the slip model and blockage factors at the impeller inlet and exit. Additionally, the new method provides an analytical approach for the selection of diffusion factor for the optimal design. The workflow for impeller preliminary design using relative diffusion effectiveness is shown in Fig. 6. For a given set of design requirements and preselected parameters, the procedure starts with a wide range of diffusion factors and calculates the associated value of relative diffusion effectiveness for each diffusion factor. The optimal preliminary design is achieved as the diffusion factor gives the maximum relative diffusion effectiveness. With completion of the preliminary design, the detailed impeller geometry, including the meridional contour and blade profile, are obtained in the following two-dimensional and three-dimensional design loop. With the impeller geometry now available, detailed aerodynamic analyses can be performed to check if the design meets the requirements.\n\nFig. 6\nFig. 6\nClose modal\n\nThis new approach was applied to three classic impellers available in the open literature with the objective being to compare the optimum designs obtained from the new method based on relative diffusion effectiveness to the original design choices of three classic impellers. Furthermore, a sensitivity analysis of the new method for different slip models and different blockage factors was performed.\n\n### Validation of the Method.\n\nValidation of the present method was performed using the preliminary design information of three impellers available in the open literature. The impellers are representative applications of centrifugal compressors in gas turbines and turbochargers, and their dimensionless specific speed varies between 0.53 and 0.81. The first impeller was designed by Came [32]. The second impeller is the CC3 impeller scaled up from an Allison Engine Design [33]. The third impeller is the SRV2-O impeller [34] designed at the German Aerospace Center (DLR). Came's impeller and the SRV2-O impeller are high-speed and high-pressure-ratio machines, while the CC3 impeller is an intermediate-pressure ratio machine. The inlet condition is subsonic for Came's impeller and the CC3 impeller, and it is transonic for the SRV2-O impeller. The principle design parameters for the three impellers are listed in Table 1.\n\nTable 1\n\nPrinciple design parameters for the three selected impellers\n\nDescriptionParameterCame's impellerCC3 impellerSRV2-O impeller\nRotational speedN (rpm)40,00021,78950,000\nMass flow rate$m\u02d9$ (kg\/s)1.814.542.55\nTotal-total pressure ratio\u03c0127.654.16.1\nIsentropic Efficiency\u03b7t120.870.920.84\nInlet prewhirl\u03b11 (deg)000\nDescriptionParameterCame's impellerCC3 impellerSRV2-O impeller\nRotational speedN (rpm)40,00021,78950,000\nMass flow rate$m\u02d9$ (kg\/s)1.814.542.55\nTotal-total pressure ratio\u03c0127.654.16.1\nIsentropic Efficiency\u03b7t120.870.920.84\nInlet prewhirl\u03b11 (deg)000\n\nCame's impeller has a design mass flow rate of 1.81\u2009kg\/s and rotational speed of 40,000\u2009rpm. The dimensionless specific speed of the impeller is 0.53, and the total-to-total pressure ratio is 7.65 at design point. The estimated impeller isentropic efficiency is 0.87. The impeller inlet hub radius was set at 30.48\u2009mm. The CC3 impeller was designed to produce a stage pressure ratio of 4.0 at a corrected mass flow rate of 4.54\u2009kg\/s and corrected speed of 21,789\u2009rpm when coupled with a vaned diffuser. The dimensionless specific speed of the impeller is 0.6. The impeller total pressure ratio at design point is 4.1, and the estimated isentropic efficiency is 92%. The impeller total pressure ratio and isentropic efficiency were tabulated from the validated CFD results [35]. As to the SRV2-O impeller, it produces a total-to-total pressure ratio of 6.1 at the design speed of 50,000\u2009rpm. The dimensionless specific speed of the impeller is 0.81, and its estimated efficiency is 0.84. The impeller inlet hub radius was 30\u2009mm. All of the impellers are of advanced design with backswept trailing edges and splitter blades. A blade count of 17 full blades and 17 splitter blades was selected in Came's design. A blade count of 13 full blades and 13 splitter blades was selected for SRV2-O impeller, and CC3 impeller features 15 full blades and 15 splitter blades. The impeller exit blade angles were provided for Came's impeller (30\u2009deg backsweep) and the CC3 impeller (50\u2009deg backsweep). However, instead of blade angle, an impeller exit radius of 112\u2009mm was selected for SRV2-O impeller. There is zero prewhirl for all the three impellers.\n\nBased upon the design duty listed in Table 1, selection of Wiesner slip model [5], and chosen values of blockage factor, the preliminary design program using the method based on relative diffusion effectiveness calculated the remaining preliminary parameters necessary for the basic overall design. The output obtained using the new method is compared to the original design choices, and the results are listed in Table 2. The impeller inlet calculations utilized in the new method follow the standard practice for centrifugal compressor design [36]: minimize the inlet tip relative Mach number at the design mass flow rate and inlet conditions. The values of the impeller inlet blockage were adjusted to match the inlet tip relative Mach number provided in the original designs, and this renders an inlet blockage of 0.095 for Came's impeller and an inlet blockage of 0.09 for CC3 and SRV2-O impeller. The geometry and velocity triangles at the impeller exit were optimized based on the relative diffusion effectiveness parameter, and an exit blockage factor of 0.19 was selected for all three impellers.\n\nTable 2\n\nComparison between results from the new method and the original design selections\n\nCame's impellerCC3 impellerSRV2-O impeller\nDescriptionParameterOriginal designNew methodOriginal designNew methodOriginal designNew method\nDiffusion factorDF1.61.61.41.4N\/A1.7\nInlet tip relative Mach numberMrel_1t0.980.980.850.851.301.30\nExit wheel speedU2 (m\/s)575.4583.6492.2489.2586586\nExit absolute flow angle\u03b12 (deg)71.270.767.1\nCame's impellerCC3 impellerSRV2-O impeller\nDescriptionParameterOriginal designNew methodOriginal designNew methodOriginal designNew method\nDiffusion factorDF1.61.61.41.4N\/A1.7\nInlet tip relative Mach numberMrel_1t0.980.980.850.851.301.30\nExit wheel speedU2 (m\/s)575.4583.6492.2489.2586586\nExit absolute flow angle\u03b12 (deg)71.270.767.1\na\n\nRepresents preselected parameters.\n\nAt the impeller inlet, the new method calculates a tip radius of 68.9\u2009mm and tip blade angle of 55.6\u2009deg for Came's impeller, which are very close to the values of 67.3\u2009mm and 53.8\u2009deg adopted in the original design. In the case of CC3 impeller, the new method calculates a tip radius of 109.2\u2009mm which is 4.2\u2009mm different from the value adopted in CC3 impeller. As to the SRV2-O impeller, the new method calculates the tip radius of 74.4\u2009mm and blade angle of 65\u2009deg. Those values are similar to the design choices from DLR, which are 78\u2009mm for inlet tip radius and 63.5\u2009deg for the tip blade angle.\n\nAs to the impeller exit geometry and flow properties, the new method gives an optimum diffusion factor of 1.6, which is also the choice in Came's design. Additionally, the new method calculates an impeller exit radius of 139.3\u2009mm, with less than 2.0\u2009mm difference from the selection in Came's design. In the case of the CC3 impeller, the method based on relative diffusion effectiveness calculated the same diffusion factor (1.40) as the design choice of the CC3 impeller. Additionally, the new method gives an optimum exit radius of 214.4\u2009mm, with less than a 1.2\u2009mm difference from the CC3 impeller design. As to the SRV2-O impeller, the new method gives an optimum diffusion factor of 1.7, leading to an exit blade angle of 39\u2009deg. The value of diffusion factor was not presented in the literature for the SRV2-O impeller. However, the backswept angle was 38\u2009deg, which is very close to the value optimized with the new approach.\n\nFurthermore, in all the three cases, the values for the optimum absolute flow angle at the impeller exit stay fairly close, within a tight range between 67.1\u2009deg and 71.2\u2009deg. This agrees well with the empirical knowledge that the optimal impeller exit flow angles are between 60\u2009deg and 70\u2009deg, as recommended by Rodgers and Sapiro [6]. In summary, the new method based on relative diffusion effectiveness provides designs similar to those achieved via traditional preliminary design methods, as indicated by the similar impeller geometries and velocity triangles calculated for the three classic impellers available in the open literature.\n\n### Sensitivity Study.\n\nA sensitivity analysis of the present method to different slip models and to different blockage factors was performed. The effect of slip models was investigated using three different models: Stodola [1], Stanitz [37], and Wiesner [5]. The sensitivity study of impeller inlet and exit blockage factors was performed over a wide range of blockage values, from 0 to 0.25 in increments of 0.05.\n\nFigure 7 shows the effect of the slip models on screening the optimum preliminary design for Came's impeller. The optimum diffusion factor and the corresponding relative diffusion effectiveness from each slip model are represented by symbols. In these calculations using different slip models, the blockage factors were set the same, with a value of 0.095 at impeller inlet and 0.19 at impeller exit. The optimum diffusion factors and maximum diffusion effectiveness calculated from Wiesner's and Stodolar's slip models are very close to each other. However, the slip model from Stanitz gave significantly higher values for both the optimum diffusion factor and maximum relative diffusion effectiveness. The optimum diffusion factor and the corresponding relative diffusion effectiveness from Stanitz's slip model are 12% and 5% higher than the values from Wiesner's slip model. Recalling that the slip model from Stanitz was derived from an impeller with radial blades (zero backsweep), this is likely the reason for the differences.\n\nFig. 7\nFig. 7\nClose modal\n\nIn addition to the diffusion factor and relative diffusion effectiveness, other preliminary design parameters from each slip model are listed in Table 3. Since slip model does not affect the calculations at impeller inlet, the inlet properties and geometry are the same for all three slip models and are not included. The geometry and flow properties at the impeller exit vary slightly with respect to the slip model. The impeller exit radius, blade height, and absolute flow angle from Stodola's model are almost identical to those obtained from Wiesner's model, with less than a 0.2% difference in magnitude. In contrast, the impeller exit blade height calculated from Stanitz's model is about 12% larger than the value calculated using Wiesner's model, which is similar to the difference in diffusion factor. Additionally, the differences in the impeller exit radius and absolute flow angle between Stanitz's and Wiesner's models are much smaller, with a 1.9% reduction in impeller exit radius and 2.3% increase in the exit absolute flow angle using Stanitz's model.\n\nTable 3\n\nSensitivity study of slip models for Came's impeller\n\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.591.791.60\nExit absolute flow angle (deg)71.172.971.2\nSlip factor0.910.940.92\nRelative diffusion effectiveness0.640.670.64\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.591.791.60\nExit absolute flow angle (deg)71.172.971.2\nSlip factor0.910.940.92\nRelative diffusion effectiveness0.640.670.64\n\nA sensitivity study of the new method to different slip models for the CC3 impeller was performed, and the results are shown in Fig. 8. The blockage factors in the calculations were set to be 0.09 at the impeller inlet and 0.19 at the impeller exit. As indicated by symbols in the figure, the slip models from Stodola and Stanitz give very similar values in the optimum diffusion factor and peak relative diffusion effectiveness. Comparing to the model from Stodola, the slip model from Wiesner gave a reduced optimum diffusion factor by 2.7% and reduced relative diffusion effectiveness by 1.9%. In addition, the detailed data output using various slip models is listed in Table 4. The geometry and flow properties at the impeller exit vary slightly for the different slip models, with less than a 2.0\u2009mm variation in exit radius, less than 0.5\u2009mm variation in blade height, and less than a 1\u2009deg variation in the absolute flow angle.\n\nFig. 8\nFig. 8\nClose modal\nTable 4\n\nSensitivity study of slip models for CC3 impeller\n\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.441.451.40\nExit absolute flow angle (deg)71.171.270.7\nSlip factor0.930.930.93\nRelative diffusion effectiveness0.740.740.72\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.441.451.40\nExit absolute flow angle (deg)71.171.270.7\nSlip factor0.930.930.93\nRelative diffusion effectiveness0.740.740.72\n\nFigure 9 shows the effect of the slip model on screening the optimum preliminary design for SRV2-O impeller, with the optimum design from each slip model indicated by the symbols. The blockage factors in the calculations were set the same as CC3 impeller, with a value of 0.09 at the impeller inlet and 0.19 at the impeller exit. For the case of the SRV2-O impeller, slip models from Stanitz and Wiesner give peak values (local maxima) in the relative diffusion effectiveness, which are considered as the optimum design case. However, the slip model from Stodola does not give such a peak value but an inflection point.\n\nFig. 9\nFig. 9\nClose modal\n\nThe relative diffusion effectiveness obtained from Stodola's slip model increases quickly for diffusion factors between 1.2 and 1.5, flattens out around 1.6, and then starts to increase sharply. Since it is impossible for relative diffusion effectiveness to continue increasing with the increase of relative diffusion, the inflection point in the case of Stodola's slip model is considered as the optimum design case. The slip models from Wiesner and Stodola give very similar values for the optimum diffusion factor and relative diffusion effectiveness. However, the Stanitz model gives a slightly lower (1%) value for the optimum diffusion factor and a higher (3%) value of the relative diffusion effectiveness, compared to the values obtained using Wiesner's slip model.\n\nAdditionally, the rest of the output data at the impeller exit obtained from different slip models are listed in Table 5. The variations in the geometry and flow properties at the impeller exit associated with the change of slip model are very small. There is less than a 3% variation in the impeller exit blade height, and the variations in the impeller exit blade angle and absolute flow angle are less than 1\u2009deg, showing that the approach is relatively insensitive to the choice of slip model.\n\nTable 5\n\nSensitivity study of slip models for SRV-O impeller\n\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.711.681.70\nExit absolute flow angle (deg)67.466.567.1\nSlip factor0.910.920.91\nRelative diffusion effectiveness0.570.590.57\nDescriptionStodolaStanitzWiesner\nDiffusion factor1.711.681.70\nExit absolute flow angle (deg)67.466.567.1\nSlip factor0.910.920.91\nRelative diffusion effectiveness0.570.590.57\n\nFurthermore, a sensitivity study of impeller inlet blockage factor was performed for a wide range of blockage values, from 0 to 0.25 in increments of 0.05, and the results are listed in Table 6 (Came's impeller), Table 7 (CC3 impeller), and Table 8 (SRV2-O impeller). A constant blockage factor of 0.19 at the impeller exit and the Wiesner slip model are used for all these calculations.\n\nTable 6\n\nSensitivity study of B1 for Came's impeller\n\nParameterValues\nB100.050.10.150.20.25\nDF1.551.571.61.631.671.7\nR1t (mm)66.867.969.170.371.773.2\n\u00df1t (deg)57.557.557.657.557.757.7\nMrel,1t0.950.960.980.991.011.04\nb2 (mm)6.06.06.06.06.16.0\n\u03b12 (deg)71.271.171.271.271.371.3\nR2 (mm)139.3139.4139.3139.3139.3139.3\nParameterValues\nB100.050.10.150.20.25\nDF1.551.571.61.631.671.7\nR1t (mm)66.867.969.170.371.773.2\n\u00df1t (deg)57.557.557.657.557.757.7\nMrel,1t0.950.960.980.991.011.04\nb2 (mm)6.06.06.06.06.16.0\n\u03b12 (deg)71.271.171.271.271.371.3\nR2 (mm)139.3139.4139.3139.3139.3139.3\n\nNote: Boldface rows represent the parameters that change with respect to the change of inlet blockage factor.\n\nTable 7\n\nSensitivity study of B1 for CC3 impeller\n\nParameterValues\nB100.050.10.150.20.25\nDF1.361.381.41.431.451.49\nR1t (mm)105.7107.5109.4111.6113.7116.4\n\u00df1t (deg)60.860.860.961.061.061.2\nMrel,1t0.830.840.850.870.890.91\nb2 (mm)16.116.016.016.015.916.1\n\u03b12 (deg)70.770.770.670.770.570.7\nR2 (mm)214.3214.4214.5214.4214.6214.4\nParameterValues\nB100.050.10.150.20.25\nDF1.361.381.41.431.451.49\nR1t (mm)105.7107.5109.4111.6113.7116.4\n\u00df1t (deg)60.860.860.961.061.061.2\nMrel,1t0.830.840.850.870.890.91\nb2 (mm)16.116.016.016.015.916.1\n\u03b12 (deg)70.770.770.670.770.570.7\nR2 (mm)214.3214.4214.5214.4214.6214.4\nTable 8\n\nSensitivity study of B1 for SRV2-O impeller\n\nParameterValues\nB100.050.10.150.20.25\nDF1.641.681.721.741.781.82\nR1t (mm)72.073.374.776.277.979.7\n\u00df1t (deg)64.764.965.165.165.465.5\nMrel,1t1.261.281.311.331.361.39\nb2 (mm)10.410.610.710.510.610.6\n\u00df2b (deg)38.739.239.639.139.239.3\n\u03b12 (deg)66.967.267.567.167.267.2\nParameterValues\nB100.050.10.150.20.25\nDF1.641.681.721.741.781.82\nR1t (mm)72.073.374.776.277.979.7\n\u00df1t (deg)64.764.965.165.165.465.5\nMrel,1t1.261.281.311.331.361.39\nb2 (mm)10.410.610.710.510.610.6\n\u00df2b (deg)38.739.239.639.139.239.3\n\u03b12 (deg)66.967.267.567.167.267.2\n\nThe results show that the impeller inlet blockage factor has a negligible effect on the geometry and flow properties at the impeller exit. The impeller exit radius, blade height, and absolute flow angle remain fairly constant over the entire range of impeller inlet blockage, with less than a 0.3\u2009mm variation in impeller exit radius and blade height, and less than 1\u2009deg variation in impeller exit blade angle and absolute flow angle. The blade angle at the impeller inlet also remains fairly constant, with less than a 1.0\u2009deg variation in magnitude.\n\nThe variation in impeller inlet blockage factor primarily affects the value of the inlet tip radius and relative Mach number. There is an approximately 10% increase in the impeller inlet tip radius as the inlet blockage factor increases from 0 to 0.25. This increase in the impeller tip radius renders a higher inlet tip relative Mach number and a higher relative diffusion ratio across the impeller. The increase in the impeller inlet tip relative Mach number and diffusion factor is about the same as the increase in the impeller inlet tip radius.\n\nAt last, a sensitivity study on the blockage factor at the impeller exit was performed over the same range (from 0 to 0.25), and the results from the three impellers are listed in Table 9. The same slip model from Wiesner was selected, and a constant inlet blockage factor of 0.09 was used. The impeller exit blockage factor only influences the blade height at the impeller exit. As the impeller exit blockage increases, the percentage of effective flow area decreases, and thus, requires an increased blade height at the impeller exit. There is an approximately 35% (Came's impeller) and 33% (CC3 and SRV2-O impeller) increase in the blade height as the blockage factor increases from 0 to 0.25 at the impeller exit.\n\nTable 9\n\nSensitivity study of B2 for the three selected impellers\n\nParameterCameCC3SRV2-O\nB2b2 (mm)b2 (mm)b2 (mm)\n04.813.08.5\n0.055.113.79.0\n0.15.414.59.5\n0.155.715.310.0\n0.26.116.310.6\n0.256.517.311.3\nParameterCameCC3SRV2-O\nB2b2 (mm)b2 (mm)b2 (mm)\n04.813.08.5\n0.055.113.79.0\n0.15.414.59.5\n0.155.715.310.0\n0.26.116.310.6\n0.256.517.311.3\n\nIn summary, the slip model does not affect the calculations at the impeller inlet, and its effect on the calculations at impeller exit is also minor. In all three cases, the method provides similar results of flow angles at the impeller inlet and exit regardless of the choice of slip models. As to the effect of blockage factors, the dominant effect from inlet blockage is on the inlet tip radius and associated relative Mach number, and the blockage factor at the impeller exit is the primary driver for impeller exit blade height. Thus, some intelligent choice of blockage must be exercised, but this choice will not affect the velocity triangles, and thus, the optimization of the velocity triangles is not affected by blockage either.\n\n## Conclusions\n\nThis paper introduces a new approach for the preliminary design and aerothermal analysis of centrifugal impellers using a relative diffusion effectiveness parameter. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller. The parameter allows for a new optimization method for centrifugal impeller 1D preliminary design, in which the optimum design is selected to achieve the maximum relative diffusion effectiveness. Additionally, it could also be used to compare the performance of various impellers.\n\nThe relative diffusion effectiveness was correlated with impeller isentropic efficiency using experimental data acquired on a single-stage centrifugal compressor. The results show positive correlation between impeller efficiency and relative diffusion effectiveness at both design and off design conditions. Furthermore, after incorporating slip, the correlation between impeller efficiency and relative diffusion effectiveness could be represented by a single linear fit over the entire operating range from 80% to 100% corrected speed, which greatly reduces the complexity of its application in analyzing impeller performance. The positive correlation between relative diffusion effectiveness and isentropic efficiency at both design and off-design conditions indicates that relative diffusion effectiveness could be used as an alternate parameter of impeller isentropic efficiency in evaluating impeller performance. Compared with isentropic efficiency, the relative diffusion effectiveness does not include the benefits associated with the centrifugal effect and thus, provides a direct indication of the impeller aerodynamic performance. Additionally, relative diffusion effectiveness could be calculated from static pressure measurements acquired at the impeller exit and thus, greatly reduces the instrumentation cost for total pressure\/temperature rakes required for the calculation of isentropic efficiency.\n\nRelative diffusion effectiveness is used in a new approach to impeller preliminary design, and it was applied to three classic impellers available in the open literature. The selected impellers [3234] are representative applications of centrifugal compressors in gas turbines and turbochargers. The results obtained from the new method based on relative diffusion effectiveness agree very well with the original design selections obtained from traditional methods based on well-tuned empirical correlations. Furthermore, sensitivity studies show that the new approach is robust and provides similar results of flow angles at impeller inlet and exit despite a wide range of choices for impeller inlet and exit blockage factors, as well as different slip models.\n\nThis new approach is useful since it does not require empirical loss models. It provides an analytical approach for selection of the optimal diffusion factor. Since it can be challenging to establish a tuned set of empirical loss models or explore designs outside of past design envelopes, the analytical approach of the new method greatly reduces the complexity of its application and provides a useful tool for centrifugal impeller preliminary design.\n\n## Acknowledgment\n\nThis research has been sponsored by Honeywell, Inc., and this support is most gratefully acknowledged. The authors also wish to thank Honeywell for granting permission to publish this work. Additionally the guidance and advice offered by Mr. Darrell James and Dr. Rakesh Srivastava of Honeywell were very valuable to this project. Assistance from Herbert Harrison and Amelia Brooks at the Purdue Compressor Research Laboratory during data acquisition was also very much appreciated.\n\n## Nomenclature\n\n\u2022 a =\n\nspeed of sound\n\n\u2022\n\u2022 A =\n\neffective area\n\n\u2022\n\u2022 b =\n\n\u2022\n\u2022 B =\n\nblockage factor\n\n\u2022\n\u2022 DF =\n\ndiffusion factor, $DF=W1t\/W2$\n\n\u2022\n\u2022 h =\n\nenthalpy\n\n\u2022\n\u2022 I =\n\nrothalpy\n\n\u2022\n\u2022 $m\u02d9$ =\n\nmass flow rate\n\n\u2022\n\u2022 M =\n\nMach number\n\n\u2022\n\u2022 N =\n\nrotational speed in rpm\n\n\u2022\n\u2022 P =\n\npressure\n\n\u2022\n\u2022 R =\n\n\u2022\n\u2022 RMR =\n\nrelative Mach number ratio\n\n\u2022\n\u2022 s =\n\nentropy\n\n\u2022\n\u2022 T =\n\ntemperature\n\n\u2022\n\u2022 U =\n\nwheel velocity\n\n\u2022\n\u2022 V =\n\nabsolute velocity\n\n\u2022\n\u2022 W =\n\nrelative velocity\n\n\u2022\n\u2022 Z =\n\n\u2022\n\u2022 \u03b1 =\n\n\u2022\n\u2022 \u03b2 =\n\n\u2022\n\u2022 \u03b5 =\n\nrelative diffusion effectiveness\n\n\u2022\n\u2022 \u03b7 =\n\nisentropic efficiency, total to total\n\n\u2022\n\u2022 \u03c0 =\n\ntotal-total pressure ratio\n\n\u2022\n\u2022 \u03c1 =\n\ndensity\n\n### Subscripts\n\nSubscripts\n\n\u2022 b =\n\n\u2022\n\u2022 com =\n\nproperties obtained from equation for conservation of mass\n\n\u2022\n\u2022 cor =\n\nproperties obtained from equation for conservation of rothalpy\n\n\u2022\n\u2022 f =\n\nproperties associated with flow, full blade\n\n\u2022\n\u2022 h =\n\nhub\n\n\u2022\n\u2022 i =\n\nideal\n\n\u2022\n\u2022 j =\n\njth iteration\n\n\u2022\n\u2022 k =\n\nkth iteration\n\n\u2022\n\u2022 m =\n\nproperties derived from conservation of mass\n\n\u2022\n\u2022 no_slip =\n\nproperties derived assuming zero slip\n\n\u2022\n\u2022 p =\n\nprimary flow\n\n\u2022\n\u2022 rel =\n\nproperties in relative frame coordinate\n\n\u2022\n\u2022 s =\n\n\u2022\n\u2022 slip =\n\nproperties incorporating slip\n\n\u2022\n\u2022 t =\n\nstagnation properties, tip\n\n\u2022\n\u2022 \u03b8 =\n\ntangential\n\n\u2022\n\u2022 1 =\n\nimpeller inlet\n\n\u2022\n\u2022 2 =\n\nImpeller exit\n\n## References\n\n1.\nStodola\n,\nA.\n,\n1927\n,\nSteam and Gas Turbines\n, Vol.\n2\n,\nMcGraw-Hill\n,\nNew York\n.\n2.\nCordier\n,\nO.\n,\n1955\n, \u201cSimilarity Considerations in Turbomachines,\u201d Verlag, D\u00fcsseldorf, Germany, VDI Report No. 3.\n3.\nHerbert\n,\nM. V.\n,\n1980\n, \u201c\nA Method of Performance Prediction for Centrifugal Compressors\n,\u201d\nAeronautical Research Council Reports and Memoranda\n, Vol.\n3843\n,\nH. M. Stationery Office\n,\nLondon\n.\n4.\nRodgers\n,\nC.\n,\n1964\n, \u201c\nTypical Performance Characteristics of Gas Turbine Radial Compressors\n,\u201d\nASME J. Eng. Power\n,\n86\n(\n2\n), pp.\n161\n170\n.\n5.\nWiesner\n,\nF. J.\n,\n1967\n, \u201c\nA Review of Slip Factors for Centrifugal Impellers\n,\u201d\nASME J. Eng. Power\n,\n89\n(\n4\n), pp.\n558\n566\n.\n6.\nRodgers\n,\nC.\n, and\nSapiro\n,\nL.\n,\n1972\n, \u201cDesign Considerations for High-Pressure-Ratio Centrifugal Compressors,\u201d\nASME\nPaper No. 72-GT-91.\n7.\nRodgers\n,\nC.\n,\n1991\n, \u201cThe Efficiencies of Single Stage Centrifugal Compressors for Aircraft Applications,\u201d\nASME\nPaper No. 91-GT-77.\n8.\nRodgers\n,\nC.\n,\n1977\n, \u201c\nImpeller Stalling as Influenced by Diffusion Limitations\n,\u201d\nASME J. Fluids Eng.\n,\n99\n(\n1\n), pp.\n84\n93\n.\n9.\nYoung\n,\nL. R.\n,\n1977\n, \u201c\nDiscussion: \u2018Impeller Stalling as Influenced by Diffusion Limitations\u2019 (Rodgers, C., 1977, ASME J. Fluids Eng., 99, pp. 84\u201393)\n,\u201d\nASME J. Fluids Eng.\n,\n99\n(\n1\n), pp.\n94\n95\n.\n10.\nBenvenuti\n,\nE.\n,\n1978\n, \u201cAerodynamic Development of Stages for Industrial Centrifugal Compressors\u2014Part 1: Testing Requirements and Equipment-Immediate Experimental Evidence,\u201d\nASME\nPaper No. 78-GT-4.\n11.\nBenvenuti\n,\nE.\n,\n1978\n, \u201cAerodynamic Development of Stages for Industrial Centrifugal Compressors\u2014Part 2: Test Data Analysis, Correlation and Use,\u201d\nASME\nPaper No. 78-GT-5.\n12.\nWhitfield\n,\nA.\n, and\nBaines\n,\nN. C.\n,\n1990\n,\n,\nLongman\n,\nLondon\n.\n13.\nDean\n,\nR. C.\n, and\nSenoo\n,\nY.\n,\n1960\n, \u201c\nRotating Wakes in Vaneless Diffusers\n,\u201d\nASME J. Basic Eng.\n,\n82\n(\n3\n), pp.\n563\n570\n.\n14.\nEckardt\n,\nD.\n,\n1976\n, \u201c\nDetailed Flow Investigation Within a High-Speed Centrifugal Compressor Impeller\n,\u201d\nASME J. Fluids Eng.\n,\n98\n(\n3\n), pp.\n390\n399\n.\n15.\nKrain\n,\nH.\n,\n1988\n, \u201c\nSwirling Impeller Flow\n,\u201d\nASME J. Turbomach.\n,\n110\n(\n1\n), pp.\n122\n128\n.\n16.\nSkoch\n,\nG. J.\n,\nPrahst\n,\nP. S.\n,\nWernet\n,\nM. P.\n,\nWood\n,\nJ. R.\n, and\nStrazisar\n,\nA. J.\n,\n1997\n, \u201cLaser Anemometer Measurements of the Flow Field in a 4:1 Pressure Ratio Centrifugal Impeller,\u201d\nASME\nPaper No. 97-GT-342.\n17.\nJapikse\n,\nD.\n,\n1985\n, \u201cAssessment of Single- and Two-Zone Modeling of Centrifugal Compressors Studies in Component Performance\u2014Part 3,\u201d\nASME\nPaper No. 85-GT-73.\n18.\nRodgers\n,\nC.\n,\n1998\n, \u201cCentrifugal Compressor Inducer,\u201d\nASME\nPaper No. 1998-GT-032.\n19.\nRodgers\n,\nC.\n,\n2003\n, \u201cHigh Specific Speed, High Inducer Tip Mach Number, Centrifugal Compressor,\u201d\nASME\nPaper No. GT2003-38949.\n20.\nSenoo\n,\nY.\n,\nHayami\n,\nH.\n,\nKinoshita\n,\nY.\n, and\nYamasaki\n,\nH.\n,\n1979\n, \u201c\nExperimental Study on Flow in a Supersonic Centrifugal Impeller\n,\u201d\nASME J. Eng. Power\n,\n101\n(\n1\n), pp.\n32\n39\n.\n21.\nHayami\n,\nH.\n,\nSenoo\n,\nY.\n, and\nUeki\n,\nH.\n,\n1985\n, \u201c\nFlow in the Inducer of a Centrifugal Compressor Measured With a Laser Velocimeter\n,\u201d\nASME J. Eng. Power\n,\n107\n(\n2\n), pp.\n534\n540\n.\n22.\nKrain\n,\nH.\n,\nHoffmann\n,\nB.\n, and\nPak\n,\nH.\n,\n1995\n, \u201cAerodynamics of a Centrifugal Compressor Impeller with Transonic Inlet Conditions,\u201d\nASME\nPaper No. 95-GT-079.\n23.\nKrain\n,\nH.\n,\nKarpinski\n,\nG.\n, and\nBeversdorff\n,\nM.\n,\n2001\n, \u201cFlow Analysis in a Transonic Centrifugal Compressor Rotor Using 3-Component Laser Velocimetry,\u201d\nASME\nPaper No. 2001-GT-0315.\n24.\nIbaraki\n,\nS.\n,\nHigashimori\n,\nH.\n, and\nMatsuo\n,\nT.\n,\n2001\n, \u201c\nFlow Investigation of a Transonic Centrifugal Compressor for Turbocharger\n,\u201d\n23nd CIMAC Congress Proceedings\n, Hamburg, Germany, May 7\u201310, pp.\n339\n346\n.\n25.\nIbaraki\n,\nS.\n,\nMatsuo\n,\nT.\n,\nKuma\n,\nH.\n,\nSumida\n,\nK.\n, and\nSuita\n,\nT.\n,\n2003\n, \u201c\nAerodynamics of a Transonic Centrifugal Compressor Impeller\n,\u201d\nASME J. Turbomach.\n,\n125\n(\n2\n), pp.\n346\n351\n.\n26.\nHigashimori\n,\nH.\n,\nHasagawa\n,\nK.\n,\nSumida\n,\nK.\n, and\nSuita\n,\nT.\n,\n2004\n, \u201c\nDetailed Flow Study of Mach Number 1.6 High Transonic Flow With a Shock Wave in a Pressure Ratio 11 Centrifugal Compressor Impeller\n,\u201d\nASME J. Turbomach.\n,\n126\n(\n4\n), pp.\n473\n481\n.\n27.\nLemmon\n,\nE. W.\n,\nHuber\n,\nM. L.\n, and\nMcLinden\n,\nM. O.\n,\n2013\n, \u201cNIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties\u2014REFPROP, Version 9.1,\u201d Standard Reference Data Program, National Institute of Standards and Technology, Gaithersburg, MD,\nReport\n.\n28.\nLou\n,\nF.\n,\nHarrison\n,\nH. M.\n,\nFabian\n,\nJ. C.\n,\nKey\n,\nN. L.\n,\nJames\n,\nD. K.\n, and\nSrivastava\n,\nR.\n,\n2016\n, \u201cDevelopment of a Centrifugal Compressor Facility for Performance and Aeromechanics Research,\u201d\nASME\nPaper No. GT2016-56188.\n29.\nSimon\n,\nH.\n,\nWallmann\n,\nT.\n, and\nMonk\n,\nT.\n,\n1987\n, \u201c\nImprovements in Performance Characteristics of Single-Stage and Multistage Centrifugal Comrpessors by Simultaneous Adjustments of Inlet Guide Vanes and Diffuser Vanes\n,\u201d\nASME J. Turbomach.\n,\n109\n(\n1\n), pp.\n41\n47\n.\n30.\nBerdanier\n,\nR. A.\n,\nSmith\n,\nN. R.\n,\nFabian\n,\nJ. C.\n, and\nKey\n,\nN. L.\n,\n2014\n, \u201c\nHumidity Effects on Experimental Compressor Performance\u2014Corrected Conditions for Real Gases\n,\u201d\nASME J. Turbomach.\n,\n137\n(\n3\n), p.\n031011\n.\n31.\nLou\n,\nF.\n,\nFabian\n,\nJ. C.\n, and\nKey\n,\nN. L.\n,\n2013\n, \u201c\nThe Effect of Gas Models on Compressor Efficiency Including Uncertainty\n,\u201d\nASME J. Eng. Gas Turbines Power\n,\n136\n(\n1\n), p.\n012601\n.\n32.\nCame\n,\nP. M.\n,\n1978\n, \u201c\nThe Development, Application and Experimental Evaluation of a Design Procedure for Centrifugal Compressors\n,\u201d\nProc. Inst. Mech. Eng.\n,\n192\n(\n1\n), pp.\n49\n67\n.\n33.\nMcKain\n,\nT. F.\n, and\nHolbrook\n,\nG. J.\n,\n1997\n, \u201cCoordinates for a High Performance 4:1 Pressure Ratio Centrifugal Compressor,\u201d National Aeronautics and Space Administration, Cleveland, OH, NASA Report No.\nNASA-CR-204134\n.\n34.\nEisenlohr\n,\nG.\n,\nKrain\n,\nH.\n,\nRichter\n,\nF. A.\n, and\nTiede\n,\nV.\n,\n2002\n, \u201cInvestigations of the Flow Through a High Pressure Ratio Centrifugal Impeller,\u201d\nASME\nPaper No. GT2002-30394.\n35.\nLarosiliere\n,\nL. M.\n,\nSkoch\n,\nG. J.\n, and\nPrahst\n,\nP. S.\n,\n1999\n, \u201c\nAerodynamic Synthesis of a Centrifugal Impeller Using Computational Fluid Dynamics and Measurement\n,\u201d\nJ. Propul. Power\n,\n15\n(\n5\n), pp.\n623\n632\n.\n36.\nJapikse\n,\nD.\n,\n1996\n,\nCentrifugal Compressor Design and Performance\n, Concepts ETI, Inc., White River Junction, VT.\n37.\nStanitz\n,\nJ. D.\n,\n1952\n, \u201c\nSome Theoretical Aerodynamic Investigations of Impellers in Radial and Mixed Flow Centrifugal Compressors\n,\u201d\nTrans. 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\section{Introduction}
Standard Model of particle physics is a low energy effective theory which has been astonishingly successful in explaining the dynamics of fundamental particles and their interactions. The discovery of Higgs Boson in 2012 at CERN LHC has strengthen our belief in this incredible theory. Despite its tremendous success there still remain unanswered questions such as origin of neutrino mass, dark matter and matter-antimatter asymmetry, to name a few. Neutrino oscillation experiments have been very instrumental in our quest to understand Standard Model(SM) predictions for the leptonic sector. They have shown at high level of statistical significance that neutrinos have non-zero but tiny mass, and flavor and mass eigenstates mix giving rise to quantum mechanical phenomena of neutrino oscillations.
