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\section{Introduction}
\label{sec:intro}
The definition of D-branes as allowed endpoints for open strings \cite{1.}, generalizes the notion of quarks on which the QCD string can terminate. In contrast to the quarks of QCD, D-branes are intrinsic excitations of the fundamental theory. D-particles can probe distances much smaller than the size of the fundamental string quanta. D-branes played a crucial role in the 'second string revolution' , the way to reconcile quantum mechanics and Einstein gravity. The D-brane concept \cite{2., 3.} is powerfull because of the relations between supersymmetric gauge theories and geometry.
The purpose of our article is connected with the searches of a higher-dimensional breaking mechanism in the context of D-branes, which is connected or associated with four-dimensional Grand Unification Theory Higgs multiplets and symmetry breaking higgs mechanism.
\section{The Category of D-branes as Derived Category of Coherent Sheaves}
Due to the important development in string theory through the discovery of D-branes we can use a compactification model: the string theory has a target space $R^{1, 3}\times X$ for compact space $X$ and focus on $X$. Let $X$ be a topological space. On such space we can construct locally free sheaf ${\cal {E}}$. If we have embedding $i:S\rightarrow X$ and a sheaf ${\cal {E}}$ on $S$, than we can define a sheaf $i_{*}{\cal {E}}$ on $X$, through the following construction: a map $f:X\rightarrow Y$ between two algebraic varieties and a sheaf ${\cal {F}}$ on $X$ define the sheaf $f_{*}{\cal {F}}$ on $Y$ by $f_{*}{\cal {F}}(U)={\cal {F}}(f^{-1}U)$. For embedding $i:X\rightarrow P^{n}$ the sheaf $i_{*}{\cal {E}}$ is given by $i_{*}{\cal {E}}(U)={\cal {E}}(U\cap P^{n})$ for all open subsets $U\subset P^{n}$. This embeds the sheaves on $X$ into the sheaves on $P^{n}$. So we have locally free sheaf on $P^{n}$, which is associated with the D-brane.
From \cite{4.} it follows that more generally we can consider a complex of locally-free sheaves:
\[\ldots\stackrel{d_{n-1}}\longrightarrow{\cal{E}}^n\stackrel{d_{n}}\longrightarrow{\cal{E}}^{n+1}\stackrel{d_{n+1}}\longrightarrow{\cal{E}}^{n+2}\stackrel{d_{n+2}}\longrightarrow\ldots ,\]
where morphisms $d_n:{\cal{E}}^{n}\rightarrow{\cal{E}}^{n+1}$,
$d_n\in\mbox{Ext}^0({\cal{E}}^{n},{\cal{E}}^{n+1})=\mbox{Hom}({\cal{E}}^{n},{\cal{E}}^{n+1})$, $d_{n+1}d_{n}=0$ for all $n$ are morphisms between locally-free sheaves ${\cal{E}}^{n}$ and ${\cal{E}}^{n+1}$.\\
For a further aim we must use the notion of a category.
A category ${\cal{L}}$ consists of the following data:
1) A class $\mbox{Ob}\ {\cal{L}}$ of objects $A, B, C, \cdots$;
2) A\hspace*{2mm} family\ of\hspace*{2mm} disjoint\hspace*{2mm}
sets\hspace*{2mm}
of\hspace*{2mm} morphisms\ $\mbox{Hom}(A, B)$
one for each ordered pair $A, B$ of objects;
3) A family of maps
\[\mbox{Hom}(A, B)\times \mbox{Hom}(B, C)\rightarrow \mbox{Hom}(A, C)\ ,\]
one for each ordered triplet $A, B, C$ of objects.
These data obey the axioms:
a) If $f\ : A\rightarrow B,\ g\ : B\rightarrow C,\ h\ : C\rightarrow D$,
then composition of morphisms is associative, that is, $h(gf)=(hg)f\ ;$
b) To each object $B$ there exists a morphism $1_B\ :\ B\rightarrow B$
such that $1_Bf=f\ ,\ g1_B=g$ for $f\ : A \rightarrow B$ and
$g\ : B \rightarrow C$\ .
Thus, the category of D-branes is the
derived category of locally-free sheaves.
Locally-free sheaves and morphisms don't form an abelian category.
So we should replace the category of locally-free sheaves
by the abelian category of coherent sheaves. Abelian category is characterized by the existence of an exact sequences.
We can form the category of D-branes, which is the derived category of coherent sheaves $D(X)$.
On a smooth space $X$, any coherent sheaf ${\cal {A}}$ has a locally-free resolution
\[0\rightarrow{\cal {F}}^{-3}\rightarrow{\cal {F}}^{-2}
\rightarrow{\cal {F}}^{-1}\rightarrow{\cal {F}}^{0}\rightarrow{\cal {A}}\rightarrow 0 \ , \]
where ${\cal {F}}^{k}$ is locally free. This is a quasi-isomorphism
${\cal {F}}^{\bullet}\rightarrow{\cal {A}}$ between
a complex of locally-free sheaves and a coherent sheaf.
Thus, an abelian category of coherent sheaves $D(X)$ of $X$
consists of objects ${\cal{E}}^{\bullet}$ - exact complexes of sheaves:
\[\ldots\stackrel{d_{n=2}}\longrightarrow{\cal {E}}^{n-1}\stackrel{d_{n=1}}\longrightarrow{\cal {E}}^{n}\stackrel{d_{n}}\longrightarrow
{\cal {E}}^{n+1}\stackrel{d_{n+1}}\longrightarrow \ldots \ \]
and morphisms between them ${\cal{E}}^{\bullet}\rightarrow{\cal{F}}^{\bullet}$:
\[\begin{tabular}{ccccccccc}
$\ldots$&$\stackrel{d_{n=2}}\longrightarrow$&${\cal {E}}^{n-1}$&$\stackrel{d_{n=1}}\longrightarrow$&${\cal {E}}^{n}$&$\stackrel{d_{n}}\longrightarrow$&${\cal {E}}^{n+1}$&$\stackrel{d_{n+1}}\longrightarrow$&$ \ldots$\\
&&&&&&&&\\
&&$\downarrow$&&$\downarrow$&&$\downarrow$&&\\
&&&&&&&&\\
$\ldots$&$\stackrel{d_{n=2}}\longrightarrow$&${\cal {F}}^{n-1}$&$\stackrel{d_{n=1}}\longrightarrow$&${\cal {F}}^{n}$&$\stackrel{d_{n}}\longrightarrow$&${\cal {F}}^{n+1}$&$\stackrel{d_{n+1}}\longrightarrow$&$ \ldots$
\end{tabular} \]
\section{Triangulated Category and Central Charge}
For physical purposes we will work in future with Calabi-Yau threefolds $X$. According to \cite{4.}, if $X$ and $Y$ are mirror Calabi-Yau threefolds then the derived category $D(X)$ is equivalent to the triangulated category Tr${\cal{F}}(Y)$. So, in such category $D(X)$ objects are distinguished triangles:
\begin{center}
\hspace*{4cm}\begin{tabular}{ccc}
&$C$&\\
&\hspace*{-4mm}${\mbox{\tiny[1]}}\hspace*{-1mm}\swarrow$\hspace*{1mm}$\nwarrow$& \hspace*{15mm}$C={\mbox{Cone}}(f)$ \hspace*{42mm} (1)\\
&$\hspace*{-2mm}A$\hspace*{1mm}$\stackrel{f}{\longrightarrow}$\hspace*{1mm}$B$&\\
\end{tabular}
\end{center}
and morphisms of this category are morphisms of distinguished triangles \cite{4.}.
Than the derived category is additive category, where exact sequences are exchanged by distinguished triangles.
According to Douglas \cite{5., 6.} instead of physical D-branes, living in the boundary conformal field theory, we can work with topological D-branes, and the
relationship between them is the notion of $\Pi$-stability.
A topological D-brane is physical if it is $\Pi$-stable. For the precise
definition of $\Pi$-stability we must compute the central charge of objects
${\cal{E}}^{\bullet}$ in $D(X)$:
\[Z({\cal{E}}^{\bullet})=\int\limits_X e^{-(B+iJ)}ch({\cal{E}}^{\bullet})
\sqrt{td(X)},\]
where $B+iJ$ is the complexified K$\ddot{a}$hler form.
For a given ${\cal{E}}^{\bullet}$ we may choose a grading
$\xi({\cal{E}}^{\bullet})$:
\[\xi({\cal{E}}^{\bullet})=\frac{1}{\pi}\mbox{arg}Z({\cal{E}}^{\bullet}) \ \ \mbox{(mod 2)}.\]
If we have a distinguished triangle in $D(X)$ of the form (1)
with $A$ and $B$ stable, than $C$ is stable with respect to the decay represented by this triangle if and only if
$\xi(B)\prec \xi(A)+1$. Also, if $\xi(B)= \xi(A)+1$ then
$C$ is marginally stable and we may state that
\[\xi(C)=\xi(B)= \xi(A)+1 .\]
We may generalize this to the case of decays into any
number of objects. For any object $E$ we may
define the set of distinguished triangles
\begin{center}
\hspace*{5cm}\begin{tabular}{ccccc}
&\hspace*{-2mm}$E_{0}$ \ $\longrightarrow$ \ \ $E_{1}$&
\hspace*{-2mm}$\longrightarrow$\ \ \ $\ldots$&
$\longrightarrow$ $E_{n}$ \ $=E$&\\
&\hspace*{-4mm}${\mbox{\tiny[1]}}$\hspace*{-1mm}$\nwarrow$\hspace*{1mm}$\swarrow$&
\hspace*{-8mm}${\mbox{\tiny[1]}}$\hspace*{-1mm}$\nwarrow$\hspace*{1mm}$\swarrow$&
\hspace*{-20mm}${\mbox{\tiny[1]}}$\hspace*{-1mm}$\nwarrow$\hspace*{1mm}$\swarrow$& \hspace*{2.3cm} (2)
\\
&$A_{1}$&\hspace*{-5mm}$A_{2}$&\hspace*{-20mm}$A_{n}$&\\
\end{tabular}
\end{center}
Then $E$ decays into $A_{1}, A_{2},\ldots A_{n}$
so long as
\[\xi(A_{1})\succ \xi(A_{2})\succ \ldots \xi(A_{n}).\]
As we have pointed out, $X$ and $Y$ are mirror Calabi-Yau
threefolds. According to \cite{7.} mirror map is defined
between the complex moduli space of a Calabi-Yau
manifold $X$ and the K$\ddot{a}$hler moduli space of its
mirror manifold $Y$. So we can work with K$\ddot{a}$hler moduli space, that is characterized by the complexified K$\ddot{a}$hler form $B+iJ$. Then we can use the Teichm$\ddot{u}$ller space
${\cal{T}}$ as the universal cover of the moduli space of
$B+iJ$ and we expect the set of stable D-branes to be well-defined
at any point in ${\cal{T}}$. \\
The following rules are applied:\\
$\bullet$ We begin with a stable set of D-branes with value of
grading $\xi$ for each D-brane.\\
$\bullet$ During the moving along a path in moduli space the
gradings will change continuously.\\
$\bullet$ Two stable D-branes may bind to form a new stable state.\\
$\bullet$ A stable D-brane may decay into other stable states.\\
So, we can write {\bf Conjecture} from \cite{4.}.\\
{\bf Conjecture.} At every point in the Teichm$\ddot{u}$ller space of $B+iJ$ there is a set of stable objects in $D(X)$ such that
every object $E$ can be written in the form (2) for some $n$ and
for stable objects $A_{k}$.
Every object in $D(X)$ is stable or unstable for a given
point in the Teichm$\ddot{u}$ller space of $B+iJ$. Thus the
open string corresponding to $f$ in (1) go from tachyonic to massive
as we pass in the Teichm$\ddot{u}$ller space.
Now as we work in ten-dimensional space-time $R^{3+1}\times {\newfont{\blackboard}{msbm8 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}/G$
and knowing that after blowing up the singularity of additional six-dimensional space-time - orbifold
${\newfont{\blackboard}{msbm8 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}/Z_3 \rightarrow
{\cal{O}}_{P_{2}}(-3)$ we have the sheaf ${\cal{O}}_{P_{2}}(-3)$
on two-dimensional projective space $P_{2}$. Here we must consider {\bf Theorem} from \cite{4.}:\\
{\bf Theorem}. Suppose $X$ is a smooth resolution of the orbifold
${\newfont{\blackboard}{msbm8 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}/G$ with
$G$ a finite subgroup of $SU(d)$ and $d\leq 3$. Then the derived
category $D(X)$ is equivalent to the derived category of
$G$-equivariant sheaves on ${\newfont{\blackboard}{msbm8 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}^d$.\\
Now we are dealing with the derived category of sheaves on $P_2$ and
we can use the statement \cite{4.}:
D-branes on the orbifold
${\newfont{\blackboard}{msbm8 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}/G$
and open strings between them are described by the
derived category of McKay quiver representations.\\
In future we will work with the derived category
of distinguished triangles over the abelian category
of McKay quivers. Objects of this category
are distinguished triangles (Figure 1)
\begin{figure}[htbp]
\centering{\includegraphics[width=.45\textwidth]{1.eps}}
\caption{\label{fig:1} \small Construction of distinguished triangle.}
\end{figure}
(numbers $a, b, c$ and
$a^{\prime}, b^{\prime}, c^{\prime}$ denote orbifold charges characterizing McKay quivers); morphisms of this category are morphisms of
distinguished triangles.
\section{Grand Unification Theory Breaking and Dimensional Reduction}
Our further work will be connected with the efforts to take many attractive features of the basic Grand Unification Theory and implement this ideas in four-dimensional models, for example, in the minimal four-dimensional supersymmetric SU(5) GUT with standard Higgs content. Moreover, because no appropriate four-dimensional GUT Higgs field is typically available to break the GUT group to the Standard Model gauge group, it is necessary to employ a higher-dimensional breaking mechanism. For type IIB theories, the corresponding vacua are realized as compactifications of F-theory on Calabi-Yau fourfolds. We will consider the left part of Figure 2 of the general overview of how GUT breaking constrains the type of GUT model \cite{7.}.
\begin{figure}[htbp]
\centering
\includegraphics[width=.45\textwidth]{2.eps}
\caption{\label{fig:2} \small General overview of how GUT breaking constrains the type of GUT model.}
\end{figure}
We will consider elliptically fibered Calabi-Yau fourfold $X$ with the base $B$ and elliptic fiber $\varepsilon$:
\begin{center}
\begin{tabular}{ccc}
$\varepsilon$\hspace*{-2mm}&$\rightarrow$&\hspace*{-2mm}$X$\\
&&\hspace*{-2mm}$\downarrow$\\
&&\hspace*{-2mm}$B$
\end{tabular}
\end{center}
This Calabi-Yau fourfold can be represented by the Figure 3.
\begin{figure}[htbp]
\centering
\includegraphics[width=.45\textwidth]{3.eps}
\caption{\label{fig:3} \small Depiction of F-theory compactified on a local model of a Calabi-Yau fourfold. }
\end{figure}
We shall assume that there exists a Calabi-Yau fourfold which contains the corresponding local enhancement in singularity type. When a del Pezzo surface $S$ intersect $S^{'}$ on a Riemann surface $\Sigma$, the singularity type enhances further. In this case, additional six-dimensional hypermultiplets localize along $\Sigma$. In terms of four-dimensional superfields, the matter content, localized on a curve $\Sigma$, consists of chiral superfields. Schematically this can be represented by the\\ Table 1.
\begin{table}[tbp]
\centering
\begin{tabular}{|clc|}
\hline
Dimension& Space& Ingredient\\
\hline
10 & $R^{3+1}\times B$ &Gravity\\
8 & $R^{3+1}\times S$ & Gauge Theory\\
6 & $R^{3+1}\times \Sigma$ & Chiral Matter\\
4& $R^{3+1}$& Cubic Interaction Terms\\
\hline
\end{tabular}
\caption{\label{tab:i} Dimensional reduction and matter content of the corresponding space-time.}
\end{table}
As we can see, we have two points of view, connected with Calabi-Yau fourfolds. One is that Calabi-Yau is the sheaf on $P_{2}$ and as the object in $D(X)$ is stable or unstable in the Teichm$\ddot{u}$ller space of $B+iJ$. The other is that it is the fibered bundle, that can be reduced to four-dimensional theory with the corresponding matter content. After implementation of a higher-dimensional breaking mechanism to obtain four-dimensional models, we can receive the minimal four-dimensional supersymmetric $SU(5)$ Grand Unification Theory with standard Higgs content.
The moduli space of an open superstring \cite{8.} which is described by $\mbox{Ext}^{i}(Q,Q^{'})$ groups and determined by the diagram \cite{4.} in Figure 4
\begin{figure}[htbp]
\centering
\includegraphics[width=.25\textwidth]{4.eps}
\caption{\label{fig:4} \small Open superstring that is described by $\mbox{Ext}^{i}(Q,Q^{'})$ group.}
\end{figure}
has the form
\[\hspace*{4.3cm}\begin{array}{ccc}
\mbox{Ext}^{0}(Q,Q^{'})&=&{\newfont{\blackboard}{msbm10 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}^{\hspace*{1mm} aa^{'}+bb^{'}+cc^{'}}\ \ \ \ , \\
\mbox{Ext}^{1}(Q,Q^{'})&=&{\newfont{\blackboard}{msbm10 scaled\magstep2}
\newcommand{\Ce}{\mbox{\blackboard\symbol{'103}}}\Ce}^{\hspace*{1mm} 3ab^{'}+3bc^{'}+3ca^{'}}\ .
\end{array}\hspace*{4.5cm} (3)\]
Substituting in (3) orbifold charges
\[a=b=c=a^{\prime}=b^{\prime}=c^{\prime}=4\]
and using the Langlands hypothesis \cite{9.}, we obtain the realization of (3)
in terms of $SU(5)$ multiplets
\[3\times(24+5_H+\overline{5}_H+5_M+\overline{5}_M+10_M+\overline{10}_M)\ ,\]
where $5_H$ and $\overline{5}_H$ are Higgs multiplets, $\overline{5}^{(i)}_M$ and $10^{(j)}_M$ are multiplets of quark and lepton superpartners.
As the transition in the Teichm$\ddot{u}$ller space is connected with stability of the object and this stability is characterized by the moduli space of an open superstring connected with Higgs multiplets, we can see how a higher-dimensional breaking mechanism is connected or associated with four-dimensional GUT Higgs multiplets and symmetry breaking higgs mechanism.
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{"url":"http:\/\/uncyclopedia.wikia.com\/wiki\/Computer_Science","text":"# Computer Science\n\nComputer science is a science that involves calculating the time it takes to get the mouse over to an icon if pushed with a certain force in the windows operating system. After extensive research, computer scientists came up with the following formula:\n\n$\\frac{force!}{mass} \\cdot 3\\cdot (speed\\; of\\; mouse) \\cdot (distance)$\n=\n$(amount\\; of\\; scribbles\\; on\\; page\\; when\\; trying\\; to\\; calculate\\; the\\; time\\; it\\; takes\\; to\\; get\\; the\\; mouse\\; over\\; to\\; an\\; icon\\; in\\; the\\; windows\\; operating\\; system)$\n\nThe above formula, often claimed to be the base of computer science, was used to derive boredman's paradox, which explains the inability to move the windows mouse between 8 AM and 8 PM GMT.\n\nAfter much discussion, the scientists got no further. Sensing imminent defeat, they went out to Starbucks for a stimulating drink, after which they switched to FreeBSD.\n\n## editOrigin\n\nThe term 'computer science' is a frequently-bandied euphemism for a methodology which attempts to predict outcomes of Software Development with a series of desperate guesses, often made in the middle of the night and usually on the late side of a deadline.\n\nComputer Science, sometimes referred to as 'software voodoo', derives from a set of absolute truths:\n\n\u2022 1. People who finance software do not have a clue as to its nature.\n\u2022 2. People who specify the characteristics of software do not have a clue as to its nature.\n\u2022 3. People who create timelines for Software Development do not have a clue as to its nature.\n\u2022 4. People who write software are aware of the fact that 1, 2 and 3 are full of shit and that, no matter how often you tell them that they're full of shit, and even attempt to explain to them with charts, graphs and numbers exactly why they're full of shit, 1, 2 and 3 will never stop believing that they know more about software than the people who write it.\n\n## editEducation\n\nCambridge University invented the term 'Degree in Computer Science' as an excuse for nerds to play Everquest in their dorm rooms. This practice has evolved into a cult in which victims (called 'students') are duped into paying large amounts of money ('tuition') for a useless piece of paper ('degree'). Major corporations such as IBM fell for this trap and began to hire these victims, creating a cycle that dominates the computer world to this day. Females, although generally unintelligent, have somehow developed an immunity to the lures of the cult, and have thus avoided becoming victims.\n\nComputer science students can be recognized by the following warning signs:\n\n\u2022 Unwashed hair (in which very small animals sometimes dwell)\n\u2022 Glasses\n\u2022 Acne\n\u2022 Poor fashion sense\n\u2022 Lack of friends\n\u2022 Prolonged computer use\n\u2022 Annoying laugh\n\u2022 Heavy consumption of Jolt Cola but will settle for Mountain Dew or Red Bull\n\u2022 Hairy palms\n\u2022 Totally BOSS\n\u2022 TOTALLY BEAST\n\n## editWindows\n\nWindows has been abandoned by most cutting-edge computer scientists and is now almost exclusively a subject of cyberarchaeological study. After having been added onto by different Microsoft programmers for years, no one person, and possibly not even any collection of people, actually know how deep and convoluted it is. As a result, cyberarchaeologists have a wonderful time digging into the tangled, murky depths of its code and trying to figure out how it really works.\n\nAs a result all Windows have been replaced by Doors. However there have been problems in the progress. When some 3rd story Windows have been replaced by doors, there have been reports of students falling out without the intention of committing suicide.\n\n## editMain concepts\n\n### editComputability\n\nSome computer scientists simply refuse to occupy themselves with real computer-related issues and will not stop going on about computability theory. This is known as the halting problem and has plagued computer science for epochs. ll\n\n### editRecursion\n\nRecursion theory is defined as the field of computer science that studies recursion. For a more in-depth analysis, see Recursion.\n\n### editInfinite loops\n\nProgress in this field of study has ground to a halt.\n\n### editSegmentation Faults\n\nThis is how God instills fear into all computer scientist kind - its workings are incomprehensible, far beyond mortal knowledge.\n\n### editNP Hard\/Complete Problems\n\nProblems which are too much effort to solve such that computer scientists get lazy and decide to brand them \"NP-hard\" or for that matter \"NP-Complete\". Experts in their field of escapism Computer Scientists have come up with a novel terminology. Not surprisingly, the otherwise awesome technique known as \"lazy evaluation\" (consisting of not evaluating the problem at all) has proven useless for 98% of the tested NP Hard problems. The other 2% just caused the machine to spontaneously combust.","date":"2013-12-08 13:42:01","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\": 2, \"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\": 0.3849083483219147, \"perplexity\": 2447.719159301111}, \"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-2013-48\/segments\/1386163065790\/warc\/CC-MAIN-20131204131745-00011-ip-10-33-133-15.ec2.internal.warc.gz\"}"} | null | null |
Q: Redirect Struts Action Result to a New Tab based on Form Value Our Struts Action class should redirect output to a new tab depending on a form value -- not always.
The standard suggestion is to add
<input type="button" onclick=openNewWindow('validate.do?param=myParam');"
JS:
openNewWindow: function(url) {
window.open(url, "_blank");
}
But this won't work for us. We should not automatically open a new tab in every case; it depends on a certain form value, which we have to check on the server-side in the Action class.
Any thoughts on how to approach this?
A: No need to use Javascript, form element has target attribute:
<s:form target="${param.paramAttr eq 'myParam' ? '_blank' : '' }" action="validate.do"
method="post"
>
..
</s:form>
| {
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\section{ Introduction }
Let $G$ be a finite abelian group. We define some central invariants in zero-sum theory which have been studied since the 1960s. Let
\begin{itemize}
\item[$\bullet$] $\mathsf D(G)$ denote the smallest integer $l\in \mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a nonempty zero-sum subsequence.
\item[$\bullet$] $\mathsf s(G)$ denote the smallest integer $l\in \mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a zero-sum subsequence $T$ of length $|T|=\exp (G)$.
\item[$\bullet$] $\eta(G)$ denote the smallest integer $l\in \mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a nonempty zero-sum subsequence $T$ of length $|T| \in [1, \exp (G)]$.
\end{itemize}
$\mathsf D(G)$ is called the \emph{Davenport constant} of $G$ and $\mathsf s(G)$ the \emph{Erd{\H{o}}s-Ginzburg-Ziv} (EGZ) constant of $G$. For the historical development of the field and the contributions of many authors we refer to the surveys \cite{GG, GE}. Here we can only provide a brief summary. There is the following chain of inequalities (\cite[5.7.2 and 5.7.4]{Ge-HK06a})
\begin{equation} \label{LOWETA}
\mathsf D (G) \le \eta (G) \le \mathsf s (G) - \exp (G) + 1 \le |G| \,.
\end{equation}
Clearly, equality holds throughout for cyclic groups which implies the classical Theorem of Erd{\H{o}}s-Ginzburg-Ziv dating back to 1961 and stating that $\mathsf s (G) = 2 |G|-1$ (\cite{EGZ}). Since about ten years the precise value of all three invariants is known for groups having rank at most two. We have (\cite[Theorem 5.8.3]{Ge-HK06a})
\medskip
\textbf{Theorem A.} {\it Let \ $G = C_{n_1} \oplus C_{n_2}$ with $1 \le n_1 \t n_2$. Then
\[
\mathsf s (G) = 2n_1 + 2n_2 - 3\,, \ \ \eta (G) = 2 n_1 + n_2 - 2 \,, \quad \text{and} \quad \mathsf D(G) = n_1 + n_2 - 1 \,.
\]
}
\medskip
In groups of higher rank precise values for any of the three invariants are known only in very special cases. We briefly sketch the state of knowledge with a focus on groups of the form $C_2^r \oplus C_n$, where $r, n \in \mathbb N$, which have found special attention in all these investigations.
To begin with the Davenport constant, suppose that $G \cong C_{n_1} \oplus \ldots \oplus C_{n_r}$, where $1 < n_1 \mid \ldots \ldots \mid n_r$, and set $\mathsf D^* (G) = 1 + \sum_{i=1}^r(n_i-1)$. An example shows that $\mathsf D^* (G) \le \mathsf D (G)$, and equality holds for $p$-groups. It is open whether or not equality holds for groups of rank three (for recent progress see \cite{Bh-SP07a}). Not even the special case where $G=C_n^3$ is known, but we know that $\mathsf D^* (C_2 \oplus C_{2n}^4) < \mathsf D (C_2 \oplus C_{2n}^4)$ \cite[Theorem 3.1]{Ge-Li-Ph12}. If $n \ge 3$ is odd and $r \in \mathbb N$, then $\mathsf D^* (C_2^r \oplus C_{2n})=\mathsf D (C_2^r \oplus C_{2n})$ if and only if $r \le 3$. If $r \le 2$, then the structure of the minimal zero-sum sequences over $C_2^3 \oplus C_{2n}$ is well-known (\cite{Sc11b}).
The only groups $G$ with $\mathsf D^* (G) < \mathsf D (G)$, for which the precise value of $\mathsf D (G)$ is known are groups of the form $G=C_2^4 \oplus C_{2n}$ for odd $n \ge 70$ (\cite[Theorem 5.8]{Sa-Ch14a}).
\smallskip
A simple example shows that $\eta (G) \le \mathsf s (G) - \exp (G) + 1$, and the standing conjecture (due to Weidong Gao \cite{Ga03a}) states that equality holds for all groups $G$. This has been confirmed for a variety of groups (see \cite{FYS} without knowing the precise value of $\eta (G)$ or $\mathsf s (G)$). If $G$ is an elementary $2$-group, then it can be seen right from the definitions that $\eta (G)=|G|$ and that $\mathsf s (G)=|G|+1$. If $G=C_3^r$, then $(\mathsf s (G)-1)/2$ is the maximal size of a cap in the $r$-dimensional affine space over $\mathbb F_3$ (\cite[Lemma 5.2]{Edel2007}). This invariant has been studied in finite geometry since decades, but precise values are known only for $r \le 6$ (\cite{Po08a}). For arbitrary primes $p$, the precise values of $\eta (C_p^3)$ and $\mathsf s (C_p^3)$ are unknown. However, there are standing conjectures which have been verified in very special cases (see \cite{G-H-S-T07}), and also the structure of sequences of length $\mathsf D (G)-1$ (resp. $\eta (G)-1$ or $\mathsf s (G)-1)$ that do not have a zero-sum subsequence (of the required length) has been studied for groups of the form $C_n^r$ (\cite[Theorem 3.2]{Ga-Ge-Sc07a}). For recent precise results for $\eta (G)$ and $\mathsf s (G)$ we refer the reader to \cite{Sc-Zh10a} and to \cite{FGZ}.
In the present paper we focus on the EGZ constant $\mathsf s (G)$ and on $\eta (G)$ for groups of the form $C_2^{r-1}\oplus C_{2n}$, where $n\ge 2$ is an integer. Our first result provides the best upper bound on $\mathsf s(C_2^r\oplus C_n)$ for $n\ge 3$ odd, which is known so far.
\begin{theorem}\label{Th2} $\mathsf s(C_2^{r-1}\oplus C_{2n})\leq 4n+2^r-5$ where $r\geq 3$ is a positive integer and $n\geq 3$ is an odd integer.
\end{theorem}
\medskip
If $G = C_2^{r-1} \oplus C_{2n}$ and $r \in [3,4]$, then we can provide precise results.
\begin{theorem} \label{Th1}
Let $n \ge 2$.
\begin{enumerate}
\item $\eta(C_2^2\oplus C_{2n})=2n+4$ \ and \ $\mathsf s(C_2^2\oplus C_{2n})=4n+3$.
\smallskip
\item $\eta(C_2^3\oplus C_{2n})=2n+6$, \ and if $n \ge 36$, then \ $\mathsf s(C_2^3\oplus C_{2n})=4n+5$.
\end{enumerate}
\end{theorem}
\section{ Notations and Terminology}
Our notations and terminology are consistent with \cite{GG} and \cite{GE}. Let $\mathbb{N}$ denote the set of positive integers, $\mathbb{P}\subseteq \mathbb{N}$ the set of prime numbers and $\mathbb{N}_0=\mathbb{N}\cup\{0\}$. For real numbers $a, b\in \mathbb{R}$, we set $[a, b]=\{x\in \mathbb{Z}\mid a\leq x\leq b\}$. Throughout this paper, all abelian groups will be written additively, and for $n\in \mathbb{N}$, we denote by $C_n$ a cyclic group with $n$ elements.
Let $G$ be a finite abelian group. We know that $|G|=1$ or $G\cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $1<n_1\mid \cdots \mid n_r\in \mathbb{N}$, where $r=\mathsf{r}(G)\in \mathbb{N}$ is the \emph{rank} of $G$ and $n_r={\exp}(G)$ is the \emph{exponent} of $G$. We denote $|G|$ the \emph{cardinality} of $G$, and $\mathsf{ord}(g)$ the \emph{order} of elements $g\in G$. For convenience, denote $C_n^r=C_{n_1}\oplus \cdots \oplus C_{n_r}$ if $n_1=\cdots =n_r=n\in \mathbb{N}$.
Let $\mathscr{F}(G)$ be the free abelian monoid, multiplicatively written, with basis $G$. The elements of $\mathscr{F}(G)$ are called \emph{sequences }over $G$. A sequence $S\in \mathscr{F}(G)$ will be written in the form
$$S=g_1\cdot \ldots \cdot g_l=\prod_{g\in G}g^{\mathsf v_{g}(S)}$$
with $\mathsf v_{g}(S)\in \mathbb{N}_0$ for all $g\in G$. We call $\mathsf v_{g}(S)$ the multiplicity of $g$ in $S$, and if $\mathsf v_{g}(S)>0$ we say that $S$ contains $g$. If for all $g\in G$ we have $\mathsf v_{g}(S)=0$, then we call $S$ the \emph{empty sequence} and denote $S=1\in \mathscr{F}(G)$. A sequence $S$ is called squarefree if $\mathsf v_{g}(S)\leq 1$ for all $g\in G$. Apparently, a squarefree sequence over G can be considered as a subset of $G$. Let $g_0\in G$, we denote
$$g_0+S=(g_0+g_1)\cdot \ldots \cdot (g_0+g_l).$$
A sequence $S_1\in \mathscr{F}(G)$ is called a subsequence of $S$ if $\mathsf v_{g}(S_1)\leq \mathsf v_{g}(S)$ for all $g\in G$, and denoted by $S_1\mid S$. If $S_1\mid S$, we denote
$$S\cdot S_{1}^{-1}=\prod_{g\in G}g^{\mathsf v_g(S)-\mathsf v_g(S_1)}\in \mathscr{F}(G).$$
If $S_1$ is not a subsequence of $S$, we write $S_1\nmid S$.
Let $S_1, S_2\in \mathscr{F}(G)$, we set
$$S_1\cdot S_2=\prod_{g\in G}g^{\mathsf v_{g}(S_1)+\mathsf v_{g}(S_2)}\in \mathscr{F}(G).$$
Furthermore, we call $S_1,\ldots,S_t\, (t\ge 2)$ are disjoint subsequences of $S$, if $S_1\cdot\ldots\cdot S_t\t S$.
For a sequence
$$S=g_1\cdot \ldots \cdot g_l=\prod_{g\in G}g^{\mathsf v_g(S)}\in \mathscr{F}(G),$$
we list the following definitions
\begin{itemize}
\item[] $|S|=l=\sum_{g\in G}\mathsf v_g(S)\in \mathbb{N}_0$ the \emph{length} of $S$,
\item[] $\mathsf{supp}(S)=\{g\in G\mid \mathsf v_g(S)>0\}\subseteq G$ the \emph{support} of $S$,
\item[] $\sigma(S)=\sum_{i=1}^{l}g_i=\sum_{g\in G}\mathsf v_g(S)g\in G$ the \emph{sum} of $S$,
\item[] $\sum(S)=\{\sum_{i\in I}g_i\mid I\subseteq [1, l] \mbox{ with } 1\leq |I|\leq l\}$ the set of all \emph{subsums} of $S$,
\end{itemize}
The sequence $S$ is called
\begin{itemize}
\item[$\bullet$] \emph{zero-sum free} if $0\notin \sum(S)$,
\item[$\bullet$] a \emph{zero-sum sequence} if $\sigma(S)=0$,
\item[$\bullet$] a \emph{short zero-sum sequence} if $\sigma(S)=0$ and $|S| \in [1, {\exp}(G)]$.
\end{itemize}
Every map of abelian groups $\phi: G\rightarrow H$ extends to a map from the sequences over $G$ to the sequences over $H$ by setting $\phi(S)=\phi(g_1)\cdot \ldots \cdot \phi(g_l)$. If $\phi$ is a homomorphism, then $\phi(S)$ is a zero-sum sequence if and only if $\sigma(S)\in \mathsf{ker}(\phi)$.
Let $G=H\oplus K$ be a finite abelian group. Let $\phi$ denote the projection from $G$ to $H$ and $\psi$ denote the projection from $ G$ to $K$. If $S\in \mathscr{F}(G)$ such that $\sigma(\phi(S))=0$, then $\sigma(S)=\sigma(\psi(S))\in \ker(\phi)=K$.
\begin{lemma} \label{cyclic}
Let $G$ be a cyclic group of order $n\geq 2$.
\begin{enumerate}
\item A sequence $S\in \mathscr{F}(G)$ is zero-sum free of length $|S|=n-1$ if and only if $S=g^{n-1}$ for some $g\in G$ with $\mathsf{ord}(g)=n$.
\smallskip
\item Let $S$ be a zero-sum free sequence over $G$ with length greater than $n/2$. Then there exists an element $g\in G$ of order $n$ such that $$S=(k_1g)\cdot\ldots\cdot(k_{|S|}g),$$ where $1\le k_1\le\cdots\le k_{|S|}$, $k=k_1+\cdots+k_{|S|}<n$, and $\Sigma(S)=\{g,2g,\ldots,kg\}$.
\smallskip
\item Let $S\in \mathscr{F}(G)$ a sequence of length $|S|=\mathsf s(G)-1$. Then the following statements are equivalent:
\begin{itemize}
\item[(a)] $S$ has no zero-sum subsequence of length $n$.
\item[(b)] $S=(gh)^{n-1}$ where $g, h\in G$ with $\mathsf{ord}(g-h)=n$.
\end{itemize}
\end{enumerate}
\end{lemma}
\begin{proof}
See \cite[Cor. 2.1.4, Th.5.1.8, Prop.5.1.12]{GE}.
\end{proof}
\begin{lemma}(\cite[Theorem 1.2]{FYS})\label{ETAF}
Let $G=H \oplus C_{mn}$ be a finite abelian group where $H \subseteq G$ is a subgroup with ${\exp}(H)=m\geq 2$ and $n \in \N$. If $n\geq \max\{m|H|+1, 4|H|+2m\}$, then $\mathsf s(G)=\eta(G)+{\exp}(G)-1$.
\end{lemma}
\begin{lemma} \label{ETA}
Let $G=H\oplus C_n$ be a finite abelian group where $H\subseteq G$
is a subgroup with $\exp (H) \mid \exp (G) = n$. Then
\[
\eta (G) \ge 2(\mathsf D(H)-1) + n \quad \text{ and} \quad \mathrm \mathsf s(G) \ge 2(\mathsf D(H)-1) + 2n-1 \,.
\]
In particular, we have for all $ r, n\in \mathbb{N}$,
\[
\eta(C_2^{r-1}\oplus C_{2n})\geq 2n+2r-2 \quad \text{and} \quad \mathsf s(C_2^{r-1}\oplus C_{2n})\geq 4n+2r-3 \,.
\]
\end{lemma}
\begin{proof}
The main statement follows from \cite[Lemma 3.2]{Edel2007}. If $H = C_2^{r-1}$, then $\mathsf D (H)=r$ and the second statement follows.
\end{proof}
\begin{lemma}\label{SUM} Let $W$ be a squarefree sequence of length $|W|\geq 9$ over $C_2^4\setminus \{0\}$ . Then for any element $w\t W$, there exist $|W|-8$ disjoint subsequences $R_1, \ldots, R_{|W|-8}$ of $Ww^{-1}$ such that $\sigma(R_i)=w$ and $|R_i|=2$ for all $i\in [1, |W|-8]$. In particular, $W$ has at least $|W|-8$ distinct zero-sum subsequences of length $3$ containing $w$.
\end{lemma}
\begin{proof} For any element $w\t W$, we can always find a set $A=\{a_1,\ldots,a_7\}\subseteq C_2^4\setminus\{0\}$ satisfying $C_2^4\setminus\{0\}=\{w\}\cup A\cup w+A$. Let $T_i=a_i\cdot(w+a_i)$ for each $i\in [1,7]$. Therefore $w=\sigma(T_1)=\ldots=\sigma(T_7)$ and $\mathsf{supp}(wT_1\cdot\ldots\cdot T_7)=C_2^4\setminus \{0\}$.
Since \begin{align*}
&\left|\left\{T_i \ \Big|\ i\in [1,7]\text{ and } T_i \nmid W\right\}\right|\le \left|\left\{g\in C_2^4\setminus \{0\}\ \Big|\ g\t T_i \text{ for some }i\in [1,7] \text{ and } g\nmid W \right\}\right|\\
\le& |C_2^4\setminus(\{0\}\cup \supp(W))|= 15-|W|,
\end{align*}
we have that $\left|
\left\{T_i \ \Big|\ i\in [1,7]\text{ and } T_i \t W\right\}\right|\ge 7-(15-|W|)=|W|-8\ge 1$.
In particular, for each $i\in [1,7]$, if $T_i\t W$, then $wT_i$ is a zero-sum subsequence of $W$.
\end{proof}
\section{The proof of Theorem \ref{Th2} and Theorem \ref{Th1}.1}
\begin{lemma}\label{Supp} Let $G=H\oplus K$ be a finite abelian group, where $H\cong C_2^r$ with $r\geq 3$ a positive integer and $K\cong C_n$ with $n\geq 3$ an odd integer. Denote $\phi_r$ to be the projection from $G$ to $H$ and $\psi_r$ to be the projection from $ G$ to $K$.
Let $S_r$ be a sequence over $C_2^r\oplus C_n$ such that $\phi_r(S_r)$ is a squarefree sequence with $\mathsf{supp}(\phi_r(S_r))=H\setminus \{0\}$. If the following property {\bf P1} holds,
\begin{itemize}
\item[\bf P1.] For any two distinct subsequences $T_1, T_2$ of $ S_r$ with $|T_1|=|T_2|=4$ and $\phi_r(\sigma(T_1))=\phi_r(\sigma(T_2))=0$, we have that $\sigma(T_1)=\sigma(T_2)$.
\end{itemize}
then $|\mathsf{supp}(\psi_r(S_r))|=1$.
\end{lemma}
\begin{proof}
We proceed by induction on $r$.
Suppose that $r=3$. Let $(e_1,e_2,e_3)$ be a basis of $H$, and $e$ be a basis of $K$.
Since $\phi_3(S_3)$ is a squarefree sequence with $\mathsf{supp}(\phi_3(S_3))=H\setminus \{0\}$, we can assume $S_3=g_1g_2g_3g_4g_5g_6g_7$,
where
\begin{align*}g_1=e_1+a_1e,&\quad g_2=e_2+e_3+a_2e,\quad g_3=e_2+ a_3e, \quad g_4=e_1+e_3+ a_4e ,\\
g_5=e_3+ a_5e ,&\quad g_6=e_1+e_2+a_6e ,\quad g_7=e_1+e_2+e_3+ a_7e,\text{ and }\ a_1, \ldots , a_7\in [0,n-1].
\end{align*}
By the property {\bf P1} and
\begin{align*}
&\phi_3(g_1+g_3+g_5+g_7)
=\phi_3(g_1+g_4+g_6+g_7)
=\phi_3(g_2+g_3+g_6+g_7)
=\phi_3(g_2+g_4+g_5+g_7)\\
=&\phi_3(g_1+g_2+g_3+g_4)
=\phi_3(g_1+g_2+g_5+g_6)
=\phi_3(g_3+g_4+g_5+g_6)=0,
\end{align*}
we have that \begin{align*}
g_1+g_3+g_5+g_7
&=g_1+g_4+g_6+g_7
=g_2+g_3+g_6+g_7
=g_2+g_4+g_5+g_7\\
&=g_1+g_2+g_3+g_4
=g_1+g_2+g_5+g_6
=g_3+g_4+g_5+g_6.
\end{align*}
By easily calculation, we obtain that $a_1=a_2=a_3=a_4=a_5=a_6=a_7$, which implies that $|\mathsf{supp}(\psi_3(S_3))|=1$.
\medskip
Suppose that the conclusion is correct for $r=d\geq 3$.
We want to prove the conclusion is also correct for $r=d+1$.
Let $(e_1,\dots,e_{d+1})$ be a basis of $H$ and $e$ be a basis of $K$. Denote by $A_i=\left<e_1,\ldots,e_i\right>\setminus \{0\}$, where $i\in[1,d+1]$. Then $A_{d+1}=A_d\cup (e_{d+1}+A_d)\cup \{e_{d+1}\}$. Since $\phi_{d+1}(S_{d+1})$ is a squarefree sequence with $\mathsf{supp}(\phi_{d+1}(S_{d+1}))=H\setminus \{0\}$, we can assume $S_{d+1}=\prod_{u\in A_{d+1}}u+c_ue$, where $c_u\in [0,n-1]$ for each $u\in A_{d+1}$.
Let
$$W_1=\prod_{u\in A_d}u+c_ue, \quad W_2=\prod_{u\in e_{d+1}+A_d}u-e_{d+1}+c_ue,\quad H'=\left<e_1,\ldots,e_d\right>, \text{ and } G'=H'\oplus K .$$
Then $W_1,W_2$ are sequences over $G'$ and $S_{d+1}=W_1\cdot (e_{d+1}+W_2)\cdot (e_{d+1}+c_{e_{d+1}}e)$. Since $\mathsf r(H')=d$, $\phi_{d+1}\big|_{G'}$ is the projection from $G'$ to $H'$, and $\psi_{d+1}\big|_{G'}$ is the projection from $G'$ to $K$, we obtain that
$W_1, W_2$ satisfy the property {\bf P1} for $r=d$ which implies that
$|\supp(\psi_{d+1}(W_1))|=|\supp(\psi_{d+1}(W_2))|=1$. Therefore we can assume that $c_u=x$ for all $u\in A_d$ and $c_u=y$ for all $u\in e_{d+1}+A_d$ where $x,y\in [0,n-1]$.
Let $B=\{0,e_1,e_2,e_1+e_2\}$ and $T_1=\prod_{u\in e_3+ B}u+c_ue$, $T_2=\prod_{u\in e_{d+1}+e_3+B}u+c_ue$, $T_3=\prod_{u\in e_{d+1}+B}u+c_ue$. Then $T_1T_2T_3\t S_{d+1}$ and $\phi_{d+1}(\sigma(T_1))=\phi_{d+1}(\sigma(T_2))=\phi_{d+1}(\sigma(T_3))=0$. By the property {\bf P1} and $|T_1|=|T_2|=|T_3|=4$, we obtain that $\psi_{d+1}(\sigma(T_1))=\psi_{d+1}(\sigma(T_2))=\psi_{d+1}(\sigma(T_3))$ and hence $$4x\equiv 4y\equiv 3y+c_{e_{d+1}}\pmod n.$$ It follows that $x=y=c_{e_{d+1}}$ which implies that $|\supp(\psi(S_{d+1}))|=1$.
\end{proof}
\medskip
\begin{proof}[\bf Proof of Theorem \ref{Th2}] Let $G=H\oplus K$ be a finite abelian group, where $H\cong C_2^r$ with $r\geq 3$ a positive integer and $K\cong C_n$ with $n\geq 3$ an odd integer. Denote $\phi$ to be the projection from $G$ to $H$ and $\psi$ to be the projection from $ G$ to $K$.
Let $S$ be a sequence over $G$ with $|S|=4n+2^r-5$. Assume to the contrary that $S$ contains no zero-sum subsequence of length $2n$.
Suppose that $\phi(G)=H=\{h_0,h_1,\ldots,h_{2^r-1}\}$. We can assume that
$$\phi(S)=h_0^{n_0}\cdot \ldots \cdot h_{2^r-1}^{n_{2^r-1}}\mbox{ and }S=W_0\cdot \ldots \cdot W_{2^r-1}$$
where $n_0, \ldots, n_{2^r-1}\in \mathbb{N}_0$, and $\phi(W_i)=h_{i}^{n_i}$ for all $i\in [0, 2^r-1]$.
Then $S$ allows a product decomposition $S=S_1\cdot \ldots \cdot S_{k}\cdot S_0$ satisfying that $\phi(S_0)$ is squarefree, and for each $i\in [1, k]$, $|S_i|=2$ and $\sigma(\phi(S_i))=0$. Therefore $S_i\t W_j$ for some $j\in [0,2^r-1]$.
Since $4n+2^r-5=|S|=2k+|S_0|\le 2k+2^r$, we obtain that $k\ge 2n-2$.
By our assumption, $\sigma(S_1)\cdot\ldots\cdot \sigma(S_k)\in \mathscr F(\ker(\phi))$ has no subsequence of length $n$. Therefore by Lemma \ref{cyclic}.3, we have that $k=2n-2$, $|S_0|=2^r-1$ and $$\sigma(S_1)\cdot \ldots \cdot \sigma(S_{2n-2})=g^{n-1}{g_1}^{n-1},$$
where $g, g_1\in \mathsf{ker}(\phi)$ and $\mathsf{ord}(g-g_1)=n$.
Since $|\supp(\phi(S_0))|=|S_0|=2^r-1$, let $\{b\}=\phi(G)\setminus\supp(\phi(S_0))$ and $$S'=S+b-\frac{n+1}{2}g_1=(S_1+b-\frac{n+1}{2}g_1)\cdot \ldots \cdot( S_{2n-2}+b-\frac{n+1}{2}g_1)\cdot (S_{0}+b-\frac{n+1}{2}g_1).$$
Therefore $S'$ has no zero-sum subsequence of length $2n$ and $0\notin \supp(\phi(S_0+b))=\supp(\phi(S_0+b-\frac{n+1}{2}g_1))$,
$$\sigma(S_1+b-\frac{n+1}{2}g_1)\cdot \ldots \cdot\sigma( S_{2n-2}+b-\frac{n+1}{2}g_1)=(g-g_1)^{n-1}\cdot 0^{n-1}.$$
Then without loss of generality, we can assume that $0\notin \supp(\phi(S_0))$ and $$\sigma(S_1)=\cdots =\sigma(S_{n-1})=g_1=0\mbox{ and }\sigma(S_n)=\cdots =\sigma(S_{2n-2})=g.$$
\medskip\noindent
{\bf Case 1.} There exists $T\mid S_0$ such that $|T|=4$, $\sigma(\phi(T))=0$ and $\sigma(\psi(T))\neq g$.
\smallskip
Then $\psi(\sigma(T))=tg$ where $t\in [2, n]$. By calculation we get
$$\sigma(S_1\cdot\ldots\cdot S_{t-2}\cdot S_{n}\cdot \ldots\cdot S_{2n-1-t}\cdot T)=0, \quad \text{ and }$$
$$|S_1\cdot\ldots\cdot S_{t-2}\cdot S_{n}\cdot \ldots\cdot S_{2n-1-t}\cdot T|
=2(t-2)+2(n-t)+4
=2n,$$
a contradiction.
\medskip\noindent
{\bf Case 2.} For any subsequence $T\t S_0$ satisfying $|T|=4$ and $\phi(\sigma(T))=0$, we have $\psi(\sigma(T))=g$.
\smallskip
By $\mathsf{supp}(\phi(S_0))=\phi(G)\setminus \{0\}$ and Lemma \ref{Supp}, we have $\mathsf{supp}(\psi(S_0))=\{a\}$ for some $a\in \psi(G)$. Let $T$ be a subsequence of $S_0$ satisfying $|T|=4$ and $\phi(\sigma(T))=0$, then $\psi(\sigma(T))=4a=g$ which implies that
\begin{equation}\label{e3.1}
a=\{\frac{n+1}{4}g\} \quad \text{ if } n\equiv 3\pmod 4 \qquad \text{ and } \qquad a=\{\frac{3n+1}{4}g\} \quad \text{ if } n\equiv 1\pmod 4\,.
\end{equation}
Without loss of generality, we can assume $h_0=0\notin \supp(\phi(S_0))$ and
$|W_1|\geq \cdots \geq |W_{2^r-1}|$.
Then $2\nmid |W_i|$ for all $i\in [1, 2^r-1]$.
We can distinguish the following two cases.
\medskip\noindent
{\bf Subcase 2.1.} $|W_1|\geq 3$.
\smallskip
Without loss of generality, we can assume that $a_1a_2a_0\t W_1$ and $S_i=a_1a_2$, $a_0\t S_0$, where $i\in [1,2n-2]$. Then $\sigma(\psi(S_i))=\psi(a_1)+\psi(a_2)\in \{0, g\}$ and hence $\psi(a_1)\neq \psi(a_0)$ or $\psi(a_2)\neq \psi(a_0)$ by Equation \eqref{e3.1}. Without loss of generality, we can assume that $\psi(a_1)\neq \psi(a_0)$. Let $S_i'=a_0a_2$ and $S_0'=S_0a_0^{-1}a_1$. Since $\sigma(\phi(S_1))\cdot\ldots \cdot \sigma(S_{i-1})\cdot\sigma(S_i')\cdot \sigma(S_{i+1})\cdot\ldots\cdot \sigma(S_{2n-2})$ has no subsequence of length $n$, by Lemma \ref{cyclic}.3 we obtain that $\sigma(\phi(S_i'))=\sigma(\phi(S_i))$ which implies that $\psi(a_1)= \psi(a_0)$, a contradiction.
\medskip\noindent
{\bf Subcase 2.2.} $|W_1|\le 2$.
\smallskip
Since $2\geq |W_1|\geq \cdots \geq |W_{2^r-1}|$ and $2\nmid |W_i|$ for all $i\in [1, 2^r-1]$, we have that $|W_1|=\cdots =|W_{2^r-1}|=1$. Then
$$|W_0|=|S|-\sum_{i=1}^{2^r-1}|W_i|=4n+2^r-5-(2^r-1)=4n-4\ge 3n-1.$$
Since $W_0$ is a sequence over $\ker(\phi)\cong C_n$, there are two disjoint zero-sum subsequences $V_1,V_2$ of length $n$ by $\mathrm \mathsf s(C_n)=2n-1$(see Theorem {\bf A}). Therefore $V_1V_2$ is a zero-sum subsequence of length $2n$, a contradiction.
\end{proof}
\medskip
\begin{proof}[\bf{Proof of Theorem \ref{Th1}.1}]
By Lemma \ref{ETA} and Inequality \ref{LOWETA}, we only need to prove that $\mathrm \mathsf s(G)\le 4n+3$. If $n$ is odd, it follows immediately by Theorem \ref{Th2}. Thus we can always assume that $n$ is even.
Let $(e_1,e_{2},e)$ be a basis of $G=C_2^{2}\oplus C_{2n}$ with $\ord(e_1)=\ord(e_{2})=2$ and $\ord(e)=2n$ and $S$ be any sequence over $G$ with $|S|=4n+3$. Assume to the contrary that $S$ contains no zero-sum subsequence of length $2n$.
Let $\theta: G\rightarrow G$ be the homomorphism defined by $\theta(e_1)=e_1$, $\theta(e_2)=e_2$ and $\theta(e)=ne$. Then $\mathsf{ker}(\theta)=\left<2e\right>\cong C_n$ and $\theta(G)=\left<e_1,e_{2}, ne\right>\cong C_2^3$.
Let $\theta(G)=\{h_0,h_1,\ldots,h_{7}\}$. We can assume that
$$\theta(S)=h_0^{n_0}\cdot \ldots \cdot h_{7}^{n_{7}}\mbox{ and }S=W_0\cdot \ldots \cdot W_{7}$$
where $n_0, \ldots, n_{7}\in \mathbb{N}_0$, and $\theta(W_i)=h_{i}^{n_i}$ for all $i\in [0, 7]$.
Then $S$ allows a product decomposition $S=S_1\cdot \ldots \cdot S_{k}\cdot S_0$ satisfying that $\theta(S_0)$ is squarefree, and for each $i\in [1, k]$, $|S_i|=2$ and $\sigma(\theta(S_i))=0$. Therefore $S_i\t W_j$ for some $j\in [0,7]$.
Since $4n+3=|S|=2k+|S_0|\le 2k+8$, we obtain that $k\ge 2n-2$.
By our assumption, $\sigma(S_1)\cdot\ldots\cdot \sigma(S_k)\in \mathscr F(\ker(\theta))$ has no subsequence of length $n$. Therefore by Lemma \ref{cyclic}.3, we have that $k=2n-2$, $|S_0|=7$ and $$\sigma(S_1)\cdot \ldots \cdot \sigma(S_{2n-2})=(2ke)^{n-1}(2k_1e)^{n-1},$$
where $k, k_1\in [0,n-1]$ and $\gcd(k-k_1,n)=1$. Since $n$ is even, without loss of generality, we can assume that $k_1$ is even.
Since $|\supp(\theta(S_0))|=|S_0|=7$, we let $\{b\}=\theta(G)\setminus\supp(\theta(S_0))\in G$ and let $$S'=S+b-k_1e=(S_1+b-k_1e)\cdot \ldots \cdot( S_{2n-2}+b-k_1e)\cdot (S_{0}+b-k_1e).$$
Therefore $S'$ has no zero-sum subsequence of length $2n$ and $0\notin \supp(\theta(S_0+b))=\supp(\theta(S_0+b-k_1e))$,
$$\sigma(S_1+b-k_1e)\cdot \ldots \cdot\sigma( S_{2n-2}+b-k_1e)=(2(k-k_1)e)^{n-1}\cdot 0^{n-1}.$$
Then without loss of generality, we can assume that $0\notin \supp(\theta(S_0))$ and $$\sigma(S_1)=\cdots =\sigma(S_{n-1})=2k_1e=0\ \mbox{ and }\ \sigma(S_n)=\cdots =\sigma(S_{2n-2})=2ke=2e.$$
\medskip\noindent
{\bf Case 1.} There exists $T\mid S_0$ such that $|T|=4$, $\theta(\sigma(T))=0$ and $\sigma(T)\neq 2e$.
\smallskip
Then $\sigma(T)=2te$ where $t\in [2, n]$. By calculation we get
$$\sigma(S_1\cdot\ldots\cdot S_{t-2}\cdot S_{n}\cdot \ldots\cdot S_{2n-1-t}\cdot T)=0 \quad \text{ and }$$
$$|S_1\cdot\ldots\cdot S_{t-2}\cdot S_{n}\cdot \ldots\cdot S_{2n-1-t}\cdot T|
=2(t-2)+2(n-t)+4
=2n,$$
a contradiction.
\medskip\noindent
{\bf Case 2.} For any subsequence $T\t S_0$ satisfying $|T|=4$ and $\theta(\sigma(T))=0$, we have $\sigma(T)=2e$.
\smallskip
Since $\mathsf{supp}(\theta(S_0))=\theta(G)\setminus \{0\}$, we can assume $S_0=g_1g_2g_3g_4g_5g_6g_7$,
where
\begin{align*}g_1=e_1+2a_1e,&\quad g_2=e_2+e+2a_2e,\quad g_3=e_2+ 2a_3e, \quad g_4=e_1+e+ 2a_4e ,\\
g_5=e+ 2a_5e ,&\quad g_6=e_1+e_2+2a_6e ,\quad g_7=e_1+e_2+e+ 2a_7e,\text{ and }a_1, \ldots , a_7\in [0,n-1].
\end{align*}
Since $
\theta(g_1+g_3+g_5+g_7)
=\theta(g_2+g_4+g_5+g_7)
=\theta(g_1+g_2+g_3+g_4)
$,
we obtain the following equations
\begin{align*}
g_1+g_3+g_5+g_7
=g_2+g_4+g_5+g_7
=g_1+g_2+g_3+g_4
=2e,
\end{align*}
which implies that
$$2e+2(a_1+a_3+a_5+a_7)e=4e+2(a_2+a_4+a_5+a_7)e=2e+2(a_1+a_2+a_3+a_4)e=2e.$$
Therefore \begin{align*}
& a_1+a_3+a_5+a_7\equiv 0 \pmod n,\\
& a_2+a_4+a_5+a_7 \equiv -1 \pmod n, \\
& a_1+a_2+a_3+a_4\equiv 0 \pmod n.
\end{align*}
Thus $2(a_1+a_3)+a_2+a_4+a_5+a_7\equiv 0\pmod n$, which implies that $2(a_1+a_3)\equiv 1\pmod n$, a contradiction to $n$ is even.
\end{proof}
\section{Preparatory results about $C_2^3\oplus C_{2n}$}
In the whole section, we consider the group $G=C_2^3\oplus C_{2n}$, where $n\ge 3$ is an odd integer. Thus $G\cong C_2^4\oplus C_n$.
Let $G=H\oplus K$, where $H,K$ are subgroups of $G$ with $H\cong C_2^4$ and $K\cong C_n$. Denote $\phi$ to be the projection from $G$ to $H$ and $\psi$ to be the projection from $ G$ to $K$.
\begin{lemma}\label{IMP} Let $G,H,K$ and $\phi,\psi$ be as above.
If $S$ is a sequence of length $10$ over $G$ such that $\phi(S)$ is a squarefree sequence over $H\setminus\{0\}$, then $S$ has two distinct subsequences $T_1$ and $T_2$ of length $\{ |T_1|, |T_2|\}
\subseteq [3,4]$ satisfying $\sigma(\phi(T_1))=\sigma(\phi(T_2))=0$ but $\sigma(\psi(T_1))\neq \sigma(\psi(T_2))$.
\end{lemma}
\begin{proof}
By Lemma \ref{SUM}, $S$ has at least two distinct subsequences $W_1$, $W_2$ of length $3$ such that $\phi(\sigma(W_1))=\phi(\sigma(W_2))=0$.
Assume to the contrary that for any zero-sum subsequence $\phi(T)$ of $\phi(S)$ with length $3$ or $4$, we have $\sigma(\psi(T))=e$, where $e\in K\setminus \{0\}$.
By Lemma \ref{SUM}, there exists a subsequence $T$ of $S$ such that $|T|=3$ and $\sigma(\phi(T))=0$.
Then $\sigma(\psi(T))=e$ and hence there exists an element $u\t T$ such that $\psi(u)\neq \frac{n+1}{2}e$.
By Lemma \ref{SUM} again, there exist disjoint subsequences $R_1, \, R_2$ of $Su^{-1}$ such that $\sigma(\phi(R_1))=\sigma(\phi(R_2))=\phi(u)$ and $|R_1|=|R_2|=2$.
Thus $\phi(R_1R_2)$, $\phi(R_1u)$, and $\phi(R_2u)$ are zero-sum sequences which implies that $\sigma(\psi(R_1R_2))=\sigma(\psi(R_1u))=\sigma(\psi(R_2u))=e$.
It follows that $\psi(u)=\frac{n+1}{2}e$, a contradiction to the choice of $u$.
\end{proof}
\begin{lemma}\label{SHO} Let $G,H,K$ and $\phi,\psi$ be as above.
If $n=3$ and $S$ is a sequence of length $12$ over $G$ such that $\phi(S)$ is a squarefree sequence, then $S$ contains a short zero-sum subsequence.
\end{lemma}
\begin{proof}
Assume to the contrary that $S$ contains no short zero-sum subsequence. Thus $0\not\in \mathsf{supp}(S)$.
If $0\in \mathsf{supp}(\phi(S))$, then there exists $g\mid S$ such that $\phi(g)=0$ and hence $\psi(g)\neq 0$. By $|Sg^{-1}|=11$ and Lemma \ref{IMP}, $Sg^{-1}$ have a subsequence $T$ of length $|T|\in\{3,4\}$ such that $\sigma(\phi(T))=0$ and $\sigma(\psi(T))\neq \psi(g)$. Since $\sigma(\psi(T))\neq 0$, we obtain $\sigma(\psi(T))=2\psi(g)$ which implies that $Tg$ is a short zero-sum subsequence of $S$, a contradiction.
\medskip
Therefore $0\notin \mathsf{supp}(\phi(S))$. Let $S=g_1\cdot\ldots\cdot g_{12}$. We distinguish the following four cases to finish the proof.
\medskip
\noindent\textbf{Case 1.} $\mathsf v_0(\psi(S))\geq 5$.
\smallskip
Without loss of generality, we can assume that $\psi(g_1\cdot\ldots\cdot g_5)=0^5$. Since $\phi(g_1\cdot\ldots\cdot g_5)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $X\mid g_1\cdot\ldots\cdot g_5$ such that $\sigma(\phi(X))=0$ which implies that $X$ is a short zero-sum subsequence of $S$, a contradiction.
\medskip
\noindent\textbf{Case 2.} $\mathsf v_0(\psi(S))=4$.
\smallskip
Without loss of generality, we can assume that $\psi(g_1g_2g_3g_4)=0^4$ and $\psi(g_5g_6g_7g_8)=e^4$ for some $e\in \psi(G)\cong C_3$. Thus $g_1, g_2, g_3, g_4, g_5+g_6+g_7, g_5+g_6+g_8\in \phi(G)\cong C_2^4$ and $g_5+g_6+g_7\neq g_5+g_6+g_8$. Choose $R=g_5+g_6+g_7$ or $g_5+g_6+g_8$ such that $\sigma(R)\neq g_1+g_2+g_3+g_4$. Then $g_1g_2g_3g_4\sigma(R)$ has a zero-sum subsequence of length $\le 4$ which implies that $S$ contains a short zero-sum subsequence, a contradiction.
\medskip
\noindent\textbf{Case 3.} $\mathsf v_0(\psi(S))=3$.
\smallskip
Without loss of generality, we can assume that $\psi(S)=0^3\cdot e^u\cdot (2e)^v$, $u+v=9$, $u\geq v$, and $\psi(g_1g_2g_3)=0^3$ for some $e\in \psi(G)\cong C_3$.
Suppose that $v=0$. We assume that $\psi(g_4\cdot \ldots\cdot g_{12})=e^9$. Then by Lemma \ref{SUM}, $\phi(g_4\cdot \ldots\cdot g_{12})$ contains a zero-sum subsequence of length $3$ which implies that $S$ contains a short zero-sum subsequence of length $3$, a contradiction.
Suppose that $v=1$. We assume that $\psi(g_4\cdot \ldots\cdot g_{11})=e^8$ and $\psi(g_{12})=2e$. Then $g_1,g_2,g_3, g_{11}+g_{12}, g_4+g_5+g_j\in \phi(G)\cong C_2^4$ for any $j\in [6,10]$. If $\sigma(g_1g_2g_3g_{11}g_{12})=0$, then $g_1g_2g_3g_{11}g_{12}$ is a short zero-sum subsequence of $S$, a contradiction. Thus $\sigma(g_1g_2g_3g_{11}g_{12})\neq 0$. Since $|[6,10]|=5$, there exists an $i\in [6, 10]$ such that $$g_4+g_5+g_i\notin \{\sigma(g_1g_2g_3(g_{11}g_{12})), \sigma(g_1g_2(g_{11}g_{12})), \sigma(g_1g_3(g_{11}g_{12})), \sigma(g_2g_3(g_{11}g_{12}))\}.$$
Therefore $g_1g_2g_3\sigma(g_{11}g_{12})\sigma(g_4g_5g_i)$ contains a zero-sum subsequence of length $\le 3$ which implies that $S$ contains a short zero-sum subsequence, a contradiction.
Suppose that $v\geq 2$. we assume that $\psi(g_4\cdot \ldots\cdot g_{8})=e^{5}$ and $\psi(g_{11} g_{12})=(2e)^2$. Then $g_4+g_{11}, g_5+g_{12}, g_6+g_{12}\in \phi(G)\cong C_2^4$ and $g_5+g_{12}\neq g_6+g_{12}$. Choose $R=g_5+g_{12}$ or $g_6+g_{12}$ such that $\sigma(R)\neq g_{4}+g_{11}+g_1+g_2+g_3$. Therefore $g_1g_2g_3\sigma(g_{4}g_{11})\sigma(R)$ contains a zero-sum subsequence of length $\le4$ which implies that $S$ contains a short zero-sum subsequence, a contradiction.
\medskip
\noindent\textbf{Case 4.} $\mathsf v_0(\psi(S))\le 2$.
\smallskip
Without loss of generality, we can assume that $\psi(S)=0^t\cdot e^u\cdot (2e)^v$, $t\in \mathbb{N}_0$, $u+v\ge 10$, and $u\geq v$ for some $e\in \psi(G)\cong C_3$.
Suppose that $u\ge 9 $. We assume that $\psi(g_1\cdot\ldots\cdot g_9)=e^9$. Then by Lemma \ref{SUM}, $\phi(g_1\cdot \ldots\cdot g_{9})$ contains a zero-sum subsequence of length $3$ which implies that $S$ contains a short zero-sum subsequence of length $3$, a contradiction.