\noindent On the contrary, within the SM, neutrinos are massless because the Higgs field cannot couple to the neutrinos due to the absence of right-handed(RH) neutrinos. The extension of SM with RH neutrino, however, require unnatural fine tuning of the Yukawa couplings to generate sub-eV neutrino masses. Dimension five Weinberg operator can generate the tiny Majorana mass for neutrinos with the SM Higgs field. In fact, there exist several beyond the Standard Model(BSM) scenarios, e.g. seesaw mechanisms, which may explain the origin of such dimension five operator and can account for dynamical origin of tiny Majorana neutrino masses by appropriately extending the field content of the SM. For example in type-I, type-II and type-III seesaw RH neutrinos, scalar triplet(s) and fermion triplet(s) are introduced to the particle content of the SM, respectively\cite{s1,s2,s3,s4,s5,s6}.
\noindent Seesaw mechanisms provide the most elegant and natural explanation of the smallness of neutrino masses. The fundamental basis of seesaw mechanism is the existence of lepton number violation at some high energy scale. In order to have neutrino mass at sub-eV, the new physics scale must be of the order of GUT scale, 10$^{16}$ GeV. Therefore, the seesaw mechanisms explain the tiny non-zero neutrino masses, but they introduce new physics scale, which is beyond the reach of current and near future accelerator experiments.
\noindent On the other hand, there are several astonishing astrophysical observations such as (i) galaxy cluster investigation\cite{gc1}, (ii) rotation curves of galaxy\cite{rc1}, (iii) recent observations of bullet cluster\cite{bc1}, and (iv) the latest cosmological data from Planck collaboration\cite{planck2018}, which have proven the existence of non-luminous and non-baryonic anatomy of matter known as ``Dark Matter"(DM). Apart from the astrophysical environments, it is very difficult to probe the existence of DM in terrestrial laboratory. Alternatively, one can accredit a weak interaction property to the DM through which it get thermalized in the early Universe, which also can be examined at the terrestrial laboratories. According to the Planck data the current DM abundance is\cite{planck2018}
\begin{center}
$\Omega_{DM}h^2 = 0.120\pm0.001$.
\end{center}
\noindent Apart from relic abundance, the particle nature of DM is still unknown. Within the SM of particle physics, there is no suitable candidate for DM, as it should be stable on the cosmological time scale. The uncertainty in the nature of DM and possible mechanisms of neutrino mass generation have opened up a window to explore new models in a cohesive way.
Out of several neutrino mass models, the texture zero models are very interesting due to their rich phenomenology and high predictability\cite{t1,t2,t3,t4,t5,t6,t7,t8,t9,t10}. In fact, texture zero models have also been successfully investigated to generate observed matter-antimatter asymmetry in different seesaw settings\cite{t11} and scotogenic scenarios\cite{sctt}. The DM phenomenology within the framework of texture zeros using type-I seesaw and scotogenesis have been studied in Refs. \cite{type1} and \cite{sct}, respectively.
\noindent Another possibility amongst the phenomenological approaches is the existence of scaling structure in the neutrino mass matrix wherein third column is scaled with respect to the second column by some model dependent parameter(s)\cite{rnm,svrma}. The scaling ansatz has been studied in Ref.\cite{mnd} where neutrino mass is generated using inverse and type-II seesaw frameworks. In Ref.\cite{mnd}, the implementation of inverse seesaw resulted in scaled neutrino mass matrix which predicts vanishing lightest mass eigenvalue(inverted neutrino mass ordering) and vanishing reactor mixing angle($\theta_{13}=0$). Subsequently, within an effective field theory approach, in order to have non-zero reactor mixing angle($\theta_{13}\neq 0$) type-II seesaw perturbation has been incorporated in the Lagrangian. We, in this work, confine to tree level dimension-4 and focus on the possible realization of texture one-zero ansatz which can be embedded in more general framework of grand unified theories wherein quarks and leptons belong to the same multiplet. For example, in Refs.\cite{fuku, svrma1}, texture zero(s) in fermion mass matrices have been investigated under SO(10) environment.
\noindent In this work, we investigate a well-motivated possibility for simultaneous explanation of DM and non-zero neutrino mass using $A_4$ non-Abelian discrete symmetry within the framework of inverse seesaw(ISS)\cite{iss1,iss2,iss3} wherein small neutrino masses emanate from new physics at TeV scale, which is within the reach of accelerator experiments. The stability of DM is assured by $Z_2$ symmetry. Type-II seesaw has been implemented to have one-zero in the effective neutrino mass matrix. Within ISS mechanism, neutrino masses are generated assuming three right-handed neutrinos $N_{T}$ and three additional SM singlet neutral fermions $S_{T}$($T=1,2,3$). The fermionic singlets ($N_{4,5}$ and $S_{4,5}$) are assumed to have Yukawa couplings with the scalar fields $H,\phi,\phi_R$ and $\phi_{S}$, which after spontaneous symmetry breaking provide diagonal Majorana mass matrix($\mu$). The scalar triplets $\Delta_1$ and $\Delta_2$ are incorporated, so that $M_{\nu}$ contain one vanishing element after type-II seesaw implementation. Along with DM abundance, we have also obtained prediction of the model for effective Majorana mass ($|M_{ee}|$) appearing in neutrinoless double beta($0\nu\beta\beta$) decay.
\noindent The paper is structured as follows. In section {\ref{sec:2}}, we have discussed the inverse seesaw mechanism based on $G_f\equiv A_4\times Z_{2}^{'}\times Z_3\times Z_4$ symmetry group and resulting neutrino mass matrices. Section {\ref{sec:3}} is devoted to the investigation of relic density of the DM. In section {\ref{sec:4}}, the prediction of the model for neutrinoless double beta decay is discussed. Finally, conclusions are summarized in section {\ref{sec:5}}.
\section{The Model}\label{sec:2}
In order to explain the smallness of neutrino mass, different versions of the seesaw mechanism play an important role. As discussed above the ISS mechanism is a viable scenario to get the mass of right-handed neutrino near the TeV scale. This scale is much below the scale, which we get from the canonical seesaw. As a requirement of ISS, the fermion sector is extended by three right-handed neutrinos $N_{i} (i = 1,2,3)$ and three extra singlet fermions $S_{j}(j=1,2,3)$. Within ISS mechanism the mass Lagrangian is written as\\
\begin{equation}
L = -\bar{\nu}_{\alpha L}m_{D}N_{i} - \bar{S_{j}}mN_{i}-\frac{1}{2}\bar{S_{j}}\mu S^C_{k} + h.c. \label{eq:1}
\end{equation}
where $m_D$, $m$ and $\mu$ are the 3$\times$3 complex mass matrices and $\alpha = (e,\mu,\tau)$, $k=(1,2,3)$. Here $m$ represents lepton number conserving interaction between neutral fermions and right-handed neutrinos and $\mu$ gives the Majorana mass terms for neutral fermions. Assuming lepton number as approximate symmetry, the Majorana mass term for right-handed neutrino is vanishing. However, it is mildly violated through singlet fermions $S$ having small mass $\mu$ in consonance with 't Hooft's naturalness criterion\cite{hooft}. The lepton number symmetry can be restored as $\mu\rightarrow 0$. Consequent to spontaneous symmetry breaking (SSB), the Lagrangian in Eqn.(\ref{eq:1}) leads to 9$\times$9 neutrino mass matrix\\
\begin{equation}
M_{\nu} = \begin{pmatrix}
0 & m_{D} & 0\\
m_D^T & 0 & m\\
0 & m^T & \mu \\
\end{pmatrix}, \\
\label{eq:2}
\end{equation}
in the basis ($\nu_L,N,S$). We can obtain standard model neutrinos at sub-eV scale from $m{_D}$ at electroweak scale, $\mu$ at keV scale and $m$ at TeV scale as explained in \cite{iss3,33}. Thus, if we consider the order $\mu << m_D << m$, then after the block diagonalization of above matrix, the 3$\times$3 effective neutrino
mass matrix is obtained as\\
\begin{equation}
m_{\nu} = m_{D} (m{^T})^{-1}\mu m^{-1} m_{D}^{T}.\\
\label{eq:3}
\end{equation}
\noindent It is clear from Eqn.(\ref{eq:3}) that there is a double suppression by mass term associated with $m$, which results in the scale that is much below to the one obtained by canonical seesaw. The essence of inverse seesaw mechanism lies in the fact that we can bring down the mass of right-handed neutrinos to TeV scale by assuming that $\mu$ should be at keV scale \cite{34,35,36}.
\noindent The symmetry group $A_4$ has played an important role in understanding particle physics \cite{37,38,39,40,41}. $A_4$ is a non-Abelian discrete symmetry group of even permutations of four objects.
Order of this group is 12. All the 12 elements are generated from two elements, S and T which satisfies: $S^2$ = $T^3$ = $(ST)^3$. It is a symmetry group of regular tetrahedron. It has four conjugacy classes, therefore, four irreducible representations: 1, 1$'$, 1$''$ and 3. The multiplication rules of irreducible representations in T basis are \cite{41,42}: {\bf1}$'\otimes${\bf1}$'$={\bf1}$''$,
{\bf1}$''\otimes${\bf1}$''$={\bf1}$'$, {\bf1}$'\otimes${\bf1}$''$={\bf1},
{\bf3}$\otimes${\bf3}={\bf1}$\oplus${\bf1}$'\oplus${\bf1}$''\oplus${\bf3}$_s\oplus${\bf3}$_a$ where,
\begin{eqnarray}
\nonumber
&&\left(\bf{3}\otimes\bf{3}\right)_{\bf{1}} =a_1b_1+a_2b_2+a_3b_3,\\ \nonumber
&&\left(\bf{3}\otimes\bf{3}\right)_{\bf{1'}}=a_1b_1+\omega a_2b_2+\omega^2 a_3b_3,\\ \nonumber
&&\left(\bf{3}\otimes\bf{3}\right)_{\bf{1''}}=a_1b_1+\omega^2 a_2b_2+\omega a_3b_3,\\ \nonumber
&&\left(\bf{3}\otimes\bf{3}\right)_{\bf{3_s}}=\left(a_2b_3+b_2a_3,a_3b_1+a_1b_3,a_1b_2+a_2b_1\right),\\ \nonumber
&&\left(\bf{3}\otimes\bf{3}\right)_{\bf{3_a}}=\left(a_2b_3-b_2a_3,a_3b_1-a_1b_3,a_1b_2-a_2b_1\right). \nonumber
\end{eqnarray}
\noindent Here $a_{i}$ and $ b_{i}$ (i = 1,2,3) are the basis vectors of the two triplets and $\omega$ = $e^{\frac{2\pi i}{3}}$.
\noindent In the model we have taken five right-handed neutrinos, three of which $N_T = (N_1,N_2,N_3)$ are transforming as triplet under $A_4$ and rest of the two i.e., $N_4$ and $N_5$ are transforming as singlets $1, 1'$, respectively. Singlet fermions $S_T = (S_1,S_2,S_3)$ and $(S_4, S_5)$ transforming as triplet and singlets $(1, 1')$ under $A_4$, respectively, are also introduced. The standard model Higgs doublet $H$ and three additional Higgs doublets $\eta_i$ transform as singlet and triplet $\eta$ under $A_4$, respectively. In addition, we have extended the scalar sector with three $SU(2)_L$ singlet scalar fields i.e. $\phi$, $\phi_R$ and $\phi_S$. After spontaneous symmetry breaking (SSB), the vacuum expectation values ($vev$) acquired by ($H$, $\eta$) and ($\phi$, $\phi_R$, $\phi_S$) give $m_D$ and ($m$, $\mu$), respectively, with minimal number of parameters. In order to have possible lepton number violation via $S_i (i=T,4,5)$ only, we distinguish the Yukawa interactions of $N_i$ and $S_i$ through $Z_2^{'}$ symmetry. Also, using $Z_3$ symmetry, all possible $N_iN_j(i,j=T,4)$ Majorana terms are inhibited. There can be possibility of Yukawa interactions like $N_TN_T\phi_{S}^{*}$, $N_4N_4\phi_{S}^{*}$ which are suppressed by $Z_4$ symmetry in the model. The fermionic and scalar field content along with respective charge assignments are shown in Table \ref{table1} and Table \ref{table2}, respectively.
\begin{center}
\begin{table}[t]
\centering
\begin{tabular}{ccccccccccccc}
Symmetry & $\bar{L}_{e}$ & $\bar{L}_{\mu}$ & $\bar{L}_{\tau}$ & $e_{R}$ & $\mu_{R}$ & $\tau_{R}$ & $N_{T}$ & $N_{4}$ & $N_{5}$ & $S_{T}$ & $S_{4}$ & $S_{5}$\\
\hline
$SU(2)_L$ & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$A_{4}$ & 1 & 1$'$ & 1$''$ & 1 & 1$''$ & 1$'$ & 3 & 1 & 1$'$ & 3 & 1 & 1$'$ \\
\hline
$Z'_{2}$ & 1 & 1 & 1
& 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1& -1 \\
\hline
$Z_{3}$ & $\omega^2$ & $\omega^2$ & $\omega^2$ & 1 & 1 & 1 & $\omega^2$ & $\omega^2$ & $\omega^2$ & $\omega$ & $\omega$ & $\omega $\\
\hline
$Z_{4}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1& 1 & 1 & $i$ & $i$ & $i$\\
\hline
\end{tabular}
\caption{Fermion field content and respective charge assignments used in the model.}
\label{table1}
\end{table}
\end{center}
\begin{center}
\begin{table}[h]
\centering
\begin{tabular}{cccccccc}
Symmetry & H & $\eta$ & $\Phi$ & $\Phi_{R}$ & $\Phi_{S}$ & $\Delta_{1}$ & $\Delta_{2}$ \\
\hline
$SU(2)_L$ & 2 & 2 & 1 & 1 & 1 & 3 & 3 \\
\hline
$A_{4}$ & 1 & 3 & 3 & 1 & 1 & 1 & 1$'$ \\
\hline
$Z'_{2}$ & 1 & 1 & -1 & -1 & 1 & 1 & 1 \\
\hline
$Z_{3}$ & $\omega$ & $\omega$ & 1 & 1 & $\omega $ & $\omega $ & $\omega $ \\
\hline
$Z_{4}$ & 1 & 1 & $-i$ & $-i$ & - 1 & 1 & 1 \\
\hline
\end{tabular}
\caption{Scalar field content and respective charge assignments used in the model.}
\label{table2}
\end{table}
\end{center}
\noindent The leading Yukawa Lagrangian is
\begin{eqnarray}
\nonumber
\mathcal{L^I} =&&y_{e}\Bar{L}_{e}e_{R}H + y_{\mu}\Bar{L}_{\mu}e_{\mu}H + y_{\tau}\Bar{L}{_\tau}e_{\tau}H + y_{1}^{\nu}\Bar{L}_{e}[N_{T} \Tilde{\eta}]_{1} + y_{2}^{\nu}\Bar{L}_{\mu}[N_{T} \Tilde{\eta}]_{1''}+\\
\nonumber
&&y_{3}^{\nu}\Bar{L}_{\tau}[N_{T} \Tilde{\eta}]_{1'} + y_{4}^{\nu}\Bar{L}_{e}N_{4}\Tilde{H} + y_{5}^{\nu}\Bar{L}_{\tau}N_{5}\Tilde{H} + y_{R}^{1}[N_{T} S_{T}]_{1} \phi_{R}+\\
\nonumber
&&y_{R}^{0}[N_{4} S_{4}]_{1} \phi_{R} + y_{\phi}^{1}[N_{T} \phi]_{1} S_{4} + y_{\phi}^{2}[N_{T} \phi]_{1''} S_{5} + y_{\phi}^{3}N_{4} [\phi S_{T}]_{1} + y_{\phi}^{4}N_{5} [\phi S_{T}]_{1''} + \\
\nonumber
&& y_{\phi}^{5}[N_{T} \phi]_{3} S_{T} + y_{s}^1 S_{T}S_{T} \phi_{S} + y_{s}^2 S_{4}S_{4} \phi_{S} + h.c.,\\
\label{eq:4}
\end{eqnarray}
\noindent where $\Tilde{H}=i\tau_3H$, $\Tilde{\eta}=i\tau_3\eta$ and $y_q(q = e,\mu,\tau)$, $y_{i}^{\nu} (i = 1,2,3,4,5)$, $y_{R}^{j}(j=0,1)$, $y_{\phi}^{k}(k=1,2,3,4,5)$, $y_{s}^p(p=1,2)$ are Yukawa coupling constants.
We have chosen the following vacuum alignments
\begin{center}
$\langle \eta \rangle \sim v_{\eta} (1,0,0)$,
$\langle \phi \rangle \sim v_{\phi}(1,0,0)$,
$\langle H \rangle = v_{h},\langle \phi_{S} \rangle = v_{S}, \langle \phi_{R} \rangle = v_{R}$.
\end{center}
\noindent The symmetry is broken down to $Z_2$ subgroup i.e. $G_f\equiv A_4\times Z_{2}^{'}\times Z_3\times Z_4\rightarrow Z_2$ by the $vev$ $\langle \eta \rangle \sim v_{\eta} (1,0,0)$ \cite{ddm}. Since the $vev$ alignment (1,0,0) remains invariant under the A$_4$ generator S = $Diag(1,-1,-1)$, the residual Z$_2$ symmetry is
\begin{center}
N$_2\rightarrow$ -N$_2$, N$_3\rightarrow$ -N$_3$,
S$_2\rightarrow$ -S$_2$, S$_3\rightarrow$ -S$_3$,\\
$\eta_2\rightarrow -\eta_2$, $\eta_3\rightarrow -\eta_3$,
$\phi_2\rightarrow -\phi_2$, $\phi_3\rightarrow -\phi_3$.
\end{center}
It is to be noted that only residual symmetry $Z_2$ is responsible for lightest dark matter candidate stability while $Z_2^{'}$ is employed to restricts the unwanted Yukawa couplings. $Z_2^{'}$ does not play any role in dark matter stabilization as the DM candidate is $Z_2^{'}$ even. From Eqn.(\ref{eq:4}), it is evident that lepton conserving interactions of $N_T$ and $S_T$ take place through the $A_4$ triplet $\phi$ introduced in model setup. Since inverse seesaw formula in Eqn.(\ref{eq:3}) assumes hierarchy of mass scale $\mu << m_D << m$, which implies that lepton conserving interaction of right-handed neutrinos $N_T$ and neutral fermion $S_T$ takes place at high scale. Also, Dirac mass term emanates from the $A_4$ triplet of SU(2)$_L$ doublet Higgs field $\eta$ having smaller mass as compared to interactions involving $N_T$, $S_T$. So, lightest $Z_2$ stabilized candidate will be $\eta_{2,3}$. It couples only with the right-handed neutrinos and not with the charged leptons. As a consequence, we have obtained a diagonal charged lepton mass matrix as\begin{equation}
m_{l}= Diag(y_{e},y_{\mu},y_{\tau})v_{h}.\\
\end{equation}
\noindent The other mass matrices which we have obtained are shown as below :\\
\begin{equation}
m_{D} =\begin{pmatrix}
A & 0 & 0 & F & 0\\
B & 0 & 0 & 0 & 0\\
C & 0 & 0 & 0 & H\\
\end{pmatrix}, \mu = \begin{pmatrix}
y & 0 & 0 & 0 & 0\\
0 & y & 0 & 0 & 0\\
0 & 0 & y & 0 & 0\\
0 & 0 & 0 & n & 0\\
0 & 0 & 0 & 0 & 0\\
\end{pmatrix} ,
m = \begin{pmatrix}
x & 0 & 0 & l & v\\
0 & x & h & 0 & 0\\
0 & h & x & 0 & 0\\
l & 0 & 0 & z & 0\\
v & 0 & 0 & 0 & 0\\
\end{pmatrix}, \\
\end{equation}
\noindent where $A=y_{1}^{\nu}v_{\eta}, B = y_{2}^{\nu}v_{\eta}, C=y_{3}^{\nu}v_{\eta}, F = y_{4}^{\nu}v_{h}, H = y_{5}^{\nu}v_{h}, $
$y = y_{S}^{1}v_{S}, n = y_{S}^{2}v_{S},$
$x= y_{R}^{1}v_{R}, z= y_{R}^{0}v_{R},l = y_{\phi}^{1}v_{\phi}+y_{\phi}^{3}v_{\phi}, v = y_{\phi}^{2}v_{\phi}+y_{\phi}^{4}v_{\phi}, h = y_{\phi}^{3}v_{\phi}$. Within ISS mechanism, the above matrices lead to the light neutrino mass matrix as follow\\
\begin{equation}
m_{\nu_{I}} = \begin{pmatrix}
X & 0 & \Delta \\
0 & 0 & 0 \\
\Delta & 0 & \Delta^{''} \\
\end{pmatrix} ,\\
\end{equation}
\noindent where $X = \frac{F^2n}{z^2}$ , $\Delta = -\frac{FHln}{vz^2}$ and $\Delta^{''} = \frac{H^2(l^2n+yz^2)}{v^2z^2}$.
\noindent When we choose a flavor basis in which we are obtaining a diagonal charged lepton mass matrix, only those neutrino mass matrix are allowed where we have at most two zeros. These neutrino mass matrices are consistent with neutrino oscillation results \cite{texture01}. Since we are getting three zeros in our neutrino mass matrix, we have introduced type-II seesaw to reduce the number of zeros in the neutrino mass matrix. The type-II seesaw contribution to the Lagrangian is given as
\begin{eqnarray}
\nonumber
\mathcal{L^{II}} = &&f_{1}(L_{e}L_{e}+L_{\mu}{L}_{\tau}+L_{\tau}{L}_{\mu})\Delta_{1} +\\
\nonumber
&&f_{2}(L_{e}{L}_{\tau}+{L}_{\tau}{L}_{e}+{L}_{\mu}{L}_{\mu})\Delta_{2} + h.c.,\\
\label{eq:8}
\end{eqnarray}
where, $f_1$ and $f_2$ are coupling constants. Therefore, the ISS + type-II seesaw Lagrangian for our model is given as
\begin{eqnarray}
\nonumber
\mathcal{L} =&&y_{e}\Bar{L}{_e}e_{R}H + y_{\mu}\Bar{L}{_\mu}\mu_{R}H + y_{\tau}\Bar{L}{_\tau}\tau_{R}H + y_{1}^{\nu}\Bar{L}_{e}N_{T} \Tilde{\eta} + y_{2}^{\nu}\Bar{L}_{\mu}N_{T}\Tilde{\eta} +y_{3}^{\nu}\Bar{L}_{\tau}N_{T} \Tilde{\eta} +\\
\nonumber
&&y_{4}^{\nu}\Bar{L}_{e}N_{4}\Tilde{H} + y_{5}^{\nu}\Bar{L}_{\tau}N_{5}\Tilde{H} +y_{R}^{0}N_{4} S_{4} \phi_{R} + y_{R}^{1}N_{T} S_{T} \phi_{R} + y_{\phi}^{1}N_{T} \phi S_{4} +\\
\nonumber
&& y_{\phi}^{2}N_{T} \phi S_{5} + y_{\phi}^{3}N_{4}\phi S_{T} + y_{\phi}^{4}N_{5} \phi S_{T} +
y_{\phi}^{5}N_{T} \phi S_{T} + y_{s}^1 S_{T}S_{T} \phi_{S} + y_{s}^2 S_{4}S_{4} \phi_{S} +\\
\nonumber
&&f_{1}(L_{e}L_{e}+L_{\mu}{L}_{\tau}+L_{\tau}{L}_{\mu})\Delta_{1} +f_{2}(L_{e}{L}_{\tau}+{L}_{\tau}{L}_{e}+{L}_{\mu}{L}_{\mu})\Delta_{2} + h.c.\\
\label{eq:9}
\end{eqnarray}
\noindent The SU(2) triplets $\Delta_{1}$ and $\Delta_{2}$ are transforming as singlets 1 and 1$'$, respectively. The vacuum expectation values $ \langle \Delta_{1} \rangle = v_{\Delta_{1}} , \langle \Delta_{2} \rangle = v_{\Delta_{2}} $ gives
\begin{equation}
m_{{\nu}_{II}} = \begin{pmatrix}
X^{'} & 0 & \Delta^{'} \\
0 & \Delta^{'} & X^{'} \\
\Delta^{'} & X^{'} & 0 \end{pmatrix},
\end{equation}
\noindent where, $X^{'} = f_1 v_{\Delta_{1}}, \Delta^{'}=f_2 v_{\Delta_{2}}. $
\noindent Finally, the neutrino mass matrix is $M_{\nu}=m_{\nu_{I}}+m_{\nu_{II}}$ and can, explicitly, be written as\\\\
\begin{equation}
M_{\nu}=
\begin{pmatrix}
X+X^{'}& 0 & \Delta + \Delta^{'} \\
0 & \Delta^{'} & X^{'} \\
\Delta + \Delta^{'} & X^{'} & \Delta^{''}\
\end{pmatrix}.
\label{eq:11}
\end{equation}
\noindent In literature, there are several techniques used to reduce the parameters of neutrino mass matrix, and texture zeros is one of them \cite{texture01,texture02,texture03,texture04,texture05,text006,text07,text08,text09,text010,text011,text012,text013,text014,text015,text016,text017,text018,text019,text020}. Interestingly, the mass matrix in Eqn.(11) corresponds to texture one-zero neutrino mass model. The phenomenological implications of these class of models have been extensively studied in the literature \cite{text015,text016,text017,text018}. Rather, in the present work we study the prediction of the current setup for beyond neutrino sector observable like DM and $0\nu\beta\beta$ decay discussed below.
\section{Relic Density of Dark Matter}\label{sec:3}
In the early universe, the particles were in thermal equilibrium i.e. the processes in which the lighter particles combine to form heavy particles and vice-versa happened at same rate. At some point of time, the conditions required for thermal equilibrium were contravened because the density of some particle species became too low. These particles are stated as ``freeze-out" and they have a constant density which is known as relic density, because the abundance of particle remains same. In the process, if any particle $\chi$ was in thermal equilibrium, then its relic abundance can be obtained by using Boltzmann equation\cite{relic,relic1,relic2,relic3}
\begin{equation}
\frac{dn_{\chi}}{dt} +3\mathcal{H}n_{\chi}= -<\sigma v>(n_{\chi}^2 - (n_{\chi}^{eqb})^2),
\label{eq:12}
\end{equation}
\noindent where $\mathcal{H}$ is the Hubble constant and $ n_{\chi}$ is the number density of the DM particle $ \chi$. Here, $n_{\chi}^{eqb}$ is the number density of particle $ \chi$ when it was in thermal equilibrium. However, $<\sigma v>$ is the thermally averaged annihilation cross-section of the DM particle. For a DM particle with electroweak scale mass, the solution of above Eqn.(\ref{eq:12}) gives \cite{approx}
\begin{equation}
\Omega_{\chi}h^2 = \frac{3 \times 10^{-27} cm^3 s^{-1 }}{<\sigma v>},\\
\label{eq:13}
\end{equation}
\noindent where $\Omega_{\chi}h^2$ gives the relic density of DM particle.
\noindent From the Lagrangian in Eqn.(\ref{eq:9}), the interaction of dark matter particle with right-handed neutrinos is as shown in Fig.\ref{fig1}. The cross-section formula for this kind of process is given as \cite{cross}
\begin{equation}
<\sigma v> = \frac{v^2 y^4 m_{\chi}^2}{48 \pi (m_{\chi}^2+m_{\psi}^2)^2},
\label{eq:14}
\end{equation}
\noindent where $y$ is Yukawa coupling of the interaction between DM and fermions, $m_\psi$ and $m_{\chi}$ represent the mass of Majorana fermion and relic particle mass respectively. Here, $ v$ is the relative velocity of two relic particles and is taken to be 0.3c at freeze out temperature. In case of m$_{DM}<M_W$, which indicates the low mass scale of relic particle, $\eta_2$,$\eta_3$ self annihilates via SM Higgs into the SM particles as shown in Fig.\ref{fig2}. The self annihilation cross-section is thus given as below\cite{relic2,cross1}\\
\begin{equation}
\sigma_{xx} = (\frac{|Y_f|^2 |\lambda_x|^2}{16 \pi s})(\frac{(s-4m_{f}^2)^{3/2}}{(s-4m_{x}^2)^{1/2}((s-4m_{h}^2)^2 + m_{h}^2 \Gamma_h^2)}),
\label{eq:15}
\end{equation}
\noindent where $ Y_f$ is Yukawa coupling of fermions and we have used its recent value as 0.308\cite{das}. Here, $ \Gamma_h$ is the SM Higgs decay width and its value used is 4.15 MeV. The $m_h$ is Higgs mass, that is 126 GeV, and $ x$ in Eqn.(\ref{eq:15}) represents $\eta_2,\eta_3$ and coupling of $ x$ with SM Higgs is represented as $\lambda_x$. Here, $s$ is thermally averaged center of mass squared energy and is given as \cite{cross1}\\
\begin{equation}
s = 4m_{\chi}^2 + m_{\chi}^2v^2.\\
\label{eq:16}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.8]{1.eps}
\end{center}
\caption{Scattering of DM particle $\eta_{2,3}$.}
\label{fig1}
\end{figure}
\begin{figure}[t]
\begin{center}
{\epsfig{file=2.eps,height=3.2cm,width=7.0cm}
\epsfig{file=3.eps,height=3.0cm,width=7.0cm}}
\end{center}
\caption{\label{fig2}Self annihilation of DM particle $\eta_{2,3}$ \cite{fig}.}
\end{figure}
\noindent The neutral component of scalar triplet $ \eta$ is our DM candidate as considered in \cite{52,53}. In this work we fixed our parameters in Eqn.(\ref{eq:13}) to obtain the recently updated constraints on relic abundance as reported by PLANCK 2018 data. To obtain the correct relic density of DM, we need to constrain the parameters like, Yukawa coupling, relic mass and mediator mass(right-handed neutrinos in our case). As stated above, we chose the relic mass much less than the mass of W-Boson. We have done our analysis for different values of relic mass and obtained different mediator masses and Yukawa couplings, which are shown in Fig.\ref{fig3}. This type of studies have already been done in \cite{cross,55}. In order to get correct relic abundance, we did our analysis for DM particle mass around 50 GeV, as suggested by many experimets like XENON1T\cite{Xenon}, PandaX-11\cite{Panda}, LUX\cite{Lux}, SuperCDMS\cite{CDMS} etc. For DM particle mass 45 GeV, 50 GeV and 55 GeV, we obtained the mass of right-handed neutrinos ranging from 138 GeV to 155 GeV. Yukawa coupling is obtained in the range 0.995-1. The results are shown in Table \ref{table3}.
\begin{figure}[t]
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[scale=.6]{dark1.eps}
\caption{}
\end{subfigure}
~\qquad
\hspace{.5cm}
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[scale=.6]{dark2.eps}
\caption{}
\end{subfigure}
\vspace{1cm}
\begin{center}
\hspace{-2.5cm}
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[scale=.6]{dark3.eps}
\caption{}
\end{subfigure}
\end{center}
\caption{Relic density of DM vs Yukawa coupling plots for (a)DM mass(m$_{\chi})$ = 50 GeV, (b) DM mass(m$_{\chi})$ = 45 GeV and (c) DM mass(m$_{\chi})$ = 55 GeV.}
\label{fig3}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
S.No. & Relic Mass(m$_{\chi}$) & Mediator Mass(m$_{\psi}$) & Yukawa Coupling($y$) \\
\hline
1 & 45 GeV & 138 GeV & 0.995-1\\
\hline
2 & 50 GeV & 146 GeV & 0.998-1 \\
\hline
3 & 55 GeV & 155 GeV & 0.996-1\\
\hline
\end{tabular}
\caption{Constraints on relic(DM) mass, right-handed neutrino mass and Yukawa Coupling.}
\label{table3}
\end{center}
\end{table}
\section{Neutrinoless Double Beta($0\nu\beta\beta$) Decay }\label{sec:4}
In general, the complex symmetric low energy effective neutrino mass matrix is given by
\begin{equation}
M_\nu=
\begin{pmatrix}
M_{ee} & M_{e\mu} & M_{e\tau} \\
M_{e\mu} & M_{\mu\mu} & M_{\mu\tau} \\
M_{e\tau} &M_{\mu\tau}& M_{\tau\tau} \\
\end{pmatrix}.
\label{eq:17}
\end{equation}
On comparing Eqn.(\ref{eq:17}) with the mass matrix in Eqn.(\ref{eq:11}) the effective Majorana neutrino mass appearing in $0\nu\beta\beta$ decay is
\begin{equation}
M_{ee}=X+X^{'},
\label{eq:18}
\end{equation}
and
\begin{equation}
M_{\mu\tau}=X^{'},
\label{eq:19}
\end{equation}
\begin{equation}
M_{e\mu}=0,
\label{eq:20}
\end{equation}
where $X$ and $X^{'}$ are inverse and type-II seesaw contributions to $0\nu\beta\beta$ decay. Eqn.(\ref{eq:20}) is our constraining condition to ascertain the allowed parameter space of the model. The elements of the mass matrix in Eqn.(\ref{eq:17}) are functions of low energy variables such as neutrino mass and mixing parameters and $CP$ violating phases. Using available data (Table \ref{table4}) for the known parameters and freely varying the unknown phases, we calculate $|M_{ee}|$ employing the formalism discussed below. Also, knowing $M_{\mu\tau}$ using data given in Table \ref{table4}, type-II contribution ($X^{'}$) can, independently, be calculated with the help of Eqn.(\ref{eq:19}). In this way, using Eqns.(\ref{eq:18}) and (\ref{eq:19}) we have been able to find the inverse ($X$) and type-II seesaw ($X^{'}$) contributions to $0\nu\beta\beta$ decay amplitude $|M_{ee}|$.
\noindent In the charged lepton basis, the Majorana neutrino mass matrix can be written as
\begin{equation}
M_{\nu}=UM_{d}U^T,
\end{equation}
where $M_{d}$ is diagonal mass matrix containing mass eigenvalues of neutrinos\\
$diag(m_{1}, m_{2}, m_{3})$.
$ U$ is neutrino mixing matrix defined as $U=V.P$
where $ P$ is diagonal phase matrix $diag(1,e^{i\alpha},e^{i(\beta+\delta)})$. In PDG representation, $V$ is given by
\begin{equation}
\begin{pmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\
-s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i\delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i\delta} & s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i\delta} & -c_{12} s_{23} -s_{12} c_{23} s_{13} e^{i\delta} & c_{23} c_{13} \\
\end{pmatrix},
\end{equation}
where $\delta$ is Dirac $CP$ violating phase and $\alpha$, $\beta$ are Majorana type $CP$ violating phases. The constraining condition (Eqn.(\ref{eq:20})) can be written as
\begin{equation}
c_{13} (e^{i(2 \beta + \delta)} m_3 s_{13} s_{23} -
c_{12} m_1 (c_{23} s_{12} + c_{12} e^{i\delta} s_{13} s_{23}) +
e^{2i\alpha} m_2 s_{12} (c_{12} c_{23} - e^{i\delta} s_{12} s_{13} s_{23}))=0.
\end{equation}
The above complex constraining equation gives two real constraints which can be solved for ratios $m_{2}/m_{1} \equiv R_{21}$ and $m_{3}/m_{1} \equiv R_{31}$ as
\begin{equation}
R_{21}=\frac{c_{12} (c_{12} s_{13} s_{23} \sin2 \beta+ c_{23} s_{12} \sin(2 \beta + \delta))}{s_{12} (s_{12} s_{13} s_{23} \sin2 (\alpha - \beta) -
c_{12} c_{23} \sin(2 \alpha - 2 \beta - \delta))},
\end{equation}
and
\begin{equation}
R_{31}=\frac{c_{12} (c_{12} s_{12} (-c_{23}^2 + s_{13}^2 s_{23}^2) \sin2 \alpha +
c_{23} s_{13} s_{23} (-c_{12}^2 \sin(2 \alpha - \delta) +
s_{12}^2 \sin(2 \alpha + \delta)))}{s_{13} s_{23} (s_{12} s_{13} s_{23} \sin2 (\alpha - \beta) -
c_{12} c_{23} \sin(2 \alpha - 2 \beta - \delta))}.
\end{equation}
Using these mass ratios, we have two different values of lowest eigenvalue $m_1$, equating them gives us
\begin{equation}
R_{\nu}\equiv\frac{\Delta m_{21}^2}{|\Delta m_{32}^2|}=\frac{R_{21}^2-1}{|R_{31}^2+R_{21}^2-2|}.
\end{equation}
The parameter space is constrained using the 3$\sigma$ range of $ R_{\nu}$. Neutrino oscillation parameters ($\theta_{12}$, $\theta_{23}$, $\theta_{13}$, $\Delta m_{21}^2$, $\Delta m_{32}^2$) are randomly generated with Gaussian distribution while $CP$ phases ($\delta$,$\alpha$,$\beta$) are varied randomly in their full range with uniform distribution. \\
The neutrino masses can be obtained using mass-squared differences as\\
\begin{center}
$ m_2=\sqrt{m_1^{2}+\Delta m_{21}^{2}}$,\hspace{.4cm}$m_3=\sqrt{m_1^{2}+\Delta m_{31}^{2}}$ \hspace{0.3cm} for Normal ordering (NO),\\
\end{center}
and
\begin{center}
$m_1=\sqrt{m_{3}^{2}-\Delta m_{31}^{2}}$ ,\hspace{.4cm} $m_2=\sqrt{m_3^{2}-\Delta m_{31}^{2}+\Delta m_{21}^{2}}$ \hspace{0.3cm} for Inverted ordering (IO),
\end{center}
whereas the lightest neutrino mass ($m_1$(NO), $m_3$(IO)) is obtained from mass ratios.
Also, we calculate the effective mass parameter
\begin{equation}
|M_{ee}|=\left|m_{1} c_{12}^2 c_{13}^2 + m_{2} s_{12}^2 c_{13}^2 e^{2i \alpha}+ m_{3} s_{13}^2 e^{2i \beta}\right|,
\end{equation}
and $|M_{\mu\tau}|$ for the allowed parameter space which are, further, used to find inverse ($|X|$) and type-II ($|X^{'}|$) seesaw contribution to $|M_{ee}|$.
\begin{table}[t]
\begin{center}
\begin{tabular}{c|c|c}
\hline \hline
Parameter & Best fit $\pm$ \( 1 \sigma \) range & \( 3 \sigma \) range \\
\hline \multicolumn{2}{c} { Normal neutrino mass ordering \( \left(m_{1}<m_{2}<m_{3}\right) \)} \\
\hline \( \sin ^{2} \theta_{12} \) & $0.304^{+0.013}_{-0.012}$ & \( 0.269-0.343 \) \\
\( \sin ^{2} \theta_{13} \) & $0.02221^{+0.00068}_{-0.00062}$ & \( 0.02034-0.02420 \) \\
\( \sin ^{2} \theta_{23} \) & $0.570^{+0.018}_{-0.024}$ & \( 0.407-0.618 \) \\
\( \Delta m_{21}^{2}\left[10^{-5} \mathrm{eV}^{2}\right] \) & $7.42^{+0.21}_{-0.20}$& \( 6.82-8.04 \) \\
\( \Delta m_{31}^{2}\left[10^{-3} \mathrm{eV}^{2}\right] \) & $+2.541^{+0.028}_{-0.027}$ & \( +2.431-+2.598 \) \\
\hline \multicolumn{2}{c} { Inverted neutrino mass ordering \( \left(m_{3}<m_{1}<m_{2}\right) \)} \\
\hline \( \sin ^{2} \theta_{12} \) & $0.304^{+0.013}_{-0.012}$ & \( 0.269-0.343 \)\\
\( \sin ^{2} \theta_{13} \) & $0.02240^{+0.00062}_{-0.00062}$ & \( 0.02053-0.02436 \) \\
\( \sin ^{2} \theta_{23} \) & $0.575^{+0.017}_{-0.021}$& \( 0.411-0.621 \) \\
\( \Delta m_{21}^{2}\left[10^{-5} \mathrm{eV}^{2}\right] \) & $7.42^{+0.21}_{-0.20}$ & \( 6.82-8.04 \) \\
\( \Delta m_{32}^{2}\left[10^{-3} \mathrm{eV}^{2}\right] \) & $-2.497^{+0.028}_{-0.028}$ & \( -2.583--2.412 \) \\
\hline \hline
\end{tabular}
\end{center}
\caption{Neutrino oscillations experimental data NuFIT 5.0 used in the numerical analysis\cite{data}.}
\label{table4}
\end{table}
\begin{figure}[t]
\begin{center}
{\epsfig{file=4af.eps,height=7.0cm,width=7.0cm}
\epsfig{file=4bf.eps,height=7.0cm,width=7.0cm}}
\end{center}
\caption{\label{fig:4} Variation of effective Majorana neutrino mass $|M_{ee}|$ with Contribution of inverse seesaw and type-II seesaw model parameters ($|X|$ and $|X'|$), for both normal and inverted ordering.}
\end{figure}
\noindent In Fig.\ref{fig:4}, we have plotted effective mass parameter $|M_{ee}|$ with inverse seesaw contribution (X) and type-II seesaw contribution (X$'$). The sensitivity reach of 0$\nu\beta\beta$ decay experiments like SuperNEMO \cite{nemo}, KamLAND-Zen \cite{zen}, NEXT \cite{next1,next2},
nEXO\cite{nexo} is, also, shown in Fig.\ref{fig:4}. In Fig.\ref{fig:4}(a), it is clear that for NO, higher density of points for type-II seesaw indicate contribution of $\mathcal{O}(0.01eV)$, whereas inverse seesaw contribution is of $\mathcal{O}(0.001eV)$. On the other hand, for IO, different seesaw contributions are of same order as can be seen in Fig.\ref{fig:4}(b). Hence, type-II seesaw contribution to $|M_{ee}|$ is large as compared to inverse seesaw contribution for NO for the texture one-zero model within the framework of inverse seesaw and type-II seesaw. For normal ordering neutrino mass spectrum, the $|M_{ee}|$ goes below upto the $\mathcal{O}(10^{-4}eV)$ and we did not obtain a clear lower bound as shown in Fig.\ref{fig:4}(a). On contrary, for inverted ordering, there is a clear cut lower bound for the $|M_{ee}|$ which can be probed in future $0\nu\beta\beta$ decay experiments. It can be seen from Fig.\ref{fig:4}(b), that the $0\nu\beta\beta$ decay experiments like SuperNEMO, KamLAND-Zen can probe the inverted ordering spectrum.
\begin{figure}[H]
\begin{center}
{\epsfig{file=5af.eps,height=7.0cm,width=7.0cm}
\epsfig{file=5bf.eps,height=7.0cm,width=7.0cm}}
\end{center}
\caption{\label{fig:5} Variation of effective Majorana neutrino mass $|M_{ee}|$ with lightest neutrino mass $m_1$($m_3$) for both normal(inverted) ordering of neutrino masses.}
\end{figure}
\begin{figure}[H]
\begin{center}
{\epsfig{file=6af.eps,height=7.0cm,width=7.0cm}
\epsfig{file=6bf.eps,height=7.0cm,width=7.0cm}}
{\epsfig{file=6cf.eps,height=7.0cm,width=7.5cm}
\epsfig{file=6df.eps,height=7.0cm,width=7.0cm}}
\end{center}
\caption{\label{fig:6} Correlation plots of ($\alpha$-$R_{21}$) and ($\delta$-$\beta$). First(second) row depicts the correlations for normal(inverted) ordering of neutrino masses.}
\end{figure}
\noindent In Fig.\ref{fig:5}, we have depicted the correlation of effective Majorana mass parameter $|M_{ee}|$ with the lightest neutrino mass $m_1 (m_3)$ for normal (inverted) ordering of neutrino masses. $|M_{ee}|$ goes below up to 0.001eV for normal ordering whereas there exist a lower bound near 0.04eV for inverted ordering. The sensitivity reaches of various current and future $0\nu\beta\beta$ decay experiments have, also, been shown in Fig.\ref{fig:5}. Although, for normal ordering, $|M_{ee}|$ is beyond the sensitivity reach of experiments but non-observation $0\nu\beta\beta$ decay shall rule out the inverted ordering in this model.
\noindent Furthermore, we have studied the correlations amongst the Majorana phases($\alpha,\beta$), Dirac phase($\delta$) and mass ratio($R_{21}\equiv\frac{m_2}{m_1}$). Some representative plots are given in Fig.\ref{fig:6}. The correlation plots ($\alpha-R_{21}$) and ($\delta-\beta$) are shown in Fig.\ref{fig:6}(a)(\ref{fig:6}(c)) and Fig.\ref{fig:6}(b)(\ref{fig:6}(d)), respectively for normal(inverted) mass ordering. For normal ordering, there is no preferable range of CP-phase(Dirac or Majorana) whereas, in case of inverted ordering, Majorana CP-phase $\alpha$ is constrained in the range($0^{\circ}-20^{\circ}) \cup (160^{\circ}-180^{\circ})$ and $\delta$ is found to lie in $(40^{\circ}-150^{\circ})\cup (200^{\circ}-325^{\circ}$) range. In Fig.\ref{fig:7}, we have depicted the correlation of neutrino mixing angles with mass ratio $R_{21}$ for normal(inverted) ordering. The model satisfies the neutrino oscillation data for both mass ordering. The model predictions are consistent with the model independent analysis performed in Ref.\cite{text015}.