Suppose that $u\le 8$. Then $v\ge 2$. We assume that $\psi(g_1\cdot \ldots\cdot g_{u})=e^{u}$ and $\psi(g_{u+1}\cdot \ldots\cdot g_{u+v})=(2e)^v$. If there exist $i_1, i_2\in [1, u]$ and $j_1, j_2\in [u+1, u+v]$ such that $i_1\neq i_2$, $j_1\neq j_2$, and $g_{i_1}+g_{j_1}=g_{i_2}+g_{j_2}$, then $g_{i_1}g_{j_1}g_{i_2}g_{j_2}$ is a short zero-sum subsequence of $S$, a contradiction. Therefore
$$|\{g_i+g_j\in \phi(G)\mid i\in [1,u] \text{\ and\ } j\in[u+1,u+v] \}|\ge uv\ge v(10-v)\ge 16=|\phi(G)|.$$
It follows that there exist an $i\in [1,u]$ and a $j\in [u+1,u+v]$ such that $\phi(g_i)=\phi(g_j)$,
a contradiction to $\phi(S)$ is squarefree.
\end{proof}
\begin{lemma}\label{L1} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S=h_1\cdot\ldots\cdot h_8$ be a sequence over $G\setminus\{0\}$ with $\phi(h_1)+\phi(h_2)=\phi(h_3)+\phi(h_4)=\phi(h_5)+\phi(h_6)=\phi(h_7)+\phi(h_8)$.
If $\phi(S)$ is a squarefree sequence with $0\not\in \supp(\phi(S))$ and satisfies the following property $(*)$:
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
then $\supp(\psi(S))=\{\frac{n+1}{4}e\}$ if $n\equiv 3 \pmod{4}$ and $\supp(\psi(S))=\{\frac{3n+1}{4}e\}$ if $n\equiv 1 \pmod{4}$.
\end{lemma}
\begin{proof}
Since $\sigma(\phi(h_1h_2h_3h_4))=\sigma(\phi(h_3h_4h_5h_6))=\sigma(\phi(h_5h_6h_1h_2))=0$, we obtain that $\sigma(\psi(h_1h_2h_3h_4))=\sigma(\psi(h_3h_4h_5h_6))=\sigma(\psi(h_5h_6h_1h_2))=e$ which implies that $\psi(h_1)+\psi(h_2)=\psi(h_3)+\psi(h_4)=\psi(h_5)+\psi(h_6)=\frac{n+1}{2}e$. With the same reason, we can prove that $\psi(h_7)+\psi(h_8)=\frac{n+1}{2}e$.
Let $\psi(h_i)=k_ie$ where $1\le i\le 8$ and $0\le k_i\le n-1$. Without loss of generality, we can assume that $k_1\le k_2,\, k_3\le k_4,\, k_5\le k_6,\,k_7\le k_8$. We consider the sequence $W=h_1h_2h_3h_5h_7$ (see Figure \ref{f1}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.7 mm
\begin{picture}(76.76,37.24)(0,0)
\put(7.00,27.04){\circle*{1.80}}
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\thinlines\lbezier(2.00,35.24)(74.76,35.24)\lbezier(74.76,35.24)(74.76,22.48)\lbezier(74.76,22.48)(12.19,22.48)\lbezier(12.19,22.48)(12.19,2.00)\lbezier(12.19,2.00)(2.00,2.00)\lbezier(2.00,2.00)(2.00,35.24)
\end{picture
\end{center}
\caption{}\label{f1}
\end{figure}
Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
\smallskip
\noindent\textbf{Case 1.} $|V|=3$.
\smallskip
Since $0\not\in \supp(\phi(S))$, we obtain that $h_1h_2\nmid V$.
By symmetry, we only need to consider $V=h_1h_3h_5$ or $V=h_2h_3h_5$.
Suppose that $V=h_1h_3h_5$. Then $\sigma(\phi(h_1h_3h_5))=0$ and hence $\sigma(\phi(h_1h_4h_6))=0$. Thus \[\sigma(\psi(h_1h_3h_5))=\sigma(\psi(h_1h_4h_6))=e\]
which implies that $\sigma(\psi(h_3h_5))=\sigma(\psi(h_4h_6))=\frac{n+1}{2}e$ and $\psi(h_1)=\frac{n+1}{2}e$ by $\sigma(\psi(h_3h_4h_5h_6))=e$. Therefore $\psi(h_2)=0$, a contradiction to $k_1\le k_2$.
Suppose that $V=h_2h_3h_5$. Then $\sigma(\phi(h_2h_3h_5))=0$ and hence $\sigma(\phi(h_1h_4h_5))=0$. Thus \[\sigma(\psi(h_2h_3h_5))=\sigma(\psi(h_1h_4h_5))=e\] which implies that $\sigma(\psi(h_2h_3))=\sigma(\psi(h_1h_4))=\frac{n+1}{2}e$ and $\psi(h_5)=\frac{n+1}{2}e$ by $\sigma(\psi(h_1h_2h_3h_4))=e$. Therefore $\psi(h_6)=0$, a contradiction to $k_5\le k_6$.
\smallskip
\noindent\textbf{Case 2.} $|V|=4$.
\smallskip
Since $\phi(S)$ is squarefree, we obtain that $h_1h_2\nmid V$.
Thus there are only two cases: $V=h_1h_3h_5h_7$ and $V=h_2h_3h_5h_7$.
\medskip
Suppose that $V=h_1h_3h_5h_7$. Since $\sigma(\phi(h_1h_3h_5h_7))=\sigma(\phi(h_2h_4h_5h_7))=0$, we have that \[\sigma(\psi(h_1h_3h_5h_7))=\sigma(\psi(h_2h_4h_5h_7))=e\] which implies that $\psi(h_1+h_3)=\psi(h_2+h_4)$. By $\psi(h_1+h_2)=\psi(h_3+h_4)=\frac{n+1}{2}e$ and $k_1\le k_2,\, k_3\le k_4$, we obtain that \begin{align*}
\psi(h_1)&=\psi(h_3)=\psi(h_2)=\psi(h_4)=\frac{n+1}{4}e \,\quad \text{ if } n\equiv 3 \pmod{4}\,,\\ \psi(h_1)&=\psi(h_3)=\psi(h_2)=\psi(h_4)=\frac{3n+1}{4}e \quad \text{ if } n\equiv 1 \pmod{4}\,.
\end{align*}
With the same reason, we can prove that
\begin{align*}
\psi(h_5)&=\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{n+1}{4}e \,\quad \text{ if } n\equiv 3 \pmod{4}\,, \\ \psi(h_5)&=\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{3n+1}{4}e \quad \text{ if } n\equiv 1 \pmod{4}\,.
\end{align*}
Therefore $\supp(\psi(S))=\{\frac{n+1}{4}e\}$ if $n\equiv 3 \pmod{4}$ and $\supp(\psi(S))=\{\frac{3n+1}{4}e\}$ if $n\equiv 1 \pmod{4}$.
\medskip
Suppose that $V=h_2h_3h_5h_7$. Since $\sigma(\phi(h_2h_3h_5h_7))=\sigma(\phi(h_2h_3h_6h_8))=0$, we have that \[\sigma(\psi(h_2h_3h_5h_7))=\sigma(\psi(h_2h_3h_6h_8))=e\] which implies that $\sigma(\psi(h_5+h_7))=\sigma(\psi(h_6+h_8))$. By $\psi(h_5+h_6)=\psi(h_7+h_8)=\frac{n+1}{2}e$ and $k_5\le k_6,\, k_7\le k_8$, we obtain that
\begin{align*}
\psi(h_5)&=\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{n+1}{4}e \quad \text{ if } n\equiv 3 \pmod{4}\,,\\ \psi(h_5)&=\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{3n+1}{4}e\quad \text{ if } n\equiv 1 \pmod{4}\,.
\end{align*}
With the same reason, we can prove that
\begin{align*}
\psi(h_3)&=\psi(h_4)=\psi(h_5)=\psi(h_6)=\frac{n+1}{4}e \,\quad \text{ if } n\equiv 3 \pmod{4}\,, \\ \psi(h_3)&=\psi(h_4)=\psi(h_5)=\psi(h_6)=\frac{3n+1}{4}e \quad \text{ if } n\equiv 1 \pmod{4}\,.
\end{align*}
Thus $\sigma(\psi(V))=\sigma(\psi(h_2h_3h_5h_7))=e$ implies that $\psi(h_2)=\psi(h_3)=\psi(h_5)=\psi(h_7)$ and hence $\psi(h_1+h_2)=\frac{n+1}{2}e$ implies that $\psi(h_1)=\psi(h_2)$.
Therefore $\supp(\psi(S))=\{\frac{n+1}{4}e\}$ if $n\equiv 3 \pmod{4}$ and $\supp(\psi(S))=\{\frac{3n+1}{4}e\}$ if $n\equiv 1 \pmod{4}$.
\smallskip
\noindent\textbf{Case 3.} $|V|=5$. Then $V=h_1h_2h_3h_5h_7$.
\smallskip
It follows that $\sigma(\phi(h_4h_6h_8))=0$ and hence $\sigma (\phi(h_4h_5h_7))=\sigma(\phi(h_3h_6h_7))=0$. Thus by Property (*), we obtain $\sigma(\psi(h_4h_5h_7))=\sigma(\psi(h_3h_6h_7))=e$ which implies that $\sigma(\psi(h_4h_5))=\sigma(\psi(h_3h_6))=\frac{n+1}{2}e$ and $\psi(h_7)=\frac{n+1}{2}e$ by $\sigma(\psi(h_3h_4h_5h_6))=e$. Therefore $\psi(h_8)=0$, a contradiction to $k_7\le k_8$.
\end{proof}
\begin{lemma}\label{L2} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S=h_1\cdot\ldots\cdot h_8$ be a sequence over $G\setminus\{0\}$ with $\phi(h_1)=\phi(h_2)+\phi(h_3)=\phi(h_4)+\phi(h_5)=\phi(h_6)+\phi(h_7)$.
If $\phi(S)$ is a squarefree sequence with $0\not\in \supp(\phi(S))$, then the following property $(*)$ does not hold.
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
\end{lemma}
\begin{proof}
Assume to the contrary that the property $(*)$ holds.
Since $\sigma(\phi(h_1h_2h_3))=\sigma(\phi(h_2h_3h_4h_5))=\sigma(\phi(h_4h_5h_1))=\sigma(\phi(h_4h_5h_6h_7))=\sigma(\phi(h_6h_7h_1))=0$, we obtain that $\sigma(\psi(h_1h_2h_3))=\sigma(\psi(h_2h_3h_4h_5))=\sigma(\psi(h_4h_5h_1))=\sigma(\psi(h_4h_5h_6h_7)))=\sigma(\psi(h_6h_7h_1))=e$ which implies that $\psi(h_1)=\psi(h_2)+\psi(h_3)=\psi(h_4)+\psi(h_5)=\psi(h_6)+\psi(h_7)=\frac{n+1}{2}e$.
Let $\psi(h_i)=k_ie$ where $1\le i\le 8$ and $0\le k_i\le n-1$. Without loss of generality, we can assume that $k_2\le k_3,\, k_4\le k_5,\, k_6\le k_7$. We consider the sequence $W=h_1h_2h_4h_6h_8$ (see Figure \ref{f2}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.7 mm
\begin{picture}(102.66,34.98)(0,0)
\put(30.73,22.97){\circle*{1.80}}
\put(30.39,2.62){\circle*{1.80}}
\put(50.74,22.97){\circle*{1.80}}
\put(50.39,2.62){\circle*{1.80}}
\put(70.40,22.97){\circle*{1.80}}
\put(90.40,22.97){\circle*{1.80}}
\put(10.38,12.45){\circle*{1.80}}
\put(7.20,14.90){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_1$\strut}}
\put(26.76,24.75){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_2$\strut}}
\put(26.90,5.09){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_3$\strut}}
\put(46.60,24.75){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_4$\strut}}
\put(46.87,5.09){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_5$\strut}}
\put(65.91,24.75){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_6$\strut}}
\put(67.13,5.09){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
\put(86.49,24.75){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_8$\strut}}
\put(70.47,2.62){\circle*{1.80}}
\thinlines\lbezier(20.76,32.78)(100.47,32.78)\lbezier(100.47,32.78)(100.47,18.12)\lbezier(100.47,18.12)(22.50,18.12)\lbezier(22.50,18.12)(10.48,5.64)\lbezier(10.48,5.64)(2.00,13.93)\lbezier(2.00,13.93)(20.76,32.78)
\end{picture
\end{center}
\caption{}\label{f2}
\end{figure}
Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
\smallskip
\noindent\textbf{Case 1.} $|V|=3$.
\smallskip
Obviously $h_1\nmid v$. Then by symmetry, we only need to consider $V=h_2h_4h_6$ or $h=h_2h_4h_8$.
Suppose that $V=h_2h_4h_6$. Then $\sigma(\phi(h_2h_4h_6))=\sigma(\phi(h_3h_5h_6))=0$ and hence $\sigma(\psi(h_2h_4h_6))=\sigma(\psi(h_3h_5h_6))=e$. Therefore $\sigma(\psi(h_2h_4))=\sigma(\psi(h_3h_5))=\frac{n+1}{2}e$ by $\sigma(\psi(h_2h_3h_4h_5))=e$. It follows by that $\psi(h_6)=\frac{n+1}{2}e$, a contradiction to $k_6\le k_7$ and $\psi(h_6)+\psi(h_7)=\frac{n+1}{2}e$.
Suppose that $V=h_2h_4h_8$. Then $\sigma(\phi(h_2h_4h_8))=\sigma(\phi(h_3h_5h_8))=0$ and hence $\sigma(\psi(h_2h_4h_8))=\sigma(\psi(h_3h_5h_8))=e$. Therefore $\sigma(\psi(h_2h_4))=\sigma(\psi(h_3h_5))=\frac{n+1}{2}e$ by $\sigma(\psi(h_2h_3))=\sigma(\psi(h_4h_5))=\frac{n+1}{2}e$. It follows by $k_2\le k_3$ and $k_4\le k_5$ that $\psi(h_2)=\psi(h_3)=\psi(h_4)=\psi(h_5)$ and $\psi(h_8)=\frac{n+1}{2}e$. Therefore by $\sigma(\phi(h_1h_3h_4h_8))=\sigma(\phi(h_2h_4h_8))=0$, we obtain that $\sigma(\psi(h_1h_3h_4h_8))=e$. But $\sigma(\psi(h_1h_3h_4h_8))=\frac{n+1}{2}e+\frac{n+1}{2}e+\frac{n+1}{2}e\neq e$, a contradiction.
\smallskip
\noindent\textbf{Case 2.} $|V|=4$.
\smallskip
By symmetry,
we only need to consider $V=h_1h_2h_4h_6$ or $V=h_2h_4h_6h_8$ or $V=h_1h_2h_4h_8$.
Suppose that $V=h_1h_2h_4h_6$. Then $\sigma(\phi(h_1h_2h_4h_6))=\sigma(\phi(h_3h_4h_6))=\sigma(\phi(h_2h_4h_7))=0$. Thus we obtain that $\sigma(\psi(h_3h_4h_6))=\sigma(\psi(h_2h_4h_7))=e$ which implies that $\sigma(\psi(h_3h_6))=\sigma(\psi(h_2h_7))=\frac{n+1}{2}e$ by $\sigma(\psi(h_2h_3h_6h_7))=e$. Therefore $\psi(h_4)=\frac{n+1}{2}e$, a contradiction to $k_4\le k_5$ and $\psi(h_4+h_5)=\frac{n+1}{2}e$.
Suppose that $V=h_2h_4h_6h_8$. Then $\sigma(\phi(h_2h_4h_6h_8))=\sigma(\phi(h_3h_5h_6h_8))=0$ and hence $\sigma(\psi(h_2h_4h_6h_8))=\sigma(\psi(h_3h_5h_6h_8))=e$. Thus $\sigma(\psi(h_2h_4))=\sigma(\psi(h_3h_5))=\sigma(\psi(h_6h_8))=\frac{n+1}{2}e$. By $k_2\le k_3,k_4\le k_5$, we obtain that $\psi(h_2)=\psi(h_3)=\psi(h_4)=\psi(h_5)$. Since $\sigma(\phi(h_1h_3h_4h_6h_8))=\sigma(\phi(h_2h_4h_6h_8))=0$, we obtain that $\sigma(\psi(h_1h_3h_4h_6h_8))=\psi(h_1)+e\in\{e,2e\}$. Thus $\psi(h_1)\in \{0,e\}$, a contradiction to $\psi(h_1)=\frac{n+1}{2}e$.
Suppose that $V=h_1h_2h_4h_8$. Then $\sigma(\phi(h_1h_2h_4h_8))=\sigma(\phi(h_1h_3h_5h_8))=0$ and hence $\sigma(\psi(h_1h_2h_4h_8))=\sigma(\psi(h_1h_3h_5h_8))=e$. Thus $\sigma(\psi(h_2h_4))=\sigma(\psi(h_3h_5))=\sigma(\psi(h_1h_8))=\frac{n+1}{2}e$. By $k_2\le k_3,k_4\le k_5$, we obtain that $\psi(h_2)=\psi(h_3)=\psi(h_4)=\psi(h_5)$. Since $\psi(h_1)=\frac{n+1}{2}e$, we obtain that $\psi(h_8)=0$. It follows that $\sigma(\phi(h_3h_4h_8))=\sigma(\phi(h_1h_2h_4h_8))=0$ and $\sigma(\psi(h_3h_4h_8))=\frac{n+1}{2}e+0\neq e$, a contradiction.
\smallskip
\noindent\textbf{Case 3.} $|V|=5$. Then $V=h_1h_2h_4h_6h_8$.
\smallskip
Then $\sigma(\phi(h_1h_2h_4h_6h_8))=\sigma(\phi(h_2h_4h_7h_8))=\sigma(\phi(h_3h_5h_7h_8))=0$ and hence $\sigma(\psi(h_2h_4h_7h_8))=\sigma(\psi(h_3h_5h_7h_8))=e$. Thus $\sigma(\phi(h_2h_4))=\sigma(\phi(h_3h_5))=\frac{n+1}{2}e$. By $k_2\le k_3,k_4\le k_5$, we obtain that $\psi(h_2)=\psi(h_3)=\psi(h_4)=\psi(h_5)$. Since $\sigma(\phi(h_1h_2h_4h_6h_8))=\sigma(\phi(h_3h_4h_6h_8))=0$, we obtain that $\sigma(\psi(h_1h_2h_4h_6h_8))=\psi(h_1)+\sigma(\psi(h_3h_4h_6h_8))=\frac{n+1}{2}e+e\notin\{e,2e\}$, a contradiction.
\end{proof}
\begin{lemma}\label{L3} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S=h_1\cdot\ldots\cdot h_8$ be a sequence over $G\setminus\{0\}$ with $\phi(h_1)=\phi(h_2)+\phi(h_3)=\phi(h_4)+\phi(h_5)$ and $\psi(h_3)=\psi(h_5)=\frac{n+1}{2}e$.
If $\phi(S)$ is a squarefree sequence with $0\not\in \supp(\phi(S))$, then the following property $(*)$ does not hold.
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
\end{lemma}
\begin{proof}
Assume to the contrary that the property $(*)$ holds.
Since $\sigma(\phi(h_1))=\sigma(\phi(h_2h_3))=\sigma(\phi(h_4h_5))$, we obtain that $\sigma(\phi(h_1h_2h_3))=\sigma(\phi(h_2h_3h_4h_5))=\sigma(\phi(h_4h_5h_1))=0$ which implies that $\sigma(\psi(h_1h_2h_3))=\sigma(\psi(h_2h_3h_4h_5))=\sigma(\psi(h_4h_5h_1))=e$. Therefore $\psi(h_1)=\frac{n+1}{2}e$ and $\psi(h_2)=\psi(h_4)=0$ by $\psi(h_3)=\psi(h_5)=\frac{n+1}{2}e$.
We distinguish the following two cases to get contradictions to our assumption.
\medskip
\noindent{\bf Case 1.} $\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{n+1}{4}e$ if $n\equiv 3\pmod 4$ and $\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{3n+1}{4}e$ if $n\equiv 1\pmod 4$.
\smallskip
Consider the sequence $W=h_2h_4h_6h_7h_8$ (see Figure \ref{f3}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.7 mm
\begin{picture}(118.00,34.33)(0,0)
\put(26.32,22.85){\circle*{1.80}}
\put(25.97,2.50){\circle*{1.80}}
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\put(45.98,2.50){\circle*{1.80}}
\put(65.98,22.85){\circle*{1.80}}
\put(85.99,22.85){\circle*{1.80}}
\put(106.34,22.85){\circle*{1.80}}
\put(5.97,12.33){\circle*{1.80}}
\put(2.00,14.28){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_1$\strut}}
\put(22.52,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_2$\strut}}
\put(22.49,4.96){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_3$\strut}}
\put(42.18,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_4$\strut}}
\put(41.46,4.79){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_5$\strut}}
\put(61.50,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_6$\strut}}
\put(82.71,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
\put(103.41,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_8$\strut}}
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\end{picture
\end{center}
\caption{}\label{f3}
\end{figure}
Suppose that $n\equiv 3\pmod 4$.
Then $\psi(W)=0^2(\frac{n+1}{4}e)^3$. Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. If $|V|=5$, then $\sigma(\psi(V))=3\frac{n+1}{4}e\notin\{e,2e\}$, a contradiction. If $|V|=4$, then $\sigma(\psi(V))=e\in \{2\frac{n+1}{4}e,\,3\frac{n+1}{4}e\}$, a contradiction. Thus $|V|=3$ and $\sigma(\psi(V))=e\in \{\frac{n+1}{4}e,\,2\frac{n+1}{4}e,3\frac{n+1}{4}e\}$ which implies that $n=3$ and $h_2h_4\t V$. But $\sigma(\phi(Vh_3h_5(h_2h_4)^{-1}))=0$ and $\sigma(\psi(Vh_3h_5(h_2h_4)^{-1})))=2\frac{n+1}{2}e+\frac{n+1}{4}e=2e\neq e$, a contradiction.
Suppose that $n\equiv 1\pmod 4$.
Then $\psi(W)=0^2(\frac{3n+1}{4}e)^3$. Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. If $|V|=5$, then $\sigma(\psi(V))=3\frac{3n+1}{4}e\notin\{e,2e\}$ which implies that $n=5$. Since $\sigma(\phi(h_3h_5h_6h_7h_8))=0$, we obtain that $\sigma(\psi(h_3h_5h_6h_7h_8))=3e+3e+4e+4e+4e=3e\notin\{e,2e\}$, a contradiction. If $|V|=4$, then $\sigma(\psi(V))=e\in \{2\frac{3n+1}{4}e,\,3\frac{3n+1}{4}e\}$, a contradiction. Thus $|V|=3$ and $\sigma(\psi(V))=e\in \{\frac{3n+1}{4}e,\,2\frac{3n+1}{4}e,3\frac{3n+1}{4}e\}$, a contradiction.
\medskip
\noindent{\bf Case 2.} There exist distinct $i,j\in [6,8]$ such that $\psi(h_i)+\psi(h_j)\neq \frac{n+1}{2}e$, say $\psi(h_6)+\psi(h_7)\neq \frac{n+1}{2}e$.
\smallskip
Consider the sequence $W=h_1h_2h_4h_6h_7$ (see Figure \ref{f4}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.7 mm
\begin{picture}(102.66,34.98)(0,0)
\put(30.73,22.97){\circle*{1.80}}
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\put(7.41,14.40){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_1$\strut}}
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\put(26.90,5.09){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_3$\strut}}
\put(46.60,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_4$\strut}}
\put(46.87,4.92){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_5$\strut}}
\put(65.91,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_6$\strut}}
\put(86.49,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
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\end{picture
\end{center}
\caption{}\label{f4}
\end{figure}
Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
Suppose that $|V|=5$. Then $\sigma(\psi(V))\neq \frac{n+1}{2}e+0+0+\frac{n+1}{2}e=e$ which implies that $\sigma(\psi(V))=2e$. Thus $\sigma(\psi(h_6h_7))=\frac{n+3}{2}e$. It follows that $\sigma(\psi(h_1h_3h_5h_6h_7))=3e\notin\{e,2e\}$, a contradiction to $\sigma(\phi(h_1h_3h_5h_6h_7))=\sigma(\phi(V))=0$.
Suppose that $|V|=4$. If $h_2h_4\t V$, then $\sigma(\phi(V(h_2h_4)^{-1}h_3h_5))=0$ and $\sigma(\psi(V(h_2h_4)^{-1}h_3h_5))=2e$, a contradiction. Thus $h_2h_4\nmid V$. By symmetry, we only need to consider $V=h_1h_2h_6h_7$. But $\sigma(\psi(V))\neq \frac{n+1}{2}e+0+\frac{n+1}{2}e=e$, a contradiction.
Suppose that $|V|=3$. By symmetry, we only need to consider $V=h_1h_6h_7$, $V=h_2h_6h_7$, $V=h_2h_4h_6$ or $V=h_1h_2h_4$. If $V=h_1h_6h_7$, then $\sigma(\psi(V))\neq e$, a contradiction.
If $V=h_2h_6h_7$, then $\sigma(\phi(h_1h_3h_6h_7))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_1h_3h_6h_7))=e$. It follows that $\psi(h_6)+\psi(h_7)=0$ which implies that $\sigma(\psi(V))=0$, a contradiction. If $V=h_2h_4h_6$, then $\sigma(\psi(V))=e$ which implies that $\psi(h_6)=e$. Thus $\sigma(\psi(h_3h_5h_6))=2e$, a contradiction to $\sigma(\phi(h_3h_5h_6))=\sigma(\phi(V))=0$. If $V=h_1h_2h_4$, then $\phi(h_3)=\phi(h_4)$, a contradiction.
\end{proof}
\begin{lemma}\label{L4} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S=h_1\cdot\ldots\cdot h_8$ be a sequence over $G\setminus\{0\}$ with $\phi(h_1)=\phi(h_2)+\phi(h_3)=\phi(h_4)+\phi(h_5)$ and $\psi(h_6)=\psi(h_7)=\frac{n+1}{2}e$.
If $\phi(S)$ is a squarefree sequence with $0\not\in \supp(\phi(S))$, then the following property $(*)$ does not hold.
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
\end{lemma}
\begin{proof}
Assume to the contrary that the property $(*)$ holds.
Since $\phi(h_1)=\phi(h_2)+\phi(h_3)=\phi(h_4)+\phi(h_5)$, we obtain that $\sigma(\phi(h_1h_2h_3))=\sigma(\phi(h_2h_3h_4h_5))=\sigma(\phi(h_4h_5h_1))=0$ which implies that $\sigma(\psi(h_1h_2h_3))=\sigma(\psi(h_2h_3h_4h_5))=\sigma(\psi(h_4h_5h_1))=e$. Therefore $\psi(h_1)=\psi(h_2)+\psi(h_3)=\psi(h_4)+\psi(h_5)=\frac{n+1}{2}e$.
Let $\psi(h_i)=k_ie$ where $1\le i\le 8$ and $0\le k_i\le n-1$. Without loss of generality, we can assume that $k_2\le k_3,\, k_4\le k_5$. Consider the sequence $W=h_1h_2h_4h_6h_7$ (see Figure \ref{f5}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.7 mm
\begin{picture}(102.66,34.98)(0,0)
\put(30.73,22.97){\circle*{1.80}}
\put(30.39,2.62){\circle*{1.80}}
\put(50.74,22.97){\circle*{1.80}}
\put(50.39,2.62){\circle*{1.80}}
\put(70.40,22.97){\circle*{1.80}}
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\put(10.38,12.45){\circle*{1.80}}
\put(7.41,14.40){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_1$\strut}}
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\put(26.90,5.09){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_3$\strut}}
\put(46.60,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_4$\strut}}
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\put(65.91,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_6$\strut}}
\put(86.49,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
\thinlines\lbezier(20.76,32.78)(100.47,32.78)\lbezier(100.47,32.78)(100.47,18.12)\lbezier(100.47,18.12)(22.50,18.12)\lbezier(22.50,18.12)(10.48,5.64)\lbezier(10.48,5.64)(2.00,13.93)\lbezier(2.00,13.93)(20.76,32.78)
\end{picture
\end{center}
\caption{}\label{f5}
\end{figure}
Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
\medskip
\noindent{\bf Case 1.} $|V|=3$.
\smallskip
By symmetry, we only need to consider $V=h_1h_6h_7$, $V=h_2h_4h_6$ or $V=h_4h_6h_7$.
Suppose that $V=h_1h_6h_7$. Then $\sigma(\psi(V))=\frac{n+3}{2}e\neq e$, a contradiction.
Suppose that $V=h_2h_4h_6$. Then $\sigma(\phi(h_3h_5h_6))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_3h_5h_6))=\sigma(\psi(V))=e$. Thus $\sigma(\psi(h_3h_5))=\sigma(\psi(h_2h_4))=\frac{n+1}{2}e$ which implies that $\psi(h_3)=\psi(h_4)=\psi(h_5)=\psi(h_6)$ by $k_2\le k_3, k_4\le k_5$. Therefore $\sigma(\psi(h_1h_3h_4h_6))=3\frac{n+1}{2}\neq e$, a contradiction to $\sigma(\phi(h_1h_3h_4h_6))=\sigma(\phi(V))=0$.
Suppose that $V=h_4h_6h_7$. Then $\sigma(\phi(h_1h_5h_6h_7))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_1h_5h_6h_7))=\sigma(\psi(h_4h_6h_7))=e$. Thus $\psi(h_5)=\frac{n-1}{2}e$ and $\psi(h_4)=0$, a contradiction to $\psi(h_4)+\psi(h_5)=\frac{n+1}{2}e$.
\medskip
\noindent{\bf Case 2.} $|V|=4$.
\smallskip
By symmetry, we only need to consider $V=h_2h_4h_6h_7$, $V=h_1h_4h_6h_7$ or $V=h_1h_2h_4h_6$.
Suppose that $V=h_2h_4h_6h_7$. Then $\sigma(\phi(h_3h_5h_6h_7))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_3h_5h_6h_7))=\sigma(\psi(V))=e$. Thus $\sigma(\psi(h_3h_5))=\sigma(\psi(h_2h_4))=0$, a contradiction to $\sigma(\psi(h_2h_3h_4h_5))=e$.
Suppose that $V=h_1h_4h_6h_7$. Then $\sigma(\phi(h_5h_6h_7))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_5h_6h_7))=e$. Thus $\psi(h_5)=0$, a contradiction to $k_5\ge k_4$ and $\psi(h_4)+\psi(h_5)=\frac{n+1}{2}e$.
Suppose that $V=h_1h_2h_4h_6$. Then $\sigma(\phi(h_1h_3h_5h_6))=\sigma(\phi(V))=0$ and hence $\sigma(\psi(h_1h_3h_5h_6))=\sigma(\psi(V))=e$. Thus $\sigma(\psi(h_3h_5))=\sigma(\psi(h_2h_4))=0$, a contradiction to $\sigma(\psi(h_2h_3h_4h_5))=e$.
\medskip
\noindent{\bf Case 3.} $|V|=5$.
\smallskip
Then $\sigma(\phi(h_3h_4h_6h_7))=\sigma(\phi(h_2h_5h_6h_7))=\sigma(\phi(V))=0$ which implies that $\sigma(\psi(h_3h_4h_6h_7))=\sigma(\psi(h_2h_5h_6h_7))=e$. Thus $\sigma(\psi(h_3h_4))=\sigma(\psi(h_2h_5))=0$ which implies that $\sigma(\psi(h_2h_3h_4h_5))=0$, a contradiction to $\sigma(\phi(h_2h_3h_4h_5))=0$.
\end{proof}
\begin{prop}\label{LL1} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S$ be a sequence over $G\setminus\{0\}$.
If $\phi(S)$ is a squarefree sequence of length $|\phi(S)|=8$ with $0\not\in \supp(\phi(S))$ and satisfies the following property $(*)$:
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
then $\frac{n+1}{2}e\not\in\supp(\psi(S))$.
\end{prop}
\begin{proof}
For any $v\in \phi(G)\setminus \{0\}=H\setminus\{0\}$, we define
$$\mathsf N_v(\phi(S))=|\{T\t \phi(S) \ :\ |T|=2 \text{ and } \sigma(T)=v \}|+\delta_v,$$
where
\[
\delta_v =\left \{ \begin{array}{ll} 1 ,\quad \mbox{ if } v\in \mathsf{supp}(\phi(S)); \\ 0,\quad \mbox{ if } v\notin \mathsf{supp}(\phi(S)).
\end{array} \right.
\]
We distinguish the following four cases.
\medskip\noindent{\bf Case 1.} There exists $v\in H\setminus\{0\}$
such that $\mathsf N_v(\phi(S))=4$ and $\delta_v=0$.
\smallskip
Without loss of generality, we can assume that $v=\phi(h_1+h_2)=\phi(h_3+h_4)=\phi(h_5+h_6)=\phi(h_7+h_8)$ where $S=h_1\cdot\ldots\cdot h_8$. By Lemma \ref{L1}, we obtain that $\frac{n+1}{2}e\not\in\supp(\psi(S))$.
\medskip\noindent{\bf Case 2.} There exists $v\in H\setminus \{0\}$
such that $\mathsf N_v(\phi(S))=4$ and $\delta_v=1$.
\smallskip
Without loss of generality, we can assume that $v=\phi(h_1)=\phi(h_2+h_3)=\phi(h_4+h_5)=\phi(h_6+h_7)$ where $S=h_1\cdot\ldots\cdot h_8$. By Lemma \ref{L2}, we obtain that $\frac{n+1}{2}e\not\in\supp(\psi(S))$.
\medskip
Now, we can assume that, for each $v\in H\setminus \{0\}$,
$\mathsf N_v(\phi(S))\le 3$.
Since $\sum_{v\in H\setminus \{0\}}\mathsf N_v(\phi(S))=\frac{8\times 7}{2}+8=36$ and $|H\setminus \{0\}|=15$, by simple calculation, we obtain that $|\{v\in H\setminus \{0\}\mid \mathsf N_v(\phi(S))= 3\}|\ge 6$. We continue with further case distinctions.
\medskip\noindent{\bf Case 3.} There exist three distinct $v_1, v_2, v_3\in H\setminus \{0\}$
such that $\mathsf N_{v_1}(\phi(S))=\mathsf N_{ v_2}(\phi(S))=\mathsf N_{ v_3}(\phi(S))=3$ and $\delta_{ v_1}=\delta_{ v_2}=\delta_{ v_3}=1$.
\smallskip
Let $S=h_1\cdot\ldots\cdot h_8$. For each $i\in [1,3]$, we denote by $A_i=\{ v_i\} \cup \supp(\phi(T_{i_1}T_{i_2}))$, where $|T_{i_1}|=|T_{i_2}|=2$, $T_{i_1}T_{i_2}\t S$ , and $ v_i=\sigma(\phi(T_{i_1}))=\sigma(\phi(T_{i_2}))$. Thus $|A_i|=5$ for each $i\in[1,3]$. By symmetry, we can distinguish the following two cases.
\medskip
\noindent{\bf Subcase 3.1. }There exists $i\in [1,3]$, say $i=1$, such that $ v_2\notin A_1$ and $ v_3\notin A_1$.
\smallskip
Then we can assume that $ v_1=\phi(h_1)=\phi(h_2+h_3)=\phi(h_4+h_5)$, $ v_2=\phi(h_6)$, and $v_3=\phi(h_7)$. It follows that $\psi(h_1)=\psi(h_6)=\psi(h_7)=\frac{n+1}{2}e$, a contradiction to Lemma \ref{L4}.
\medskip
\noindent{\bf Subcase 3.2.}For each $i\in [1,3]$, there exists $j\in [1,3]$ such that $j\neq i$ and $ v_j\in A_i$.
\smallskip
For $i=1$, we can assume that $ v_2\in A_1$. Then $ v_1\in A_2$.
For $i=3$, we obtain that $ v_1\in A_3$ or $ v_2\in A_3$ which implies that $ v_3\in A_1$ or $ v_3\in A_2$. By symmetry, we can assume that $ v_3\in A_1$ and hence $ v_2, v_3\in A_1$.
Without loss of generality, we can assume that $ v_1=\phi(h_1)=\phi(h_2+h_3)=\phi(h_4+h_5)$, $ v_2=\phi(h_3)$, and $ v_3=\phi(h_5)$. It follows that $\psi(h_1)=\psi(h_3)=\psi(h_5)=\frac{n+1}{2}e$, a contradiction to Lemma \ref{L3}.
\medskip\noindent{\bf Case 4.} There exist three distinct $ v_1, v_2, v_3\in H\setminus \{0\}$
such that $\mathsf N_{ v_1}(\phi(S))=\mathsf N_{ v_2}(\phi(S))=\mathsf N_{ v_3}(\phi(S))=3$ and $\delta_{ v_1}=\delta_{ v_2}=\delta_{ v_3}=0$.
\smallskip
Let $S=h_1\cdot\ldots\cdot h_8$. For each $i\in [1,3]$, we denote by $A_i=\supp(\phi(R_{i_1}R_{i_2}R_{i_3}))$, where $|R_{i_1}|=|R_{i_2}|=|R_{i_3}|=2$, $R_{i_1}R_{i_2}R_{i_3}\t S$ , and $ v_i=\sigma(\phi(R_{i_1}))=\sigma(\phi(R_{i_2}))=\sigma(\phi(R_{i_3}))$. Thus $|A_i|=6$ for each $i\in[1,3]$ and hence $|A_i\cap A_j|\ge 6+6-8= 4$ for distinct $i,j$ where $1\le i,j\le 3$.
We proceed by the following two claims
\begin{enumerate}
\item[]\noindent{\bf Claim A. } For each $i\in [1,3]$ and each $k\in [1,3]$, $\sigma(\psi(R_{i_k}))=\frac{n+1}{2}e$.
\smallskip
\noindent
{\it Proof of \,{\bf Claim A}. }
For each $i\in [1,3]$, $\sigma(\phi(R_{i_1}R_{i_2}))=\sigma(\phi(R_{i_1}R_{i_3}))=\sigma(\phi(R_{i_2}R_{i_3}))=0$ implies that $\sigma(\psi(R_{i_1}R_{i_2}))=\sigma(\psi(R_{i_1}R_{i_3}))=\sigma(\psi(R_{i_2}R_{i_3}))=e$. Thus $\sigma(\psi(R_{i_k}))=\frac{n+1}{2}e$ for all $k\in [1,3]$.
\qedhere{(Proof of Claim A)}
\item[]
\noindent{\bf Claim B. } For each $j\in [2,3]$, there exist $1\le s<t\le 3$ such that $\supp(\phi(R_{1_s}R_{1_t}))\subseteq A_j$.
Furthermore, there exist distinct $1\le x,y\le 3$ such that $R_{1_s}R_{1_t}=R_{j_x}R_{j_y}$.
\smallskip
\noindent
{\it Proof of \,{\bf Claim B}. }
Without loss of generality, we can assume that $j=2$. Let $R_{1_1}=g_1g_2, R_{1_2}=g_3g_4$, and $ R_{1_3}=g_5g_6$.
Since $|A_1\cap A_2|\ge 4$, by symmetry, we only need to consider two cases: $\supp(\phi(g_1g_2g_3g_4))\subseteq A_1\cap A_2$ and $\supp(\phi(g_1g_2g_3g_5))= A_1\cap A_2$.
Suppose that $\supp(\phi(g_1g_2g_3g_5)) =A_1\cap A_2$. Then there exists $x\in [1,3]$ such that $R_{2_x}\t g_1g_2g_3g_5$. By symmetry, there are only three cases: $R_{2_x}=g_1g_2$, $R_{2_x}=g_1g_3$, and $R_{2_x}=g_3g_5$. If $R_{2_x}=g_1g_2$, then $ v_1= v_2$, a contradiction.
If $R_{2_x}=g_1g_3$, then $ v_2=\sigma(\phi(R_{2_x}))=\sigma(\phi(g_2g_4))$ which implies that $\phi(g_4)\in A_1\cap A_2$, a contradiction.
If $R_{2_x}=g_3g_5$, then $ v_2=\sigma(\phi(R_{2_x}))=\sigma(\phi(g_4g_6))$ which implies that $\supp(\phi(g_4g_6))\subseteq A_1\cap A_2$, a contradiction.
Suppose that $\supp(\phi(g_1g_2g_3g_4))\subseteq A_1\cap A_2$. Then $\supp(\phi(R_{1_1}R_{1_2}))\subseteq A_2$. Furthermore, there must exist $x\in[1,3]$ such that $R_{2_x}\t R_{1_1}R_{1_2}$. Thus $\sigma(\phi(R_{2_x}))=\sigma(\phi(R_{1_1}R_{1_2}R_{2_x}^{-1}))$ which implies that $R_{1_1}R_{1_2}R_{2_x}^{-1}=R_{2_y}$ for some $y\in [1,3]\setminus \{x\}$.
\qed{(Proof of Claim B)}
\end{enumerate}
\medskip
Without loss of generality, we can assume that $S=h_1\cdot\ldots\cdot h_8$, $ v_1=\phi(h_1)+\phi(h_2)=\phi(h_3)+\phi(h_4)=\phi(h_5)+\phi(h_6)$.
If $|A_1\cap A_2|=|A_1\cap A_3|=4$, then $ v_2=\sigma(\phi(h_7h_8))= v_3$, a contradiction. Thus by symmetry and Claim B, we can assume that $|A_1\cap A_2|=5$ and $ v_2=\phi(h_1)+\phi(h_3)=\phi(h_2)+\phi(h_4)=\phi(h_5)+\phi(h_7)$ which implies that $\phi(h_1)+\phi(h_4)=\phi(h_2)+\phi(h_3)=\phi(h_6)+\phi(h_7)$
(See Figure \ref{f6}).
\smallskip
\begin{figure}[ht]
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Then we have \begin{align*}
\psi(h_1\cdot\ldots\cdot h_7)&=(\frac{n+1}{4}e)^7 \quad \text{ if } n\equiv 3 \pmod 4\,,\\
\psi(h_1\cdot\ldots\cdot h_7)&=(\frac{3n+1}{4}e)^7 \quad \text{ if } n\equiv 1 \pmod 4\,.\\
\end{align*}
Assume to the contrary that $\frac{n+1}{2}e\in \mathsf{supp}(\psi(S))$. Then
$\psi(h_8)=\frac{n+1}{2}e$.
Consider the sequence $W=h_1h_2h_5h_7h_8$. Since $\phi(W)\in \mathscr{F}(\phi(G))$ and $\mathsf D(\phi(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\phi(V))=0$ and $|V|\in \{3, 4, 5\}$. If $|V|=4$, by $\sigma(\psi(V))=e$ we obtain that $V=h_1h_2h_5h_7$ which implies that $\phi(h_6)=\phi(h_7)$, a contradiction. If $|V|=5$, then $\sigma(\psi(V))=\frac{n+3}{2}e\notin \{e, 2e\}$, a contradiction.
Therefore $|V|=3$. By $\sigma(\psi(V))=1$, we obtain that $h_8\t V$. Since $h_1h_2\nmid V $ and $h_5h_7\nmid V$, by symmetry, we only need to consider $V=h_1h_5h_8$. Then $\sigma(\phi(h_2h_3h_4h_5h_8))=0$ and $\sigma(\psi(h_2h_3h_4h_5h_8))\notin \{e, 2e\}$, a contradiction.
\end{proof}
\begin{cor}\label{xx1} Let $G,H,K$ and $\phi,\psi$ be as above.
Let $ K=\left<e\right>$ and $S$ be a sequence over $G\setminus\{0\}$.
If $\phi(S)$ is a squarefree sequence with $0\not\in \supp(\phi(S))$ and satisfies the following property $(*)$:
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\qquad (*)\right.
\]
then $|S|\le 8$.
\end{cor}
\begin{proof}
Assume to the contrary that $|S|\ge 9$. Without loss of generality, we can assume that $|S|=9$.
If $\supp(\psi(S))=\{0\}$, then $S\in \mathscr F(\ker(\psi))$. By $\mathsf D(\ker(\psi))=\mathsf D(C_2^4)=5$, $S$ has a subsequence $V$ of length $|V|\in [3,5]$ such that $\sigma(\phi(V))=0$ and $\sigma(\psi(V))=0$, a contradiction.
Thus we can choose $w\t S$ such that $\psi(w)\neq0$. By Lemma \ref{SUM}, there exist distinct $g_1,g_2\in \supp(Sw^{-1})$ such that $\phi(w)=\phi(g_1)+\phi(g_2)$.
If there exist another $g_3,g_4\in \supp(S(wg_1g_2)^{-1})$ such that $\phi(w)=\phi(g_3)+\phi(g_4)$, then $\sigma(\phi(wg_1g_2))=\sigma(\phi(wg_3g_4))=\sigma(\phi(g_1g_2g_3g_4))=0$ which implies that $\sigma(\psi(wg_1g_2))=\sigma(\psi(wg_3g_4))=\sigma(\psi(g_1g_2g_3g_4))=e$. Therefore $\psi(w)=\frac{n+1}{2}e$, a contradiction to Proposition \ref{LL1}.
Otherwise, choose $g_3\in \supp(S(wg_1g_2)^{-1})$. Then $\phi(S(w+g_3)w^{-1})$ is a squarefree sequence of length $9$. By Lemma \ref{SUM}, there exist distinct $g_i,g_j\in \supp(Sw^{-1})$ such that $\phi(w+g_3)=\phi(g_i)+\phi(g_j)$. Clearly, $g_ig_j\t S(wg_1g_2g_3)^{-1}$ or $|\{g_1,g_2\}\cap\{g_i,g_j\}|=1$.
Suppose that $g_ig_j\t S(wg_1g_2g_3)^{-1}$. Then $\sigma(\phi(wg_1g_2))=\sigma(\phi(wg_3g_ig_j))=\sigma(\phi(g_1g_2g_3g_ig_j))=0$ which implies that $\sigma(\psi(wg_1g_2))=\sigma(\psi(wg_3g_ig_j))=e$ and $\sigma(\psi(g_1g_2g_3g_ig_j))\in \{e,2e\}$. Therefore $\psi(w)\in\{\frac{n+1}{2}e, 0\}$. It follows by the choice of $w$ that $\psi(w)=\frac{n+1}{2}e$, a contradiction to Proposition \ref{LL1}.
Supppose that $|\{g_1,g_2\}\cap\{g_i,g_j\}|=1$. By symmetry, we can assume that $g_i=g_1$ and $g_j\in \supp(S(wg_1g_2g_3)^{-1})$. Then $\sigma(\phi(wg_1g_2))=\sigma(\phi(wg_1g_3g_j))=\sigma(\phi(g_1g_2g_1g_3g_j))=\sigma(\phi(g_2g_3g_j))=0$ which implies that $\sigma(\psi(wg_1g_2))=\sigma(\psi(wg_1g_3g_j))=\sigma(\psi(g_2g_3g_j))=e$. Therefore $\psi(g_2)=\frac{n+1}{2}e$, a contradiction to Proposition \ref{LL1}.
\end{proof}
\section{The proof of Theorem \ref{Th1}.2}
\begin{prop}\label{RE1} $\eta(C_2^3\oplus C_{2n})=2n+6$, where $n\geq 3$ is an odd integer.
\end{prop}
\begin{proof} Let $G=H\oplus K$ be a finite abelian group, where $H\cong C_2^4$ and $K\cong C_n$ with $n\geq 3$ an odd integer. Denote $\phi$ to be the projection from $G$ to $H$ and $\psi$ to be the projection from $ G$ to $K$.
In order to prove that $\eta(G)=2n+6$, by Lemma \ref{ETA} we only need to prove that $\eta(G)\le 2n+6$.
Assume to the contrary that there exists a sequence $S$ of length $2n+6$ over $G$ containing no short zero-sum subsequence.
Since $|\phi(S)|=|S|=2n+6=2(n-5)+16$, $\eta(C_2^4)=16$, and $\mathsf D(C_n)=n$,
we obtain that $S$ allows a product decomposition as
$$S=S_1\cdot \ldots \cdot S_{r}\cdot S_0,$$
where $S_1, \ldots, S_{r}, S_0$ are sequences over $G$ and, for every $i\in [1, r]$, $\phi(S_i)$ has sum zero and length $|S_i|\le 2$. Therefore $\phi(S_0)$ is squarefree over $H\setminus\{0\}$ and $n-4\leq r\leq n-1$.
We distinguish the following four cases depending on $r$ to get contradictions.
\medskip
\noindent\textbf{Case 1.} $r=n-1$. Then $|S_0|\ge 8$.
\smallskip
We proceed by the following assertion first
\begin{itemize}
\item[]
\noindent
{\bf Assertion $A$. }There exists an element $e\in \ker(\phi)=K$ such that $\sigma(S_1)\cdot\ldots\cdot \sigma(S_{n-1})=e^{n-1}$.
Furthermore, for any element $h\t SS_0^{-1}$, the sequence $S_0h$ has the following property:
\[
\left\{\begin{aligned} & \text{For any subsequence $V$ of $S_0h$ with $\sigma(\phi(V))=0$, \text{ we have that }}\\
&\sigma(\psi(V)) =\left \{ \begin{aligned}& e ,&\quad \quad \quad& \mbox{ if } |V|=3\mbox{ or }4, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=5.
\end{aligned} \right.
\end{aligned}\right.
\]
\medskip
\noindent
{\it Proof of \,{\bf Assertion A}. } By our assumption and Lemma \ref{cyclic}.1, $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-1})$ is zero-sum free over $K$. Then there exists an element $e\in K\setminus \{0\} $ such that $\sigma(S_1)=\cdots =\sigma(S_{n-1})=e$.
Without loss of generality, we can assume that $h\t S_{n-1}$.
If $\sigma(\psi(V))=0$, then $V$ is a short zero-sum subsequence of $S$, a contradiction. Thus $\sigma(\psi(V))\neq0$.
If $\sigma(\psi(V))=ke$ with $k\in [2,n-1]$ and $|V|\in \{3,4\}$, then
$S_1\cdot \ldots \cdot S_{n-k}\cdot V$ is a short zero-sum subsequence of $S$ , a contradiction.
If $\sigma(\psi(V))=ke$ with $k\in [3,n-1]$ and $|V|=5$, then
$S_1\cdot \ldots \cdot S_{n-k}\cdot V$ is a short zero-sum subsequence of $S$ , a contradiction.
\qed{(Proof of Assertion A)}
\end{itemize}
\medskip
If $|S_0|> 8$, we obtain a contradiction to Corollary \ref{xx1}. Thus we can assume that $|S_0|=8$ and hence $|S_i|=2$ for each $i\in [1,n-1]$.
If $\supp(\phi(S))\nsubseteq \supp(\phi(S_0))$, there exists $h\t SS_0^{-1}$, such that $\phi(S_0h)$ is squarefree. By Corollary \ref{xx1}, $|S_0h|\le 8$, a contradiction.
Thus $\supp(\phi(S))\subseteq \supp(\phi(S_0))$. Without loss of generality, we can assume that $S_{n-1}=h_1h_2$, $h_3\t S_0$ and $\phi(h_1)=\phi(h_2)=\phi(h_3)$. If there exist distinct $1\le i,j\le 3$ such that $\psi(h_i+h_j)\neq e$, then $S_1\cdot\ldots\cdot S_{n-2}\cdot h_ih_j$ has a short zero-sum subsequence, a contradiction. Therefore
$\psi(h_1+h_2)=\psi(h_1+h_3)=\psi(h_2+h_3)=e$ which implies that $\psi(h_1)=\psi(h_2)=\psi(h_3)=\frac{n+1}{2}e$ and hence $\frac{n+1}{2}e\in \supp(\phi(S_0))$, a contradiction to Proposition \ref{LL1}.
\medskip
\noindent\textbf{Case 2.} $r=n-2$. Then $|S_0|\ge 10$.
\smallskip
By Lemma \ref{IMP}, $S_0$ has a subsequence $T$ of length $|T|\in \{3,4\}$ such that $\sigma(\phi(T))=0$ and $\sigma(\psi(T))\neq \sigma(\psi(S_1))$. By our assumption, the sequences $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-2})\sigma(T)$ is zero-sum free over $\ker(\phi)=K$. By Lemma \ref{cyclic}.1, we obtain that
$$\sigma(S_1)=\cdots=\sigma(S_{n-2})=\sigma(T),$$
a contradiction.
\medskip
\noindent\textbf{Case 3.} $r=n-3$. Then $|S_0|\ge 12$.
\smallskip
If $n=3$, then by Lemma \ref{SHO}, $S$ contains a short zero-sum subsequence, a contradiction. We can assume that $n\ge 5$.
\smallskip
By Lemma \ref{SUM}, there exist disjoint $T_1, T_2\mid S_0$ such that $\sigma(\phi(T_1))=\sigma(\phi(T_2))=0$ and $|T_1|=|T_2|=3$. By our assumption, the sequence $\sigma(S_1)\cdot \ldots \cdot \sigma(S_{n-3})\cdot \sigma(T_1)\cdot \sigma(T_2)$ contains no zero-sum subsequence over $\ker(\phi)=K\cong C_n$, therefore by Lemma \ref{cyclic}.1,
$$\sigma(S_1)=\cdots=\sigma(S_{n-3})=\sigma(T_1)=\sigma(T_2)=e,$$ for some $e\in \ker(\phi)=K$ of order $n$.
\begin{itemize}
\item[]
\noindent\textbf{Assertion $B$. } Let $V$ be a subsequence of $S_0$ with $\sigma(\phi(V))=0$. Then
\[
\sigma(\psi(V)) =\left \{ \begin{aligned}
& e ,&& \mbox{ if } |V|=3, \\
& e\mbox{ or }2e,&\quad &\mbox{ if } |V|=4\mbox{ or }5.
\end{aligned} \right.
\]
\noindent
{\it Proof of \,{\bf Assertion B}. } If $|V|=3$, then $|S_0V^{-1}|=12-3=9$.
By Lemma \ref{SUM}, there exists $V_1\mid S_0V^{-1}$ such that $\sigma(\phi(V_1))=0$ and $|V_1|=3$.
By our assumption, the sequence $\sigma(S_1)\cdot \ldots \cdot \sigma(S_{n-3})\cdot \sigma(V)\cdot \sigma(V_1)$ contains no zero-sum subsequence over $K$. Therefore by Lemma \ref{cyclic}.1,
$$\sigma(S_1)=\cdots=\sigma(S_{n-3})=\sigma(V)=\sigma(V_1)=e.$$
If $|V|=4$ or $5$, by our assumption, $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-3})\sigma(V)$ is zero-sum free over $K$. Since $\sigma(S_1)=\cdots=\sigma(S_{n-3})=e$, we obtain that $\Sigma(\sigma(S_1)\cdot \ldots\cdot\sigma(S_{n-3}))=\{e,\ldots,(n-3)e\}$. It follows that $\sigma(\psi(V))\in \{e, 2e\}$.
\qed{(Proof of Assertion B)}
\end{itemize}
Suppose that $\mathsf{supp}(\psi(S_0))\setminus \{0, \frac{n+1}{2}e\}\neq \emptyset$.
Choose $u\mid S_0$ such that $\psi(u)\notin \{0, \frac{n+1}{2}e\}$. By Lemma \ref{SUM}, there exists a set $\{u_1, u_2, u_3, u_4\}\subseteq \mathsf{supp}(S_0u^{-1})$ such that $\sigma(\phi(uu_1u_2))=\sigma(\phi(uu_3u_4))=\sigma(\phi(u_1u_2u_3u_4))=0$. Then by Assertion $B$, we deduce that $\sigma(\psi(uu_1u_2))=\sigma(\psi(uu_3u_4))=e$ and $\sigma(\psi(u_1u_2u_3u_4))\in \{e, 2e\}$. Therefore $\psi(u_1+u_2)=\psi(u_3+u_4)\in \{e, \frac{n+1}{2}e\}$ and hence $\psi(u)\in \{0, \frac{n+1}{2}e\}$, a contradiction.
\smallskip
Suppose that $\mathsf{supp}(\psi(S_0))\subseteq \{0, \frac{n+1}{2}e\}$.
If there exists $v\mid S_0$ such that $\psi(v)=\frac{n+1}{2}e$, by Lemma \ref{SUM}, there exists a set $\{ v_1, \ldots, v_8\}\subseteq \mathsf{supp}(S_0v^{-1})$ such that
$$\sigma(\phi(v v_1 v_2))=\sigma(\phi(v v_3 v_4))=\sigma(\phi(v v_5 v_6))=\sigma(\phi(v v_7 v_8))=0.$$
Thus
$$\sigma(\psi(vv_1v_2))=\sigma(\psi(v v_3 v_4))=\sigma(\psi(v v_5 v_6))=\sigma(\psi(vv_7 v_8))=e$$
and
$$\psi( v_1+ v_2)=\psi( v_3+ v_4)=\psi( v_5+v_6)=\psi( v_7+ v_8)=\frac{n+1}{2}e.$$
Since $\mathsf{supp}(\psi(S_0))\subseteq \{0, \frac{n+1}{2}e\}$, we have $\psi( v_1\cdot \ldots \cdot v_8)=0^4(\frac{n+1}{2}e)^4$ which implies that $0^4\t \psi(S_0)$. Then we can always assume that $0^4\t \psi(S_0)$.
Choose $R\mid S_0$ such that $0^4\t\psi(R)$ and $|R|=5$.
By $\mathsf D(C_2^4)=5$, there exists $R_1\t R$ such that $\sigma(\phi(R_1))=0$. By our assumption, $\sigma(\psi(R_1))\neq 0$. It follows that $\sigma(\psi(R_1))=\frac{n+1}{2}e\notin \{e, 2e\}$ by $n\ge 5$, a contradiction.
\medskip
\noindent\textbf{Case 4.} $r=n-4$. Then $|S_0|\ge14$ and $n\ge 5$.
\smallskip
By Lemma \ref{SUM}, there exists a subsequence $T_1$ of $S_0$ such that $\sigma(\phi(T_1))=0$ and $|T_1|=3$. Since $|S_0T_1^{-1}|=11$, there exists a subsequence $T_2$ of $S_0T_1^{-1}$ such that $\sigma(\phi(T_2))=0$, $|T_2|\in \{3,4\}$, and $\sigma(\psi(T_2))\neq \sigma(\psi(T_1))$ by Lemma \ref{IMP}. By our assumption, the sequence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-4})\sigma(T_1)\sigma(T_2)$ contains no zero-sum subsequence. Therefore by Lemma \ref{cyclic}.2, there exists an element $e\in K$ such that
$$\sigma(S_1)\cdot\ldots\cdot\sigma(S_{n-4})\cdot\sigma(T_1)\cdot\sigma(T_2)=e^{n-3}(2e),$$
which implies that $\sigma(S_1)=\ldots=\sigma(S_{n-4})=e$.
Again by Lemma \ref{IMP}, there exists a subsequence $T_3$ of $S_0$ such that $\sigma(\phi(T_3))=0$, $|T_3|\in \{3,4\}$, and $\sigma(\psi(T_3))\neq e$. Therefore $\sigma(\psi(T_3))=2e$ or $3e$.
Suppose that $\sigma(\psi(T_3))=2e$.
Since $|S_0T_3^{-1}|\ge 10$, there exists a subsequence $T_4$ of $S_0T_3^{-1}$ such that $\sigma(\phi(T_4))=0$, $|T_4|\in \{3,4\}$, and $\sigma(\psi(T_4))=te$ with $t\in[2,n]$. If $t\ge 4$, then $S_1\cdot \ldots\cdot S_{n-t}\cdot T_4$ is a short zero-sum subsequence of $S$, a contradiction. Otherwise $2\le t\le 3$. Then $S_1\cdot \ldots\cdot S_{n-t-2}\cdot T_3\cdot T_4$ is a short zero-sum subsequence of $S$, a contradiction.
Suppose that $\sigma(\psi(T_3))=3e$.
Since $|S_0T_3^{-1}|\ge 10$, there exists a subsequence $T_4$ of $S_0T_3^{-1}$ such that $\sigma(\phi(T_4))=0$, $|T_4|\in \{3,4\}$, and $\sigma(\psi(T_4))=te$ with $t\in[1,n]\setminus\{3\}$. If $t\ge 4$, then $S_1\cdot \ldots\cdot S_{n-t}\cdot T_4$ is a short zero-sum subsequence of $S$, a contradiction. Otherwise $1\le t\le 2$. Then $S_1\cdot \ldots\cdot S_{n-3-t}\cdot T_3\cdot T_4$ is a short zero-sum subsequence of $S$, a contradiction.
\end{proof}
\begin{lemma}\label{EIMP1} Let $(e_1,e_2,e_3,e)$ be a basis of $G=C_2^3\oplus C_{2n}$ with $\ord(e_1)=\ord(e_2)=\ord(e_3)=2$ and $\ord(e)=2n$, where $n\geq 2$ is an even integer. Suppose that $\theta:G\rightarrow G$ is the homomorphism defined by $\theta(e_1)=e_1$, $\theta(e_2)=e_2$, $\theta(e_3)=e_3$, $\theta(e)=ne$ and $\zeta:G\rightarrow G$ is the homomorphism defined by $\zeta(e_1)=\zeta(e_2)=\zeta(e_3)=0$, $\zeta(e)=e$.
If $S$ is a sequence of length $|S|=8$ over $G$ such that $\theta(S)$ is a squarefree sequence with $0\notin \mathsf{supp}(\theta(S))$, then for any $k\in [1,n-1]$ and $\gcd(k,n)=1$, there exists a subsequence $T$ of $S$ with length $|T|\in [3,4]$ such that $\sigma(T)\in \ker(\theta)$ and $\sigma(T)\neq
2ke$.
\end{lemma}
\begin{proof}Without loss of generality, we can assume that $k=1$. Otherwise choose $(e_1,e_2,e_3,ke)$ to be a basis of $G$.
Assume to the contrary that for all subsequences $T$ of $S$ with $|T|\in [3,4]$ and $\sigma(T)\in \ker(\theta)$, we have that $\sigma(T)=
2e$.
For any $v\in \theta(G)\setminus \{0\}$,
we define
$$\mathsf N_v(\theta(S))=|\{T\t \theta(S) \ :\ |T|=2 \text{ and } \sigma(T)=v \}|+\delta_v,$$
where
\[
\delta_v =\left \{ \begin{array}{ll} 1 ,\quad \mbox{ if } v\in \mathsf{supp}(\theta(S)); \\ 0,\quad \mbox{ if } v\notin \mathsf{supp}(\theta(S)).
\end{array} \right.
\]
Then $\sum_{v\in \theta(G)\setminus \{0\}}\mathsf N_v(\theta(S))=\frac{8\times 7}{2}+8=36$ and $|\theta(G)\setminus \{0\}|=15$ which implies that there exists an element $v\in \theta(G)\setminus \{0\}$ such that $\mathsf N_v(\theta(S))\ge3$. Therefore we can distinguish the following two cases.
\medskip\noindent{\bf Case 1.} There exists $v\in \theta(G)\setminus\{0\}$
such that $\mathsf N_v(\theta(S))\ge 3$ and $\delta_v=1$.
\smallskip
Without loss of generality, we can assume that $\sigma(\theta(h_1))=\sigma(\theta(h_2h_3))=\sigma(\theta(h_4h_5))$. Then we have $\sigma(\theta(h_1h_2h_3))=\sigma(\theta(h_2h_3h_4h_5))=\sigma(\theta(h_4h_5h_1))=0$ which implies that $\sigma(\zeta(h_1h_2h_3))=\sigma(\zeta(h_2h_3h_4h_5))=\sigma(\zeta(h_4h_5h_1))=2e$. Therefore $\zeta(h_1)=\zeta(h_2+h_3)=\zeta(h_4+h_5)=e$ or $(n+1)e$.
Let $\zeta(h_i)=k_ie$, where $k_i\in [0,2n-1]$ for each $i\in [1,8]$. Then $k_2+k_3,k_4+k_5$ are odd and there exist distinct $i,j\in [6,8]$ such that $k_i\equiv k_j\pmod 2$. Without loss of generality, we can assume that $k_2,k_4,k_6+k_7$ are even and hence $k_1,k_3,k_5$ are odd.
Consider the sequence $W=h_1h_2h_4h_6h_7$ (see Figure \ref{ff3}).
\begin{figure}[ht]
\begin{center}
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\put(65.91,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_6$\strut}}
\put(86.49,25.00){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
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\end{picture
\end{center}
\caption{}\label{ff3}
\end{figure}
Since $\theta(W)\in \mathscr{F}(\theta(G))$ and $\mathsf D(\theta(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\theta(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
Suppose that $|V|=5$. Then $\sigma(\zeta(V))=(2k+1)e$ for some $k\in [0,n-1]$, a contradiction to $\sigma(\theta(V))=0$.
Suppose that $|V|=4$. If $h_2h_4\t V$, then $\sigma(\theta(V))=\sigma(\theta(V(h_2h_4)^{-1}h_3h_5))=\sigma(\theta(h_2h_3h_4h_5))=0$ and hence $\sigma(\zeta(V))=\sigma(\zeta(V(h_2h_4)^{-1}h_3h_5))=\sigma(\zeta(h_2h_3h_4h_5))=2e$. Therefore $\zeta(h_2+h_4)=e$ or $(n+1)e$, a contradiction to $k_2,k_4$ are even. Thus, without loss of generality, we only need to consider $V=h_1h_2h_6h_7$. Then $\sigma(\zeta(V))=(2k+1)e$ for some $k\in [0,n-1]$, a contradiction to $\sigma(\theta(V))=0$.
Suppose that $|V|=3$. By symmetry, we only need to consider $V=h_1h_6h_7$, $V=h_2h_6h_7$, $V=h_2h_4h_6$ or $V=h_1h_2h_4$. If $V=h_1h_6h_7$, then $\sigma(\zeta(V))=(2k+1)e$ for some $k\in [0,n-1]$, a contradiction to $\sigma(\theta(V))=0$.
If $V=h_2h_6h_7$, then $\sigma(\theta(h_1h_3h_6h_7))=\sigma(\theta(V))=\sigma(\theta(h_1h_2h_3))=0$ and hence $\sigma(\zeta(h_1h_3h_6h_7))=\sigma(\zeta(V))=\sigma(\zeta(h_1h_2h_3))=2e$. It follows that $\zeta(h_2)=\zeta(h_1+h_3)=\zeta(h_6+h_7)=e$ or $(n+1)e$, a contradiction. If $V=h_2h_4h_6$, then $\sigma(\zeta(V))=\sigma(\zeta(h_3h_5h_6))=\sigma(\zeta(h_2h_3h_4h_5))=2e$ which implies that $\zeta(h_2+h_4)=e$ or $(n+1)e$, a contradiction to $k_2,k_4$ are even. If $V=h_1h_2h_4$, then $\theta(h_3)=\theta(h_4)$, a contradiction.
\medskip\noindent{\bf Case 2.} There exists $v\in \theta(G)\setminus\{0\}$ such that
$\mathsf N_v(\theta(S))\ge 3$ and $\delta_v=0$.
\smallskip
Without loss of generality, we can assume that $\theta(h_1+h_2)=\theta(h_3+h_4)=\theta(h_5+h_6)$. Then $\sigma(\theta(h_1h_2h_3h_4))=\sigma(\theta(h_3h_4h_5h_6))=\sigma(\theta(h_5h_6h_1h_2))=0$ and hence $\sigma(\zeta(h_1h_2h_3h_4))=\sigma(\zeta(h_3h_4h_5h_6))=\sigma(\zeta(h_5h_6h_1h_2))=2e$. Therefore $\zeta(h_1+h_2)=\zeta(h_3+h_4)=\zeta(h_5+h_6)=e$ or $(n+1)e$.
Let $\zeta(h_i)=k_ie$, where $k_i\in [0,2n-1]$ for each $i\in [1,8]$. Without loss of generality, we can assume that $k_2,k_4,k_6$ are even and $k_1,k_3,k_5$ are odd.
Therefore we can distinguish the following two cases.
\medskip\noindent{\bf Subcase 2.1.} $k_7,k_8$ are odd.