\begin{figure}[t]
\begin{center}
{\epsfig{file=MixNO.eps,height=7.0cm,width=7.0cm}
\epsfig{file=MixIO.eps,height=7.0cm,width=7.0cm}}
\end{center}
\caption{\label{fig:7} Correlation plots of ($\theta_{ij}$-$R_{21}$) for normal(inverted) ordering of neutrino masses. Horizontal dashed lines are $3\sigma$ experimental range of respective mixing angles. }
\end{figure}
\section{Conclusions}\label{sec:5}
In conclusion, we have proposed a model based on $A_4$ discrete flavor symmetry implementing inverse and type-II seesaw mechanisms to have LHC accessible TeV scale right-handed neutrino mass and texture one-zero in the resulting Majorana neutrino mass matrix, respectively. The symmetry is broken down to $Z_2$ subgroup i.e. $G_f\equiv A_4\times Z_{2}^{'}\times Z_3\times Z_4\rightarrow Z_2$ by the $vev$ $\langle \eta \rangle \sim v_{\eta} (1,0,0)$ which, further stabilizes the DM candidate $\eta_{2,3}$. We scanned the ranges of Yukawa coupling ($y$), right-handed neutrino mass ($m_\psi$) and DM mass ($m_{\chi}$) to obtain observed relic abundance of DM $\eta_{2,3}$. We find that to have observed relic abundance the $m_\chi$, $m_\psi$ and $y$ should be around 50 GeV, 146 GeV and 0.998-1, respectively. Also, we have obtained the prediction of the model for $0\nu\beta\beta$ decay amplitude $|M_{ee}|$. In particular, we calculated inverse ($X$) and type-II ($X^{'}$) seesaw contributions to $|M_{ee}|$, while texture one-zero model being consistent with low energy experimental constraints. The type-II seesaw contribution to $|M_{ee}|$ is found to be large as compared to inverse seesaw contribution for normal ordering neutrino masses. The model predicts a robust lower bound on $|M_{ee}|$ for inverted ordering neutrino masses which can be probed in future $0\nu\beta\beta$ decay experiments like SuperNEMO, KamLAND-Zen. The correlation of $|M_{ee}|$-$m_1(m_3)$ shows that inverted ordering is ruled out in case of non-observation of $0\nu\beta\beta$ decay while normal ordering may still be allowed. Also, contradistinction to normal ordering, the CP-violating phases $\delta$ and $\alpha$ are constrained in the range ($0^{\circ}-20^{\circ}) \cup (160^{\circ}-180^{\circ})$ and $(40^{\circ}-150^{\circ})\cup (200^{\circ}-325^{\circ}$), respectively, for inverted ordering of neutrino masses.\\
\hspace{-.4cm}\textbf{\Large{Acknowledgments}}
\vspace{.3cm}\\
R. Verma acknowledges the financial support provided by the Central University of Himachal Pradesh. B. C. Chauhan is thankful to the Inter University Centre for Astronomy and Astrophysics (IUCAA) for providing necessary facilities during the completion of this work. M. K. acknowledges the financial support provided by Department of Science and Technology, Government of India vide Grant No. DST/INSPIRE Fellowship/2018/IF180327.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 9,973
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{"url":"https:\/\/cyclostationary.blog\/2021\/01\/07\/sptk-interconnection-of-linear-systems\/","text":"# SPTK: Interconnection of Linear Systems\n\nReal-world signal-processing systems often combine multiple kinds of linear time-invariant systems. We look here at the general kinds of connections.\n\nPrevious Post: Frequency Response Next Post: Convolution\n\nIt is often the case that linear time invariant (or for discrete-time systems, linear shift invariant) systems are connected together in various ways, so that the output of one may be the input to another, or two or more systems may share the same input. In such cases we can often find an equivalent system impulse response that takes into account all the component systems. In this post we focus on the serial and parallel connections of LTI systems in both the time and frequency domains.\n\n[Jump straight to \u2018Significance of Connected LTI Systems in CSP\u2019 below.]\n\n### Time-Domain Analysis\n\nLet\u2019s first look at a serial connection of two systems, then we\u2019ll move on to a parallel connection. A serial connection of two systems just means that the output of the first system forms the input to the second system.\n\nLet\u2019s go through the mathematics that describes an equivalent system for the serially connected systems in Figure 1. As we progress we\u2019ll present results with a bit less detail, but for this first example of interconnected systems, we\u2019ll look at all the details.\n\nThe basic mathematical problem is to find an expression for the response $z(t)$ in terms of the input $x(t)$ and the two impulse response functions $h_j(t)$. Using what we\u2019ve already established about linear time-invariant systems in the Signal Processing ToolKit posts, we can write the following two expressions\n\n$\\displaystyle y(t) = h_1(t) \\otimes x(t) \\hfill (1)$\n\nand\n\n$\\displaystyle z(t) = h_2(t) \\otimes y(t) \\hfill (2)$\n\nwhere $\\otimes$ denotes convolution. We are looking for an expression that does not explicitly contain $y(t)$. Let\u2019s do the obvious and substitute $y(t)$ from (1) into (2).\n\n$\\displaystyle z(t) = \\int_{-\\infty}^\\infty y(v) h_2(t-v)\\, dv \\hfill (3)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty \\left[ \\int_{-\\infty}^\\infty x(u) h_1(v-u)\\, du \\right] h_2(t-v)\\, dv \\hfill (4)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty x(u) h_1(v-u) h_2(t-v)\\, du dv \\hfill (5)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty x(u) h_1(v^\\prime) h_2(t-v^\\prime -u)\\, dv^\\prime du \\hfill (6)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty x(u) \\left[ \\int_{-\\infty}^\\infty h_1(v^\\prime) h_2((t-u) - v^\\prime) \\, dv^\\prime \\right] \\, du \\hfill (7)$\n\nand we notice that the inner integral is the function\n\n$\\displaystyle g(t^\\prime) = h_1(t^\\prime) \\otimes h_2(t^\\prime)$\n\nevaluated at $t^\\prime = t-u$. So our expression becomes\n\n$\\displaystyle z(t) = \\int_{-\\infty}^\\infty x(u) g(t-u) \\, du \\hfill (8)$\n\n$\\displaystyle = x(t) \\otimes g(t) \\hfill (9)$\n\nwhere\n\n$\\displaystyle g(t) = h_1(t) \\otimes h_2(t) \\hfill (10)$\n\nThe equivalent system has an impulse-response function that is the convolution of the two individual impulse-response functions $h_1(t)$ and $h_2(t)$, as illustrated in Figure 2.\n\nWe can use induction to prove that the serial connection of $N$ LTI systems has an impulse-response function that is equal to the convolution of the $N$ component-system impulse-response functions, as shown in Figure 3.\n\nNow let\u2019s turn to the parallel connection of two linear time-invariant systems, as depicted in Figure 4.\n\nThe math is simpler here:\n\n$\\displaystyle y_1(t) = \\int_{-\\infty}^\\infty x(u) h_1(t-u) \\, du \\hfill (11)$\n\n$\\displaystyle y_2(t) = \\int_{-\\infty}^\\infty x(u) h_2(t-u) \\, du \\hfill (12)$\n\n$\\displaystyle z(t) = y_1(t) + y_2(t) \\hfill (13)$\n\nThe equivalent system is found by adding $y_1(t)$ to $y_2(t)$,\n\n$\\displaystyle z(t) = \\int_{-\\infty}^\\infty x(u) h_1(t-u) \\, du + \\int_{-\\infty}^\\infty x(u) h_2(t-u) \\, du \\hfill (14)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty x(u) \\left[ h_1(t-u) + h_2(t-u) \\right] \\, du \\hfill (15)$\n\n$\\displaystyle = \\int_{-\\infty}^\\infty x(u) g(t-u) \\, du \\hfill (16)$\n\nwhere\n\n$\\displaystyle g(t) = h_1(t) + h_2(t) \\hfill (17)$\n\nSo the equivalent system for a parallel connection is simply the sum of the two component systems\u2019 impulse response functions. For the parallel connection of $N$ linear time-invariant systems $h_k(t)$, we have the equivalent impulse response\n\n$\\displaystyle g(t) = \\sum_{k=1}^N h_k (t) \\hfill (18)$\n\nwhich is illustrated in Figure 5.\n\nThe equivalent systems for serial and parallel connections of discrete-time linear shift-invariant systems follow the same rules as those above for continuous-time linear time-invariant systems. The mathematics is nearly identical, with convolution integrals replaced by convolution sums.\n\n### Frequency-Domain Analysis\n\nWe can gain more insight into the nature of the output of an interconnected set of systems if we look at the input-output relations in the frequency domain. Starting again with the serial connection of two linear time-invariant systems (Figure 1), we have the temporal input-output relation\n\n$\\displaystyle z(t) = h_1(t) \\otimes h_2(t) \\otimes x(t) \\hfill (19)$\n\nwhere $\\otimes$ denotes convolution. We\u2019ve already established in the Signal Processing Toolkit that the Fourier transform of a signal that is equal to the convolution of two functions is the product of the Fourier transforms of those two functions: convolution in the time domain is multiplication in the frequency domain. This means that (19) can be easily transformed to yield the input-output relationship for the serial connection of two linear time-invariant systems given by\n\n$\\displaystyle Z(f) = H_1(f)H_2(f)X(f) = G(f)X(f) \\hfill (20)$\n\nwhich is illustrated in Figure 6.\n\nUsing the result shown in Figure 3, we find that the equivalent transfer function for the serial connection of $N$ LTI systems with transfer functions $H_k(f)$ is simply their product,\n\n$\\displaystyle G(f) = \\prod_{k=1}^N H_k(f) \\hfill (21)$\n\nas illustrated in Figure 7.\n\nIt should now come as no surprise (Equation (18)) that the parallel connection of $N$ linear time-invariant systems has an equivalent transfer function that is simply the sum of the individual-system transfer functions, as shown in Figure 8.\n\n### Interpretations\n\n#### Serial ConnECTiON\n\nA linear time-invariant system is completely characterized (in terms of determining the output for any input) by its impulse-response function. The Fourier transform of the impulse-response function also, therefore, completely characterizes the system, and that function is called the transfer function. The output $Y(f)$ is related to the input $X(f)$ through multiplication by the transfer function,\n\n$\\displaystyle Y(f) = H(f) X(f) \\hfill (22)$\n\nSo a spectral component of the input, $X(f_0)$, is scaled by the transfer function evaluated at the frequency of the spectral component, $H(f_0)$, to produce the spectral component at the output $Y(f_0)$. Since $H(f_0)$ can be a complex number, both the magnitude and phase of the spectral component in $x(t)$ can be modified by the system, delivering (\u2018transferring\u2019) the spectral component $H(f_0)X(f_0)$ to $y(t)$.\n\nIn the case of the serial connection of $N$ linear time-invariant systems, the scaling of the input spectral component $X(f_0)$ is simply the multiplication of all the individual scaling functions from the connected systems, or $\\displaystyle \\prod_{k=1}^N H_k(f_0)$. To annihilate the spectral component at $f=f_0$ in the output, for example, all that is required is that at least one of the component systems has a transfer function that is zero at $f_0$.\n\n#### Parallel Connection\n\nIn the case of the parallel connection of $N$ linear time-invariant systems, the equivalent impulse-response function is the sum of the component-system impulse-response functions, and so the equivalent transfer function is the sum of the component-system transfer functions. The spectral component in the input $x(t)$ with frequency $f_0$ is scaled by the sum of the component-system transfer functions, or $\\displaystyle \\sum_{k=1}^N H_k(f_0)$. To annihilate a spectral component of the input so that it does not appear in the output requires that this sum be zero. A sufficient, but not necessary, way to do that is to make sure each $H_k(f_0)$ is zero.\n\n### Examples\n\n#### Bandpass Filter\n\nA bandpass filter is a linear time-invariant system that has a transfer function that is zero (or very small) for all input frequencies $f$ except those in the two intervals\n\n$\\displaystyle [-f_{h}, -f_{l}]$\n\nand\n\n$\\displaystyle [f_l, f_h]$\n\nwhere $f_h > f_l$. Only a band of frequencies is passed through the system, not all frequencies. We can construct a bandpass filter from a lowpass filter and a highpass filter in serial connection. A lowpass filter passes all frequencies in the interval $[-f_{lpf}, f_{lpf}]$ and a highpass filter passes all frequencies in the two intervals\n\n$\\displaystyle [-\\infty, -f_{hpf}]$\n\nand\n\n$\\displaystyle [f_{hpf}, \\infty]$\n\nConsider the serial connection of a lowpass filter $h_3(t)$ and a highpass filter $h_4(t)$ shown in Figure 9. The maximum frequency that is passed by the lowpass filter $h_3(t)$ (the `cutoff frequency\u2019) is $f_3$, and the minimum frequency that is passed by the highpass filter $h_4(t)$ (its cutoff frequency) is $f_4$. Crucially, $f_3 > f_4$. When we connect these two linear time-invariant systems serially, we multiply their transfer functions, obtaining the transfer function $G(f) = H_3(f)H_4(f)$ shown in the figure. Here, $f_l = f_4$ and $f_h = f_3$.\n\n#### Notch Filter\n\nThe inverse of a bandpass filter is called a notch filter or bandstop filter. Such a linear time-invariant system passes all frequencies except those in the two intervals\n\n$\\displaystyle [-f_{h}, -f_{l}]$\n\nand\n\n$\\displaystyle [f_l, f_h]$\n\nThe notch filter can be created by the parallel connection of a suitable lowpass filter with a highpass filter, as shown in Figure 10. Here we must have $f_1 < f_2$. The equivalent system is the sum of the two transfer functions, or $G(f) = H_1(f) + H_2(f)$.\n\n#### Complex Arrangement of Filters\n\nIn this final example of the interconnection of linear time-invariant systems, consider the arrangement of seven different systems shown in Figure 11. Just think if we had to figure out the equivalent system by dealing with the time-domain convolutions. It would be a mess of integrals.\n\nInstead, let\u2019s look at it through the frequency-domain lens of the transfer function.\n\n$\\displaystyle Y_1(f) = X(f) H_1(f)H_2(f) \\hfill (23)$\n\n$\\displaystyle Y_2(f) = [H_3(f) + H_4(f)]Y_1(f) \\hfill (24)$\n\n$\\displaystyle \\Rightarrow Y_2(f) = X(f) \\left[ H_1(f)H_2(f)\\left[ H_3(f) H_4(f)\\right] \\right] \\hfill (25)$\n\n$\\displaystyle Y_3(f) = H_1(f)H_5(f)H_6(f) \\hfill (26)$\n\n$\\displaystyle Y_4(f) = Y_3(f) + X(f)H_1(f)H_7(f) \\hfill (27)$\n\n$\\displaystyle \\Rightarrow Y_4(f) = X(f)H_1(f)H_5(f)H_6(f) + X(f)H_1(f)H_7(f) \\hfill (28)$\n\n$\\displaystyle Z(f) = Y_4(f) + Y_2(f) \\hfill (29)$\n\n$\\displaystyle = X(f) \\left( H_1(f) H_2(f) [H_3(f) + H_4(f)] \\right)$\n\n$\\displaystyle + X(f) \\left( H_1(f) H_5(f) H_6(f) + H_1(f)H_7(f) \\right) \\hfill (30)$\n\n$\\displaystyle \\Rightarrow G(f) = Z(f)\/X(f) = H_1(f) \\left[ H_2(f)(H_3(f) + H_4(f)) + H_5(f) H_6(f) + H_7(f) \\right] \\hfill (31)$\n\n### Significance of Connected LTI Systems in CSP\n\nCSP is mostly about nonlinear, as opposed to linear, mathematical operations, so that the role of LTI systems and their interconnected versions mostly has to do with their effects on a signal prior to application of CSP. A canonical example involves a pulse-amplitude-modulated signal (such as PSK and QAM), with a pulse-shaping LTI system (filter) $P(f)$, a propagation-channel LTI system $H(f)$, and a reception LTI system $R(f)$. These systems are connected in series (the output of one forms the input of the next), so that the equivalent LTI system that connects the modulated impulse train at the transmitter to the received signal is $G(f) = P(f)H(f)R(f)$. This equivalent system can then be used to determine the spectral correlation function or the cyclic cumulants.\n\nPrevious Post: Frequency Response Next Post: Convolution","date":"2022-07-03 08:15:57","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\": 0, \"img_math\": 228, \"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\": 0.7095014452934265, \"perplexity\": 348.20622792296723}, \"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-2022-27\/segments\/1656104215805.66\/warc\/CC-MAIN-20220703073750-20220703103750-00144.warc.gz\"}"}
| null | null |
\section{\label{sec:1} Introduction}
As the development of optical clocks is currently pursued in many
laboratories worldwide \cite{Poli2013,Bloom2014,Chou2010,
Hinkley2013}, a broad range of applications is taking shape with
optical clocks on ground and in space employed for high precision
tests of fundamental physics \cite{Schiller2009, Wolf2009},
chronometric levelling-based geodesy \cite{Chou2010a}, improved RF
standards for navigation \cite{Fortier2011} and observations of
cosmic radio sources with very-long baseline interferometry (VLBI)
\cite{Rogers1983}. For all these purposes today's complex and
bulky optical-clock experimental setups need to be re-engineered
into more compact and power efficient systems, ensuring at the
same time a high stability and accuracy, but also high operation
reliability in critical environments \cite{Leibrandt2011}, such as
application in the field or even satellites.
A first step in the engineering challenge leading to space based
optical clocks is to demonstrate transportable clocks. These are
interesting because frequency comparisons of today's best optical
clocks cannot be done through satellite links and tests with
optical fiber links require dedicated equipment
\cite{Predehl2012,Williams2008,Calonico2014}. A frequency transfer
standard will allow for comparing clocks at the accuracy level
provided by the transportable clock \cite{Bize2005}.
In this paper we present the realization of a transportable
strontium optical clock, based on the integration of two
subsystems, a compact atomic source including a compact cooling
and trapping laser set-up, and a transportable clock laser,
providing the laser-cooled sample of strontium atoms and the
ultra-stable radiation source for interrogating the clock
transition, respectively. The clock laser, transported by van from
PTB in Braunschweig to LENS in Florence was employed to perform
high resolution spectroscopy of the clock transition
$^1$S$_0\,$-${}^3$P$_0$ of $^{88}$Sr in Lamb-Dicke regime and to
characterize systematic frequency shifts of the optical
transition.
In the following are presented in detail novel design solutions
employed for each subsystem that allowed us to reduce size, weight
and power consumption with respect to a traditional laser cooling
apparatus.
\section{\label{sec:2} The transportable clock setup}
\subsection{\label{subsec:strontium_source} Compact system for cooling and trapping strontium atoms}
The compact laser-cooling strontium source
mainly consists of the following modules: the cooling laser
sources, a frequency distribution breadboard and a vacuum system.
The three modules are connected through optical fibers and mounted
directly on a 120$\,$cm$\,\times\,$90$\,$cm breadboard. The
control electronics is hosted under the main breadboard in a 19"
rack (size 60$\,$cm$\,\times\,$60$\,$cm$\,\times\,$180$\,$cm).
Cooling and trapping of strontium atoms is performed through a
two-stage magneto-optical-trap (MOT) on the dipole allowed
$^1$S$_0\,$-${}^1$P$_1$ transition and on the intercombination
line $^1$S$_0\,$-${}^3$P$_1$, respectively (see Fig.
\ref{fig.LivelliSr}). All the employed lasers are based on
semiconductor diodes \cite{Poli2007, Poli2009}. The laser set is
composed of a frequency doubled 300$\,$mW 461$\,$nm laser, a
50$\,$mW 689 nm laser, a 420$\,$mW 813$\,$nm (master + tapered
amplifier) laser and two $\sim$20$\,$mW 679$\,$nm and 707$\,$nm
repumper lasers. These sources are used for the two-stage
laser-cooling and subsequent optical trapping in a one-dimensional
(1D) lattice at the magic wavelength \cite{Katori2009}.
\begin{figure}[t]\begin{center}
\includegraphics[width=0.5\textwidth]{livelliSrMOD2Rev2.eps}
\caption{Energy levels and relevant optical transition for neutral
Sr. The number near the level is the total angular momentum
$J$.\label{fig.LivelliSr}}
\end{center}
\end{figure}
All the optical beams for the first laser-cooling stage at 461 nm
are produced in a compact module (size
30$\,$cm$\,\times\,$40$\,$cm$\,\times\,$8$\,$cm).
The module contains ultra-stable mountings for mirrors,
polarization cubes, plates, lenses and acousto-optic-modulators
(AOMs) to which the 461 nm radiation is delivered by fibre
\cite{SchioppoPhDThesis}.
In order to reduce the number of beam shaping optical elements
used in the breadboard while maintaining a high diffraction
efficiency on AOMs, an input beam $1/e^2$ diameter has been set to
$0.5\,$mm. As a result we obtain high diffraction efficiency on
AOMs ($>$80 \% in single pass configuration) without the use of
any additional telescope, while still keeping the possibility of
resolving the diffraction orders at short distances. Moreover with
this choice a coupling efficiency into optical fibers of about 65
\% is achieved. Then, the overall efficiency of a typical $\sim
25\,$cm long optical path inside the breadboard, including the AOM
diffraction efficiency (in single-pass configuration) and the
coupling efficiency into the output fiber, is typically
$\sim50\,\%$. The frequency detunings from the
$^1$S$_0\,$-${}^1$P$_1$ resonance for Zeeman slower and MOT beams,
of $-320\,$MHz and $-40\,$MHz respectively, are obtained by using
two AOMs in single-pass configuration, driven at 170 MHz ($-$
order) and 110 MHz ($+$ order), respectively. All output beams can
also be completely shut off by fast and compact mechanical
shutters.
A beam for frequency stabilization of the first stage cooling
light is obtained from a double-pass configuration through an AOM
driven at 75$\,$MHz and sent to the atomic beam used to load the
MOT (see Fig. \ref{fig:vacuum_system}). In order to extract an
error signal from spectroscopy on the cooling transition, the
driving RF signal is also frequency modulated at $\sim$ 10$\,$kHz
with a peak to peak deviation of $\sim$10 MHz. A low power ($\sim$
1 mW) resonant beam for absorption imaging is generated in a
similar double pass configuration. For both double-pass AOMs a
cat's-eye configuration has been realized by focusing the optical
beam on the retro-reflecting mirror with a lens at distance f = 60
mm from AOM and retro-reflection mirror.
The atomic sample is trapped in a vacuum system (size
1200$\,$cm$\,\times\,$40$\,$cm $\times$ 36$\,$cm), see Fig.
\ref{fig:vacuum_system}) consisting of the oven region, where a
collimated Sr atomic beam is produced, a Zeeman slower and the
science region where the Sr atoms are subsequently Zeeman slowed,
trapped in a two-stage MOT and eventually transferred in a 1D
vertical optical lattice for clock spectroscopy. The oven region
is pumped by a 40 l/s ion pump (pressure during operation $\sim
10^{-6}$ Pa), the science region is evacuated by a 55 l/s ion pump
and a titanium sublimation pump (pressure $\sim10^{-7}$ Pa). A
differential pumping tube (internal diameter 5 mm, length 7.5 cm)
ensures to maintain the pressure difference between the two
regions.
In the oven region, atoms are sublimated by a compact and
efficient dispenser \cite{Schioppo2012} based on a heater placed
in vacuum, providing an atomic flux intensity of
$1.7\times10^{13}$ s$^{-1}$sr$^{-1}$ ($^{88}$Sr atoms) at the oven
temperature of $380\,^{\circ}$C with a total power consumption of
26 W. The atomic beam is collimated by a nozzle of about 120
capillaries (internal diameter 200$\,\mu$m, length $8\,$mm). The
high collimation ($\sim 40$ mrad) and flux allow the laser
stabilization to be performed with high signal-to-noise employing
a simple transverse fluorescence spectroscopy on the atomic beam
in the oven chamber (see Fig. \ref{fig:vacuum_system}).
The atomic beam propagates along a 23$\,$cm long tube externally
wrapped with coils for Zeeman slowing. The magnetic field shape
for Zeeman slowing has been designed to smoothly match the
off-axis component of the MOT coils' field (see Fig.
\ref{fig:slowing_dynamics_final}) \cite{SchioppoPhDThesis}. The
first part of the Zeeman slower (before the inversion of the field
given by the MOT coils) has a length of $18\,$cm and it operates
with a power consumption of 17$\,$W.
\begin{figure}[t]
\includegraphics[width=0.6 \textwidth]{vacuum_system_assembly_5.eps}
\caption{\label{fig:vacuum_system} Technical drawing of the
vacuum system (upper view and section side view) showing the main
part of the system (the oven region, on the left and the science
region, on the right.).}
\end{figure}
The decelerated atomic beam is trapped in a MOT at the center of a
science cell with eight CF40 and sixteen CF16 optical windows. Two
custom CF150 flanges host the pair of coils (in anti-Helmholtz
configuration as needed for MOT operation) outside the MOT chamber
2.6$\,$cm away from the atoms. In this configuration a magnetic
field gradient of $\sim 500$ mT/m is obtained with a total power
consumption of $\sim60$ W. With these levels of power consumption
for oven, Zeeman slower and MOT coils we could avoid the
complication of water cooling, with a typical chamber temperature
of 318 K.
To make the alignment of the MOT beams long-term stable, the
cooling beams at 461 nm and 689 nm are delivered by three dichroic
fiber-coupled beam expanders fastened to the science cell.
A system of setting screws and counter screws allows fine
alignment and locking of the MOT beams. Similarly, the MOT beam
retroreflectors, repumping, imaging and clock spectroscopy
output-couplers are fastened onto the science cell. The alignment
of the MOT beams has been maintained for more than one year
without any further adjustment.
\subsection{\label{subsec:clock_laser} Transportable clock laser}
The main clock laser breadboard is based on two diode lasers in a
master-slave setup. The master is a filter stabilized extended
cavity laser with resonator length 10 cm \cite{Baillard2006}.
About 0.5 mW of the master-laser power is used to injection lock
the slave laser. The breadboard has a total size of 60 cm $\times$
45 cm $\times$ 10 cm, its design is similar to frequency
distribution breadboard presented in the previous section. A 200
MHz AOM in double-pass configuration is used to bridge the
frequency gap between the optical resonator with free spectral
range of 1.5 GHz and the frequency of the clock transition. This
AOM is used to scan the frequency of the laser and also to enable
frequency feedback when the laser is locked to the clock
transition. Two output ports for clock spectroscopy and for
counting of the laser frequency provide a maximum optical power of
about 2 mW each. Both ports are provided with AOMs for switching
which can also be used to stabilize the optical path length of the
fibers. The interferometric setup for this stabilization is
included on the breadboard \cite{Falke2012}. The laser was locked
to a transportable high finesse cavity whose transportability had
already been demonstrated within the SOC project \cite{Vogt2011}.
\section{\label{sec:results} Experimental Results}
\subsection{Cooling and trapping}
In order to develop a more compact and low power consumption
experimental system for the production of ultra-cold strontium
atoms, several design solutions have been adopted, together with
an optimization of the efficiency of each cooling and trapping
step. In order to slow down the atoms sublimating from the oven
from 430$\,\text{m}/\text{s}$ to below 130$\,\text{m}/\text{s}$,
an optical beam at $461\,$nm, circularly polarized, is sent
counter propagating to the atomic beam. The laser beam has an
optical power of $30\,$mW, an initial $1/e^2$ radius of $r =
5\,$mm, focused after 1$\,$m to compensate the absorption of
photons in the slowing process. Atoms are kept in resonance during
the slowing by compensating the variation of Doppler shift with
the proper Zeeman shift from the magnetic field provided by the
Zeeman slower. The sample of cold $^{88}$Sr atoms is produced
through two laser-cooling stages. The first stage consists of a
MOT operating on the $^1$S$_0\,$-${}^1$P$_1$ transition at
$461\,$nm. This so called blue MOT is realized by three pairs of
counter-propagating optical beams, circularly polarized, detuned
by $\delta_L=-2\pi\times40\,$MHz, with a saturation parameter for
each beam of $s\sim1$ and a magnetic field gradient of $500\,$
mT/m. The blue MOT capture velocity is $
v_{\text{c}}=\sqrt{2a_{\text{max}}r}\simeq100\,\text{m/s}$ where
$a_{\text{max}}=\frac{1}{M}\hslash
k_{L}\frac{\Gamma}{2}\frac{s}{1+s}$ is the maximum acceleration
exerted by cooling light, with $k_L=2\pi/\lambda$ the wavevector
of the photons at $\lambda=461\,$nm, $\Gamma=2\pi\times32\,$MHz
the natural linewidth of the $^1$S$_0\,$-${}^1$P$_1$ transition
and $M$ the atomic mass of strontium 88. As expected we find that
the Zeeman slower increases the loading rate of the MOT, resulting
in a net increase of the blue MOT population by a factor of
$\sim40$ (see Fig. \ref{fig:BlueMOT_loading_cropped}).
\begin{figure}[t]
\includegraphics[width=0.6 \textwidth]{Blue_MOT_loading_cropped_3Rev.eps}
\caption{\label{fig:BlueMOT_loading_cropped} Comparison among blue
MOT loading curves and final blue MOT population obtained with
repumpers and Zeeman slower beam (red dash), without Zeeman slower
magnetic field (green dash), without repumpers (blue line) and
without Zeeman slower beam (black dash).}
\end{figure}
The $^1$S$_0\,$-${}^1$P$_1$ transition used for the blue MOT is
not perfectly closed due to the decay channel of the
5p$\,{}^1$P$_1$ state towards the 4d$\,{}^1$D$_2$ state, which has
a lifetime of $0.5$ ms \cite{Xu2003}. Atoms in the latter state
may decay to the ground state through the 5p$\,{}^3$P$_1$ within
less than 1$\,$ms or may decay to the metastable 5p$\,{}^3$P$_2$
state and be lost. In order to recycle the atoms stored in the
metastable 5p$\,^3$P$_2$ state a 10 mW repumper laser at 707$\,$nm
is used to pump these atoms in the 6s$\,^3$S$_1$ state. An
additional 10 mW laser at 679$\,$nm is necessary to deplete the
$^3$P$_0$ state since it is also coupled to the 6s$\,^3$S$_1$
state. The repumping laser beams are superimposed, expanded to 10
mm of diameter and sent to the blue MOT and retroreflected. The
repumping increases the atom number in the blue MOT by a factor
$\sim60$. We studied the loading dynamics of the blue MOT
population $N$, given by $dN/dt=\phi_{c}-\Gamma_{L}N\,-\beta'
N^2$, where $\phi_{c}$ is the effective loading rate of atoms in
the MOT, $\Gamma_{L}$ is the linear loss rate (mainly due to
background collisions when repumpers are operating) and $\beta'$
is the the coefficient for two-body collisional loss
\cite{Dinneen1999}. The previous equation with the initial
condition $N(t=0)=0$ has the standard solution $
N(t)=N_{st}(1-\exp(-t/\tau))/(1-\xi\exp(-t/\tau))\,$ where
$N_{st}$ is the stationary number of trapped atoms, $\tau$ is the
effective trap loading time and $\xi$ is the collisional loss
fraction \cite{Dinneen1999}. With repumpers we measured
$N_{st}=4.5\times10^7$ $^{88}$Sr atoms, corresponding to an atomic
density of $n_0=8\times10^9$ cm$^{-3}$, $\tau=282\,$ms and
$\beta'=1\times10^{-8}$ s$^{-1}$, leading to a rate of captured
atoms of
$\phi_{c}=N_{st}/\tau+\beta'N{_{st}}^2=9.7\times10^{7}\,\text{s}^{-1}$.
Considering that the rate of atoms effused by the oven in the
solid angle covered by the Zeeman slowing beam is
$6.5\times10^8\,\text{s}^{-1}$, about $15$ \% of the atoms are
actually trapped into the blue MOT. This efficiency is mostly
determined by the optical pumping into the 4d$\,{}^1$D$_2$ state,
where atoms can be considered lost from the slowing dynamics since
the state lifetime is comparable to the slowing time scale. The
probability of an atom of decaying into the ground state after the
absorption of $n$ photons is
\begin{figure}[t]
\includegraphics{slowing_dynamics_final.eps}
\caption{\label{fig:slowing_dynamics_final} Simulation of the
slowing dynamics with the field profile given by the Zeeman slower
coils and the off-axis quadrupole field of the MOT coils.}
\end{figure}
\begin{equation}
\left(\frac{\Gamma_{\text{b}}}{\Gamma_{\text{b}}+\Gamma_{\text{d}}}\right)^{n}\simeq\left(1-\frac{\Gamma_{\text{d}}}{\Gamma_{\text{b}}}\right)^{n}\simeq\exp\left(-\frac{\Gamma_{\text{d}}}{\Gamma_{\text{b}}}n\right)\,.
\end{equation}
where $\Gamma_b=2.0\times10^8$ s$^{-1}$ and
$\Gamma_d=3.9\times10^3$ s$^{-1}$ are the decay rates of the state
$^1$P$_1$ into $^1$S$_0$ and $^1$D$_2$ states, respectively.
Therefore the actual fraction of slowed atoms can be estimated
through
\begin{equation}
\eta\equiv\int_{v_{\text{min}}}^{v_{\text{max}}}
\exp\left(-\frac{\Gamma_{\text{d}}}
{\Gamma_{\text{b}}}n(v)\right)f(v)dv\simeq20\,\%\;,
\label{eq:Zeeman_slower_efficiency}
\end{equation}
where $v_{\text{min}}=40\,\text{m/s}$ is the minimum final
velocity of atoms at the end of the slowing dynamics,
$v_{\text{max}}=390\,\text{m/s}$ is the maximum atom velocity that
can be slowed down (according to the results of numerical
simulation shown in Fig. \ref{fig:slowing_dynamics_final}),
$n(v)=(v-v_{\text{min}})/v_{\text{rec}}$ is the number of photons
necessary to reduce the atom velocity from $v$ to
$v_{\text{min}}$, $v_{\text{rec}}=h/\lambda
M\simeq1\,\text{cm}/\text{s}$ is the recoil velocity due to the
absorption of a photon at $461\,$nm and $f(v)$ is the velocity
distribution in an effusive atomic beam at $T=380\,^{\circ}$C
\begin{equation}
f(v)=2\left(\frac{M}{2k_{\textrm{B}}T}\right)^{2}v^{3}\exp\left(-\frac{Mv^{2}}{2k_{\textrm{B}}T}\right)\;.
\end{equation}
Therefore the estimation of the Zeeman slower efficiency
$\eta\simeq20\,\%$ given by Eq. \ref{eq:Zeeman_slower_efficiency}
is close to the measured value. Additionally the Blue MOT
population enhancement factor due to the operation of Zeeman
slower can be estimated from the ratio between the actual fraction
of slowed atoms $\eta$ and the fraction of trapped atoms without
Zeeman slower
\begin{equation}
\eta\;/\int_{0}^{v_{\text{c}}}f(v)dv\simeq70\;,
\end{equation}
which is close to the measured value of $\sim40$.
The final temperature of the blue MOT is minimized by continuously
reducing the optical power of the cooling beams from the initial
total saturation parameter of $s=1$ to a final value of
$s=5\times10^{-3}$ in $10\,$ms. This power-reduced phase lasts for
50$\,$ms, leading to a final temperature of the blue MOT of about
$\sim2\,$mK measured with absorption imaging (see Fig.
\ref{fig:laser_cooling}).
The second laser-cooling stage is performed by using the
$^1$S$_0\,$-${}^3$P$_1$ intercombination transition at 689$\,$nm
(red MOT). The optical beams needed for this phase are
superimposed on the blue MOT beams through the three dichroic beam
expanders. The $1/e^2$ diameter of the beams for the red MOT is
10$\,$mm and the total intensity incident on the atoms is
$60\,\text{mW}/\text{cm}^2$. The atomic sample at the end of the
first cooling stage ($T\sim2\,$mK) has a Doppler width of
$\sim2\,$MHz, too large to be efficiently transferred into the red
MOT operating on a $7\,$kHz natural linewidth transition. For this
reason the spectrum of the 689$\,$nm laser is artificially
broadened \cite{Katori1999,Loftus2004}. This is realized by
modulating the radio frequency driving of the AOM setting the
frequency detuning of the cooling beams. The modulation frequency
is $50\,$kHz, with a span of $4\,$MHz, leading to an optical
spectrum of $\sim$ 80 sidebands, with a saturation parameter of
400 for each one, with the closest to resonance by $-300\,$kHz.
This so called red MOT broadband cooling phase lasts for 60$\,$ms
and more than 70$\,\%$ of the blue MOT population is transferred
into the red MOT. At the beginning of this phase the Zeeman slower
field is turned off and the MOT magnetic field gradient is only 20
mT/m in order to have all the atomic velocity classes resonant
with the cooling light and it is linearly increased to 70 mT/m in
a time interval of $30\,$ms to compress the atomic sample. With a
blue MOT loading time of $400$ ms the final population of the
broadband red MOT is $2\times10^7$ atoms, with an atomic density
of $1.5\times10^{11}\,\text{cm}^{-3}$ and a temperature of
$\sim15\,\mu$K.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\textwidth]{laser_cooling_2.eps}
\caption{\label{fig:laser_cooling} Absorption imaging
\emph{in-situ} of the atomic sample at the relevant phases of the
laser cooling sequence lasting a time $t$, with a constant blue
MOT loading time of 400 ms. a) blue MOT full power. b) blue MOT
low power. c) red MOT broadband. d) red MOT single frequency.}
\end{center}
\end{figure}
The second laser-cooling phase is completed by employing a single
frequency red MOT, with a detuning of $-300\,$kHz, a constant
magnetic field gradient of 20 mT/m and reducing the total
intensity of the cooling beams down to
$500\,\mu\text{W}/\text{cm}^2$ in $50\,$ms. The single frequency
red MOT phase produces a sample of $1.5\times10^7$ atoms, with an
atomic density of $2\times10^{11}\,\text{cm}^{-3}$ and a
temperature of $\sim2.5\,\mu$K. Since the gravity force is
comparable to the radiation force of the red MOT beams, the atomic
sample sags down from the center of the MOT quadrupole field,
assuming an half-disk-like shape with a vertical and horizontal
diameter of $500\,\mu$m \cite{Loftus2004}.
In order to have a long Doppler-free interrogation time of the
clock transition, the laser-cooled strontium sample is trapped
into a vertical lattice realized by retroreflecting a $290\,$mW
laser beam near the magic wavelength for $^{88}$Sr
$\lambda_\text{magic}=813.42757(62)\,$nm \cite{Akatsuka2010}. At
this wavelength the light shifts of the $^1$S$_0$ and $^3$P$_0$
level are equal, so that the frequency of the clock transition
$^1$S$_0\,$-${}^3$P$_0$ is not light-shifted by the lattice. The
lattice laser output is coupled into a single mode fiber
delivering an optical beam with $1/e^2$ diameter of $1.2\,$mm,
which is expanded and focused into the red MOT by a $300\,$mm
focal length lens. The latter is mounted on a 3-axis translation
stage with micro-metric actuators, so that the alignment of the
beam onto the red MOT can be finely tuned. The resulting beam
waist radius on the atomic cloud is $w_0\sim40\,\mu$m. After the
focus, the divergent lattice beam is then collimated and
retroreflected by means of a dichroic mirror. The latter is
employed to couple into the lattice the clock probe beam at
$\lambda_c=698\,$nm. The resulting beam diameter of the clock
light on the atoms is $74\,\mu$m. Taking into account the power
losses due to the telescope, focusing optics and cell windows the
estimated lattice trap depth is $U_0\simeq76\,E_{\text{rec}}$
(corresponding to $\sim12\,\mu$K, in temperature units), where
$E_{\text{rec}}=\hslash^2 k^2/2m$ is the photon recoil energy with
the wavevector $k=2\pi/\lambda_\text{magic}$ of the lattice light
and $m$ the atomic mass of $^{88}$Sr. At this depth of the
potential the estimated longitudinal trap frequency is
$\nu_z=2E_{\text{rec}}\sqrt{U_{0}/E_{\text{rec}}}/h=60\,$kHz.
The lattice is continuously operating and at the end of the second
laser-cooling stage about $3\times10^5$ atoms remain trapped,
populating $\sim1000$ sites, for a total extension of
$\sim400\,\mu$m corresponding to the vertical size of the red MOT.
The $1/e^2$ radius of the atomic cloud trapped in the lattice is
measured with absorption imaging to be $\sigma_r=11\,\mu$m. The
longitudinal $1/e^2$ radius of a single site, of the order of
$\lambda_L/2$, cannot be resolved by absorption imaging and is
estimated from the size
$\sigma_z=\sqrt{\hslash/m\omega_z}\sim45\,\text{nm}$ of the wave
function of atoms populating the ground longitudinal vibrational
state (is a good approximation since we have
$k_{\text{B}}T/h\nu_{z}\sim0.9$). Thus the trapping volume for
lattice site is
$V_{\text{site}}=(2\pi)^{3/2}\sigma_{r}^{2}\sigma_{z}\simeq5\times10^{-10}\,\text{cm}^{3}$.
In order to keep the shift and broadening effects on the clock
transition due to atomic collisions \cite{Lisdat2008}, we reduced
the number of trapped atoms to below $10^4$ (less than 10 atoms
per site) by reducing the blue total MOT loading time to 100 ms,
leading to a peak atomic density per site below
$2\times10^{-10}\,\text{cm}^{-3}$. The number of atoms is
controlled by varying the duration of the blue MOT stage and by
finely tuning the position of the gate valve separating the oven
from the science region.
The lifetime of atoms trapped into the lattice is measured to be
$1.4\,$s, limited by background gas collisions. Both power and
frequency of the lattice laser are not stabilized. The relative
RMS power fluctuation is below the $1$ \% level. The lattice
wavelength is tuned near to the magic wavelength of $^{88}$Sr
\cite{Akatsuka2010}, monitored by a wave-meter and measured to be
stable at $\lambda=813.4280(1)\,$nm over a time scale of several
hours.
\subsection{Lattice clock spectroscopy}
As a first test of the performance of the transportable clock
laser after the two-days, 1300 km-long transportation, the laser
was compared with a stationary clock laser \cite{Tarallo2011}.
Fig. \ref{fig:beatclocklaser} shows the beat note recorded just
after re-installation of the clock laser in the lab. The
installation process took about one day, mainly do to
re-thermalization of the transportable cavity, which was not
actively temperature stabilized during transportation. The beat
note shows a linewidth of the order of 1 Hz compatible with the
laser frequency stability of $2-3\times10^{-15}$ at 1 s.
\begin{figure}[t]
\includegraphics[width=0.6\textwidth]{beatnoteRev.eps}
\caption{\label{fig:beatclocklaser} Beatnote of the transportable
laser with a stationary clock laser at 698 nm. The estimated
emission linewidth for each laser is 1 Hz.}
\end{figure}
The comparison with the stationary clock has also been used to
estimate the absolute frequency of the transportable clock laser
with an uncertainty of less than $1\,$MHz.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{ClockSpectraRev.eps}
\caption{\label{fig:clock_spectra} Spectra of the $^{88}$Sr clock
transition for different values of the mixing magnetic field
$\left|\mathbf{B}\right|$, clock probe beam intensity $I$ and
excitation pulse length $\Delta t$. The frequency axes have
arbitrary offsets. The clock resonance is fitted with a Lorentzian
function and the obtained FWHM is compared to the Fourier limit
linewidth $\Delta\nu_{\text{F}}\simeq0.8/\Delta t$ and to the Rabi
linewidth
$\Delta\nu_{\text{R}}=0.35\sqrt{\left|\Delta_{B}\Delta_{L}\right|}$,
where $\Delta_{B}$ and $\Delta_{L}$ are the second order Zeeman
shift and clock probe light shift, respectively (see the text). a)
shows a typical search-scan spectrum with the maximum number of
atoms loaded into the lattice $N\simeq2\times10^5$. b) is taken
with clock interrogation $\pi$ pulses and a lattice population of
$N=7\times10^{3}$.}
\end{center}
\end{figure}
Considering that the single photon $^{88}$Sr clock transition
$^1$S$_0\,$-${}^3$P$_0$ is forbidden at any order, the
magnetic-field-induced spectroscopy method is used to controllably
allow the clock transition, by means of an external magnetic field
coupling the $^3$P$_0$ to the $^3$P$_1$ state
\cite{Taichenachev2006}. The search-scan is performed by using a
mixing magnetic field of $\left|\mathbf{B}\right|=19\,$mT and a
clock probe beam intensity of $I=5.7\,\text{W}/\text{cm}^2$,
leading to a Rabi frequency of $275\,$Hz, given by
\begin{equation}
\Omega_{\text{R}}=\alpha\sqrt{I}\left|\mathbf{B}\right|
\label{eq:Rabi}
\end{equation}
where $\alpha
(\text{Sr})=198\,\text{Hz}/(\text{T}\sqrt{\text{mW}/\text{cm}^{2}})$
\cite{Taichenachev2006}. In order to have a high-contrast spectrum
the excitation pulse length is $\Delta t=100\,$ms, thus
overdriving the clock transition having an estimated $\pi$ pulse
duration of $\Delta t_{\pi}=1.8\,$ms. In the search-mode the clock
laser frequency is changed by $4\,$kHz, covering a span of
$200\,$kHz in about 50$\,$s (experiment cycle 1$\,$s). A typical
search-mode scan is shown in Fig. \ref{fig:clock_spectra}. The
excitation probability is given by the ratio
$n(^{3}\text{P}_{0})/[n(^{1}\text{S}_{0})+n(^{3}\text{P}_{0})]$,
where $n(^{3}\text{P}_{0})$ and $n(^{1}\text{S}_{0})$ is the
atomic population of the $^{3}\text{P}_{0}$ and $^{1}\text{S}_{0}$
state, respectively. The $^{1}\text{S}_{0}$ population is obtained
through absorption imaging of the atomic sample after the clock
transition interrogation. Atoms in the $^{1}\text{S}_{0}$ state
are blown away from the lattice by the resonant 461$\,$nm imaging
beam. The atoms excited into the $^{3}\text{P}_{0}$ level by the
clock probe beam are pumped back into the $^{1}\text{S}_{0}$ state
by means of a 50$\,$ms pulse at $679\,$nm and $707\,$nm. The
$^{1}\text{S}_{0}$ population is then measured through an
additional absorption imaging sequence. From the spectra in Fig.
\ref{fig:clock_spectra}a, the motional sidebands are measured at
$\sim65$ kHz from the carrier
, thus corresponding to a Lamb-Dicke parameter
$\eta=\sqrt{\nu_{R}/\nu_{z}}\simeq0.3$, where
$\nu_R=h/(2m\lambda^2)$ is the atomic recoil frequency shift
associated to the absorption of a photon with $\lambda=698\,$nm.
The excitation probability in the search-mode scan is only
$\sim30\,\%$ since the actual clock transition is under-sampled
because of the need of covering a large scanning span in reduced
time. The excitation pulse duration in this high-resolution mode
is chosen to realize an effective $\pi$ pulse for each
configuration of magnetic field and probe intensity. For this
purpose the actual atomic Rabi frequency is measured through the
observation of the Rabi oscillations (see Fig.
\ref{fig:Rabi_oscillations}). On the carrier the excitation
probability approaches the $70$ \% level. We find that the fitted
Rabi frequency is about $50$ \% of the one calculated from Eq.
\ref{eq:Rabi}, an effect we attribute, together with the reduced
excitation probability of $70$ \%, to the inhomogeneous
distribution of the Rabi frequency among the atoms that could be
given by a residual spatial inhomogeneity of the clock probe, by a
residual misalignment between the probe and lattice axes
(estimated to ~1 mrad) and by the thermal transverse atomic motion
in the lattice potential \cite{Blatt2006}. This effect can be
significantly reduced by employing a probe beam with larger
diameter and a colder atomic sample.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{Rabi1Rev.eps}
\includegraphics[width=0.4\textwidth]{Rabi3Rev.eps}
\caption{\label{fig:Rabi_oscillations} Measurement of the Rabi
oscillations in condition of resonance with the clock transition
for different values of the mixing magnetic field and clock probe
beam intensity. The data are fitted with the function
$a(1-\cos(2\pi\Omega_{\text{FIT}}\Delta t)\exp(-\Delta
t/\tau_{\text{FIT}}))$, where $\Omega_{\text{FIT}}$ is the actual
atomic Rabi frequency and $\tau_{\text{FIT}}$ gives the
decoherence time-scale. The measured atomic Rabi frequency
$\Omega_{\text{FIT}}$ is $\sim50\,\%$ of the estimated Rabi
frequency $\Omega_{\text{R}}$ calculated from the mixing magnetic
field and clock laser intensity (see Eq. \ref{eq:Rabi}). The
corresponding $\pi$ pulse interrogation length $\Delta
t_{\pi}=1/2\Omega_{\text{FIT}}$ was employed for the clock spectra
b) of Fig. \ref{fig:clock_spectra}.}
\end{center}
\end{figure}
By reducing both the mixing magnetic field and the clock probe
intensity the low drift rate of the clock laser (few
$\text{mHz}/\text{s}$) allowed us to reliably measure Rabi
oscillations lasting for more than $100\,$ms and Fourier limited
spectra below the $10\,$Hz FWHM linewidth (see Fig.
\ref{fig:Rabi_oscillations}).
\begin{figure}
\includegraphics[width=0.49\textwidth]{zeeman3.eps}
\includegraphics[width=0.4\textwidth]{probe2Rev.eps}
\vspace{0.1cm}
\includegraphics[width=0.49\textwidth]{residualszeeman3.eps}
\includegraphics[width=0.45\textwidth]{residualsprobe2.eps}
\caption{\label{fig:shifts} Systematic shifts of the
$^1$S$_0\,$-${}^3$P$_0$ clock transition in $^{88}$Sr optical
lattice clock. a) Measurement of the second order Zeeman shift as
a function of the Hall probe voltage.
b) Measurement of the probe AC linear Stark shift as a function of
the photodiode voltage.
The respective curves are used to calibrate the Hall probe and the
photodiode to estimate the applied magnetic field $\mathbf{B}$ and
the probe light intensity $I$ through the knowledge of the shift
coefficients \cite{Taichenachev2006}. The two calibration
coefficients are $\gamma=29.0(4)$ mV/mT for the Hall probe and
$\eta=$ 104(1) mV/(W cm$^{-2}$) for the photodiode, respectively.}
\end{figure}
A study of the systematics has been carried out and values of
estimated shifts on the $^1$S$_0\,$-${}^3$P$_0$ clock transition
are reported in Table \ref{tab:final_budget}.
The 2$^{\mathrm{nd}}$ order Zeeeman and probe Stark shifts have
been evaluated by scanning around the clock transition and
interleaving measurements with different values of the magnetic
field and probe power. As frequency reference we rely on the clock
cavity resonance whose typical 1 Hz/s linear drift was compensated
to better than 10 mHz/s with a programmable feed-forward AOM
driver.
Magnetic field and clock probe power are monitored at $1\,\%$
precision level with a Hall probe and a photodiode, respectively.
They are calibrated through the known coefficient $\beta$ and $k$
\cite{Taichenachev2006} by measuring the clock frequency shift as
a function of the magnetic field ($2.2-4.1\,$mT range, also
inverting the direction of the field) and probe intensity
($0.9-3.2\,\text{W}/\text{cm}^2$ range), as shown in Fig.
\ref{fig:shifts}. Values are calculated for the $8.0(0.1)\,$Hz
linewidth clock transition (see Table \ref{tab:final_budget}),
obtained with a magnetic field of
$\left|\mathbf{B}\right|=1.19(4)\,$mT and probe beam intensity of
$I=0.922(13)\,\text{W}/\text{cm}^2$, corresponding to a second
order Zeeman shift and probe AC Stark shift of
$\Delta_B=\beta\left|\mathbf{B}\right|^2=-33.5(2.4)\,$Hz and
$\Delta_L=kI=-16.6(2)\,$Hz respectively, where
$\beta=-23.3\,\text{Hz}/\text{mT}^2$ and
$k=-18\,\text{Hz}/(\text{W}/\text{cm}^2)$ \cite{Taichenachev2006}.
The uncertainties on the second order Zeeman shift and probe light
shift are given by the quadratic sum of the standard error from
the fit and the error associated by the voltage reading on the
Hall probe and the photodetector.
\begin{table}[t]
\caption{\label{tab:final_budget} Systematic frequency shift and
uncertainty budget for the $^{88}$Sr optical lattice clock as
extracted from spectra of the clock transition. The values are
reported for the operating conditions of the $8.0(0.1)\,$Hz clock
transition linewidth (Fig. \ref{fig:clock_spectra} b)).}
\begin{ruledtabular}
\begin{tabular}{lcc}
Contributor & Shift (Hz) & Uncertainty (Hz)\\
\hline
2nd order Zeeman & -33.5 & 2.4\\
Clock light & -16.6 & 0.2\\
Collisions & 1.0 & 0.4\\
Blackbody radiation & -2.5 & 0.5\\
AC Stark (lattice) & -0.7 & 0.9\\
\hline
Total Uncertainty & & 2.8\\
\end{tabular}
\end{ruledtabular}
\end{table}
The effects of collisions on the $^{88}$Sr clock transition have
been studied in detail in \cite{Lisdat2008}. In our optimal
conditions the number of atoms trapped into the lattice is kept at
$7.0(7)\times10^3$ ($\sim7$ atoms per site), so that the
corresponding peak atomic density per lattice site of
$\rho=1.39(14)\times10^{10}\,\text{cm}^{-3}$ leads to a
collisional shift and broadening of $\rho\times
\Delta\nu_{\rho}=1.0(4)\,$Hz and $\rho\times
\gamma_{\text{dep}}/\pi=1.4(6)\,$Hz respectively, with the
coefficients
$\Delta\nu_{\rho}=(7.2\pm2.0)\times10^{-11}\,\text{Hz}/\text{cm}^3$
and $\gamma_{\text{dep}}=(3.2\pm1.0)\times10^{-10}
\text{cm}^3/\text{s}$ \cite{Lisdat2008}. The uncertainties on
collisional shift and broadening take into account our
experimental $10\,\%$ fluctuation of the number of atoms
shot-to-shot and the uncertainties of the $\Delta\nu_{\rho}$ and
$\gamma_{\text{dep}}$ coefficients and density determination.
During the clock spectroscopy measurement the vacuum chamber
temperature is monitored with one thermistor directly attached to
the main chamber. No water cooling is employed to cool the MOT
coils and after a warm up of about three hours, the temperature of
the whole cell stabilizes to $318\,$K. While a more detailed study
of temperature gradients is mandatory for accuracy level below the
10$^{-16}$ level, this lies beyond the scope of the present
article. We estimated the blackbody radiation shift to $-2.5\,$Hz
with an uncertainty of 0.5 Hz, including the uncertainty due to
temperature gradients up to $\pm$ 10 K inside the main cell. The
AC Stark shift induced by the lattice light has been evaluated
considering the effect of the detuning of 0.40 pm with respect to
the magic wavelength for $^{88}$Sr (with a scalar coefficient $8$
Hz/nm/$E_R$). Taking also into account the uncertainty in the
knowledge of the magic wavelength and the smaller contribution
from hyperpolarizability effect \cite{Brusch2006}, the shift
amount to 0.3(4) Hz. The amplified spontaneous emission (ASE) from
the tapered amplifier used for lattice laser is about 40 nm wide
and symmetric around the emission wavelength. With a typical ASE
intensity 40 dB below the carrier, we estimated an additional
shift of -1.0(5) Hz due to this effect. The total value for the AC
Stark shift has been evaluated to be -0.7(9)Hz. At $8\,$Hz clock
transition linewidth the achieved total fractional uncertainty is
$\sim7\times10^{-15}$, with a transition quality factor of
$Q\sim5\times10^{13}$. With a current clock cycle time of $T_c=1$
s (mainly limited by technical delays in the absorption imaging
detection) and an interrogation time $\Delta t=130$ ms, the
stability of the clock is limited at $4\times 10^{-15}$ at 1 s by
the Dick effect \cite{Quessada2003}. With minor changes to the
detection system a reduction of the clock cycle time to $\sim 400$
ms can be implemented, maintaining the same number of atoms into
the lattice, thus reducing the contribution of the Dick noise
below $2\times 10^{-15}$ level.
\section{\label{sec:conclusions} Conclusions}
We presented a compact and transportable optical clock based on
strontium atoms. Novel design solutions allowed us also to reduce
the volume, weight and power consumption with respect to
traditional laser cooling apparatus. As a result the experimental
physics package is contained in a volume of less than 2 m$^3$ and
no water cooling is needed to operate the clock. Furthermore, a
modular architecture ensured a high degree of operation
reliability of the apparatus both in stationary condition and
after a transportation of the experimental set up. To ensure high
clock frequency stability, cooling and trapping stages have been
optimized to allow high efficiency transfer among different
cooling and trapping stages, thus allowing for faster clock cycle
time with high duty cycle. Spectroscopy on $^1$S$_0$ -$^3$P$_0$
clock transition on bosonic $^{88}$Sr isotopes has been
demonstrated with an 8 Hz resolution. Eventually, an evaluation of
the main systematic frequency shifts on the clock transition has
been done and the fractional uncertainty of the clock is
$7\times10^{-15}$.
The authors acknowledge financial support from ESA and the
European Union Seventh Framework Programme (FP7/2007-2013 grant
agreement 263500, project ``Space Optical Clocks'') and the
European Metrology Research Programme (EMRP) under IND14. The EMRP
is jointly funded by the EMRP participating countries within
EURAMET and the European Union. We also acknowledge support by the
DFG RTG 1729 ``Fundamentals and applications of ultra-cold
matter''. We thank D. V. Sutyrin for useful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 856
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\section{Introduction}
Despite the wealth of over 4000 verified exoplanets known today\footnote{https://exoplanetarchive.ipac.caltech.edu (30 April 2020)} there are still many unanswered questions concerning the formation mechanisms and system evolution which has led to the observed distribution of these objects. In particular, planetary migration, dynamical interactions with nearby stars and stellar evolution all can have major effects on the distribution and architecture of the final exoplanet population. This is especially important early in a planet's life (e.g. <1Gyr), where phenomena such as accretion \citep{Marley2007OnJupiters,Manara2019ConstrainingRates}, ionising radiation causing atmospheric loss \citep{Baraffe2003Evolutionary209458,Owen2019AtmosphericExoplanets} and dynamical interactions with other forming planetary system bodies \citep{Ida2010TowardStars,Schlichting2015AtmosphericImpacts} can significantly change the mass, radius and orbital parameters of early exoplanets. While traditional exoplanet studies have been biased towards older exoplanets due to their relatively quiet host stars, there is a strong case to be made for the search for younger exoplanets, where planets are undergoing the majority of their evolution \citep{Adams2006LongTermInteractions,Ida2010TowardStars,Spiegel2012SpectralScenarios,Ida2013TowardGiants}. Discovering exoplanets <1 Gyr old will thus help to probe the main causes of planetary evolution, and help to fill a key gap in our knowledge of exoplanet history.
However, the host stars of young exoplanets provide significant challenges for discovery due to their typically increased activity, rotation rates and relative proximity to neighbouring stars \citep[e.g.][]{Koeltzsch2009VariabilityRegion,Sergison2015UntanglingAnalysis,Rivilla2015Short-Cloud,Mascareno2016MagneticSeries,Briceno2018ThePopulations,Cegla2019StellarDiagnostics}. The combination of these factors can result in periodic stellar variability that can be on the order of planetary signals in both period and intensity \citep{Armstrong2015K20,Cody2018AMissions}.
Despite these challenges a small number of young exoplanets have been found. In line with early exoplanet discoveries, the first exoplanets around young stars were found using the radial velocity method, including four Hot Jupiters found in the Hyades \citep{Sato2007ATauri,Quinn2014HDCLUSTER} and Praesepe \citep{Quinn2012TWOCLUSTER} open clusters. However, for planets less massive than Hot Jupiters, the radial velocity method was found to be severely hampered by the inherent stellar variability and radial velocity jitter of the young host stars \citep{Saar1997Activity-RelatedStars,Paulson2004SearchingActivity,Brems2019Radial-velocityAge}. Recent methods such as those employed by \citet{Korhonen2015StellarCycles,Rajpaul2015AData} have pushed this limit down to approximately Neptune-sized planets for some solar-like stars, however Earth-sized planets are currently still beyond the capabilities of these methods for young active stars.
The high photometric precision offered by the launch of the \textit{Kepler} satellite proved crucial in increasing this sample of young exoplanets, especially when deliberately pointed at young open clusters in the \textit{K2} extended mission. Most discoveries in this era were made by the \textit{Zodiacal Exoplanets In Time (ZEIT)} team, who found 17 planets between Feb 2016 and Oct 2018 \citep[e.g.][]{Mann2016ZodiacalCluster,Rizzuto2017ZodiacalK2,Rizzuto2018Zodiacal16}.
Other interesting discoveries of young exoplanets in K2 have included K2-33b - a Neptune-sized planet in the 5-10 Myr Upper Scorpius stellar association \citep{David2016AStar}, K2-136A c in the Hyades - the first Neptune-sized planet orbiting a binary system in an open-cluster \citep{Ciardi2018K2-136:Planet}, and the closely-packed system of four planets around a $\sim$23 Myr pre-main sequence star in the Taurus-Auriga star forming region \citep{David2019ATaub,David2019FourTau}.
The launch of the \textit{TESS} is now ushering in a new era of exoplanet discovery, including the discovery of two new young exoplanets: DS Tuc Ab in the 45 Myr Tucana-Horologium association \citep{Benatti2019AA,Newton2019TESSAssociation}, and the ~24 Myr AU Mic b in the Beta Pictoris Moving Group \citep{Plavchan2020AUPaper}. These recent new discoveries highlight the fact that it is now possible to find planets across a wide range of the early evolution of these systems, but considerably more planets are needed at these ages before reliable statistics can be generated.