\smallskip
Consider the sequence $W=h_1h_3h_5h_7h_8$ (see Figure \ref{f8}).
\begin{figure}[ht]
\begin{center}
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\put(63.52,28.85){\fontsize{8.53}{10.24}\selectfont \makebox(7.5, 3.0)[l]{$h_7$\strut}}
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\lbezier(2.00,22.48)(2.00,35.24)
\end{picture
\end{center}
\caption{}\label{f8}
\end{figure}
Since $\theta(W)\in \mathscr{F}(\theta(G))$ and $\mathsf D(\theta(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\theta(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish two cases depending on $|V|$.
Suppose that $|V|=5$ or $3$. Then $\sigma(\zeta(V))=(2k+1)e$ for some $k\in[0,n-1]$, a contradiction to $\sigma(\theta(V))=0$.
Suppose that $|V|=4$. By symmetry, we only need to consider $V=h_1h_3h_5h_7$ or $V=h_1h_3h_7h_8$. For both cases, $h_1h_3\t V$. Since $\sigma(\theta(V))=\sigma(\theta(V(h_1h_3)^{-1}h_2h_4))=\sigma(\theta(h_1h_2h_3h_4))=0$, we obtain that $\sigma(\zeta(V))=\sigma(\zeta(V(h_1h_3)^{-1}h_2h_4))=\sigma(\zeta(h_1h_2h_3h_4))=2e$ which implies that $\zeta(h_1+h_3)=e$ or $(n+1)e$, a contradiction to $k_1,k_3$ are odd.
\medskip\noindent{\bf Subcase 2.2.} $k_7$ or $k_8$ is even. Say, $k_7$ is even.
\smallskip
Consider the sequence $W=h_1h_2h_4h_6h_7$ (see Figure \ref{f9}).
\begin{figure}[ht]
\begin{center}
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\begin{picture}(76.76,37.24)(0,0)
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\caption{}\label{f9}
\end{figure}
Since $\theta(W)\in \mathscr{F}(\theta(G))$ and $\mathsf D(\theta(G))=\mathsf D(C_2^4)=5$, there exists a subsequence $V\mid W$ such that $\sigma(\theta(V))=0$ and $|V|\in \{3, 4, 5\}$. We distinguish three cases depending on $|V|$.
Suppose that $|V|=5$. Then $\sigma(\zeta(V))=(2k+1)e$ for some $k\in[0,n-1]$, a contradiction to $\sigma(\theta(V))=0$.
Suppose that $|V|=4$. Since $\sigma(\theta(V))=0$, we obtain that $V=h_2h_4h_6h_7$. Then $\sigma(\theta(V))=\sigma(\theta(h_1h_3h_6h_7))=\sigma(\theta(h_1h_2h_3h_4))=0$ and hence $\sigma(\zeta(V))=\sigma(\zeta(h_1h_3h_6h_7))=\sigma(\zeta(h_1h_2h_3h_4))=2e$. Therefore $\zeta(h_1+h_3)=e$ or $(n+1)e$, a contradiction to $k_1,k_3$ are odd.
Suppose that $|V|=3$. Since $\sigma(\theta(V))=0$, we obtain that $h_1\nmid V$. By symmetry, we only need to consider $V=h_2h_4h_6$ or $V=h_2h_4h_7$. For both cases, $h_2h_4\t V$. Since $\sigma(\theta(V))=\sigma(\theta(V(h_2h_4)^{-1}h_1h_3))=\sigma(\theta(h_1h_2h_3h_4))=0$, we obtain that $\sigma(\zeta(V))=\sigma(\zeta(V(h_2h_4)^{-1}h_1h_3))=\sigma(\zeta(h_1h_2h_3h_4))=2e$ which implies that $\zeta(h_1+h_3)=e$ or $(n+1)e$, a contradiction to $k_1,k_3$ are odd.
\end{proof}
\begin{prop}\label{RE2} $\eta(C_2^3\oplus C_{2n})=2n+6$, where $n\geq 2$ is an even integer.
\end{prop}
\begin{proof} Let $G=C_2^3\oplus C_{2n}$, where $n\ge 2$ is an even integer. Suppose that $\theta:G\rightarrow G$ is the homomorphism defined by $\theta(e_1)=e_1$, $\theta(e_2)=e_2$, $\theta(e_3)=e_3$, $\theta(e)=ne$ and $\zeta:G\rightarrow G$ is the homomorphism defined by $\zeta(e_1)=\zeta(e_2)=\zeta(e_3)=0$, $\zeta(e)=e$.
In order to prove that $\eta(G)=2n+6$, by Lemma \ref{ETA} we only need to prove that $\eta(G)\le 2n+6$.
Assume to the contrary that there exists a sequence $S$ of length $2n+6$ over $G$ containing no short zero-sum subsequence.
Since $|\theta(S)|=|S|=2n+6=2(n-5)+16$, $\eta(C_2^4)=16$, and $\mathsf D(C_n)=n$,
we obtain that $S$ allows a product decomposition as
$$S=S_1\cdot \ldots \cdot S_{r}\cdot S_0,$$
where $S_1, \ldots, S_{r}, S_0$ are sequences over $G$ and, for every $i\in [1, r]$, $\theta(S_i)$ has sum zero and length $|S_i|\le 2$. What's more, $\theta(S_0)$ has no zero-sum subsequence of length $\le 2$ and $n-4\leq r\leq n-1$.
We distinguish the following four cases depending on $r$ to get contradictions.
\medskip
\noindent\textbf{Case 1.} $r=n-1$. Then $|S_0|\ge 8$.
\smallskip
Since $S$ has no short zero-sum subsequence, we obtain that
$\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-1})$ is zero-sum free over $\ker(\theta)\cong C_n$. By Lemma \ref{cyclic}.1, there exists an element $k\in [1,n-1] $ such that $\sigma(S_1)=\cdots =\sigma(S_{n-1})=2ke$ and $\gcd(k,n)=1$. Without loss of generality, we can assume that $k=1$.
Since $\theta(S_0)$ has no zero-sum subsequence of length $\le 2$, by Lemma \ref{EIMP1} we obtain that there exists a subsequence $V$ of $S_0$ with length $|V|\in [3,4]$ such that $\sigma(V)=2te$ where $t\in [2,n]$.
Then by calculation we get
$$\sigma(S_1\cdot\ldots\cdot S_{n-t}\cdot V)=0, \quad \text{ and }$$
$$|S_1\cdot\ldots\cdot S_{n-t}\cdot V|
\le 2(n-t)+4
\le 2n,$$ which implies that $S_1\cdot\ldots\cdot S_{n-t}\cdot V$ is a short zero-sum subsequence of $S$,
a contradiction.
\medskip
\noindent\textbf{Case 2.} $r=n-2$. Then $|S_0|\ge 10$.
\smallskip
Since $\theta(S_0)$ is squarefree, by Lemma \ref{SUM}, $S_0$ has a subsequence $T$ of length $3$ such that $\sigma(\theta(T))=0$. By our assumption, the sequence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-2})\sigma(T)$ is zero-sum free over $\ker(\theta)\cong C_n$. By Lemma \ref{cyclic}.1, we obtain that
$$\sigma(S_1)=\cdots=\sigma(S_{n-2})=\sigma(T)=2ke, \quad \text{ for some $k\in [1,n-1]$ and $\gcd(k,n)=1$.}$$
Without loss of generality, we can assume that $k=1$.
By Lemma \ref{EIMP1}, $S_0$ has a subsequence $T'$ of length $|T'|\in \{3,4\}$ such that $\sigma(\theta(T'))=0$ and $\sigma(T')=2te$ with $t\in [2,n]$. Therefore the sequence $S_1\cdot\ldots\cdot S_{n-t}\cdot T'$ is a short zero-sum subsequence of $S$, a contradiction.
\medskip
\noindent\textbf{Case 3.} $r=n-3$. Then $|S_0|\ge 12$ and $n\ge 4$.
\smallskip
\smallskip
By Lemma \ref{SUM}, there exist two subsequences $T_1,T_2$ of $ S_0$ such that $\sigma(\theta(T_1))=\sigma(\theta(T_2))=0$ and $|T_1|=|T_2|=3$.
By our assumption, the sequence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-3})\cdot\sigma(T_1)\cdot \sigma(T_2)$ contains no zero-sum subsequence. Therefore by Lemma \ref{cyclic}.1, there exists an element $k\in [1,n-1]$ and $\gcd(k,n)=1$ such that
$$\sigma(S_1)\cdot\ldots\cdot\sigma(S_{n-3})\cdot\sigma(T_1)\cdot\sigma(T_2)=(2ke)^{n-1}.$$
Without loss of generality, we can assume that $k=1$.
By Lemma \ref{EIMP1}, there exists $T_1'\mid S_0$ such that $\sigma(\theta(T_1'))=0$, $|T_1'|\in [3,4]$, and $\sigma(T_1')=2t_1e$ with $t_1\in[2,n]$.
By Lemma \ref{EIMP1} again, there exists $T_2'\mid S_0(T_1')^{-1}$ such that $\sigma(\theta(T_2'))=0$, $|T_2'|\in [3,4]$, and $\sigma(T_2')= 2t_2e$ with $t_2\in[2,n]$.
If $t_1\ge 3$, then $S_1\cdot\ldots\cdot S_{n-t_1}\cdot T_1'$ is a short zero-sum subsequence of $S$, a contradiction.
If $t_2\ge 3$, then $S_1\cdot\ldots\cdot S_{n-t_2}\cdot T_2'$ is a short zero-sum subsequence of $S$, a contradiction.
Otherwise $t_1+t_2=4\le n$. Then $S_1\cdot\ldots\cdot S_{n-4}\cdot T_1'\cdot T_2'$ is a short zero-sum subsequence of $S$, a contradiction.
\medskip
\noindent\textbf{Case 4.} $r=n-4$. Then $|S_0|\ge 14$ and $n\ge 4$.
\smallskip
We distinguish two cases depending on $n$.
\noindent\textbf{Subcase 4.1.} $n\ge 6$.
\smallskip
By Lemma \ref{SUM}, there exist two disjoint subsequences $T_1,T_2$ of $ S_0$ such that $\sigma(\theta(T_1))=\sigma(\theta(T_2))=0$ and $|T_1|=|T_2|=3$.
By our assumption, the sequence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-4})\cdot\sigma(T_1)\cdot \sigma(T_2)$ contains no zero-sum subsequence. Therefore by Lemma \ref{cyclic}.2, there exists an element $k\in [1,n-1]$ and $\gcd(k,n)=1$ such that
$$\sigma(S_1)\cdot\ldots\cdot\sigma(S_{n-4})\cdot\sigma(T_1)\cdot\sigma(T_2)=(2ke)^{n-3}4ke\text{ or }(2ke)^{n-2}.$$
Without loss of generality, we can assume that $k=1$ and $\sigma(T_1)=2e$. By Lemma \ref{EIMP1}, there exists $T_3\mid S_0(T_1)^{-1}$ such that $\sigma(\theta(T_3))=0$, $|T_3|\in [3,4]$, and $\sigma(T_3)\neq 2e$. Then the sequence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-4})\cdot\sigma(T_1)\cdot \sigma(T_3)$ contains no zero-sum subsequence and hence $\sigma(S_1)\cdot \ldots\cdot \sigma(S_{n-4})=(2e)^{n-4}$.
By Lemma \ref{EIMP1}, there exists $T_1'\mid S_0$ such that $\sigma(\theta(T_1'))=0$, $|T_1'|\in [3,4]$, and $\sigma(T_1')=2t_1e$ with $t_1\in[2,n]$.
By Lemma \ref{EIMP1} again, there exists $T_2'\mid S_0(T_1')^{-1}$ such that $\sigma(\theta(T_2'))=0$, $|T_2'|\in [3,4]$, and $\sigma(T_2')= 2t_2e$ with $t_2\in[2,n]$.
If $t_1\ge 4$, then $S_1\cdot\ldots\cdot S_{n-t_1}\cdot T_1'$ is a short zero-sum subsequence of $S$, a contradiction.
If $t_2\ge 4$, then $S_1\cdot\ldots\cdot S_{n-t_2}\cdot T_2'$ is a short zero-sum subsequence of $S$, a contradiction.
Otherwise $t_1+t_2\le 6\le n$. Then $S_1\cdot\ldots\cdot S_{n-t_1-t_2}\cdot T_1'\cdot T_2'$ is a short zero-sum subsequence of $S$, a contradiction.
\smallskip\noindent\textbf{Subcase 4.2.} $n=4$. Then $S=S_0$.
\smallskip
By Lemma \ref{SUM}, there exist two disjoint subsequences $T_1,T_2$ of $ S_0$ such that $\sigma(\theta(T_1))=\sigma(\theta(T_2))=0$ and $|T_1|=|T_2|=3$.
If $\sigma(T_1)\neq \sigma(T_2)$, since $T_1T_2$ can not be zero-sum, without loss of generality, we can assume that $\sigma(T_1)=2e$ and $\sigma(T_2)=4e$. By Lemma \ref{EIMP1}, there exists $T_3\mid S_0(T_1T_2)^{-1}$ such that $\sigma(\theta(T_3))=0$, $|T_3|\in [3,4]$, and $\sigma(T_3)= 2te$ with $t\in[2,4]$. Thus, one of the sequences $T_3, T_1T_3, T_2T_3$ must be a short zero-sum subsequence of $S$, a contradiction.
Then $\sigma(T_1)=\sigma(T_2)$, since $T_1T_2$ can not be zero-sum, without loss of generality, we can assume that $\sigma(T_1)=\sigma(T_2)=2e$.
We claim that for any subsequence $T$ of $S$ satisfying that $|T|=3$ and $\sigma(\theta(T))=0$, we have $\sigma(T)=2e$.
In fact, $T_1$ or $T_2$ must be disjoint with $T$. We can assume that $T_1$ and $T$ are disjoint. If $\sigma(T)=6e$, then $T_1T$ is a short zero-sum subsequence, a contradiction. If $\sigma(T)=4e$, we can do it as before to obtain a contradiction. Then $\sigma(T)=2e$.
Since $\sigma(T_1)=2e$, we can
choose $g\t T_1$ such that $\zeta(g)\not\in \{e,5e\}$. By Lemma \ref{SUM}, there exist subsequences $R_1,\ldots, R_4$ of $ST_1^{-1}$ such that $\theta(g)=\sigma(\theta(R_1))=\ldots=\sigma(\theta(R_4))$ and $|R_1|=\ldots=|R_4|=2$. Since for each $i\in [1,6]$, $\sigma(\theta(gR_i))=0$, we obtain that $\sigma(gR_i)=2e$. Thus $\sigma(\zeta(R_i))=2e-\zeta(g)$ for each $i\in [1,4]$. By $\sigma(\theta(R_1R_2))=\sigma(\theta(R_1R_2))=0$, we have $\sigma(R_1R_2)=\sigma(R_1R_2)=4e-2\zeta(g)$. If $\sigma(R_1R_2)=2e$, then $\zeta(g)\in \{e,5e\}$, a contradiction. If $\sigma(R_1R_2)=4e$, then $R_1R_2R_3R_4$ is a short zero-sum subsequence of $S$, a contradiction. Otherwise $\sigma(R_1R_2)=6e$. Then $T_1R_1R_2$ is a short zero-sum subsequence of $S$, a contradiction.
\end{proof}
\begin{proof}[\bf Proof of Theorem \ref{Th1}.2]
By Proposition \ref{RE1} and \ref{RE2}, it follows that
$\eta(G)=2n+6$.
If $n\geq 36=\max\{2|C_2^3|+1, 4|C_2^3|+4\}$, by Lemma \ref{ETAF}, we have that $\mathsf s(C_2^3\oplus C_{2n})=\eta(C_2^3\oplus C_{2n})+{\exp}(C_2^3\oplus C_{2n})-1=2n+6+2n-1=4n+5$.
\end{proof}
\subsection*{Acknowledgements}
The authors would like to thank Professor Alfred Geroldinger of University of Graz for his many helpful suggestions.
This research was supported by NSFC (grant no. 11401542), the Fundamental Research Funds for the Central Universities (grant no. 2652014033), and
the Austrian Science Fund FWF (project no. M1641-N26).
\bigskip
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} | 8,645 |
Q: DB2 Optimize for n rows I'm learning DB2, and I came across this clause: OPTIMIZE FOR 1 ROW right after FETCH FIRST 100 ROWS ONLY.
I understand that FETCH FIRST 100 ROWS ONLY would give me the first 100 rows that qualified. But I don't understand what the OPTIMIZE FOR 1 ROW really doing here. I read this DB2 documentation, it says
Use OPTIMIZE FOR 1 ROW clause to influence the access path. OPTIMIZE FOR 1 ROW tells Db2 to select an access path that returns the first qualifying row quickly.
and this DB2 documentation, it says
In general, if you are retrieving only a few rows, specify OPTIMIZE FOR 1 ROW to influence the access path that Db2 selects.
But I'm still confused. Is using OPTIMIZE FOR n ROWS would make a query more efficient?
I also found this post on SO and it seems like OPTIMIZE FOR n ROWS is equivalent to FETCH FIRST n ROWS ONLY per the accepted answer.
But when I experimented it myself using OPTIMIZE FOR n ROWS instead of FETCH FIRST n ROWS ONLY, the result set was not the same. With OPTIMIZE FOR n ROWS, the query returns all qualifying rows.
Could someone please explain it to me what OPTIMIZE FOR n ROWS really does? Thanks!
A:
Is using OPTIMIZE FOR n ROWS would make a query more efficient?
Not necessarily. However, it might cause your application to start receiving rows earlier than it otherwise would, if there is an access plan alternative that can find the first row matching the query criteria faster although the entire query will as a result run longer.
There's this bit in the Db2 for LUW docs that gives some examples specific to that platform:
Try specifying OPTIMIZE FOR n ROWS along with FETCH FIRST n ROWS ONLY, to encourage query access plans that return rows directly from the referenced tables, without first performing a buffering operation such as inserting into a temporary table, sorting, or inserting into a hash join hash table.
Applications that specify OPTIMIZE FOR n ROWS to encourage query access plans that avoid buffering operations, yet retrieve the entire result set, might experience poor performance. This is because the query access plan that returns the first n rows fastest might not be the best query access plan if the entire result set is being retrieved.
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"redpajama_set_name": "RedPajamaStackExchange"
} | 8 |
{"url":"https:\/\/chemistry.stackexchange.com\/questions\/26738\/which-is-more-dissociated-ions-in-water-or-undissociated?noredirect=1","text":"# Which is more: dissociated ions in water or undissociated? [closed]\n\nWhile studying equilibrium we get to know that there are $1*10^{-7} \\ce{~~OH-}$ and $\\ce{H+}$ ions in pure water. However, I think there are more undissociated molecules.\n\n\u2022 \u2013\u00a0Mithoron Feb 28 '15 at 15:12\n\u2022 Wait.... What exactly are you asking? \u2013\u00a0M.A.R. Feb 28 '15 at 17:36\n\n[\u2026] there are $1\\cdot10^{-7}\\ \\ce{OH-}$ [\u2026] ions\nAre you sure that this is an absolute number or rather a concentration in $\\mathrm{\\frac{mol}{L}}$? IN the latter case, what would be the concentration of undiscociated $\\ce{H2O}$ molecules in 1 liter of water?","date":"2021-02-28 23:29:08","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\": 0.7401840090751648, \"perplexity\": 991.6296425986474}, \"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-2021-10\/segments\/1614178361776.13\/warc\/CC-MAIN-20210228205741-20210228235741-00417.warc.gz\"}"} | null | null |
{"url":"https:\/\/phys.libretexts.org\/Bookshelves\/Quantum_Mechanics\/Book%3A_Introductory_Quantum_Mechanics_(Fitzpatrick)\/8%3A_Central_Potentials\/8.2%3A_Infinite_Spherical_Potential_Well","text":"$$\\require{cancel}$$\n\n# 8.2: Infinite Spherical Potential Well\n\nConsider a particle of mass $$m$$ and energy $$E>0$$ moving in the following simple central potential:\n\n$V(r) = \\left\\{\\begin{array}{lcl} 0&\\,&\\mbox{for 0\\leq r\\leq a}\\\\[0.5ex] \\infty&&\\mbox{otherwise} \\end{array}\\right..$\n\nClearly, the wavefunction $$\\psi$$ is only non-zero in the region $$0\\leq r \\leq a$$. Within this region, it is subject to the physical boundary conditions that it be well behaved (i.e., square-integrable) at $$r=0$$, and that it be zero at $$r=a$$. (See Section [s5.2].) Writing the wavefunction in the standard form\n\n$\\label{e9.27} \\psi(r,\\theta,\\phi) = R_{n,l}(r)\\,Y_{l,m}(\\theta,\\phi),$\n\nwe deduce (see the previous section) that the radial function $$R_{n,l}(r)$$ satisfies\n\n$\\frac{d^{\\,2} R_{n,l}}{dr^{\\,2}} + \\frac{2}{r}\\frac{dR_{n,l}}{dr} + \\left[k^{\\,2} - \\frac{l\\,(l+1)}{r^{\\,2}}\\right] R_{n,l} = 0$ in the region $$0\\leq r \\leq a$$, where\n\n$\\label{e9.29} k^{\\,2} = \\frac{2\\,m\\,E}{\\hbar^{\\,2}}.$Defining the scaled radial variable $$z=k\\,r$$, the previous differential equation can\n\nbe transformed into the standard form\n\n$\\frac{d^{\\,2} R_{n,l}}{dz^{\\,2}} + \\frac{2}{z}\\frac{dR_{n,l}}{dz} + \\left[1 - \\frac{l\\,(l+1 )}{z^{\\,2}}\\right] R_{n,l} = 0.$\n\nThe two independent solutions to this well-known second-order differential equation are called spherical Bessel functions, and can be written\n\n\\begin{aligned} j_l(z)&= z^{\\,l}\\left(-\\frac{1}{z}\\frac{d}{dz}\\right)^l\\left(\\frac{\\sin z}{z}\\right),\\\\[0.5ex] y_l(z)&= -z^{\\,l}\\left(-\\frac{1}{z}\\frac{d}{dz}\\right)^l\\left(\\frac{\\cos z}{z}\\right).\\end{aligned}\n\nThus, the first few spherical Bessel functions take the form \\begin{aligned} j_0(z) &= \\frac{\\sin z}{z},\\\\[0.5ex] j_1(z)&=\\frac{\\sin z}{z^{\\,2}} - \\frac{\\cos z}{z},\\\\[0.5ex] y_0(z) &= - \\frac{\\cos z}{z},\\\\[0.5ex] y_1(z) &= - \\frac{\\cos z}{z^{\\,2}} - \\frac{\\sin z}{z}.\\end{aligned}\n\nThese functions are also plotted in Figure [sph]. It can be seen that the spherical Bessel functions are oscillatory in nature, passing through zero many times. However, the $$y_l(z)$$ functions are badly behaved (i.e., they are not square integrable) at $$z=0$$, whereas the $$j_l(z)$$ functions are well behaved everywhere. It follows from our boundary condition at $$r=0$$ that the $$y_l(z)$$ are unphysical, and that the radial wavefunction $$R_{n,l}(r)$$ is thus proportional to $$j_l(k\\,r)$$ only. In order to satisfy the boundary condition at $$r=a$$ [i.e., $$R_{n,l}(a)=0$$], the value of $$k$$ must be chosen such that $$z=k\\,a$$ corresponds to one of the zeros of $$j_l(z)$$. Let us denote the $$n$$th zero of $$j_l(z)$$ as $$z_{n,l}$$. It follows that\n\n$k\\,a = z_{n,l},$ for $$n=1,2,3,\\ldots$$. Hence, from Equation ([e9.29]), the allowed energy levels are $\\label{e9.39} E_{n,l} = z_{n,l}^{\\,2}\\,\\frac{\\hbar^{\\,2}}{2\\,m\\,a^{\\,2}}.$ The first few values of $$z_{n,l}$$ are listed in Table [tsph]. It can be seen that $$z_{n,l}$$ is an increasing function of both $$n$$ and $$l$$.\n\nThe first few zeros of the spherical Bessel function $$j_l(z)$$.\n$$n=1$$ $$n=2$$ $$n=3$$ $$n=4$$\n$$l=0$$ 3.142 6.283 9.425 12.566\n[0.5ex] $$l=1$$ 4.493 7.725 10.904 14.066\n[0.5ex] $$l=2$$ 5.763 9.095 12.323 15.515\n[0.5ex] $$l=3$$ 6.988 10.417 13.698 16.924\n[0.5ex] $$l=4$$ 8.183 11.705 15.040 18.301\n\nWe are now in a position to interpret the three quantum numbers\u2014 $$n$$, $$l$$, and $$m$$\u2014which determine the form of the wavefunction specified in Equation ([e9.27]). As is clear from Chapter [sorb], the azimuthal quantum number $$m$$ determines the number of nodes in the wavefunction as the azimuthal angle $$\\phi$$ varies between 0 and $$2\\pi$$. Thus, $$m=0$$ corresponds to no nodes, $$m=1$$ to a single node, $$m=2$$ to two nodes, et cetera. Likewise, the polar quantum number $$l$$ determines the number of nodes in the wavefunction as the polar angle $$\\theta$$ varies between 0 and $$\\pi$$. Again, $$l=0$$ corresponds to no nodes, $$l=1$$ to a single node, et cetera. Finally, the radial quantum number $$n$$ determines the number of nodes in the wavefunction as the radial variable $$r$$ varies between 0 and $$a$$ (not counting any nodes at $$r=0$$ or $$r=a$$). Thus, $$n=1$$ corresponds to no nodes, $$n=2$$ to a single node, $$n=3$$ to two nodes, et cetera. Note that, for the case of an infinite potential well, the only restrictions on the values that the various quantum numbers can take are that $$n$$ must be a positive integer, $$l$$ must be a non-negative integer, and $$m$$ must be an integer lying between $$-l$$ and $$l$$. Note, further, that the allowed energy levels ([e9.39]) only depend on the values of the quantum numbers $$n$$ and $$l$$. Finally, it is easily demonstrated that the spherical Bessel functions are mutually orthogonal: that is, $\\int_0^a j_l(z_{n,l}\\,r\/a)\\,j_{l}(z_{n',l}\\,r\/a) \\,r^{\\,2}\\,dr = 0$ when $$n\\neq n'$$ . Given that the $$Y_{l,m}(\\theta,\\phi)$$ are mutually orthogonal (see Chapter [sorb]), this ensures that wavefunctions ([e9.27]) corresponding to distinct sets of values of the quantum numbers $$n$$, $$l$$, and $$m$$ are mutually orthogonal.\n\n# Contributors\n\n\u2022 Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)\n","date":"2020-02-20 04:44:20","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\": 0.9720087647438049, \"perplexity\": 215.66741711752175}, \"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\/1581875144637.88\/warc\/CC-MAIN-20200220035657-20200220065657-00330.warc.gz\"}"} | null | null |
\section{INTRODUCTION}
\glsreset{gp}\Glspl{gp} \cite{Rasmussen2004} are a ubiquitous tool in machine learning. They provide a flexible non-parametric approach to non-linear data modelling. \Glspl{gp} have been used in a number of machine learning problems, including latent variable modelling \cite{Lawrence2005}, dynamical time-series modelling \cite{Wang2005} and Bayesian optimisation \cite{Snoek2012}.
At present, a gap exists in the literature; we fill that gap by providing the first analysis of the moments of the arc length of a vector-valued \gls{gp}. Previous work tackles only the univariate case, making it inapplicable to important applications in, for example, medical vision (or brain imaging) \cite{Hauberg2015} and path planning \cite{Moll2004}. The authors believe that an understanding of the arc length properties of a \gls{gp} will open up promising avenues of research. Arc length statistics have been used to analyze multivariate time series modelling \cite{Wickramarachchi2015}. In \cite{Tosi2014}, the authors minimize the arc length of a deterministic curve which then implicitly defines a \gls{gp}. In another related paper \cite{Hennig2014a}, the authors use \glspl{gp} as approximations to geodesics and compute arc lengths using a na\"ive Monte Carlo method.
We envision the arc length as a cost function in Bayesian optimisation, as a tool in path planning problems \cite{marchant2014bayesian} and a way to construct meaningful features from functional data.
Consider a Euclidean space $X = \mathbb{R}^n$ and a differentiable injective function $\gamma:[0,T] \to \mathbb{R}^n$. Then the image of the curve, $\gamma$, is a curve with length:
\begin{equation}
\text{length}(\gamma) = \int^{T}_{0}|\gamma'(t)|\mathrm{d}t.
\end{equation}
Importantly, the length of the curve is independent of the choice of parametrization of the curve \cite{docarmo:1992}. For the specific case where $X = \mathbb{R}^2$, with the parametrization in terms of $t$, $\gamma = (y(t), x(t))$, we have:
\begin{equation}
\text{length}(\gamma) = \int^{T}_{0}|\gamma'(t)|\mathrm{d}t = \int^{T}_{0}\sqrt{y'(t)^2 + x'(t)^2}\mathrm{d}t.
\end{equation}
If we can write $y = f(x)$, $x=t$, then our expression reduces to the commonly known expression for the arc length of a function:
\begin{equation}
\text{length}(\gamma) = s = \int^{T}_{0}|\gamma'(t)|\mathrm{d}t = \int^{T}_{0}\sqrt{1 + \left(f'(t)\right)^2}\mathrm{d}t,
\end{equation}
where we have introduced $s$ as a shorthand for the length of our curve. In some cases the exact form of $s$ can be computed, for more complicated curves, such as Beizer and splines curves, we must appeal to numerical methods to compute the length.
Our interest lies in considering the length of a function modelled with a \gls{gp}. Intuition suggests that the length of a \gls{gp} will concentrate around the mean function with the statistical properties dictated by the choice of kernel and the corresponding hyperparameters.
Previous work \cite{BARAKAT1970} considered the derivative process $f'(t)$, which is itself a \gls{gp}. A direct calculation was performed and an exact form for the mean was obtained in terms of modified Bessel functions; a form for the variance is also presented. This result is also derived in \cite{Miller1956,Corrsin1961}. Analysis has been presented on the arc length of a high-level excursion from the mean \cite{NOSKO1985}.
However, other important questions have not been explored within the literature. In particular, the shape of the distribution has not been communicated, computing the arc length of a posterior \glspl{gp} has not been addressed, nor has anyone considered the arc length of \glspl{gp} in anything other than $\mathbb{R}$. These issues are addressed within this paper; we present a new derivation of the mean of a one dimensional \gls{gp} and derive the moments of a \gls{gp} in $\mathbb{R}^n$.
The paper is structured as follows. In Section \ref{sec:theory} we review the theory of \glspl{gp} and introduce the notation necessary to deal with \gls{gp}s defined on $\mathbb{R}^n$. In Section \ref{sec:one_d} we examine the one dimensional case, deriving a distribution over the arc length increment before computing the mean and variance of the arc length. In Section \ref{sec:general_case} we consider the general case. A closed form distribution is not possible for the increment, therefore we provide a moment-matched approximation that proves high-fidelity to the true distribution. This distribution allows us to compute the corresponding moments for the arc length. Section \ref{sec:numerical} presents numerical simulations demonstrating the theoretical results. Finally we conclude with thoughts on the use of arc length priors.
\section{GAUSSIAN PROCESSES}
\label{sec:theory}
\subsection{Single Output Gaussian Processes}
Consider a stochastic process from a domain $f: \mathcal{X} \to \mathbb{R}$. Then if $f$ is a \gls{gp}, with mean function $\mu$ and kernel $k$, we write
\begin{align}
f \sim \mathrm{GP}(\mu,k).
\end{align}
We can think of a \gls{gp} as an extension of the multivariate Gaussian distribution for function values and as the multivariate case, a \gls{gp} is completely specified by its mean and covariance function.
For a detailed introduction see \cite{Rasmussen2004}.
Given a set of observations $S = \{x_i, y_i\}_{i=1}^{N}$ with Gaussian noise $\sigma^2$ the posterior distribution for an unseen data, $x_{*}$, is
\begin{align}
p(f(x_{*})|S,x_{*},\phi) & = \mathcal{N}(f(x_*),m(x_*),k(x_{*},x_{*})).
\end{align}
Letting $X = [x_1, \dots, x_n]$, and defining $k_{x_{*}} = K(X,x_{*})$, the posterior mean and covariance are:
\begin{align}
m(x_{*}) & = k_{x_*}^T(k(X,X) + \sigma^2\mathbf{I})^{-1}y\\
C_{*}(x_{*},x_{*}) & = k(x_{*},x_{*}) - k_{x_*}^T(k(X,X) + \sigma^2\mathbf{I})^{-1}k_{x_{*}}
\end{align}
The derivative of the posterior mean can be calculated:
\begin{align}
\frac{\partial m_*}{\partial x_*} & = \frac{\partial k(x_*,X)}{\partial x_{*}}(k(X,X)+ \sigma^2\mathbf{I})^{-1}y,\label{eqn:post_mean_deriv}
\end{align}
wherever the derivative of the kernel function can be calculated. The covariance for the derivative process can likewise be derived and hence a full distribution over the derivative process can be specified.
Alternatively, a derivative \gls{gp} process can be defined for any twice-differentiable kernel in terms of our prior distribution. If $f \sim \mathrm{GP}(\mu, k)$, then we write the derivative process as
\begin{align}
f' \sim \mathrm{GP}(\partial \mu,\partial^2k).
\end{align}
The auto-correlation, $\rho(\tau)$, of the derivative process is related to the autocorrelation of the original process via the relation \cite{Middleton1960}:
\begin{align}
\rho_{f'}(\tau) & = -\frac{\mathrm{d}^2}{\mathrm{d}\tau^2}\rho_{f}(\tau) = \frac{\partial^2}{\partial x\partial x'}k(x-x'),
\end{align}
where $\tau = x-x'$. The variance of $f'$ is therefore given by $\sigma_{f'}^2 = \rho_{f'}(0)$.
\subsection{Vector Valued Gaussian Processes}
The development of the multi-output \glspl{gp} proceeds in a manner similar to the single output case; for a detailed review see \cite{Alvarez2012}.
The outputs are random variables associated with different processes evaluated at potentially different values of $\mathbf{x}$.
We consider a vector valued \gls{gp}:
\begin{align}
\mathbf{f} & \sim \mathrm{GP}(\mathbf{m}, \mathbf{K}),
\end{align}
where $\mathbf{m} \in \mathbb{R}^D$ is the mean vector where $\{m_d(x) \}_{d=1}^{D}$ are mean functions associated with each output and $\mathbf{K}$ is now a positive definite matrix valued function. $(\mathbf{K}(\mathbf{x},\mathbf{x}'))_{d,d'}$ is the covariance between $f_d(x)$ and $f_{d'}(x')$. Given input $\mathbf{X}$, our prior over $\mathbf{f}(\mathbf{X})$ is now
\begin{align}
\mathbf{f}(\mathbf{X}) \sim \mathcal{N}(\mathbf{m}(\mathbf{X}), \mathbf{K}(\mathbf{X},\mathbf{X})).
\end{align}
$\mathbf{m}(\mathbf{X})$ is a $DN$-length vector that concatenates the mean vectors for each output and $\mathbf{K}(\mathbf{X},\mathbf{X})$ is a $ND \times ND$ block partitioned matrix. In the vector valued case the predictive equations for an unseen datum, $\mathbf{x_{*}}$ become:
\begin{align}
\mathbf{m}(\mathbf{x}_{*}) & = \mathbf{K}_{\mathbf{x}_*}^T(\mathbf{K}(\mathbf{X},\mathbf{X}) + \mathbf{\Sigma})^{-1}\mathbf{y}\\
\mathbf{C}_{*}(\mathbf{x}_{*},\mathbf{x}_{*}) & = \mathbf{K}(\mathbf{x}_{*},\mathbf{x}_{*}) - \mathbf{K}_{x_*}^T(\mathbf{K}(\mathbf{X},\mathbf{X}) + \mathbf{\Sigma})^{-1}\mathbf{K}_{\mathbf{x}_{*}},
\end{align}
where $\mathbf{\Sigma}$ is block diagonal matrix with the prior noise of each output along the diagonal. The problem now focuses on specifying the form of the covariance matrix $\mathbf{K}$. We are interested in separable kernels of the form:
\begin{align}
\mathbf{K}(\mathbf{x},\mathbf{x}')_{d,d'} & = k(\mathbf{x},\mathbf{x}')k_T(d,d'),
\end{align}
where $k$ and $k_T$ are themselves valid kernels. The kernel can then be specified in the form:
\begin{align}
\mathbf{K}(\mathbf{x},\mathbf{x}') & = \mathbf{k}(\mathbf{x},\mathbf{x}')\mathrm{B}
\end{align}
where $\mathrm{B}$ is a $D \times D$ matrix. For a data set $\mathbf{X}$:
\begin{align}
\mathbf{K}(\mathbf{X},\mathbf{X}) & = \mathrm{B}\otimes k(\mathbf{X},\mathbf{X}),
\end{align}
with $\otimes$ representing the Kronecker product. $\mathrm{B}$ specifies the degree of correlation between the outputs. Various choices of $\mathrm{B}$ result in what is known as the Intrinsic Model of Coregionalisation (IMC) or Linear Model of Coregionalisation (LMC).
\subsection{Kernel Choices}
A \gls{gp} prior is specified by a choice of kernel, which encodes our belief about the nature of our function behaviour. In the case of infinitely differentiable functions we might choose the exponentiated quadratic, for periodic functions, the periodic kernel, or for cases where we wish to control the differentiability of our function we might select the Mat\'{e}rn class of kernels \cite{Rasmussen2004}. Samples from each kernel result in distinct sample curve behaviour.
We demonstrate how the choice of kernel impacts the statistical behaviour of our arc length and relate this to the kernel hyperparameters of several popular kernels. For the vector-valued \gls{gp} we show the statistical properties are related to the choice of the spatial kernel coupled with the choice of the output dependency matrix $\mathrm{B}$ and in particular, its eigenvalues. This highlights how kernel choice affects not only the shape but the length of our functions or conversely how knowledge of the prior curve length could be used to inform kernel selection.
\section{ONE DIMENSIONAL ARC LENGTH}
\label{sec:one_d}
First we consider the one dimensional case, where we develop a new method to derive the expected length; an approach that can be used in the vector case. Consider a \gls{gp}, $f \sim \mathrm{GP}(0,\mathrm{K})$ with a corresponding derivative process $f' \sim \mathrm{GP}(0, \partial^2\mathrm{K})$. Then the arc length is the quantity:
\begin{align}
s & = \int_{a}^{b}\sqrt{1 + (f')^2}\mathrm{d}t \label{eqn:oned_length}.
\end{align}
We are interested in computing $\mathbb{E}[s]$ and $\mathbb{V}[s]$, which require integrating $s$ and $s^2$ against the distribution over $f'$.
Instead of attempting to compute these quantities directly we sidestep the problem and first determine the probability distribution over the arc length integrand $(1+ (f')^2)^{1/2}$.
\subsection{Integrand Distribution}
We present a new method for deriving the mean and variance of the arc length of a one-dimensional \gls{gp} by first considering the transformation of a normal distribution variable under the non-linear transformation $g(x) = (1+x)^{1/2}$.
Specifically, we can consider the distribution of a normally distributed random variable under the transformation $g$:
\begin{equation}
Y = g(X) = \sqrt{1+X^2}, \quad X\sim \mathcal{N}(\mu, \sigma^2).
\end{equation}
We consider the more general case where $\mu \neq 0$.
Intuitively, we expect our distribution for $Y$ to be a skewed Chi distribution.
We are able to directly compute the probability density function for $Y$ by considering the cumulative distribution and using the standard rules for the transformation of probability functions:
\begin{align}
P(Y < y) & = F_{X}(\sqrt{y^2 - 1} + \mu) \notag \\
& \quad- (1 - F_{X}(\sqrt{y^2-1}-\mu)),
\end{align}
where $F_{X}$ is the cumulative probability distribution of $X$; details in the Supplementary Material.
The probability density function (pdf) of $Y$ is obtained by taking the derivative of $P(Y<y)$ with respect to $y$; further details in the Supplementary Material:
\begin{align}
p_Y(y) & = \frac{1}{\sqrt{2\pi}\sigma}\left[\exp\left(-\frac{(\sqrt{y^2 - 1} + \mu)^2}{2\sigma^2}\right) \right. \notag \\ & \quad + \left.\exp\left(-\frac{(\sqrt{y^2 - 1} - \mu)^2}{2\sigma^2}\right) \right]\frac{y}{\sqrt{y^2-1}}.
\end{align}
This probability distribution is valid for $y>1$ and a straightforward calculation shows that $\int_{y \in Y}p_{Y}(y)\mathrm{d} y = 1$.
Computation of the expectation of the integrand distribution can now be done in closed; the process is outlined in the Supplementary Material. The final expression is:
\begin{align}
\mathbb{E}[y] & = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{\mu^2}{2\sigma^2}\right)\notag\\
& \!\!\!\!\!\sum_{l=0}^{\infty}\frac{\Gamma\left(l+\frac{1}{2}\right)}{(2l)!}\left(\frac{\mu}{\sigma^2}\right)^{2l}U\left(l+\frac{1}{2},l+2,\frac{1}{2\sigma^2}\right)\!. \label{eqn:oned_inc_mean}
\end{align}
Here $\Gamma(n)$ is the gamma function and $U(a,b,z)$ is the confluent hypergeometric function of the second kind, defined by the integral expression:
\begin{align}
U(a,b,z) & = \int^{\infty}_{0}\exp\left(-zt\right)t^{a-1}(1+t)^{b-a-1}\mathrm{d}t.
\end{align}
A similar process allows us to derive an exact expression for $\mathbb{E}_{p_Y(y)}[y^2]$ and hence $\mathbb{V}_{p_{Y}(y)}[y]$. Figure~\ref{fig:1d_inc_disst} shows draws of $g(X)$, overlaid with $p_{Y}(y)$ for a range of $\mu$ and $\sigma$.
\begin{figure*}
\vspace{.3in}
\includegraphics{dist.png}
\caption{Histogram of samples from $\sqrt{1 + X^2}$, where $X\sim \mathcal{N}(\mu, \Sigma)$, overlaid with the corresponding distribution. We display the effects of varying $\mu$ and $\sigma$.}
\label{fig:1d_inc_disst}
\end{figure*}
\subsection{Arc Length Statistics}
Having derived expressions for the arc length integrand distribution we are able to evaluate the moments of the arc length.
Specifically, we consider a zero mean \gls{gp} with kernel $\mathrm{K}$:
\begin{align}
f \sim \mathcal{N}(0,\mathrm{K})
\end{align}
The derivative process is a \gls{gp} \cite{Rasmussen2004} defined by:
\begin{align}
f'\sim \mathcal{N}(0, \partial^2 \mathrm{K})
\end{align}
Taking the expectation of the arc length, noting that the integrand is non-negative, therefore by Fubini's Theorem \cite{Fubini1907} we can interchange the expectation and integral:
\begin{align}
\mathbb{E}[s] & = \mathbb{E}\left[\int_{0}^{T}\sqrt{1 + (f')^2}\mathrm{d}t\right]\\
& = \int_{0}^{T}\mathbb{E}\left[\sqrt{1 + (f')^2}\right]\mathrm{d}t.
\end{align}
The variance of $f'$ is given by $\sigma_{f'}^2 = R_{f'}(0)$.
At each point along the integral the expectation of the integrand is the same, therefore we arrive at:
\begin{align}
\mathbb{E}[s] & = \int_{0}^{T}\left[ \frac{1}{\sqrt{2\pi}\sigma_{f'}}\Gamma\left(\frac{1}{2}\right)U\left(\frac{1}{2} ,2 ,\frac{1}{2\sigma_{f'}^2}\right) \right]\mathrm{d}t\\
& =T\frac{1}{\sqrt{2\pi}\sigma}\Gamma\left(\frac{1}{2}\right)U\left(\frac{1}{2} ,2 ,\frac{1}{2\sigma_{f'}^2}\right),
\end{align}
where we have used the expectation of the integrand for the zero-mean case.
Using identities related to the Confluent Hypergeometric we can rewrite the mean as:
\begin{align}
\mathbb{E}[s] = \frac{ T \exp (1/4 \sigma_{f'}^2)}{2\sqrt{2\pi}\sigma_{f'}}\left[\mathrm{BF}_{0}\left(\frac{1}{4\sigma_{f'}^2}\right) + \mathrm{BF}_{1}\left(\frac{1}{4\sigma_{f'}^2}\right)\right],
\end{align}
where $\mathrm{BF}_i$ is the modified Bessel function of the second kind of order $i$.
For a posterior distribution of the arc length, given data observations, we would use Eqn~\ref{eqn:oned_inc_mean} along with the the posterior derivative mean, $\mu_{f'} = \frac{\partial m_*}{\partial x_*} $ and variance function of the posterior \gls{gp}, $\sigma_{f'}^2$ to compute the expected length;
\begin{align}
\mathbb{E}[s] & = \sum_{l=0}^{\infty}\frac{\Gamma\left(l+\frac{1}{2}\right)}{(2l)!}\int^{T}_{0}\frac{1}{\sqrt{2\pi}\sigma_{f'}}\exp\left(-\frac{\mu_{f'}^2}{2\sigma_{f'}^2}\right)\notag\\
& \!\!\!\!\!\left(\frac{\mu_{f'}}{\sigma_{f'}^2}\right)^{2l}U\left(l+\frac{1}{2},l+2,\frac{1}{2\sigma_{f'}^2}\right)\mathrm{d}t\!, \label{eqn:oned_inc_mean_post}
\end{align}
where $\mu_{f'} $ and $\sigma_{f'}$ depend on $t$.
We have derived a closed form expression for the mean of the arc length of a one dimensional zero mean \gls{gp}, reproducing the original result from \cite{BARAKAT1970} whilst providing a way to compute the arc length mean of a \gls{gp} posterior distribution.
The variance involves the computation of the second moment, a calculation involving the bi-variate form of the integrand distribution; we do not derive that in this paper.
An alternate derivation is reported in \cite{BARAKAT1970}.
\subsubsection{Kernel Derivatives}
The value of the mean arc length is determined solely by the derivative variance, $\sigma_{f'}^2$. For stationary kernels, $k(x,x') = k(x-x')$, this equates to:
\begin{align}
\sigma_{f'}^2 & = \frac{\partial^2}{\partial x \partial x'}k(x-x')\bigg{|}_{x=x'}.
\end{align}
Table~\ref{tab:kernderivs} summarises a table of common kernels \cite{Rasmussen2004} and the variance of the effective length scale in terms of their hyperparameters. The effect of the choice of hyperparameters on the expected length is shown in Figure~\ref{fig:se_heat_two}.
\begin{table}[h]
\caption{Derivative process variance, $\sigma_{\dot{f}}^2$, in terms of kernel hyperparameters for a range of common kernels. In each case $\lambda^2$ is the output (signal) variance hyperparameter and $\sigma$ is the input dimension length scale hyperparameter.}
\label{tab:kernderivs}
\begin{center}
\begin{tabular}{p{2cm}p{1.5cm}p{1.5cm}p{1.5cm}}
\toprule
Square Exponential & Mat\'{e}rn, $\nu= \frac{3}{2}$ & Mat\'{e}rn, $\nu= \frac{3}{2}$ & Rational Quadratic\\
\midrule
$\lambda^2/\sigma^2$ & $3\lambda^2/\mathbf{\sigma}^2$ & $5\lambda^2/3\sigma^2$ & $\lambda^2/\sigma^2$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{center}
\begin{figure}[h]
\vspace{.3in}
\includegraphics{length_heatmap_two.png}
\caption{Values of the expected arc length (colour shading) for various values of the SE kernel parameters. The plot shows the heat map of the log of the arc length to show sufficient detail. The length is dominated by the input scale parameter.}
\label{fig:se_heat_two}
\end{figure}
\end{center}
\section{MULTI-DIMENSIONAL ARC LENGTH}
\label{sec:general_case}
In this section we present the first treatment of the arc length of a \gls{gp} in more than one output dimension. We present an approximation to the arc length integrand distribution and use this to compute the moments of the arc length. For the vector case, we now consider a vector \gls{gp} and its corresponding derivative process:
\begin{align}
\mathbf{f} \sim \mathrm{GP}(0, \mathbf{K}), \quad \mathbf{f}' \sim \mathrm{GP}(0, \partial^2\mathbf{K}),
\end{align}
where $\mathbf{K} = \mathrm{B}\otimes \mathbf{k}$, with a coregionalised matrix $\mathrm{B}$ and a stationary kernel $\mathbf{k}$. The arc length for the vector case is given by:
\begin{align}
s = \int_{a}^{b}|\mathbf{f}'|\mathrm{d}t.
\end{align}
As we did in the one-dimensional case we first consider the distribution of the arc length integrand $|\mathbf{f}'|$ and then use this to derive the moments of the arc length itself.
\subsection{Integrand Distribution}
We are interested in the distribution over the arc length.
Ultimately we are interested in $\mathbb{R}^3$, however, the theory we present is valid for any $\mathbb{R}^n$.
We consider the random variable $\mathrm{W}$, defined by:
\begin{align}
\mathrm{W} & = |\mathbf{x}| = (\mathbf{x}^T\mathbf{x})^{1/2} = \sqrt{\sum^n_{i}\mathbf{x}_{i}^2} \\
\mathbf{x} & \sim \mathcal{N}(\mu, \Sigma),
\end{align}
with $\mathbf{x}, \mu \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{n\times n}$ is a full-rank covariance matrix.
This is the square root of the sum of squares of correlated normal variables.
It is well know that the sum of squares of independent identically distributed normal variables is Chi-squared distributed and that the corresponding square root is Chi distributed \cite{Johnson2}.
At first glance it seems that we should easily be able to identify this transformed distribution, however, the full-covariance between the elements of $\mathbf{x}$ hinder the derivation of a straightforward distribution.
Substantial work has been done on the distribution of quadratic forms, $Q(\mathbf{x})= \mathbf{x}^TA\mathbf{x}$ \cite{Mathai1992}, where $\mathbf{x}$ is an $n \times 1$ normal vector defined previously and $A$ is a symmetric $n \times n$ matrix. It is possible to write:
\begin{align}
Q(\mathbf{x}) & = \mathbf{x}^TA\mathbf{x} = \sum_{i}^{n} \lambda_i(U_i + b_i)^2, \label{eqn:quadratic_form_expan}
\end{align}
where the $U_i$ are $ $ i.i.d. normal variables with zero mean and unit variance, the $\lambda_i$ are the eigenvalues of $\Sigma$ and $b_i$ is the $i$th component of $b = P^T\Sigma^{\frac{1}{2}}\mu$, with $P$ a matrix that diagonalises $\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}$ .
Observing the summation of the quadratic form in Eqn \ref{eqn:quadratic_form_expan}, we see that our distribution is a weighted sum of Chi-squared variables. Unfortunately, there exists no simple closed-form solution for this distribution, however, it is possible to express this distribution via power-series of Laguerre polynomials and some approximations have been used \cite{Mathai1992}.
We note that a Chi-squared distribution is a gamma distributed variable for the case where the shape parameter is $v/2$ and the scale factor is 2.
Therefore we will approximate $Q(\mathbf{x}) = \mathbf{x}^T\mathbf{x}$ with a single gamma random variable by moment matching the first two moments.
The mean and variance of $Q(\mathbf{x})$ are given by:
\begin{align}
\mathbb{E}[Q(\mathbf{x})] & = \text{tr}(\Sigma ) + \mu^T\mu, \\
\mathbb{V}[Q(\mathbf{x})] & = 2\text{tr}(\Sigma\Sigma) + 4 \mu^T\Sigma \mu,
\end{align}
where $\text{tr}()$ denotes that trace of a matrix. The pdf of a gamma distribution with shape $k_G$ and scale $\theta_G$ is given by
\begin{align}
p_{G}(x: k_G, \theta_G) = \frac{x^{k_{G}-1}\exp\left(-\frac{x}{\theta_G}\right)}{\theta_{G}^{k_G} \Gamma(k_G)}.
\end{align}
The first two moments are:
\begin{align}
\mu_G & = k_G\theta_G, \quad \sigma_G^2 = k_G\theta_G^2.
\end{align}
Solving for $k_G$ and $\theta_G$:
\begin{equation}
k_G = \frac{\mu_G^2}{\sigma_G^2}, \quad \theta_G = \frac{\sigma_G^2}{\mu_G}.
\end{equation}
Equating moments, we set $\mu_G = \mathbb{E}[Q(\mathbf{x})] $ and $\sigma_G^2 = \mathbb{V}[Q(\mathbf{x})]$. Thus, $Q$ is approximated as a gamma random variable and we write, $Q(\mathbf{x}) \sim \text{Gamma}(k_G, \theta_G)$.
\\\\
Now we are in a position to consider the quantity $\sqrt{Q}$.
Here we use that fact that if a random variable $Q \sim \text{Gamma}(k_G,\theta_G)$, then the random variable $\mathrm{W} = \sqrt{Q}$ is a Nakagami random variable $\mathrm{W} \sim \text{Nagakami}(m, \Omega)$, with parameters given by $m = k_G$ and $\Omega = k_G\theta_G$.
The nagakami distribution \cite{Hoffman2013} is:
\begin{align}
p_{\mathrm{Nak}}(x; m, \theta) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right).
\end{align}
Using the value for $k$ and $\theta$ obtained via our moment matched approximation and transforming to the Nakagami distribution we say $\sqrt{Q}$ is approximated as a Nakagami distribution with parameters:
\begin{equation}
m = \frac{\mu_{G}^2}{\sigma_{G}^2}, \quad \Omega = \mu_{G}.
\end{equation}
In terms of our original distribution $\mathbf{x} \sim \mathcal{N}(\mu,\Sigma)$, we therefore have $\mathrm{W} = \sqrt{\mathbf{x}^T\mathbf{x}} \sim \text{Nakagami}(m,\Omega)$, with:
\begin{equation}
m = \frac{[ \text{tr}(\Sigma ) + \mu^T\mu]^2}{2\text{tr}(\Sigma\Sigma) + 4 \mu^T\Sigma \mu}, \quad \Omega = \text{tr}(\Sigma ) + \mu^T\mu.
\end{equation}
The mean and variance are:
\begin{align}
\mathbb{E}[\mathrm{W} ] & = \frac{\Gamma(m + \frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^\frac{1}{2}\\
\mathbb{V}[\mathrm{W}] & = \Omega\left( 1 - \frac{1}{m}\left( \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)} \right)^2 \right).
\end{align}
The method we have used to derive the distribution of the arc length integrand is summarised in Eqn \ref{eqn:flow}:
\begin{align}
\mathcal{N}(\mu, \Sigma) \underset{\text{Approximate}}{\overset{Q}{\rightarrow}} \text{Gamma}(k_G, \theta_G) \underset{\text{Exact}}{\overset{\sqrt{Q}}{\rightarrow}} \text{Nakagami}(m, \Omega)
\label{eqn:flow}
\end{align}
Numerical samples of $Q(\mathbf{x})$ and $\sqrt{Q(\mathbf{x})}$ and the pdf of the corresponding gamma and Nakagami distributions are show in Figure~\ref{fig:3d_mom_approx} for $d=3$.
The approximated distributions show a reasonable approximation for a range of $\mu$ and $\Sigma$.
The quadratic form approximated to the gamma distribution is exact when all the eigenvalues of the covariance are identical, in that case we have only a single gamma random variable.
\begin{figure*}
\vspace{.3in}
\includegraphics[trim={0 0 0 10}]{moment_approx.png}
\caption{Samples from $Q(\mathbf{x})$ and $\sqrt{Q(\mathbf{x})}$ overlaid with the approximated gamma and Nakagami distributions. $\mathbf{x} \sim (0, \Sigma)$ in the top row, and $\mathbf{x} \sim (\mu, \Sigma)$ in the bottom row with $\mu$ and $\Sigma$ randomly generated. Similar plots are obtained for different values of $\mu$ and $\Sigma$. The gamma and Nakagami distributions provide a reasonable approximation to the shape of the distribution, whilst capturing the true mean and variance. }
\label{fig:3d_mom_approx}
\end{figure*}
\subsection{Arc Length Statistics}
We are now in a position to consider the arc length directly.
Taking the expectation of the arc length, recalling that expectation is a linear operator and using Fubini's theroem:
\begin{align}
\mathbb{E}[s
& = \int_{0}^{T}\mathbb{E}\left[(\mathbf{f}'^T\mathbf{f}')^{\frac{1}{2}}\right]\mathrm{d}t.
\end{align}
Recalling the form of our kernel as $\mathbf{K}(x,x') = \mathrm{B}\otimes k(x,x')$, the infinitesimal distribution of $\mathbf{f}'$ is constant with respect to $t$ with covariance given by:
\begin{align}
\Sigma_{f'} = \mathrm{B}\otimes \frac{\partial^2}{\partial x \partial x'} k(x,x')\bigg{|}_{x=x'} = \mathrm{B}~\sigma_{f'}^2.
\end{align}
Therefore the expected length of the arc length is
\begin{align}
\mathbb{E}[s]
& \approx T\frac{\Gamma(m_{f'}+\frac{1}{2})}{\Gamma(m_{f'})}\left(\frac{\Omega_{f'}}{m_{f'}}\right)^{\frac{1}{2}},\label{eqn:first_moment}
\end{align}
with,
\begin{align}
m_{f'} = \frac{[\text{tr}(\Sigma_{f'})]^2}{2\text{tr}(\Sigma_{f'}\Sigma_{f'})}, \quad \Omega_{f'} = \text{tr}(\Sigma_{f'}),
\end{align}
where we have used the Nakagami approximation to the arc length integrand to evaluate the mean.
As in the one-dimensional case, the expected length of the \gls{gp} is determined solely by the choice of kernel and the length of the interval.
The calculation of the variance requires the second moment:
\begin{align}
\mathbb{E}[s^2
& = \int\int\mathbb{E}\left[|\mathbf{f}'_{t_1}|~|\mathbf{f}'_{t_2}|\right]\mathrm{d}t_1\mathrm{d}t_2.
\end{align}
Making use of the Nakagami approximation to our integrand we need the mixed moment of two correlated Nakagami variables.
Let us write $|\mathbf{f}'_{t_1}| \approx \mathrm{W}_1$, $|\mathbf{f}'_{t_2}| \approx \mathrm{W}_2$, with $\mathrm{W}_1 \sim \text{Nakagami}(m_{f'}, \Omega_{f'})$ and $\mathrm{W}_2 \sim \text{Nakagami}(m_{f'}, \Omega_{f'})$.
The mixed moments of two correlated Nakagami variables with the same parameters is given by \cite{Reig2002}:
\begin{align}
\mathbb{E}[\mathrm{W}_1^n\mathrm{W}_2^l] & = \frac{\Omega}{m}\frac{[\Gamma(m+n/2)]^2}{[\Gamma(m)]^2}{}_{2}F_{1}\left(-\frac{n}{2},-\frac{l}{2},m: \rho(\tau) \right),
\end{align}
where $\rho(\tau)$ is the correlation between the gamma variables that the Nakagami distribution was derived from and ${}_2F_{1}$ is the hypergeometric function:
\begin{align}
{}_{2}F_{1}(a,b,c:z) & = \sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!
\end{align}
with the Pochhammer $(q)_n$ symbol defined as $(q)_n = q(q+1)\dots(q+n-1) $ and $(q)_0= 1$. The second moment can now be expressed as a power series in $\rho$:
\begin{align}
\mathbb{E}[s^2]
& = \frac{\Omega}{m}\frac{[\Gamma(m+1/2)]^2}{[\Gamma(m)]^2}\sum_{n=0}^{\infty}\frac{\left(-\frac{1}{2}\right)_n\left(-\frac{1}{2}\right)_n}{(m)_n}\frac{1}{n!}\notag \\ &\qquad\int_0^T\int_0^T\rho(t_1-t_2)^n\mathrm{d}t_1\mathrm{d}t_2. \label{eqn:second_moment}
\end{align}
We derive the correlation function:
\begin{align}
\rho(t-t') & = \left[\frac{\partial^2}{\partial t \partial t'} k(t,t')\right]^2 \frac{1}{\sigma_{f'}^4}
\end{align}
Eqn~\ref{eqn:second_moment} can be solved numerically (noting that the two-dimensional integral is readily tackled using traditional methods of quadrature) and the variance is then computed by $\mathbb{V}[s] = \mathbb{E}[s^2] - \mathbb{E}[s]^2$.
\subsection{Arc Length Posterior}
The moments of the arc length of a \gls{gp} posterior follow a similar derivation.
The posterior mean of the arc length is:
\begin{align}
\mathbb{E}[s] & \approx \int_{0}^{T}\frac{\Gamma(m_{f'}+\frac{1}{2})}{\Gamma(m_{f'})}\left(\frac{\Omega_{f'}}{m_{f'}}\right)^{\frac{1}{2}}\mathrm{d}t,
\label{eqn:arc_mean_post}
\end{align}
where $m_{f'}$ and $\Omega_{f'}$ are the Nakagami parameters which now depend on the mean and covariance functions of the \gls{gp} posterior, which themselves are functions of $t$. This non tractable expression now requires an integration (which, again, can be efficiently approximated with quadrature). The posterior second moment is given by:
\begin{align}
\mathbb{E}[s^2] & \approx \sum_{n=0}^{\infty}\frac{\left(-\frac{1}{2}\right)_n(-\frac{1}{2})_n}{n!}\int_0^T\int_0^T\left(\frac{\Omega_1}{m_1}\right)^{\frac{1}{2}}\left(\frac{\Omega_2}{m_2}\right)^{\frac{1}{2}}\notag\\ &\quad\frac{\Gamma(m_1 + 1/2)\Gamma(m_2+1/2)}{\Gamma(m_1)\Gamma(m_2)(m_{2})_n}\rho(|t_1-t_2|)^n\mathrm{d}t_1\mathrm{d}t_2,
\label{eqn:arc_var_post}
\end{align}
where $m_i$ and $\Omega_i$ again depend on the mean and covariance functions of the \gls{gp} posterior and are evaluated at $t_i$.
\section{SIMULATIONS}
\label{sec:numerical}
In this section we generate samples from our \gls{gp} prior and compute the arc length, focusing on the vector case.
We show the effect of the kernel choice and show the fidelity of our theoretical results.
To generate our curves we specify a zero mean \gls{gp} kernel, $\mathrm{K} = \mathrm{B}\otimes k(t,t')$, with fixed $\mathrm{B}$ and we use the Mat\'{e}rn Kernel with $\nu = 3/2$, which we call the M32 kernel:
\begin{align}
k(t,t') & = \lambda^2\left( 1 + \frac{\sqrt{3}||t-t'||}{\sigma} \right)\exp\left(-\frac{\sqrt{3}||t-t'||}{\sigma}\right).
\end{align}
We draw a sample $f_i = (x_i, y_i, z_i)$ evaluated at evenly spaced $t$.
The arc length of the \gls{gp} draw is then computed numerically
Unit variance and length scale parameters are chosen and the arc length is computed over the interval $t = [0,1]$.
Figure~\ref{fig:3d_samples_arc} shows the sample lengths, the theoretical mean and variance, and the Nakagami distribution of a single arc length integrand.
Our theoretical results are close to the numerically generated values.
The plot of the Nakagami distribution demonstrates the wide variance of an individual arc length integrand with respect to the overall variance.
We see that the integration over the input domain has a sort of `shrinking' effect on overall variance when compared to the individual variance.
\begin{figure}
\vspace{.3in}
\includegraphics[trim={0 0 0 10}]{Vector_samples.png}
\caption{Histogram of GP Lengths. The theoretical and empirical mean are shown and the corresponding variance. The Nakagami distribution of the integrand is also shown. We can see the integral over integrands has the effect of shrinking the variance relative to a single integrand.}
\label{fig:3d_samples_arc}
\end{figure}
No estimation methods are required to calculate the arc length statistics.
Our approximated equations are closed form (for the mean) and a quadrature problem (for the variance).
\section{CONCLUSION}
In this paper we derive the moments of a vector valued \gls{gp}.
To the best of the authors' knowledge, this is the first treatment of the arc length in more than one dimension.
The increment distribution was approximated via its moments to a Nakagami distribution which provide a closed form for the mean, Eqn \ref{eqn:first_moment} and an expression for the second moment, Eqn \ref{eqn:second_moment}, of the arc length.
Importantly, we are also able to derive the first, Eqn \ref{eqn:arc_mean_post}, and second, Eqn \ref{eqn:arc_var_post}, moment of the arc length of a \gls{gp} posterior, conditioned on observations of the function.
The moments were shown to depend on the choice of kernel, the hyperparameters and the length of the interval. Numerical experiments confirmed the fidelity of our approximation to the arc length integrand and the arc length moments. We also provide a visual understanding of the distribution.
We see knowledge of the arc length as a valuable tool which will allow us to encode more information into our prior over kernel choices. The explicit relation between the arc length moments and the kernel hyperparameters allow us to use prior information to better initialize and constrain our models, in particular in cases where lengths correspond to interpretable quantities, such as a path trajectory.
Potential avenues of future research include analysis of the non-stationarity of curves, curve minimisations problems, and generating curves of a given length. Furthermore, we see potential application in Bayesian optimization, as a path planning tool and for constructing interpretable features from functional data.
\subsubsection*{Acknowledgements}
AT and MO are grateful for the support of funding from the Korea Institute of Energy Technology Evaluation and Planning (KETEP).
The authors are grateful for initial conversations with Tom Gunter who highlighted the gap in the literature.
\bibliographystyle{siam}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,224 |
Q: Adding print friendly link on specific nodes In my Drupal 7 site I use the Printer, email and PDF versions module. I want to show the printer link on some specific pages of a specific content type. So I have enabled the link in that content type, which include a boolean field (field_printable) where the admin should be able to choose to include the link or not.
In my template.php file I try to figure out how to work this out. This is what I have got so far in my template.php file:
function mytheme_preprocess_node(&$variables) {
if($variables['type'] == "mycontenttype"){
// boolean field that returns 1 if checked
if($node->field_printable['und'][0]['value'] == 1){
what to put here ...?
}
}
}
I should probably use print_insert_link(); to insert the link, but I cannot figure out how. Can anyone point me in the right direction? I have spent hours googling on similar questions, but I am stuck right now.
Any help would be very appreciated, thanks in advance.
Edited: Screen goes blank after I try this...
function mytheme_preprocess_node(&$variables) {
if($node->nid == 408){
$variables['print_custom_link'] = print_insert_link();
}
}
I also added this in my node template file:
print render($content);
if(!empty($print_custom_link)){
print render($print_custom_link);
}
A: OK, so I thought it could be a good idea to post an answer to my own question, in case others struggling with the same problem is looking at this post. And thanks again Laurent!
Turns out that my boolean field 'field_printable" is available directly in $variables. And $node was not available at all.
So in my template.php I ended up with this:
function mytheme_preprocess_node(&$variables) {
if($variables['type'] == "mycontentpage"){
if($variables['field_printable']['und'][0]['value'] == 1){
$variables['print_custom_link'] = print_insert_link();
}
}
}
In my node template (right after the row "print render($content);"):
if(!empty($print_custom_link)){
print $print_custom_link;
}
That's it. Now it works exactly as I wanted :-)
A: The fist step should be to configure your node type to be printable in structure>content type>the node type you want to print.
Then by default a print link will appears when you render your node content.
If you want to display a print link in a particular area and/or for a specific node (or even a view) then you can decide to use print_insert_link() to display the print link : https://www.drupal.org/node/306888
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 388 |
Applying Agile Methods to Engineering Management - Managers in fast-paced technology companies deal with constantly changing environments - from hyper-growth to employee turnover. As a result, managers need to be agile enough to respond to anything that comes their way. This talk is about how to manage teams more effectively by adopting some of the principles used in agile software development.