A particularly promising place to look for these young exoplanets is within stellar associations. These are groups of gravitationally unbound stars which have a common origin, so still retain a common proper motion across the sky. Their relatively diffuse nature compared to similarly young star clusters makes them far more suited to study with \textit{TESS} given the relatively large pixels of the satellite \citep[angular resolution $\sim$21",][]{Vanderspek2018TESSHandbook}. Stars in these associations share similar ages, positions, compositions and kinematics, meaning that precise stellar properties can be determined \citep{Torres2006AstronomyMethod}. This in turn provides a significant advantage for precise determination of exoplanet characteristics. Furthermore, the ages of stars within these associations are typically very well constrained, which will allow a more detailed timeline of planetary evolution to be assembled. The most extensive census of 'bona-fide' members of these stellar associations was assembled by \citet{Gagne2018BANYAN150pc} in the development of their BANYAN $\Sigma$ Bayesian membership tool, but has recently been further expanded by works such as \citet{Gagne2018BANYAN.Data} and \cite{Esplin2019AMasses} thanks to the release of Gaia DR2 \citep{GaiaCollaboration2018GaiaProperties}. The extremely precise astrometric data from \textit{Gaia} will doubtless allow for further expansions to the membership lists of these associations as the satellite's mission continues. Note that \citet{Bouma2019Cluster7} recently assembled a larger list of young stars in clusters and associations (of which the BANYAN sample is a subset) by considering a wide array of sources in the literature, however they admit that this sample was designed for "completeness, not accuracy". For the initial design of the pipeline constructed in this work the homogeneous BANYAN selection criteria focused solely on stellar associations was preferred.
In order to discover any exoplanets within these associations, a key challenge is dissociating true transit signals from instrumental systematics or stellar 'noise' (activity or variability of the host star). While instrumental trends are commonly removed using techniques like cotrending basis vectors \citep{Thompson2016KeplerManual}, spacecraft pointing-based decorrelation \citep{Vanderburg2014AMission,Aigrain2016K2SC:Regression} or subtracting trends shared by simultaneously observed nearby stars \citep{Kovacs2005ASurveys,Kim2009DetrendingSurveys}, the problem of dissociating stellar noise from transit signals remains a challenging one. A large array of different methods have been developed in an attempt to solve this problem. These methods can be largely separated into three major approaches: sliding filters (e.g. \citet{Savitzky1964SmoothingProcedures.}), fitting splines/polynomials (e.g. \citet{Vanderburg2016PLANETARYMISSION}) and gaussian processes (e.g. \citet{Aigrain2015PreciseMission}; Gillen \& Aigrain, in prep).
An in-depth review and comparison of the most commonly used stellar activity detrending methods can be found in \citet{Hippke2019WotanPython}.
The earliest wide-field survey, HATNet \citep{Bakos2004Wide-FieldDetection}, used the Trend Fitting Algorithm \citep{Kovacs2005ASurveys} and External Parameter Decorrelation \citep{Bakos2010HAT-P-11b:Field} to detrend its light-curves, while WASP \citep{Pollacco2006TheCameras} shortly after preferred the \texttt{sysrem} algorithm of \citet{Tamuz2005CorrectingCurves} - which removes common time- and position-dependent trends by taking into consideration the weighted average magnitude residuals for all stars - coupled with a boxcar-smoothing technique to handle some of the more variable stars observed \citep{CollierCameron2006ASearches}. More recent ground-based surveys such as KELT \citep{Pepper2007TheSurveys} and NGTS \citep{Wheatley2018TheNGTS} have developed these techniques further, and even introduced more advanced detrending methods such as gaussian processes as they target more active stars \citep[e.g.][]{Costes2019NGTS-8bHot-Jupiters,Gillen2019NGTS1}.
However, the largest source of light-curves in the search for exoplanets has been space-based surveys, the volume and variety of which has required the development of more complex and robust detrending techniques.
The most prolific transiting exoplanet surveys thus far have been the \textit{Kepler} \citep{Borucki2010KeplerResults} and \textit{K2} \citep{Howell2014TheResults} missions and the ongoing \textit{TESS} mission \citep{Rickeretal.2014TheSatellite}, all of which share braodly similar detrending techniques. Overviews of the \textit{Kepler}, \textit{K2} and \textit{TESS} SPOC pipelines can be found in \citet{Jenkins2010OVERVIEWPIPELINE}, at https://keplerscience.arc.nasa.gov/k2-pipeline-release-notes.html and in \citet{Jenkins2016TheCenter} respectively.
While these state-of-the-art detrending methods have been extremely effective at discovering new exoplanets around older stars, they can still struggle to disassociate true transiting signals from the complex activity of young host stars. This was particularly well illustrated by \citet{Hippke2019WotanPython} in construction of their WOTAN tool, where only ~35$\%$ of 0.5$R_{Jup}$ exoplanet signals injected into a sample of 316 young stars were recovered (irrespective of the detrending method chosen), compared to an almost 100$\%$ recovery rate for similar planets around less noisy stars. Interestingly however, each of the methods tested recovered slightly different populations of the injected signals, so combining all of the tested methods increased the percentage of recovered planets to 43.8$\%$. This discrepancy was further highlighted by \citet{Rodenbeck2018RevisitingB}, who showed that different detrending methods resulted in quite different conclusions when considering the potential super-moon around \textit{Kepler-1625 b}. It is thus clear that not only is there a need for new detrending methods specifically focused on young stars, but also that there is a distinct benefit of using multiple detrending methods on the same data-set, thus providing dual motivations for the construction of an alternative detrending method in this work. The development and current design of the resulting detrending pipeline used in this work is described in detail in section \ref{detrending}.
The remainder of this paper is organised as follows. Section \ref{Methods} discusses the young star target selection, observations and the choice of a FFI pipeline, before describing the methods used to clean additional systematics from the \textit{TESS} data, detrend stellar variability, search for transits and inject model transits for a sensitivity analysis. Section \ref{Results} then discusses important results seen as a result of the developed detrending techniques, including the recovery of known young exoplanets, TOIs, and eclisping binaries, followed by an overview of the sorts of rotation and activity seen in this sample and the results of the conducted sensitivity analysis. The implications of these results for the future of exoplanetary searches around young stars are then discussed in detail in Section \ref{Discussion}, before a summary and conclusion in Section \ref{Conclusions}.
\section{Target Selection, Observations and Detrending Methods} \label{Methods}
\subsection{Target Selection}
While stellar association membership is now being expanded on a cluster by cluster basis thanks to the increased astrometric precision of the \textit{Gaia} satellite \citep{GaiaCollaboration2016TheMission,GaiaCollaboration2018GaiaProperties} (see for example \citet{Kuhn2019KinematicsDR2,Damiani2019TheData,Zari2019StructureRegion}), a key list of 'bona-fide' stellar association members were assembled by \citet{Gagne2018BANYAN150pc} during preparation of the BANYAN $\Sigma$ Bayesian cluster membership tool. \citet{Gagne2018BANYAN150pc} consider a star to be a 'bona-fide' member if it has galactic XYZ UVW values consistent with those known for a given stellar association and exhibits an independent sign of youth. Such signs can include mid-infrared excess, G-J vs GALEX NUV-G colours consistent with youth, X-ray emission with HR1 $\geq$ -0.15, lithium absorption above 100 mArmstrongs or a compatible luminosity class \citep{Gagne2018BANYAN.Data}. Combining \citet{Gagne2018BANYAN150pc}'s initial census of bona-fide/high probability stellar association members with new high-probability members added in the two following BANYAN $\Sigma$ papers \citep{Gagne2018BANYAN.Data,Gagne2018BANYAN.2} yielded a total of 2977 objects spread over the 27 nearest known young stellar associations. \citet{Gagne2018VOLANS-CARINA:Pc} later expanded the BANYAN $\Sigma$ tool to also include the two new Argus \citep{Zuckerman2018TheDisks} and Volans-Carina associations using the same membership criteria, so these were also added to the initial target list, to give the final distribution of 3076 young stars in stellar associations illustrated in Figure \ref{fig:sky_distribution}. For clarity an association by association breakdown of targets in the original BANYAN sample and the final sample of the sector 1-5 targets analysed in this work is presented in Table \ref{tab:Assos_breakdown}. This census remains the most extensive assemblage of bona-fide and high probability stellar association members with a common membership criterion, and hence represents a valuable starting place for the search for exoplanets around young stars in this work.
\begin{figure*}
\includegraphics[width = \textwidth]{Figures/Hammer_Aitoff_final.pdf}
\caption{Equatorial sky distribution of initial young star sample, based on the bona-fide and high probability members of the 29 nearest stellar associations built into the BANYAN $\Sigma$ tool. The entire sample is shown in the background with the 256 sector 1-5 targets analysed in this work presented in solid colour. Abbreviations after those chosen in \citet{Gagne2018BANYAN150pc}.}
\label{fig:sky_distribution}
\end{figure*}
\begin{table}
\centering
\caption{Breakdown of association distribution for the complete BANYAN sample (centre) and final set of targets analysed in this work (right). The final targets consist those targets in Sectors 1-5 with available 30min-cadence light-curves from the \citet{Oelkers2018PrecisionApproach} pipeline, as discussed in Section \ref{pipeline choice}. Association abbreviations are based on those presented in \citet{Gagne2018BANYAN150pc} with the addition of ARGUS for the Argus association and VCA for the Volans-Carina Association.}
\label{tab:Assos_breakdown}
\begin{tabular}{ccc}
\hline
Association & BANYAN Targets & S1-S5 Analysed Targets\\
\hline
118TAU & 12 & 0\\
ABDMG & 298 & 55\\
ARGUS & 40 & 0\\
BPMG & 135 & 13\\
CAR & 85 & 19\\
CARN & 110 & 18\\
CBER & 76 & 0\\
COL & 107 & 43\\
CRA & 14 & 0\\
EPSC & 42 & 0\\
ETAC & 16 & 0\\
HYA & 241 & 7\\
IC2391 & 14 & 0\\
IC2602 & 16 & 0\\
LCC & 310 & 0\\
OCT & 101 & 40\\
PL8 & 32 & 0\\
PLE & 204 & 0\\
ROPH & 182 & 0\\
TAU & 178 & 1\\
THA & 92 & 49 \\
THOR & 46 & 0\\
TWA & 50 & 0\\
UCL & 410 & 0\\
UCRA & 16 & 0\\
UMA & 13 & 0\\
USCO & 161 & 0\\
VCA & 59 & 0\\
XFOR & 16 & 11\\
\hline
Total & 3076 & 256\\
\hline
\end{tabular}
\end{table}
The various BANYAN survey results were combined into a single tabular target list using the \textit{TOPCAT} table handling software \citep{Taylor2005TOPCATSoftware}. This list was then cross-matched with version 8 of the \textit{TESS} input catalog (TIC) \citep{Stassun2019TheList} using a 3 arcsecond radius. The Web \textit{TESS} Viewing Tool\footnote{https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py} on the NASA \textit{TESS} website was used to determine which sector each target would be observed in and in turn compile target lists for each individual sector. Of the original 3076 objects, 1832 of them were forecast to be observed in \textit{TESS}'s first year of observations, with the breakdown of sources to be viewed in each Southern Hemisphere sector outlined in Table \ref{tab:BANYAN_year1}. For this work, only those viewed in Sectors 1 to 5 were considered.
\begin{table}
\centering
\caption{Overview of BANYAN bona-fide members of stellar associations viewed in the first year of \textit{TESS} observations. Note that sources in the overlapping regions of the \textit{TESS} sectors, such as those in the continuous viewing zone, are counted multiple times.}
\label{tab:BANYAN_year1}
\begin{tabular}{cc}
\hline
Sector & Number of Sources observed\\
\hline
1 & 118\\
2 & 130\\
3 & 124\\
4 & 186\\
5 & 336\\
6 & 258\\
7 & 163\\
8 & 166\\
9 & 272\\
10 & 399\\
11 & 750\\
12 & 396\\
13 & 200\\
\hline
\end{tabular}
\end{table}
\subsection{Observations}
\textit{TESS} is the successor to the highly successful \textit{Kepler} space telescope, which is designed to survey $>$85\% of the sky to search for planets transiting bright host stars \citep{Rickeretal.2014TheSatellite}. \textit{TESS} launched on 18 April 2018 and is now is into year two of its primary mission, having completed observations of the 13 Southern Hemisphere sectors. Sectors 1-5 considered in this work were observed between 25 July and 11 December 2018.
Two main data products are produced by the \textit{TESS} mission: 2min cadence light-curves (detrended and raw) and 30min cadence FFIs. 2min cadence light-curves are generated by Science Processing and Operations Center (SPOC) pipeline \citep{Jenkins2016TheCenter} for approximately 300,000 of the most promising stars (prioritized by the smallest transiting planets that can be detected) from the \textit{TESS} Candidate Target List \citep[CTL,][]{Stassun2018TheList,Stassun2019TheList}, and are then made accessible to the public via the Mikulski Archive for Space Telescopes (MAST\footnote{https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html}). However, while these primary data products are very powerful for the main \textit{TESS} mission, because of the limited data transfer rates of the \textit{TESS} primary mission far from all of the stars in \textit{TESS}'s field of view will have 2min light-curves generated. Instead, light-curves from these stars can be retrieved from the 30min cadence Full-Frame Images. In addition, because of the increased activity and rotation rate of young stars discussed above, the standard detrending methods built for the 2min light-curves have difficulty flattening the light-curves for transit searches, and may even introduce confusing additional artefacts as a result of detrending \citep{Hippke2019WotanPython}. Because of the greater coverage offered by the 30min cadence data and concerns about detrending artefacts, the 30min cadence data were thus chosen for the initial transit search, with the simple aperture photometery (SAP) 2min cadence light-curves (where available) used as a secondary check.
An additional challenge common to both the 2min and 30min cadence data is the reasonably large pixels of the \textit{TESS} focal plane \citep{Vanderspek2018TESSHandbook}, which leads to significant blending of stars in crowded regions. This precludes the use of \textit{TESS} photometry alone for confirming transits in the dense centres of young clusters without confirmation from higher resolution instruments such as the \textit{SPITZER} Space Telescope \citep{Werner2004TheSpitzerTelescopeMission} or the cameras of the \textit{Next Generation Transit Survey - NGTS} \citep{Wheatley2013NextNGTS}. The power of \textit{NGTS}'s comparative resolution for identifying the true photometric source for \textit{TESS} objects is well-illustrated by \citet{Jackman2019NGTS-7Ab:Dwarfb} in the discovery of NGTS-7ab.
\subsection{Choice of FFI pipeline} \label{pipeline choice}
A number of methods currently exist for extracting light-curves from the \textit{TESS} FFIs. The simplest approach is to perform simple aperture photometry on the raw FFIs by overlaying an aperture over a cut-out around the object of interest, and summing up the flux under the target aperture for each cadence. This can be performed easily using the prebuilt \textit{lightkurve}\footnote{https://github.com/KeplerGO/lightkurve} \texttt{Python} package \citep{LightkurveCollaboration2018Lightkurve:Python}, and observing the resulting target-pixel-file provides an instructive look at the area around the star of interest. However, the simplistic nature of this anaylsis yields relatively noisy light-curves uncorrected for issues such as spacecraft pointing, jitter and localised scattered light. A greatly improved simple aperture photometry pipeline which accounts for these issues has been built in \texttt{Python} by \citet{Feinstein2019Eleanor:Images}. Christened \textit{eleanor}\footnote{https://github.com/afeinstein20/eleanor}, this package performs background subtraction, removal of spacecraft systematics such as jitter and pointing drift, and aperture/psf photometry. In addition, it provides tools to complete further systematics-removal via principal component analysis or psf-modelling. Light-curves from \textit{eleanor} will eventually be hosted on \textit{MAST}, but at the time of writing is only available as an open-source tool designed to work for \textit{TESS} sectors 1 and 2.
An alternative difference imaging approach for 30min light-curve generation has been pursued by \citet{Oelkers2018PrecisionApproach}. Attempting to overcome the challenges posed to aperture photometry by \textit{TESS}'s large pixel sizes, this approach extracts light-curves via difference imaging analysis (DIA). In this method, one frame is blurred to the seeing conditions of the next before the two are subtracted from each other in order to retain only the variation in flux of stars between frames. Highlighting stars of interest within this process aids removal of contaminating stars in crowded regions and hence improves the light-curve extraction compared to standard aperture photometry methods. A similar technique has been used in ground-based surveys such as the Kilodegree Extremely Little Telescope \citep{Pepper2007TheSurveys,Siverd2012KELT-1b:Star}, though with a simpler Gaussian kernel than the Dirac $\delta$-function kernel used here. A full description of the DIA technique as applied to \textit{TESS} light-curves in sectors 1 and beyond can be found in \citet{Oelkers2018PrecisionApproach,Oelkers2019Beyond}. Light-curves extracted via this pathway are accessible from the Filtergraph data visualization service.\footnote{https://filtergraph.com/tess\_ffi}. It is further worth noting that as this document was being prepared \citet{Bouma2019Cluster7} released an alternative difference imaging pipeline for extraction of 30min FFI light-curve images which they applied to sectors 6 and 7, however these sectors are outside the scope of this work.
One further pipeline which can be used to extract 30-minute light-curves from the raw FFIs is the TASOC pipeline, under development by the \textit{TESS Asteroseismic Science Consortium} (\textit{TASC}). Heavily based on the K2P$^2$ pipeline \citep{Lund2015K2P2Mission}, this pipeline aims to supply \textit{TESS} photometry data for use in asteroseismology. Data from this source can be accessed online\footnote{http://tasoc.dk} after joining the consortium. So far raw light-curves extracted from the FFIs are available for sectors 1 and 2 \citep{Handberg2019TDA1+2}, while the open-source code used to extract these images from the FFIs can be found on github.\footnote{https://github.com/tasoc} However, one must be somewhat careful when using this source for transit-searches, as the primary goal of this group is asteroseismology, so at times light-curves may have been extracted in a way that prioritises the variability of the stellar signal over potential transit-like events. While this pipeline is still currently under development, a useful overview of its design and aims has been written by \citet{Lund2017DataTESS}.
Given the increased availability and comparatively clean light-curves provided by the DIA pipeline of Oelkers and Stassun \citep{Oelkers2019Beyond,Oelkers2018PrecisionApproach}, this pipeline was chosen to extract light-curves from the \textit{TESS} FFIs in this work, resulting in light-curves for 256 individual objects.
\subsection{Removal of additional systematics from 30min light-curves}
Unlike the 2min PDCSAP light-curves retrieved from MAST, the 30min light-curves supplied by \citet{Oelkers2019Beyond} have not undergone the in-depth quality analysis completed by the SPOC pipeline, and as such still include some less-trustworthy epochs of increased pointing jitter, regular spacecraft momentum dumps and known data anomalies. It was thus necessary to remove these systematics before activity detrending and transit searches could begin.
The first step undertaken was to cut any epochs where fine pointing was known to have been lost, or other spacecraft anomalies affected the data. This was achieved by consulting the \textit{TESS} data release notes\footnote{Available at https://archive.stsci.edu/tess/tess\_drn.html} for each sector. This primarily affected sectors 1, 3 and 4. In Sector 1 a period of anomalously high pointing jitter was seen between approximately TJD 1347-1349 due to problems with the fine-pointing calibration. This was observed to be particularly bad between TJD 1348-1349.29, so all epochs between these times were masked from the analysis. Similarly in sector 3 a few experiments on the attitude control system (ACS) were undertaken by the \textit{TESS} team, dramatically increasing the scatter at these times. As a result, only data between TJD 1385.8966-1395.4800 in orbit 13 and TJD 1396.6050-1406.2925 in orbit 14 are scientifically useful, and all data at other epochs in this sector was cut. Sector 4 on the other hand was plagued by an instrument anomaly between TJD 1418.54 and TJD 1421.21, where communication was lost between the instrument and satellite. As a result, no data or telemetry was collected for this period, and some systematic trends were introduced following activation of the on-board heaters. Additional strong glints between TJD 1422.2297 - 1423.5020 (orbit 15) and TJD 1436.1047 - 1436.8353 (orbit 16) also plague some of the light-curves in sector 4 (particularly those on camera 4), however the amplitude and duration of these appear to vary between different targets, so may be better examined on a case by case basis.
Following the removal of these sections of unreliable data, a more automated method was required to identify and remove additional epochs of increased spacecraft scatter on an epoch-by-epoch time-frame, such as those around the regular spacecraft momentum dumps. This was completed by generating scattering quality masks based on the engineering "quaternion" data, using a similar method to that described by \citet{Vanderburg2019TESS858} to prepare sector 3 30min cadence data in the discovery of multiple super-Earths around HR 858. The \textit{TESS} quaternion data\footnote{Available at https://archive.stsci.edu/missions/tess/engineering/} consists of 2s-cadence time-series data for each sector, describing attitude changes in three primary vectors (Q1, Q2, Q3) based on deviations from a selection of local guide stars. This provides a sector-specific overview of the spacecraft attitude and thus allows the generation of scatter-based quality masks for all targets in the \textit{TESS} aperture. An example of this data (both raw and binned into 30min bins) can be seen in Fig \ref{fig:Quaternions_compilation}. To identify epochs with excessive scatter, the standard deviation for each vector was calculated, and any epoch with pointing scatter $\geq$5 standard deviations from the mean was flagged for removal. By combining the results from all three vectors into a single mask, all epochs within 0.01d of these points were removed from the data-set. This step efficiently removed all spurious signals relating to momentum dumps, as well as any remaining short periods of overly large scatter that were not picked up in the initial wider cuts.
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/Quaternions_S1_Compilation.png}
\caption{Q1 Engineering quaternion data for Sector 1, Camera 1 showing a clear increase in scatter around the 2.5d momentum dumps and the loss of fine pointing between TJD 1347-1349. Top: Raw 2s Engineering Quaternion data. Middle: Quaternion data binned into 30min bins to show typical number of affected data points. Bottom: Final quaternions after the removal of known periods of increased scatter and quaternion-based cleaning.}
\label{fig:Quaternions_compilation}
\end{figure}
For sources of particular interest which also possess \textit{TESS} 2min quality flags, it may also be advisable to implement the quality flags from the SPOC pipeline to the 30min data (perhaps in a similar manner to the conservative approach taken by \citet{Bouma2019Cluster7} in building their difference-imaging extraction pipeline for sectors 6 and 7) in order to remove the effects of additional systematics such as cosmic rays. However, given the lack of 2min light-curve availability for many of the objects in this sample, this feature was not implemented in the development of this initial pipeline.
Following the removal of such sector-specific effects, the data was split into separate sections wherever there was a data gap of more than 0.1 days (to reduce the effect of flux jumps often seen after data gaps), except where doing so would result in fewer data points than the prescribed detrending window length discussed below.
\subsection{Detrending of Stellar Variability} \label{detrending}
\subsubsection{Choice of base-detrending method}
Given the challenging range of activity-related and intrinsic stellar variability seen in the light-curves of the young star sample (and the associated difficulty of modelling so many individual light-curves in an initial transit search), it was considered wise to approach the detrending problem from the ground up, rather than necessarily relying on more traditional methods such as the Savitzky-Golay filter \citep{Savitzky1964SmoothingProcedures.}. A number of different detrending techniques were trialled early in the process, including simple low-order polynomial fitting, sinusoidal modelling, Savitky-Golay filtering and general smoothing over a range of window sizes. Of these, low-order polynomial fitting and the smoothing methods proved most successful at recovering injected transit signals (see Section \ref{injected_transits} for more information on the injected transit method used).
A search for a method to combine and improve both of these methods led to LOWESS smoothing, or Locally Weighted Scatterplot Smoothing. This method, developed originally by \citet{Cleveland1979RobustScatterplotsb}, is a local polynomial regression method which works by fitting a low-order polynomial to a subset of the data (the width of which is set by a user-defined window) at each point along the x-axis using weighted least-squares regression. Under the weighted least-squares regression method, points nearer to the data-point being estimated are given more weight than those further away in the window. This weighting is one of the key differences between this method and the more commonly used Savitzky-Golay filter \citep{Savitzky1964SmoothingProcedures.}, and is particularly important for this application given the often swift evolution of young star light-curves. Indeed following the choice of this LOWESS-smoothing method, it was independently highlighted to be one of the best-performing detrending methods for the young-star sample tested by \citet{Hippke2019WotanPython}. In exoplanet literature however no other mention of LOWESS smoothing for exoplanet searches was found, with even the related LOESS smoothing method appearing quite rarely, despite having been used in detrending the TRAPPIST-1 system \citep{Luger2017ATRAPPIST-1} and in the Autoregressive Planet Search of \citet{Caceres2019AutoRegressiveMethodology}.
For this work, the standard tricube weighting function ($w = (1-|x|^3)^3$) was used, and the number of residual-based re-weightings retained as the default value of 3. The delta parameter was retained as $delta = 0.0$ as the sector-by-sector datasets were not overly large, but this can be adjusted if further speed enhancements are desired. Through experimentation a window size of 30 FFI data points (15hrs) was found to yield a good compromise between preserving the shape of injected transits and smoothing the stellar activity and variability of the host stars, except in more rapidly evolving light-curves, where a window size of 20 data points (10hrs) was found to be more appropriate.
\subsubsection{Removal of Peaks and Troughs} \label{peak_cutting}
One of the other key challenges present in using a Box Least Squares (BLS) search for light-curves from young active stars is that the troughs of regular stellar activity (e.g. from the rotation of star-spots) are often picked up as the largest peaks in the BLS periodograms, even after LOWESS detrending. This is particularly the case for rapidly rotating stars, or those with short-period intrinsic variability. Such sources typically exhibit sharp peaks and troughs in the extracted 30min light-curve, which is often highlighted in the BLS search. In an attempt to combat this problem for light-curves with particularly sharp stellar activity, the effect of cutting the peaks and troughs of this stellar activity was tested. To begin with, the peaks were located using the \texttt{find\_peaks} function from the \texttt{scipy.signal} library\footnote{https://docs.scipy.org/doc/scipy/reference/signal.html} \citep{Virtanen2019SciPyPython} using a required prominence of 0.001 and width of 15 data points (7.5hrs). This helped to ensure that only wider peaks generally associated with stellar activity/variability were flagged as peaks or troughs, but the settings may need adjusting for some more complex light-curves. The activity/variability troughs were located using a similar \texttt{find\_peaks} search on a negative version of the light-curve flux. Data points within 0.1d either side of each peak and trough were then cut from the light-curve before LOWESS-detrending was applied. An example of this in practice is shown in Fig \ref{fig:Peak_cut_example} for HIP 32235, a rotationally variable G6V star in the Carina stellar association.
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/Peak_cut_fig.pdf}
\caption{Example of the peak cutting technique in use on the rapidly evolving light-curve for HIP 32235. The applied 20-box partial LOWESS-smoothing model used for detrending the light-curve is also over-plotted in colour. Note that the change in colour between subsequent sections of the LOWESS-detrending shows where the light-curve has been split after any 0.1d gap in the data. This typically occurs after each peak or trough, or where significant detrimental scatter has been removed.}
\label{fig:Peak_cut_example}
\end{figure}
The power of this peak-cutting method is illustrated for the same source in Fig \ref{fig:Peak_cut_periodograms}. When a 0.04 $R_P/R_*$ radius ratio planet is injected into the light-curve (using the method described in Section \ref{injected_transits}) and the light-curve is detrended using the LOWESS-based method described above (boxsize = 20), any signal of the planet is clearly overwhelmed by the 3.84d period rotational variability of the star, as is shown in the top of Fig \ref{fig:Peak_cut_periodograms}. However, applying the described peak-cutting technique to the data easily recovers the signal of the injected 8-day 0.04 $R_P/R_*$ planet, as illustrated in the lower half of Fig \ref{fig:Peak_cut_periodograms}. This technique is thus another powerful tool for pushing down to lower radii in the search for exoplanets around young active stars. In general, this peak-cutting method was found particularly effective for light-curves with sharp oscillations/rotations, or ones which were strongly periodic. However, it is important to note that this technique did not always offer improvements over the non-peak-cut method. A further discussion of when this technique is most effective can be found in section \ref{Peak_cutting_comparison}.
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/HIP_32235_Peak_cut_comparison.pdf}
\caption{BLS periodograms for HIP 32235 following 20-bin LOWESS detrending without (top) and with (bottom) peak cutting implemented. In this case peak cutting displays a clear benefit to the recovery of an injected $0.04 R_P/R_*$ 8d planet in the presence of rapidly evolving (3.84d period) stellar activity. The highest power period is highlighted in blue, with aliases of the same period shown by dotted blue lines.}
\label{fig:Peak_cut_periodograms}
\end{figure}
\subsubsection{Transit masking and light-curve interpolation} \label{transit masking}
Transit masking is a commonly used method in planetary and eclipsing binary science to independently detrend long-term variability in stellar light-curves without changing the shape of the transit curve, and is implemented into most common light-curve manipulation tools (e.g. \citet{Luger2016CURVES,LightkurveCollaboration2018Lightkurve:Python}). However, due to the often rapid evolution of young star light-curves, simply cutting out the data near a suspected transit before detrending can lead to spurious variability signals over the duration of the transit. In order to combat this issue in the developed detrending method a two-step transit masking approach was used which accounts for the brightness variation of the host star during the transit. Firstly, a mask is generated based on a period, epoch and duration for the suspected transit (either user-defined or from a previous BLS search), and the new light-curve generated by removing any points within the selected transit duration. Then, in order to account for the stellar flux variability over the course of the transit, the new holes in this light-curve are refilled using a quadratic interpolation (using \texttt{scipy.interpolate.interp1d}) between the cut points. It is this new flux array featuring interpolated sections over the transit mask that is then used in the LOWESS-detrending step, and in turn divided out to form the final detrended light-curve. The importance of using such a method is illustrated using the source DS Tuc A in Fig \ref{fig:Transit_mask}. It is immediately obvious that not taking into consideration the variation in stellar flux around the epoch of the first cut transit (approx TJD 1132) would lead to a considerably different transit shape after the main LOWESS-based detrending was is undertaken. A full discussion of the full detrending of DS Tuc A and the resulting recovery of the young exoplanet DS Tuc Ab can be found below in Section \ref{Recovery of DS Tuc A b}.