This is the fourth of a twelve-part series covering the sessions from Calibrate, an engineering leadership conference held September, 2015 in San Francisco.
Oftentimes that sense of accomplishment fades quickly and is replaced by feelings of uncertainty and thoughts like, "Great, exactly how do I manage people?", or "There are so many things to do, where do I start?" If you come from a software development background, Kate Heddleston would argue that you have all the tools you need to tackle people management. During her talk at Calibrate, she shed some light on what she called Agile Management. If you're wondering whether or not it's anything like Agile Development, spoiler alert - it is.
Taking an agile approach to management forces you into action, because just like agile development, your goal is to deliver a solution to a problem quickly. One of the nice things about agile is the ability to iterate, which eliminates the burden of coming up the perfect solution on the first try. As managers, we often feel the need to solve the problem on the first try. This can lead to doing nothing, for fear of doing the wrong thing. Kate reminds us that inaction is one of the worst things a manager can do, as it conveys the message that the manager doesn't care about problems facing the team. If you're interested in learning more about Agile Management, have a listen to Kate's talk. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,388 |
Kerala, the better known and most visited Southern Indian state, with its coconut fringed beaches, beautiful landscapes and backwaters, sets it apart from its neighbours. Kerala holidays are perfect for a first-time visitor to India and an excellent holiday option for families. The slow flowing calm found in the network of lagoons and rivers known as the backwaters is wonderfully relaxing and a visit here offers an insight into a unique way of life.
Another fascinating aspect of Kerala is that beneath its relatively westernised surface, lies an ancient world of shamanistic beliefs that are celebrated each year in dramatic and colourful festivals. Kerala's perfect climate suits the unique blend of culture and scenery and will never fail to entice visitors old and new.
Wander the streets of ram shackled Fort Kochi with its mixing pot of cultures and religions illustrating a fascinating past.
Walk the lush green paddy fields of the plains, through spindly rubber plantations and blooming coffee plantations.
The tea estates in the hill side villages of Thekkady and Munnar are an incredible place to explore.
Take time to spend a day relaxing on a traditional house boat on the Vembanad Lake.
Go shopping at the spice market, the ginger and clove factory or visit the weekly antiques market.
Spend a few days as a guest in the home of a plantation owner, experiencing the cultivation of spices and coffee first hand, followed by relaxation on the beach at Neeleshwar.
Kerala offers a humid climate with a shorter season than the dry north. The best time to holiday in Kerala is between October and April, however do be aware of the increasing humidity from March onwards. Maximum temperatures rarely rise above 32 degrees while minimum temperatures at sea level are almost never below 20 degrees. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,434 |
Q: How to write nested query in jooq with SUM operation from the subquery output? Let's say I want to find out the sum of one of the table from the nested subquery. Let's take the subquery given below. How to calculate the status count using DSL.sum(when(condition))
Table<Record4<Integer, Integer, String, String>> orgWiseCountTable = dsl.select(
field("master_id", Integer.class).as(masterId),
field("transaction_id", Integer.class).as(transactionId),
field("api_status", String.class).as(status),
field("Organisations.organisation_name",String.class).as(organisationName)
)
.from(TestTable)
.leftJoin(Organisations)
.on(TestTable.organsiationId.eq(Organisations.organsiationId)
.where(condition)
.groupBy(TestTable.master_id).asTable("orgWiseCount");
Using the above Table record how can we calculate the sum of api_status depending on whether or not the value is COMPLETED or not.
Trying the below approach but it isn't working.
transactions = dsl.select(
orgWiseCountTable.field(organisationName).as("OrganisationName"),
orgWiseCountTable.field("SUM(status = 'COMPLETED')").as(
"SuccessCount"),
orgWiseCountTable.field("SUM(kyc_api_transaction_master.api_status <> 'COMPLETED')").as(
"FailureCount"),
orgWiseCountTable.field(totalCount).as("TotalCount")
)
.from(orgWiseCountTable)
.fetch()
.into(Count.class);
Getting Null pointer exception while calculating sum. How can this be fixed?
A: Using derived tables
As I answered to your follow-up question, I don't think you really need the derived table. You'll simplify your life greatly, if you just remove it.
Dereferencing columns from a derived table
If your actual query is more complex than the one you've presented here, you'll have to dereference columns from your derived table like this:
// Dereference first:
Field<String> status = orgWiseCountTable.field("status", String.class);
// Alternatively, dereference using generaetd code for added convenience:
Field<String> status = orgWiseCountTable.field(THE_TABLE.STATUS);
// Then use in expressions:
DSL.count().filterWhere(status.eq("Completed")).as("SuccessCount");
Aggregate filter clause
Note, I'm using FILTER instead, which is a standard SQL feature. This will be a bit more convenient to write in jOOQ, than using the MySQL specific approach of mapping booleans to integers in order to sum them. A.eq(B) is a Condition in jOOQ, which extends Field<Boolean>, and you cannot really sum booleans.
Code generation
This isn't necessary relevant to your question, but I highly recommend you start using jOOQ's code generator. A lot of features will be unlocked by that, and your queries will be much simpler to write.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,204 |
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Artificial Intelligence Artificial Intelligence Workstations. Rainfinity FMA is the first step in a complete file virtualization strategy that balances storage utilization while providing a single view across multiple servers, NAS and CAS devices. FMA key features include:. It ensures scalability, data integrity and ease of deployment for extremely large and complex network file environments. Based in San Jose, California, privately held Rainfinity is a leading provider of virtualization solutions for heterogeneous networked attached storage NAS and file system environments. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,115 |
\section{Introduction}
One of the fastest growing areas within graph theory is the study of domination.
Many variants of the basic concepts of domination have appeared in the literature.
We refer to \cite{hhs1} for a survey of the area.
As many other graph invariants, domination has been studied on different graph products.
Several papers have been published in the last fifteen years concerning
various types domination in the lexicographic product of two graphs, including
domination (Nowakowski and Rall \cite{nr}, \u{S}umenjak et al. \cite{spt} and G\"{o}z\"{u}pek et al. \cite{GHM1}),
total and restrained domination (Zhang et al. \cite{zlm}),
Roman domination (\u{S}umenjak et al. \cite{spt}),
rainbow domination (\u{S}umenjak et al. \cite{srt}),
super domination (Dettlaff et al. \cite{dlrz}) and double domination (Cabrera Mart\'inez et al.\cite{ccr}).
However, to the best knowledge of the author,
there are no studies related to domination in graphs representable as generalized lexicographic products.
This fact motivates us to begin an exploration of several domination-related parameters
(among which are the total, restrained, total restrained, outer connected
and total outer connected domination numbers)
in the generalized lexicographic product of graphs.
We give basic terminologies and notations in the rest of this section.
All graphs in this paper will be finite, simple, and undirected.
We use \cite{hhs1} as a reference for terminology and notation which are not explicitly defined here.
In a graph $G$, for a subset $S \subseteq V (G)$ the {\em subgraph induced} by $S$ is the graph
$\left\langle S \right\rangle$ with vertex set $S$ and edge set $\{xy \in E(G) \mid x, y \in S\}$.
The {\em complement} $\overline{G}$ of $G$ is the graph whose
vertex set is $V (G)$ and whose edges are the pairs of nonadjacent vertices of $G$.
We write $K_n$ for the {\em complete graph} of order $n$ and $P_n$ for the {\em path}
on $n$ vertrices. Let $C_m$ denote the {\em cycle} of length $m$.
For any vertex $x$ of a graph $G$, $N_G(x)$ denotes the set of all neighbors
of $x$ in $G$, $N_G[x] = N_G(x) \cup \{x\}$ and the degree of $x$ is $deg_G(x) = |N_G(x)|$.
The {\em minimum} and {\em maximum} degrees
of a graph $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively.
For a subset $S \subseteq V (G)$, let $N_G[S] = \cup_{v \in S}N_G[v]$.
The distance between vertices $x$ and $y$ of a graph $G$ is denoted by $dist_G(x,y)$.
An {\em isomorphism} of graphs $G$ and $H$ is a bijection
$f \colon V(G)\to V(H)$ such that any two vertices $u$ and $v$ of $G$
are adjacent in $G$ if and only if $f(u)$ and $f(v)$ are adjacent in $H$.
If an isomorphism exists between two graphs, then the graphs are called isomorphic
and denoted as $ G\simeq H$. We use the notation $[k]$ for $\{1,2,..,k\}$.
Let $G$ be a graph with vertex set
$V(G) = \{\textbf{1}, \textbf{2},..,\textbf{n}\}$ and let
$\Phi = (F_1,F_2,..,F_n)$ be an ordered $n$-tuple of paired disjoint graphs.
Denote by $G[\Phi]$ the graph with vertex set $\cup_{i=1}^n V(F_i)$
and edge set defined as follows: (a) $F_1, F_2,.., F_n$ are induced subgraphs of $G[\Phi]$,
and (b) if $x \in V(F_i)$, $y \in V(F_j)$, $i,j \in [n]$ and $i \not= j$, then
$xy \in E(G[\Phi])$ if and only if \textbf{ij} $\in E(G)$.
A graph $G[\Phi]$ is called the {\em generalized lexicographic product} of $G$ and $\Phi$.
If $F_i \simeq F$ for every $i=1,2,..,n$, then $G[\Phi]$
becomes the standard lexicographic product $G[F]$.
Each subset $U = \{u_1,u_2,..,u_n\} \subseteq V(G[\Phi])$ such that
$u_i \in V(F_i)$, for every $i \in [n]$, is called a $G$-layer.
From the definition of $G[\Phi]$ it immediately follow:
\begin{itemize}
\item[(A)] (folklore) $G[\Phi] \simeq G$ if and only if $G[\Phi] = G[K_1]$.
$G[F] \simeq F$ if and only if $G \simeq K_1$.
If $G$ has at least two vertices, then $G[\Phi]$ is connected if and only if $G$ is connected.
If $G$ is edgeless, then $G[\Phi] = \cup_{i=1}^nF_i$.
For any $G$-layer $U = \{u_1,u_2,..,u_n\}$
the bijection $f \colon V(G)\to U$ defined by $f(\textbf{i}) = u_i \in V(F_i)$ is an
isomorphism between $G$ and $\left\langle U \right\rangle$.
For any $x \in V(F_i)$ and $y \in V(F_j)$, $i \not= j$, is fulfilled
$dist_{G[\Phi]}(x,y) = dist_G(\textbf{i},\textbf{j})$.
\end{itemize}
The equality $dist_{G[\Phi]}(x,y) = dist_G(\textbf{i},\textbf{j})$
will be used in the sequel without specific references.
Since for any domination-related parameter $\mu$, which we consider in this work,
and for any two disjoint graphs $G_1$ and $G_2$ is fulfilled
$\mu(G_1 \cup G_2) = \mu(G_1) + \mu(G_2)$
(when it is known that at least one of the left or right sides of this equality exists),
we restrict our attention only on
connected generalized lexicographic products. Therefore,
in what follows when a graph $G[\Phi]$ is under consideration
we assume that $G$ is a connected graph of order $n \geq 2$.
Unless otherwise stated, we also assume that always $\Phi = (F_1,F_2,..,F_n)$.
Let $\mathcal{I}$ denote the set of all mutually nonisomorphic graphs.
A {\em graph property} is any non-empty subset of $\mathcal{I} $.
We say that a {\em graph $G$ has property} $\mathcal{P}$ whenever
there exists a graph $H \in \mathcal{P}$ which is isomorphic to $G$.
For example we list some graph properties:
\medskip
$\bullet$ $\mathcal{T} = \{H \in \mathcal{I}$ : $\delta(H) \geq 1\}$;
$\bullet$
$\mathcal{F} = \{H \in \mathcal{I}$ : $H$ is a forest$\}$;
$\bullet$ $\mathcal{M} = \{H \in \mathcal{I}$ : $H$ has a perfect matching $\}$;
$\bullet$ $\mathcal{S}_k = \{H \in \mathcal{I}$ : $\Delta(G) \leq k \}$, $k \geq 0$.
$\bullet$ $\mathcal{C} = \{H \in \mathcal{I}$ : $H$ is connected$\}$.
\medskip
Any set $S \subseteq V(G)$ such that $\left\langle S \right\rangle$
possesses the property $\mathcal{A} \subseteq \mathcal{I}$ and
$\left\langle V(G)-S \right\rangle$ possesses the property $\mathcal{B} \subseteq \mathcal{I}$
is called an $(\mathcal{A}, \mathcal{B})$-{\em set}.
A {\em dominating set} for a graph $G$ is a set of vertices $D \subseteq V(G)$
such that every vertex of $G$ is either in $D$ or is adjacent to an element of $D$.
A dominating $(\mathcal{A}, \mathcal{B})$-set $S$ of a graph $G$ is
a {\em minimal dominating $(\mathcal{A}, \mathcal{B})$-set}
if no set $S^{'} \subsetneq S$ is a dominating $(\mathcal{A}, \mathcal{B})$-set.
The set of all minimal dominating $(\mathcal{A}, \mathcal{B})$-sets of a graph $G$
is denoted by $MD_{(\mathcal{A}, \mathcal{B})} (G)$.
The {\em domination number with respect to the pair} $(\mathcal{A}, \mathcal{B})$,
denoted by $\gamma_{(\mathcal{A}, \mathcal{B})} (G)$,
is the smallest cardinality of a dominating $(\mathcal{A}, \mathcal{B})$-set of $G$.
The {\em upper domination number with respect to the pair} $(\mathcal{A}, \mathcal{B})$,
denoted by $\Gamma_{(\mathcal{A}, \mathcal{B})} (G)$,
is the maximum cardinality of a minimal dominating $(\mathcal{A}, \mathcal{B})$-set of $G$.
A $\gamma_{(\mathcal{A}, \mathcal{B})}$(resp., $\Gamma_{(\mathcal{A}, \mathcal{B})}$)-{\em set}
of a graph $G$ is every set in $MD_{(\mathcal{A}, \mathcal{B})} (G)$
having cardinality $\gamma_{(\mathcal{A}, \mathcal{B})} (G)$ (resp., $\Gamma_{(\mathcal{A}, \mathcal{B})}(G)$).
Note that:
\begin{itemize}
\item[\rm (a)] $\gamma_{(\mathcal{I}, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{I}, \mathcal{I})} (G)$ are known as
the domination and upper domination numbers $\gamma(G)$ and $\Gamma(G)$ of $G$, respectively,
\item [\rm (b)] $\gamma_{(\mathcal{S}_0, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{S}_0, \mathcal{I})} (G)$ are known as
the independent domination number $i(G)$ and the independence number $\beta_0(G)$,
\item [\rm (c)] $\gamma_{(\mathcal{T}, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{T}, \mathcal{I})} (G)$ are known as
the total domination and upper total domination numbers $\gamma_t(G)$ and $\Gamma_t(G)$ (\cite{cdh}),
\item [\rm (d)] $\gamma_{(\mathcal{I}, \mathcal{T})} (G)$ and $\Gamma_{(\mathcal{I}, \mathcal{T})} (G)$ are known as
the restrained domination and upper restrained domination numbers $\gamma_r(G)$ and $\Gamma_r(G)$ (\cite{t}),
\item [\rm (e)] $\gamma_{(\mathcal{T}, \mathcal{T})} (G)$ and $\Gamma_{(\mathcal{T}, \mathcal{T})} (G)$ are known as
the total restrained domination and upper total restrained domination numbers $\gamma_{tr}(G)$ and $\Gamma_{tr}(G)$ (\cite{clm}).
\item [\rm (f)] $\gamma_{(\mathcal{I}, \mathcal{C})} (G)$ and $\Gamma_{(\mathcal{I}, \mathcal{C})} (G)$ are known as
the outer-connected domination and upper outer-connected domination numbers
$\gamma^{oc}(G)$ and $\Gamma^{oc}(G)$ (\cite{cy}),
\item [\rm (g)] $\gamma_{(\mathcal{T}, \mathcal{C})} (G)$ and $\Gamma_{(\mathcal{T}, \mathcal{C})} (G)$ are known as
the total outer-connected domination and upper total outer-connected domination numbers
$\gamma^{oc}_t(G)$ and $\Gamma^{oc}_t(G)$ (\cite{cyt}),
\item [\rm (h)] $\gamma_{(\mathcal{M}, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{M}, \mathcal{I})} (G)$ are known as
the paired domination and upper paired domination numbers $\gamma_p(G)$ and $\Gamma_p(G)$ (\cite{hs}).
\end{itemize}
The following inequalities are folklore: $\gamma(G) \leq \gamma_t(G) \leq \min\{\gamma_p(G), \gamma_{tr}(G), \gamma_t^{oc}(G)\}$,
$\gamma_r(G) \leq \gamma_{tr}(G)$, $\gamma^{oc}(G) \leq \gamma_t^{oc}(G)$ and $\gamma(G) \leq \min \{\gamma_r(G), \gamma^{oc}(G)\}$.
Not much work has been done on finding relationships between
(a) the value of a given domination parameter in the standard lexicographic product and that of its factors, and
(b) the values of two given domination parameters in the standard lexicographic product.
We list those results that relate to the domination parameters above defined.
\begin{them}\label{known}
Let $G_1$ and $G_2$ be graphs with at least two vertices.
\begin{itemize}
\item[(i)] (Zhang et al. \cite{zlm}) If $\gamma(G_2)=1$, then $\gamma(G_1[G_2]) = \gamma(G_1)$.
\item[(ii)] (Zhang et al. \cite{zlm}) $\gamma_t(G_1[G_2]) = \gamma_t(G_1)$.
\item[(iii)] (Zhang et al. \cite{zlm}) If $\delta(G_1) \geq 1$, then $\gamma_r(G_1[G_2]) = \gamma_r(G_1)$.
\item[(iv)] (\u{S}umenjak et al. \cite{spt}) If $G_1$ and $G_2$ are connected and $\gamma(G_2) \geq 2$, then
$\gamma(G_1[G_2]) = \gamma_t(G_1[G_2]) = \gamma_t(G_1)$.
\end{itemize}
\end{them}
In Section $2$ we obtain results on the parameters $\gamma$, $\gamma_t$, $\gamma_r$, $\gamma_{tr}$, $\gamma_p$,
$\gamma^{oc}$ and $\gamma_t^{oc}$ in a generalized lexicographic product.
Some of them generalize or/and extend those stated in the above theorem.
To continue we need the following definition.
\begin{definition}\label{abwell}
Let $\mathcal{A}, \mathcal{B} \subseteq \mathcal{I}$.
A graph $G$ is said to be well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated if
$\gamma_{(\mathcal{A}, \mathcal{B})} (G) = \Gamma_{(\mathcal{A}, \mathcal{B})} (G)$.
\end{definition}
In a 1970 paper, Plummer \cite{p} introduced the notion of considering graphs in
which all maximal independent sets have the same size;
he called a graph having this property a {\em well-covered graph}. Equivalently,
a well-covered graph is one in which the greedy algorithm for constructing
independent sets yields always maximum independent sets.
Clearly well-covered graphs form the class of all well $i$-dominated graphs.
Topp and Volkmann \cite{tv2} gave a characterization of
well covered generalized lexicographic product of graphs.
Well $\gamma$-dominated graphs were introduced by Finbow et al. \cite{fhn}.
Obviously each well $\gamma$-dominated graph is well covered.
Recall the characterization of the well-dominated nontrivial lexicographic product of two graphs,
which was recently obtained by G\"{o}z\"{u}pek, Hujdurovi\'c and Milani\v{c} in \cite{GHM1}.
\begin{them} \label{wdf} \cite{GHM1}
A nontrivial lexicographic product, $G[F]$, of a connected graph $G$ and
a graph $H$ is well-dominated if and only if one of the following conditions holds:
\begin{itemize}
\item[(i)] $G$ is well-dominated and $F$ is complete, or
\item[(ii)] $G$ is complete and $F$ is well-dominated with $\gamma(F) = 2$.
\end{itemize}
\end{them}
In Section $3$ we present results on well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated graphs;
in particular we characterize well $\gamma$-dominated and well $\gamma_t$-dominated
generalized lexicographic product of graphs.
We conclude in Section 4 with some open problems.
\section{Seven domination parameters}
Recall that the equality $\gamma_t(G) = \gamma_t(G[F])$ was proven by X. Zhang et al. \cite{zlm}.
The next theorem shows that the equality remains valid if we remove $F$ by $\Phi$.
\begin{theorem}\label{t=t=}
If $I$ is a $\gamma_t$-set of some $G$-layer of $G[\Phi]$, then $I$ is a $\gamma_t$-set of $G[\Phi]$.
In particular, $\gamma_t(G) = \gamma_t(G[\Phi])$.
\end{theorem}
\begin{proof}
Let $U$ be a $G$-layer of $G[\Phi]$ and $I$ a $\gamma_t$-set of $\left\langle U \right\rangle$.
By the definition of a graph $G[\Phi]$ we immediately obtain that
$I$ is a total dominating set of $G[\Phi]$.
Since $\left\langle U \right\rangle \simeq G$, $\gamma_t(G) \geq \gamma_t(G[\Phi])$.
Now if the equality holds, then clearly $I$ is a $\gamma_t$-set of $G[\Phi]$.
For each $\gamma_t$-set $T$ of $G[\Phi]$ denote by $s_T$
the number of all $i$ for which $T$ and $V(F_i)$ have at least two elements in common.
Choose now a $\gamma_t$-set $R$ of $G[\Phi]$ so that $s_R$ is minimum.
Suppose $s_R\not=0$ and $x,y \in V(F_m) \cap R$ for some $m \in [n]$.
Then $|R \cap V(F_l)|=0$ for all $F_l$'s such that $\textbf{l}\textbf{m} \in E(G)$.
Consider now the set $R_1 = (R-\{y\}) \cup \{z\}$, where $z\in V(F_l)$.
Obviously $R_1$ is a $\gamma_t$-set $G[\Phi]$ with $s_R > s_{R_1}$,
which contradicts the choice of $R$. Thus $s_R=0$ and then there
is a $G$-layer of $G[\Phi]$, say $H$, which contains $R$. Since clearly
$R$ is a total dominating set of $\left\langle H \right\rangle$ and $\left\langle H \right\rangle \simeq G$,
we obtain $\gamma_t(G) \leq \gamma_t(G[\Phi])$.
\end{proof}
\begin{theorem}\label{gamma=1=}
$\gamma(G) \leq \gamma(G[\Phi])$.
The equality holds if and only if there is a $\gamma$-set $I$ of $G$ such that if $\textbf{i}_\textbf{j} \in I$
is an isolated vertex of $\left\langle I \right\rangle$, then $\gamma(F_j) = 1$.
If $\gamma(G) = \gamma(G[\Phi])$, then $|D \cap V(F_i)| \leq 1$, $i=1,2,..,n$,
for each $\gamma$-set $D$ of $G[\Phi]$.
\end{theorem}
\begin{proof}
Let $D$ be a $\gamma$-set of $G[\Phi]$ and let $F_{i_1},F_{i_2},..,F_{i_k}$ be all $F_j$'s
each of which has a common vertex with $D$.
Choose a $G$-layer $U = \{u_1,u_2,..,u_n\}$ so that $u_{i_s} \in V(F_{i_s})\cap D$ for $s=1,2,..,k$.
Clearly $D_1 = \{u_{i_1}, u_{i_2},.., u_{i_k}\}$ is a dominating set of $U$.
Since $\left\langle U \right\rangle \simeq G$, we have
$\gamma(G) = \gamma(U) \leq |D_1| \leq |D| = \gamma(G[\Phi])$.
Assume now that $\gamma(G) = \gamma(G[\Phi]) =k$.
Then $D_1$ is a $\gamma$-set of $U$ and $D_1=D$.
This immediately implies $|D \cap V(F_i)| \leq 1$ for all $i \in [n]$.
Since $\left\langle U \right\rangle \simeq G$, $I = \{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},.., \textbf{i}_\textbf{k}\}$ is a $\gamma$-set of $G$.
If $\textbf{i}_\textbf{r}$ is an isolated vertex of $\left\langle I \right\rangle$, then
$u_{i_r}$ is an isolated vertex of $\left\langle D_1 \right\rangle$.
Since $D=D_1$, $u_{i_r}$ is an isolated vertex of $\left\langle D \right\rangle$
and therefore $u_{i_r}$ must dominate all vertices in $F_{i_r}$.
Let there be a $\gamma$-set $I = \{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},.., \textbf{i}_\textbf{k}\}$ of $G$
such that for each isolated vertex $\textbf{i}_\textbf{s}$ in $\left\langle I \right\rangle$, $\gamma(F_{i_s}) = 1$.
But then the set $R = \{x_{i_1}, x_{i_2},.., x_{i_k}\}$, where $x_{i_s} \in V(F_{i_s})$
has maximum degree in $F_{i_s}$, $s=1,2,..,k$, is a dominating set of $G[\Phi]$.
Hence $\gamma(G) \geq \gamma(G[\Phi])$ and the required follows.
\end{proof}
\begin{corollary}\label{g=1=}
If $\gamma(F_1) = \gamma(F_2) = ... = \gamma(F_n) = 1$,
then $\gamma(G) = \gamma(G[\Phi])$.
\end{corollary}
By Theorem \ref{t=t=}, Theorem \ref{gamma=1=} and the well known inequalities $\gamma(G) \leq \gamma_t(G) \leq 2\gamma(G)$,
we obtain the next inequality chain.
\begin{corollary}\label{ggt}
$\gamma(G) \leq \gamma(G[\Phi]) \leq \gamma_t(G[\Phi]) = \gamma_t(G) \leq 2\gamma(G) \leq 2\gamma(G[\Phi])$.
\end{corollary}
An immediate consequence of this corollary is the following.
\begin{corollary}\label{g=gt=}
If $\gamma(G) = \gamma_t(G)$, then $\gamma(G[\Phi]) = \gamma_t(G[\Phi])$.
If $\gamma_t(G[\Phi]) = 2\gamma(G[\Phi])$, then $\gamma_t(G) =2\gamma(G)$.
\end{corollary}
Now we concentrate on the case when all $F_i$'s have at least two vertices.
We need the following key lemma for our purpose in this work.
\begin{lemma}\label{lexlemaanew}
Let $\mu \in \{\gamma,\gamma_t\}$,
$D$ be a $\mu$-set of $G[\Phi]$ and $|V(F_i)| \geq 2$ for all $i \in [n]$.
Then the following assertions hold.
\begin{itemize}
\item[(i)] $|D \cap V(F_i)| \leq 2$ for all $i=1,2,..,n$.
If $|D \cap V(F_s)|=2$ for some $s \in [n]$, then no vertex of $F_s$
is adjacent to a vertex of $D-V(F_s)$ and $D \cap V(F_s)$ is a $\mu$-set of $F_s$.
If $|D \cap V(F_i)| = |D \cap V(F_j)| = 2$, $i \not= j$,
then the distance between any vertex of $F_i$ to any vertex of $F_j$ is at least three.
\item[(ii)] Let $R = \{i \mid 2=|D \cap V(F_i)|\} \not= \emptyset$.
For all $i \in R$ let $D \cap V(F_i) = \{z_{i1}, z_{i2}\}$
and $x_i \in N(z_{i2}) -V(F_i)$.
Then the set $D^* =( D -\cup_{i \in R}\{z_{i2}\}) \cup \cup_{i \in R}\{x_i\}$
is a $\mu$-set of $G[\Phi]$ and $|D^* \cap V(F_r)| \leq 1$ for all $r \in [n]$.
\item[(iii)] $G[\Phi]$ has a $\mu$-set $U$ such that (a) $|U \cap V(F_i)| \leq 1$ for all $i \in [n]$,
(b) if $\mu = \gamma$, then $U$ is both a $\gamma_r$-set and a $\gamma^{oc}$-set of $G[\Phi]$, and
(c) if $\mu = \gamma_t$, then $U$ is both a $\gamma_{tr}$-set and a $\gamma_t^{oc}$-set of $G[\Phi]$.
\item[(iv)] Let $\mu = \gamma$ and $\gamma(F_i) \geq 2$ for all $i \in [n]$.
Then a $\gamma$-set $U$ (see (iii)) is a $\nu$-set for all $\nu= \gamma_r, \gamma_t, \gamma_{tr}, \gamma^{oc}, \gamma_t^{oc}$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) Let for some $j \in [n]$ is fulfilled $D \cap V(F_j) = \{u_1,u_2,..,u_r\}$, where $r\geq 2$.
Then $N[\{u_1,w\}] \supseteq N[\{u_2,..,u_r\}]$ for any neighbor $w$ of $u_1$ which is outside $V(F_j)$.
As $D$ is a $\mu$-set of $G[\Phi]$, we have $w \not\in D$ and $r=2$.
Since $w$ was chosen arbitrarily, $N[\{u_1,u_2\}] \cap D = \{u_1,u_2\}$.
But then $\{u_1,u_2\}$ is a $\mu$-set of $F_j$.
Let $|D \cap V(F_i)| = |D \cap V(F_s)| = 2$ for some $i,s \in [n], i \not= s$.
Suppose $z_1,z_2,z_3$ is a shortest path in $G[\Phi]$, where $z_1 \in V(F_i)$ and $z_3 \in V(F_s)\cap D$.
Clearly $z_2 \not\in V(F_i) \cup V(F_j) \cup D$.
Then $D^\prime = (D-\{z_3\}) \cup \{z_2\}$ is a $\mu$-set and $z_2$ is a
common neighbor of the vertices in $V(F_i) \cap D$, a contradiction.
(ii) By (i) all $x_i$'s are paired distinct and outside $D$,
and no two of them belong to the same $F_j$, $j \in [n]$.
Hence $|D^*| = |D|$, $|D^* \cap V(F_r)| \leq 1$ for all $r \in [n]$
and since $N[\{z_{i1}, z_{i2}\}] \subset N[\{z_{i1}, u_i\}]$ and $z_{i1}$ and $u_i$ are adjacent,
$D^*$ is a $\mu$-set of $G[\Phi]$.
(iii) Define a $\mu$-set $U$ such that $U=D$ when $|D \cap V(F_i)| \leq 1$ for all $i \in [n]$, and
$U=D^*$ otherwise (see (ii)). Hence $|U \cap V(F_i)| \leq 1$
and since $G$ is connected of order $n \geq 2$ and $|V(F_i)| \geq 2$ for all $i \in [n]$,
a graph $\left\langle V(G[\Phi])-D \right\rangle$ is connected of order at least two.
(iv) Since $\gamma(F_i) \geq 2$ for all $i \in [n]$, $U$ is a total dominating set of $G[\Phi]$
and since $\gamma(G[\Phi]) \leq \gamma_t(G[\Phi])$,
$U$ is a $\gamma_t$-set of $G[\Phi]$.
The required now immediately follows by (iii).
\end{proof}
\begin{theorem}\label{two}
Let $|V(F_i)| \geq 2$ for all $i \in [n]$. Then
\begin{itemize}
\item[(i)] $\gamma(G[\Phi]) = \gamma_r(G[\Phi]) = \gamma^{oc}(G[\Phi])$, and
\item[(ii)] $\gamma_t(G) = \gamma_t(G[\Phi]) = \gamma_{tr}(G[\Phi]) = \gamma_t^{oc}(G[\Phi])$.
\end{itemize}
\end{theorem}
\begin{proof}
Immediately by Lemma \ref{lexlemaanew}(iii) and Theorem \ref{t=t=}.
\end{proof}
Let $\mu, \nu \in \{\gamma, \gamma_r, \gamma^{oc}, \gamma_t, \gamma_{tr}, \gamma_t^{oc}\}$.
If the sets of all $\mu$-sets and all $\nu$-sets of a graph $H$ coincide, then we say
$\mu(H)$ {\em strongly equal} to $\nu(H)$, written $\mu(H) \equiv \nu(H)$.
\begin{theorem}\label{three}
Let $|V(F_i)| \geq 3$ for all $i \in [n]$. Then
\begin{itemize}
\item[(i)] $\gamma(G[\Phi]) \equiv \gamma_r(G[\Phi]) \equiv \gamma^{oc}(G[\Phi])$, and
\item[(ii)] $\gamma_t(G[\Phi]) \equiv \gamma_{tr}(G[\Phi]) \equiv \gamma_t^{oc}(G[\Phi])$.
\end{itemize}
\end{theorem}
\begin{proof}
By Lemma \ref{lexlemaanew}(i) we know that for any $\mu$-set $D$, $\mu \in \{\gamma, \gamma_t\}$,
of $G[\Phi]$ is fulfilled $|D \cap V(F_i)| \leq 2$ for all $i=1,2,..,n$.
Since $n \geq 2$ and $|V(F_i)| \geq 3$ for all $i \in [n]$, $D$ is both restrained and outer-connected.
The rest immediately follows by the previous theorem.
\end{proof}
\begin{theorem}\label{gamma=2}
If $\gamma(F_i) \geq 2$ for all $i \in [n]$, then
\[
\gamma(G[\Phi]) = \gamma_r(G[\Phi]) = \gamma_t(G[\Phi]) = \gamma_{tr}(G[\Phi]) =
\gamma^{oc}(G[\Phi]) = \gamma_t^{oc}(G[\Phi]).
\]
If $\gamma(F_i) \geq 3$ for all $i \in [n]$, then
\[
\gamma(G[\Phi]) \equiv \gamma_r(G[\Phi]) \equiv \gamma_t(G[\Phi]) \equiv \gamma_{tr}(G[\Phi])
\equiv \gamma^{oc}(G[\Phi]) \equiv \gamma_t^{oc}(G[\Phi]).
\]
\end{theorem}
\begin{proof}
The first equality chain immediately follows by Lemma \ref{lexlemaanew}(iv).
Assume now $\gamma(F_i) \geq 3$ for all $i\in [n]$ and let $D$ be any $\gamma$-set of $G[\Phi]$.
Lemma \ref{lexlemaanew}(i) implies that $D$ must be total dominating.
The required now follows by Theorem \ref{three}.
\end{proof}
A dominating set $D$ of a graph $G$ is {\em efficient dominating} if
every vertex in $V(G)$ is dominated by exactly one vertex of $D$.
Note that each efficient dominating set is a $\gamma$-set.
\begin{theorem}\label{eff}
Let $D$ be a $\gamma$-set of $G[\Phi]$, $|V(F_i)| \geq 2$ and $|D\cap V(F_i)| \not= 1$ for all $i \in [n]$.
Let all $F_i$'s for which $|D\cap V(F_i)| \not= 0$ be $F_{i_1},F_{i_2},..,F_{i_s}$.
Then $|D \cap V(F_{i_r})| = 2$ for all $r \in [s]$, $\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},.., \textbf{i}_\textbf{s}\}$
is an efficient dominating set of $G$ and
\begin{align}
2\gamma(G)
& = \gamma_t(G) = \gamma(G[\Phi]) = \gamma_r(G[\Phi]) = \gamma_t(G[\Phi]) = \gamma_{tr}(G[\Phi]) = \gamma^{oc}(G[\Phi]) \nonumber \\
& = \gamma_t^{oc}(G[\Phi]) = \gamma_p(G[\Phi]). \nonumber
\end{align}
\end{theorem}
\begin{proof}
By Lemma \ref{lexlemaanew}(i) the following hold:
(a) $|D \cap V(F_i)| \in \{0, 2\}$ for all $i \in [n]$,
(b) $|D\cap V(F_{i})| = 2$ if and only if $i \in \{i_1,i_2,..,i_s\}$,
(c) $\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\}$ is an efficient dominating set of $G$ and
(d) $D\cap V(F_{i_r})$ is a $\gamma$-set of $F_{i_r}$ for all $r \in [s]$.
Therefore $2\gamma(G) = \gamma(G[\Phi])$
and by $\gamma_t(G[\Phi]) = \gamma_t(G) \leq 2\gamma(G) = \gamma(G[\Phi]) \leq \gamma_t(G[\Phi])$,
we have $\gamma_t(G[\Phi]) = \gamma(G[\Phi])$.
In view of Theorem \ref{two}, it remains to prove that $\gamma(G[\Phi]) = \gamma_p(G[\Phi])$.
Let $D \cap V(F_{i_r}) = \{z_{r1}, z_{r2}\}$ and $x_r \in N(z_{r2}) -V(F_{i_r})$, $r \in [s]$.
Since $\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\}$ is an efficient dominating set of $G$,
the set $D^* =( D -\cup_{r \in [s]}\{z_{r2}\}) \cup \cup_{r\in [s]}\{x_r\}$
is a dominating $(\mathcal{M}, \mathcal{C})$-set of $G[\Phi]$ of cardinality $|D^*| = |D| = \gamma(G[\Phi])$.
Since each dominating $(\mathcal{M}, \mathcal{C})$-set is a dominating $(\mathcal{M}, \mathcal{I})$-set,
$\gamma_p(G[\Phi]) = \gamma(G[\Phi])$.
\end{proof}
\begin{remark}\label{roc}
In the end of the proof of the above theorem we obtain that a set $D^*$
is a dominating $(\mathcal{M}, \mathcal{C})$-set of $G[\Phi]$.
Hence under the assumptions of Theorem \ref{eff},
$2\gamma(G) = \gamma_{(\mathcal{M},\mathcal{C})}(G[\Phi])$.
It is quite natural to call the numbers $\gamma_{(\mathcal{M},\mathcal{C})}(G)$ and
$\Gamma_{(\mathcal{M},\mathcal{C})}(G)$ the paired outer connected and
upper paired outer connected domination numbers of a graph $G$.
\end{remark}
\section{Well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated graphs}
We begin with an obvious but very useful observation.
\begin{observation}\label{eqchains}
Given a graph $G[\Phi]$ and properties $\mathcal{A}, \mathcal{B} \subseteq \mathcal{I}$.
Assume that $F_l$ has a dominating $(\mathcal{A}, \mathcal{B})$-set for all $l \in [n]$,
and denote by $\mathscr{D}_{G[\Phi]}(\mathcal{A}, \mathcal{B} )$ the family of all subsets $U$ of $V(G[\Phi])$
such that $U = \cup_{r=1}^s D_{l_r}$, where
$\{\textbf{l}_\textbf{1}, \textbf{l}_\textbf{2},..,\textbf{l}_\textbf{s}\}$
is a maximal independent set of $G$ and $D_{l_i}$ is a minimal dominating $(\mathcal{A}, \mathcal{B})$-set
of $F_{l_i}$, $i=1,2,..,s$.
Let all elements of $\mathscr{D}_{G[\Phi]}$ be minimal dominating $(\mathcal{A}, \mathcal{B})$-sets of $G[\Phi]$.
Then
\begin{align}
\label{eq:n1}
\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi])
& \leq \min\{|U| \mid U \in \mathscr{D}_{G[\Phi]} (\mathcal{A}, \mathcal{B})\} \nonumber \\
& = \min\{ \Sigma_{r=1}^s\gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\}\nonumber \\
&\leq \min\{\Sigma_{r=1}^k\gamma_{(\mathcal{A}, \mathcal{B})} (F_{j_r}) \mid
\{\textbf{j}_\textbf{1}, \textbf{j}_\textbf{2},..,\textbf{j}_\textbf{k}\} \mbox{\ is an $i$-set of $G$}\} \\
& \leq i(G)\max\{\gamma_{(\mathcal{A}, \mathcal{B})}(F_j) \mid j \in [n]\} \nonumber
\end{align}
and
\begin{align}
\label{eq:n2}
\Gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi])
& \geq \max\{|U| \mid U \in \mathscr{D}_{G[\Phi]}(\mathcal{A}, \mathcal{B})\} \nonumber \\
& = \max\{ \Sigma_{r=1}^s\Gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\}\nonumber \\
&\geq \max\{\Sigma_{r=1}^s\Gamma_{(\mathcal{A}, \mathcal{B})} (F_{i_r}) \mid
\{\textbf{j}_\textbf{1}, \textbf{j}_\textbf{2},..,\textbf{j}_\textbf{k}\} \mbox{\ is a $\beta_0$-set of $G$}\} \\
& \geq \beta_0(G)\min\{\Gamma_{(\mathcal{A}, \mathcal{B})}(F_j) \mid j \in [n]\}, \nonumber
\end{align}
where $In(G)$ is the set of all maximal independent sets of $G$.
\end{observation}
As a first consequence of this observation
a necessary condition for a generalized lexicographic product of graphs to be
well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated follows.
\begin{corollary}\label{necessary}
Under the conditions and notation of Observation \ref{eqchains},
if $G[\Phi]$ is well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated, then
all $F_i$'s are well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated and
$\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]) = \Sigma_{r=1}^s\gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r})$
for each $\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)$.
\end{corollary}
\begin{proof}
By (\ref{eq:n1}) and (\ref{eq:n2}) we have
$\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]) \leq \min\{ \Sigma_{r=1}^s\gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\}$ and
$\max\{ \Sigma_{r=1}^s\Gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\}
\leq \Gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi])$, respectively.
Since $G[\Phi]$ is well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated,
$\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]) = \min\{ \Sigma_{r=1}^s\gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\} =
\max\{ \Sigma_{r=1}^s\Gamma_{(\mathcal{A}, \mathcal{B})}(F_{i_r}) \mid
\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\} \in In(G)\} =
\Gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi])$.
This equality chain immediately implies the required.
\end{proof}
\begin{corollary}\label{corchain}
Under the conditions and notation of Observation \ref{eqchains}, assume that
$\gamma_{(\mathcal{A}, \mathcal{B})}(F_1) = \gamma_{(\mathcal{A}, \mathcal{B})}(F_2) =..=\gamma_{(\mathcal{A}, \mathcal{B})}(F_n)$
and $\Gamma_{(\mathcal{A}, \mathcal{B})}(F_1) = \Gamma_{(\mathcal{A}, \mathcal{B})}(F_2) =..=\Gamma_{(\mathcal{A}, \mathcal{B})}(F_n)$.
Then
\begin{equation}\label{eq:n3}
\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]) \leq i(G)\gamma_{(\mathcal{A}, \mathcal{B})}(F_1)
\leq \beta_0(G)\Gamma_{(\mathcal{A}, \mathcal{B})}(F_1) \leq \Gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]).
\end{equation}
If $G[\Phi]$ is well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated,
then $G$ is well covered.
\end{corollary}
\begin{proof}
The middle inequality is a consequence of $i(G) \leq \beta_0(G)$ and
$\gamma_{(\mathcal{A}, \mathcal{B})}(G) \leq \Gamma_{(\mathcal{A}, \mathcal{B})}(G)$.
The left and right inequalities follow immediately by (\ref{eq:n1}) and (\ref{eq:n2}), respectively.
If $G[\Phi]$ is well $\gamma_{(\mathcal{A}, \mathcal{B})}$-dominated,
then the inequality chain (\ref{eq:n3}) becomes equality chain
implying $i(G) = \beta_0(G)$.
\end{proof}
To formulate our next result, we need the following domination parameters.
\begin{itemize}
\item [\rm (h)] $\gamma_{(\mathcal{F}, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{F}, \mathcal{I})} (G)$ are
the acyclic domination and upper acyclic domination numbers $\gamma_a(G)$ and $\Gamma_a(G)$ (\cite{hhr}),
\item [\rm (i)] $\gamma_{(\mathcal{S}_k, \mathcal{I})} (G)$ and $\Gamma_{(\mathcal{S}_k, \mathcal{I})} (G)$ are
the $k$-dependent domination and upper $k$-dependent domination numbers $\gamma^k(G)$ and $\Gamma^k(G)$ (\cite{fhhr}).
\end{itemize}
\begin{remark}\label{appl}
Let the pair $(\mathcal{A}, \mathcal{B})$ be one of
$(\mathcal{I}, \mathcal{I})$, $(\mathcal{F}, \mathcal{I})$, $(\mathcal{S}_k, \mathcal{I})$, $(\mathcal{I}, \mathcal{T})$,
$(\mathcal{T}, \mathcal{I})$, $(\mathcal{T}, \mathcal{T})$ and $(\mathcal{M}, \mathcal{I})$.
In addition if $(\mathcal{A}, \mathcal{B})$ is one of the last four pairs, let $\delta(F_i) \geq 1$ for all $i \in [n]$.
Then the assumptions of Observation \ref{eqchains} are fulfilled.
Therefore the inequality chains (\ref{eq:n1}) and (\ref{eq:n2})
as well as Corollary \ref{necessary} and Corollary \ref{corchain} are valid
for such a pair $(\mathcal{A}, \mathcal{B})$.
\end{remark}
The next result shows that the left and right inequalities in the chain (\ref{eq:n3}) become
equalities in the case when $(\mathcal{A}, \mathcal{B}) = (\mathcal{S}_0, \mathcal{I})$
or equivalently, when
$\gamma_{(\mathcal{A}, \mathcal{B})} = i$
and $\Gamma_{(\mathcal{A}, \mathcal{B})} = \beta_0$.
\begin{theorem}\label{ind} (\cite{nr} when $G[\Phi]=G[F]$)
If $i(F_1) = i(F_2) =..= i(F_n)$, then $i(G[\Phi]) = i(G)i(F_1)$.
If $\beta_0(F_1) = \beta_0(F_2) =..= \beta_0(F_n)$, then $\beta_0(G[\Phi]) = \beta_0(G)\beta_0(F_1)$.
\end{theorem}
\begin{proof} By Remark \ref{appl} and Corollary \ref{corchain} we know that
$i(G[\Phi]) \leq i(G)i(F_1)$ and $\beta_0(G[\Phi]) \geq \beta_0(G)\beta_0(F_1)$.
Let now $I$ be any maximal independent set of $G[\Phi]$ and let
$F_{i_1}, F_{i_2},..,F_{i_s}$ be all $F_i$'s each of which has a vertex in $I$.
Choose $u_{i_r} \in V(F_{i_r})$ arbitrarily, $r=1,2,..,s$, and consider any $G$-layer $H$
containing all vertices of $U = \{u_{i_1}, u_{i_2},..,u_{i_s}\}$.
Clearly $U$ is a maximal independent set of $H \simeq G$;
hence $\beta_0(G) \geq |U| \geq i(G)$.
It remains to note that obviously $I \cap V(F_{i_r})$ is maximal independent set of $F_{i_r}$, $r \in [s]$.
Therefore $i(G[\Phi]) \geq i(G)i(F_1)$ and $\beta_0(G[\Phi]) \leq \beta_0(G)\beta_0(F_1)$.
\end{proof}
The following characterization of well covered generalized lexicographic product of graphs
is due to Topp and Volkmann \cite{tv2}.
\begin{them}\label{wellcovered}\cite{tv2}
The generalized lexicographic product $G[\Phi]$
is a well covered graph if and only if all $F_i$'s are well covered and
$\Sigma_{r=1}^s\beta_0(F_{i_r}) = \Sigma_{p=1}^l\beta_0(F_{j_p})$
for every two maximal independent sets
$\{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\}$
and
$\{\textbf{j}_\textbf{1}, \textbf{j}_\textbf{2},..,\textbf{j}_\textbf{l}\}$
of $G$.
\end{them}
Next we present a characterization of well $\gamma$-dominated generalized lexicographic product of graphs.
For a graph $G[\Phi]$ and any minimal dominating set $R$ of $G$
let $I_R = \{\mathbf{i} \mid deg_{\left\langle R \right\rangle}(\mathbf{i}) = 0 \ \mbox{and} \ \gamma(F_i)\geq 2\}$.
\begin{theorem}\label{neces}
Let $G[\Phi]$ be such that $|V(F_i)| \geq 2$ for all $i \in [n]$.
Then $G[\Phi]$ is a well $\gamma$-dominated graph if and only if
the following assertions hold.
\begin{itemize}
\item[(i)] $F_i$ is well $\gamma$-dominated with $\gamma(F_i) \leq 2$ for all $i \in [n]$.
\item[(ii)] there is a number $k$ such that for each minimal dominating set $R$ of $G$, $|R| + |I_R| = k$.
\end{itemize}
\end{theorem}
\begin{proof}
$\Rightarrow$
Let $D_j$ be a $\Gamma$-set of $F_j$ and $D_j^\prime$ a $\gamma$-set of $G- N[V(F_j)]$ for some $j \in [n]$.
Then $D_j \cup D_j^\prime$ is a minimal dominating set of $G[\Phi]$.
Since $G[\Phi]$ is well dominated, $D_j \cup D_j^\prime$ is a $\gamma$-set of $G[\Phi]$.
Now using Lemma \ref{lexlemaanew}(i), we obtain that $|D_j| \leq 2$ and $D_j$ is a $\gamma$-set of $F_j$.
Thus, (i) is satisfied.
Let $R= \{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\}$ be
an arbitrary minimal dominating set of $G$ and let
$U = \{u_1, u_2,.., u_s\}$ be a $G$-layer of $G[\Phi]$ such that
$u_i$ belongs to some minimal dominating set of $F_i$, $i=1,2,..,n$.
Since $\left\langle U \right\rangle \simeq G$,
$R_1 = \{u_{i_1}, u_{i_2},..,u_{i_s}\}$ is a minimal dominating set of $U$.
If $R_1$ is a dominating set of $G[\Phi]$, then clearly $R_1$ is a minimal dominating set of $G[\Phi]$;
hence $I_R= \emptyset$ and $|R| + |I_R| = |R_1|$. Since $G[\Phi]$ is well $\gamma$-dominated,
$|R| + |I_R| = \gamma(G[\Phi]) = \Gamma(G([\Phi])$.
Now let $I_R = \{\textbf{j}_\textbf{1}, \textbf{j}_\textbf{2},..,\textbf{j}_\textbf{l}\}$.
Since $\gamma(F_{j_r}) = \Gamma(F_{j_r}) = 2$, for all $r \in [l]$ (by (i)),
there is $v_{j_r} \in V(F_{j_r})$ such that $\{u_{j_r} , v_{j_r} \}$ is a
$\Gamma$-set of $F_{j_r}$, $r=1,2,..,l$. But then
$R_1 \cup \{v_{j_1}, v_{j_2},..,v_{j_l}\}$ is a minimal dominating set of $G[\Phi]$
and as $G[\Phi]$ is $\gamma$-well dominated, $\gamma(G([\Phi]) = \Gamma(G([\Phi]) = |R_1| + l = |R| + |I_R|$.
$\Leftarrow$
Let $D$ be an arbitrary minimal dominating set of $G[\Phi]$ and $F_{i_1},F_{i_2},..,F_{i_s}$ be all $F_i$'s
each of which has an element in common with $D$. Clearly
$R = \{\textbf{i}_\textbf{1}, \textbf{i}_\textbf{2},..,\textbf{i}_\textbf{s}\}$
is a minimal dominating set of $G$.
Assume $D$ and $F_{i_r}$ have more than one element in common.
By (i), there are exactly two vertices belonging to both $D$ and $F_{i_r}$.
But then for each $\textbf{j} \in N(\textbf{i}_\textbf{r})$, $D \cap F_j$ is empty.
Therefore $\textbf{i}_\textbf{r} \in I_R$, which implies $|D| = |R| + |I_R|$.
Now by (ii), $|D| = k$ and since $D$ was chossen arbitrarily, $G[\Phi]$ is well $\gamma$-dominated.
\end{proof}
By the proof of the above theorem we obtain the next result.
\begin{corollary}\label{gPhi}
If $G[\Phi]$ is well dominated and $|V(F_i)| \geq 2$ for all $i \in [n]$, then
for each minimal dominating set $R$ of $G$, $|R| + |I_R| = \gamma(G[\Phi])$.
\end{corollary}
\begin{theorem}\label{wd2} (\cite{GHM1} when $G[\Phi]=G[F]$)
Let $G[\Phi]$ be such that $|V(F_i)| \geq 2$ for all $i \in [n]$,
$\gamma(F_1) = \gamma(F_2) =..=\gamma(F_n)$ and $\Gamma(F_1) = \Gamma(F_2) =..=\Gamma(F_n)$.
Then $G[\Phi]$ is well $\gamma$-dominated if and only if one of the following conditions holds:
\begin{itemize}
\item[(i)] $G$ is well-dominated and all $F_i$'s are complete, or
\item[(ii)] $G$ is complete and $F_i$ is well $\gamma$-dominated with $\gamma(F_i) = 2$ for all $i \in [n]$.
\end{itemize}
\end{theorem}
\begin{proof}
$\Rightarrow$
Assume first that $G[\Phi]$ is well $\gamma$-dominated.
Using Remark \ref{appl} by Corollary \ref{necessary} and Corollary \ref{corchain} we have
$i(G)= \beta_0(G)$ and $\Gamma(F_1) = \gamma(F_1)$.
Let $I = \{\textbf{l}_\textbf{1}, \textbf{l}_\textbf{2},..,\textbf{l}_\textbf{s}\}$
be an arbitrary $i$-set of $G$ and $D_j$ an arbitrary $\gamma$-set of $F_j$, $j=1,2,..,n$.
Then clearly $D = \cup_{r=1}^s D_{l_r}$ is a $\gamma$-set of $G[\Phi]$.
Now by Lemma \ref{lexlemaanew}(i) it follows that $\gamma(F_1) \leq 2$.
If $\gamma(F_1) = 2$, then by Lemma \ref{lexlemaanew}(i) it follows that
each $i$-set of $G$ is efficient dominating.
This fact allow us to conclude that a graph $G$ is complete.
So, let $\gamma(F_1) = 1$. We already know that $\Gamma(F_1) = \gamma(F_1)$.
Hence all $\left\langle F_i \right\rangle$'s are complete.
Let $U$ be a $G$-layer of $G[\Phi]$, $R_1$ a $\gamma$-set of $U$ and
$R_2$ a $\Gamma$-set of $U$. Since all $\left\langle F_i \right\rangle$'s are complete,
both $R_1$ and $R_2$ are minimal dominating sets of $G[\Phi]$ and since
$G[\Phi]$ is well dominated,
$R_1$ and $R_2$ have the same cardinality. Thus, $U$ is well $\gamma$-dominated.
It remains to note that $G \simeq U$.
$\Leftarrow$ If (ii) is valid, then obviously $\gamma(G[\Phi]) = \Gamma(G[\Phi]) = 2$.
So, suppose (i) is true and let $T_1$ and $T_2$ be different minimal dominating sets
of $G[\Phi]$. Since all $\left\langle F_i \right\rangle$'s are complete,
there are two $G$-layers, say $U_1$ and $U_2$, which contain $T_1$ and $T_2$,
respectively.
Clearly $T_i$ is a minimal dominating set of $\left\langle U_i \right\rangle \simeq G$, $i=1,2$.
Since $G$ is well covered, $|T_1| = |T_2|$ and we are done.
\end{proof}
A characterization of well $\gamma_t$-dominated generalized lexicographic product of graphs follows.
\begin{theorem}\label{welltotal}
Given a graph $G[\Phi]$ with $\delta(F_i) \geq 1$ for all $i \in [n]$.
Then $G[\Phi]$ is a well $\gamma_t$-dominated graph if and only if
$G$ is complete and for all $i \in [n]$, $F_i$ is well $\gamma_t$-dominated with $\gamma_t(F_i)=2$.
Moreover, if $G[\Phi]$ is a well $\gamma_t$-dominated, then $\gamma_t(G[\Phi])=2$.
\end{theorem}
\begin{proof}
$\Rightarrow$
Let $I = \{\textbf{l}_\textbf{1}, \textbf{l}_\textbf{2},..,\textbf{l}_\textbf{s}\}$
be an arbitrary maximal independent set of $G$ and $D_j$ an arbitrary $\Gamma_t$-set of $F_j$, $j=1,2,..,n$.
Then clearly $D = \cup_{r=1}^s D_{l_r}$ is a $\gamma_t$-set of $G[\Phi]$.
Lemma \ref{lexlemaanew}(i) now implies that $I$ is an efficient dominating set of $G$ and
$D_{l_r}$ is a $\gamma_t$-set of $F_{l_r}$ with $\gamma_t(F_{l_r}) =2$ for all $r \in [s]$.
Since $I$ was chosen arbitrarily and each vertex of $G$ belongs to some maximal independent set of $G$,
we can conclude that (a) all $F_i$'s are well $\gamma_t$-dominated graphs with $\gamma_t(F_i) =2$, and
(b) all maximal independent sets of $G$ are efficient dominating.
The latter means that a graph $G$ is complete.
Finally, by Theorem \ref{t=t=}, $\gamma_t(G[\Phi]) = \gamma_t(G) = 2$.
$\Leftarrow$ Obviously each minimal total dominating set of $G[\Phi]$ has cardinality $2$.
\end{proof}
Now we need the following obvious but useful observation.
\begin{observation}\label{delta=0}
Given a graph $G[\Phi]$ with $\delta(F_i) = 0$ and $|V(F_i)| \geq 2$ for all $i \in [n]$.
Then a set $T$ is a minimal total dominating set of $G[\Phi]$
if and only if $T$ is a minimal total dominating set of some $G$-layer of $G[\Phi]$.
In particular, $\Gamma_t(G) = \Gamma_t(G[\Phi])$.
\end{observation}
\begin{theorem}\label{welltotal2}
Given a graph $G[\Phi]$ with $\delta(F_i) = 0$ and $|V(F_i)| \geq 2$ for all $i \in [n]$.
Then $G[\Phi]$ is a well $\gamma_t$-dominated graph if and only if
$G$ is well $\gamma_t$-total dominated.
\end{theorem}
\begin{proof}
By Theorem \ref{t=t=} we have $\gamma_t(G[\Phi]) = \gamma_t(G)$, and
by Observation \ref{delta=0} - $\Gamma_t(G) = \Gamma_t(G[\Phi])$.
Therefore $\gamma_t(G[\Phi]) = \Gamma_t(G[\Phi])$ if and only if $\gamma_t(G) = \Gamma_t(G)$.
\end{proof}
In \cite{GHM1} G\"{o}z\"{u}pek, Hujdurovi\'c and Milani\v{c} posed the following problem.
\begin{prob}\label{m} \cite{GHM1}
Characterize the nontrivial lexicographic product graphs that are well $\gamma_t$-dominated.
\end{prob}
The previous two theorems together give us the following characterization result.
\begin{theorem}\label{wtd}
Let $G[F]$ be such that $|V(G)|, |V(F)| \geq 2$ and $G$ connected.
Then $G[F]$ is well-$\gamma_t$-dominated if and only if one of the following conditions holds:
\begin{itemize}
\item[(i)] $G$ is complete and $F$ is well $\gamma_t$-dominated with $\gamma_t(F)=2$.
\item[(ii)] $G$ is well $\gamma_t$-dominated and $\delta(F) = 0$.
\end{itemize}
\end{theorem}
\section{Open problems}
We conclude the paper by listing some interesting problems and directions for further research.
\begin{itemize}
\item[$\bullet$] Find results on well $\mu$-dominated graphs, where $\mu$ is at least one of
$\gamma_r, \gamma^{oc}, \gamma_{tr}, \gamma_t^{oc}, \gamma_a, \gamma_p, \gamma^k, k \geq 1$.
In particular, characterize the generalized lexicographic product graphs that are
well $\mu$-dominated.
\end{itemize}
\begin{itemize}
\item[$\bullet$] Characterize/describe those graphs $G$ having an efficient dominating set
of cardinality $\gamma_t(G)/2$ (see Theorem \ref{eff}). Such graphs are
all circulants $C(4k+2; \{1,2,..,k\} \cup \{n-1,n-2,..,n-k\}$, $k \geq 1$.
\end{itemize}
\begin{itemize}
\item[$\bullet$]
Find results on dominating $(\mathcal{M}, \mathcal{C})$-sets (see Remark \ref{roc}).
\end{itemize}
\begin{itemize}
\item[$\bullet$]
Characterize/describe those generalized lexicographic product of graphs $G[\Phi]$ for which
at least one of the following holds:
$\gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi]) = i(G)\gamma_{(\mathcal{A}, \mathcal{B})}(F_1)$
and
$\beta_0(G)\Gamma_{(\mathcal{A}, \mathcal{B})}(F_1) = \Gamma_{(\mathcal{A}, \mathcal{B})}(G[\Phi])$
(see Corollary \ref{corchain}).
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,414 |
\section{Introduction}
\label{Introduction}
The physical layer is the basis for ensuring the quality of communication services for a wireless communication system. As a critical technology in physical layer for the fifth generation (5G), the massive multiple-input multiple-output (MIMO) will also be integral in the sixth generation (6G) to meet the growing needs of the mobile data traffic. Specifically, the channel state information (CSI) feedback is indispensable in massive MIMO to ensure that the base station (BS) can accurately obtain the channel quality and perform further resource allocation, data transmission, etc.
To fully explore the potential of massive MIMO, accurate CSI feedback can be conducted based on the codebook based mechanism including TypeI and enhanced TypeII (eTypeII), which have been standardized in release 16 (R16) by 3rd Generation Partnership Project (3GPP) \cite{1,2,3} and widely used for CSI feedback in current 5G new radio (NR) system. However, since the performance of such a scheme depends on the codebook, beam vector quantization error by codebook is inevitable, and the complexity of codebook design and the corresponding feedback overhead will increase significantly with the growing number of antennas and subcarriers, which brings considerable challenges to codebook based methods.
The core contributions are summarized as follows.
\begin{itemize}
\item With 2nd Wireless Communication Artificial Intelligence (AI) Competition (WAIC) as the background, a framework of eigenvector-based CSI feedback is introduced, where the system-level channel model for dataset and eigenvector-based system model provide a strong and effective guidance for future system design, performance evaluation and protocol specification in task of deep learning (DL) based CSI feedback.
\item Based on the provided framework, a Transformer backbone for CSI feedback referred to EVCsiNet-T is proposed, which processes all eigenvectors of the CSI as a squence using self-attention mechanism and can potentially be a plausible and convincing baseline for research of CSI feedback in academia and recent study item of 3GPP.
\item Beyond the backbone of EVCsiNet-T, a series of potential enhancements including data augmentation, loss function design, training strategy and model ensemble which can be exploited to further improve the performance of the backbone are introduced.
\item To demonstrate the advanced performance and a promising prospect of Transformer on DL-based CSI feedback, we conduct various numerical experiments involving the comparison between the EVCsiNet-T and traditional codebook methods over different channels.
\end{itemize}
The structure of this paper is as follows, the background including the relevant works and the 2nd WAIC with the task of DL-based eigenvector-based CSI feedback are introduced in Section \ref{Background}. The framework system model and channel model involved in the 2nd WAIC are depicted in Section \ref{System Model and Channel Model}. A basic Transformer backbone for CSI feedback named EVCsiNet-T is proposed in Section \ref{Basic Transformer Solution for CSI Feedback}. Moreover, a series of potential enhancements for DL-based CSI feedback is introduced in Section \ref{Enhancing Schemes}. Finally, experimental results and a brief conclusion are provided in Section \ref{Experiments} and \ref{CONCLUSION}, respectively.
\section{Background}
\label{Background}
\subsection{Relevant Works}
\label{Relevant Methods}
Since the DL has gained great successes in the fields of computer vision (CV) and natural language processing (NLP), the combination of wireless communication and DL has also attracted great attention in recent years \cite{Xiao2021AIEW, xiao2022channelgan, liu2021evcsinet}. The study item (SI) named `study on artificial intelligence (AI)/machine learning (ML) for NR air interface` has been established in 3GPP release 18 (R18) in which DL-based CSI feedback is regarded as an important use case \cite{213599, 212927, 210235, 210236}. In academia, DL-based CSI feedback has been widely studied since it can simultaneously achieve higher compression and recovery accuracy and lower feedback overhead \cite{wang2017deep,wen2018deep,sun2020ancinet,lu2020multi,chen2020deep,mashhadi2020distributed,cao2021lightweight,guo2020deep,guo2021canet,lu2018mimo,wang2018deep,guo2020convolutional,li2020spatio,chen2019novel,lu2019bit}.
CsiNet \cite{wen2018deep} is firstly proposed whose encoder and decoder are constructed by convolutional neural network (CNN) to conduct CSI compression and recovery at the user equipment (UE) and BS, respectively. Furthermore, a series of follow-up studies designed based on various kinds of CNNs \cite{sun2020ancinet,lu2020multi,chen2020deep,mashhadi2020distributed,cao2021lightweight,guo2020deep,guo2021canet} and long short-term memory (LSTM) \cite{lu2018mimo} have been proposed to further improve the CSI feedback performance. It can be noticed that most studies originated from CsiNet mainly focus on the full channel state information (F-CSI) feedback, where the whole downlink channel matrix is compressed and recovered at encoder and decoder, respectively. However, the solution for CSI feedback discussed in 3GPP is based on the compression and feedback of the eigenvector of the channel matrix, which is the main application in current 5G system. Therefore, besides the DL-based F-CSI feedback, it is necessary to study the eigenvector-based CSI feedback using DL methods, which can achieve direct fair performance comparison with the existing codebook based solutions. Different from the DL-based F-CSI feedback, EVCsiNet \cite{liu2021evcsinet} is introduced which concentrates on eigenvector-based CSI feedback. However, the current research on eigenvector-based CSI feedback is still relatively preliminary calling for more exploration, which thus is the main focus of this paper.
\subsection{Wireless Communication AI Competition}
\label{WAIC2nd}
In order to further explore the practical application of AI in wireless communication systems, IMT-2020(5G) Promotion Group 5G+AI Work Group held the 2nd WAIC in July 2021 with a topic of AI Enlightens Wireless Communication which is still committed to promoting the deep integration and mutual promotion of the wireless communication and AI. The 2nd WAIC focuses on the task of Performance Improvement of DL-based CSI feedback and DL-based channel estimation. As one of the tasks, DL-based CSI feedback aims to evaluate the performance of DL-based CSI recovery and feedback overhead reduction with classic data-set on 3GPP system-level channel model. Since the high degree of integration between the task and the industry, there are more than $300$ teams involving the contestants from related companies, universities and research institutes participate in the competition. Focusing on the schedule of the competition, the two-month online results submission for participants and the one-day offline seminar for interested researchers provide a good opportunity for technical exploring and sharing.
As for the task of DL-based CSI feedback in 2nd WAIC, the participants achieve excellent results and there are six over top ten teams utilize the Transformer \cite{Vaswani2017AttentionIA} architecture as the backbone for model design. The Transformer architecture was firstly proposed to solve problems in the field of natural language processing (NLP) and computer vision (CV) \cite{Dosovitskiy2021AnII}, and the successful utilization in 2nd WAIC shows great potential of Transformer in the DL-based CSI feedback task.