The authors make available a basic version of this detrending code online. \footnote{https://github.com/mbattley/YSD}
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/Transit_mask_new.pdf}
\caption{Example of interpolation over transit masked areas of the light-curve for DS Tuc A in order to best preserve the stellar activity signature through these segments. Original data points are shown in black, while interpolation over the removed transits is shown in red. The red box on the right shows detail of the area in the vicinity of the first transit.}
\label{fig:Transit_mask}
\end{figure}
\subsection{Transit searching algorithm} \label{bls}
The standard Box Least Squares method was used to search for periodic transit-like signals in the detrended light-curves. This method, originally developed by \citet{Kovacs2002ATransits}, fits a series of box-shaped dips at a range of periods in order to generate a periodogram comparing the relative strengths of the different period hypotheses. The specific implementation used in this code is the \texttt{astropy.timeseries.BoxLeastSquares} method\footnote{https://docs.astropy.org/en/stable/api/astropy.timeseries.BoxLeastSquares.html}, which is a \texttt{python} implementation of the computational method described by \citet{Hartman2016VARTOOLS:Data}. In this work the strongest BLS peak and the two next strongest non-harmonic peaks were investigated.
\subsection{Injected transits} \label{injected_transits}
In order to test the sensitivity of this new detrending method to finding planets around young stars, a series of model planet transits were injected into the light-curves before detrending. For this analysis eight different orbital periods were tested (1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0 and 14.0 days), with the epoch chosen randomly for each injection. Note that unlike ground-based surveys, \textit{TESS} does not exhibit 1-day systematic errors due to the Earth's rotation which could adversely affect these period choices. Rather than injecting specific planet sizes, five planet to star ratios $R_p/R_*$ were tested in this sensitivity analysis: 0.1, 0.075, 0.05, 0.04 and 0.03, beyond which recovery was observed to be quite rare. For reference, around a sun-like star a radius ratio of 0.1 corresponds to an approximately Jupiter-sized planet and $R_p/R_*$ = 0.03 corresponds to a sub-Neptune-sized planet. Stellar parameters for each star were retrieved from TIC v8 \citep{Stassun2019TheList}. Where possible, orbital separation for each planet was then derived from Kepler's Third Law. If information on a star's mass or radius was not available, the corresponding planet was assigned an orbital separation of 17.0$R_*$, representing the average orbital separation for planets with 8d periods (in the middle of the period range) on the NASA Exoplanet Archive.\footnote{https://exoplanetarchive.ipac.caltech.edu/} Planet transits were generated using \citet{Kreidberg2015Batman:Python}'s \texttt{batman} python implementation of \citet{Mandel2002AnalyticSearches}'s transit model, assuming non-linear limb-darkening with coefficients [0.5, 0.1, 0.1, -0.1]. After injection, each light-curve was detrended using a standard 30-bin run of the LOWESS-smoothing pipeline (without peak-cutting implemented). Injected signals were considered to have been 'recovered' if they appeared as one of the three highest peaks in the BLS periodogram, ignoring harmonics of the maximum peak.
\section{Results from sectors 1-5} \label{Results}
Despite 30min DIA light-curves only being available for 256 of the BANYAN objects within the first five sectors of \textit{TESS} data, a wide range of interesting activity was observed, from the recovery of known young exoplanets and \textit{TESS} objects of interest (TOIs) to an eccentric eclipsing binary system and a large variety of unusual rotation and activity profiles. Even this relatively small sample clearly demonstrates the unusually large variation in light-curves of young stars compared to their older counterparts, and consequently helps to explain why far fewer planets have thus far been found around stars of these ages. The conducted sensitivity analysis goes one step further, investigating the comparative recovery rates for different combinations of injected period and planetary radius. Some of the most interesting results are summarised below in this section.
\subsection{Recovery of confirmed young exoplanets} \label{Recovery of DS Tuc A b}
One of the most promising initial results from the application of this pipeline on the 30min data was the recovery of both of the known transiting exoplanets found around young stars in sectors 1-5: DS Tuc A b \citep{Benatti2019AA,Newton2019TESSAssociation} and AU Mic b \citep{Plavchan2020AUPaper}. The recovery of DS Tuc A b is described in detail here as an example of the full pipeline in use.
DS Tuc A (TIC 410214986/TOI 200.01/HIP 116748 A) was observed in Sector 1 of the \textit{TESS} observations, carried out between 25th July - 22nd August 2018. It is a G6V type star known to be associated with the 45 Myr Tucana-Horologium association \citep{Zuckerman2000IdentificationFormation}. DS Tuc A fell on camera 3 of the instrument, and yielded approximately 27 days of photometry.
Interestingly, \citet{Newton2019TESSAssociation} admit in their work that the candidate signal was actually found by human eyeballing around a spurious activity-induced periodicity peak flagged by a run of the SPOC Transiting Planet Search (TPS) module on the 2min PDC-SAP data. However, a later archival TPS run of the SPOC pipeline (after the planet candidate was announced to the community as TOI 200.01) was seen to detect a periodic transit crossing event which passed all of \citet{Newton2019TESSAssociation}'s false positive tests. The planetary candidate was later confirmed using additional photometry, spectroscopic methods and high contrast imaging.
\citet{Benatti2019AA} independently reprocessed the TESS data for this object using improved stellar parameters (including crucially accounting for dilution from DS Tuc B) and fitted two different models to determine planetary parameters. The first model involved modelling only the first two transits (on account of the large pointing jitter around the 3rd transit) using \texttt{PyOrbit} \citep{Malavolta2016TheCluster} to complete a simultaneous fit for modulation and a transit signal, along with \texttt{emcee} \citep{Foreman-Mackey2012Emcee:Hammer} and \texttt{PyDE} \citep{Storn1997DifferentialSpaces} to establish the most likely planetary parameters. Alongside this they tested modelling all three transits with the \texttt{batman} package \citep{Kreidberg2015Batman:Python} after applying a 0.55d 3rd order running polynomial to flatten the light-curve and analysed the posterior with \texttt{emcee}. While both methods yielded consistent results, they eventually adopted the planetary parameters from the first method and found a best-fit solution of a 0.5$R_{Jup}$ planet. \citet{Benatti2019AA} confirmed the planetary nature of this object using radial velocities from the HARPS spectrograph.
In this work, the planetary signal was highlighted as the highest peak of the BLS periodogram in a 20-bin LOWESS-partial run of the standard detrending pipeline, however the transits were also clearly visible by eye after the 30-bin LOWESS-partial run. Noting that the third transit fell in the period of heightened pointing jitter between TJD 1347-1349, this section of the data was unmasked for this target. The recovered period was 8.14d, in agreement with the accepted value from \citet{Newton2019TESSAssociation}. Using the period and epoch derived from the BLS search, the transit masking and light-curve interpolation technique described in section \ref{transit masking} was applied, completing the clear detrending and transit recovery of DS Tuc A b presented in Fig \ref{fig:DS_Tuc_A_detrend}. Note that in the bottom panel the light-curve has been folded by 8.138d, the accepted planet period from \citep{Newton2019TESSAssociation}. Interestingly, the use of the peak-cutting technique for this object was found to increase the significance of the true transit period in the 30-bin case (changing the 8.14d period-peak in the BLS periodogram from insignificant to the 6th strongest after peak-cutting), but decreased its significance for the 20-bin LOWESS-detrend.
\begin{figure}
\includegraphics[width = \columnwidth]{Figures/DS_Tuc_A_detrending_new.pdf}
\caption{Example of the developed detrending pipeline in use, showing the recovery of the 45Myr exoplanet DS Tuc A b. Top: The original sector 1 light-curve for DS Tuc A with overplotted 20-bin LOWESS detrending in blue, including interpolation over suspected transits. Middle: BLS Periodogram for the the light-curve after detrending, with the peak period of 8.138d highlighted in blue (n.b. aliases of this period are shown by the blue dotted lines). Bottom: the resulting light-curve after the detrending pipeline has been applied, folded by the maximum peak of the BLS Periodogram}
\label{fig:DS_Tuc_A_detrend}
\end{figure}
The clear recovery of this Neptune to Saturn-sized planet using the 30min data alone bodes well for future exoplanet candidate discoveries from the Full Frame Images, thus demonstrating the wealth of knowledge to be gained from these images.
Unfortunately data for AU Mic was not extracted by \citep{Oelkers2018PrecisionApproach}'s 30min pipeline, however application of this pipeline to the 2min data for AU Mic easily recovered the signal of the 8.46d planet proposed by \citet{Plavchan2020AUPaper}, as shown in Fig \ref{fig:AU_Mic_Detrend}. In this case a 20bin-lowess smoothing run revealed an alias 8.46d period as the third highest peak in the periodogram. In this particular case the peak-cutting option did not aid recovery of the planet substantially since the improvements of cutting the sharp troughs were balanced by the inadvertent cutting of the first transit. Nonetheless, the 8.5d signal remained the third highest peak after 20bin-LOWESS smoothing was applied to the peak-cut light-curve.
\begin{figure}
\includegraphics[width = \columnwidth]{Figures/AU_Mic_Detrending.pdf}
\caption{A 20bin-LOWESS smoothing run on the \textit{TESS} 2min cadence data for AU Mic, showing the recovery of the 8.46d planetary signal first identified by \citet{Plavchan2020AUPaper}. Format similar to that explained for Figure \ref{fig:DS_Tuc_A_detrend}, except that the final light-curve is folded by the third highest period, as highlighted in blue in the BLS periodogram.}
\label{fig:AU_Mic_Detrend}
\end{figure}
\subsection{Retrieval of other \textit{TESS} objects of interest}
A number of other interesting signals were independently found using this pipeline, including a pair of additional \textit{TESS} Objects of Interest (TOIs). In the interest of independence these signals were found with no prior knowledge of the TOIs in these sectors, but instead candidate signals highlighted in the BLS search in this work were cross-matched with the TOI list at a later date. This resulted in the recovery of two known TOIs: TOI 447.01 and TOI 450.01.
For TOI 447.01, a repeated dip of approximately 15mmag was highlighted by the BLS search in sector 5 with a period of 5.528 days around the F5/6V star HD 33512/TIC 14091633 (see Fig \ref{fig:Other_EB1}). This signal was uncovered using the a standard 30-bin LOWESS run of the pipeline, without peak-cutting. HD 33512 is a relatively bright (T=8.8) F5/6V star in the Octans association, giving it an approximate age of 35$\pm$5 Myr \citep{Gagne2018BANYAN150pc}, which would make this an interesting target for follow-up of young evolving systems if the planetary hypothesis was correct. As a TOI this target underwent follow-up by the TFOP team, and though initially seemed positive from the \textit{TESS} data alone, it was eventually flagged as a False Positive (FP) by the TFOP working group due to the observed linear drift in RV measurements, relatively large radius, and changing width of FWHM.\footnote{https://exofop.ipac.caltech.edu/tess/target.php?id=14091633} Nonetheless, this object is a likely long-period binary, so may still be an interesting system for the study of young binary systems.
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/HD_33512_overview.pdf}
\caption{Recovery of TOI 447.01/HD 33512, a likely long-period binary in the 35Myr Octans Association. In this case the maximum period was found to be 5.528d. Format after that explained in Fig \ref{fig:DS_Tuc_A_detrend}.}
\label{fig:Other_EB1}
\end{figure}
The second TOI independently recovered was TOI 450.01, around the source 2MASS J05160118-3124457/TIC 77951245. This source was observed in sectors 5 and 6 of \textit{TESS} observations, however alike to TOI 447.01 only sector 5 light-curves were available for it from the DIA 30min pipeline. In this work the candidate signal was initially revealed by a 30-bin LOWESS-partial run of the main detrending pipeline on sector 5. Two approximately 40mmag transits were observed in the flattened light-curve, with a recovered period of 10.74 days and initial epoch of 1443.2d, as shown in Fig \ref{fig:Other_EB2}. The parameters derived from this 30min data are slightly different to those derived from the 2min data by the SPOC TOI analysis (P = 10.7148$\pm$0.0001, depth = 63.6$\pm$0.8, epoch = 2458443.1686 $\pm$ 0.0003), however this is not surprising given the much lower density of points in each transit signal (15 times fewer). TIC 77951245 is an M4 star of T = 12.375 in the 42 Myr Columba association. Based on the depth of the observed signals and the radius of the star, this signal would correspond to an approximately Jupiter-sized planet if the planetary hypothesis was correct. However, follow-up of this object by the TFOP working group eventually revealed it to be a near-equal mass spectroscopic binary, based on HRS measurements on the 10m SALT telescope on 10th August 2019.\footnote{https://exofop.ipac.caltech.edu/tess/target.php?id=77951245}
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/EB2_J0516-3124_overview.pdf}
\caption{Recovery of TOI 450.01/J0516-3124, a spectroscopic binary in the 42 Myr Columba Association. In this case the maximum period was found to be 10.7d. Format after that explained in Fig \ref{fig:DS_Tuc_A_detrend}, except that overplotted LOWESS-detrending in the top panel has bins of 30 rather than 20.}
\label{fig:Other_EB2}
\end{figure}
These two TOIs along with the new confirmed exoplanet DS Tuc A b discussed in Section \ref{Recovery of DS Tuc A b} represent all three of the TOIs highlighted by the main 2min SPOC pipeline for this selection of young stars. This illustrates the fact that this pipeline is working as effectively as the main SPOC 2min pipeline for these young objects, further emphasising its potential for finding candidate signals around stars present only in the 30min cadence data.
\subsection{Eclipsing Binaries}
A number of clear young eclipsing binaries were also revealed by this survey, perhaps the most dramatic of which being HD 28982. This highly eccentric eclisping binary system was revealed in this work by a 30-bin run of the LOWESS-based detrending (without peak cutting) applied to sector 5 of the \textit{TESS} 30min DIA data (see Fig \ref{fig:full_page_lc_table}, panel a). A period of 5.97 days was clearly highlighted as the maximum power period in the BLS periodogram, and two distinctly different duration and depth transit signals were visible. HD 28982 is associated with the AB Doradus Moving Group (ABDMG), which is approximately 150Myr old \citep{Gagne2018BANYAN150pc}.
\begin{figure*}
\includegraphics[width =\textwidth]{Figures/Full_page_lc_table_larger_font.pdf}
\caption{Sector 1-5 light-curve zoo. Column 1 shows the original light-curve for each star with a 20-bin LOWESS-smoothed fit overplotted in blue. Column 2 shows the BLS periodogram for each source after LOWESS-detrending, with the highest power period highlighted in blue (except for J0635-5737 - panel d - where a shorter alias is chosen for clarity), and aliases of this period shown with dotted blue lines. Column 3 shows the flux after 30-bin LOWESS-detrending for each source folded by the highest power period. Individual sources from top to bottom: Panel a: HD 28982, an eccentric EB in ABDMG (approx 150Myr) - DIA 30min data, Sector 5; Panel b: HD 28982 in \textit{TESS} 2min data; Panel c: J0529-2852/TIC 31281820 in COL (approx 42 Myr), demonstrating strongly periodic drops - DIA 30min data, Sector 5; Panel d: J0635-5737/TIC 348839788 in ABDMG (aprox 150 Myr), demonstrating 'double variation' in its rotation - DIA 30min data, Sector 1; Panel e: J0552-5929/TIC 350712873 in THA (approx 45 Myr), demonstrating very fast rotation - DIA 30min data, Sector 1.}
\label{fig:full_page_lc_table}
\end{figure*}
Because it has long been known to be a 'bona-fide' member of this moving group it was included in the initial \textit{TESS} CTL \citep{Stassun2019TheList} and received 2min coverage in the main \textit{TESS} survey. This provides an interesting opportunity to compare the 2min and 30min data for this young object. After retrieval from MAST, the 2min data was subjected to the same detrending as the 30min data by expanding the number of bins for each LOWESS-based detrending step to 450, 15 times as many as in the 30min analysis. This revealed the detrended light-curve shown in panel b of Fig \ref{fig:full_page_lc_table}. Aside from the obvious increase in signal to noise and visually more obvious signal provided by the 2min data, the 30min data clearly provides sufficient information to constrain the EB period and durations. However alike to the TOIs, the 30min data suffers in terms of depth accuracy, likely due to exposure smearing from the longer 30min cadence data. In addition, the 30min cadence data from the DIA pipeline is more affected by scatter than the 2min data, likely due to a combination of pointing scatter and improperly corrected scattered light. Care must therefore be taken when evaluating planetary radii for planet candidates based on 30min data alone, and may be better left to higher-cadence photometry in follow-up observations.
\subsection{Rotation and activity} \label{rotation}
Rotation and spurious stellar activity were the cause of the strongest peaks in many of the stars viewed in this sample, which helps to explain why searching for planets around young stars is so much harder than around many older stellar hosts. As a first step in characterisation of this rotation and activity, the period of each star's main stellar variability/activity was also recorded using the generalized Lomb-Scargle method \citep{Lomb1976Least-squaresData} implemented into \texttt{python} as the \texttt{LombScargle} function in the \texttt{python} \texttt{astropy library}\footnote{https://docs.astropy.org/en/stable/api/astropy.timeseries.LombScargle.html} \citep{TheAstropyCollaboration2013Astropy:Astronomy,AstropyCollaboration2018ThePackage}. Furthermore, the main amplitude of the primary variability was determined in the peak-cutting step discussed in Section \ref{peak_cutting}, using the peaks and troughs identified for removal. A wide variety of rotation and activity curves were observed in this work, broadly separated into four categories: near-uniform periodic, periodic but evolving, aperiodic and fast rotators. Amplitudes of oscillation varied from 0.02\% to 9\%, while primary periods of flux variation varied from 0.122d (e.g. J0552-5929 - panel e of Fig \ref{fig:full_page_lc_table}) to non-variable over the 27-day time-period.
The first of these categories, those light-curves which exhibit near-uniform variation in amplitude over a set period, are well illustrated by the sinusoidal curve of sources such as HIP 1993 and HIP 1113 (Fig \ref{fig:sinusoidal_var}), alongside strongly periodic drops in sources like 2MASS 05292529-2852274 (see panel c of Fig \ref{fig:full_page_lc_table}). These types of sources are theoretically ideal for detrending, as they can be easily modelled by quasi-periodic smoothing functions, especially when the period of flux variation is over periods of two or more days. Furthermore, the peak-cutting method discussed in Section \ref{peak_cutting} was often observed to aid the retrieval of injected transits for these periodic sources. However, very fast rotation of this type still provides difficulties, as discussed below.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/Combined_fig_for_HIP_1993_and_HIP_1113_no_titles.pdf}
\caption{Examples of rotation/activity type 1: Near-uniform sinusoidal variation - HIP 1993 (top) and HIP 1113 (bottom) in Sector 1}
\label{fig:sinusoidal_var}
\end{figure}
Another very important category of variation seen are those light-curves with strong periods but obvious variation in flux amplitude. This type of light-curve is well illustrated by sources such as HIP 105388 (observed in \textit{TESS} sector 1) and AB Pic (observed in all southern hemisphere sectors - 1-13) - see Fig \ref{fig:HIP_105388_AB_Pic}. The oscillation amplitudes of these sources are both observed to considerably change over the course of a single sector. Indeed, in the case of AB Pic, the amplitude of the primary oscillations changes from 0.01\% to >0.03\% over the course of the five sectors of data for which the 30min DIA data exists. Other sources such as HIP 1481 appear to exhibit 'beating'-like behaviour in their light-curves, going through periods of more and less intense oscillations yet with similar periods throughout. Thus for these objects, while the strong periodic nature of their oscillations makes the periods easy to identify, blind removal of these varying oscillations is difficult on a wide-scale basis. Nonetheless, the peak-cutting technique discussed in Section \ref{peak_cutting} particularly beneficial for sources of this type, such as HIP 32235 (Fig \ref{fig:Peak_cut_example}) and HIP 105388 (top, Fig \ref{fig:HIP_105388_AB_Pic}).
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/Combined_fig_for_HIP_105388_and_AB_Pic_no_titles.pdf}
\caption{Examples of rotation/activity type 2: Periodic with rapidly evolving amplitudes - HIP 105388 (top) and AB Pictoris (bottom) in Sector 1}
\label{fig:HIP_105388_AB_Pic}
\end{figure}
Alongside these periodic light-curves, a smaller number of aperiodic variations (or perhaps those with rotation or variation periods much longer than the 27 day observation time) were observed. Most of these variations were sufficiently long-period to be easily removed by the base 30-bin LOWESS-smoothing method (as was the case for J0449-5741 and HD 35289 (Fig \ref{fig:Aperiodic})), or are sufficiently aperiodic that any activity would not overcome true periodic transit signals on the BLS periodogram. However, some more complex aperiodic cases exist, such as TIC 348839788 (Panel d, Fig \ref{fig:full_page_lc_table}), where two separate rotation and activity profiles seem to be apparent - one aperiodic larger amplitude evolution clearly evident in the original light-curve, and one much faster rotation-based signal with a period of approximately 0.27 days (see panel d, Fig \ref{fig:full_page_lc_table}. This 'double variation' is present in a small number of other sources too, so should be considered carefully in future detrending efforts. For such sources this will likely require at least two detrending steps (a wider smoothing to remove the large-scale activity, followed by smaller-scale smoothing/modelling of the rapid rotation) in order to yield a flat pre-BLS-search light-curve.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/Combined_fig_for_J0449_5741_and_HD_5289_no_titles.pdf}
\caption{Examples of rotation/activity type 3 - Aperiodic variations. DIA 30min-cadence light-curves for J0449-5741 (TIC 220425740, top) and HD 35289 (bottom) in \textit{TESS} Sector 5}
\label{fig:Aperiodic}
\end{figure}
Perhaps the greatest challenge in the search for young exoplanets is the case of rapidly rotating stars such as HIP 22295 (Panel e, Fig \ref{fig:full_page_lc_table}) and CD-46 287 (bottom panel, Fig \ref{fig:Sensitivity_Comparison}. These fast rotators are unfortunately quite common in stars of such young ages (indeed at 52 stars (20.3$\%$) in this sample had rotation periods of less than 1 day), likely due to leftover angular momentum from their formation \citep{Prialnik2009AnEvolution}. For the fastest rotators, 30min data alone struggles to untangle rotation from any potential transit signals, especially when rotation periods of less than 1 day are coupled with significant changes in amplitude. In order to more effectively dissociate such rapid rotation from transit signals, using more sensitive 2min cadence data is preferred, likely coupled with more intelligent modelling than simple smoothing methods. The planned \textit{TESS} 20s cadence data would also aid this effort, as rotation profiles will become more carefully defined and more detail of transit ingresses/egresses may become apparent. However, techniques to overcome this fast rotation challenge still need to be developed.
\subsection{Sensitivity analysis results}
\subsubsection{Overall results}
The conducted sensitivity analysis revealed a number of interesting results, for which an overall summary is presented in Table \ref{tab:Sensitivity_overall} and Figure\ref{fig:Overall_Heatmap}. What is immediately apparent is the steep drop-off in recovery as the $R_P/R_*$ radius ratio decreases, falling from 77.6$\%$ at a radius ratio of 0.1 down to 20.4$\%$ at 0.03. This is to be expected as the inherent scatter and leftover variability amplitude of the light-curves steadily overcomes the signal of the smaller injected planets.
\begin{table}
\addtolength{\tabcolsep}{-2pt}
\centering
\caption{Results from complete Sector 1-5 sensitivity analysis for the 256 stars in the BANYAN young star sample with DIA FFI data. Percentage recovery is presented both overall and in each individual sector. Note that results from the 14d injections were excluded from the percentage recoveries due single transits frequently being present at this period. Total number of individual sources for each sector are: Sector 1: 74; Sector 2: 77; Sector 3: 75; Sector 4: 120; Sector 5: 138.}
\label{tab:Sensitivity_overall}
\begin{tabular}{ccccccc}
\hline
$R_P/R_*$ & Overall ($\%$) & S1 ($\%$) & S2 ($\%$) & S3 ($\%$)& S4 ($\%$) & S5 ($\%$)\\
\hline
0.1 & 77.6 & 79.7 & 80.3 & 73.1 & 69.8 & 84.3\\
0.075 & 61.7 & 61.6 & 64.0 & 56.8 & 51.4 & 72.2\\
0.05 & 40.9 & 43.8 & 36.9 & 39.6 & 31.6 & 50.31\\
0.04 & 30.2 & 31.3 & 30.4 & 31.1 & 20.0 & 38.0\\
0.03 & 20.4 & 24.3 & 18.4 & 22.1 & 11.1 & 26.6\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[width =\columnwidth]{Figures/Overall_heatmap.pdf}
\caption{Overall sensitivity analysis results (Sectors 1-5) by injected period and planetary radius. The numerical contents of each box corresponds to the overall percentage recovery of injected planets for each particular combination, with colour-gradient of the same added for clarity.}
\label{fig:Overall_Heatmap}
\end{figure}
Interestingly the recovery of injected signals was not entirely consistent across the different sectors viewed in this analysis. This discrepancy appears to result from two major factors: comparative sector systematics and evolving light-curve amplitudes for a small subset of stars. Regarding the first factor, Sector 4 was the worst affected, with recovery rates for all radius ratios consistently far below the average. This is likely due to the strong reflected light glints seen at the end of each orbit for some sources (particularly those on camera 4), which are harder to systematically remove without also removing useful data from other light-curves. Conversely, recovery was significantly better in Sector 5 compared to the other sectors, with 26.6$\%$ of injected planets still retrieved down to radius ratios of 0.03. This inter-sector variation is well illustrated by the light-curves for J0455-6051/TIC 55651278 (an M5 star in the AB Doradus Moving Group), where injected planetary signals were recovered down to radius ratios of 0.075, 0.05, 0.075, 0.1 and 0.03 in sectors 1, 2, 3, 4 and 5 respectively in the 2-6 day period cases. In this case such variation was caused by a combination of evolving flaring-type stellar variability coupled with extra noise in Sector 4. On the other hand, AB Pictoris (a K1V star in the Carina Association) demonstrates the second source of sector-dependent sensitivity, with injected planetary signals recovered down to steadily larger radius ratios of 0.04, 0.04, 0.075, 0.075 and 0.1 in sectors 1-5 due to the evolving stellar variability increasing in amplitude over time. While differences in the systematics of each sector can reasonably be expected to reduce over time as the \textit{TESS} satellite pointing is refined, the evolution of individually active sources will remain a challenge not only for \textit{TESS} but also for future missions such as \textit{PLATO}.
A slightly more complex trend was observed when varying the injected planetary periods, as is highlighted by the overall heat-map in Figure \ref{fig:Overall_Heatmap}, and the sector-by-sector heat-maps in the Appendix. For all sectors except Sector 2 (where a 2-day period was preferred), the recovery peaked around a 4-day injected period for the deepest (0.1$R_P/R_*$) transits, unlike the shortest 1.0 day period that may be initially expected. However as the injected planet radii decreased a more standard drop in recovery from 1-14d was observed. Such behaviour suggests that this form of analysis is most sensitive to larger planets in the 2-6 day period regime, and most sensitive to closer-orbiting planets as radii decrease further. In case the observed drop for 1-day planets was due to 1-day Earth-related systematics such as Earthshine, the injection/recovery analysis was repeated with a 1.1d period planet, however similar recovery rates were observed to the 1-day case. Instead this discrepancy may be related to a combination of the increased activity of young stars coupled with the small number of data points per transit for such short-period planets. The primary reason for the much lower recovery of planets with injected periods of 14-days was that data gaps and randomly injected epochs frequently led to single transits appearing in the data-set. This same effect was also observed for some 12-day injections. Some interesting sector-to-sector variations are also clear in the sector-specific heat-maps (see Figures \ref{fig:Sensitivity_S1}-\ref{fig:Sensitivity_S5}), with Sector 4 exhibiting the lowest recovery rates (especially for the smallest radius planets), and Sector 5 the highest.
Another useful product of this sensitivity analysis is the individual sensitivity to planet detection for each star. An overview of this table is presented in Table \ref{tab:Full_Sensitivity_Table}, with a full version available online.
\begin{table*}
\addtolength{\tabcolsep}{-1pt}
\centering
\caption{Full sensitivity analysis table for each of the 256 stars with DIA light-curves. Includes information on the highest likelihood period recovered for each star in every sector it appears, with injected1-14 day period planets and $R_P/R_*$ radius ratios from 0.1 to 0.03. Full table available online.}
\label{tab:Full_Sensitivity_Table}
\begin{tabular}{ccccccccccc}
\hline
Target ID & RA (deg) & Dec (deg) & Sector & Injected Period (d) & Radius Ratio & ... & Log likelihood & Max Period (d) & Recovered? & Notes\\
\hline
2M0123-6921 & 20.79 & -69.36 & 1 & 1.0 & 0.1 & ... & 0.00121 & 2.65 & TRUE & Alias \\
2M0123-6921 & 20.79 & -69.36 & 1 & 1.0 & 0.075 & ... & 0.00125 & 3.98 & TRUE & Alias\\
... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ...\\
WX Col B & 84.30 & -42.72 & 5 & 14.0 & 0.04 & ... & 2.57e-5 & 12.52 & FALSE & - \\
WX Col B & 84.30 & -42.72 & 5 & 14.0 & 0.03 & ... & 1.95e-5 & 11.22 & FALSE & -\\
\hline
\end{tabular}
\end{table*}
\subsubsection{Rotation period vs recovery}
\begin{figure*}
\includegraphics[width =\textwidth]{Figures/Rot_P_vs_depth_larger.pdf}
\caption{Stellar rotation period vs recovery depth for a 4-day injected period planet across all targets and sectors. Design similar to Figure \ref{fig:Overall_Heatmap}}
\label{fig:Rot_P_vs_depth}
\end{figure*}
As has been seen in Section \ref{rotation}, a wide range of rotation periods was observed for stars in this sample. This provides an interesting opportunity to test the relationship between rotation rate and recovery, which is especially important given concerns about finding planets around swiftly rotating stars and those with rotation periods near the injected planetary period. In order to investigate this relationship, a injection/recovery analysis of rotation rate vs depth recovered for each target was conducted for each injected period. A typical example of one of these plots is shown in Figure \ref{fig:Rot_P_vs_depth} with an injected period of 4.0 days. Somewhat counter-intuitively, while there is a slight skew towards larger radius ratios below rotation periods of 5 days, there is no significant evidence that the overall recovery depth of injected transits is a function of rotation period, with recoveries down to radius ratios of 0.03 (and complete non-recoveries) across the entire period range. In addition, recovery depth does not seem to be detrimentally affected by being close to the injected period (4 days in this case), as similar recovery was observed for every injected period. Note that many variations between individual period ranges can be explained by the relatively small number of targets in this sample, especially in the 10-11d period case where only eight targets were present with periods in this range. However, it should be noted that 60$\%$ of all unrecovered signals were from targets with rotation periods of less than 1 day. This constituted 40$\%$ of all injected signals into targets with these rotation periods. Furthermore the vast majority (79$\%$) of the recovered light-curves with rotation periods <1 day were only found down to radius ratios of 0.075$R_P/R_*$). This suggests that searching for small planets (<0.075$R_P/R_*$) around stars with rotation periods less than 1 day is likely futile until better techniques are developed to detrend this fast rotation. However, since failures for all other rotation periods were consistent at 0-3 light-curves per 1-day period interval (e.g. 4-5d), searching for planets around stars with rotation periods longer than 1 day shows promise, even down to 0.03$R_P/R_*$ planets.
\subsubsection{Effectiveness of the peak-cutting technique} \label{Peak_cutting_comparison}
Since the peak-cutting technique exhibited variable effectiveness according to the shape of the light-curves in building the pipeline, much of the prior analysis was undertaken without the peak-cutting option applied. However, to test the wider effectiveness of this option and evaluate where its application was most useful, a comparison study was undertaken for all targets in Sector 1, both with and without peak-cutting. For this comparison test planets with a set period of 8.0d and radius ratios of $R_P/R_* = 0.1-0.03$ were injected into each of the light-curves in Sector 1. Light-curves were then detrended using the standard 30-bin LOWESS-detrending method described above and searched through using the standard BLS method.
Of the 74 Sector 1 targets in the sample with light-curves available from \citet{Oelkers2018PrecisionApproach}'s DIA FFI pipeline, the basic peak-cutting technique failed for 13-15 of the objects (depending on the injected radius ratio). In all cases this was due to peaks and/or troughs not being located by the automatic \texttt{find\_peaks} function, typically because the light-curves were simply too flat to exhibit any significant peaks or troughs. Of the remaining 59-61 light-curves, recovery with peak-cutting was in general comparable, or slightly better than, the recovery of planets when peak-cutting wasn't applied (as is summarised in Table \ref{tab:Peak_cutting_comparion}). The one exception to this was the 0.1$R_P/R_*$ case, which was caused by the peak-cut analysis failing for three of the original 0.1 radius ratio light-curves.
\begin{table}
\centering
\caption{Comparison between percentage recovery of injected planets in Sector 1 for both in the original and peak-cut light-curves.}
\label{tab:Peak_cutting_comparion}
\begin{tabular}{cccc}
\hline
Radius Ratio & Number of lcs & Original ($\%$) & Peak-Cut ($\%$) \\
\hline
0.1 & 61 & 90.2 & 82.0\\
0.075 & 59 & 62.7 & 64.4\\
0.05 & 59 & 39.0 & 40.7\\
0.04 & 60 & 31.7 & 35.0\\
0.03 & 60 & 25.0 & 25.0\\
\hline
\end{tabular}
\end{table}
However, it is when looking at individual targets that the power of the peak-cutting technique is most evident. For seven of the targets in Sector 1, the use of the peak-cutting technique yielded significant improvements in the recovery of smaller injected radius ratios. The most significant improvements were seen for the object J0247-6808, where injected planets were recovered down to a radius ratio of 0.03$R_P/R_*$, despite not being recovered for any radius ratio in the non-peak-cutting case. Similarly in the case of HIP 32235 the depth recovered dropped to 0.03 from 0.075 in the non-peak-cut analysis. Similar improvements (though with less significant drops in recovered depth) were observed for RBS 38, TYC 8895-112-1, J0346-6246, J0414-7025 and J2231-5709.
There were two different reasons why recovery was improved for these objects - one showing the technique working as designed and the other a fortuitous side-effect. The former can be seen in HIP 32235, RBS 38, TYC 8895-112-1 and J2231-5709 where each light-curve exhibits sharp variability peaks with periods of order 3-8 days and amplitudes greater than 2.5$\%$. In this case the technique aids recovery of the planets by successfully cutting the sharp turning points of these light-curves which were previously leftover as false-transits after the LOWESS-smoothing step. It is these types of light-curve variablity (approximately $3 < P_{rot} < 8$ days; Amplitude $> 2.5\%$) which are best-handled by using the peak-cutting technique. All light-curves with rotation periods of 3-8 days which weren't improved by peak-cutting were later found to have had at least one of their peaks cut, however their depths reached were not affected.
The other three light-curves improved by the use of this technique (J0247-6808, J0346-6246 and J0414-7025) were aided accidentally, having sections of increased scatter masked as a result of the applied cuts. This is a convenient side-effect of searching for planets among light-curves with intrinsically higher scatter, but less scientifically interesting.
The limitations of this technique were identified by investigating the small selection of stars detrimentally affected by the peak-cutting. The three main failure modes were:
\begin{enumerate}
\item Effectively flat light-curves dominated by scatter
\item Light-curves with activity/variablity periods of $\sim$ 2 days
\item HIP 33737 - a star with a flat-bottomed rotation activity
\end{enumerate}
In the first case, all peak-cut does is remove useful data, since no significant activity-based peaks and troughs were present. Meanwhile in the 2-day activity/variability period case, a significant portion of the data is cut, with the remaining intervals between each cut too short to be effectively flattened by the 20-30bin (10-15hr) LOWESS-smoothing. These two failure modes effectively represent the two limits of usefulness for this technique, of order 15 days and 2 days respectively. The final case (HIP 33737) was a unique one, where the light-curve exhibited unusually flat-bottomed troughs well-handled by the LOWESS-smoothing technique, and thus the peak-cutting only removed useful data, alike to the first failure mode.
Crucially however, for all light-curves which did not fall into one of the three failure modes identified above, the use of the peak-cutting technique was not found to affect the shallowest depth of transit recovered. This alleviates the chief concern about the use of the peak-cutting technique: that the recovery depth may be reduced if transits near peaks and troughs are inadvertently cut. Overall then, this peak-cutting technique shows the greatest effectiveness for stars exhibiting activity/variability of periods 3-8 days, but may be applied to all stars with periods of approximately 3-15 days without significant detriment, unless those light-curves are particularly flat.
\section{Discussion} \label{Discussion}
As this work has shown, young host stars present many extra challenges in comparison to the generally older, less active stars previous exoplanetary searches have been biased towards. Quicker rotation, increased amplitude activity and other strong stellar-based periodicities wash out candidate planetary signals in BLS searches and make finding transiting signals harder using the traditional automated exoplanet-searching tools. Furthermore, by eye the large amplitude variation of many sources such as HIP 105388 effectively hide real transit signals unless careful detrending is applied first.
Nonetheless, the methods presented in this work have shown promise at pushing down to lower-radius planets around young stars. The base LOWESS-detrending method provides a useful combination of smoothing and polynomial fitting and, as demonstrated by \citep{Hippke2019WotanPython}, generally outperforms more traditional exoplanet smoothing methods such as Savitzky-Golay filters for younger stars due to its weighted approach to smoothing. Furthermore, for sharp but evolving oscillations such as those seen for HIP 33235, cutting the peaks and troughs of these oscillations yields a significant improvement in the recovery of injected planets, especially for periods of 3-8 days. After initial recovery, the shape and depth of the retrieved transit is then greatly improved by incorporating the developed activity interpolation over transit gaps.
However, as demonstrated by the sensitivity analysis, the large range in activity type, period and amplitude coupled with the variation in scatter for each source results in significantly different recovery rates for injected planets overall, ranging from sources like CD-46 287 (a K6Ve star in the 35 Myr Octans association, viewed in sector 2) where no planet was recovered even for the largest radius ratio, to HD 202969 (an F8/G0V star also from Octans, viewed in sector 1), where the injected planet was easily recovered across the whole radius ratio range. Comparing these two light-curves (shown in Fig \ref{fig:Sensitivity_Comparison}), the reason for this discrepancy is immediately clear; while the period of the rotational variation in the flux from HD 202969 is much longer than that of the transit duration (and easy to smooth with the 30-bin LOWESS-detrending technique), the rotation period of CD-46 287 is 0.37d, well within the realm of the duration of a planetary transit. Since this pipeline is designed to only remove variation with periods longer than a transit it struggles to remove such short period variations, and as such cannot recover injected planets from such light-curves.
\begin{figure}
\includegraphics[width = \columnwidth]{Figures/Rotation_Detrending_Comparison.pdf}
\caption{Light-curve comparison for HD 202969 (above) and CD-46 287 (below) with an 8.0d, 0.03 $R_P/R_*$ radius ratio injected planet. While the LOWESS-detrending pipeline has no trouble modelling HD 202969 due to its relatively long duration variation, it struggles to detrend the sharp 0.37d period variability of CD-46 287.}
\label{fig:Sensitivity_Comparison}
\end{figure}
Observing other sources for which no injected planet sizes were ever recovered, the major challenges appear to be rotation with periods < 1 day, excess scatter, large amplitude flares or other spurious outlying points in the light-curve. These latter two problems present a problem to automatic detrending due to their unique light-curve by light-curve behaviour. Nonetheless these could conceivably be flagged by shape or sigma-clipping and handled on a case-by-case basis. On the other hand, scatter falls into two categories: time-localised scatter such as the scattered light glints seen at the end of some sector 4 light-curves, and non-localised scatter which affects the entirety of the light-curve. The first of these problems is soluble, and is likely best handled in a similar way to flares and instrumental scatter, by flagging and removing specific time periods which are affected. However, excess scatter which extends to the full extent of the light-curve (which can be in excess of 2\%, as in the case of J0122-2566) likely precludes the use of these light-curves for planet searches. This latter type of scatter was most common for very dim stars in the sample, largely with \textit{TESS} T magnitudes in excess of 13.
The problem of fast rotation is a more systematic issue and is arguably the most important detrending-related challenge presented by young stars. In this work the detrending pipeline struggled to recover injected planetary signals in light-curves with large amplitude flux changes coupled with rotation periods of less than 1 day, largely due to the width of the LOWESS-smoothing being at least 20-bins, or 10hrs wide. Since any variability with a period of 1 day or less will involve at the very least one turning point, such widely-spaced windows can not reasonably be expected to accurately trace and smooth such activity/rotation. Indeed, as discussed in section \ref{rotation}, some of the fastest rotation observed had periods of less than 0.4 days, or less than 10hrs. However, it is not unheard of for verified planets such as Kepler-1283 b \citep{Morton2016POSITIVES} to have transit durations as long as 0.4 days, so dropping the LOWESS-smoothing window to any shorter than this could result in significant distortion of potential transit shapes along with removal of stellar activity signals. It is thus clear that a more targeted method is required for such fast rotators, likely with more intelligent modelling-based methods such as Gaussian Processes \citep[e.g.][]{Gillen2019NGTS1}. Unfortunately however GP methods are much more intensive than a simple smoothing or polynomial based methods, and require a more informed knowledge of a star's characteristics. This makes GP-based techniques very well suited to in-depth followup of candidates, but perhaps applying such methods to the entire stellar sample would be less efficient than simply flagging them in an initial BLS periodogram and analysing them separately. At these early stages it is still worth considering other modelling methods as well (such as fitting a sum of sinusuoids as in \citet{Gillen2019NGTS1}), especially since \citet{Hippke2019WotanPython} found such a range in effectiveness in young star detrending methods. If a simpler algorithmic method can be found to model or detrend these very fast rotators then it could help to speed up wide-field transit searches around young, rapidly rotating stars.
Another interesting challenge highlighted by the sensitivity analysis is the rapid evolution of some young star light-curves. This evolution makes modelling the light-curves more difficult and also means that detrending is more effective in some regions of the light-curve than others. As demonstrated for AB Pictoris, this changes how easy it is to recover injected (or no doubt real) exoplanet signals depending on when in the activity cycle one views the star. If however one gains a better understanding of a star's activity cycle through long-term monitoring and asteroseismology, this evolving activity can become an opportunity for increasing the effectiveness of planet searches by targeting quiet sections of the stellar activity cycle. This knowledge would be crucial for radial velocity follow-up of young host stars, since as \citet{Oshagh2017UnderstandingMeasurements} have shown, for very active stars radial-velocity jitter is correlated with photometric variation in stellar light-curves. Hence being able to predict epochs of low stellar activity based on knowledge gained from photometric monitoring of these young stars will prove crucial for characterising any discovered planets through radial-velocity followup.
One thing that was made quite clear from this initial survey of young star light-curves is that the 'one size fits most' approach of large-scale photometric surveys such as \textit{Kepler}, \textit{K2} and the main \textit{TESS} SPOC pipeline is often not appropriate for young stars given the large range in shape, amplitudes and periods of light-curve variability observed. Indeed, as discussed in section \ref{rotation}, even in this relatively small number of young stars surveyed periods were observed to vary between 0.27d to non-variable over the 27d observation window. Meanwhile activity and rotation behaviour varied from near-uniform to constantly evolving, and variability shape changed from beating sinusoids to almost flat aside from significant flaring activity. In order to more comprehensively search for planets around such active stars, it is recommended that future detrending pipelines focus on more effectively matching detrending techniques to the primary type of light-curve variability observed. This could be achieved through an initial automated variability-type assignment, similar to - but more advanced than - the current 'variable' vs 'non-variable' assignment implemented into the \textit{Kepler/K2/TESS} pipelines. By creating defined groups of similarly shaped light-curves and types of variability/activity, machine-learning techniques could then be used to assign light-curves the most effective detrending technique based on their perceived 'group' of variability. Particularly important will be separating known types of intrinsic variability, quick rotators and rapidly evolving light-curves, however many other important groups may become obvious with time. A more in-depth look into different types of variability in young stars over all sectors would thus be very beneficial, and may inform future detrending methods in missions like \textit{PLATO} \citep{Rauer2014TheMission}.
Attempting to understand these stars in more detail raises the important question of whether 30min data is enough to characterise any discovered variability or exoplanet candidate signals, or whether 2min cadence data is required. In this work 30min cadence light-curves were shown to effectively find the period of primary variability (and often second and third variability periods), and to recover all of the currently known TOIs identified through the 2min data, so it is undeniably a very powerful data source for the general stellar sample. However, for those TOIs recovered, exposure smearing resulted in smaller transit depths and thus less accurate transit parameters compared to the 2min data. This highlights the importance of the 2min data (or alternative follow-up photometry) for accurate characterisation of any discovered signals. Furthermore, while the 30min data was sufficient for characterisation of longer-period variability and activity in the light-curves, as the community pushes towards shorter period rotation (especially that with periods of less than 1 day), dissociating transit signals from stellar activity and rotation signals becomes increasingly difficult. This is the realm where 2min data may be crucial in the search for young exoplanets. Furthermore, as the community attempts to understand the causes and evolution of activity in young stars, 2min cadence data would significantly aid asteroseismological efforts for these young stars. However, a full comparison between 2min and 30min data-sets still needs to be undertaken before significant conclusions can be made.
One further major stumbling-lock remains in the way of finding significant numbers of exoplanets around young stars: so far, a large enough sample of 'bona-fide' young stars does not exist. Even considering basic transit statistics, only approximately 0.47\% of sun-like stars are expected to host Hot Jupiters \citep{Haswell2010TransitingExoplanets}, so considering that only 3076 stars exist in the extended BANYAN sample (some of which will not even be viewed by \textit{TESS}'s primary mission), at most 15 such planets could be expected to be found around these young stars. \citet{Bouma2019Cluster7} attempted to remedy this situation somewhat by concatenating 13 different catalogs of young stars and cluster members from literature, yielding 1,061,447 individual target stars. However, because of the mix of catalogs used and the non-homogenous membership criterion applied, this may include some stars that are not truly young. Furthermore, because it contains many members from within clusters, \textit{TESS} will struggle to create accurate light-curves for many of these stars due to blending on its relatively large 21" pixels. Another promising method that has come into its own with the release of a significant amount of data from the \textit{Gaia} satellite \citep{GaiaCollaboration2016TheMission,GaiaCollaboration2018GaiaProperties} has been expanding cluster and association membership through proper- and galactic-motion relationships. Groups such as \citet{Damiani2019TheData} and \citet{Lodieu2019APopulation} have used these effectively already to expand the Sco-OB2 and Hyades associations, however each of these stars still needs an independent sign of youth before it can be accepted as a 'bona-fide' member of a stellar association. This independent sign of youth is currently the biggest bottle-neck in expanding the known sample of young stars (especially since many confirmation methods require individual spectroscopic follow-up), so searching for new signs of youth in photomtery or astrometry has the potential to have a significant impact.
In summary, although searching for planets around young, active stars undeniably presents several extra challenges compared to searching around older host stars, the techniques developed in this and other recent work on young stars are beginning to delve more effectively into this age range. However, two major stumbling-blocks remain before the search for young exoplanets can come into fruition: the relatively small number of bona-fide young stars and the very fast rotation of many young stars. Very important groundwork is thus still required to increase this sample of known young stars, and to develop techniques which can more effectively probe young potential host stars with shorter period stellar variability.
\section{Summary} \label{Conclusions}
In this work techniques have been developed to aid the search for transiting exoplanets around young, active stars in the 30min cadence \textit{TESS} FFI data. Young exoplanets (<1Gyr in age) inhabit a very important part of the exoplanet evolutionary timescale, where formation mechanisms, accretion, migration and dynamical interactions can significantly change the shape of observed planetary systems. However, they are also typically situated around young, active and often rapidly rotating host stars, severely hampering the discovery of new planets using the transit method. The developed method attempts to detrend these spurious stellar activity signatures using a 20-30 bin LOWESS method of \citet{Cleveland1979RobustScatterplots} at its base, combined with automated peak-cutting and activity interpolation over transit gaps in order to more effectively differentiate activity from transit signals and preserve the transit shape. A basic version of this pipeline is made available online.\footnote{https://github.com/mbattley/YSD} It is hoped that using this method in tandem with other detrending/transit-search pipelines such as the main \textit{TESS} SPOC pipeline \citep{Jenkins2016TheCenter} and Gaussian-Process based methods \citep[e.g.][]{Gillen2019NGTS1} will expand the number of young planets that can be found.
These techniques were applied to young stars in stellar associations from the extended BANYAN sample \citep{Gagne2018BANYAN150pc,Gagne2018BANYAN.Data,Gagne2018BANYAN.2}, using the \textit{TESS} sector 1-5 light-curves derived from the Difference Imaging Approach (DIA) pipeline of \citet{Oelkers2018PrecisionApproach}. Lacking the data quality-flags of the \textit{TESS} 2min cadence data, periods of excess pointing scatter were instead removed by considering the \textit{TESS} data release notes and the engineering quaternion data.
While no new exoplanet candidates were revealed in this work, results from this initial survey revealed a variety of interesting objects, including the retrieval of the new young exoplanet DS Tuc Ab, TOI 447.01, TOI 450.01, a number of eclipsing binaries and a large array of interesting rotation and activity. In order to test the sensitivity of the developed detrending techniques to different planetary sizes, model \texttt{batman} transits \citep{Kreidberg2015Batman:Python} were injected into each of the young star light-curves at a range of $R_P/R_*$ radius ratios from 0.1 to 0.03 and periods from 1-14 days. The percentage of recovered transit signals from the injected planets dropped from 77.6\% at a radius ratio of 0.1$R_P/R_*$ to 20.4\% at a ratio of 0.03, however was seen to vary considerably between different targets and sectors. Meanwhile while increasing the injected planet period was seen to result in a decreasing recovery rate for smaller planets, the recovery rate was actually observed to peak around periods of 2-6 days for larger planets. An investigation into the relationship between rotation period and recovery depth did not suggest that the two were significantly correlated, aside from a slight skew towards larger planets at short rotation periods and the known difficulty of very short-period (<1day) rotation.
These results alongside deeper examination of light-curves in this sample lead to a number of interesting conclusions. The sensitivity of specific young star light-curves to transit searches appears to be most limited by fast rotation (<1 day rotation periods), excess scatter, scattered light glints and significant flaring activity. Meanwhile the rapid evolution of many young star light-curves offers both a detrending challenge and a potential opportunity to search more efficiently for exoplanets at less active times. Given the vast array of different types of young star light-curves seen in this initial survey, in the future a multi-faceted detrending approach which first classifies light-curves according to their broad activity/variability is perceived as beneficial. It is clear that the 30min cadence data shows particular promise for additional detections to the main SPOC 2min pipeline, as it is capable of retrieving all of the TOIs highlighted in the sectors analysed. However, planet parameters derived from the 30min light-curves can be less reliable due to phenomena such as exposure smearing. Thus the acquisition of 2min light-curves for young stars remains desirable, especially since this will also aid in-depth astroseismic characterisation of these sources.
Overall, the two largest challenges remaining before significant progress can be made in the field of young exoplanet science are those of very rapidly rotating stars and the relatively small numbers of confirmed young stars. Solving these two problems has the potential to gift the exoplanetary community with a significant advancement in understanding of how exoplanets form and develop into stable exoplanetary systems.
\section*{Acknowledgements}
The authors would like to thank the anonymous referee for their comments which improved the quality and robustness of this paper.
This paper includes data collected by the \textit{TESS} mission. We acknowledge the use of public TOI Release data from pipelines at the \textit{TESS} Science Office and at the \textit{TESS} Science Processing Operations Center. Funding for the \textit{TESS} mission is provided by NASA's Science Mission directorate. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. \textit{TESS} data were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. MPB acknowledges support from the University of Warwick via the Chancellor's International Scholarship. DLP acknowledges support from STFC and also the Royal Society. DJA acknowledges support from the STFC via an Ernest Rutherford Fellowship (ST/R00384X/1).
\bibliographystyle{mnras}
|
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| 2,286
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'use strict';
// Soluzsessions controller
angular.module('soluzsessions').controller('SoluzsessionsController', ['$scope', '$stateParams', '$location', 'Authentication', 'Soluzsessions',
function($scope, $stateParams, $location, Authentication, Soluzsessions) {
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| 9,219
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import Earth from './assets/earth.svg?sprite';
import Map from './assets/map.svg?sprite';
import People from './assets/people.svg?sprite';
export const sgfBenefits = [
'Grant awards between $10K - $40K.',
'Individualized training and support.',
'Form part of the GFW partnership.',
];
export const fellowshipBenefits = [
'A monthly stipend, and access to additional discretionary funds for feld visits, equipment purchases, professional development, or workshop costs.',
'All expenses paid (airfare, visa costs, meals and accommodation) to participate in tech camp and the GFW User Summit in Washington, DC.',
'Training by experts in forest monitoring, land use planning, advocacy and enforcement, and project design and implementation.',
'Participation in the GFW partnership and SGF/Fellowship alumni community.',
'Communications coverage in WRI's newsletters, blogs and social media.',
];
export const results = [
{
icon: Earth,
label: '<b>44 projects</b> from <b>30 countries</b>',
},
{
icon: Map,
label: 'Over <b>1.8 billion hectares</b> of forests monitored using GFW',
},
{
icon: People,
label: 'Over <b>1,800</b> people trained in using GFW',
},
];
|
{
"redpajama_set_name": "RedPajamaGithub"
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Franz Ihle (* 17. März 1933 in Illerberg; † 20. September 2021) war ein deutscher Politiker der CSU.
Leben
Ihle besuchte die Volks- und die Oberrealschule und erreichte die Mittlere Reife. Daraufhin studierte er an der Staatsbauschule München und machte 1953 die Prüfung als staatlich geprüfter Bauingenieur. Anschließend war er im elterlichen Bauunternehmen tätig und übernahm 1963 dieses Unternehmen. Ferner war er Aufsichtsratsvorsitzender der Raiffeisenbank Neu-Ulm/Weißenhorn und Präsident des Bayerischen Squashverbands.
Politik
1966 wurde Ihle Mitglied der CSU und zugleich Mitglied des Kreistags Illertissen. Ab 1972 war er Mitglied des Kreistags Neu-Ulm sowie des Stadtrats in Vöhringen. Am 25. Januar 1977 rückte er für Stefan Höpfinger in den Bayerischen Landtag nach. In diesen wurde er auch 1978 zunächst gewählt, am 17. Januar 1979 musste er aber nach einer Berichtigung des Wahlergebnisses wieder ausscheiden und sein Mandat an Georg Fickler abgeben. Am 26. September 1984 durfte Ihle wieder in den Landtag einziehen, diesmal rückte er für den verstorbenen Rudolf Kluger nach. Bei den drei darauffolgenden Landtagswahlen wurde er im Stimmkreis Neu-Ulm direkt gewählt. Er war somit bis 1998 Abgeordneter des Bayerischen Landtags.
Auszeichnungen
Ihle ist Träger des Bayerischen Verdienstordens und des Verdienstkreuzes am Bande.
Weblinks
Einzelnachweise
Landtagsabgeordneter (Bayern)
Träger des Bundesverdienstkreuzes am Bande
Träger des Bayerischen Verdienstordens
CSU-Mitglied
Deutscher
Geboren 1933
Gestorben 2021
Mann
|
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| 3,094
|
Сало́нии () — плебейский род в Древнем Риме. Представители данного рода впервые упоминаются в античных источниках с середины IV века до н. э., но лишь немногим удалось достичь курульных магистратур. Некоторые из Салониев в конце существования Римской империи стали христианскими проповедниками и епископами. Происхождение родового имени (где словообразующим является корень salo-), возможно, указывает на профессию солеторговца основателя рода; по-видимому, общего происхождения с номеном Салоний было родовое прозвище первых Ливиев — Салинатор ().
В числе наиболее известных представителей данного рода можно назвать следующих:
Публий Салоний, первый из упоминающихся античными авторами Салониев. Военный трибун в 342 году до н. э., который почти ежегодно был попеременно то военным трибуном, то первым центурионом ();
Гай Салоний, член посольства триумвиров, направленного в 194 до н. э. сенатом в Бруттий для выведения колонии в Темпсу. В 173 году он был в составе другой коллегии, целью которой являлось распределение незанятых земель в Лигурии и Цизальпийской Галлии между новыми поселенцами;
Квинт Салоний Сарра, первым в роду достиг претуры в 192 до н. э. и, согласно жеребьёвке, управлял Сардинией;
Салоний, младший писец Катона Цензора, дочь которого последний, являясь вдовцом, взял в жёны;
Салония, вторая супруга Катона Старшего, родившая ему Марка Порция Катона, прозванного Салонианом;
Салония, римская матрона, сын которой, Матидий, предполагаемый зять Траяна, был включён в сенат Клавдием во время исполнения последним цензорских полномочий;
Гай Салоний Матидий Патруин (ум. 78), вероятный легат-пропретор Верхней Германии () времён правления императора Веспасиана, а также претор в неустановленном году. Состоял в коллегии арвальских братьев. Был женат на старшей сестре Траяна и приходился дедом императрице Вибии;
Салония Палестрис, жена Марка Ульпия Гермии, вольноотпущенника Траяна, который был похоронен в Ампелуме (Дакия) в возрасте 55-и лет и на могиле которого Салония вместе с его либертином Диогеном установила памятник, датируемый I-й пол. II века;
Марк Салоний Лонгин Марцелл, римский сенатор, живший в императорское время и занимавший различные должности, включая плебейский трибунат, эрарную префектуру и прокураторство в Африке и Мёзии;
Салоний, генуэзский епископ (Лигурия) в середине V века, отцом которого являлся Евгерий, епископ Лугдуна. В своё время был учеником Сальвиана Массилийского.
Примечания
Литература
Древнеримские роды
|
{
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| 2,129
|
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