In the following, based on the background of 2nd WAIC, the details of the i) dataset construction method, ii) the Transformer solution for CSI feedback and iii) the noteworthy enhancing schemes for CSI feedback are introduced. The corresponding experimental results are also provided. With the success of the competition, it becomes an unprecedented way to promote the integration of 5G and AI by involving academic and industrial research and development to solve typical wireless problems with AI tools.
\begin{table}[tbp]\small
\centering
\caption{Dataset parameter settings for 2nd WAIC.}
\label{tabChannelSeting}
\begin{tabular}{cc}
\hline
Parameter & Value \\
\hline
Channel model & UMa \& NLoS\\
\hline
Carrier frequency $F_{\rm c}$ & 3.5GHz\\
\hline
Subcarrier spacing $B_{\rm sc}$ & 15KHz\\
\hline
Number of resource blocks $N_{\rm RB}$ & 48\\
\hline
Number of subbands $N_{\rm sb}$ & 12\\
\hline
Number of Tx antennas $N_{\rm t}$ & 32 \\
\hline
Number of Rx antennas $N_{\rm r}$ & 4\\
\hline
UE speed & 3km/h\\
\hline
Number of cells $N_{\rm c}$ & 57\\
\hline
Number of UEs in training set $N_{\rm train}$ & 3000 \\
\hline
Number of UEs in testing set $N_{\rm test}$ & 400 \\
\hline
Number of sampling slots $N_{\rm slot}$ & 200 \\
\hline
Number of interval slots $T$ & 50\\
\hline
\end{tabular}
\end{table}
\section{Framework of Eigenvector-Based CSI Feedback}
\label{System Model and Channel Model}
\subsection{Channel Model}
\label{Channel Model}
3GPP technical report (TR) 38.901 \cite{4} entitled `Study on channel model for frequencies from 0.5 to 100 GHz` helps to properly model and evaluate the performance of physical layer techniques using the appropriate channel model. Since the channel models in TR 38.901 are recognized and typical, the study of CSI feedback based on these channel models is appropriate and persuasive. As a benchmark channel for simulation and performance evaluation in 3GPP, the system level three-dimensional channel model is adopted in this paper to reflect the channel characteristics in realistic environments. Specifically, four types of channel models including urban-macro (UMa), urban-micro indoor-hotspot, and rural-macro are defined in TR 38.901, and we selectively consider the UMa model with pure non-line-of-sight (NLoS). The system-level time domain downlink channel matrix $\mathbf{H}_{\rm{t}}$ can be written as
\begin{equation}
\mathbf{H}_{\rm{t}} = \sum_{d=1}^{N_{\rm d}} \widetilde{\mathbf{H}}_d = \sum_{d=1}^{N_{\rm d}}\sum_{l=1}^{L_d} \widetilde{\mathbf{H}}_{d,l}
\label{eqHsys}
\text{,}
\end{equation}
where $\widetilde{\mathbf{H}}_{d,l}$ and $L_d$ denote the channel of $l$-th sub-path and the total number of sub-paths in the $d$-th cluster, respectively. Note that the channel defined in TR 38.901 is considered, where each sub-path $\widetilde{\mathbf{H}}_{d,l}$ is generated with different angle-of-departures, angle-of-arrivals, angle-spreads in both azimuth and zenith domains, power and delay distributions, and initial phases, etc, therefore the small-scale fading of frequency selective also occurs. Moreover, due to the frequency selection characteristics of the channel, CSI of different subcarriers are different, which introduces the subband-based CSI feedback. Moreover, $N_{\rm slot}\times N_{\rm UE}$ channel samples are provided in related dataset, where $N_{\rm slot}$ slots are uniformly sampled with $T$ continuous interval slots for each UE, $N_{\rm UE} = N_{\rm train} + N_{\rm test}$ denotes the number of randomly distributed UEs in $N_{\rm c}$ cells, $N_{\rm train}$ and $N_{\rm test}$ represent the numbers of UEs in training and testing set, respectively. The corresponding channel and dataset parameter settings in 2nd WAIC are shown in Table \ref{tabChannelSeting}. Note that in order to balance the difficulty of the competition for the contestants and make the score distribution of the leaderboard reasonable, frequency selective fading of the channel is further reduced. The dataset (Data-WAIC2nd.zip) and an example code (reference model\_WAIC2nd\_EVCsiNet-T.zip) involved in 2nd WAIC are released in the page of 'News' of Wireless-Intelligence \footnotemark[1]\footnotetext[1]{https://wireless-intelligence.com/} which is a channel database website provided for AI-based wireless communication research. Since the dataset are constructed and modified based on 3GPP specification, the results on this channel dataset have strong guiding significance for further research on the future communication system design and protocol specification.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.55]{gray-eps-converted-to.pdf}
\caption{Illustration of gray scale of energy for one sample.}
\label{gray}
\end{figure}
\subsection{System Model}
We consider a typical MIMO system with $N_{\rm t}$ transmitting antennas at BS and $N_{\rm r}$ receiving antennas at UE. Due to the frequency selection characteristics of the channel, CSI of different subcarriers are different, which introduces the subband-based CSI feedback by dividing the full bandwidth into multiple subbands for feedback. By conducting fast Fourier transform (FFT) on the time domain downlink channel matrix $\mathbf{H}_{\rm{t}}$ described in Section \ref{Channel Model}, the downlink channel in the frequency domain can be given as
\begin{equation}\label{Hf}
\mathbf{H}_{\rm{f}}= \big[\mathbf{H}_1,\mathbf{H}_2,\cdots,\mathbf{H}_{N_{\rm{sb}}}\big],
\end{equation}
where $N_{\rm{sb}}$ represents the number of subbands consisting of 4 resource blocks (RBs) as the basic feedback granularity, and $\mathbf{H}_k\in\mathbb{C}^{N_{\rm r}\times N_{\rm t}}, 1\leq k\leq N_{\rm{sb}}$ indicates the downlink channel of the $k$th subband. According to the application of the current 5G system, the solution for CSI feedback is based on the compression and feedback for the eigenvector of the channel matrix. However, most existing academia researches \cite{wang2017deep,wen2018deep,sun2020ancinet,lu2020multi,chen2020deep,mashhadi2020distributed,cao2021lightweight,guo2020deep,guo2021canet,lu2018mimo,wang2018deep,guo2020convolutional,li2020spatio,chen2019novel,lu2019bit} for DL-based CSI feedback mainly concentrate on full CSI compression and recovery, which is incomparable with the current Type I and eType II codebook based mechanisms with eigenvector CSI operation defined by 3GPP. Moreover, since the study item named 'study on artificial intelligence (AI)/machine learning (ML) for NR air interface' has been established in 3GPP R18 in which DL-based CSI feedback is regarded as an important use case \cite{213599, 212927, 210235, 210236}, the study for advanced structure of neural network (NN) and comparison between DL-based methods and codebook based methods are significant and indispensable. These motivate the study of this paper, and the DL-based CSI feedback for eigenvector considered in this paper can also achieve fair comparison with the traditional codebook based mechanism including TypeI and eTypeII in the 5G system. Assuming ideal channel estimation at UE and implementing the single layer downlink transmission, the corresponding eigenvector for the $k$th subband $\mathbf{w}_k\in\mathbb{C}^{N_{\rm t}\times 1}$ with normalization $||\mathbf{w}_k||^2=1$, can be utilized as the downlink precoding vector and calculated using eigenvector decomposition, i.e.,
\begin{equation}
\mathbf{H}_k^{\rm H}\mathbf{H}_k \mathbf{w}_k =\lambda_k \mathbf{w}_k,
\end{equation}
where $\lambda_k$ represents the maximum eigenvalue of $\mathbf{H}_k^{\rm H}\mathbf{H}_k$. As all $N_{\rm{sb}}$ eigenvectors should be feeded back to the BS for downlink precoding, total $N_{\rm{sb}}N_{\rm t}$ complex coefficients for $N_{\rm{sb}}$ subbands should be compressed and recovered using NN. Thus a CSI sample $i$ across total $N_{\rm{sb}}$ subbands can be writen as
\begin{equation}\label{Wi}
\mathbf{W}_i= \big[\mathbf{w}_{1,i},\mathbf{w}_{2,i},\cdots,\mathbf{w}_{N_{\rm{sb}},i}\big] \in \mathbb{C}^{N_{\rm t}\times N_{\rm{sb}}}.
\end{equation}
The gray scale of $\mathbf{W}$ is also described in Fig. \ref{gray}, where x-axis indicates the subband index and y-axis indicates the corresponding Tx antenna index, and deeper color means the smaller energy of the corresponding channel element. For CSI compressing and feedback, a DL-based CSI feedback scheme is implemented as shown in Fig. \ref{basicScheme}, where a CSI sample $\mathbf{W}$ is compressed to a bitstream $\mathbf{s}$ of length $M$ using a DL encoder at UE. Then the CSI $\mathbf{W}$ is recovered from the feedback bitstream $\mathbf{s}$ exploiting the DL decoder at the BS. Moreover, denoting the test set $\mathcal{W}$ consisting of $|\mathcal{W}|$ CSI samples where $|\cdot|$ denotes the cardinality of a set, the average squared generalized cosine similarity (SGCS) is usually utilized to evaluate the CSI compression and recovery accuracy as follows
\begin{figure}[tb]
\centering
\includegraphics[scale=0.8]{basicScheme-eps-converted-to.pdf}
\caption{Illustration of DL-based E-CSI feedback scheme.}
\label{basicScheme}
\end{figure}
\begin{equation}\label{score_function}
\setlength{\abovedisplayskip}{-5pt}
\setlength{\belowdisplayskip}{5pt}
\begin{split}
\rho(\mathcal{W},\mathcal{W}') = \frac{1}{|\mathcal{W}|N_\textrm{sb}}\sum_{i=1}^{|\mathcal{W}|}\sum_{k=1}^{N_\textrm{sb}}\Big(\frac{\|\mathbf{w}_{k,i}^{\rm H}\mathbf{w}'_{k,i}\|_2}{\|\mathbf{w}_{k,i}\|_2\|\mathbf{w}'_{k,i}\|_2}\Big)^2
\end{split}
\text{,}
\end{equation}
where $\rho(\mathcal{W},\mathcal{W}')\in [0,1]$, $\|\cdot\|_2$ denotes the $\ell_{2}$ norm, $\mathcal{W}'$ represents the predicted eigenvector CSI set, $\mathbf{w}_{k,i}$ and $\mathbf{w}'_{k,i}$ represent the $k$th eigenvectors of $i$th sample in $\mathcal{W}$ and $\mathcal{W}'$, respectively. A higher SGCS performance indicates higher CSI compression and recovery accuracy. Based on the basic requirements of wireless communication, the 2nd WAIC focuses on the SGCS performance under different feedback overhead in a complex channel environment. The model design with $M = 48$ and $M = 128$ bits are required which correspond to low feedback overhead scenario and high feedback overhead scenario, respectively. The final score is obtained by averaging the SGCS performance of the two scenarios.
\section{A Transformer Backbone for CSI Feedback}
\label{Basic Transformer Solution for CSI Feedback}
Recently, Transformer \cite{Vaswani2017AttentionIA} based solution shows great potential in CSI feedback, whose natural encoder-decoder structure is quite suitable for respectively implementing at the UE and BS. By implementing Transformer for CSI feedback, a specific domain of the channel such as paths, antenna pairs, or subbands can be treated as the sequence inputs and processed by self-attention block, which is similar to the sequential processing issue in NLP and benefits the extraction of unique characteristics of the wireless channel. Here, a Transformer backbone referred to EVCsiNet-T for solving CSI feedback is introduced, where all subbands are regarded as a sequence. By splitting the real and imaginary parts of a complex CSI sample, the input of the EVCsiNet-T can be written as
\begin{equation}
\widehat{\mathbf{W}} = [{\rm Re}(\mathbf{W})^{\rm T},{\rm Im}(\mathbf{W})^{\rm T}]^{\rm T} \in \mathbb{R}^{2N_{\rm t}\times N_{\rm{sb}}},
\end{equation}
where the subscript $i$ indicating the index of a sample is omitted to simplify the representation, and ${\rm Re}(\cdot)$ and ${\rm Im}(\cdot)$ denote the real and imaginary parts of a matrix, respectively. To implement the CSI feedback, the EVCsiNet-T solves the following problem during the training phase, i.e.,
\begin{equation}
\begin{split}
\mathop{\min}_{\Theta_{\rm E},\Theta_{\rm D}} L(\widehat{\mathbf{W}},\widehat{\mathbf{W}}')
\end{split}
\text{,}
\end{equation}
where $\Theta_{\rm E}$ and $\Theta_{\rm D}$ denote the weights of encoder and decoder of EVCsiNet-T, respectively, and $L(\cdot)$ denotes the loss function such as mean square error (MSE), minus cosine similiarity, etc.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.8]{basicTransformer-eps-converted-to.pdf}
\caption{Illustration of architecture of EVCsiNet-T.}
\label{basicTransformer}
\end{figure}
The architecture of EVCsiNet-T is depicted in Fig. \ref{basicTransformer} in detail which consists of encoder and decoder. Each vector of subband in the input CSI matrix is firstly processed by an embedding layer with embedding dimension of $N_{\rm{e}}$ and positionally encoded by adding a learnable position vector. Then $N_{\rm{b}}$ basic blocks are sequentially introduced to conduct the feature extraction. Moreover, a flatten layer and a dense layer with $M / B$ units are implemented in order to adapt the length of feedback bits, where $B$ is the number of quantization bits and $M$ denotes the total length of feedback bits. Finally, a uniform quantization layer is employed to transform the floating number vector with length of $M / B$ to the bitstream.
As for the decoder, the fed back bitstream is converted to a vector of float numbers using the dequantization layer. Next, a dense layer with $N_{\rm{e}}N_{\rm{sb}}$ units using gaussian error linear units (GELU) activation function is employed, whose output is reshaped to a $N_{\rm{e}} \times N_{\rm{sb}}$ matrix. After that, $N_{\rm{b}}$ basic blocks are sequentially deployed to extract the features, whose output is flattened and processed by a dense layer with $2N_{\rm{t}}N_{\rm{sb}}$ units and activation function of GELU. Finally, the reshape layer is implemented to obtain the output with the shape of the original CSI.
The basic block can be the core component in both encoder and decoder, in which the input is firstly processed by a multi-head attention layer \cite{Vaswani2017AttentionIA} with the number of heads $N_{\rm{head}}$ and then added to the shortcut from the input of the block. A layer normalization and feedforward network are sequentially conducted where the feedforward network is constructed by two dense layers with the number of units $\{ k_{\rm h}N_{\rm{e}}, N_{\rm{e}}\}$ sandwiching a GELU, where $k_{\rm h}$ denotes the scaling factor of hidden layer. The shortcut from the input of feed forward network is also added to the output and a layer normalization is implemented finally.
\section{Potential Enhancements}\label{Enhancing Schemes}
Different from the image data and sequence data respectively in CV and NLP, the channel data obtained from wireless communication system has some unique features, which call for more suitable tactics for performance enhancing. Specifically, based on the backbone introduced in Section \ref{Basic Transformer Solution for CSI Feedback}, there are also a series of enhancing schemes for DL-based CSI feedback during the 2nd WAIC, which can further improve the performance of the model and are introduced in more detail in this section.
\subsection{Data Augmentation}\label{Data Augmentation}
The data augmentation approach is widely utilized to improve the diversity of the training data and alleviate the overfitting problem. There are a number of data augmentation approaches exploited in the competition including noise injection, flipping, shift, and rotation, which are introduced in detail in this subsection.
\subsubsection{Noise Injection}
Noise injection refers to adding noise to the training data which is shown in Fig. \ref{flipAndShift} (a). The noise is typically drawn from the various statistical distributions. In general, the noise is injected with a zero-mean Gaussian distribution, under which condition the injection process can be mathematically written as
\begin{equation}\label{noise_injectioin}
\mathbf{W}_{\rm{aug}} =
\mathbf{W} + \alpha \Upsilon, \quad \eta \sim \mathcal{N}(0,\sigma^2),
\end{equation}
where $\mathbf{W}_{\rm{aug}}$ denotes the augmented CSI tensor, $\mathbf{W}$ denotes the noise-free CSI tensor, $\Upsilon$ represents the noise tensor with the same shape as the CSI tensor $\mathbf{W}$ and with elements sampled from a Gaussian distribution $\mathcal{N}(0,\sigma^2)$, and $\alpha$ is the coefficient that scales the magnitude of the injected noise, respectively.
\subsubsection{Flipping}
The flipping method for CSI considers flip the CSI tensor according to the dimension of subband or antenna pairs. As shown in Fig. \ref{flipAndShift} (b), for a CSI tensor
\begin{equation}
\mathbf{W}= \big[\mathbf{w}_{1},\mathbf{w}_{2},\cdots,\mathbf{w}_{N_{\rm{sb}}}\big] \in \mathbb{C}^{N_{\rm t}\times N_{\rm{sb}}}
\end{equation}
as an example, where $\mathbf{w}_k\in\mathbb{C}^{N_{\rm t}\times 1}$ denotes the corresponding eigenvector for the $k$th subband, the flipping procesure according to the dimension of subband can be written as
\begin{equation}\label{aug_Flipping}
\mathbf{W}^{\rm{aug}} = \big[\mathbf{w}_{N_{\rm{sb}}},\mathbf{w}_{N_{\rm{sb}-1}},\cdots,\mathbf{w}_{1}\big] \in \mathbb{C}^{N_{\rm t}\times N_{\rm{sb}}},
\end{equation}
where $\mathbf{W}_{\rm{aug}}$ denotes the augmented CSI tensor and flipping in antenna paire dimensions can be easily generalized..
\begin{figure}[tb]
\centering
\includegraphics[scale=0.6]{flipAndShift-eps-converted-to.pdf}
\caption{Data augmentation schemes.}
\label{flipAndShift}
\end{figure}
\subsubsection{Shifting}
Shifting method considers to shift along a certain dimension of the tensor according to the step size, which can change the position of the content of the tensor. Fig. \ref{flipAndShift} (c) shows the cyclic shifting of a CSI tensor $\mathbf{W}$ on the dimension of subband. It shifts the last subband to the first position for $p$ times, which can be written as
\begin{equation}\label{aug_circular_shift}
\mathbf{W}^{\mathbf{aug}}= \big[\mathbf{w}_{N_{\rm{sb}}-p+1}, \cdots,\mathbf{w}_{N_{\rm{sb}}}, \mathbf{w}_{1},\cdots,\mathbf{w}_{N_{\rm{sb}}-p}\big],
\end{equation}
where $\mathbf{w}_k\in\mathbb{C}^{N_{\rm t}\times 1}$ denotes the corresponding eigenvector for the $k$th subband and $N_{\rm sb}$ denotes the number of subbands. Different from the cyclic shifting, there can also be another method refered to random shift achieves data augmentation by randomly shuffling the subbands, which is shown in Fig. \ref{flipAndShift} (d). In more detail, cyclic shift can generate at most $N_{\rm sb} - 1$ samples based on one raw data sample while random shift does at most $N_{\rm sb}! - 1$ samples. Therefore random shift can provide more diversity of data augmentation than cyclic shift. However, since the relationship of adjacent subbands does not change when cyclic shifting, thus certain frequency selectta ive fading features are still retained in the generated data.
\subsubsection{Rotation}
Rotation augmentation shown in Fig. \ref{flipAndShift} (e) utilizes sine and cosine functions to randomly rotate the eigenvector by a certain angle, which can be written as
\begin{equation}
\begin{split}
\operatorname{Re}\left\{\mathbf{w}_{\rm aug}\right\} =\cos\left(\theta\right)\operatorname{Re}\left\{\mathbf{w}\right\} -\sin\left(\theta\right)\operatorname{Im}\left\{\mathbf{w}\right\} ,
\\
\operatorname{Im}\left\{\mathbf{w}_{\rm aug}\right\}
=\sin\left(\theta\right)\operatorname{Re}\left\{\mathbf{w}\right\} +\cos\left(\theta\right)\operatorname{Im}\left\{\mathbf{w}\right\} ,
\end{split}
\end{equation}
where $\operatorname{Re}\{\cdot\}$ and $\operatorname{Im}\{\cdot\}$ are the real and imaginary part of the input, $\mathbf{w}$ is one subband eigenvector of the CSI tensor $\mathbf{W}$, $\theta$ is the rotation degree parameter which can be different for defferent eigenvectors, respectively.
\subsection{Loss Function Design}
\label{Loss Function Design}
Loss function design is a significant factor affecting the training. The minus cosine similarity and MSE which are commonly used always fail to strictly satisfy the final goal or unique features of the task, calling for novel method for designing the loss function. Here, we will give an introduction on enhanced loss function design for DL-based CSI feedback.
\subsubsection{Scoring Loss}
Since the final goal of the task is the pursuit of a high score according to (\ref{score_function}), a design method can be directly adapt the scoring indicator function as the loss function, i,e.,
\begin{equation}
L = -\rho,
\end{equation}
where $\rho$ is the scoring function in (\ref{score_function}) to evaluate the accuracy of the CSI compression and recovery.
\subsubsection{Quantization Error Compensation Loss}
The quantization error can be up to half the quantization interval and are always unavoidable. Therefore the quantization error compensation loss $L_{\rm quant}$ can be designed to reduce the error between the vector before quantization $\mathbf{v}$ and the vector after dequantization $\mathbf{v}'$ caused by quantization procedure, i.e.,
\begin{equation}
L_{\rm{quant}} = L_{\rm{base}} (\mathbf{v}, \mathbf{v}'),
\end{equation}
where $L_{\rm{base}}$ is differentiable and can be implemented using MSE, normalized MSE (NMSE) and so on, and $\mathbf{v}$ and $\mathbf{v}'$ are the input and output vectors of the quantization layer, respectively. Note that the quantization layer consisting of quantization and dequantization processes is generally non-differentiable in mathematics, but can still be implemented during training where same gradient is applied before and after the quantization layer. The quantization error compensation loss can be added to the original loss function forming the global loss to enhance the recovery performace, i.e.,
\begin{equation}
L = L_{\rm{original}} +L_{\rm{quant}},
\end{equation}
where $L_{\rm{original}} = L_{\rm{base}}(\widehat{\mathbf{W}},\widehat{\mathbf{W}}')$ reduces the error between the original CSI $\widehat{\mathbf{W}}$ and the recovered CSI $\widehat{\mathbf{W}}'$, $L_{\rm{quant}}$ reduces the quantization error to further improve the CSI recovery accuracy, and $L_{\rm{base}}$ can also be defined as MSE, NMSE, etc.
\subsection{Training Strategy}
\label{Tricks for Training}
\subsubsection{Learning Rate Warm-up and Decay}
Since the weights of the model are randomly initialized, a large learning rate may bring instability and oscillation of the model at the beginning of training. The warmup method can make the learning rate small during the first several training epochs, where the model can gradually become stable. After that, the training can continue under a learning rate set according to certain strategies leading to a better convergence of the model. One of the prevalent learning rate warmup methods is the gradual warmup method, using which the learning rate is updated by
\begin{equation}
\alpha_t = \dfrac{t}{T_{\rm warmup}}\alpha_{\rm max},
\end{equation}
where $t \in [1,T_{\rm warmup}]$ denotes the $t$th epoch, $T_{\rm warmup}$ denotes the number of warm-up epochs, and $\alpha_{\rm max}$ denotes the final warmed up learning rate, respectively.
After warm-up, learning rate decay can be further utilized which starts with the warmed up learning rate $\alpha_{\rm max}$ and then decaying it $T_{\rm decay}$ epochs in subsequent training. This can be beneficial for the optimization and generalization of the model \cite{You2019HowDL}. As an example, the cosine decay strategy can be written as
\begin{equation}
\begin{split}
\alpha_{t}=\alpha_{\min }&+\frac{1}{2}\left(\alpha_{\max }-\alpha_{\min }\right)\\
&\cdot\left(1+\cos \left(\frac{(t-T_{\rm warmup}) \pi}{T_{\rm decay}}\right)\right),
\end{split}
\end{equation}
where $t \in [T_{\rm warmup},T_{\rm decay}]$ denotes the $t$th epoch and $\alpha_{\min}$ represents the minimum values of the learning rate, respectively.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.85]{ProgressiveResizing-eps-converted-to.pdf}
\caption{Staged training for shrinking the number of feedback bits in different stages.}
\label{Progressive_Resizing}
\end{figure}
\subsubsection{Staged Training}
Another training strategy referred to staged training considers dividing the entire training procedure into several stages. where different factors, e.g., loss function, batch size, learning rate, datasets, can be adjusted in different training stages. Hence the design of the staged training is highly personalized. For example, progressive resizing is a powerful staged training method in CV for improving the training efficiency and effect, which gradually expands the size of image in consequent training stages \cite{Liang2021GuidanceNW}. Similar idea can also be applied in DL-based CSI feedback. As shown in Fig. \ref{Progressive_Resizing}, one can train with a large number of feedback bits for certain epochs and then shrinks to smaller feedback bits in the next re-training stage, in which way a more effective and stable convergence can be obtained.
\subsection{Model Ensemble}
\label{Model Ensembling}
\begin{figure}[tb]
\centering
\includegraphics[scale=0.78]{modelEnsembling-eps-converted-to.pdf}
\caption{Illustration of the method for multi-model ensembling.}
\label{modelEnsembling}
\end{figure}
The model ensemble is another optional enhancing method, which can integrate the ability of multiple models and improve the accuracy performance of the ensemble model during training or testing period. For the method of model ensemble during testing period as an example which is shown in Fig. \ref{modelEnsembling}. $V$ encoder-decoder pairs are constructed with different conditions such as model structure, random seed and batch size for training, etc. The $V$ encoder-decoder pairs are jointly implemented at the UE, wherein the corresponding decoders are also adapted at BS. At the UE side, the CSI $\mathbf{W}$ is firstly processed by $V$ encoder-decoder pairs obtaining encoded bitstreams $\{\mathbf{s}'_1, \dots, \mathbf{s}'_V,\}$ and the decoded CSI $\{\mathbf{W}'_1, \dots, \mathbf{W}'_V \}$, respectively. The decoded CSI can be utilized to select the encoder-decoder pair with the best SGCS performance, whose index is encoded by
\begin{equation}
\begin{split}
\mathbf{b} = C(j = \mathop{\arg\max}\limits_{1 \leq i \leq V} \rho(\mathbf{W}, \mathbf{W}_i'))
\end{split}
\text{,}
\end{equation}
where $j$ denotes the index of best encoder-decoder pair, $C(\cdot)$ denotes index encoding function and $b \in \{0,1\}^{\lceil \log_2^{V} \rceil}$ represents the bitstream indicating the index of the best pair, respectively. The joint bitstream for index and CSI, $\mathbf{s} = [\mathbf{b}^{\rm{T}}, \mathbf{s}_j^{\rm{T}}]^{\rm{T}}$ is fed back to the BS side, where the BS selects the decoder according to $\mathbf{b}$ and decodes the CSI by the selected decoder, obtaining the final recovered CSI $\mathbf{W}'$. The above model ensemble method can combine the advantages of multiple models. However, deploying multiple encoder-decoder pairs on the UE brings storage overhead at the UE side and the feedback overhead for encoding index. Moreover, considering a fixed number of total feedback bits $M = \lceil \log_2^{V} \rceil + M_s$, where $\lceil \log_2^{V} \rceil$ and $M_s$ respectively represent the length of bits overhead for encoding index and CSI tensor, the increasing number of the total models $V$ improves the diversity of ability of the ensemble model, but decreases the length of bits for encoding CSI $M_s$ resulting in lower performance of each single model. Thus the number of models $V$ is the hyperparameter that needs to be carefully tuned.
\section{Experiments}
\label{Experiments}
\subsection{SGCS Performance Comparison}
In this section, the experimental results are provided to verify the superiority of the EVCsiNet-T compared with traditional TypeI (with mode 1) and eTypeII (with the number of orthogonal beams $L=2$ and $4$) codebook based methods \cite{2}, where the system parameters are listed in Table \ref{tabChannelSeting}. The i) channel data in 2nd WAIC, and ii) link-level channel data of clustered delay line (CDL) A and C with delay spread of 30 and 300 ns respectively, are considered.
Moreover, the embedding dimension, the number of basic blocks, quantization bits, heads in multi-head attention layer and scaling factor of hidden layer of EVCsiNet-T are set to $N_{\rm e} = 512$, $N_{\rm b = 10}$, $B = 2$, $N_{\rm head} = 16$ and $k_{\rm h} = 2$, respectively. The cosine similarity loss function and the default adaptive momentum (Adam) optimizer with the learning rate of 0.001 are adopted to train the EVCsiNet-T for 300 epochs.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.68]{CDLA-eps-converted-to.pdf}
\caption{SGCS comparison on CDL-A30 for different solutions.}
\label{CDLA}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[scale=0.68]{CDLC-eps-converted-to.pdf}
\caption{SGCS comparison on CDL-C300 for different solutions.}
\label{CDLC}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[scale=0.68]{UMA-eps-converted-to.pdf}
\caption{SGCS comparison on the dataset in 2nd WAIC for different solutions.}
\label{UMA}
\end{figure}
Fig. \ref{CDLA}, \ref{CDLC} and \ref{UMA} show the SGCS performance comparison between EVCsiNet-T and traditional codebook based TypeI and eTypeII methods over different channels including CDL-A with delay spread of 30 ns (CDL-A30), CDL-C with delay spread of 300 ns (CDL-C300), and the channel in 2nd WAIC. Note that to reveal the superiority of the backbone itself the enhancing methods described in Section \ref{Enhancing Schemes} are not implemented in basic EVCsiNet-T in Fig. \ref{CDLA}, \ref{CDLC} and \ref{UMA}. It can be noticed that under low feedback overhead condition the EVCsiNet-T with $M=32$ and $M=48$ outperform TypeI with $M=32$ and eTypeII ($L=2$) with $M=49$. As for the configuration of high feedback overhead, the EVCsiNet-T with $M=120$ also performs higher SGCS than eTypeII ($L=4$) with $M=128$. Moreover, taking CDL-C300 as an example, the EVCsiNet-T with $M=32$ achieves $\rho = 0.903$ which even outperforms the $\rho = 0.868$ of eTypeII ($L=4$) with higher feedback bits of $M=128$. This indicates that the EVCsiNet-T can significantly reduce the feedback overhead without the cost of performance loss in comparison with traditional codebook based counterparts. Meanwhile, three different types of channels lead to different correlation features between subbands in the frequency domain. This property can be well exploited by EVCsiNet-T for CSI compression and recovery since the different subbands are treated as a sequence of inputs so that the cross-correlated features can be extracted effectively by attention mechanism, which further brings performance gain in comparison with the traditional codebook solution with relative bias in the selection of broadband beam group and subband beams.
\begin{table*}[tb]
\Large
\caption{Transformer based solutions with different enhancing schemes.}
\begin{threeparttable}
\resizebox{\textwidth}{!}{
\renewcommand{\arraystretch}{1}
\begin{tabular}{|m{0.8cm}<{\centering}|m{1.6cm}<{\centering}|m{1.6cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|m{1.9cm}<{\centering}|}
\hline
\multirow{3}{*}{\begin{tabular}[m]{@{}c@{}}\\ \\Solu-\\ tion\\ ID\end{tabular}} &\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\\ \\SGCS \\(48bits)\end{tabular}}&\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\\ \\SGCS \\(128bits)\end{tabular}} &\multicolumn{9}{c|}{Scheme}\\ \cline{4-12}
& & &\multicolumn{4}{c|}{Data Augmentation} &\multicolumn{2}{c|}{Loss Function Design} &\multicolumn{2}{c|}{Training Strategy} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\\ \\ Model \\Ensemble\end{tabular}} \\ \cline{4-11}
& & &Noise Injection & Flipping &Shifting &Rotation & Scoring Loss & Quan. Error Compensation & Learning Rate Warm-up and Decay &Staged Training & \\ \hline
1 & 0.877 & 0.938& & & & & $\surd$ & &$\surd$ & & $\surd$ \\ \hline
2 & 0.865 & 0.935& & & & & $\surd$ & $\surd$ & $\surd$ & & $\surd$ \\ \hline
3 & 0.850 & 0.930& & & & & $\surd$ & $\surd$ & &$\surd$ & \\ \hline
4 & 0.849 & 0.922& & $\surd$ & $\surd$ & $\surd$ & $\surd$ & & $\surd$ & & \\ \hline
5 & 0.843 & 0.922& $\surd$ & & $\surd$ & $\surd$ & $\surd$ & &$\surd$ &$\surd$ & \\ \hline
6 & 0.842 & 0.921& & & & & & $\surd$ & &$\surd$ & \\ \hline
\end{tabular}
}
\end{threeparttable}
\label{Capability composition of participating models table}
\end{table*}
Beyond the superior performance of EVCsiNet-T compared to the traditional codebook based methods, there are also some enhancing schemes which can empower EVCsiNet-T and further improve the performance of this backbone. For solving DL-based CSI feedback, in 2nd WAIC there are six over top ten teams use this backbone with various of enhancing schemes introduced in Section \ref{Enhancing Schemes}. Note that since the effectiveness of enhancing scheme is different in the case of different datasets, backbone hyperparameters and joint implementation of enhancing schemes, the simple evaluation of single enhancing scheme is relatively unilateral. Therefore, in Table \ref{Capability composition of participating models table} we also provide the evaluation of good-performing solutions using various combinations of enhancement schemes on the dataset in 2nd WAIC setting the number of feedback bits $M=48$ and $128$, and check mark indicates the utilization of corresponding enhancing scheme.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.47]{BLER-eps-converted-to.pdf}
\caption{Link-level BLER performance comparison on CDL-C300 for
different solutions..}
\label{BLER}
\end{figure}
\subsection{BLER Performance Comparison}
As shown in Fig. \ref{BLER} to further reveal the effectiveness of the proposed method, the link-level block error rate (BLER) performace comparison on CDL-C300 channels for different solutions is provided with the signal-to-noise (SNR) from -1 to 6 dB over $10^{4}$ realizations. Moreover, without considering adaptive modulation and coding, the modulation and coding scheme (MCS) is configured as 19 \cite{2}. The ideal channel estimation is assumed where the BS knows the perfect CSI with SGCS = 1. We can notice that the proposed EVCsiNet-T with $M = 120$ can achieve BLER \textless $10^{-3}$ with SNR = 6 dB and significantly outperforms eTypeII of $L = 4$ with overhead of 128 bits. Moreover, the EVCsiNet-T method can greatly reduce the feedback overhead with similar BLER performance, since the EVCsiNet-T with $M = 48$ and $32$ slightly outperform eTypeII $L=4$ and $2$ with feedback bits of 128 and 49, respectively.
\subsection{Complexity Analysis}
The complexity analysis of EVCsiNet-T with the number of feedback bits $M = 32, 48$ and $120$ is provided in Table \ref{tabFLOPs} from the perspective of floating point operations (FLOPs) and trainable parameters.
\begin{table}[!ht]
\small
\centering
\caption{FLOPs and number of trainable parameters.}
\label{tab4}
\setlength{\tabcolsep}{2.4mm}{
{\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{$M$} & \multicolumn{2}{c|}{FLOPs ($\times 10^{7}$) } & \multicolumn{2}{c|}{Trainable Par. ($\times 10^{7}$) }\\
\cline{2-5} & Encoder & Decoder & Encoder & Decoder \\ \hline
$32$ & $4.2099$ & 4.2099 & $2.1107$ & 2.1108 \\ \hline
$48$ & 4.2111 & 4.2111 & 2.1113 & 2.1114 \\ \hline
$120$ & 4.2166 & 4.2166 & 2.1141 & 2.1142 \\ \hline
\end{tabular}}
\label{tabFLOPs}}
\end{table}
Obviously, the EVCsiNet-T with $M=32$ requires $8.4198 \times 10^7$ FLOPs for inference. Since the feedback delay of the existing communication system is required to be less than 1 ms, the inference time using a common device of NVIDIA Tesla V100 SXM2 with double-precision performance of $7.8 \times 10^{12}$ floating point operations per second (FLOPS) can theoretically be 10.8 $\mu$s which meets the requirement well. It can also be noticed that the FLOPs and trainable parameters are quite close for different number of feedback bits, which indicates that the the inference time can not be affected too much by different number of feedback bits.
\section{Conclusion}
\label{CONCLUSION}
In this paper, we first give a description of the framework of CSI feedback and its corresponding channel model involved in the 2nd WAIC. Then a Transformer backbone for CSI feedback referred to EVCsiNet-T is proposed. Moreover, a series of enhancement schemes in 2nd WAIC including i) data augmentation, ii) loss function design, iii) training strategy, and iv) model ensemble are introduced. The experimental results involving the comparison between EVCsiNet-T and traditional codebook methods over different channels are further provided, which show the advanced performance and a promising prospect of Transformer on CSI feedback problem. Since the significance and indispensability of interpretable ML in the AI-enhanced wireless communication, we will study on this field with more theoretical interpretation and analysis in future research. Finally we sincerely hope that this article and 2nd WAIC can broaden the thinking and insights for interested researchers.
\section*{Acknowledgement}
\label{ACKNOWLEDGEMENT}
We sincerely thank China Academy of Information and Communications Technology, Guangdong OPPO Mobile Telecommunications Corp., Ltd, National Mobile
Communications Research Laboratory of Southeast University and vivo Mobile Communication Co., Ltd for their great help and support on the research and the holding of the 2nd WAIC. We also would like to express our gratitude to all contestants for their participation and sharing.
\bibliographystyle{gbt7714-numerical}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,261 |
Timimus is a genus of small coelurosaurian theropod dinosaur from the Early Cretaceous of Australia. It was originally identified as an ornithomimosaur, but now it is thought to be a different kind of theropod, possibly a tyrannosauroid.
Discovery and species
In 1991, two femora (thighbones), one from an adult and one from a juvenile, were found within a metre of each other at the Dinosaur Cove East site, in the small "Lake Copco" quarry, at the southern tip of Australia.
The type species, Timimus hermani, was formally named and shortly described by Dr Thomas Rich and his wife Patricia Vickers-Rich in 1993/1994. The generic name means "Tim's Mimic" and combines the name of both the discoverers' son Timothy Rich and palaeontologist Tim Flannery with a Latin mimus, "mimic", a reference to the presumed affinity of the species with the Ornithomimosauria. The specific name honours volunteer John Herman who, for many years, assisted the Dinosaur Cove project.
The holotype specimen, NMV P186303, was found in a layer of the Eumeralla Formation, dating to the Albian faunal stage in the early Cretaceous, some 106 million years ago. It consists of a left femur of an adult individual.
In 1994, Dr. Thomas Rich commented that, while it would have been more ideal to have had the most complete specimen possible as a holotype, it was highly unlikely that future material of Timimus would be found, due to the limited nature of sites to be explored in the area. Also, the holotype would have had characteristics which both identified it as an ornithomimosaur and a new genus within that group. Thus the name would serve as a reference point for the material within paleontological literature. Rich stated: "By themselves, the names of dinosaurs are like telephone numbers - they are labels that go with specimens and the ideas that flow from the analysis of the material. Confusing labels, like an inaccurate telephone book, lead to an unworkable system, so one must be careful in putting names or labels on things. But the act of doing so is not creating those specimens or the ideas associated with them; it is merely creating a convenient "handle" for purposes of communication".
The second femur, that of a juvenile, was assigned as the paratype, specimen NMV P186323. Some vertebrae from the site have been referred to the species, as well as some other South Australian material.
Description
The holotype thighbone is 44 centimetres long. From this, a total length of the animal of 2.5 metres has been extrapolated. The slenderness of the bone suggest a lithe animal. The paratype femur is 19.5 centimetres long. The femora show several features that were considered diagnostic. There is no extensor groove between the condyles of the lower joint, which would have been a basal trait for an ornithomimosaur. The femoral head is anteroposteriorally flattened. The anterior trochanter is in a high position and reaches the level of the major trochanter. In 2016 the NMV P186303 specimen was estimated to be 4.3 meters (14 ft) long, and 200 kg (441 lbs) in weight.
Phylogeny
In 1994, the describers assigned Timimus to the "Ornithomimosauridae", with which the Ornithomimidae were meant. Ornithomimosaur remains from Gondwana are rare and dubious; Timimus was thus presented as proof that the group was indeed present in the Southern Hemisphere and would even have originated there. Immediately, however, a position within the Ornithomimosauria was doubted by Thomas Holtz. Today, it is recognised that Timimus shares no derived traits, synapomorphies, with the Ornithomimosauria and thus any proof it would belong to this group is lacking. It perhaps belongs to some coelurosaurian group; some workers consider it a nomen dubium. A 2012 study found it to be a valid tyrannosauroid, a conclusion supported by Delcourt and Grillo (2018).
Paleobiology
The habitat of Timimus consisted of polar forests with mild summers but cold and dark winters due to the closer proximity of the area to the South Pole during the Early Cretaceous. In 1996, Anusuya Chinsamy, an expert on the microstructure of fossil bones, examined bone material from Timimus and Leaellynasaura and discovered they exhibited different bone histology. The ornithischian showed a continuous rate of bone deposition, while the coelurosaur had a cyclical pattern of bone formation, which suggested Timimus may have hibernated in colder months. A possible Timimus hermani or related form from the Strzelecki Group near Inverloch, Victoria left a fossil of the first phalanx of its third toe with a depressed fracture on the plantar surface.
Notes
References
External links
Dann's Dinosaur Info: Timimus
Early Cretaceous dinosaurs of Australia
Tyrannosaurs
Fossil taxa described in 1993
Taxa named by Patricia Vickers-Rich
Taxa named by Tom Rich
Paleontology in Victoria | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,518 |
From Grace to Glory: Music of Todd Agnew
by Todd Agnew
CD|Jun 2017
Out of Print. Limited Availability
Glory Defined: Biggest Hits
Live From the Woods (2 Cds)
Need To Breathe
Steve Grace
Coming Up to Breathe
Phil Wickham
Laura Story
Restart (2014)
Is this product missing content?
Product Code 766887900016
Department Music
Category Pop Rock
Sub-Category General
Publisher Ardent Records
Release Date Jun 2017
Todd Agnew
Praise & worship singer and songwriter Todd Agnew was born March 15, 1971, in Dallas, TX. With a kind of indie rock approach, pop-oriented style, and unique voice, Agnew's spiritually directed songs aren't quite like anything else in the contemporary Christian market. Basing himself in Memphis, TN, and working through Ardent Records, he has released several albums with the label, including Grace Like Rain (2003); Reflection of Something (2005); his ambitious retelling of the Christmas story, Do You See What I See? (2006); and Better Questions (2007). Agnew relocated back to Texas in 2008.
Bio by Steve Leggett, www.allmusic.com. Accessed 02.04.14
Bestsellers in Pop Rock
Run Wild. Live Free. Love Strong. Anniversary Edition
I Can Only Imagine: The Very Best of Mercyme
Stars Go Dim
Wow Hits 2019 Deluxe Double CD
How Can It Be Deluxe Edition | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,822 |
Harman Patil
I love reading books and watching sci fi flicks
Allure (magazine)
Updated on Sep 11, 2022
Editor Michelle Lee
Total circulation(2013) 1,165,392
Categories Beauty
Publisher Condé Nast
Allure is an American women's beauty magazine, published monthly by Conde Nast in New York City. It was founded in 1991 by Linda Wells. Michelle Lee replaced Wells in 2015. A signature of the magazine is its annual Best of Beauty awards—accolades given in the October issue to beauty products deemed the best by magazine staff.
Awards for Allure
Awards for Linda Wells
In 1990, S.I. Newhouse Jr., chairman of Condé Nast, and then editorial director Alexander Liberman approached Linda Wells to develop a concept they had for a beauty magazine. At the time, Wells was the beauty editor and the food editor at The New York Times Magazine.
The magazine's prototype was shredded shortly before the scheduled launch date and, after overhauling everything (including the logo), Allure made its debut in March 1991 designed by Lucy Sisman. The magazine's original format was oversize, but this prevented it from fitting into slots at grocery-store checkouts and required advertisers to resize their ads or create new ones. After four issues, Allure changed to a standard-size glossy format.
Allure focuses on beauty, fashion, and women's health. Allure was the first women's magazine to write about the health risks associated with silicone breast implants, and has reported on other controversial health issues.
The magazine's circulation, initially 250,000 in 1991, is over 1 million as of 2011.
Many writers have contributed to Allure. Among them are Arthur Miller, John Updike, Jhumpa Lahiri, Michael Chabon, Kathryn Harrison, Frank McCourt, Isabel Allende, and Francine du Plessix Gray. Elizabeth Gilbert's essay "The Road to Rapture," published in Allure in 2003, was the precursor to her memoir, Eat, Pray, Love (Viking Adult). Photographers who have shot for Allure include Michael Thompson, Mario Testino, Patrick Demarchelier, Tina Barney, Marilyn Minter, Carter Smith, Steven Klein, Steven Meisel, and Helmut Newton. Cover subjects have included Demi Lovato, Jennifer Aniston, Jennifer Lopez, Julia Roberts, Angelina Jolie, Reese Witherspoon, Mary-Kate and Ashley Olsen, Victoria Beckham, Beyoncé, Fergie, Britney Spears, Naya Rivera, Jessica Simpson, Kate Hudson, Christina Aguilera, Rihanna, and Gwen Stefani. (See List of Allure cover models).
Allure began its Best of Beauty awards program 14 years ago, at the initiative of Wells, to help readers choose among the vast array of makeup, skin-care, and hair-care products on the market.
Allure has two sets of awards, one judged by the magazine's editors and the other by readers. A "winners' seal" logo, developed by Allure, appears on many of the winning products. To ensure that its judgments are neutral, Allure's ad department isn't involved in the selections.
In 2010, the magazine developed an iPhone app that highlights the winning products and tells users where they can buy them based on their location.
There was an outrage when the magazine showed Marissa Neitling with an Afro haircut.
Awards (for Allure)
The National Magazine Award for Design (1994)
The Editorial Excellence Award from Folio (2001)
The Circulation Excellence Award from Circulation Management (2001)
"Ring Leader," an essay by Natalie Kusz from the February 1996 issue of Allure, was selected for The Best American Essays 1997 (Houghton Mifflin).
The magazine has been on Adweek's Hot List in 1993, 1994, 1995, 2003, and 2007.
Allure has received 29 awards from the American Academy of Dermatology, 9 journalism awards from the Fragrance Foundation, and the Excellence in Media Award from the Skin Cancer Foundation.
Awards (for Linda Wells)
The Achiever Award from Cosmetic Executive Women (2001)
The Matrix Award for magazine leadership from New York Women in Communications, Inc. (2009)
Wells, along with Allure editors Michael Carl and Kelly Atterton, have appeared as judges on the Bravo TV series Shear Genius.
Allure editors have appeared as experts on programs such as the Today show and 60 Minutes, and Allure stories frequently receive national attention.
Hilary Duff played an Allure intern in Cheaper by the Dozen 2.
Allure (magazine) Wikipedia
Similar TopicsThe Young Nurses
Champ (film)
Jurgi Oteo
The Young Nurses | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 872 |
Community learning may refer to:
Adult education
Learning community | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,371 |
{"url":"https:\/\/learn.careers360.com\/ncert\/ncert-solutions-class-11-physics-chapter-6-work-energy-and-power\/","text":"# NCERT Solutions for Class 11 Physics Chapter 6 Work\u00a0Energy and Power\n\nNCERT Solutions for Class 11 Physics Chapter 6 Work\u00a0Energy and Power: consider the situation that you are carrying a 10 Kg stationeries in your hand and you are not moving. After some time your hand will get pain. In this situation, you are doing work physically according to you. But in Physics, there is no work or you are doing zero work. Can you find why work is zero here? The\u00a0NCERT Solutions for Class 11 Physics Chapter 6 Work\u00a0Energy and Power introduces the concepts of work and types of work, the relation between work and energy and the concept of power. In\u00a0NCERT Solutions for Class 11 Physics Chapter 6, you will study about conservation of energy (energy can be neither created nor be destroyed). For example, when water is coming through a pipe the potential energy of water stored in a tank at height is converted to kinetic energy. When we switch a fan the fan rotates, where the electrical energy is converted to mechanical energy. Can you find out more examples of conservation of energy?.\n\nTo understand briefly about the work done by a force let's consider a situation as shown in the figure\n\nHere weight mg is acting downwards and force and displacement are in the upward direction. Here the work done by the gravitational force is negative since displacement is opposite to gravity. The work done by force F is positive since F and displacement are in the same direction. The\u00a0NCERT Solutions for Class 11 Physics Chapter 6\u00a0explains\u00a0many other physical and mathematical aspects of work.\n\nThe main topics of the\u00a0NCERT Solutions for Class 11 Physics Chapter 6 Work\u00a0Energy and Power are listed below:\n\n6.1 Introduction\n\n6.2 Notions Of Work And Kinetic Energy: The Work-energy Theorem\n\n6.3 Work 6.4 Kinetic Energy\n\n6.5 Work Done By A Variable Force\n\n6.6 The Work-energy Theorem For A Variable Force\n\n6.7 The Concept Of Potential Energy\n\n6.8 The Conservation Of Mechanical Energy\n\n6.9 The Potential Energy Of A Spring\n\n6.10 Various Forms Of Energy: The Law Of Conservation Of Energy\n\n6.11 Power\n\n6.12 Collisions\n\nAnother important topic of the\u00a0NCERT Solutions for Class 11 Physics Chapter 6 is the collisions. Some of the main points of the topic collisions are listed below.\n\n\u2022 For an\u00a0\u00a0elastic collision\u00a0Law of conservation of momentum and that of Kinetic Energy holds good\n\u2022 For inelastic collision\u00a0Law of conservation of momentum hold good but kinetic energy is not conserved\n\u2022 Coefficient of restitution is the ratio of relative velocity after the collision to relative velocity before the collision\n\u2022 For a perfectly elastic collision coefficient of restitution is one\n\u2022 For inelastic collision\u00a0coefficient of restitution is less than one\n\u2022 For a perfectly inelastic collision\u00a0coefficient of restitution is zero\n\nWe provide you with\u00a0NCERT Solutions for Class 11 Physics Chapter 6 Work\u00a0Energy and Power\u00a0below:\n\nView all\u00a0NCERT Solutions for Class 11 Physics Chapter 6 Work Energy and Power here.\n\n## NCERT Solutions for class 11 Physics\n\n Chapter-1 Physical world Chapter-2 Chapter-3 Chapter-4 Chapter-5 Chapter-7 Chapter-8 Chapter-9 Chapter-10 Mechanical Properties of Fluids Chapter-11 Chapter-12 Chapter-13 Kinetic Theory Chapter-14 Chapter-15\n\n## NCERT Solutions for Class 11- Subject wise\n\n NCERT solutions for Class 11\u00a0Biology NCERT solutions for Class 11 Maths NCERT solutions for Class 11 Chemistry NCERT solutions for Class 11\u00a0Physics","date":"2019-12-13 06:17:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 7, \"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\": 0.503510057926178, \"perplexity\": 921.3709908575272}, \"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\/1575540548544.83\/warc\/CC-MAIN-20191213043650-20191213071650-00441.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/the-hodge-star-operator.958707\/","text":"# The Hodge star operator\n\n\u2022 A\nI'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts.\nLet ##\\lambda## be a ##p##-vector in ##\\bigwedge^p V## and let ##\\sigma^1,\\ldots,\\sigma^n## be a basis of ##V##. There's a unique ##*\\lambda## such that, for all ##\\mu\\in \\bigwedge^{n-p}##,$$\\lambda \\wedge \\mu = (*\\lambda, \\mu)\\sigma,$$ where ##\\sigma = \\sigma^1 \\wedge \\cdots \\wedge \\sigma^n##.\nFlanders says it's enough to consider ##\\lambda=\\sigma^1\\wedge\\cdots\\wedge\\sigma^p## because of linearity, but what about ##\\lambda=\\sigma^H## where ##h_1<\\cdots <h_p##?\nIn that case I get $$*\\lambda = (-1)^s(\\sigma^\\bar{H},\\sigma^\\bar{H})\\sigma^\\bar{H},$$ where ##H\\sqcup\\bar{H}=\\{1,\\ldots,n\\}## and ##s## is the permutation needed to permute ##H\\bar{H}=(h_1,\\ldots,h_p,\\bar{h}_1,\\ldots,\\bar{h}_{n-p})## into ##(1,\\ldots,n)##.\nI suspect this is not correct... or is it?\n\nfresh_42\nMentor\nI'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts.\nLet ##\\lambda## be a ##p##-vector in ##\\bigwedge^p V## and let ##\\sigma^1,\\ldots,\\sigma^n## be a basis of ##V##. There's a unique ##*\\lambda## such that, for all ##\\mu\\in \\bigwedge^{n-p}##,$$\\lambda \\wedge \\mu = (*\\lambda, \\mu)\\sigma,$$ where ##\\sigma = \\sigma^1 \\wedge \\cdots \\wedge \\sigma^n##.\nFlanders says it's enough to consider ##\\lambda=\\sigma^1\\wedge\\cdots\\wedge\\sigma^p## because of linearity, but what about ##\\lambda=\\sigma^H## where ##h_1<\\cdots <h_p##?\nIn that case I get $$*\\lambda = (-1)^s(\\sigma^\\bar{H},\\sigma^\\bar{H})\\sigma^\\bar{H},$$ where ##H\\sqcup\\bar{H}=\\{1,\\ldots,n\\}## and ##s## is the permutation needed to permute ##H\\bar{H}=(h_1,\\ldots,h_p,\\bar{h}_1,\\ldots,\\bar{h}_{n-p})## into ##(1,\\ldots,n)##.\nI suspect this is not correct... or is it?\nI don't understand your question. If you have a permutation of ##\\sigma_i## you get another ##*\\lambda##, with an appropriate sign. So if we have ##*\\sigma##, then we get all ##*\\lambda## by linear extension and sign corrections. So the only minor neglect was to say by \"because of linearity\" instead of \"because of alternating linearity\".\n\nkiuhnm\nI don't understand your question. If you have a permutation of ##\\sigma_i## you get another ##*\\lambda##, with an appropriate sign. So if we have ##*\\sigma##, then we get all ##*\\lambda## by linear extension and sign corrections. So the only minor neglect was to say by \"because of linearity\" instead of \"because of alternating linearity\".\n\nI wasn't sure whether my formula for ##*\\lambda## was correct. I had doubts because I got a wrong sign in example 1 on page 16. That example says that if the base of ##V## is ##(dx^1,dx^2,dx^3,dt)## where ##(dx^i,dx^i)=1## and ##(dt,dt)=-1##, then $$*(dx^i dt) = dx^j dx^k,$$ where \"##(i,j,k)## is cyclic order\". I didn't take into account the cyclicity of the order and so got a wrong sign. In particular, $$*(dx^2 dt) = -(dx^1 dx^3) = dx^3 dx^1.$$ I missed the very last step.","date":"2021-05-17 15:21:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.92198246717453, \"perplexity\": 1992.2963547154839}, \"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\/1620243991258.68\/warc\/CC-MAIN-20210517150020-20210517180020-00448.warc.gz\"}"} | null | null |
<?php
class KirbyGitHelper
{
const DEFAULT_MESSAGES = array(
'page.create' => 'create(page): %s',
'page.update' => 'update(page): %s',
'page.delete' => 'delete(page): %s',
'page.sort' => 'sort(page): %s',
'page.hide' => 'hide(page): %s',
'page.move' => 'move(page): %s',
'file.upload' => 'create(page): %s',
'file.replace' => 'update(page): %s',
'file.rename' => 'delete(page): %s',
'file.update' => 'sort(page): %s',
'file.sort' => 'hide(page): %s',
'file.delete' => 'move(page): %s',
'user.suffix' => "\n\nby %s", // appended to the commit message
);
private $repo;
private $repoPath;
private $branch;
private $pullOnChange;
private $pushOnChange;
private $commitOnChange;
private $setGitUser;
private $gitBin;
private $windowsMode;
private $commitMessages;
public function __construct($repoPath = false)
{
$this->repoPath = $repoPath ? $repoPath : c::get('gcapc-path', kirby()->roots()->content());
$this->branch = c::get('gcapc-branch', '');
$this->messages = c::get('gcapc-messages', self::DEFAULT_MESSAGES);
}
private function initRepo()
{
if ($this->repo) {
return true;
}
require_once(__DIR__ . DS . 'Git.php' . DS. 'Git.php');
$this->pullOnChange = c::get('gcapc-pull', false);
$this->pushOnChange = c::get('gcapc-push', false);
$this->commitOnChange = c::get('gcapc-commit', false);
$this->setGitUser = c::get('gcapc-set-git-user', false);
$this->gitBin = c::get('gcapc-gitBin', '');
$this->windowsMode = c::get('gcapc-windowsMode', false);
if ($this->windowsMode) {
Git::windows_mode();
}
if ($this->gitBin) {
Git::set_bin($this->gitBin);
}
$this->repo = Git::open($this->repoPath);
if (!$this->repo->test_git()) {
trigger_error('git could not be found or is not working properly: ' . Git::get_bin());
}
}
private function getRepo()
{
if ($this->repo == null) {
$this->initRepo();
}
return $this->repo;
}
public function commit($commitMessage)
{
if ($this->setGitUser) {
$user = site()->user();
// TODO use a better template mechanism
if (!empty($user->firstname()) && !empty($user->lastname())) {
$userName = $user->firstname() . ' ' . $user->lastname() . ' (' . $user->username() . ')';
} else {
$userName = $user->username();
}
$userEmail = $user->email();
$this->getRepo()->run("config user.name '${userName}'");
$this->getRepo()->run("config user.email '${userEmail}'");
}
$this->getRepo()->add('-A');
$this->getRepo()->commit($commitMessage);
}
public function push($branch = false)
{
$branch = $branch ? $branch : $this->branch;
$this->getRepo()->run("push origin $branch"); // push('origin', $branch) inserts a spurious --tags option
}
public function pull($branch = false)
{
$branch = $branch ? $branch : $this->branch;
$this->getRepo()->pull('origin', $branch);
}
public function kirbyChangePage($key, $page) {
$commitMessage = $this->getMessage($key, $page->uri());
$this->kirbyChange($commitMessage);
}
public function kirbyChangeFile($key, $file) {
$commitMessage = $this->getMessage($key, $file->page()->uri() . '/' . $file->filename());
$this->kirbyChange($commitMessage);
}
private function kirbyChange($commitMessage)
{
try {
$this->initRepo();
if ($this->branch) {
$this->getRepo()->checkout($this->branch);
}
if ($this->pullOnChange) {
$this->pull();
}
if ($this->commitOnChange) {
$this->commit($commitMessage . $this->getMessage('user.suffix', site()->user()));
}
if ($this->pushOnChange) {
$this->push();
}
} catch(Exception $exception) {
trigger_error('Unable to update git: ' . $exception->getMessage());
}
}
private function getMessage($key, ...$params)
{
$message = isset($this->messages[$key]) ? $this->messages[$key] : self::DEFAULT_MESSAGES[$key];
return sprintf($message, ...$params);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
var SuperObject = module.exports = function(native) {
BasicObject.call(this, native);
};
// To fix circular dependencies, we include this later
var BasicObject = require('./basic-object');
SuperObject.instanceType = null;
SuperObject.prototype = Object.create(BasicObject.prototype);
SuperObject.prototype.constructor = SuperObject;
module.exports = SuperObject;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,856 |
Q: Host Named Site Collections - HTTP to HTTPS redireciton on SharePoint 2010? We have a Host Named Site Collection on our SharePoint 2010 farm and its URL is - https://test.demo.com. It is accessible with - https://test.demo.com however accessing http://test.demo.com leads to "Unable to connect" error.
We want to place HTTP to HTTPS redirection, shall we make use of URL redirection?
A: what is your AAM settings?
you need to update the AAM settings, adding another internal URL with http...which will redirect it to https.
Internal URL Zone Public URL for Zone
http://test.demo.com Default https://test.demo.com
https://test.demo.com Default https://test.demo.com
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,591 |
It is a good news for those who are willing to get a job in a government department. Approx 16,000 posts are waiting for them. There are 15907 vacancies will be filled very soon in 89 Government Departments. It might be increased upto 16000, after the information of vacancies from 27 other govt departments.
The recruitment process has been started to fill 9000 posts, rest of the post will be filled soon. In state, employees are working on 80% posts in every govt department, rest 20% posts are still vacant. However, in terms of backlog posts, it decreased in comparision of last year, now the backlog seats are 1113 in numbers.
Help us to improve this article/job posting "16000 Posts are vacant in Govt Department". We would be glad if you can comment below and provide your valuable suggestions and feedback. If this page have any wrong information/list or out of context content, then you can report us as well.
when we will start apply for the vacant seats?
there is no any clu to find the apply option. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,184 |
/**
* Created by knut on 14-11-18.
*/
var proxyquire = require('proxyquire');
var log = require('../../logger').create();
var sq = require('./parser/sequenceDiagram').parser;
var newD3;
var d3 = {
select:function(){
return new newD3();
}
};
//var sd = proxyquire('./sequenceRenderer', { './d3': d3 });
var sd = proxyquire('./sequenceRenderer', { '../../d3': d3 });
var str;
describe('when parsing a sequenceDiagram',function() {
var parseError;
beforeEach(function () {
sq.yy = require('./sequenceDb');
sq.yy.clear();
parseError = function(err, hash) {
log.debug('Syntax error:' + err);
log.debug(hash);
};
sq.yy.parseError = parseError;
});
it('it should handle a sequenceDiagram defintion', function () {
str = 'sequenceDiagram\n' +
'Alice->Bob:Hello Bob, how are you?\n' +
'Note right of Bob: Bob thinks\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(3);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('Bob');
});
it('it should space in actor names', function () {
str = 'sequenceDiagram\n' +
'Alice->Bob:Hello Bob, how are - you?\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(2);
expect(messages[0].from).toBe('Alice');
expect(messages[1].from).toBe('Bob');
});
it('it should handle in async messages', function () {
var str = 'sequenceDiagram\n' +
'Alice-xBob:Hello Bob, how are you?\n';
sq.parse(str);
var actors = sq.yy.getActors();
//log.debug(actors);
expect(actors.Alice.description).toBe('Alice');
expect(actors.Bob.description).toBe('Bob');
var messages = sq.yy.getMessages();
expect(messages.length).toBe(1);
expect(messages[0].type).toBe(sq.yy.LINETYPE.SOLID_CROSS);
});
it('it should handle in async dotted messages', function () {
var str = 'sequenceDiagram\n' +
'Alice--xBob:Hello Bob, how are you?\n';
sq.parse(str);
var actors = sq.yy.getActors();
//log.debug(actors);
expect(actors.Alice.description).toBe('Alice');
expect(actors.Bob.description).toBe('Bob');
var messages = sq.yy.getMessages();
expect(messages.length).toBe(1);
expect(messages[0].type).toBe(sq.yy.LINETYPE.DOTTED_CROSS);
});
it('it should handle in arrow messages', function () {
var str = 'sequenceDiagram\n' +
'Alice->>Bob:Hello Bob, how are you?\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
expect(actors.Bob.description).toBe('Bob');
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(1);
expect(messages[0].type).toBe(sq.yy.LINETYPE.SOLID);
});
it('it should handle in arrow messages', function () {
var str = 'sequenceDiagram\n' +
'Alice-->>Bob:Hello Bob, how are you?\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
expect(actors.Bob.description).toBe('Bob');
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(1);
expect(messages[0].type).toBe(sq.yy.LINETYPE.DOTTED);
});
it('it should handle comments in a sequenceDiagram', function () {
str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n'+
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(3);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('Bob');
});
it('it should handle new lines in a sequenceDiagram', function () {
str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(3);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('Bob');
});
it('it should handle one leading space in lines in a sequenceDiagram', function () {
str = 'sequenceDiagram\n' +
' Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(3);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('Bob');
});
it('it should handle several leading spaces in lines in a sequenceDiagram', function () {
str = 'sequenceDiagram\n' +
' Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'Bob-->Alice: I am good thanks!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(3);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('Bob');
});
it('it should handle several leading spaces in lines in a sequenceDiagram', function () {
str = 'sequenceDiagram\n'+
'participant Alice\n'+
'participant Bob\n'+
'Alice->John: Hello John, how are you?\n'+
' loop Healthcheck\n'+
'John->John: Fight against hypochondria\n'+
' end\n'+
'Note right of John: Rational thoughts<br/>prevail...\n'+
' John-->Alice: Great!\n'+
' John->Bob: How about you?\n'+
'Bob-->John: Jolly good!\n';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
expect(messages.length).toBe(8);
expect(messages[0].from).toBe('Alice');
expect(messages[2].from).toBe('John');
});
it('it should handle loop statements a sequenceDiagram', function () {
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'loop Multiple happy responses\n\n' +
'Bob-->Alice: I am good thanks!\n' +
'end';
sq.parse(str);
var actors = sq.yy.getActors();
//log.debug(actors);
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(5);
expect(messages[0].from).toBe('Alice');
expect(messages[1].from).toBe('Bob');
});
it('it should handle opt statements a sequenceDiagram', function () {
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'opt Perhaps a happy response\n\n' +
'Bob-->Alice: I am good thanks!\n' +
'end';
sq.parse(str);
var actors = sq.yy.getActors();
//log.debug(actors);
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(5);
expect(messages[0].from).toBe('Alice');
expect(messages[1].from).toBe('Bob');
});
it('it should handle opt statements a sequenceDiagram', function () {
var str = 'sequenceDiagram;Alice->Bob: Hello Bob, how are you?;opt Perhaps a happy response;Bob-->Alice: I am good thanks!;end;';
sq.parse(str);
var actors = sq.yy.getActors();
//log.debug(actors);
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(4);
expect(messages[0].from).toBe('Alice');
expect(messages[1].type).toBe(sq.yy.LINETYPE.OPT_START);
expect(messages[2].from).toBe('Bob');
});
it('it should handle alt statements a sequenceDiagram', function () {
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n\n' +
'%% Comment\n' +
'Note right of Bob: Bob thinks\n' +
'alt isWell\n\n' +
'Bob-->Alice: I am good thanks!\n' +
'else isSick\n' +
'Bob-->Alice: Feel sick...\n' +
'end';
sq.parse(str);
var actors = sq.yy.getActors();
expect(actors.Alice.description).toBe('Alice');
actors.Bob.description = 'Bob';
var messages = sq.yy.getMessages();
//log.debug(messages);
expect(messages.length).toBe(7);
expect(messages[0].from).toBe('Alice');
expect(messages[1].from).toBe('Bob');
});});
describe('when checking the bounds in a sequenceDiagram',function() {
var parseError, _d3, conf;
beforeEach(function () {
sq.yy = require('./sequenceDb');
sq.yy.clear();
parseError = function(err, hash) {
log.debug('Syntax error:' + err);
log.debug(hash);
};
sq.yy.parseError = parseError;
conf = {
diagramMarginX:50,
diagramMarginY:10,
actorMargin:50,
width:150,
// Height of actor boxes
height:65,
boxMargin:10,
messageMargin:40,
boxTextMargin:15,
noteMargin:25
};
sd.setConf(conf);
});
it('it should handle a simple bound call', function () {
sd.bounds.init();
sd.bounds.insert(100,100,200,200);
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(100);
expect(bounds.starty).toBe(100);
expect(bounds.stopx ).toBe(200);
expect(bounds.stopy ).toBe(200);
});
it('it should handle an expanding bound', function () {
sd.bounds.init();
sd.bounds.insert(100,100,200,200);
sd.bounds.insert(25,50,300,400);
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(25);
expect(bounds.starty).toBe(50);
expect(bounds.stopx ).toBe(300);
expect(bounds.stopy ).toBe(400);
});
it('it should handle inserts within the bound without changing the outer bounds', function () {
sd.bounds.init();
sd.bounds.insert(100,100,200,200);
sd.bounds.insert(25,50,300,400);
sd.bounds.insert(125,150,150,200);
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(25);
expect(bounds.starty).toBe(50);
expect(bounds.stopx ).toBe(300);
expect(bounds.stopy ).toBe(400);
});
it('it should handle a loop without expanding the area', function () {
sd.bounds.init();
sd.bounds.insert(25,50,300,400);
sd.bounds.verticalPos = 150;
sd.bounds.newLoop();
sd.bounds.insert(125,150,150,200);
var loop = sd.bounds.endLoop();
expect(loop.startx).toBe(125-conf.boxMargin);
expect(loop.starty).toBe(150-conf.boxMargin);
expect(loop.stopx ).toBe(150+conf.boxMargin);
expect(loop.stopy ).toBe(200+conf.boxMargin);
// Check bounds of first loop
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(25);
expect(bounds.starty).toBe(50);
expect(bounds.stopx ).toBe(300);
expect(bounds.stopy ).toBe(400);
});
it('it should handle multiple loops withtout expanding the bounds', function () {
sd.bounds.init();
sd.bounds.insert(100,100,1000,1000);
sd.bounds.verticalPos = 200;
sd.bounds.newLoop();
sd.bounds.newLoop();
sd.bounds.insert(200,200,300,300);
// Check bounds of first loop
var loop = sd.bounds.endLoop();
expect(loop.startx).toBe(200-conf.boxMargin);
expect(loop.starty).toBe(200-conf.boxMargin);
expect(loop.stopx ).toBe(300+conf.boxMargin);
expect(loop.stopy ).toBe(300+conf.boxMargin);
// Check bounds of second loop
loop = sd.bounds.endLoop();
expect(loop.startx).toBe(200-2*conf.boxMargin);
expect(loop.starty).toBe(200-2*conf.boxMargin);
expect(loop.stopx ).toBe(300+2*conf.boxMargin);
expect(loop.stopy ).toBe(300+2*conf.boxMargin);
// Check bounds of first loop
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(100);
expect(bounds.starty).toBe(100);
expect(bounds.stopx ).toBe(1000);
expect(bounds.stopy ).toBe(1000);
});
it('it should handle a loop that expands the area', function () {
sd.bounds.init();
sd.bounds.insert(100,100,200,200);
sd.bounds.verticalPos = 200;
sd.bounds.newLoop();
sd.bounds.insert(50,50,300,300);
var loop = sd.bounds.endLoop();
expect(loop.startx).toBe(50 - conf.boxMargin);
expect(loop.starty).toBe(50 - conf.boxMargin);
expect(loop.stopx ).toBe(300 + conf.boxMargin);
expect(loop.stopy ).toBe(300 + conf.boxMargin);
// Check bounds after the loop
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(loop.startx);
expect(bounds.starty).toBe(loop.starty);
expect(bounds.stopx ).toBe(loop.stopx);
expect(bounds.stopy ).toBe(loop.stopy);
});
});
describe('when rendering a sequenceDiagram',function() {
var parseError, _d3, conf;
beforeEach(function () {
sq.yy = require('./sequenceDb');
sq.yy.clear();
parseError = function(err, hash) {
log.debug('Syntax error:' + err);
log.debug(hash);
};
sq.yy.parseError = parseError;
newD3 = function() {
var o = {
append: function (type) {
return newD3();
},
attr: function (key, val) {
return this;
},
style: function (key, val) {
return this;
},
text: function (txt) {
return this;
},
0:{
0: {
getBBox: function () {
return {
height: 10,
width: 20
};
}
}
}
};
return o;
};
conf = {
diagramMarginX:50,
diagramMarginY:10,
actorMargin:50,
width:150,
// Height of actor boxes
height:65,
boxMargin:10,
messageMargin:40,
boxTextMargin:15,
noteMargin:25
};
sd.setConf(conf);
});
it('it should handle one actor', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'participant Alice\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe( conf.width);
expect(bounds.stopy ).toBe(conf.height);
});
it('it should handle one actor and a note', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'participant Alice\n' +
'Note left of Alice: Alice thinks\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(-(conf.width/2)-(conf.actorMargin/2));
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe( conf.width );
// 10 comes from mock of text height
expect(bounds.stopy ).toBe( conf.height + conf.boxMargin + 2*conf.noteMargin +10);
});
it('it should handle one actor and a note to the right', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'participant Alice\n' +
'Note right of Alice: Alice thinks\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe( (conf.width/2) + (conf.actorMargin/2) + conf.width);
// 10 comes from mock of text height
expect(bounds.stopy ).toBe( conf.height + conf.boxMargin + 2*conf.noteMargin +10);
});
it('it should handle two actors', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe(conf.width*2 + conf.actorMargin);
expect(bounds.stopy ).toBe(0 + conf.messageMargin + conf.height);
});
it('it should draw two actors and two messages', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n'+
'Bob->Alice: Fine!\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe(0 + conf.width*2 + conf.actorMargin);
expect(bounds.stopy ).toBe(0 + 2*conf.messageMargin + conf.height);
});
it('it should draw two actors notes to the right', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n'+
'Note right of Bob: Bob thinks\n' +
'Bob->Alice: Fine!\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
var expStopX = conf.actorMargin +conf.width+ (conf.width/2) + conf.noteMargin + conf.width;
expect(bounds.stopx ).toBe(expStopX);
expect(bounds.stopy ).toBe(2*conf.messageMargin + conf.height + conf.boxMargin + 10+ 2*conf.noteMargin);
});
it('it should draw two actors notes to the left', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n'+
'Note left of Alice: Bob thinks\n' +
'Bob->Alice: Fine!\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe( -(conf.width/2)-(conf.actorMargin/2));
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe( conf.width*2 + conf.actorMargin);
expect(bounds.stopy ).toBe( 2*conf.messageMargin + conf.height + conf.boxMargin +10+ 2*conf.noteMargin);
});
it('it should draw two loops', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'Alice->Bob: Hello Bob, how are you?\n'+
'loop Cheers\n' +
'Bob->Alice: Fine!\n' +
'end\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe(0 + conf.width*2 + conf.actorMargin);
expect(bounds.stopy ).toBe(0 + 2*conf.messageMargin + conf.height + 3*conf.boxMargin + conf.boxTextMargin);
});
});
describe('when rendering a sequenceDiagram with actor mirror activated',function() {
var parseError, _d3, conf;
beforeEach(function () {
sq.yy = require('./sequenceDb');
sq.yy.clear();
parseError = function(err, hash) {
log.debug('Syntax error:' + err);
log.debug(hash);
};
sq.yy.parseError = parseError;
newD3 = function() {
var o = {
append: function (type) {
return newD3();
},
attr: function (key, val) {
return this;
},
style: function (key, val) {
return this;
},
text: function (txt) {
return this;
},
0:{
0: {
getBBox: function () {
return {
height: 10,
width: 20
};
}
}
}
};
return o;
};
conf = {
diagramMarginX:50,
diagramMarginY:10,
actorMargin:50,
width:150,
// Height of actor boxes
height:65,
boxMargin:10,
messageMargin:40,
boxTextMargin:15,
noteMargin:25,
mirrorActors:true,
// Depending on css styling this might need adjustment
// Prolongs the edge of the diagram downwards
bottomMarginAdj:1
};
sd.setConf(conf);
});
it('it should handle one actor', function () {
sd.bounds.init();
var str = 'sequenceDiagram\n' +
'participant Alice\n';
sq.parse(str);
sd.draw(str,'tst');
var bounds = sd.bounds.getBounds();
expect(bounds.startx).toBe(0);
expect(bounds.starty).toBe(0);
expect(bounds.stopx ).toBe( conf.width);
expect(bounds.stopy ).toBe(2*conf.height+2*conf.boxMargin);
});
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,852 |
<div ng-if="display && isShown('type')"
class="type">{{ variable.type }}</div>
<div ng-if="display && isShown('name')"
class="name">{{ variable.name }}</div>
<div ng-if="display && isShown('value') && isPrimitive()"
ng-class="{null: isNull()}"
class="value">
<span ng-if="isNull()"
class="null-symbol"><null></span>
{{ (variable.value === null ? '' : variable.value.toString()) }}
</div>
<div ng-if="display && isShown('value') && variable.type === 'Object'"
ng-class="{null: isNull()}"
class="value">
<a ng-click="editVariableValue()">
{{ variable.valueInfo.objectTypeName }}
</a>
</div>
<div ng-if="!display"
class="input-group editing">
<div ng-if="isShown('type')"
class="input-group-btn type">
<select class="form-control"
ng-model="variable.type"
ng-options="variableType for variableType in variableTypes track by variableType"
ng-disabled="isDisabled('type')"
required>
</select>
</div><!-- /btn-group -->
<input ng-if="isShown('name')"
type="text"
class="form-control name"
ng-model="variable.name"
placeholder="varName"
ng-disabled="isDisabled('name')"
required />
<div ng-if="isShown('value') && !isNull()"
class="value-wrapper input-group"
ng-class="{checkbox: useCheckbox()}">
<div ng-if="variable.type !== 'Null'"
class="input-group-btn">
<a ng-click="setNull()"
class="btn btn-default set-null"
ng-disabled="isDisabled('value')"
tooltip="Set value to "null"">
<span class="glyphicon glyphicon-remove"></span>
</a>
</div>
<input ng-if="isPrimitive() && !useCheckbox()"
type="text"
class="form-control value"
ng-model="variable.value"
ng-disabled="isDisabled('value')"
placeholder="Value of the variable"
cam-variable-validator="{{variable.type}}" />
<input ng-if="useCheckbox()"
type="checkbox"
class="value"
ng-model="variable.value"
ng-disabled="isDisabled('value')"
placeholder="Value of the variable"
cam-variable-validator="{{variable.type}}" />
<div ng-if="variable.type === 'Object'"
class="value form-control-static">
<a ng-click="editVariableValue()" ng-disabled="isDisabled('value')">
{{ variable.valueInfo.objectTypeName || '<undefined>' }}
</a>
</div>
</div>
<div ng-if="variable.type !== 'Null' && isShown('value') && isNull()"
ng-click="setNonNull()"
class="value-wrapper value null-value btn btn-default"
ng-disabled="isDisabled('value')"
tooltip="Re-set to previous or default value">
<span class="null-symbol"><null></span>
</div>
<div ng-if="variable.type === 'Null'"
ng-disabled="isDisabled('value')"
class="value-wrapper value btn no-click null-value">
<span class="null-symbol"><null></span>
</div>
</div>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 447 |
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Strewth! The best Aussie songs in The Sounds of Australia Archive
Rachael Rosel
From Men at Work at Cold Chisel, these are the most Aussie songs in the official archives! Source: Getty.
The Sounds of Australia Archive kicked off in 2007 as a way of creating and preserving an audio snapshot of Australia's culture through one of our most beloved mediums – music and sound. The National Film and Sound Archive of Australia adds ten new additions each year, and 2019 welcomed some all-time favourite artists such as Olivia Newton John, John Farnham and Savage Garden into the established list of legends.
With the growing list showcasing well over 100 sounds ranging from speeches to songs to sounds of nature all the way from 1896 to 2008, it's time to look back at those that made the cut. So here are just a few of the nation's best moments of sound, all of which are drenched in history and truly showcase what it's like to be an Aussie!
'Happy Little Vegemites'
You simply can't call yourself Australian without knowing this classic tune. While this TV commercial is the version that most know and love, the original from 1953 is the one to be preserved in all its glory in the archives.
'Down Under' by Men at Work
Since this addictive tune was released in the early '80s, it became an unofficial national anthem not only for locals but for those abroad as well. From the very first few notes to the flute that caused controversy regarding its similarity to fellow Aussie classic tune 'Kookaburra Sits in the Old Gum Tree', this song definitely holds its own as one of the most iconic locally-sourced tunes of all time.
'I've Been Everywhere Man' by Lucky Star
This tongue-twister has been a hit in its own right in Australia, Britain, New Zealand, Germany and the United States (just to name a few) with a new country-specific set of lyrics written for each cover. But Lucky Starr's 1962 version is the original and by far the best as he rattles off 94 Australian towns in just over two minutes!
'C'mon Aussie C'mon' by The Mojo Singers
There's surely nothing more quintessentially Aussie than cricket in the summer, and this tune encapsulates just that. What was originally just written as a jingle for a TV ad, ended up topping the charts as Australians everywhere cheered on their favourite cricketers that were called out in the song including the late great Max Walker.
'From Little Things Big Things Grow' by Paul Kelly and Kev Carmody
This powerful song explains the story of the Gurindji people and Vincent Lingiari who continuously protested their right to their land until the pivotal moment when then-Prime Minister Gough Whitlam returned what was theirs. It was a powerful piece that placed focus on the injustice Aboriginal people faced and became fuel in the fight for racial equality.
'Khe Sanh' by Cold Chisel
Jimmy Barnes knew what he was doing when he created this masterpiece as his first ever single with Cold Chisel. The classic rock tune reflects the experience and emotional aftermath felt by everyday Aussies following the Vietnam War and struck a chord with everyone who experienced the turmoil firsthand.
'Skippy the Bush Kangaroo' by Eric Jupp
What started off as simply a show about a boy and his pet kangaroo quickly became one of the most iconic representations of Australian culture of all time. The theme song is guaranteed to stick in everyone's minds with every child who watched it wishing they could have a friend like Skippy!
'Eagle Rock' by Daddy Cool
You'd be hard pressed to find something catchier than this '70s hit by Melbourne band Daddy Cool. The song has long been a staple at Aussie parties and get-togethers since it was released and has now earned lifetime status in the national archives.
'Living in the '70s' by Skyhooks
Australian music was blessed when this groovy track came on the scene. Not only did it give everyone something fun to dance to but it also gave a pretty idealistic view of what it was really like to be living in the '70s.
'I Was Only 19' by Redgum
With a history so strongly engrained in war and turmoil, it's no wonder so many Australian classics attempt to encompass the traumatic experience through music. This track by Redgum tells the story of a young boy's trials and tribulations in the Vietnam War. The sadness and struggle of a teenager forced to fight for his country is something that has definitely stuck with the Australian public even today.
'Play School Theme Song'
It's been a children's TV staple since 1966 and if you didn't watch it as a kid then you're almost sure to remember the theme song from the kids or grandkids time. While it gets updated every few years, the words always stay the same which means hearing that first line would take Australians of every generation back to a much simpler time!
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Proudly Australian owned and operated | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,194 |
Epistilbit ist ein relativ häufig vorkommendes Mineral aus der Mineralklasse der "Silikate und Germanate". Strukturell gehört er innerhalb der Gerüstsilikate zur Gruppe der Zeolithe. Epistilbit kristallisiert im triklinen Kristallsystem mit der idealisierten chemischen Zusammensetzung Ca3[Si18Al6O48]·16H2O und ist damit chemisch gesehen ein wasserhaltiges Calcium-Alumosilicat.
Epistilbit bildet prismatisch-säulige, häufig nach der c-Achse [001] gestreckte Kristalle mit rhombenförmigem Querschnitt bis zu maximal 3 cm Größe, an denen das Prisma {110} immer trachtbestimmend ist. Nahezu alle Epistilbit-Kristalle sind nach (100) verzwillingt und weisen dadurch ein pseudorhombisches Aussehen auf. Wiederholte Zwillingsbildung nach (110) und (100) führt zu zyklischen oder "V-förmigen" Zwillingen. Die Kopflächen der Kristalle sind oft matt, rau oder "gefrostet". Epistilbit findet sich auch in Form von radialstrahligen, blätterigen und seltener auch körnigen Aggregaten.
Die Typlokalität des Epistilbits ist das Gebiet Breiðdalur–Berufjörður in der Gemeinde Djúpavogshreppur, Austurland, Island, und hier wahrscheinlich der Berg "Teigarhorn" am Berufjord ().
Etymologie und Geschichte
Das heute Epistilbit genannte Mineral wurde erstmals 1826 von Gustav Rose im deutschen Wissenschaftsmagazin Poggendorffs Annalen der Physik und Chemie als ein neues, zur Familie der Zeolithe gehörendes Mineral aus "Island und den Färöer Inseln" beschrieben, wo es in den "Höhlungen eines Mandelsteins" auftreten sollte.
Von den Färöer-Inseln sind allerdings keine modernen Epistilbit-Funde bekannt, so dass wahrscheinlich alle von Rose untersuchten Stufen Material aus Island darstellten. Der wahrscheinliche Erstfundort ist das Ufer des Berufjörður ("Berufjord") am Fuße des Búlandstindur in der Nähe des Observatoriums Djúpivogur.
Rose benannte das Mineral im Hinblick auf die Ähnlichkeit des Minerals mit den Kristallen des Stilbits (). Stilbit war 1797 von Jean-Claude Delamétherie aufgrund seines Perlmutt- bzw. Glasglanzes nach , genannt worden. Der Name Epistilbit hat Mineralogen und Mineralsammler seit seiner Einführung verwirrt. Epistilbit ist eine eigene Mineralspezies und hat in keiner Hinsicht etwas mit Stilbit, Desmin (= Stilbit) oder Epidesmin (= Stellerit) zu tun.
Ein Mineral, welches heute ebenfalls als Epistilbit bezeichnet wird, hatte allerdings schon 1823 Johann Friedrich August Breithaupt als "neue Species der Zeolith-Ordnung erkannt" und als Monophan (nach , mit Rücksicht auf den Schimmer auf der Spaltfläche) beschrieben, weswegen dieser Name eigentlich prioritär wäre. Das Mineral war in der Sammlung von Abraham Gottlob Werner als Adular fehlbestimmt und später von Breithaupt mit dem Epistilbit von Rose zum "Geschlecht Monophan-Zeolith" vereinigt worden.
Als Parastilbit hatte Wolfgang Sartorius von Waltershausen Kristalle bezeichnet, die zusammen mit Chabasit, Heulandit, Desmin und Calcit bei Thyrill am Hvalfiorderstrand im Borgarfjord vorkamen und die sich von Epistilbit angeblich durch größere Härte, Dichte und andere Winkel zwischen den Kristallflächen unterschieden. Bei der Untersuchung des Originalmaterials durch Conrad Friedrich August Tenne im Jahre 1881 stellte sich aber die völlige Übereinstimmung von Parastilbit mit Epistilbit heraus.
Karl Wilhelm Georg Freiherr von Fritsch fand 1866 auf der griechischen Insel Santorin strahlenförmige Büschel aus kurzssäuligen, bis 3 mm langen Kristallen, die er zu Ehren seines Reisegefährten W. Reiss Reissit nannte. Sowohl Karl von Fritsch als auch Friedrich Hessenberg verglichen das Mineral mit Epistilbit, Parastilbit und Monophan – und hielten beide Minerale wegen der unterschiedlichen Winkel zwischen den Flächen für unterschiedliche Spezies. Nach Otto Luedecke und Carl Adolf Ferdinand Hintze sind Epistilbit und Reissit aber identisch.
Als Oryzit (nach oder Orizit) wurde 1879 von Giuseppe Grattarola ein Mineral aus dem Granit des Ganges "Masso Foresi" oder "Fonte del Prete" auf Elba, Italien, bezeichnet. Nach Andreas Arzruni und Paul Heinrich von Groth wohl identisch mit Heulandit, Stefano Merlino zufolge aber identisch mit Epistilbit.
Das vermutliche Typmaterial für Epistilbit (wahrscheinlich der Holotyp) wird in der Sammlung des Museum für Naturkunde – Leibniz-Institut für Evolutions- und Biodiversitätsforschung, Berlin, Deutschland (Katalognummer 1999-0100), aufbewahrt. Zu der 4 × 3 × 2 cm großen Stufe gehört ein von C. S. Weiss geschriebenes Etikett mit einer ausführlichen Beschreibung der Stufe. Laut der Originalbeschreibung von Rose befinden sich weitere Epistilbit-Typproben am Muséum national d'histoire naturelle in Paris, Frankreich.
Klassifikation
In der veralteten, aber teilweise noch gebräuchlichen 8. Auflage der Mineralsystematik nach Strunz gehörte der Epistilbit zur Mineralklasse der "Silikate und Germanate" und dort zur Abteilung der "Gerüstsilikate (Tektosilikate), mit Zeolithen", wo er zusammen mit Barrerit, Brewsterit-Ba, Brewsterit-Sr, Goosecreekit, Heulandit-Ba, Heulandit-Ca, Heulandit-K, Heulandit-Na, Heulandit-Sr, Klinoptilolith-Ca, Klinoptilolith-K, Klinoptilolith-Na, Stellerit, Stilbit-Ca und Stilbit-Na die Untergruppe der "Blätterzeolithe I" mit der System-Nr. VIII/J.23 innerhalb der Zeolithgruppe bildete.
Die seit 2001 gültige und von der International Mineralogical Association (IMA) verwendete 9. Auflage der Strunz'schen Mineralsystematik ordnet den Epistilbit ebenfalls in die Abteilung der "Gerüstsilikate (Tektosilikate) mit zeolithischem H2O; Familie der Zeolithe" ein. Diese ist allerdings weiter unterteilt nach der Gerüststruktur, so dass das Mineral entsprechend seinem Aufbau in der Unterabteilung "Ketten von Fünfer-Ringen" zu finden ist, wo es als einziges Mitglied die unbenannte Gruppe 9.GD.45 bildet.
Auch die vorwiegend im englischen Sprachraum gebräuchliche Systematik der Minerale nach Dana ordnet den Epistilbit in die Klasse der "Silikate und Germanate" und dort in die Abteilung der "Gerüstsilikate: Zeolith-Gruppe" ein. Hier ist er zusammen mit Mordenit, Maricopait, Dachiardit-Ca, Dachiardit-Na, Ferrierit-Mg, Ferrierit-K, Ferrierit-Na, Boggsit, Gottardiit, Terranovait, Mutinait und Direnzoit in der Gruppe "Mordenit und verwandte Arten" mit der System-Nr. 77.01.06 innerhalb der Unterabteilung der "Echten Zeolithe" zu finden.
Chemismus
Nass-chemische Analysen an einem Epistilbit vom Fossárfell am Berufjörður lieferten 57,79 % SiO2; 17,62 % Al2O3; 0,02 % Fe2O3; 0,02 % MgO; 8,21 % CaO; 0,06 % K2O; 12,21 % H2O(+) sowie 3,10 % H2O(-)(Summe 100,42 %).
Auf der Basis von 48 Sauerstoff-Atomen errechnet sich daraus die empirische Formel (Ca2,69Na0,82K0,02)Σ=3,53Al6,35Si17,67O48·15,61H2O, die sich zu Ca3[Al6Si18O48]·16H2O idealisieren lässt.
Bei der Analyse von Epistilbit verschiedener anderer Fundorte wurden SrO und MgO nie und BaO nur selten und in geringen Mengen (z. B. am Finkenhübel bei Glatz (Kłodzko), Woiwodschaft Niederschlesien, Polen, und Berufjörður am Fuße des Berges Búlandstindur, Island) angetroffen. Aus den chemischen Analysen von Epistilbit folgender Fundorte sind die anschließend aufgeführten empirischen Formeln bekannt:
"Fonte del Prete" bei San Piero in Campo, Campo nell'Elba, Elba, Italien: Ca2,2Na0,7K0,3[Al5,5Si18,6O48]·14,5H2O
"Gibelbach" bei Fiesch, Bezirk Goms, Kanton Wallis, Schweiz: Ca2,8Na0,3K0,3[Al6,2Si17,8O48]·15,8H2O
"Berufjord" am Fuße des Búlandstindur, Island: Ca2,3Na1,1K0,1[Al6,0Si18,1O48]·15,2H2O
"Kuroiwa", Präfektur Niigata, Region Chūbu, Honshū, Japan: Ca2,77Na0,35K0,01[Al6,26Si17,82O48]·15,93H2O
"Yugawara", Präfektur Kanagawa, Region Kantō, Honshū, Japan: Ca2,8Na0,4K0,1[Al6,4Si17,7O48]·15,7H2O
Epistilbit weist eine nur sehr geringe Variation bei den austauschbaren Kationen auf. Er ist immer calciumdominant und enthält moderate Mengen an Natrium. Kalium ist selten, Magnesium, Barium und Strontium sind so gut wie nie vorhanden. Ermanno Galli und Romano Rinaldi zufolge schwankt die chemische Zusammensetzung um die "mittlere Formel" Ca2,5Na0,8K0,2(Al6Si18O48)·16H2O, wobei das Verhältnis Ca/(Na + K) zwischen 9 und 2 und das Na/(Na + Ca)-Verhältnis zwischen 0,1 und 0,3 variiert.
Die Elementkombination Ca–Al–Si–H–O weisen unter den derzeit bekannten Mineralen neben Epistilbit insgesamt 29 Spezies auf, darunter befinden sich mit Klinoptilolith-Ca, Cowlesit, Epistilbit, Erionit-Ca, Gismondin-Ca, Gmelinit-Ca, Goosecreekit, Laumontit, Lévyn-Ca, Parthéit, Skolezit, Stellerit, Wairakit und Yugawaralith insgesamt 14 Zeolithminerale.
Aus chemischer Sicht ist Epistilbit ein Dimorph von Goosecreekit – beide Minerale besitzen demzufolge dieselbe chemische Formel CaAl2Si6O16·5H2O, weisen jedoch unterschiedliche Kristallstrukturen auf. Allerdings werden von der IMA zwei unterschiedliche Formeln für die beiden Minerale angegeben – sie können damit nicht dimorph sein.
Kristallstruktur
Epistilbit kristallisiert im triklinen Kristallsystem in der mit den Gitterparametern a = 9,083 Å, b = 17,738 Å; c = 10,209 Å; α = 89,95°; β = 124,58° und γ = 90,00° sowie einer Formeleinheit pro Elementarzelle.
Ältere Strukturverfeinerungen für Epistilbit wurden in der monoklinen durchgeführt.
Slaughter und Kane und Alberti und Kollegen analysierten in der Struktur des Epistilbits niedrigsymmetrische (C2) Domänen (A und B) und erkannten, dass diese gebildet werden, um energetisch ungünstige Tetraeder-Oktaeder-Tetraeder-Winkel von 180° zu vermeiden. Im Gegensatz zum Dachiardit treten diese Domänen nicht im Verhältnis 1:1 auf.
Ping Yang und Thomas Armbruster zeigten, dass die genannten Domänen durch eine zwillingsartige (010)-Spiegelebene erklärt werden können. Sie fanden ferner heraus, dass Epistilbit vom Gibelbach bei Fiesch im Resultat einer (Si,Al)-Ordnung und der Verteilung der Extraframework-Kationen triklin () kristallisiert. Bereits vorher hatten Mizuhiko Akizuki und Hirotsugu Nishido auf der Basis von optischen Untersuchungen eine trikline Symmetrie für den Epistilbit vorgeschlagen. Die Symmetrieverringerung lässt sich durch partielle Si-Al-Ordnung und durch die Verteilung des Calciums in den Kanälen der lockeren Struktur erklären.
In den älteren Strukturbeschreibungen wird ausgeführt, dass das Alumosilicat-Gerüst des Epistilbits Ketten aus Vierer-Ringen enthält, die zu Schichten parallel (010) verknüpft sind. Die Kationen-Positionen, welche drei Sauerstoff-Atome der Vierer-Ringe und sechs Wassermoleküle koordinieren, liegen auf der Spiegelebene. Nach Thomas Armbruster und Mickey Gunter weist die Kristallstruktur des Epistilbits dieselbe Orientierung der Tetraeder in den Schichten aus Sechser-Ringen auf wie Dachiardit – die Spitzen der SiO4-Tetraeder weisen analog diesem Mineral nach oben bzw. nach unten. Die Schichten parallel (010), welche die sehr vollkommene Spaltbarkeit des Epistilbits nach (010) verursachen, sind ferner parallel zur b-Achse [010] durch Vierer-Ringe verbunden, wodurch die Kanäle aus Zehner-Ringen blockiert werden. Offene Kanäle werden durch parallel [001] angeordnete Achter-Ringe begrenzt. Im triklinen Epistilbit sind vier Ca-Positionen in einem Käfig angeordnet, der durch die Zehner-Ringe aus Tetraedern begrenzt wird. Zwei dieser Positionen sind mit den beiden anderen Positionen durch eine Pseudo-Rotationsachse (zweizählige Digyre) verbunden. Dadurch können aufgrund der kurzen Ca-Ca-Distanzen nur zwei Positionen gleichzeitig besetzt werden. Ca besitzt eine tetragonal-antiprismatische Koordination mit fünf H2O-Molekülen und drei gerüstbildenden Sauerstoff-Atomen. Es besteht eine starke Korrelation zwischen der Al-Verteilung in den benachbarten Tetraedern und der Besetzung der vier möglichen Ca-Positionen.
Epistilbit ist strukturell mit Mordenit, den Vertretern der Dachiardit- und der Ferrierit-Gruppe sowie Bikitait verwandt.
Eigenschaften
Morphologie
Die morphologischen Angaben beziehen sich auf die alte monokline Aufstellung!
Epistilbit bildet prismatisch-säulige, häufig nach der c-Achse [001] gestreckte Kristalle mit rhombenförmigem Querschnitt, an denen das Prisma {110} immer dominiert und damit trachtbestimmend ist (vergleiche auch dazu die nebenstehenden Kristallzeichnungen). Dazu treten das schmale Pinakoid {010} sowie das Basispinakoid {001} oder das Pinakoid {10} als Terminierung sowie – bei Zwillingen – ein zweites {001} oder {10} auf der anderen Seite der Zwillingsebene (100). Zu den häufigen Kristallformen zählen {110}; {001}; {01}, {010}, {12}, als selten werden {011} und {11} angegeben. Sehr selten ist findet sich an Epistilbit-Kristallen die matte oder raue Form {02}.
Nahezu alle Epistilbit-Kristalle sind nach (100) verzwillingt und weisen dadurch ein pseudorhombisches Aussehen auf. Seltener ist Estilbit auch nach {110} verzwillingt und bildet so dünntafelige, flache, an Yugawaralith erinnernde Zwillingskristalle. Wiederholte Zwillingsbildung nach (110) und (100) führt zu zyklischen oder "V-förmigen" Zwillingen. Die Kopflächen sind oft matt oder "gefrostet". Die Kristalle des Epistilbits sind mit Längen von 3 bis 10 mm für ein Zeolithmineral relativ klein, können in Ausnahmefällen aber Größen bis zu maximal 3 cm erreichen.
Epistilbit findet sich auch in Form von radialstrahligen, blätterigen und seltener auch körnigen Aggregaten.
Physikalische und chemische Eigenschaften
Die Kristalle des Epistilbits sind meist weiß; aber auch farblos bis weiß, gelblich oder bläulich; blassrosa oder rosarot bis rot. Ihre Strichfarbe ist hingegen immer weiß. Die Oberflächen der durchscheinenden bis durchsichtigen Kristalle zeigen einen charakteristischen glasartigen Glanz sowie auf (010) einen starken Perlmuttglanz. Epistilbit besitzt entsprechend diesem Glasglanz eine mittelhohe Lichtbrechung (nα = 1,485 bis 1,505; nβ = 1,497 bis 1,515; nγ = 1,497 bis 1,519) und eine mittelhohe Doppelbrechung (δ = 0,012 bis 0,014). Im durchfallenden Licht ist der zweiachsig negative Epistilbit farblos und zeigt keinen Pleochroismus.
Epistilbit besitzt eine "ziemlich vollkommene" Spaltbarkeit nach (010). Aufgrund seiner Sprödigkeit bricht das Mineral aber ähnlich wie Amblygonit, wobei die Bruchflächen uneben ausgebildet sind. Epistilbit weist eine Mohshärte von 4,5 auf und gehört damit zu den mittelharten Mineralen. Seine Härte liegt zwischen denen der Referenzminerale Fluorit (Härte 4) und Apatit (Härte 5) – er lässt also mit dem Taschenmesser mehr oder weniger leicht ritzen. Die gemessene Dichte für Epistilbit beträgt je nach Autor 2,22 bis 2,68 g/cm³, die berechnete Dichte 2,266 g/cm³.
Epistilbit zeigt im langwelligen UV-Licht keine Fluoreszenz. Im kurzwelligen UV-Licht (254 nm) kann er eine sehr schwache weiße Fluoreszenz aufweisen.
Das Mineral ist durch konzentrierte Salzsäure, HCl, unter Abscheidung pulveriger Kieselsäure langsam löslich, aber nicht vollkommen zersetzbar. Vor dem Lötrohr ist es zu blasigem Email schmelzbar, ohne sich zur Perle zu runden. Im Kölbchen entweicht Wasser. Geglühter Epistilbit wird nicht mehr angegriffen.
Epistilbit muss normalerweise nur in dem Bleichmittel "Biz bleach" oder einer Seifenlösung eingeweicht und anschließend mit Druckwasser oder Ultraschall behandelt werden, um restliche Lehm- oder Gesteinsteilchen zu entfernen. Da das Mineral nur schwach in Salzsäure löslich ist, kann es zur Entfernung von störendem Calcit für kurze Zeit (2 bis 5 Minuten) in HCl eingelegt werden. Da dadurch aber andere Begleitminerale ebenfalls entfernt werden könnten, ist warme Essigsäure zur Entfernung von Calcit geeigneter. Oxalsäure kann zur Entfernung störender Eisenoxidbeläge (Limonit) und Ascorbinsäure zur Entfernung störender Manganoxide verwendet werden.
Epistilbit ist piezoelektrisch – ungeachtet seiner zentrosymmetrischen Struktur. Er soll ferner auch pyroelektrische Eigenschaften besitzen.
Identifizierung und Unterschiede gegenüber ähnlichen Mineralen
Epistilbit wird gelegentlich mit Stilbit (Epidesmin), Heulandit, Goosecreekit, Yugawaralith und der Orthoklas-Varietät Adular verwechselt. Die Verwechslung mit Stilbit beruht meist nicht auf morphologischen Ähnlichkeiten, sondern lediglich aufgrund der ähnlichen Namen. Trotzdem sind Verwechslungen von Stilbit mit Epistilbit z. B. aus der Grube "Gelbe Birke" bei Schwarzenberg und aus dem Antrim County in Nordirland bekannt. Beide Minerale sind in keiner Beziehung miteinander verwandt. Während Stilbit (bzw. die Vertreter der Stilbit-Gruppe) immer einen quadratischen Querschnitt aufweist und von vier rautenförmigen Flächen oder vier dreieckigen Flächen mit dem Pinakoid {100} terminiert wird, zeigt Epistilbit einen rautenförmigen Querschnitt und nur zwei rautenförmigen Flächen im Bereich der Terminierung. Stilbit-Kristalle sind plattig nach der b-Achse [010] – in derselben Richtung wie seine Spaltbarkeit – entwickelt, während Epistilbit-Kristalle nach der a-Achse [100] abgeplattet sind und eine Spaltbarkeit nach {010} aufweisen. Auch weist Stilbit einen stärkeren Perlmuttglanz auf Spaltflächen auf. Das Röntgendiffraktogramm von Stilbit unterscheidet sich deutlich von dem des Epistilbits.
Heulandit (bzw. die Vertreter der Heulandit-Gruppe) wird mit Epistilbit nur aufgrund der engen Assoziation beider Minerale, der ähnlichen Mineralvergesellschaftung und der gleichen Spaltbarkeit verwechselt. Heulandit ist generell plattig nach der b-Achse, parallel zu seiner Spaltbarkeit, während Epistilbit plattig nach der a-Achse ausgebildet ist und eine Spaltbarkeit nach {010} aufweist. Die Diffraktogramme beider Minerale zeigen deutliche Unterschiede.
Goosecreekit besitzt zwar eine ähnliche Morphologie und Spaltbarkeit wie Epistilbit, allerdings sind die Flächen des Goosecreekits im Gegensatz zu denen des Epistilbits meist charakteristisch gekrümmt, gebogen oder parkettiert. Goosecreekit-Kristalle sind generell isometrisch oder nach der b-Achse gestreckt, während Epistilbit-Kristalle generell eine Streckung nach der c-Achse aufweisen. Auch hier zeigen die Diffraktogramme beider Minerale charakteristische Unterschiede.
Flache lattenförmige Epistilbit-Zwillinge ähneln morphologisch Yugawaralith. Zwar sollte eine sorgfältige Untersuchung eine Unterscheidung zwischen beiden Mineralen ermöglichen, jedoch hilft auch hier eine röntgendiffraktometrische Analyse bei der positiven Identifizierung.
Die Orthoklas-Varietät Adular besitzt eine sehr ähnliche Morphologie sowie vergleichbare optische und chemische Eigenschaften wie Epistilbit. Allerdings weist Adular eine Spaltbarkeit nach {001} auf, während Epistilbit eine Spaltbarkeit nach {010} besitzt. Adular ist zwar in Vulkaniten selten, jedoch häufig in mirolithischen Hohlräumen sowie auf Klüften in granitischen Gesteinen und Gneisen, wo auch Epistilbit auftreten kann. Außerdem ist Adular deutlich härter als Epistilbit. Auch hier sollte für eine positive Identifizierung auf eine röntgendiffraktometrische Analyse zurückgegriffen werden.
Verwechslungen von Adular-Kristallen mit Epistilbit kennt man z. B. aus dem "Waterworks Valley" bei St Lawrence auf der Kanalinsel Jersey sowie aus alpinen Klüften.
Bildung und Fundorte
Bildungsbedingungen
Das Zeolithmineral Epistilbit findet sich fast ausschließlich in Hohlräumen vulkanischer Gesteine wie siliciumreicher tholeiitischer Basalte und dichter Olivinbasalte sowie auf Alpinen Klüften in Gneisen. Möglicherweise handelt es sich deshalb um hydrothermale Bildungen. Häufig stellt Epistilbit eine Bildung zu Beginn einer Zeolith-Kristallisationssequenz dar, wenn sowohl Siliciumgehalt als auch pH-Wert hoch sind.
Ferner wird Epistilbit in Geothermiebohrungen in Basalten auf Island, bei Temperaturen zwischen 80 °C und 160 °C gefunden – dieser Temperaturbereich ähnelt dem von Heulandit, Stilbit und Mordenit. Exotische Vorkommen sind ein Aplitpegmatit auf der Insel Elba sowie ein Dolerit am Mount Adamson, Viktorialand in der Antarktis. Sedimentär gebildete Epistilbite sind unbekannt.
Typische Begleitminerale des Epistilbits sind andere siliciumreiche Zeolithe wie Heulandit, Stilbit und Mordenit sowie Quarz. Als weitere Parageneseminerale des Epistilbits werden Dachiardit, Skolezit, Lévyn, Laumontit, Chabasit, Gyrolith, Pumpellyit, Pyrit und Sphalerit sowie Calcit, Chabasit-Ca und weitere Vertreter der Chabasit-Reihe, Amethyst und Chalcedon, Fluorapophyllit-(K) und Okenit genannt.
Als relativ häufige Mineralbildung wurde der Epistilbit bisher (Stand 2019) von rund 150 Fundpunkten beschrieben.
Die Typlokalität des Epistilbits befindet sich im Gebiet zwischen dem Tal Breiðdalur und dem Fjord Berufjörður in der Gemeinde Djúpavogshreppur, Austurland, Island. Es handelt sich um den Berg Teigarhorn, der eine der berühmtesten Zeolithfundstellen auf Island darstellt und seit 1976 als Naturdenkmal geschützt ist. Epistilbit findet sich hier zusammen mit anderen Zeolithmineralen in tholeiitischen Kliffs nahe dem Bauernhof Teigarhorn.
Fundorte
Angesichts der sehr großen Anzahl an Fundorten für Epistilbit können hier nur einige wenige, vor allem schöne Kristalle liefernde Lokalitäten erwähnt werden. Die schönsten Epistilbit-Stufen stammen von Fundstellen auf Island, aus Indien sowie aus dem Staat Washington, USA. Zu beachten ist, dass das Auftreten von Epistilbit an vielen Fundorten unsicher ist, da flach terminierter Stilbit sehr häufig fälschlich als Epistilbit bezeichnet worden sind.
Europa
Auf Island wurde Epistilbit neben dem "Teigarhorn" im Gebiet Breiðdalur und Berufjörður auch im Bereich der Bergrücken "Rauðafell" und "Fossárfell" sowie bei "Krossanes" südöstlich des Hamarsfjörður, alle in der Gemeinde Djúpavogshreppur, gefunden. Ferner von "Topagata" am Berg Tunga, Hvitarsiða, Gemeinde Borgarbyggð, sowie "Hvalstöðin" (Hvalstöd) und dem Berg Þyrill (Thyrill), beide am Hvalfjörður in der Gemeinde Hvalfjarðarsveit, alle in Vesturland.
Die Färöer gelten, obwohl keine spezifische Fundstelle angegeben wurde, zwar als Cotyp-Lokalität, jedoch hat Volker Betz auf den Färöern keinen Epistilbit finden können. Rudy Tschernich zufolge sollen die als von den Färöern stammend etikettierten Epistilbit-Stufen in den Sammlungen ursprünglich auf Island während Reisen in beide Gebiete geborgen worden sein. Epistilbit von den Färöern wurde von Ermanno Galli und Romano Rinaldi sowie im Jahre 2006 von Ole Jørgensen beschrieben. Epistilbit ist danach in den Oberflächenaufschlüssen auf den Färöern selten, wurde aber im Material aus der Bohrung Lopra-1/1A häufig angetroffen.
In Deutschland wurde Epistilbit aus den Greifensteinen bei Ehrenfriedersdorf sowie aus der Grube "Gelbe Birke" bei Schwarzenberg, beide im Erzgebirge in Sachsen, beschrieben. Allerdings hat sich der Epistilbit aus der "Gelben Birke" als flach terminierter Stilbit erwiesen. Allerdings soll EDX- und XRD-Analysen zufolge doch Epistilbit vorliegen. Zweifelhaft ist ein bereits von Gustav von Leonhard erwähntes Epistilbit-Vorkommen im Basalt bei "Gierswiese unfern Honnef" im Siebengebirge.
In der Schweiz wurden bis zu 4 mm lange Kristalle zusammen mit Heulandit, Stilbit, Laumontit, Chabasit und Chlorit auf Quarz-Kristallen und farblosen, weißen oder grünen Fluorit-Oktaedern gefunden, die in Lettenklüften sitzen, welche durch Gneise am "Gibelbach" bei Fiesch im Tal der Rhone, Bezirk Goms, Kanton Wallis, Schweiz setzen. Epistilbit-Kristalle bis zu 4 mm Länge finden sich, selten in Begleitung von Skolezit, Heulandit, Laumontit, Prehnit, Fluorit, Chlorit, Epidot, Titanit, Albit, Calcit und Apatit, in Hohlräumen der Granite und Gneise bei Biasca, Valle Leventina, Misox, Kanton Tessin, sowie in den Gneissteinbrüchen bei Arvigo, Calanca, Calancatal, Misox, Kanton Graubünden.
In unregelmäßigen, quarzfreien Auslaugungshohlräumen im Eklogit-Amphibolit am "Gradischkogel" auf der Koralpe bei St. Vinzenz nördlich Soboth, Kärnten, Österreich, bildet Epistilbit Kristallrasen, die aus weißen, durchscheinenden, lattigen, maximal 1 mm großen Kristallen aufgebaut sind, welche gelegentlich auf Heulandit sitzen.
Der auf Santorin (ehemals Thera) an der Südküste von Akrotiri in Griechenland am Meer gefundene, in zersetzen trachytischen Laven zusammen mit Calcit und Stilbit auf Quarz vorkommende "Reissit" hat sich als Epistilbit erwiesen.
In Italien bei "Fonte del Prete" unweit San Piero in Campo, Campo nell'Elba, Elba, in Pegmatitgängen, die den miozänen Granodiorit-Stock "Monte Capanne" durchsetzen. Von hier stammt die reiskornartige Varietät "Orizit", die von Dachiardit, Mordenit, Stilbit, Heulandit, Apatit, Pollucit, Quarz, Feldspat und Turmalin begleitet wird. Prismatische, rötliche Epistilbit-Kristalle bis 15 mm Länge, die von Quarz, Calcit und der Chabasit-Varietät Phakolith – sowie seltener Analcim, Heulandit und Laumontit – begleitet werden, stammen aus Vulkaniten bei Osilo unweit Sassari auf Sardinien. Kleine, gestreckte Epistilbit-Kristalle in Begleitung von Chabasit, Calcit, Stilbit, Laumontit und Heulandit stammen von "Punta Santa Vittoria", Pula, Metropolitanstadt Cagliari, Sardinien. Kleinere Epistilbite wurden in Montecchio Maggiore, Provinz Vicenza, Venetien, sowie bei Palagonia, Nördliche Iblea-Hochebene, Metropolitanstadt Catania, Autonome Region Sizilien, geborgen. Schließlich mit Skolezit, Heulandit und Laumontit in Calciumcarbonat/-silicat-Gesteinen bei Novate Mezzola, Sondrio, Region Lombardei.
Weiße bis farblos-durchsichtige, nach {100} und {110} verzwillingte Epistilbit-Kristalle bis zu 2 mm Länge stammen aus mit Quarz ausgekleideten ehemaligen Gasblasen bis zu 10 cm Durchmesser in einem dunkelroten Vulkanit bei Finkenhübel (Mrówieniec), Suszyna unweit Kłodzko (ehemals Glatz), Gmina Radków, Powiat Kłodzki, sowie aus dem Granitsteinbruch "Graniczna III" bei Strzegom (ehemals Striegau) in der gleichnamigen Stadtgemeinde, Powiat Świdnicki, beide in der Woiwodschaft Niederschlesien, Polen.
Aus Rumänien wird Epistilbit als späthydrothermale Bildung zusammen mit Baryt, Montmorillonit und Pb-Zn-Erzen in argillitisierten Andesiten oder basaltischen Andesiten bei Vorța, Kreis Hunedoara im Siebenbürgischen Erzgebirge (Munții Metaliferi) in Siebenbürgen sowie aus ehemaligen Gasblasen in mesozoischen Basalten z. B. bei Vața de Sus unweit Brad im Kreis Hunedoara beschrieben.
In der Slowakei wurde Epistilbit zusammen mit Stilbit, Skolezit, Laumontit, Chabasit, Heulandit, Calcit und Apophyllit in Hohlräumen in einem Andesit bei Šiatorská Bukovinka unweit Fiľakovo im Cerová vrchovina, Okres Lučenec, Banskobystrický kraj.
In Ungarn am Kontakt von granitischen mit andesitischen Gesteinen bei Nadap im Kreis Gárdony, Velence-Gebirge, Komitat Fejér, sowie in Spalten in einem Andesit bei Sátoros, Komitat Nógrád. Ebenfalls aus Spalten im Andesit vom Berg Csák unweit Szob im gleichnamigen Kreis im Komitat Pest.
Im Vereinigten Königreich kennt man Epistilbit von der Insel Rathlin sowie aus Portrush, County Antrim, Nordirland, obwohl es sich bei diesem Material möglicherweise um flach terminierte Stilbit-Kristalle handelt. Aus Schottland in kleinen, blass fleischfarbenen Kristallen aus blasenreichen Basalten in der "Talisker Bay", Island of Skye, sowie zusammen mit Skolezit bei "Dearg Sgeir" auf der Insel Mull, Argyll and Bute. Bei farblos-durchsichtigen, flächenreichen Epistilbit-Kristallen von maximal 2 mm Länge sowie bis 2 cm großen, aber flächenärmeren Kristallen in einem blasenreichen Basalt aus einem Straßenbelag bei "Castle Eden", Hartlepool, County Durham, England, handelt es sich möglicherweise um Material, welches als Schiffsballast, wohl aus Island, nach England gekommen ist.
Asien
In Indien ist "Epistilbit … unter den berühmten 'Poona-Zeolithen' eher selten".
Farblose bis rötliche Kristalle bis zu 2 cm Länge stammen aus mit Quarz ausgekleideten Hohlräumen in grünlichen Pillowbasalten im "Bombay Quarry" bei Khandivali nördlich Mumbai (ehemals Bombay), Mumbai City, Maharashtra. Der Epistilbit wird von Chabasit, Babingtonit, Prehnit, Heulandit, Calcit und Laumontit begleitet, während andere Hohlräume Okenit, Gyrolith, Hydroxyapophyllit, Stilbit, Skolezit und Yugawaralith enthalten können. Milchweiße Epistilbit-Kristalle fand Matthew Forster Heddle in mit Achat (Kascholong) gefüllten Hohlräumen in Basalten bei Igatpuri, nordöstlich Mumbai.
Tafelige Kristalle bis 1,5 cm Länge stammen aus dem Basalt von Aklahare bei Nashik nordöstlich von Mumbai in Maharashtra. Villhori in den "Pandulena Hills" bei Nashik (vgl. dazu die Pandavleni-Höhlen) lieferte außergewöhnliche, bis 3 cm große, farblose bis milchigweiße Kristalle und plattige Zwillinge auf drusigem Quarz. Kleine farblose Epistilbit-Kristalle wurden zusammen mit Goosecreekit und Heulandit in mit Quarz ausgekleideten Hohlräumen im Basalt bei Alibag unweit Mumbai gefunden. Für diesen Fundort sind auch rote, mehrfach verzwillingte Epistilbit-Kristalle typisch.
In den Basalt-Steinbrüchen im Gebiet von Pune im gleichnamigen Distrikt in Maharashtra enthalten in mit Quarz/Chalcedon ausgekleideten Hohlräumen weiße bis rosafarbene, "gefrostete" Epistilbit-Kristalle bis zu 1,5 cm Länge, die meist von cremefarbenem Gyrolith begleitet werden. In den Brüchen von Pune wurde die Sukzession weißer Chalcedon → terminierter farbloser Quarz → Mesolith → Epistilbit → Stilbit → grünlichweißer Fluorapophyllit angetroffen. Im nordöstlich von Pune liegenden Sirur sind kleine Epistilbite und winzige Thomsonit-Aggregate auf haarfeinen Okenit-Nadeln gefunden worden.
Farblose, rosa- und orangefarbene Epistilbit-Kristalle bis zu 1 cm Länge sowie radialstrahlige Aggregate bis zu 1,5 cm Durchmesser treten zusammen mit Stilbit, Heulandit, Mesolith, Fluorapophyllit, Mordenit, Chlorit, Calcit und drusigem Quarz in Basalt in verschiedenen Steinbrüchen bei Savada (Sawda) unweit Jalgaon im Distrikt Aurangabad, Maharashtra, auf.
Weiße, durchsichtige, gestreckte Epistilbit-Kristalle bis 2,5 mm Länge wurden zusammen mit Quarz in Hohlräumen magmatischer Gesteine in den Lokalitäten "Hirogawara", "Okuyugawara" und "Yugawara Hot Spring" bei Yugawara, Präfektur Kanagawa, Region Kantō, Honshū, Japan, gefunden. Aus der Goldlagerstätte der "Hishikari Mine" bei Kagoshima in der Präfektur Kagoshima, Kyūshū. Zusammen mit Skolezit, Laumontit und Lévyn in Andesiten bei "Kuroiwa", Präfektur Niigata, Region Chūbu, Honshū. Kleine, glasglänzende, durchsichtige Epistilbit-Kristalle fanden sich zusammen mit Chabasit und Quarz bei Kumomi unweit Matsuzaki, Präfektur Shizuoka, Region Chūbu, Honshū. Schließlich auch in milchweißen Kristallen in Begleitung von Heulandit und Pumpellyit aus Hohlräumen in Basalten bei Takahagi, sowie bei Furuyada unweit Mitama, beide in der Präfektur Yamanashi.
Ozeanien
Epistilbit aus Basalten nahe North Cape auf der Nordinsel in Neuseeland hat sich als Thomsonit und/oder Stilbit mit flachen Endflächen erwiesen. Auf der Südinsel ist Epistilbit der häufigste Zeolith in untermeerischen gasblasenreichen Andesitflüssen in den nördlichen Takitimu Mountains im westlichen Southland, die aus dem unteren Perm stammen. Epistilbit findet sich in zeolitfaziellen Gesteinen, die aus Laumontit, Analcim, Calcit und Chlorit (mit untergeordneten Gehalten an Prehnit, Pumpellyit und Epidot) bestehen und in allmählich in höhertemperierte Vergesellschaftungen der Prehnit-Pumpellyit-Fazies übergehen. Epistilbit findet sich selten auch zusammen mit Heulandit und Stilbit bei "Stew Point Station" im Tal des Rangitata River, Ashburton District, Region Canterbury.
Amerika
Epistilbit, Wairakit und Lévyn kleiden gasblasenreiche Basalte am "Cerro China Muerta" bei La Amarga, Departamento Catán Lil, Provinz Neuquén im südlichen Argentinien aus.
In Kanada findet sich Epistilbit in British Columbia in Form von farblos-transparenten Kristallen bis zu 2mm Länge am "Gold Pan Camp" bei Spences Bridge im Canyon des Fraser River, Kamloops Mining Division, Goldpan Provincial Park. In Nova Scotia von der Halbinsel Partridge Island bei Parrsboro im Cumberland Co. In den Vulkaniten der Bay of Fundy am "Cape Blomidon", in "Morden" und am "Ross Creek" sowie im stillgelegten Basaltsteinbruch "Arlington Quarry", alle im Kings County, Nova Scotia. In Ontario zusammen mit Analcim, Chabasit, Cowlesit, Heulandit, Faujasit, Garronit, Gismondin, Harmotom, Phillipsit, Laumontit, Mesolith, Stilbit, Thomsonit, Apophyllit, Datolith, Prehnit und Calcit in Gängen im präkambrischen Metapyroxenit bei "Davis Hill Locality", Dungannon Township, Hastings County. In Québec wurde Epistilbit zusammen mit Analcim, Chabasit, Cowlesit, Heulandit, Faujasit, Garronit, Gismondin, Harmotom, Phillipsit, Laumontit, Mesolith, Stilbit, Thomsonit, Apophyllit, Datolith, Prehnit und Calcit in Gängen, die durch präkambrische Metapyroxenite in einem unbenannten Straßenaufschluss bei Laurel sowie bei "Hincks Bridge" und "Notre Dam de la Salette" setzen.
Vereinigte Staaten
Alaska: Winzige, farblose Epistilbit-Kristalle und tafelige Zwillinge bis zu 3 mm Länge stammen von der Kupreanof-Insel südwestlich von Petersburg. Es handelt sich um mit Quarz ausgekleidete Hohlräume in Basalt, in denen auch Calcit und Chabasit auftreten.
Kalifornien: Epistilbit (möglicherweise auch nur flach terminierter Stilbit) wird aus einer Vergesellschaftung mit Kohlenwasserstoffen, Cinnabarit, Metacinnabarit, Pyrit, Markasit, Chalcedon und Calcit in hydrothermalen Quecksilber-Lagerstätten in einem detritischen Serpentinit-Tonstein-Sandstein nahe einer heißen Quelle im "Sulphur Creek District" (Wilbur Springs District) im Colusa County im Bereich der Kalifornischen Küstengebirge beschrieben. Epistilbit tritt ferner zusammen mit Skolezit, Sulfiden und Chlorit in einer Kontaktzone zwischen metamorphosierten oberpaläozoischen Kalksteinen sowie intrusiven Quarz-Monzoniten und Pegmatiten im "Commercial Limestone Quarry" bei Crestmore Heights unweit Riverside im Riverside County. Das anstehende Gestein in der Nähe enthält u. a. Stilbit, Phillipsit, Chabasit, Natrolith, Laumontit, Thomsonit, Skolezit, Gonnardit und Mordenit. Epistilbit-Kristalle bis zu 8 mm Länge finden sich zusammen mit Stilbit und Heulandit in miozänen blasenreichen Basalten im "Little Sycamore Canyon" in den Santa Monica Mountains, unweit Thousand Oaks im Ventura County.
Colorado: Farblos-wasserklare Epistilbit-Kristalle bis zu 4 mm Länge wurden in mit Quarz und Chalcedon ausgekleideten Blasen in Basalten am Südhang des Uncompahgre Peak im San-Juan-Gebirge unweit Lake City im Hinsdale County beobachtet.
Connecticut: Bis zu 2 mm lange, farblose Epistilbite werden in Begleitung von Chabasit und Calcit auf Quarz in Hohlräumen in grünen Vulkaniten entlang des "Wilbur Crass Parkway", Tolland County. Von Brewsterit begleitete, mikroskopisch kleine Epistilbit-Kristalle finden sich in einem Hohlraum in Pyroxen in einer kontaktmetamorphen Zone bei Danbury, Fairfield County.
Hawaii: Exzellente Stufen mit glänzenden, farblos-durchsichtigen, gestreckten Epistilbit-Kristallen bis zu 3 mm Größe stammen aus einem rötlichbraunen, blasenreichen Basalt aus dem stillgelegten "Puu o Ehu Quarry" am Nordufer des Enchanted Lake in den Lanikai Hills, Kailua, Insel Oʻahu, Honolulu City and County. Farblos-durchsichtige Epistilbit-Kristalle bis zu 3 mm Länge fanden sich zusammen mit Chabasit, Analcim, Skolezit, Calcit, Quarz, Pyrit und Aragonit in blasenreichen Olivinbasalten aus dem "Kapaa Quarry", Kailua, Oʻahu. In Vulkaniten aus dem "Kaena Quarry" bei Mokulēʻia auf Oʻahu fand sich Epistilbit in Begleitung von Phillipsit und Aragonit.
New Jersey: Epistilbit, angeblich aus "Bergen Hill", hat sich als Thomsonit erwiesen. Bei "Epistilbiten" aus "Paterson", "Summit" und "Upper Montclair" handelt es sich häufig um flach terminierte Stilbite. Der "Upper New Street Quarry" (Burger's Quarry) bei Paterson, Passaic County, hat hingegen wirklich Epistilbit geliefert.
New York: Aus dem "Baylis Quarry" bei Bedford im Westchester County stammender Epistilbit hat sich als flach terminierter Stilbit herausgestellt.
Oregon: Farblose halbkugelige Epistilbitaggregate bis zu 1 cm Durchmesser, die auf Mordenit sitzen und von radialem Heulandit und massivem Garranit überzogen werden, finden sich in blasenreichen eozänen Basalten am "Neer Road Pit" bei Goble, Columbia County. In den Hohlräumen kristallisierten die Minerale in der Sukzession gediegen Kupfer → Tonminerale → Okenit → Tschernichit-Boggsit → Lévyn-Offretit → Erionit → Heulandit → Mordenit → Epistilbit → Heulandit → Opal → Chalcedon → Seladonit → Okenit → Apophyllit-Stilbit-Apophyllit →Analcim-Cowlesit-Analcim → Garronit-Phillipsit → Lévyn-Thomsonit-Lévyn → Mesolith-Thomsonit → Chabasit → Calcit. Farblos-durchsichtige, flache, nach (110) und (100) verzwillingte Epistilbite wurden zusammen mit Siderit in blasenreichen Basalten entlang des "Clackamas River", Estacada, Clackamas County, beobachtet.
Pennsylvania: "Epistilbit" aus Perkiomenville im Montgomery County ist lediglich flach terminierter Stilbit.
Washington: Außergewöhnliche Stufen mit glänzenden, milchigweißen Epistilbit-Kristallen bis zu 2,5 cm Länge fanden sich auf nadeligem Mordenit und reiskornartigem Quarz in tertiären Vulkaniten entlang des Nordufers des Riffe Lake, Kosmos bei Morton, Lewis County. In bis zu 5 cm Durchmesser aufweisenden ehemaligen Gasblasen in Basalten entlang des "Big Tree Creek", Yacolt, Clark County, wurde Epistilbit in verschiedenen Generationen beobachtet. Zuerst entstandener Epistilbit bildet opake, cremefarbene, traubig-nierige Hohlraumauskleidungen oder bis zu 1 cm große Einzelkristalle auf rosa- bis lachsfarbenem Heulandit und Mordenit. Eine spätere, farblose Generation überwächst präexistente Epistilbit-Kristalle und bildet so Phantomkristalle mit den Formen {110}, {001}, {010} und {12} oder winzige, farblose, flache Zwillinge bis zu 4 mm Größe. Auf dem Epistilbit kristallisierten Lévyn, Gonnardit und Thomsonit. Die Bildungsreihenfolge ist Tonminerale → Mordenit → farbiger Heulandit → Chalcedon → Quarz → Epistilbit → Skolezit → Laumontit → farbloser Heulandit → Stilbit → Epistilbit-Lévyn → Gonnardit → Phillipsit → Thomsonit → Mesolith → Chabasit → Calcit. Winzige farblose Epistilbite wurden selten in mit Quarz ausgekleideten Hohlräumen in blasenreichen Basalten entlang des "First Creek" bei Liberty, Kittitas County beobachtet. Das Mineral tritt auch in alterierten Basalten südlich von Vale auf der Grenze der Counties Thurston und Lewis sowie auf Mordenit im südlich von Kalama, Cowlitz County, gelegenen Steinbruch "Todd Road Quarry" auf. Schließlich wurde farblos-transparenter Epistilbit auch in Begleitung von Heulandit, Stilbit, Fluorit, Datolith, Pyrit und Calcit in Gängen sowie Gasblasen in Vulkaniten bei "Quartz Creek" am Lewis River im Skamania County gefunden.
Verwendung
Obwohl Zeolithe auf unterschiedlichste Weise – z. B. als Ionenaustauscher, Molekularsiebe, für Waschmittel oder als Katalysatoren in der chemischen Industrie – Verwendung finden, besitzt Epistilbit keine wirtschaftliche Bedeutung. So ist er aufgrund seiner Seltenheit nur für den Sammler von Mineralen von Interesse. Neuerdings existieren modifizierte Materialien aus säurebehandeltem Epistilbit, die in der Lage sind, Treibhausgase wie Kohlenstoffdioxid, CO2, und Stickstoffdioxid, NO2 zu adsorbieren.
Ungeachtet seiner geringen Härte und der nur sehr selten ausgeprägten Farben ist Epistilbit auch verschliffen worden, allerdings zumeist nur als Kuriosität. Aus größeren indischen Kristallen sind 4 mm große und 0,12 ct schwere Steine geschliffen worden.
Eventuell stellt Epistilbit ein Gesundheitsrisiko dar, da er möglicherweise sowohl faserbildend und damit Lungenerkrankungen verursachend als auch mutationsauslösend ist.
Siehe auch
Systematik der Minerale
Liste der Minerale
Literatur
Weblinks
Mineralienatlas: Epistilbit (Wiki)
Einzelnachweise
Anerkanntes Mineral
Triklines Kristallsystem
Zeolithe
Calciummineral
Aluminiummineral
Siliciummineral | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,190 |
Versend (; ) község Baranya vármegyében, a Bólyi járásban.
Fekvése
Mohácstól nyugatra helyezkedik el, közvetlenül Bóly északi szomszédságában. A további szomszédos települések: észak felől Liptód, kelet felől Szajk, nyugat felől Monyoród, északnyugat felől pedig Máriakéménd.
Megközelítése
A település kiváló közlekedési adottságokkal rendelkezik, mivel közvetlenül a belterületének déli szélén halad el a Mohácsról Pécsig vezető 57-es főút, és nem sokkal attól délebbre az M60-as autópálya, melynek csomópontja is van itt. A község központjába azonban csak az 56 116-os számú mellékút vezet be, mely kevéssel a 13. kilométere előtt ágazik ki az 57-es útból, északi irányban.
Bóly városával, és azon keresztül Villány, Siklós és Harkány térségével az 5701-es út köti össze – ennek az M60-assal alkotott kereszteződésénél létesült a sztráda itteni csomópontja is –; határszélét délkeleten érinti még a Szajk és Bóly közt húzódó 5714-es út is.
Népesség
A település népességének változása:
A 2011-es népszámlálás során a lakosok 96,7%-a magyarnak, 52,9% cigánynak, 10,4% horvátnak, 6,2% németnek mondta magát (2% nem nyilatkozott; a kettős identitások miatt a végösszeg nagyobb lehet 100%-nál). A vallási megoszlás a következő volt: római katolikus 87,3%, református 1,4%, felekezeten kívüli 4,5% (5,6% nem nyilatkozott).
Története
Ősi magyar település, mely az Árpád-korban már állt.
Nevét az oklevelek 1268-ban említik először Wirsindh néven, 1276-ban Wersund, 1288-ban Wersend, 1292-ben Versundyként írták. 1268-ban hercegi ember nevében tűnt fel. 1276-ban Tanch fia János ispán, - hűtlensége esetére leköti versendi szőlőjét urának, Óvári Konrádnak. 1276 és 1297 között nemesek lakták, 1288-ban baranyai várjobbágyok faluja volt.
A török hódoltság alatt az itt átvonuló csatározások miatt elnéptelenedett. Később német telepesek érkeztek a faluba.
2001-ben a lakosság 12,3%-a horvát, 28% cigány, 7,2% pedig német volt.
Oktatás
A helyi általános iskolába a 2019/2020-as tanévben 106 diák járt.
Közélete
Polgármesterei
1990–1994: Harnisfőger Ádám (független)
1994–1998: Körtési Károly (független)
1998–2002: Körtési Károly (független)
2002–2006: Körtési Károly (független)
2006–2010: Körtési Károly (független)
2010–2014: Kárász István (MSZP)
2014–2019: Kárász István (független)
2019-től: Kárász István (független)
Nevezetességei
Római katolikus templom.
Jegyzetek
Források
Kapcsolódó szócikkek
Baranya megye települései
Baranya vármegye települései | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,202 |
As the origin of an increasing proportion of cultural touchstones, so too has the Internet spawned its own genre of memorabilia. Inspired by "the way designers showcased their work by holding it in front of them," Nadia Ahmad's "Handvas" is among the more successful examples we've seen—a clever way to display a poster or print, modeled after a popular trope of product photography.
In fact, Ahmad isn't a product designer by training or trade: the Sydney-based art director works in advertising by day and simply wanted to make her idea a reality. "I didn't have the skills or knowledge to produce it," she noted by e-mail. "So I went in search of a company that could help [me] bring my idea to life."She eventually teamed up with local design agency Vert Design, working closely with principal Andrew Simpson and industrial designer Eric Siu, who introduced her to 3D printing. It took them nearly a year to bring the concept into production—they experimented with plaster and casting along the way, but Ahmad ultimately concluded that 3D printing offered the highest degree of quality and detail.
Vert Design elaborated on their working relationship:Nadia came to us with such a fun and playful concept that we fully embraced the idea. We begun by creating an articulated model of a human hand which was then digitally manipulated until we were happy with the clamp of the fingers and wrists. This was integral to the outcome as the designs could have been done in a multitude of ways but to stay anatomically true to how a designer would actually hold the posters we employed the digital model.
To produce the final outcome, we 3D-printed the models to ensure all of the detail would be retained and were really pleased with the outcome. We're looking forward to working together with Nadia on more projects in the future.
Handvas is available now at Handvas.com.
I really love these. I also really love had grenade art. It is super fresh. Anyone know of any other hand grenade / street art with molotov cocktails artwork that is out there? It seems pretty rare. I want to collect this because it looks the it could be the first outsider artwork to approach the subject matter of hand grenade. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,049 |
//
// Permission is hereby granted, free of charge, to any person obtaining
// a copy of this software and associated documentation files (the
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// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
// OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
//
#if NET_2_0
using System;
using System.Collections;
using System.Web;
using System.Web.UI;
namespace System.Web.UI.WebControls
{
public sealed class DataKeyArray : ICollection, IEnumerable, IStateManager
{
private ArrayList keys;
bool trackViewState;
public DataKeyArray (ArrayList keys)
{
this.keys = keys;
}
public int Count {
get { return keys.Count; }
}
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get { return false; }
}
public bool IsSynchronized {
get { return false; }
}
public DataKey this [int index] {
get { return (DataKey) keys [index]; }
}
public object SyncRoot {
get { return keys.SyncRoot; }
}
public void CopyTo (DataKey[] array, int index)
{
foreach (DataKey current in this)
array [index++] = current;
}
void ICollection.CopyTo(Array array, int index)
{
foreach(object current in this)
array.SetValue(current, index++);
}
public IEnumerator GetEnumerator()
{
return keys.GetEnumerator();
}
void IStateManager.LoadViewState (object savedState)
{
if (savedState == null) return;
object[] data = (object[]) savedState;
for (int n=0; n<data.Length && n<keys.Count; n++)
((IStateManager)keys[n]).LoadViewState (data [n]);
}
object IStateManager.SaveViewState ()
{
if (keys.Count == 0) return null;
object[] data = new object [keys.Count];
for (int n=0; n<keys.Count; n++)
data [n] = ((IStateManager)keys[n]).SaveViewState ();
return data;
}
void IStateManager.TrackViewState ()
{
trackViewState = true;
foreach (IStateManager k in keys)
k.TrackViewState ();
}
bool IStateManager.IsTrackingViewState {
get { return trackViewState; }
}
}
}
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,957 |
\section{Introduction}
\label{sec:introduction}
In industrial control systems (ICSs), significant effort has been made to generate
network communication whitelists to detect suspicious communications.
However, despite a large number of false alerts caused by a whitelist-based detection,
there have been no methods thus far that quantify the anomalies of communications not present on a whitelist.
Therefore, we propose a new framework for scoring communications not present on a whitelist
to determine whether the communications are normal or anomalous.
We propose a graph convolutional network-based suspicious communication pair estimation (GCN SCOPE), a framework using relational graph convolutional networks (R-GCNs).
The proposed method regards the problem of scoring communications not present on a whitelist as a link prediction problem
in multigraphs (graphs that are permitted to have multiple edges, that is, edges that have the same end nodes),
where the nodes of graphs represent the IP addresses observed in the network,
and the edges of the graphs represent TCP/UDP port numbers used between the two IP addresses.
The importance of ICS protection, including their critical infrastructures such as power equipment
and water processing facilities, has rapidly increased in recent years.
ICSs used to be considered safe against malware or cyberattacks because
they were isolated from enterprise IT systems or the Internet.
However, the growing requirements of remote monitoring, remote operations, and big data management
have rapidly introduced the concept of the Internet of Things (IoT), and hence an increasing number of ICS networks are connecting to IT networks or the Internet.
As a result, many cases of malware infection in ICS networks, resulting in major damage, have been reported.
There have also been cases of social engineering attacks or attacks using removable devices.
Stuxnet\cite{Stuxnet}, which was discovered in 2010, is a type of malware that invades a stand-alone computer system
through USB storage. Targeting Iran's nuclear facilities, it caused severe damage.
Stuxnet is the first well-known example of an ICS being targeted. Moreover, because many general-purpose
PCs are currently being used in ICSs, there are many cases in which they have been accidentally infected with malware
that did not target the ICS system but forced a suspension of the operations.
Unlike a general IT environment, network-based monitoring solutions are preferred in an ICS.
Differing from most general IT environments, ICS environment requires a
continuous and stable device operation.
A general PC receives frequent security updates, but in an ICS environment, down time during an updating procedure is not allowed.
Therefore, in many cases, older system versions continued to run.
Moreover, because some devices are used for a long time (up to 10 to 20 years),
security products may not be supported in certain cases.
In an ICS environment, although a whitelist-based detection method is thought to be effective,
it may cause a significant number of false detections.
With this method, whitelists consist of triplets (server IP address, TCP/UDP port number, and client IP address, hereinafter called a communication triplet), and alerts are raised when communication triplets that have not been previously observed
are noticed\cite{Barbosa13,stouffer11}.
As a result of many false detections, security operators are forced to deal with false alerts, which makes the method impractical.
To solve this problem, we proposed a new framework, shown in Figure \ref{fig:overview}, to score the anomaly of
unobserved communication triplets by learning communication triplets observed in the training data.
The output scores allow us to filter out unimportant detection events, and
focus only on fatal alerts.
Thus, the proposed method enables security operators to concentrate on significant alerts.
We propose GCN SCOPE, a method based on the framework,
utilizing R-GCNs\cite{R-GCN}.
The flow of the proposed method is shown in Figure \ref{fig:prediction_flow}.
An R-GCN is a model proposed by Schlichtkrull et al., that enables accurate link prediction in multigraphs.
The proposed method interprets the communication situation of ICS networks as multigraphs,
the nodes of which represent IP addresses, and the edges of which represent TCP/UDP port numbers,
and estimates the possibility of the emergence of unobserved links.
This method enables us to avoid many false detections that cannot be avoided if we use
a simple whitelisting method, and correctly detect genuinely anomalous communications.
We independently collected network traffic for three manufacturing plants for 2 weeks each.
We use 1 week of data for training and 1 week of data for testing.
To investigate how well GCN SCOPE can distinguish between normal and anomalous triplets,
we use the test triplets as negative samples, and randomly extract triplets as positive samples,
and quantify the performance based on the distinguishability.
The proposed method achieved a receiver operating characteristic (ROC) area under the curve (AUC) of 0.957,
and outperforms baselines including DistMult and two other heuristics.
\noindent
\textbf{Contribution.}
The main contributions of this paper are as follows:
\begin{itemize}
\item We propose a new framework to quantify the anomalies of unobserved communication triplets
(consisting of tuples of the server IP address, TCP/UDP port number, and client IP address) by
learning communication triplets observed in the training data.
\item We developed a method based on the framework above using R-GCNs, and
demonstrated that this method can distinguish communication triplets observed as test data
from randomly extracted anomalous triplets while outperforming the accuracy of
baselines such as DistMult, which uses graph embedding such as in an R-GCN, and heuristics,
which score the triplets using the first- and second-order proximities of the graphs.
\end{itemize}
\noindent
\textbf{Paper Organization.}
The rest of this paper is organized as follows:
Related studies are described in Section \ref{sec:related_works}.
The problem statement is outlined in Section \ref{sec:problem_statement}.
Section \ref{sec:preliminaries} then describes the R-GCNs
used as components of our proposed approach.
We then show the details of our method in Section \ref{sec:proposed_method}.
Section \ref{sec:experiment} describes the data collected
and presents an evaluation of our experiment.
Finally, Section \ref{sec:conclusion} concludes this paper.
\section{Related Studies}
\label{sec:related_works}
Network-based monitoring methods are roughly divided into three types:
signature-based detection, rule-based detection, and anomaly detection.
Although signature-based detection has few false positives,
it can only detect known attacks.
However, an increase in malware variants has decreased the signature detection rate.
Therefore, it is necessary to utilize other monitoring methods.
In rule-based detection, for example, information on the communication server and client pair,
the protocols used, and the time of occurrence
during a specific period are stored as a whitelist,
and communications that are not present on the whitelist are detected.
Rules can also be manually created by referring to such information as specifications.
Because this method stores only normal communications as a whitelist,
it can detect unknown attacks.
However, if the granularity of the rules to be created is too coarse,
there is a risk of passing an attack, and if the granularity of the rule is too fine,
there is a risk of causing many false positives.
Among the rule-based detection methods available,
an approach using communication whitelists is considered to be
particularly effective for ICS networks.
Barbosa et al. proposed a method that learns a set of tuples consisting of the server IP address, client IP address, TCP/UDP,
and port number as a whitelist in a SCADA network,
and when the learning phase ends, an alert is generated for communications
not present on the whitelist. \cite{Barbosa13}.
However, this method causes many false detections in reality,
and as a result, imposes a heavy load on security analysts, and is often impractical during an operation.
GCN SCOPE does not provide a binary output as in a simple whitelisting method
but quantifies an anomalous communication not present on a whitelist,
which allows us to focus only on genuine anomalous communications.
Choi et al. pointed out cases in which we cannot distinguish a client-server correctly,
and where the port numbers used by the clients vary and proposed a method to deal with such problems.
In this paper, we assume we can extract the server ports accurately and provide the same service.
In fact, we can extract such information from our dataset using Zeek (formerly Bro)\cite{Bro}.
Furthermore, the client port number tends to vary each time, and thus we ignore the client port number with our method.
The different types of anomaly detection methods include traffic anomaly detection methods
in which feature vectors are extracted from traffic of the entire network
and anomalies are detected using machine learning.
Yun et al. proposed an anomaly detection method in which the number of packets are observed from the traffic of the ICS network
for each pair of source and destination devices for a certain period
and are converted into feature vectors. An anomaly is detected when the pairs are far from the feature vectors observed during the training phase.
Kitsune\cite{Kitsune} efficiently extracted feature vectors
related to traffic in real-time, such as information related only to the source IP,
information related to the source and destination IPs, and information on the port number, and the
algorithm then detects anomalies using a reconstruction error of the autoencoder.
However, the traffic anomaly detection methods described above work well
only when a large deviation (number of packets, data size, and transmission interval)
occurs compared with usual traffic, and they cannot properly evaluate unobserved communication triplets.
In some cases, a cyberattack can be conducted with an extremely small number of packets or data sizes,
and in such cases, it cannot be detected by the traffic anomaly detection methods described above.
Other anomaly detection methods such as payload-based detection,
and methods based on the N-gram of the byte strings, are well-known \cite{PAYL,ANAGRAM,ZOE}.
However, these methods require a high calculation cost because it is necessary to analyze all packets in the network.
Furthermore, such methods cannot detect attack packets with payloads similar to the byte string of usually observed payloads,
and do not perform well with a protocol containing many random byte sequences\cite{Hadziosmanovic12}.
By contrast, the proposed method is extremely lightweight because it utilizes only the connectivity of the communication,
and can detect the activity of an attacker without depending on the packet payload.
\section{Problem Statement}
\label{sec:problem_statement}
In an ICS network,
whitelisting is considered to be effective as a security monitoring method,
where the whitelists consist of tuples of the server IP address,
TCP/UDP port number, and client IP address (communication triplets) that have appeared in the past,
and a rule-based detection method is applied that generates an alert when a communication triplet
that has not been previously observed is discovered.
For example, communication triplets have the form shown in Figure \ref{fig:triplets}.
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=\linewidth]{triplets.png}
\caption{An example of a communication triplet}
\label{fig:triplets}
\end{center}
\end{figure}
The reasons why this method is suitable for the ICS network are as follows:
\begin{itemize}
\item It operates on the network level, that is, there is no need to modify the devices.
\item In an ICS, new IP addresses and new communication pairs are unlikely to appear as compared with traditional IP networks,
where the number of legitimate connections is too large to be manageable.
\item It can handle proprietary protocols because it does not depend on the packet payload.
\item Although it is relatively easy to execute a lateral movement only within {\it observed IP pairs} in an ICS network
because of the large number of valid communication IP pairs,
it is difficult to execute a lateral movement within {\it observed communication triplets}, which include TCP/UDP port numbers.
\end{itemize}
However, this method causes frequent false detections.
The following patterns are factors that cause false positives:
\begin{enumerate}
\item Insufficient period of whitelist learning
\item Fundamental changes in communication patterns
\item Unusual communications owing to non-steady operation (maintenance, troubleshooting, etc.)
\end{enumerate}
False positives from 1) may be avoided by sufficiently learning the whitelists for a lengthy period.
However, if we wait until the whitelist converges, we cannot detect anomalies for a long time.
Moreover, 2) may occur and the whitelist will need to be retrained before it converges.
Avoiding problem 3) is difficult in principle for whitelisting.
To reduce false positives caused by a simple whitelist-based judgment,
we distinguish whether non-whitelisted communications are caused as the result of a normal ICS environment
operation or the result of an anomalous communication such as a cyberattack.
To realize this, we propose a framework
consisting of the preparation phase, learning phase, and scoring phase, as in Figure \ref{fig:overview}.
\begin{figure*}[!tb]
\begin{center}
\includegraphics[width=\linewidth]{overview.png}
\caption{Framework for communication triplet scoring in this study.
Communication triplet indicating tuples comprising the sets of $(s, p, c)$, where $s$ is the server's IP address,
$p$ is the TCP/UDP port number, and $c$ is the client IP address.}
\label{fig:overview}
\end{center}
\end{figure*}
Normal and anomalous communications can be separated
using an output score and the given threshold value, which is the case with whitelisting.
Therefore, the final output of the framework is the scores of the communication triplets.
We can quantify the anomalous communication triplets from normal whitelisted triplets
because none of the communications in an ICS network appear chaotic.
The protocols of communication may depend on the role of the device,
and some communications may be triggered by other communications.
\section{Preliminaries}
\label{sec:preliminaries}
Our method utilizes R-GCNs \cite{R-GCN} as
the building components and we, therefore, describe such networks in this section.
R-GCNs are an extension of graph convolutional networks (GCNs) \cite{GCN} for multigraphs.
GCNs can be applied to semi-supervised node classification or a graph auto-encoder (GAE) \cite{VGAE}, and
R-GCNs can also be used for such applications.
Herein, we show the graph convolution calculation of R-GCNs,
an embedding learning method using R-GCNs given the learned triplets,
and a link prediction method using a learned node and relation embeddings.
\subsection{Relational Graph Convolutional Networks}
R-GCNs are related to a class of neural networks operating on graphs,
and were developed specifically to deal with the highly multi-relational
data characteristics of realistic knowledge bases.
The following notations are introduced:
Directed and labeled multigraphs are denoted as $G=({\cal V}, {\cal E}, {\cal R})$
with nodes $v_i \in {\cal V}$ and labeled edges (relations) $(v_i, p, v_j) \in {\cal E}$,
where $p \in {\cal R}$ is a relation type.
Motivated by the architectures used by GCNs and other methods,
R-GCNs are defined through the following simple propagation model
for calculating the forward-pass update of the entity or node
denoted by $v_i$ in a relational multi-graph:
\begin{eqnarray}
\label{eq:rgcn}
h^{(l+1)}_i = \sigma\left(\sum_{p\in {\cal R}}\sum_{j \in {\cal N}_{i}^{p}}
\frac{1}{C_{i,p}}W_p^{(l)}h_j^{(l)} + W_0^{(l)}h_i^{(l)}\right),
\end{eqnarray}
where $h_i^{(l)} \in \mathbb{R}^{d^{(l)}}$ is the hidden state of node $v_i$
in the $l$-th layer of the neural network, ${\cal N}_i^p$ denotes
the set of neighbor indices of node $i$ under relation $p \in {\cal R}$,
and $W_p^{(l)}$ denotes the weight matrix for a simple linear transformation
depending on the relation $p$. In addition, $W_0$ is a weight matrix for a self-loop, and
$C_{i, p}$ is a problem-specific normalization constant that can either be
learned or chosen in advance (such as $C_{i,p} = |{\cal N}_i^p|$).
\subsection{Link Prediction Using R-GCN}
Link prediction deals with the prediction of new triplets (i.e., $(subject, relation, object)$).
Formally, a knowledge base is represented by a directed, labeled graph
$G=({\cal V}, {\cal E}, {\cal R})$. Rather than a full set of edges {\cal E},
only an incomplete subset $\hat{\cal E}$ is given.
The task is to assign scores $f(s, p, c)$ to possible edges $(s, p, c)$ to determine
how likely those edges are to belong to ${\cal E}$.
To tackle this problem, a graph auto-encoder model was introduced,
which is comprised of an entity encoder and a scoring function (decoder).
The encoder maps each entity $v_i \in {\cal V}$ to a real-valued vector $e_i \in \mathbb{R}^d$.
The decoder reconstructs edges of the graph relying on the vertex representations;
in other words, it scores $(s, p, c)$-triplets through a function
$f:\mathbb{R}^d \times {\cal R} \times \mathbb{R}^d \to \mathbb{R}$.
The representations through an R-GCN encoder can be computed as
$e_i = h_i^{(L)}$, similar to the graph auto-encoder model introduced by Kipf and Welling\cite{VGAE}.
As the scoring function, DistMult factorization\cite{DistMult} is used,
which is known to perform well on standard link prediction benchmarks when
used on its own. In DistMult, every relation $p$ is associated with a {\it diagonal
matrix} $R_p \in \mathbb{R}^{d \times d}$, and a triple $(s,p,c)$ is scored as follows:
\begin{eqnarray}
f(s,p,c) = e_{s}^TR_pe_c.
\end{eqnarray}
An R-GCN decoder is based on DistMult, and does not explicitly model the asymmetry in the relation; hence,
$f(s,p,c) = e_{s}^TR_pe_c = e_{c}^TR_pe_s = f(c, p, s)$ is true.
The model is trained with negative sampling:
For each observed example $\omega$, negative examples are sampled.
The negative samples are sampled by randomly corrupting either the subject or object
of each positive example.
The model is optimized for a cross-entropy loss to score observable triplets higher than
the negative triplets:
\begin{eqnarray}
\label{eq:learning_gae}
{\cal L} = - \frac{1}{(1+\omega)|\hat{\cal E}|} \sum_{(s,p,c,y)\in{\cal T}} y\log l\left( f(s,p,c)\right) + \nonumber \\
(1-y)\log (1-l\left(f(s,p,c)\right),
\end{eqnarray}
where ${\cal T}$ is the total set of real and corrupted triplets, $l$ is the logistic sigmoid function,
and $y$ is an indicator set to $y=1$ for positive triplets and $y=0$ for negative triplets.
\section{Proposed Method}
\label{sec:proposed_method}
In this section, we show the key idea of the proposed method, GCN SCOPE,
and then describe the details of our approach.
The overall flow of the scoring of communication triplets with the proposed method
is shown in Figure \ref{fig:prediction_flow}.
\begin{figure*}[!tb]
\begin{center}
\includegraphics[width=\linewidth]{prediction_flow.png}
\caption{Link prediction flow of GCN SCOPE}
\label{fig:prediction_flow}
\end{center}
\end{figure*}
\subsection{Key Idea}
As described in Section \ref{sec:problem_statement},
this study aims to detect anomalous communication triplets
based on normal communication triplets observed inside an ICS network.
This problem can be reduced to the problem of link prediction of multigraphs,
which is a task used to predict triplets
that are not clearly given but potentially exist with a high possibility.
The link between particular two devices can be estimated based on the {\it roles} of the devices as represented by their IP addresses,
where the {\it roles} indicate the types of device (such as HMI, PLC, RTU, Data historian, or SIS) and
the communication contents in the network.
We hypothesize that the {\it roles} can be recursively estimated by the connectivity among the {\it roles} of the neighbor devices.
We, therefore, focus on R-GCNs because they allow us to recursively extract embeddings expressing the {\it roles} of the devices.
For a {\it role} estimation, it is important to propagate the {\it role} information of the connected devices.
If a model does not take the features of each node into account and only considers the connectivity information,
the nodes cannot obtain the {\it role} information of the connected devices.
If a model tries to directly optimize the embeddings of each device such as in DistMult\cite{DistMult}
and not propagate the embeddings to the connected devices, then the embeddings of each node tend to overfit the training connections
because the embeddings of each node are independently optimized for the connecting devices.
In fact, as we can see in Section \ref{sec:experiment},
the optimal embedding size of DistMult is smaller than
that of the proposed method.
\subsection{Scoring Communication Triplets}
We describe the details of our method based on
the framework shown in Figure \ref{fig:overview} herein.
\subsubsection{Preparation Phase}
GCN SCOPE uses the information of the communication triplets observed in the target ICS networks.
The communication triplet is the tuple $(s, p, c)$
where $s$ indicates the server IP address, $p$ is a TCP/UDP port number, and $c$ represents a client IP address.
We have to extract all communication triplets $(s, p, c)$s observed during the learning period.
Let the set of all IP addresses, which emerge as the server IP addresses or client IP addresses, be ${\cal V}$,
the set of all the emerging TCP/UDP port numbers be ${\cal R}$, and
the set of all emerging communication triplets $(s, p, c)$ be $\hat{\cal E}$.
Note that the proposed method is based on an R-GCN, which assumes undirected multigraphs.
Therefore, if we observe one communication triplet $(s, p, c)$,
we consider the opposite triplet $(c, p, s)$ as having been observed as well.
The objective of our method is to score communication triplets in
monitored ICS networks, and hence devices outside the target networks should be filtered out.
For example, if a target network allows some devices to communicate with the Internet through a gateway,
various IP addresses on the Internet appear that should be filtered out.
\subsubsection{Learning Phase}
In this phase, a graph autoencoder model using an R-GCN learns the communication triplets observed during the training period.
A part of the inputs of the learning phase is $({\cal V}, \hat{\cal E}, {\cal R})$ obtained during the preparation phase.
In addition to the inputs above, we have to provide some hyper parameters to a graph autoencoder model
such as the dropout rate, number of hidden layer units, L2 regularization weight,
the negative sampling rate, and regularization method (i.e., basis decomposition or block-diagonal decomposition;
for more details, see \cite{R-GCN}) and the parameters described therein.
We then run the learning algorithm of the graph autoencoder using the training triplets $\hat{\cal E}$
by optimizing the loss in Equation (\ref{eq:learning_gae}).
Finally, the model learns the parameters $R_{p_i} \in \mathbb{R}^{d \times d} \ (i \in \{1, \ldots, |{\cal R}|\})$ is the set of diagonal matrices of TCP/UDP port number embeddings
and the weights of the R-GCN parameter
$W_{p_i}^{(l)} \ (i \in \{1, \ldots, |{\cal R}|\}), l$ is the number of hidden layers$)$.
We can also obtain the embedding of the IP addresses,
$e_{i} \in \mathbb{R}^{d} \ (i \in \{1, \ldots, |{\cal V}|\})$,
by calculating a forward-pass update in Equation (\ref{eq:rgcn}).
The overall learning algorithm is shown in Algorithm \ref{alg:learning}.
\begin{algorithm}
\label{alg:learning}
\caption{Communication triplet learning}
\begin{algorithmic}[1] \label{alg:learning}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE ${\cal V}$: the set of observed IP addresses \\
$\hat{\cal E}$: the set of trained triplets $(s, p, c)$ \\
${\cal R}$: the set of observed TCP/UDP port numbers \\
\ENSURE $e_{i} \in \mathbb{R}^{d} \ (i \in \{1, \ldots, |{\cal V}|\})$: the embeddings of IP addresses, \\
$R_{p_j} \in \mathbb{R}^{d \times d} \ (j \in \{1, \ldots, |{\cal R}|\})$: the embeddings of TCP/UDP port numbers
\STATE Learn $R_{p_i} \ (i\in \{1,\ldots,|{\cal R}|\})$ and $W_{p_j}^{(l)} \ (j\in \{1,\ldots,|{\cal R}|\}, l$ is the number of hidden layers$)$
by minimizing the loss in Equation (\ref{eq:learning_gae})
\STATE Calculate $e_{i} \in \mathbb{R}^{d} \ (i \in \{1, \ldots, |{\cal V}|\})$ by calculating the forward-pass update in Equation (\ref{eq:rgcn})
\RETURN $e_{i} \ (i \in \{1, \ldots, |{\cal V}|\})$, $R_{p_j} \ (j \in \{1, \ldots, |{\cal R}|\})$
\end{algorithmic}
\end{algorithm}
\subsubsection{Scoring Phase}
In this phase, we consider the scoring target communication triplet $(\tilde{s}, \tilde{p}, \tilde{c})$.
We first check if $(\tilde{s}, \tilde{p}, \tilde{c})$ is included in the communication triplets
$\hat{\cal E}$ observed during the learning period.
If $(\tilde{s}, \tilde{p}, \tilde{c})$ is included in $\hat{\cal E}$,
$(\tilde{s}, \tilde{p}, \tilde{c})$ is processed as normal and excluded from the scoring target.
Furthermore, if $\tilde{s}$ or $\tilde{c}$ is a new IP address or $\tilde{p}$ is a new TCP/UDP port number,
it is immediately processed as an anomalous triplet.
In addition, $(\tilde{s}, \tilde{p}, \tilde{c})$ is scored using an R-GCN only in cases other than the above two,
that is, cases $\tilde{s}$ and $\tilde{c}$ are in ${\cal V}$ and $\tilde{p}$ is in ${\cal R}$,
although the triplet $(\tilde{s}, \tilde{p}, \tilde{c})$ is not in $\hat{\cal E}$.
In this case, $(\tilde{s}, \tilde{p}, \tilde{c})$ are scored
using the embeddings $e_{i} \in \mathbb{R}^{d} \ (i \in \{1, \ldots, |{\cal V}|\})$
and $R_{p_i} \in \mathbb{R}^{d \times d} \ (i \in \{1, \ldots, |{\cal R}|\})$
obtained during the learning phase.
The embeddings of $(\tilde{s}, \tilde{p}, \tilde{c})$ are $(e_{\tilde s}, R_{\tilde p}, e_{\tilde c})$,
and the score of this triplet is calculated as $e_{\tilde s}^T R_{\tilde p} e_{\tilde c}$.
The overall scoring algorithm is shown in Algorithm \ref{alg:scoring}.
\begin{algorithm}
\label{alg:scoring}
\caption{Communication triplet scoring}
\begin{algorithmic}[1] \label{alg:scoring}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE $(\tilde{s}, \tilde{p}, \tilde{c})$: scoring target triplet, \\
${\cal V}$: the set of observed IP addresses, \\
$\hat{\cal E}$: the set of trained triplets $(s, p, c)$, \\
${\cal R}$: the set of observed TCP/UDP port numbers, \\
$e_{p_i} \in \mathbb{R}^{d} \ (i \in \{1, \ldots, |{\cal V}|\})$: the set of node embeddings, \\
$R_{p_i} \in \mathbb{R}^{d \times d} \ (i \in \{1, \ldots, |{\cal R}|\})$: the set of relation embeddings in a diagonal matrix
\ENSURE $score$: the score of $(\tilde{s}, \tilde{p}, \tilde{c})$
\IF {$(\tilde{s}, \tilde{p}, \tilde{c}) \in \hat{\cal E}$}
\STATE $score = \infty$
\ELSIF {$\tilde{s} \not\in {\cal V}$ or $\tilde{c} \not\in {\cal V}$ or $\tilde{p} \not\in {\cal R}$}
\STATE $score = - \infty$
\ELSE
\STATE $score = e_{\tilde s}^T R_{\tilde p} e_{\tilde c}$
\ENDIF
\RETURN $score$
\end{algorithmic}
\end{algorithm}
\section{Experiment}
\label{sec:experiment}
In this section, we first describe the dataset used in our experiments,
and then explain the baselines used as a comparison with our method.
Finally, we show the results of two experiments.
\subsection{Dataset}
In this paper, we used the network traffic of three factories owned by the company Panasonic for evaluation.
The network monitoring of an ICS is generally conducted by collecting packets
using a mirror port of an L2 switch,
and the datasets used in this paper are collected in the same manner.
Each factory produces different items, and the installed facilities, communication protocols,
and network configurations are completely different.
Along with industrial protocols such as Modbus and Ethernet/IP,
IT-based protocols such as NetBIOS, DNS, HTTP, HTTPS, FTP, SMB, RDP, SSH, MSSQL can be
observed in these factories.
In these datasets, only unicast communications are the targets of learning and scoring,
and multicast and broadcast communications are excluded.
As the reason for excluding these communications, the incorporation of multicasting or broadcasting will result in the establishment of links with IP addresses
not specifically intended for communications.
The nature of each dataset is shown in Table \ref{tb:datasets}.
\begin{table}[tb]
\begin{tabular}{|l|l|l|l|l|}
\hline
& Factory A& Factory B& Factory C \\ \hline
\# of IP addresses & 364 & 150 & 4109 \\ \hline
\# of TCP/UDP ports& 319 & 26 & 328 \\ \hline
\# of training triplets& 2241 & 2081 & 23993 \\ \hline
\# of test triplets & 764 & 558 & 4302 \\ \hline
\end{tabular}
\caption{Number of IP addresses, TCP/UDP port numbers, training triplets, and test triplets
in an ICS dataset.
The training and test triplets are only composed of unicast communications and are each collected during one week.}
\label{tb:datasets}
\end{table}
In Table \ref{tb:datasets}, the numbers of IP addresses, TCP/UDP port numbers, and training triplets
are obtained by counting the numbers of those that appeared during a specific week,
and test triplets are obtained as follows:
\begin{itemize}
\item Test triplets are composed only of triplets from data immediately following the week of training.
\item Triplets included in the training triplets are excluded from the test triplets.
\item Triplets with unobserved IP addresses or TCP/UDP port numbers are also excluded from the test triplets.
\end{itemize}
\subsection{Baselines}
We compare our method with three methods and uniform random scores.
The three methods are {\it DistMult}\cite{DistMult}, {\it first-order proximity-priority heuristic}
and {\it second-order proximity-priority heuristic}.
DistMult is a method for converting each node into the embeddings,
and the DistMult model is optimized using Equation \ref{eq:learning_gae}.
In other words, DistMult is equivalent to an R-GCN without graph convolution layers.
The two heuristics are methods considering first- and second-order proximities.
Many graph-embedding methods are designed to preserve this nature\cite{Goyal18}.
A first-order proximity to node $v_j$ from a view of node $v_i$ is higher
if more edges from $v_i$ to $v_j$ exist.
Let $s_i = [s_{i1}, \ldots, s_{in}]$ denote the first-order proximity
between $v_i$ and the other nodes. Then, the second-order proximity to node $v_j$ from a view of node $v_i$
is determined based on the similarity of $s_i$ and $s_j$.
In this paper, we use two variations as heuristics.
The first is a first-order proximity-first heuristic, and the other is a second-order proximity-first heuristic.
The former outputs a higher score to the node having higher first-order proximity,
and if two nodes have the same first-order proximity, it outputs a higher score to the node
having a higher second-order proximity.
The latter is the opposite method, which prioritizes second-order proximity.
\subsection{Results}
To evaluate GCN SCOPE, we conducted two types of experiments.
The first is a prediction performance experiment in which each method predicts
the existence of communication triplets in the test dataset.
The other is a performance experiment in which each method
distinguishes whether each given communication triplet is in the test dataset
or if anomalous triplets are randomly extracted.
With the proposed method, we utilize R-GCNs with two relational graph convolution layers,
and block diagonal decomposition regularization with a size of 10.
As the reason for using only two graph convolution layers, a
a large number of stacks of graph convolution layers is known to
degrade the performance\cite{QimaiLi18}.
The proposed method and DistMult both have multiple hyper parameters excluding those above.
Therefore, we split the data of factory A into training and validation data,
and search the hyper parameters using Bayes optimization with 50 iterations,
which achieves the best {\it mean reciprocal rank} (MRR)\cite{Bordes13} for the validation data.
The final hyper parameters adopted in the proposed method are as follows:
a dropout rate of 0.2, 100 hidden layer units,
an L2 regularization weight of 0.0, a learning rate of 0.01, and a
negative sampling rate of 10.
The final hyper parameters adopted in DistMult are as follows:
50 hidden layer units,
an L2 regularization weight of 0.01, a learning rate of 0.02,
and a negative sampling rate of 10.
\subsubsection{Link Prediction for Test Triplets}
We evaluate the performance of our method and the baselines using the {\it mean reciprocal rank} (MRR) and {\it Hits at n}(H@n),
where the models of each method learn the training triplets of each dataset and output the scores of the test triplets.
\[
MRR = \frac{1}{|Q|}\sum_{i=1}^{|Q|}\frac{1}{{\rm rank}_i},
\]
where ${\displaystyle {\text{rank}}_{i}}$ refers to the rank position of the correct answer for the i-th query.
Here, H@n is the proportion of the correct entities that are ranked within the top n.
This is the standard metric for link prediction methods.
The results are shown in Table \ref{tb:result1}.
Each metric is evaluated in the filtered settings.
\begin{table*}[tb]
\centering
\begin{tabular}{llllllllllllllllllll}
\hline
& \multicolumn{4}{l}{Factory A} & & \multicolumn{4}{l}{Factory B} & & \multicolumn{4}{l}{Factory C} &\\ \cline{2-5} \cline{7-10} \cline{12-15} \cline{17-20}
& MRR & \multicolumn{3}{l}{Hits@n} & & MRR & \multicolumn{3}{l}{Hits@n} & & MRR & \multicolumn{3}{l}{Hits@n} &\\ \cline{1-5} \cline{7-10} \cline{12-15} \cline{17-20}
Model & & 1 & 3 & 10 & & & 1 & 3 & 10 & & & 1 & 3 & 10 &\\ \hline
\begin{tabular}[c]{@{}l@{}}GCN SCOPE (proposed)\end{tabular} & {\bf 0.240} & {\bf 0.172} & {\bf 0.238} & {\bf 0.366} & & {\bf 0.210} & {\bf 0.108} & {\bf 0.222} & 0.395 & & {\bf 0.291} & {\bf 0.167} & {\bf 0.423} & {\bf 0.504} & \\
DistMult\cite{DistMult} & 0.177 & 0.096 & 0.198 & 0.334 & & 0.161 & 0.047 & 0.158 & {\bf 0.435} & & 0.149 & 0.079 & 0.174 & 0.298 &\\
1st-order proximity-first heuristic & 0.182 & 0.122 & 0.189 & 0.312 & & 0.063 & 0.006 & 0.041 & 0.159 & & 0.192 & 0.120 & 0.263 & 0.287 &\\
2nd-order proximity-first heuristic & 0.168 & 0.101 & 0.179 & 0.277 & & 0.055 & 0.009 & 0.031 & 0.116 & & 0.151 & 0.056 & 0.244 & 0.286 & \\
Random & 0.016 & 0.001 & 0.005 & 0.021 & & 0.040 & 0.007 & 0.022 & 0.070 & & 0.002 & 0.000 & 0.001 & 0.002 &\\ \hline
\end{tabular}
\caption{Link prediction results for triplets observed in test data on real ICS datasets.}
\label{tb:result1}
\end{table*}
As shown in Table \ref{tb:result1}, GCN SCOPE outperforms the baselines in almost every case,
which means that GCN SCOPE shows a high performance in link prediction for the communication triplets of ICS networks.
\subsubsection{Distinction between Normal and Anomalous Links}
To investigate how well GCN SCOPE can distinguish between the normal and anomalous triplets,
we quantify the distinguishability of each method
using the AUC of the ROC curve.
We use the test triplets as negative samples, and randomly extract triplets as positive samples.
Although it is preferable to generate anomalous triplets based on known cyberattacks or a malware strategy,
it is difficult to know the probability distribution, and thus random triplets are used instead.
We generate random communication triplets by choosing two different IP addresses and TCP/UDP port numbers separately and uniformly at random
from those observed in the training data,
A total of 500 of these anomalous communication triplets are extracted in each dataset,
and each is composed only of triplets not included in the training and test triplets.
The evaluations are executed in two ways:
One method is a score-based evaluation (Table \ref{tb:result2_score_based}),
and the other is a rank-based evaluation (Table \ref{tb:result2_rank_based}).
A score-based evaluation considers how a threshold judgment is executed using the original output scores.
A rank-based evaluation is based on the harmonic mean of $score = \frac{1}{rank_s} + \frac{1}{rank_c}$,
where $rank_s$ is calculated by finding the ranking of $(s,p,c)$ of all scores of the filtered communication triplet $(s,p,*)$,
and $rank_c$ is calculated by finding the ranking of $(s,p,c)$ of all scores of the filtered communication triplet $(*,p,c)$.
This is an evaluation metric based on the same idea as that used in an MRR.
\begin{table}[tb]
\centering
\begin{tabular}{ccccc}
\hline
Model & Factory A & Factory B & Factory C \\ \hline
\begin{tabular}[c]{@{}c@{}}GCN SCOPE (proposed)\end{tabular} & {\bf 0.962} &{\bf0.914} &{\bf 0.996}\\
DistMult\cite{DistMult} & 0.262 & 0.668 & 0.488 \\
1st-order proximity-first heuristic & 0.853 & 0.735 & 0.771 \\
2nd-order proximity-first heuristic & 0.820 & 0.632 & 0.769 \\
Random & 0.512 & 0.521 & 0.519 \\ \hline
\end{tabular}
\caption{ROC AUC of {\it score-based} link distinction on real ICS datasets.}
\label{tb:result2_score_based}
\end{table}
\begin{table}[tb]
\centering
\begin{tabular}{ccccc}
\hline
Model & Factory A & Factory B & Factory C \\ \hline
\begin{tabular}[c]{@{}c@{}}GCN SCOPE (proposed)\end{tabular} & {\bf 0.903} & {\bf 0.764} & {\bf 0.989} \\
DistMult\cite{DistMult} & 0.710 & 0.745 & 0.900 \\
1st-order proximity-first heuristic & 0.767 & 0.708 & 0.767 \\
2nd-order proximity-first heuristic & 0.768 & 0.680 & 0.766 \\
Random & 0.554 & 0.521 & 0.532 \\ \hline
\end{tabular}
\caption{ROC AUC of {\it rank-based} link distinction on real ICS datasets.}
\label{tb:result2_rank_based}
\end{table}
From these evaluations, GCN SCOPE with a score-based judgment enables us to
distinguish normal triplets from anomalous triplets with high accuracy.
The performance of a score-based evaluation of DistMult is
significantly inferior to that of a rank-based evaluation,
the reason for which is considered to be based on the case of DistMult, in which all embeddings of the devices are optimized independently,
which causes a different scale of the scores in terms of the device.
By contrast, in the case of an R-GCN,
the embeddings are jointly optimized unlike with DistMult because of the effect of the graph convolution.
\section{Conclusion}
\label{sec:conclusion}
In this study, we developed GCN SCOPE, a method that learns the embeddings of IP addresses using
the learning of R-GCNs from observed communication triplets
(consisting of tuples of server IP addresses, TDP/UDP port numbers, and client IP addresses),
and scores unobserved communication triplets.
To the best of our knowledge, the proposed method is the first to quantify anomalous communication triplets
that have not previously been observed.
With the proposed method, a multigraph is constructed using communication triplets
observed in an ICS network, R-GCNs models are learned using the extracted multigraph, and
unobserved communication triplets are scored using learned R-GCNs models.
The proposed method achieved an average AUC of ROC curve of 0.957,
which outperforms the AUCs of comparative methods such as DistMult, a method that directly optimizes the node embeddings,
and heuristics, which score triplets using the first- and second-order proximities of multigraphs.
This means that the operators of a security operation center using communication whitelisting
are unleashed from the processing of large numbers of unimportant alerts.
In future studies, the performance changes will be observed using scoring functions when
considering the direction of the edges, such as in TransE\cite{Bordes13} or ConvE\cite{ConvE}.
It also seems promising to use not only the edge types such as the TCP/UDP port numbers but
also the communication data size or communication time interval to enhance the accuracy of anomalous link detection.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,781 |
Electorate analysis: To be vacated at the coming election by Labor member Martin Whitely, the seat of Bassendean extends northwards from the Swan River to the Reid Highway, from Bassendean and Ashfield through Morley and Lockridge to Beechboro. The one-vote one-value redistribution at the 2008 election stripped the electorate of the Caversham and Swan Valley outskirts, and the current redistribution has added 3000 voters between Beechboro Road and Tonkin Highway in Morley and Beechboro for the loss of 1000 in Bayswater at the city end of the electorate, with no impact on the margin.
Bassendean has been safe for Labor since its creation in 1996, being held by former Morley MP and later Gallop government minister Clive Brown until 2005, and thereafter by Martin Whitely, previously member for Roleystone. Whitely came to the seat when the party's national executive intervened at Geoff Gallop's behest to protect all sitting members, heading off a preselection war threatened by the redistribution which abolished Whitely's seat. The process alienated Whitely from the AMWU Left faction, leaving him factionally unaligned. After retaining his endorsement in 2008 with the support of Alan Carpenter, Whitely announced in January 2012 that he would not seek another term. He also took the opportunity to call on Eric Ripper, shortly to be unseated as leader by Mark McGowan, to follow his example.
Whitely had been a critic of the concentration of preselection power in the hands of union leaders, specifically Dave Kelly of the Left faction Liquor Hospitality and Miscellaneous Workers Union (now United Voice) and Joe Bullock of the Right faction Shop Distributive and Allied Employees Association. It was Kelly who would emerge as his preselection successor, after more than a decade as the union's state secretary and figurehead of its attendant sub-faction. Whitely has nonetheless endorsed Kelly, while saying he plans to run for Senate preselection in opposition to the anticipated nomination of Bullock, who is believed to have the numbers sewn up. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,227 |
Aludel (via spanskan från arabiskan, al-uthal =kärl) är en enkel destillationsanordning, konstruerad av päronformade, i båda ändar öppna och i varandra trädda lerkärl.
Aludlar användes tidigare i Spanien, särskilt för framställning av kvicksilver.
Källor
Nationalencyklopedin multimedia plus, 2000
Laboratorieutrustning | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,492 |
Marcin Bełdowski herbu Jastrzębiec – wojski kruszwicki w latach 1668-1698, pisarz grodzki inowrocławski.
Członek konfederacji województw kujawskich w 1670 roku.
Był elektorem Michała Korybuta Wiśniowieckiego z województwa brzeskokujawskiego w 1669 roku. W 1674 roku był elektorem Jana III Sobieskiego z województwa brzeskokujawskiego.
Przypisy
Bibliografia
Suffragia Woiewodztw, y Ziem Koronnych, y Wielkiego Xięstwá Litewskiego, zgodnie na Naiaśnieyszego Jana Trzeciego Obránego Krola Polskiego, Wielkiego Xiążęćiá Litewskiego, Ruskiego, Pruskiego, Mázowieckiego, Zmudzkiego, Inflantskiego, Smolenskiego, Kijowskiego, Wołhynskiego, Podolskiego, Podláskiego, y Czerniechowskiego Dáne między Wárszawą á Wolą / Dnia Dwudziestego pierwszego Máiá / Roku 1674, [b.n.s.]
Urzędnicy kujawscy i dobrzyńscy XVI-XVIII wieku. Spisy". Oprac. Krzysztof Mikulski i Wojciech Stanek przy współudziale Zbigniewa Górskiego i Ryszarda Kabacińskiego. Kórnik 1990, s. 216.
Marcin Bełdowski
Członkowie konfederacji województw kujawskich 1670
Członkowie stanu rycerskiego I Rzeczypospolitej
Elektorzy Jana III Sobieskiego
Elektorzy Michała Korybuta Wiśniowieckiego
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Konfederaci I Rzeczypospolitej (województwo brzeskokujawskie)
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Pisarze grodzcy inowrocławscy
Urzędnicy ziemscy I Rzeczypospolitej
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\section{Introduction}
Global mobile data traffic has recently seen a significant increase. In order to address this challenge, the ultra-dense network (UDN) is being considered as one of the potential solutions, which is realized by deploying small base stations (SBSs) densely at the traffic hotspots. In cellular networks, to maintain the service, the user equipment (UE) changes its serving base station (BS) as it moves, which is called handover (HO). Traditionally, the HO process is designed for macro-cell systems and triggered by the event A3 of UE \cite{RRC}, when the difference between the periodically measured Reference Signal Received Power (RSRP)/ Reference Signal Received Quality (RSRQ) of a candidate cell (target cell) and that of the serving cell is higher than the HO hysteresis margin (HHM). Once A3 is triggered, the UE will wait for a predetermined time duration, i.e., time-to-trigger (TTT); afterwards, if the triggering condition keeps being satisfied, the HO is finalized and the UE is connected to the target cell. In the traditional macro-cell system, static HHM and TTT strategies are usually adoptable.
However, in UDNs, the HO strategy for macro-cell systems may lead to the frequent HO problem, where the HO process can be triggered even with a little movement of the UE \cite{7572176}. Frequent HOs may diminish the capacity gain obtained by network densification as the HO processes break the data flows while terminating the serving link and reestablishing the link to the target BS, which is called the HO delay \cite{7792669}. Moreover, the increased HO overheads from the serving BSs, target BSs, UEs, and core networks consume more energy and other resources. Hence, to utilize the full potential of UDN, it is essential to maintain an appropriate HO rate, defined as the expected number of HOs per unit time. In \cite{6477064,7110473}, the performance analysis over HO rates was conducted for the irregularly shaped network topologies, where no solutions were proposed for the frequent HO problem.
One way to optimize the HO process is tuning the HO parameters adaptively by implementing threshold comparisons with several specific metrics \cite{6067552,5629008,7341220}. In \cite{6067552}, an adaptive HHM approach was proposed to lower the number of HOs, which uses a predefined RSRQ threshold and path loss factor to adapt the HHM. In \cite{5629008}, the authors defined a weighted-sum cost function, which consists of key parameters related to the UE speed, cell load, and number of user connections. Then, the cost function was integrated into a typical RSRP-based procedure as an additive factor to change HHM adaptively. In \cite{7341220}, a fuzzy logic controller that can tune the HHM adaptively was proposed to decrease the signaling load caused by HO. However, the above threshold-comparison based HO-tuning methods lack systematic methodologies to optimize the algorithm-related parameters, e.g., the number of rule bases \cite{7341220}, which may not be acceptable in practical applications.
Another type of HO optimization strategies is based on dynamic programming (DP). Specifically, the HO decision process is modeled as a Markov Decision Process (MDP) \cite{712192} and the objective is to obtain the optimal policy, which is the probability distribution of actions (HO decisions) conditioned on the input state. In \cite{7725961}, an upper bound for the HO performance in UDNs was derived, where the Viterbi algorithm was used to derive the optimal HO decision policy under the ideal assumption that the positions of SBSs or the UE trajectories are available in advance. In \cite{6777395}, one HO decision policy was proposed to minimize the HO rate while maintaining certain system throughput. Specifically, with prior knowledge of the cell load transition probabilities, the state observations were first converted into belief states. Then the optimal policy to maximize the expected reward over the belief state was derived. In \cite{7314972}, a TTT selection policy was proposed by assuming the knowledge of the UE trajectory distribution and channel propagation model. All the above results \cite{7725961,6777395,7314972} adopted a similar methodology, which is to first compute certain transit probabilities with the knowledge of some network dynamics, e.g., UE trajectories \cite{7314972} or load state transition probabilities \cite{6777395}, and then use the derived transit probabilities to derive the optimal HO decision policy. In other words, the above methods to obtain the HO decision policy are falling into the traditional model-based framework. However, in practice, it is difficult to assume strong prior knowledge on network dynamics as required in the model-based methods. Hence, it is highly desirable that the method to derive the optimal HO decision policy does not depend on any prior knowledge of network dynamics, i.e., we need to seek model-free or data-driven approaches.
The most similar work to our proposed method is given in \cite{7572176,7925950}, where the authors proposed some model-free learning-based approaches to solve the frequent HO problem. In particular, the upper confidence bandit (UCB) algorithm was used to derive the optimal HO policy, which minimizes the number of HO occurrences while ensuring certain throughput, outperforming the 3GPP protocol by up to $80\%$. Moreover, the UCB algorithm samples rewards from the reward distribution of SBSs continually and tries to find the single best SBS with the largest mean reward to camp on consistently.
However, two major issues exist in applying the method mentioned above to the considered mobility optimization problem. First, in large-scale systems including different scenarios, e.g., offices vs. malls, the mobility patterns may vary dramatically across different areas, which needs different optimal controllers. Hence, when the UEs enter some new scenarios, they need to learn from random starting points under the setup in \cite{7572176,7925950}, which may degrades the performance; second, for newly arriving UEs in one scenario, the learning has to start from some possibly ill-performed initial points, such that the UEs may suffer from unacceptable performance, possibly over a long period.
Here, we propose a two-layer framework to address the above issues. Specifically, we first adopt a centralized controller that partitions UEs into clusters with unsupervised learning \cite{WARRENLIAO20051857}, where the UEs in different areas can have different mobility patterns. After the clustering, the HO processes of UEs in the same cluster are modeled as the similar MDPs. We adopt the reinforcement learning (RL) framework to learn the optimal controller for each UE, which makes HO decisions. We incorporate the situation and exploration information of UEs into states to model the HO process, where the exploring information, i.e., the serving SBS indices for UEs, can facilitate the exploitation versus exploration tradeoff to accelerate the learning. However, such an adopted state space could be large, which is hard to track. Recently, deep neural network (DNN) \cite{mnih-dqn-2015} was proposed to approximate the values of $Q-$functions\footnote{The action value function, $Q-$function, is used to generate appropriate policies, which will be introduced later. } for all the states, which is called DQN. To make the DQN more suitable to RL\footnote{The DQN works well when the data distribution is stationary, which is not the case in most RL applications. In particular, the policy improvement process, which will be introduced later, can dramatically change the state occurrence probability. Moreover, the consecutive states are correlated strongly in RL systems.}, the authors in \cite{mnih-dqn-2015} adopted the techniques of experience replay and delayed updated target networks to stabilize the learning and prevent divergence of RL algorithms. Many approaches were later proposed to improve the stability, convergence, and learning speed of DQN, e.g., the dueling network structure \cite{Wang2016} and the fast learning technique \cite{He2017}.
In this paper, our contributions are summarized as follows.
\begin{itemize}
\item Firstly, we propose the two-layer framework to learn optimal HO controllers in large-scale systems with different scenarios. In particular, the centralized controller can cluster the UEs according to their mobility patterns with unsupervised learning, where the mobility patterns are similar in the same cluster. We then use the RL framework to obtain the optimal controller for each UE with the same cluster.
\item Secondly, we adopt a model-free asynchronous advantage actor-critic (A3C) \cite{pmlr-v48-mniha16} RL framework to achieve the optimal HO policy in each cluster. Note that such an asynchronous framework in A3C can accelerate learning when the number of UEs increases, while the learning time increases at least linearly against the number of UEs with the introduced methods in \cite{7572176,7925950}. Also, we propose to use certain situation and exploration information as the states to derive better performance.
\item Thirdly, we utilize DNN to approximate the $Q-$function and generate the policy in A3C. Due to the generalization ability of DNN \cite{712192}, such function approximators can represent the whole state space with limited weights, which can avoid the degraded performance for the clustered newly-arriving UEs during the learning transitions. Specifically, after the clustering for the newly-arriving UEs, the clustered newly-arriving UEs fetch the DNN weights from global parameter servers of their clusters as the pretrained networks, which are better than random initialized networks. We then propose two methods to utilize this pretained networks: on-line vs. off-line. In the on-line method, the UEs keep learning and fetch the weights from the parameter server as the pretrained network periodically. While in the off-line method, the UEs treat the pretrained networks as static controllers.\footnote{It is worth noting that if the DNN cannot generalize the whole state space, it may lead to degraded performance for the UEs using the pretrained network.}
\item Finally, to improve the performance at the very beginning for both the on-line and off-line methods, i.e., at the time before the RL algorithm is executed to learn the DNN, we initialize the DNN with supervised learning (SL), where the training data set is from the output of the traditional HO scheme proposed in 3GPP.
\end{itemize}
The rest of the paper is organized as follows. The preliminaries about MDP and RL are presented in Section II. The system model is introduced in Section III, with the clustering process in section III-A, the asynchronous joint learning framework in Section III-B, the state vector design in III-C, and the reward signal design in Section III-D. In Section IV, the policy optimization problem is formulated in Section IV-A; the learning algorithm is discussed in Section IV-B; the algorithm implementation framework is explained in Section IV-C; the DQN structure and network initialization with supervised learning are presented in Section IV-D and Section IV-E, respectively. Simulation setups and results are provided in Section V. Finally, Section VI concludes the paper.
\section{Preliminary}
Before proceeding to describe the system model, we first describe the necessary frameworks, i.e., the Markov decision process (MDP) and the general RL algorithm frameworks, where the value function and policy gradient methods are introduced, which will be utilized in our learning scheme. Finally, a special kind of recurrent neural network (RNN), i.e., the long-short term memory (LSTM) network, is presented, which will be utilized as the DQN function approximator in the proposed RL framework.
\subsection{Markov Decision Process}
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{RLframework.eps}
\caption{ General framework of RL algorithm. }
\label{fig1}
\end{figure}
Typically, the RL problem is modeled as a MDP, i.e., $\left\langle {{\bf{S}},{\bf{A}},{\bf{P}},{\bf{\pi}}, r,\gamma } \right\rangle $, which is composed of a finite state space ${\bf{S}}$, an action space ${\bf{A}}$, a transit probability set ${\bf{P}}$ mapping each point $({\bf{s}},{\bf{a}}) \in {\bf{S}} \times {\bf{A}}$ to the next state ${\bf{s}}' \in {\bf{S}}$, a policy ${\bf{\pi}}:{\bf{S}} \to {\bf{A}} $, a reward function $r: {\bf{S}} \times {\bf{A}} \to \mathcal{R}$, and a discount factor $\gamma$ penalizing the future rewards. Specifically, at time step $t$, the agent senses the state ${\bf{s}}_{t}$ and takes action ${\bf{a}}_{t}$ according to policy $\pi({\bf{a}}_{t}|{\bf{s}}_{t})$. Then the environment feeds back the reward signal $r_t$ and the new state ${\bf{s}}_{t+1}$ to the agent. Hence, the general framework of RL is shown in Fig. \ref{fig1}. The goal of the agent is to obtain a policy maximizing the average cumulative reward in the long run. In a typical RL framework, two value functions are defined as:
\begin{align}
&Q^{\pi}({\bf{s}}_{t},{\bf{a}}_{t})=E_{\pi}\left(\sum\limits_{k = 0}^{T-t-1} {\gamma^k {r_{t+k}}} | {\bf{s}}_{t},{\bf{a}}_{t} \right)= \sum\limits_{k = 0}^{T-t-1} {{\gamma ^k}{{({P^{\pi} })}^k}} {r_{t}}\label{1},\\
&V^{\pi}({\bf{s}}_{t})=E_{\pi}\left(\sum\limits_{k = 0}^{T-t-1} {\gamma^k {r_{t+k}}} | {\bf{s}}_{t}\right)=\sum\limits_{{\bf{a}}_{t}}\pi({\bf{a}}_{t}|{\bf{s}}_{t}) Q^{\pi}({\bf{s}}_{t},{\bf{a}}_{t})\label{2}
\end{align}
with $P^{\pi}$ defined as an operator over reward $r_{t}$, i.e.,
\begin{equation}
({{P^{\pi} }})r_{t} \buildrel \Delta\over = \sum\limits_{{\bf{s}}_{t+1},{\bf{a}}_{t+1}} {P({\bf{s}}_{t+1}|{\bf{s}}_{t},{\bf{a}}_{t})\pi ({\bf{a}}_{t+1}|{\bf{s}}_{t+1})r_{t+1}},
\label{21}
\end{equation}
where $(X)^k$ denotes $k$ successive applications of operator $X$, $P({\bf{s}}_{t+1}|{\bf{s}}_{t},{\bf{a}}_{t})$ is the transmit probability and $T$ is the terminal time step\footnote{If $T<\infty$, it is a episodic task, which means that the agent-environment interaction breaks naturally into episodes and each episode ends in a special state called the terminal state. Otherwise, it is a continuing task without a terminal state. }. Moreover, $Q^{\pi}({\bf{s}}_{t},{\bf{a}}_{t})$ and $V^{\pi}({\bf{s}}_{t})$ are called action-value and state-value functions, respectively. We see that the action-value function $Q^{\pi}({\bf{s}}_{t},{\bf{a}}_{t})$ (or state-value function $V^{\pi}({\bf{s}}_{t})$) describes the expected discounted sum of rewards starting from state-action pair $({\bf{s}}_{t},{\bf{a}}_{t})$ (or state ${\bf{s}}_{t}$) over a given policy $\pi$. Hence, the goal of an RL algorithm is to obtain the optimal policy $\pi^*$ by solving following problem:
\begin{equation}
\pi^* = \arg \max\limits_\pi V^\pi({\bf{s}}_{t})\label{3}
\end{equation}
Furthermore, we define two operators as
\begin{align}
&\left( {{\Gamma ^\pi }} \right)Q({\bf{s}}_{t},{\bf{a}}_{t}) \buildrel \Delta \over = r_t + \gamma\left( {{P^\pi }} \right)Q({\bf{s}}_{t},{\bf{a}}_{t})\\
&\left( {{\Gamma ^*}} \right)Q({\bf{s}}_{t},{\bf{a}}_{t}) \buildrel \Delta \over = r_t + \gamma\max\limits_{\pi} \left( {{P^\pi }} \right)Q({\bf{s}}_{t},{\bf{a}}_t)
\end{align}
which are called the Bellman operator and the optimal Bellman operator, respectively. Moreover, $\left( {{P^\pi }} \right)Q({\bf{s}}_{t},{\bf{a}}_{t})= \sum\nolimits_{{\bf{s}}_{t+1},{\bf{a}}_{t+1}} P({\bf{s}}_{t+1}|{\bf{s}}_{t},{\bf{a}}_{t})\pi ({\bf{a}}_{t+1}|{\bf{s}}_{t+1})Q({\bf{s}}_{t+1},{\bf{a}}_{t+1})$. Note that $Q^\pi$ is the unique fixed point of operator $\Gamma^\pi$, i.e., $\Gamma^\pi Q^\pi=Q^\pi$, and $Q^*$ is the optimal action-value function under $\pi^*$ with $\Gamma^* Q^*=Q^*$. Moreover, $V^\pi$ and $V^*$ are also the fixed points of $\Gamma^\pi$ and $\Gamma^*$, respectively.
\subsection{Value-Function Prediction with Deep Neural Network }
Generally, value-function prediction is to derive the value functions with a given policy. The intuitive way to calculate the value functions is based on the definitions in (\ref{1}) and (\ref{2}). However, this is usually not feasible as the transit probability is difficult to obtain in practical systems. In \cite{712192}, function approximators were proposed to estimate the value functions. For this purpose, DNN approximators could be used to generalize the whole state space using limited parameters \cite{mnih-dqn-2015,pmlr-v48-mniha16}, when the state space is large. Hence, the prediction problem for the action-value function is equivalent to solving the following the optimization problem:
\begin{equation}
\min\limits_{\bf{w}} J({\bf{w}}) = \sum\limits_{{\bf{s}}_{t}\in{\bf{S}} } \mu({\bf{s}}_{t})\sum\limits_{{\bf{a}}_{t}\in{\bf{A}} } \pi({\bf{a}}_{t}|{\bf{s}}_{t})(Q^\pi({\bf{s}}_{t},{\bf{a}}_{t})-Q_{\bf{w}}({\bf{s}}_{t},{\bf{a}}_{t}))^2,
\end{equation}
where ${\bf{w}}$ is the approximator parameter (e.g., the DNN weights) to be determined and $\mu({\bf{s}}_{t})$ is the state distribution, which depends on transit probability and policy, thus unknown in general. To solve the above optimization problem, algorithms based on stochastic gradient decent (SGD) could be adopted. Specifically, at every time step $t$, the parameter ${\bf{w}}$ is updated as:
\begin{equation}
{\bf{w}} \leftarrow {\bf{w}} + \alpha [Q^\pi({\bf{s}}_{t},{\bf{a}}_{t})-Q_{\bf{w}}({\bf{s}}_{t},{\bf{a}}_{t})] \nabla_{\bf{w}} Q_{\bf{w}}({\bf{s}}_{t},{\bf{a}}_{t})\label{4},
\end{equation}
where $\alpha$ is a positive step-size and the state-action pair $({\bf{s}}_{t},{\bf{a}}_{t})$ is sampled from the environment following a given policy. However, the exact update in (\ref{4}) cannot be performed since the update target $Q^\pi({\bf{s}}_{t},{\bf{a}}_{t})$ is unknown. Therefore, the update rule could be modified as
\begin{equation}
{\bf{w}} \leftarrow {\bf{w}} + \alpha [G_t-Q_{\bf{w}}({\bf{s}}_{t},{\bf{a}}_{t})] \nabla_{\bf{w}} Q_{\bf{w}}({\bf{s}}_{t},{\bf{a}}_{t}),
\end{equation}
where $G_t$ is an estimated target approximating the true value $Q^\pi({\bf{s}}_{t},{\bf{a}}_{t})$. Typically, the target $G_t$ has two main choices: the Monte-Carlo vs. bootstrapping target. In particular, the Monte-Carlo target is given as $G_t \buildrel\textstyle.\over= r_t + \gamma r_{t+1} + ...\gamma^{T-t-1} r_T$ and a TD(0) \cite{712192} bootstrapping target is given as $G_t\buildrel\textstyle.\over=r_t + \gamma Q_{\bf{w}}({\bf{s}}_{t+1},{\bf{a}}_{t+1}) $.
\subsection{Policy Gradient Methods}
In the above, the value-function prediction methods estimate the value function first, based on which the policy improvement is performed. Policy gradient methods instead parameterize the policy with ${\bf{\theta}}$ and use SGD to update ${\bf{\theta}}$, thus the policy directly. In particular, the gradients over ${\bf{\theta}}$ is obtained from a derivation of the objective function in (\ref{3}) \cite{sutton2000policy}:
\begin{align}
\nabla_{{\bm{\theta}}} V^{\pi_{{\bm{\theta}}}}({\bf{s}}_{t})
=E_{\pi_{{\bm{\theta}}}}\left[{{\nabla _{{\bm{\theta}}} }\log {\pi _{{\bm{\theta}}} }({{\bf{a}}_{t}}|{\bf{s}}_{t})}Q^{\pi_{{\bm{\theta}}}}({\bf{s}}_{t},{\bf{a}}_{t}) \right].
\label{7}
\end{align}
Hence, at time step $t$, the gradient used to update parameter ${\bm{\theta}}$ is sampled from (\ref{7}) (thus stochastic gradient), given as
\begin{equation}
\Delta {\theta}={{\nabla _{{\bm{\theta}}} }\log {\pi _{{\bm{\theta}}} }({{\bf{a}}_{t}}|{\bf{s}}_{t})}Q^{\pi_{{\bm{\theta}}}}({\bf{s}}_{t},{\bf{a}}_{t}).
\label{20}
\end{equation}
To compute the terms in (\ref{20}), we could use the previously introduced value-function prediction methods to estimate $Q^{\pi_{{\bm{\theta}}}}({\bf{s}}_{t},{\bf{a}}_{t})$.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{LSTM-general.eps}
\caption{ Structure of LSTM unit. }
\label{fig11}
\end{figure}
\subsection{LSTM}
An LSTM network is capable of learning long-term dependencies. In particular, an LSTM network is considered as multiple copies of the LSTM unit, each of which passes a message to its successor. Moreover, as shown in Fig. \ref{fig11}, an LSTM unit utilizes four gates to keep long-term information from previous time steps. The information transferred between the time steps include the cell state and hidden state. The first operation in an LSTM unit is to decide what information to throw away from the cell state, which is done by multiplying the output of forgotten gate by the cell state from the last time step. The next operation is to store new information in the cell state, which has two parts: the cell gate first generates the candidate cell state value; and the input gate then weighs the candidate cell state value. The weighted candidate cell state is then added to the multiplied cell state by the forgotten gate, to update the cell state. Finally, the hidden state is decided by the updated cell state and the output of the output gate, which is considered as the output of the LSTM unit for forward-propagation and back-propagation. In our DQN approximator, we add a fully connected neuron layer to generate the input for each LSTM unit. In addition, the output of the LSTM unit is processed by two separated neuron output layers to generate the value function and the policy, respectively, as later shown in Fig. \ref{fig5}.
\section{System Model}
In the system, we consider $L$ areas, denoted as $\{\mathscr{A}_l\}, l \in \{1,2,...,L\}$, which could be nonadjacent physically and represent different scenarios, e.g., offices vs. malls. Furthermore, $K_l$ SBSs exist and $N_l$ UEs move around in area $\mathscr{A}_l$. Hence, we denote the set of all the UEs as ${\bf{Z}}=\{{\rm{UE}}_{i,l}\}$, $i\in{1,..,N_l}$, $l\in{1,..L}$. The centralized controller partitions the UEs in the $L$ areas into $H$ clusters according to the user mobility patterns, i.e., cluster ${{\bf{G}}_h} = \{{\rm{UE}}_{i,l,h}\}\subseteq{\bf{Z}}, h\in\{1,..,H\}$ and ${\rm{UE}}_{i,l,h}$ represents the UE $i$ in area $l$ physically partitioned into cluster $h$ logically. In addition, we have ${\bf{Z}} = \bigcup\nolimits_{h = 1}^H {{{\bf{G}}_h}}$ and ${{\bf{G}}_{{h_1}}}\bigcap {{{\bf{G}}_{{h_2}}} = \emptyset }$, for $h_1 \ne h_2 $. In this paper, for the convenience of analysis, we assume that the UEs share similar mobility patterns in the same area. Hence, it is reasonable to assume that the UEs in the same area will be clustered into the same cluster. At time step $t$, the $i-$th UE in area $l$ and cluster $h$ maintains its active SBS index set by a vector ${\bf{b}}_{i,l,h,t}, i \in \{1,2,...,N_l\}$, $l\in{1,..L}$ with $|{\bf{b}}_{i,l,h,t}| = M_l$. For simplicity, we assume that $M_l=K_l$, and the system operates over equal-length time slots and the HO decision of UE $i$ is made at the beginning of each time slot, where UE $i$ chooses a SBS from ${\bf{b}}_{i,l,h,t}$ to camp on. If the UE is out of the system coverage at the beginning of time slot $t$, the current state is noted as the terminal state and time $t$ thus becomes the terminal time step $T$, i.e., $T=t$, which ends the current episode. The HO controllers are learned in each cluster using an asynchronous learning framework, which will be discussed in details later. We model the HO process of UE $i$ in area $l$ and cluster $h$ as a discrete-time episodic MDP process $\left\langle {{\bf{S}}_{i,l,h},{\bf{A}}_{i,l,h},{\bf{P}}_{i,l,h},{\bf{\pi}}_{i,l,h}, r_{i,l,h},\gamma } \right\rangle $.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{systemmodel.eps}
\caption{ An example of our proposed framework is represented, where three areas, i.e., $\mathscr{A}_1$, $\mathscr{A}_2$ and $\mathscr{A}_3$ are not adjacent. We assume that the mobility patterns are similar in $\mathscr{A}_1$ and $\mathscr{A}_2$ and different from that in $\mathscr{A}_3$. Hence, the centralized controller can partition the UEs in the three areas into two clusters according to the mobility patterns. The parameter server in the same cluster enables parameter sharing among the UEs. }
\label{fig2}
\end{figure}
In Fig. \ref{fig2}, an example of our proposed two-layer framework with three areas is presented. To clarify this framework, in the following, we first introduce the centralized controller and then the asynchronous learning framework within each cluster. Afterwards, the designs of the state vector and the reward signal will be discussed.
\subsection{Clustering by Centralized Controller}
As mentioned above, the centralized controller partitions the UEs into clusters based on their mobility patterns, which are then assumed similar across UEs in the same cluster. According to \cite{7904647}, the mobility patterns can be extracted from geographic contexts, which include locations of the UEs, often enriched with speed information as well as past trajectories. However, these geographic contexts typically cannot be obtained directly in cellular networks, for which some mobility tracking algorithms have been proposed to obtain the locations and the speeds of the UEs from the RSRP \cite{1390891}. In this paper, we assume that we have already obtained the geographical contexts after implementing certain mobility tracking algorithm. Specifically, the feature vector ${\bf{d}}_{i,l}$ of UE $i$ in area $l$ is defined as
\begin{equation}
{\bf{d}}_{i,l} = \{{\bf{d}}_{i,l}^1,...,{\bf{d}}_{i,l}^{T_u}\},
\end{equation}
where $T_u$ is a fixed time period for mobility observations and ${\bf{d}}_{i,l}^t, t\in\{1,...T_u\}$ contains the 2-dimensional coordinates and speeds, i.e., ${\bf{d}}_{i,l}^t =\{x_{i,l}^{t},y_{i,l}^{t},v_{i,l}^t\}$. Therefore, the input data set to the centralized controller is ${\bf{D}} = \{{\bf{d}}_{i,l}\}, i\in\{1,..,N_l\}, l\in\{1,...,L\}$.
We then utilize a standard K-means clustering algorithm \cite{macqueen1967} to partition UEs into $H$ clusters with the input data set ${\bf{D}}$. Specifically, the objective of K-Means clustering is to minimize the total intra-cluster variance:
\begin{equation}
\min\limits_{\{{\bf{o}}_h\}} \sum\limits_{h=1}^H \sum\limits_{i=1}^{N_l}\sum\limits_{l=1}^{L} dis({\bf{d}}_{i,l}, {\bf{o}}_h),
\end{equation}
where ${\bf{o}}_h=\{{\bf{o}}_h^t\}, t \in \{1,...,T_u\} $ is the centroid of cluster $h$, ${\bf{o}}_h^t$ has the same length as ${\bf{d}}_{i,l}^t$, i.e., ${\bf{o}}_h^t=\{{o_{h,x}^t,o_{h,y}^t,o_{h,v}^t}\}$, the distance between ${\bf{d}}_{i,l}$ and ${\bf{o}}_h$ is defined as $dis({\bf{d}}_{i,l}, {\bf{o}}_h)=\tau[\sum\nolimits_{t=1}^{T_u}(x_{i,l}^t-o_{h,x}^t)^2+(y_{i,l}^t-o_{h,y}^t)^2]+(1-\tau)\sum\nolimits_{t=1}^{T_u}(v_{i,l}^t-o_{h,v}^t)^2$, and $\tau$ is a weight factor. The full algorithm is given in Algorithm \ref{alg}.
\begin{algorithm}[H]
\renewcommand{\thealgorithm}{1}
\caption{The K-means clustering algorithm}
\label{alg}
\begin{algorithmic}[1]
\State Randomly initialize $H$ centroids ${\bf{o}}_1,...,{\bf{o}}_H$
\Repeat
\For {$l \in \{1,...,L\}$ }
\For{ $i \in \{1,...,N_l\}$ }
\State Calculate cluster index $h_{i,l}$ for ${\bf{d}}_{i,l}$, i.e.,
\begin{equation}
h_{i,l} = \arg \min \limits_h dis({\bf{d}}_{i,l}, {\bf{o}}_h)
\end{equation}
\EndFor
\EndFor
\For {$h \in \{1,..H\}$}
\State Recalculate the centroids
\begin{equation}
{\bf{o}}_h = \frac {{\sum\nolimits_{i,l}}{{\bf{1}}\{h_{i,l}=h\}}{\bf{d}}_{i,l}}{{\sum\nolimits_{i,l}}{{\bf{1}}\{h_{i,l}=h\}}}
\end{equation}
\EndFor
\Until No shifts of centroids
\end{algorithmic}
\end{algorithm}
However, the optimal number of the clusters $H$ is usually unknown. To obtain its value, clustering validation \cite{5694060} could be adopted. Specifically, we first use each candidate total cluster number to get different clustering results; then, the corresponding internal validation indices of each cluster is obtained, where the maximum candidate cluster number is set as $L$ in our formulation, which is the total number of areas; finally, we choose the best clustering result with the optimal cluster number according to the validation indices.
For the newly arriving UE in an area, it fetches the parameters from the parameter server controlling the cluster that includes this area, to have a jump start. Note that we need to jointly re-cluster all the UEs periodically based on the time scale of mobility pattern changes, which is usually on the order of hours \cite{Galati}. The details for handling the above aspects is skipped in this paper, in order to focus on other core issues.
\subsection{Asynchronous Joint Learning Scheme}
After the clustering, the UEs are partitioned into clusters. In each cluster, an A3C framework is utilized to learn the optimal HO controller, as shown in Fig. \ref{fig3}.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{arlsystem.eps}
\caption{ An example of asynchronous learning scheme in area $l$, where all the UEs are in cluster $h$. }
\label{fig3}
\end{figure}
If a UE requests to update the controller, it first fetches the most recent copy of the controller parameters from the parameter server, which stores the key global controller parameters. The UE then executes the controller and interacts with the environment, where the advantage actor-critic algorithm is used to compute the gradient of the controller network (assuming DQN is used). Finally, the UE pushes the new gradient back to the controller parameter server, which updates the global network parameter. Furthermore, the global controller weights in the parameter server are updated asynchronously in a lock-free fashion\cite{pmlr-v48-mniha16}. Note that the convergence property of this asynchronous learning framework has been established in [19] and we later show that this framework accelerates the individual HO controller learning as more UEs are joining the system.
\subsection{Action}
At time step $t$, the action $a_{i,l,h,t}$ for UE $i$ in area $l$ and cluster $h$ is a scalar representing the serving SBS index in ${\bf{b}}_{i,l,h,{t}}$.
\subsection{State Vector}
As mentioned above, the state vector represents the information for the situation and exploration. We use the RSRQs from all the SBSs to each UE representing the situation information and the serving SBS index to represent the exploration information. Hence, at time step $t$, the state vector for UE $i$ in area ${l}$ and cluster $h$ is given as
\begin{equation}
{\bf{s}}_{i,l,h,t}= \{ {\bf{q}}_{i,l,h,t}\;,\;{\bf{a}}_{i,l,h,{t-1}}\}
\end{equation}
where ${\bf{s}}_{i,l,h,t} \in {\bf{S}}_{i,l,h} $, ${\bf{q}}_{i,l,h,t} = \{q_{i,l,h,t}^1,...,q_{i,l,h,t}^{M_l}\}$ contains the RSRQs from all the candidate SBSs in ${\bf{b}}_{i,l,h,t}$ to UE $i$, and ${\bf{a}}_{i,l,h,{t-1}}$ is a one-hot encoded vector\footnote{The indices of SBSs are nominal labels, where the one-hot encoding is utilized to generate nominal vectors contained in state vectors.} according to the action $a_{i,l,h,{t-1}}$. In particular, a one-hot encoding \cite{silver2016mastering} is a representation of integral variables as binary vectors. Each integer value is represented as a binary vector that is all zero values except the index of the integer, which is marked with a one. For example, the action $a_{i,l,h,{t-1}}$ is an integer in the range of $[0,5]$ and its value $a_{i,l,h,{t-1}} = 2$. Accordingly, the one-hot vector ${\bf{a}}_{i,l,h,{t-1}}=[0,0,1,0,0,0]$.
\subsection{Reward Signal}
The reward signal should encourage the learning algorithm to achieve our goal, which is to ensure certain downlink throughput while minimizing the number of HOs. Here, the throughput is measured by the averaged throughput, which is defined as the ratio between the sum rate and the total time steps in the episode. Moreover, the HO performance is measured by the averaged HO rate, defined as the ratio between the HO times and the total time steps in the episode. According to the measures defined above, we set the reward signal as a weighted sum between the averaged throughput and the HO rate, which is available to the UE at the termination of each episode. The rewards for all the non-terminal steps could be set as zero. However, the resulting delayed rewards may cause the so-called credit assignment problem {\cite{712192}}, which degrades the performance of the RL approach.
To address this issue, we utilize reward shaping \cite{ng1999policy} to make the reward more informative and accelerate the training. In particular, at every time step $t$, the reward for UE $i$ in area $l$ and cluster $h$ is defined as
\begin{equation}
r_{i,l,h,t} = \left\{
\begin{aligned}
R_{i,l,h,t}& - {\beta E,}&\text{if HO occurs}\\
&{ R_{i,l,h,t}}, &\text{otherwise}
\end{aligned}
\right.
\end{equation}
where $R_{i,l,h,t}$ is the rate of UE $i$ at time step $t$ in area $l$ and cluster $h$, $E$ is the energy consumption when a HO process occurs, and $\beta$ is a normalizing weight factor.
\section{Problem Formulation and Learning Algorithm}
In this section, we first formulate the RL problem for each UE after the clustering is done. The learning algorithm is then presented. Finally, we introduce the neural network structure of the controller and the novel idea of supervised-learning based network initialization.
\subsection{Problem Formulation and Learning Algorithm}
The policy for UE $i$ in area $l$ and cluster $h$ is defined as $\pi_{{\bm{\theta}}_{i,l,h}}$. Hence, the goal of UE $i$ in area $l$ and cluster $h$ is to obtain the optimal parameter ${\bm{\theta}}_{i,l,h}^*$ by solving the optimization problem (\ref{3}). For simplicity, we omit the subscripts $l$ and $h$ in the following derivations, unless explicitly mentioned. Hence, at time step $t$, the gradient to update parameter ${\bm{\theta}}_{i}$ is given as
\begin{equation}
\Delta {\bm{\theta}}_{i,t} ={{\nabla _{{\bm{\theta}}_{i}} }\log {\pi _{{\bm{\theta}}_{i}} }({a_{i,t}}|{\bf{s}}_{i,t})}Q^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i,t},a_{i,t}).
\end{equation}
To estimate $Q^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i,t+k},a_{i,t+k})$, we use a $n-$step prediction approach \cite{712192} combining the Monte-Carlo method \cite{sutton2000policy} and TD prediction \cite{712192} to balance the variance and bias caused by RL execution. In particular, the Monte-Carlo method alone obtains unbiased estimates but introduces a high variance, while TD learning alone could lower the variance but may introduce bias, especially when the estimation is not accurate. We here combine them: From time step $t$, UE $i$ interacts with the environment for the next $n$ time steps and obtains a trajectory $({\bf{s}}_{i,t},a_{i,t},r_{i,t},.....,{\bf{s}}_{i,t+n})$\footnote{If the terminal state occurs in the trajectory, the $n-$step prediction degrades to Monte-Carlo sampling without prediction.}.
We then use an estimator $V_{{\bf{w}}_{i}}({\bf{s}}_{i,t})$ parameterized by ${\bf{w}}_{i}$ to estimate the state-value function $V^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i,t})$, with the estimation of $Q^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i,{t}},a_{i,{t}})$ given as $Q_{{\bf{w}}_{i}}({\bf{s}}_{i,{t}},a_{i,{t}})= \sum\nolimits_{k' = t}^{n - 1} {{\gamma ^{k'-t}}{r_{i,k'}}} + {\gamma ^{n-t}}{V_{{\bf{w}}_{i}}}({{\bf{s}}_{i
,t + n}})$. Consequently, the accumulated gradient from the sampled trajectory is given as
\begin{align}
&\Delta {\bm{\theta}}_{i}=\sum\limits_{k=0}^{n-1} \Delta {\bm{\theta}}_{i,t+k}\notag\\ &= \sum\limits_{k = 0}^{n-1}{{\nabla _{{\bm{\theta}}_{i}} }\log {\pi _{{\bm{\theta}}_{i}} }({a_{i,t+k}}|{{\bf{s}}_{i,t+k}})} A_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k}),
\label{10}
\end{align}
where $\Delta {\bm{\theta}}_{i,t}$ is the gradient at time step $t$, $A_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k})=Q_{{\bf{w}}_{i}}({\bf{s}}_{i,{t+k}},a_{i,{t+k}})-V_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k})$ , and $V_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k})$ is an inserted baseline \cite{712192} to further decrease the variance. In addition, the sampled gradient $\Delta {\bf{w}}_{i}$ is calculated as
\begin{align}
\Delta {\bf{w}}_{i}=\sum\limits_{k=0}^{n-1} \Delta {\bf{w}}_{i,t+k}= \sum\limits_{k = 0}^{n-1}{\nabla _{{\bf{w}}_{i}} }V_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k})A_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k}),
\label{11}
\end{align}
where $\Delta {\bf{w}}_{i,t}$ is the gradient at time step $t$, and $A_{{\bf{w}}_{i}}({\bf{s}}_{i,t})$ is estimated by forward-propagation in the approximator, which is independent of the back-propagation over ${\bf{w}}_{i}$.
Note that the gradient $\Delta {\bm{\theta}}_{i} $ is used to update the neural network weights in the policy improvement for $\pi_{{\bm{\theta}}_{i}}$ to approach the optimal policy incrementally. Meanwhile, ${Q}_{{\bf{w}}_{i}}({\bf{s}}_{i},a_{i})$ estimates $Q^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i},a_{i})$ to evaluate the goodness of policy ${\pi_{{\bm{\theta}}_{i}}}$, such that (\ref{11}) is for the process of policy evaluation, i.e., value estimation. We see that (\ref{10}) is based on the derivative of the objective function, which is guaranteed to converge to a local optimum according to the stochastic approximation theorem \cite{712192}. To prove the convergence of policy evaluation following the gradient in (\ref{11}), we recall the Belleman operator $\Gamma ^{\pi_{{\bm{\theta}}_{i}}}: \left( {{\Gamma ^{\pi_{{\bm{\theta}}_{i}}} }} \right)Q({\bf{s}}_{i},a_{i})\buildrel \Delta \over = r_{i} + \left( P^{\pi_{{\bm{\theta}}_{i}}} \right)Q({\bf{s}}_{i},a_{i})$ \cite{NIPS2016}, where $Q({\bf{s}}_{i},a_{i})$ is an arbitrary $Q-$function for $({\bf{s}}_{i},a_{i})$, with ${\bf{Q}}^{\pi_{{\bm{\theta}}_{i}}}$\footnote{ ${\bf{Q}}^{\pi_{{\bm{\theta}}_{i}}}$ is the collection of the $Q-$fucntion $Q^{\pi_{{\bm{\theta}}_{i}}}({\bf{s}}_{i},a_{i})$ for all the state-action pairs.} defined as the unique fixed point of operator $\Gamma ^{\pi_{{\bm{\theta}}_{i}}}$, i.e., $\left(\Gamma ^{\pi_{{\bm{\theta}}_{i}}}\right){\bf{Q}}^{\pi_{{\bm{\theta}}_{i}}}={\bf{Q}}^{\pi_{{\bm{\theta}}_{i}}}$, where the operator $\Gamma ^{\pi_{{\bm{\theta}}_{i}}}$ is $\gamma-$contraction\cite{712192}, i.e., $\left\| {\left(\Gamma ^{\pi_{{\bm{\theta}}_{i}}}\right){\bf{Q}}_1-\left(\Gamma ^{\pi_{{\bm{\theta}}_{i}}}\right){\bf{Q}}_2} \right\|_\infty \le \left\| {{\bf{Q}}_1-{\bf{Q}}_2} \right\|_\infty $ and $\left\| {{\bf{Q}}_1-{\bf{Q}}_2} \right\|_\infty = \max\nolimits_{({\bf{s}}_{i},a_{i})}|Q_1({\bf{s}}_{i},a_{i})-Q_2({\bf{s}}_{i},a_{i})|$. Note that the operator ${{P^{\pi_{{\bm{\theta}}_{i}}}}} $ is defined in (\ref{21}). Hence, from the contraction mapping theorem \cite{712192}, the iterative application of operator $\Gamma^{\pi_{{\bm{\theta}}_{i}}}$ to an arbitrarily $Q-$function converges to $Q^{\pi_{i}}$. When the $n-$step prediction target is sampled from $\left( {{\Gamma ^{\pi_{{\bm{\theta}}_{i}}} }} \right)^n Q_{{\bf{w}}_{i}}({\bf{s}}_{i},a_{i})$ with $\left( {{\Gamma ^{\pi_{{\bm{\theta}}_{i}}} }} \right)^n $ being $\gamma-$contraction, the policy evaluation process (\ref{11}) converges to $Q^{\pi_{{\bm{\theta}}_{i}}}$. Note that $\left( {{\Gamma ^{\pi_{{\bm{\theta}}_{i}}} }} \right)^n$ denotes $n$ successive applications of operator $ {{\Gamma ^{\pi_{{\bm{\theta}}_{i}}} }}$. In our algorithm, we implement the policy improvement and evaluation processes iteratively. The policy is improved with respect to the $Q-$function, while the $Q-$function is driven towards the true value function for the policy. It is easy to see that if both processes stabilize, then the estimated $Q-$function and policy are at least locally optimal \cite{712192}.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{algorithmframework.eps}
\caption{ The overview of the learning algorithm implementation. }
\label{fig4}
\end{figure}
\subsection{Learning Algorithm Implementation}
When implementing the above learning algorithm, we need to follow the general RL framework as shown in Fig. \ref{fig1}. In particular, at time step $t$, the UE $i$ performs action $a_{i,t}$ according to the policy $\pi _{{\bm{\theta}}_{i}}$ given the state ${\bf{s}}_{i,{t}}$. Then, the environment generates the next state ${\bf{s}}_{i,{t+1}}$ and the reward signal $r_{i,t}$. The experience transition $({\bf{s}}_{i,{t}},a_{i,t},r_{i,t})$ is stored in the trajectory buffer; the UE receives the next state ${\bf{s}}_{i,{t+1}}$ and perform $a_{i,{t+1}}$ determined by $\pi _{{\bm{\theta}}_{i}}$, and this process continues. The UE accomplishes the interaction with $\pi _{{\bm{\theta}}_{i}}$ for $n$ time steps and the trajectory $({\bf{s}}_{i,t},a_{i,t},r_{i,t},.....,{\bf{s}}_{i,t+n})$ is stored in the buffer. Note that we could obtain ${V_{{\bf{w}}_{i}}}({{\bf{s}}_{i,t + n}})$ if ${\bf{s}}_{i,t+n}$
is given. Moreover, the state-value function at each step in the sampled trajectory can be estimated by forward-propagation in the approximator. Hence, for each state ${\bf{s}}_{i,t+k}, k \in \{0,1,...,n\}$ in the trajectory, we obtain $A_{{\bf{w}}_{i}}({\bf{s}}_{i,t+k})$, which is used to calculate the accumulated gradients $\Delta {\bm{\theta}}_{i}$ and $\Delta {\bf{w}}_{i}$ according to (\ref{10}) and (\ref{11}), respectively, by back-propagation in the neural network approximator. The whole framework is shown in Fig. \ref{fig4}.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{networkstructure.eps}
\caption{ The structure of DQN for state value estimation and policy generations. }
\label{fig5}
\end{figure}
\subsection{DQN in HO Controller}
As shown in Fig. \ref{fig5}, for UE $i$ in area $l$ and cluster $h$, we utilize a three-layer neural network to generate $\pi_{{\bm{\theta}}_{i}}$ and estimate $V_{{\bf{w}}_{i}}$, including the encoding layer, the LSTM \cite{Sutskever} layer and the output layer, where the encoding neuron layer and the output layer are fully connected and the LSTM layer consists of multiple LSTM units as we discussed earlier. In particular, at time step $t$, a repeating module includes the encoding layer, the LSTM unit, and the value and policy output layers with weights ${\bf{u}}_{i,t}^{en}$, ${\bf{u}}_{i,t}^{lstm}$, ${\bf{u}}_{i,t}^{vo}$, and ${\bf{u}}_{i,t}^{po}$, respectively. Note that the weights in the repeating modules are shared across all the time steps in the same back-propagation truncation block, which will be clarified later. Accordingly, the parameters for the actors and critics are ${{\bm{\theta}}_{i}}=[{\bf{u}}_{i}^{en},{\bf{u}}_{i}^{lstm},{\bf{u}}_{i}^{po}]$ and ${{\bf{w}}_{i}}=[{\bf{u}}_{i}^{en},{\bf{u}}_{i}^{lstm},{\bf{u}}_{i}^{vo}]$, respectively, with subscript $t$ neglected. We see that the actors and critics share the weights of the encoding and LSTM layers, and the weights of the whole network could be denoted as ${{\bf{u}}_{i}}=[{\bf{u}}_{i}^{en},{\bf{u}}_{i}^{lstm},{\bf{u}}_{i}^{vo},{\bf{u}}_{i}^{po}]$. Furthermore, the weights ${{\bm{\theta}}_{i}}$ and ${{\bf{w}}_{i}}$ for UE $i$ in cluster $h$ are the copies of the global weights ${{\bm{\theta}}_h}=[{\bf{u}}_{h}^{en},{\bf{u}}_{h}^{lstm},{\bf{u}}_{h}^{po}]$ and ${{\bf{w}}_h}=[{\bf{u}}_{h}^{en},{\bf{u}}_{h}^{lstm},{\bf{u}}_{h}^{vo}]$, which are stored and updated in the parameter servers serving the UEs in cluster $h$.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{LSTM.eps}
\caption{ The structure of repeating module. }
\label{fig10}
\end{figure}
The structure of the repeating module is shown in Fig. \ref{fig10}.
Specifically, the LSTM unit can explore a self-learned amount of long-range temporal information, and it consists of the forget, input, cell and output gates with weights of ${\bf{u}}_{i}^{fg}$, ${\bf{u}}_{i}^{ig}$, ${\bf{u}}_{i}^{cg}$ and ${\bf{u}}_{i}^{og}$, respectively. In the forward-propagation, the cell value ${\bf{c}}_{i,t}$ and the hidden state ${\bf{h}}_{i,t}$ at time step $t$ are given as
\begin{align}
&{{\bf{c}}_{i,t}} = {{\bf{c}}_{i,t - 1}} \times {\bf{f}}_{i}^{fg} + {\bf{f}}_{i}^{ig} \times {\bf{f}}_{i}^{cg},\\
&{{\bf{h}}_{i,t}} = {\bf{f}}_{i}^{og} \times \tanh ({{\bf{c}}_{i,t}}),
\end{align}
where ${\bf{f}}_{i}^{fg}$, ${\bf{f}}_{i}^{ig}$, ${\bf{f}}_{i}^{cg}$ and ${\bf{f}}_{i}^{og}$ are the outputs of forget, input, cell and output gates defined as\footnote{In this paper, we drop the bias for simplicity.}
\begin{align}
&{\bf{f}}_{i}^{fg} = \sigma ({\bf{u}}_{i}^{fg} \times [{\bf{f}}_{i}^{en} ,{{\bf{h}}_{i,t - 1}}]),\\
&{\bf{f}}_{i}^{ig} = \sigma ({\bf{u}}_{i}^{ig} \times [{\bf{f}}_{i}^{en} ,{{\bf{h}}_{i,t - 1}}] ),\\
&{\bf{f}}_{i}^{cg} = \tanh ({\bf{u}}_{i}^{cg} \times [{\bf{f}}_{i}^{en},{{\bf{h}}_{i,t - 1}}] ),\\
&{\bf{f}}_{i}^{og} = \sigma ({\bf{u}}_{i}^{cg} \times [{\bf{f}}_{i}^{en} ,{{\bf{h}}_{i,t - 1}}] ),
\end{align}
respectively, with ${\bf{f}}_{i}^{en}={\bf{u}}_{i}^{en} \times {{\bf{s}}_{i,t}}$ being the output of encoding layer. Moreover, we have the activation functions as $\sigma(x) = \frac{1}{1+e^{-x}}$ and $tanh(x) = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$, and ${\bf{u}}_{i}^{lstm} = [{\bf{u}}_{i}^{fg}, {\bf{u}}_{i}^{ig}, {\bf{u}}_{i}^{cg}, {\bf{u}}_{i}^{og}]$. Furthermore, the outputs of the actors are vectors (the policies) with dimension $M_l$ to represent the probability distribution of choosing among the $M_{l}$ SBSs, which is ${{\pi}}_{{\bm{\theta}}_{i}}(a_{i,t}|{\bf{s}}_{i,t})=softmax({\bf{u}}_{i,t}^{po}\times{{\bf{h}}_{i,t}})$ at time step $t$, with $softmax({\bf{x}}) = [\frac{e^{x_1}}{\sum\nolimits_{k=1}^{|{\bf{x}}|}e^{x_k}},...,\frac{e^{x_{|{\bf{x}}|}}}{\sum\nolimits_{k=1}^{|{\bf{x}}|}e^{x_k}}]$. In addition, the action $a_{i,t}$ is chosen according to the policy ${{\pi}}_{{\bm{\theta}}_{i}}$. The outputs of the critics are scalars to estimate the state-value function, which is $V_{{\bf{w}}_{i}}({\bf{s}}_{i,t})={\bf{u}}_{i,t}^{vo}\times{{\bf{h}}_{i,t}}$ at time step $t$.
For a communication system, utilizing LSTM has two advantages: The LSTM units can capture the moving patterns from the historical location information; and it can also accumulate the exploration information, which leads to better tradeoff between exploration and exploitation.
Furthermore, we apply the truncated back-propagation through time (BPTT) technique \cite{Sutskever} to alleviate error accumulation, where BPTT is executed every $n$ steps. The initial cell state and hidden state for the BPTT block are from the cell and hidden state outputs of the LSTM unit at the last time step in the previous BPTT block\footnote{If the terminal state occurs in the previous BPTT block, the initial cell and hidden states are both set as zero vectors for the current BPTT block.}. In a BPTT block, the gradient with respect to ${\bf{u}}_{i}$ at the time step $t$ is given as
\begin{align}
&\Delta {\bf{u}}_{i,t}= \Delta {\bm{\theta}}_{i,t}+\Delta {\bf{w}}_{i,t}\notag\\
&={\nabla _{{\bf{u}}_{i}} } [\log {\pi _{\bm{\theta}_{i}} }({a_{i,t+k}}|{{\bf{s}}_{i,t+k}})+V_{{\bf{w}}_{i}}({\bf{s}}_{i,t})] A_{{\bf{w}}_{i}}({\bf{s}}_{i,t}).
\label{16}
\end{align}
After obtaining the gradients at each time step in the BPTT block from (\ref{16}), the accumulated gradients $ \Delta {\bm{\theta}}_{i}$ and $\Delta {\bf{w}}_{i}$ are derived according to (\ref{10}) and (\ref{11}).
Note that we have adopted the asynchronous RMSProp \cite{pmlr-v48-mniha16} routine to update the parameters in cluster $h$, which is given as
\begin{align}
{\bf{g}}_{h} = \alpha^\prime {\bf{g}}_{h}+ (1 - \alpha^\prime )\Delta {\bm{\theta}_{i}} ^2
\label{12}
\end{align}
where $\alpha^\prime$ is a decay factor. In asynchronous RMSProp, ${\bf{g}}_{h}$ is used to update the DQN weights $\bm{\theta}_h$ in cluster $h$ as
\begin{align}
\bm{\theta}_h \leftarrow \bm{\theta}_h - \eta \frac{\Delta {\bm{\theta}_{i}}}{{\sqrt {{\bf{g}}_{h} + \varepsilon } }},\label{13}
\end{align}
with $\varepsilon$ a small constant to prevent the dominator from approaching 0, and ${\bm{\theta}}_h$ sharable to all the UEs in cluster $h$. Note that the multiplication, the division, the square and the square root operations in (\ref{12}) and (\ref{13}) are all element-wise. The way for handling the global update for ${\bf{w}}_h$ with $\Delta {\bf{w}}_{i}$ is the same as those in (\ref{12}) and (\ref{13}). The full algorithm for UE $i$ in area $l$ and cluster $h$ to update HO controller in cluster $h$ is given in Algorithm \ref{alg2}, where we assume that $t$ is the local time step counter for each UE, $t_{global}$ is the global time step counter for all the UEs, $t_{start}$ is the local record of the starting time step for each UE.
\begin{algorithm}
\caption{The asynchronous advantage actor-critic learning algorithm for UE $i$ in area $l$ and cluster $h$}
\label{alg2}
\begin{algorithmic}[1]
\State Initialize the local counter as $t=1$
\Repeat \State Reset accumulated gradients: $\Delta {\bm{\theta}}_{i} = \Delta {\bf{w}}_{i} =0$;
\State Fetch parameters ${\bm{\theta}}_{i}$ and ${\bf{w}}_{i}$ from parameter server, i.e., copying ${\bm{\theta}}_h$ and ${\bf{w}}_h$;
\State Set $t_{start}=t$ and get initial state ${\bf{s}}_t$
\Repeat \State Perform ${a_{i,t}}$ according to the policy ${\pi _{{\bm{\theta}}_{i}} }({a _{i,t}}|{{\bf{s}}_{i,t}})$
\State Set $t \leftarrow t+1$ and $t_{global} \leftarrow t_{global}+1$
\Until terminal ${\bf{s}}_{i,t}$ or $t-t_{start}==n$
\begin{equation}
r = \left\{
\begin{aligned}
&0, &\text{for terminal ${\bf{s}}_{i,T}$},\notag\\
&V_{{\bf{w}}_{i}}({\bf{s}}_{i,t}), &\text{otherwise},\notag
\end{aligned}
\right.
\end{equation}
\For {$j \in \{t-1,...,t_{start}\}$ }
\State $r \leftarrow r_{i,j}+\gamma r$
\State Accumulate the gradients, i.e.,
\begin{align}
&\Delta {\bm{\theta}}_{i} \leftarrow \Delta {\bm{\theta}}_{i} +
(r -V_{{\bf{w}}_{i}}({\bf{s}}_{i,j})){\nabla _{{\bm{\theta}}_{i}} }\log {\pi _{{\bm{\theta}}_{i}} }({a_{i,j}}|{{\bf{s}}_{i,j}}) \notag\\&+ \eta \sum\limits_a \pi_{{\bm{\theta}}_{i}}(a|{{\bf{s}}_{i,j}})log\pi_{{\bm{\theta}}_{i}}(a|{{\bf{s}}_{i,j}}) \\
&\Delta {\bf{w}}_{i} \leftarrow \Delta {\bf{w}}_{i}+ (r -V_{{\bf{w}}_{i}}({\bf{s}}_{i,j})){\nabla _{{\bf{w}}_{i}} }V_{{\bf{w}}_{i}}({\bf{s}}_{i,j})
\end{align}
\EndFor
\State Using (\ref{12}) and (\ref{13}) to update ${\bm{\theta}}_h$ and ${\bf{w}}_h$ in parameter server with accumulated gradients $\Delta {\bm{\theta}}_{i}$ and $\Delta {\bf{w}}_{i} $
\Until the learning is done.
\end{algorithmic}
\end{algorithm}
\subsection{Initialization with Supervised Learning }
To boost the performance of the RL framework and fully utilize what we have already known from the traditional model-based HO strategies, we further propose to initialize the involved neural networks with supervised learning. In particular, we assume that UE $i$ in area $l$ and cluster $h$ receives the RSRPs in its active set ${\bf{b}}_{i,t}$ at time step $t$. If one of the SBSs in ${\bf{b}}_{i,t}$ satisfies the following condition for a
specific time determined by the TTT:
\begin{equation}
RSRP_{b_{i,t} \in {\bf{b}}_{i,t}} > RSRP_{b_{i,t}^{serving}} + HHM,
\end{equation}
where $RSRP_{b_{i,t}^{serving}}$ is the RSRP of the serving SBS for UE $i$ in area $l$ and cluster $h$ at time step $t$, the UE hands itself over to the SBS with index $b_{i,t}$. Hence, we obtain the action $a_{i,t}$. Moreover, we obtain ${\bf{s}}_{i,t}$ over the RSRPs from all the SBSs in the ${\bf{b}}_{i,t}$ and the serving SBS index. To generate the training data, we incorporate a similar technique to that in \cite{Boje2e}, where we use five different pairs of HHMs and TTTs. Specifically, in cluster $h$, we sample five episodes as training data, i.e., $\{({\bf{s}}_{i,1}^j,a_{i,1}^j),...,({\bf{s}}_{i,T}^j,a_{i,T}^j)\}_{j=1}^5$.
Note that we can only initialize the encoding layer ${\bf{u}}_{h}^{en}$, LSTM layer ${\bf{u}}_{h}^{lstm}$, and policy softmax output layer ${\bf{u}}_{h}^{po}$, while the value output layer ${\bf{u}}_{h}^{vo}$ is initialized randomly, due to the absence of knowledge on system dynamics.
\section{Simulation}
In this section, the simulations setups are shown firstly. Then the simulation results are presented. Finally, the upper bound of number of UEs in each cluster is discussed.
\subsection{Simulation Setups}
In the experiment, we have $L=3$ nonadjacent areas, i.e., $\mathscr{A}_1$, $\mathscr{A}_2$ and $\mathscr{A}_3$. Each area has 6 SBSs, i.e., $K_1 = K_2 = K_3 = 6$. We assume that all the three areas are $16 \times 16 $ square meters and the SBSs are deployed in each area randomly. Moreover, a random walk model \cite{WCM72} is utilized to simulate the movements of UEs. In particular, at every time step, the UEs move in four directions, i.e., east, south, west and north. In addition, the probability distribution of the four directions are set as $[0.25,0.25,0.25,0.25]$ in $\mathscr{A}_1$ and $\mathscr{A}_2$, and $[0.6,0.2,0.1,0.1]$ in $\mathscr{A}_3$. The speed levels of UEs are chosen randomly from interval $[1,3]$ $m/s$. The transmit power of all SBS is set as $30 dBm$, with bandwidth $10 MHz$ and thermal noise density $-174 dBm/Hz$. The interruption time of HO is up to $50ms$ \cite{RRC}, and the energy consumption for the HO is $E=0.3$ $Joule$. The pathloss is given as $PL(dB) = 36.7\log_{10}(distance)+39.4$\cite{7987783} and the log-normal shadowfading is with zero mean and standard deviation $8 dB$. The small-scale fading is assumed to be Rayleigh distributed. The length of the BPTT block is set as $n=20$. The encoding, policy output and value output layers are fully connected with sizes $12\times8$, $8\times6$ and $8\times1$, respectively. We further set $\eta= 0.01$ for entropy regularization and $\alpha ^\prime = 0.99$ for the RMSProp decay factor. Finally, we set the time period for mobility observation in of clustering is $T_u = 150$ time slots. Note that the interferences are not considered in our paper.
\subsection{Simulation Results}
First of all, the clustering validation result is shown in Table \ref{table1}, where we utilize the Calinski-Harabaz index (CHI) \cite{5694060} as the measure to evaluate the goodness of clustering with different cluster numbers. In particular, the CHI is defined as:
\begin{equation}
{\rm{CHI}}(k) = \frac{{\sum\limits_{h = 1}^H {N_h dis({ {{{\bf{o}}_h},{\bf{o}}} })} }}{{\sum\limits_{h = 1}^H {\sum\limits_{{{\bf{G}}_h}} dis({{ {{{\bf{d}}_{i,l,h}}, {{\bf{o}}_h}} }}) } }} \times \frac{{\left| {\bf{D}} \right| - H}}{{H - 1}},
\label{15}
\end{equation}
where ${\bf{o}}$ is the centroid of the whole data set ${\bf{D}}$ and $|{\bf{D}}|$ is the size of the data set. Note that the numerator and the dominator in (\ref{15}) represent the inter-cluster separation and the intra-cluster compactness, respectively. Hence, the cluster number with the largest CHI is the optimal cluster number. Table \ref{table1} shows the CHI values with different data set sizes, where we assume that each area has 4, 40 and 400 UEs, respectively. Moreover, the CHI values are averaged over 100 random runs. From the result, we see that it is better to partition the UEs into two clusters under the current setup. In the following, we assume that each area has 4 UEs for simplicity.
\begin{table}[H]
\centering
\caption{Clustering validation results with Calinski-Harabaz Index}
\label{table1}
\resizebox{0.9\linewidth}{!}{%
\begin{tabular}{|c|c|c|ll}
\cline{1-3}
\multicolumn{1}{|l|}{\multirow{2}{*}{Total UEs}} & \multicolumn{2}{l|}{Calinski-Harabaz Index} & & \\ \cline{2-3}
\multicolumn{1}{|l|}{} & H=2 & H=3 & & \\ \cline{1-3}
12 & 31.97 & 26.09 & & \\ \cline{1-3}
120 & 300.99 & 208.50 & & \\ \cline{1-3}
1200 & 2964.97 & 1986.55 & & \\ \cline{1-3}
\end{tabular}%
}
\end{table}
Then, in Fig. \ref{fig8}, the tradeoff between the HO rate and throughput is shown by changing the weight factor $\beta$ in the reward signal. We see that the rewards with larger $\beta$ could encourage the controller to lower the HO rate by sacrificing the throughput.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{plotreward.eps}
\caption{ The tradeoff between HO rate and throughput when the weight factor $\beta=5,10,15,20,25$. }
\label{fig8}
\end{figure}
After the clustering, the UEs in areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are partitioned into the same cluster, with the others into the second cluster. In Fig. \ref{fig6}, we show the estimation of the state-value function with/without SL-based initialization for the UEs in area $\mathscr{A}_1$, where the total learning time is 119 seconds, and $\beta=25$. In particular, the learning without SL-based initialization starts with randomly initialized weights. According to the definition of state-value function, a larger state value means that the current policy gains more expected rewards in the future. Hence, SL-based initialization could help the A3C framework derive a better policy through the same length of learning period. It is worth noting that the state value may decay at the beginning, for the policy is stochastic and the exploration in the parameter space is based on the current policy. Hence, if the initial policy is bad, the exploration can cause certain performance degradation. We observe that SL-based initialization network could mitigate such degradation, since the 3GPP HO policy used in SL is typically better than a random policy. To illustrate the gain of clustering, where we could group more UEs with similar mobility patterns into one cluster such that the shared global parameter server could exploit more information, we also draw the case without clustering, i.e., only the UEs in $\mathscr{A}_1$ share one parameter server instead of all the UEs in $\mathscr{A}_1$ and $\mathscr{A}_2$ share one parameter server. The results in Fig. \ref{fig6} show that the proposed asynchronous RL framework could achieve larger state values with clustering. It is due to the fact that clustering with a global parameter server per cluster allows more UEs to update the weights in the shared parameter server, which could exploit more sampled data in the same time period. Hence, the A3C framework accelerates learning when the number of UEs increases in the cluster, which is a strong indicator that the proposed method could be applied to large systems.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=0.94\linewidth]{pretrain.eps}
\caption{ The estimation of the state-value function versus learning time.}
\label{fig6}
\end{figure}
For the UCB learning algorithm \cite{7572176,7925950} in comparison, the action and reward are set the same as those in our work. The action for UE $i$ in area $l$ and cluster $h$ at time step $t$ is given as
\begin{align}
{a_{i,l,h,t}} = \arg \mathop {\max }\limits_{k = 1,...K} \left( {{\mu _{i,l,h,k}} + \sqrt {\frac{{2\ln t}}{{{M_{i,l,h,k,t - 1}}}}} } \right)\\
{\mu _{{a_{i,l,h,t}}}} \leftarrow {\mu _{{a_{i,l,h,t}}}} + \frac{1}{{{M_{i,l,h,{a_{i,t}},t}}}}({r_{i,l,h,t}} - {\mu _{i,l,h,{a_{i,l,h,t}}}}) \label{14}
\end{align}
where ${\mu _{i,l,h,k}}$ is the estimated mean of the reward distribution of SBS $k$ for UE $i$ in area $l$ and cluster $h$, which is updated as in (\ref{14})\footnote{At time step $t$, $k=a_{i,l,h,t}$.}, and $M_{i,l,h,k,t - 1}$ is the number of times that UE $i$ selects SBS $k$ up to time step $t$.
\begin{figure}[!htbp]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics [width=\linewidth]{plotHO.eps}
\caption{ The averaged throughput and HO rate in the testing time. }
\label{fig7}
\end{figure}
In Fig. \ref{fig7}, we compare the averaged HO rates and throughputs of UCB, SL initialized A3C-online and A3C-offline RL methods with/without clustering for a newly arriving UE. Note that we randomly pick one learned network from the 100 realizations used in Fig. \ref{fig6} and consider it as the pretrained network for testing, where the testing period is 2 seconds. In particular, the pretrained network is used to control the UE HOs without further learning in the off-line method. In the on-line method, the UE keeps learning with the same A3C framework. Moreover, the newly arriving UE with UCB needs to learn with random initialization. Hence, for fairness, the HO rate and throughput for UCB are averaged over 121 seconds including the learning and testing periods. Similar to the on-line method, the UE with UCB keeps learning during testing. Furthermore, such generated HO rates and throughputs are all averaged over 500 random runs. We see that our proposed methods could gain better throughputs and lower HO rates after equal learning time compared with the UCB algorithm, which is already up to $80\%$ better than the 3GPP method \cite{7925950}. In addition, the clustering with a global parameter server per cluster further improves the performance.
\subsection{Discussion}
The asynchronous learning scheme suffers from a critical issue, which is called delayed-gradient. Specifically, before UE $i$ in area $l$ and cluster $h$ wants to push the gradient $\Delta {\boldsymbol{\theta}_{i,l,h}}$ (calculated based on the global weights ${\boldsymbol{\theta}_{h}}$ in cluster $h$ at a global time step $t_{global}$) to the parameter server of cluster $h$, several other UEs may have already pushed in their gradients, and the global weights ${\boldsymbol{\theta}_{h}}$ in the parameter server may have been updated for $\Delta t_{global}$ time steps, where $\Delta t_{global}$ is called the delay factor. Moreover, the upper bound of the delay factor $\Delta t_{global}$ is roughly proportional to the number of UEs. According to the recent results \cite{Lian15}, the number of UEs should be upper-bounded by $O(\sqrt{t_{ps}^h})$ \cite{Lian15} to ensure that the asynchronous learning scheme can accelerate the learning.
\section{Conclusions}
In this study, we proposed a two-layer framework to optimize the HO processes in the goal of lowering the HO rates and ensuring the system throughputs, where the user mobility patterns could be heterogeneous. In the proposed framework, a centralized controller first partitioned the UEs with different mobility patterns into clusters. Then, an asynchronous learning scheme is adopted in each cluster. In particular, we let the UEs fetch the most recent copy of DQN approximators from a global parameter server in each cluster, then let them execute the policy and compute the gradients using the observed states. Afterwards, the new gradients are pushed back to the parameter server for updating the global parameter. Thanks to the generalization ability of DNN, newly arriving UEs can use the pretrained neural networks from the parameter server to avoid the likely ill-performed initial points. To further improve the performance, we use SL based on data from traditional HO methods to initialize the DQN approximators before the execution of RL, to compensate the negative effects caused by exploration in the early stages of learning.
\IEEEpeerreviewmaketitle
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
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{"url":"https:\/\/www.physicsforums.com\/threads\/area-of-a-circle-and-integral.613610\/","text":"Area of a circle and integral\n\n1. Jun 13, 2012\n\nestro\n\n$$\\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} \\sqrt{r^2-x^2-y^2}dy$$\n\nI can calculate the above integral [part of a double integral] by the conventional way [somewhat long], however my book says that this integral equals to $$\\frac {\\pi}{2}(r^2-x^2)$$ because the integral is actually the area of half a circle. I have difficulty to understand how this integral has something to do with half of a circle. [The author of my book thinks this is fact is trivial, however for me it is not =(]\n\nAnd if the above integral is the area of half a cicle than what about:\n$$\\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} (r^2-x^2-y^2)dy$$\nI'll appreciate any help.\n\n[EDIT: I think now I understand why the first integral is the area of half a cirlce, now thinking what I can tell about the second integral...]\n\nLast edited: Jun 13, 2012\n2. Jun 13, 2012\n\nNewtonianAlch\n\nWell, if you look at the bounds, y = sqrt(r^2 - x^2)\n\nYou can rearrange to get x^2 + y^2 = r^2\n\n3. Jun 13, 2012\n\nInfinitum\n\nSince you are integrating with respect to y, $r^2 - x^2 = t^2$ is a constant. So the integral becomes,\n\n$$\\int_{-t}^{t} \\sqrt{t^2-y^2}dy$$\n\nNow, the function you are integrating can be taken as,\n\n$$X = \\sqrt{t^2 - y^2}$$\n\nAnd this is the equation of the semi-circle.\n\nOr, fully in circle form(the square root gives only positive solutions implying half circle),\n\n$$X^2 + y^2 = t^2$$\n\nWhere t is the radius. So, the integral you are trying to find is simply the area of a half circle of radius t.\n\n4. Jun 13, 2012\n\nestro\n\nThanks for the answers however I already understood the $$\\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} \\sqrt{r^2-x^2-y^2}dy$$ case.\n\nLooking now at $$\\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} (r^2-x^2-y^2)dy$$\n\nCan I compute this integral in similar way? I seems like this time it is an ellipse.\n\n5. Jun 13, 2012\n\nInfinitum\n\nWhy do you think it is an ellipse?\n\nUsing the same idea, you would have,\n\n$$X + y^2 = t^2$$\n\nThis doesn't resemble any of the standard shapes, I believe. ie circle, ellipse, square etc.\n\n6. Jun 13, 2012\n\nestro\n\nSo I guess my idea is wrong, I'm trying to calculate the following integral:\n\n$$\\iint\\limits_{x^2+y^2\\leq R^2}(R^2-x^2-y^2)dxdy$$\n\nAny ideas?\n\n7. Jun 13, 2012\n\nInfinitum\n\nI wonder what the idea was....\n\nHow did you try this one?\n\n8. Jun 13, 2012\n\nestro\n\nI tried a lot of things the last one was:\nhttps:\/\/dl.dropbox.com\/u\/27412797\/hard_integral.jpeg [Broken]\nBut calculating this integral is a lot of work, I was given a hint that I can use the symmetry of the area which is a circle, but still unable to find the proper way to use this symmetry.\n\nLast edited by a moderator: May 6, 2017\n9. Jun 13, 2012\n\nsharks\n\nestro, this is such a huge scan! You could have used Microsoft Paint or any other image editor to cut the white parts and downsize it for viewing convenience.\n\n10. Jun 13, 2012\n\nestro\n\nSorry for that.\nJust cropped the scan, please refresh =)\n\n11. Jun 13, 2012\n\nsharks\n\nHave you done polar coordinates? It's much simpler to evaluate this double integral if you first convert to polar coordinates.\n$$\\iint\\limits_{x^2+y^2\\leq R^2}(R^2-x^2-y^2)dxdy=\\int^{\\theta=2\\pi}_{\\theta=0} \\int^{r=R}_{r=0} (R^2-r^2).rdrd\\theta$$\nNote that $x^2+y^2=r^2$ where $r$ is the radius of the circle.\n$x^2+y^2\\leq R^2$ describes a disc of radius R.\n\n12. Jun 13, 2012\n\nestro\n\nI'm not allowed to use polar coordinates on this question. The instructor told us symmetry [of the area of integration] should be enough to solve it. =(\n\n13. Jun 13, 2012\n\nsharks\n\n$$\\iint\\limits_{x^2+y^2\\leq R^2}(R^2-x^2-y^2)dxdy$$\nFirst, try to describe the region of integration:\nFor y fixed, x varies from $x =-\\sqrt{R^2-y^2}$ to $x =\\sqrt{R^2-y^2}$\ny varies from $y =-R$ to $y=R$\nNow that you have the limits, you should be able to evaluate the double integral.\n\n14. Jun 13, 2012\n\nInfinitum\n\nHmm, the only way I can currently think of using symmetry is by splitting the term as $\\sqrt{R^2-x^2-y^2} \\cdot \\sqrt{R^2-x^2-y^2}$. Its not a very less tedious method, though(but it is interesting!)\n\n15. Jun 13, 2012\n\nestro\n\nThis is exactly what I did [see the attached scan], I'm looking for more elegant solution... =)\n\n16. Jun 13, 2012\n\nestro\n\nYes, I tried this one as well bu maybe I missed something let me recheck.\n\n17. Jun 13, 2012\n\nestro\n\nActually I have pretty nice solution, but I'm not sure my instructor meant it to be solved this way. I will post it in a hour. [I think you will like it]\n\n18. Jun 13, 2012\n\nInfinitum\n\nNice, I'll ponder over it a bit more till then, getting somewhere.\n\n19. Jun 13, 2012\n\nestro\n\nIn the following proof I pretty much used only simple tools and intuition. I'm still looking for another way to solve it let me know if you have other ideas.\n\nThe drawing is only 1\/4 of the actual 3D body. [My drawing skills are as bad as my math=)]\nhttps:\/\/dl.dropbox.com\/u\/27412797\/short_proof_2.JPG [Broken]\n\nLast edited by a moderator: May 6, 2017\n20. Jun 13, 2012\n\nInfinitum\n\nNice and elegant","date":"2018-03-21 15:42:31","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\": 2, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8777106404304504, \"perplexity\": 900.6253712045694}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257647660.83\/warc\/CC-MAIN-20180321141313-20180321161313-00177.warc.gz\"}"} | null | null |
Q: laravel 8 (errno: 150 "Foreign key constraint is incorrectly formed") i try to create forum in last php and laravel 8. I have buy course in udemy in laravel 8 i follow video from him but in my computer have error and in video doesn't have
2021_11_13_000535_create_posts_table
use Illuminate\Database\Migrations\Migration;
use Illuminate\Database\Schema\Blueprint;
use Illuminate\Support\Facades\Schema;
class CreatePostsTable extends Migration
{
/**
* Run the migrations.
*
* @return void
*/
public function up()
{
Schema::create('posts', function (Blueprint $table) {
$table->id();
$table->string('title');
$table->integer('is_deleted');
$table->integer('is_approved');
$table->string('image');
$table->unsignedBigInteger('discussion_id');
$table->foreign('discussion_id')->references('id')->on('discussions')->onDelete('cascade');
$table->unsignedBigInteger('user_id');
$table->foreign('user_id')->references('id')->on('users')->onDelete('cascade');
$table->string('slug');
$table->timestamps();
});
}
/**
* Reverse the migrations.
*
* @return void
*/
public function down()
{
Schema::dropIfExists('posts');
}
}
2021_11_19_165302_create_discussions_table
<?php
use Illuminate\Database\Migrations\Migration;
use Illuminate\Database\Schema\Blueprint;
use Illuminate\Support\Facades\Schema;
class CreateDiscussionsTable extends Migration
{
/**
* Run the migrations.
*
* @return void
*/
public function up()
{
Schema::create('discussions', function (Blueprint $table) {
$table->id();
$table->string('title');
$table->string('desc');
$table->unsignedBigInteger('forum_id');
$table->foreign('forum_id')->references('id')->on('forums')->onDelete('cascade');
$table->integer('is_deleted')->default(0);
$table->string('image')->nullable();
$table->integer('notify')->default(0);
$table->unsignedBigInteger('user_id');
$table->foreign('user_id')->references('id')->on('users')->onDelete('cascade');
$table->timestamps();
});
}
/**
* Reverse the migrations.
*
* @return void
*/
public function down()
{
Schema::dropIfExists('discussions');
}
}
when i try to do migrate table i have this error :
Migrated: 2014_10_12_000000_create_users_table (1,399.45ms)
Migrating: 2014_10_12_100000_create_password_resets_table
Migrated: 2014_10_12_100000_create_password_resets_table (2,117.91ms)
Migrating: 2019_08_19_000000_create_failed_jobs_table
Migrated: 2019_08_19_000000_create_failed_jobs_table (1,592.76ms)
Migrating: 2019_12_14_000001_create_personal_access_tokens_table
Migrated: 2019_12_14_000001_create_personal_access_tokens_table (2,125.96ms)
Migrating: 2021_11_12_234608_create_categories_table
Migrated: 2021_11_12_234608_create_categories_table (2,452.77ms)
Migrating: 2021_11_12_235039_create_forums_table
Migrated: 2021_11_12_235039_create_forums_table (2,849.71ms)
Migrating: 2021_11_13_000340_create_tags_table
Migrated: 2021_11_13_000340_create_tags_table (526.62ms)
Migrating: 2021_11_13_000535_create_posts_table
Illuminate\Database\QueryException
SQLSTATE[HY000]: General error: 1005 Can't create table `stsdb`.`posts` (errno: 150 "Foreign key constraint is incorrectly formed") (SQL: alter table `posts` add constraint `posts_discussion_id_foreign` foreign key (`discussion_id`) references `discussions` (`id`) on delete cascade)
at vendor/laravel/framework/src/Illuminate/Database/Connection.php:703
699▕ // If an exception occurs when attempting to run a query, we'll format the error
700▕ // message to include the bindings with SQL, which will make this exception a
701▕ // lot more helpful to the developer instead of just the database's errors.
702▕ catch (Exception $e) {
➜ 703▕ throw new QueryException(
704▕ $query, $this->prepareBindings($bindings), $e
705▕ );
706▕ }
707▕ }
+9 vendor frames
10 database/migrations/2021_11_13_000535_create_posts_table.php:28
Illuminate\Support\Facades\Facade::__callStatic()
+21 vendor frames
32 artisan:37
Illuminate\Foundation\Console\Kernel::handle()
someone can explain why have this error and how resolve this? because i dont understand i follow like in video in my pc have error and in video doesn't have. Someone can help resolve this? I have checked in google but i have dont find how resolve this, i follow tuts build forum laravel 8 with telegram notification
A: The problem is that your migration for the posts table is run prior to your discussions migration.
This happens because Laravel runs the migrations ordered by the timestamp in the migrations file name:
2021_11_13_000535_create_posts_table -> 13. November
2021_11_19_165302_create_discussions_table -> 19. November
Therefore the dicsussions table is not created yet as my comment suggested!
The solution is easy, change the file name to:
2021_11_20_000535_create_posts_table -> 20. November
Next time please take a look into your DB as I suggested earlier.
From their documentation:
Generating Migrations
You may use the make:migration Artisan command to generate a database
migration. The new migration will be placed in your
database/migrations directory. Each migration filename contains a
timestamp that allows Laravel to determine the order of the
migrations:
https://laravel.com/docs/8.x/migrations#generating-migrations
| {
"redpajama_set_name": "RedPajamaStackExchange"
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# **ALSO BY LARRY TYE**
_Satchel: The Life and Times of an American Legend_
_The Father of Spin: Edward L. Bernays and the Birth of Public Relations_
_Home Lands: Portraits of the New Jewish Diaspora_
_Rising from the Rails: Pullman Porters
and the Making of the Black Middle Class_
**WITH KITTY DUKAKIS**
_Shock: The Healing Power of Electroconvulsive Therapy_
Copyright © 2012 by Larry Tye
All rights reserved.
Published in the United States by Random House, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.
RANDOM HOUSE and colophon are registered trademarks of Random House, Inc.
Library of Congress Cataloging-in-Publication Data
Tye, Larry.
Superman : the high-flying history of America's most enduring hero / Larry Tye.
p. m.
eISBN: 978-1-58836-918-5
1. Superman (Fictitious character) I. Title.
PN6728.S9T94 2012
741.5′973—dc23 2011045280
www.atrandom.com
Jacket design: David Stevenson
Jacket illustration: © The Dynamic Duo/Corbis Images
Jacket photograph: © Shutterstock
SUPERMAN is a trademark of DC Comics. Used under License
v3.1
_To Lisa_
# **Preface**
**ENDURANCE**
THE MOST ENDURING AMERICAN HERO of the last century is someone who lived half his life in disguise and the other half as the world's most recognizable man. He is not Jack Kennedy or Joltin' Joe DiMaggio, Batman or Jerry Seinfeld, although all of them were inspired by him. It was on his muscle-bound back that the iconic comic book took flight and the very idea of the superhero was born. He appeared on more radio broadcasts than Ellery Queen and in more movies than Marlon Brando, who once pretended to be his father. He helped give America the backbone to wage war against the Nazis, the Great Depression, and the Red Menace. He remains an intimate to kids from Boston to Belgrade and has adult devotees who, like Talmudic scholars, parse his every utterance. And he has done it all with an innocence and confidence that allowed him to appear publicly wearing underpants over full-body tights, and to assume an alter ego who kept pursuing the prettiest girl in town even though he seldom got her.
The most enduring American hero is an alien from outer space who, once he reached Earth, traded in his foreign-sounding name Kal-El for a singularly American handle: Superman.
Ah, you say, the Man of Steel—I know him! But do you really? Do you know the wrenching story of his birth and nurturing at the hands of a parade of young creators yearning for their own absent fathers? The first was the youngest child of Lithuanian immigrants who was devastated when his dad died during a robbery. While there was no bringing back his father and role model, Jerry Siegel did bring to life a hero able not just to run fast and jump high but, as we see early on, to fend off a robber. Who would publish this fanciful tale? How about Jack Liebowitz, a hardheaded comic book entrepreneur whose own dad had died just after he was born and who needed a champion? Whitney Ellsworth, the man who wrote, edited, and produced nearly all the episodes of the 1950s TV show that introduced many baby boomers to this costumed hero, was just fourteen when he lost his forty-five-year-old father to a heart attack. George Reeves, TV's original Clark Kent and Superman, didn't even know who his real father was until he was in his twenties. Who better to create the ultimate childhood fantasy figure than men whose childhoods had been stolen from them?
Not just Superman but his rivals, too, were more than they seemed—and more than just fantasy. Many of them were real-world menaces, which made the Superman stories timely and authentic. Superman stood up to Hitler and Stalin before America did. The Metropolis Marvel used his radio broadcast to expose the savagery of the Ku Klux Klan, and in his comic books he upended slumlords and wife-beaters. Lex Luthor, Superman's most persistent foe, likely came from Jerry Siegel's boyhood. The day after Jerry's father died, his hometown newspaper published a letter denouncing the kind of vigilante justice that would become Superman's early signature. The letter writer: A. L. Luther.
The superhero never revealed how he voted, but during the Great Depression he was a New Dealer hell-bent on truth and justice, and during the Reagan Revolution he was a patriot trumpeting the American way. His sex life underwent an even more drastic about-face: from celibate to satisfied husband. There is one more thing that even his most fervent fans may not know about the Man of Steel: He is Jewish.
I have been captivated by Superman ever since I wrote my first book, the life story of the public relations pioneer and master manipulator Edward L. Bernays. Borrowing ideas from his uncle Sigmund Freud, Bernays single-handedly shaped many of our political and cultural appetites—from sweetening sour politicians like Calvin Coolidge to selling America on the European tenor Enrico Caruso. Negro Leagues strikeout king Leroy "Satchel" Paige, the subject of my most recent book, borrowed Bernays's techniques in crafting his own eye-popping legend. Paige and Bernays got me wondering: Why does America embrace the heroes it does? What do our choices say about them and, more important, about us? There's no better way to understand modern-day heroes than to look at Superman, the superhero who tapped into the American psyche more effectively than anyone else and, as a result, has lasted longer than all of them.
Clearly there was a serious story here, but for me the other appeal of writing this book was getting to be ten again. I had grown up reading Superman comics and watching nearly all 104 episodes of _Adventures of Superman_ on TV. I sat mesmerized by his movies and pondered: What would it feel like to actually take flight? Superman was comfort food for my spirit, and writing this book let me partake of that comfort all over again. It also let me imagine what Superman did between adventures, when he doffed his cape and sprawled out on the couch.
Okay, so I still cared about Superman, but did anyone else? Sure, he was a big deal when I was coming of age in the 1960s, but I assumed he was passé in the virtual realities of the new millennium. Then I started paying attention. My four-year-old nephew showed up one night wearing a Superman shirt. My sixteen-year-old stepdaughter told me how, when she was four, she had trick-or-treated dressed not as Supergirl but as Superman. My oldest friend's fourteen-year-old daughter showed me her DVD collection of _Smallville_ , a show I had never heard of that, for ten seasons, has chronicled the adventures of a young Clark Kent. The final test came on Halloween, when merchants in my hometown of Lexington, Massachusetts, hand out candy all afternoon, packing sidewalks with costumed kids who provide the perfect sampling with which to judge who's hot in the world of heroes. Spider-Man did well, with half a dozen children dressed in webbed costumes, but the hands-down winner was the blue tights, red cape, and bright yellow _S_ of Superman.
I still had doubts, but they had changed from whether anyone cared to why. Why did Senator Barack Obama pose in front of a Superman statue and later, just before his election as president, joke that he, too, came from Krypton? Surely he understood better than anyone the value of bonding to such a symbol of strength and honor. Why is the Man of Steel as popular today as he was in my boyhood and in every era back to his begetting in the 1930s? That is more than we can say for Jim Thorpe or Dwight Eisenhower, the Phantom, the Lone Ranger, or Tarzan the Ape Man. By the time of his death, even a brilliant pitcher and consummate character like Satchel Paige had become a vague if nostalgic memory for most. Heroes, understandably, are woven into their time and seldom last far beyond it. How had Superman broken the mold?
To answer those questions, I did what any journalist would: I asked smart people who ought to know. I did more than two hundred interviews, starting with historians, clerics, and psychologists who have written and lectured about Superman's godlike attributes, his corrupting influences, and why children and their grandparents continue to embrace him. I spoke with writers and artists who brought Superman to life in comic books, comic strips, novels, and graphic novels, as well as on radio, TV, film, and animation. Ninety-two-year-old ghostwriter Alvin Schwartz told me that he wasn't supposed to tell anyone that he was the man behind the newspaper strip, but then _The New York Times_ outed him. Jack Larson and Noel Neill explained why they initially did all they could to escape their Jimmy Olsen and Lois Lane personas from the wildly popular _Adventures of Superman_ , but now—like Superman's other aging midwives—they relish the attention the connection still brings. Aaron Smolinski recounted the way kids teased him for appearing naked as baby Clark in _Superman: The Movie_ and the way people still ask, "How did you lift the truck?" "I say, 'I'm Superman.' "
I visited Superman's publisher in New York. I went to his movie studio in California and his hometown of Cleveland. I talked to fanatic fans and casual ones, adolescents and octogenarians, here and abroad. They all agreed with Donald Wurzelbacher of Cincinnati that Superman is "the godfather of superheroes... the original, first, and greatest."
I listened to hour after hour of Superman's old radio broadcasts and watched his early and recent TV shows, cartoons, and films. I read everything I could find about him in books, magazines, and newspapers, where columnists loved to quote him in the financial and editorial sections as well as in the comics, in tabloids like the _New York Post_ and high-minded broadsheets like _The New York Times_. I pored over thousands of pages of public records, and others that have never been released, from the ongoing lawsuit against Superman's publisher by his creators' heirs. I read the unpublished memoirs of Jerry Siegel and Jack Liebowitz, Superman's originator and patron, and talked to their friends and relatives. I reviewed yellowing police files and coroner's reports on George Reeves, the TV Man of Steel, with forensic experts and researchers who have spent a lifetime looking into his death. I began by worrying whether, given all that has been written on Superman, I would have anything new to say. I ended by worrying how to fit into a single manuscript all I have to say on this unambiguous hero who is as much a part of our communal DNA as Paul Bunyan or Huckleberry Finn.
So how has Superman managed to thrive for nearly seventy-five years?
It starts with the intrinsic simplicity of his story. Little Orphan Annie and Oliver Twist remind us how compelling a foundling's tale can be, and Superman, the sole survivor of a doomed planet, is a super-foundling. The love triangle connecting Clark Kent, Lois Lane, and Superman has a side for everyone, whether you are the boy who can't get the girl, the girl pursued by the wrong boy, or the conflicted hero. His secret identity might have been annoying if we hadn't been let in on the joke, and if we didn't each have a hero hidden within ourselves. He was not just any hero, but one with the very powers we would like to have: the strength to lift boulders and planets, the speed to outrun a locomotive or a bullet, and, coolest on anyone's fantasy list, the gift of flight.
Superpowers are just half the equation. More essential is knowing what to do with them, and nobody has a more instinctual sense than Superman of right and wrong. He is an archetype of mankind at its pinnacle. Like John Wayne, he sweeps in to solve our problems. No thank-you needed. Like Jesus Christ, he descended from the heavens to help us discover our humanity. He is neither cynical, like Batman, nor fraught, like Spider-Man. For the religious, he can reinforce whatever faith they profess; for nonbelievers, he is a secular messiah. The more jaded the era, the more we have been lured back to his clunky familiarity. The outcome of his adventures may be as predictable as those of Sherlock Holmes—the good guy never loses—but that too is reassuring.
So is his uniform. His tights and cape, in radiant primary colors, make Superman as instantly recognizable as Santa Claus—and as comforting. That familiarity helped his handlers move him from the printed page to the airwaves, then from the small screen to the big. No need to explain who he was; everyone knew as soon as they saw him. A costume could also be electrifying, the more so when it didn't come with a mask. Just ask Robin Hood and Elvis Presley.
That does not mean he hasn't changed with the times. Superman has evolved more than the fruit fly. In the 1930s he was just the crime fighter we needed to take on Al Capone and the robber barons. In the 1940s he defended the home front while brave GIs battled overseas. Early in the Cold War he stood taller than ever for his adopted country, while in its waning days he tried single-handedly to eliminate nuclear stockpiles. For each era he zeroed in on the threats that scared us most, using powers that grew or diminished depending on the need. So did his spectacles, hairstyle, even his job title. Each generation got the Superman it needed and deserved. Each change offered a Rorschach test of that time and its dreams. Superman, always a beacon of light, was a work in progress.
Over the years, comics, too, have been transformed—from childhood entertainment to art form to mythology—and Superman helped drive that transformation. The comic book and its leading man could only have taken root in America. What could be more U.S.A. than an orphaned outsider who arrives in this land of immigrants, reinvents himself, and reminds us that we can reach for the sky? Yet today this flying Uncle Sam is global in his reach, having written himself into the national folklore from Beirut to Buenos Aires. It is that constancy and purity—knowing that he is not merely the oldest of our superheroes but the most transcendent—that has reeled back aging devotees like me and drawn in new ones like my stepdaughter and nephew. It is what makes the Man of Tomorrow timeless as well as ageless.
# **Contents**
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
_Preface: Endurance_
**CHAPTER 1** _Giving Birth_
**CHAPTER 2** _A Hero for His Times_
**CHAPTER 3** _A Matter of Faith_
**CHAPTER 4** _The Speed of Sound_
**CHAPTER 5** _Superman, Inc_.
**CHAPTER 6** _The Deadly Truth_
**CHAPTER 7** _Imagine This_
**CHAPTER 8** _Believing a Man Can Fly_
**CHAPTER 9** _Back to the Future_
**CHAPTER 10** _Till Death Do Us Part_
**CHAPTER 11** _Tights and Fights_
_Acknowledgments_
_Appendix: Superman's CV_
_Notes_
_Bibliography_
_About the Author_
_Photo Insert_
# **CHAPTER 1**
# **Giving Birth**
LEGEND HAS IT THAT SUPERMAN was born under a fiery red sun on the futuristic planet of Krypton, in a crystal tower overlooking the Jewel Mountains and the Scarlet Jungle. But the legend has it wrong. In fact, Superman was born under a hazy yellow sun in a gritty Jewish precinct of Cleveland, two blocks from the Hebrew Orthodox Old Age Home and down the street from Glenville High. Just ask Jerry Siegel. He's the one who brought him to life there in the throes of the Great Depression.
Jerry Siegel happened to have been born in a gritty Jewish precinct of Cleveland, too, in 1914. And being Jerry never was easy. His trouble began in first grade. The stubby six-year-old had proudly memorized the rules for asking to pee: You raised your hand, and the teacher acknowledged you and said it was okay to go to the bathroom. The boy behind him did it. A pigtailed girl followed. But there was no reply when Jerry raised his hand. Finally the teacher turned his way: "What do _you_ want?" He told her. "No," she said. Maybe she thought he was faking. Maybe it was that he was short, shy, wore glasses, and was the child not of refined German Jews but of unwashed immigrants from Eastern Europe. Whatever the reason, his bladder swelled and a puddle formed under his seat. With other children pointing, the teacher descended: "You are a bad, bad, bad, bad boy! Bad and disgusting! Leave the room, this very instant! Go home!" "At an early age," Jerry recalled decades later, "I got a taste of how it feels to be victimized."
That sensation became a pattern. On Valentine's Day, classmates addressed cards to one another; the teacher handed them out as the students waited anxiously. The first year Jerry got just one, from his sympathetic teacher. The next year he secretly inscribed a card to himself. Jerome the Loner, he thought. Jerome the Pariah. Jerome the Outcast. Schoolwork was equally problematic. The semester started with smiles and anticipation. "Happiness," he would say, "vibrated all over the place. But then, when the grim business of cramming knowledge into one's skull got down to business, interest in arithmetic, geography, etc. just slid off my brain, and oozed into a crack in the floor, where it gradually evaporated." He got used to _D_ s and _F_ s—and to summers repeating the failed subjects, which "was even more dismal. While other kids enjoyed summer vacation, I had my nose rubbed into education."
Recess, too, was a trial and oftentimes a terror for him. Tormenters were everywhere. Some tripped him as he tried to escape, others punched. His very name became a source of ridicule. "Siegel, Seagull, bird of an Eagle!" they would chant. If only he really could fly away. If only the girls hadn't heard. He was too bashful to say a word to pretty ones like Lois Amster, the girl he had a crush on, but even the homely ones showed zero interest. "I hadn't asked for the face or physique I was born with," he wrote. "I had not sculpted my nose, or fashioned my chin, or decided how broad my shoulders would be, or how tall I would become. I looked searchingly into the mirror for a clue. The mirror refused to commit itself." Doubts like those are part of growing up. Most kids outrun or outgrow them. Jerry's stuck like a mark of Cain from grammar school all the way through high school, where he would often turn up late, with his hair flying off in different directions and his pajamas just visible under his pant cuffs and over his shirt collar.
With the real world offering no solace, he created one built around fantasies. Mornings, he stood in the schoolyard until his classmates disappeared indoors, then he headed to the public library. Pulling his favorites from the tall stacks of books, he was transported into the dime-novel worlds of master detective Nick Carter, collegiate crime buster Frank Merriwell, and adventurers closer to his age and circumstance like the Rover Boys. Fred Rover and his cousins Jack, Andy, and Randy may have been in military school, but that never kept them from exploring wrecked submarines or prospecting for pirates' gold. On weekends, Jerry went to matinees at the motion picture theater. Western megastar Tom Mix made 336 films and Jerry saw all that his allowance would allow. He also was an insatiable consumer of movies starring Douglas Fairbanks, Sr., as Zorro, Robin Hood, and the thief of Baghdad. And watching was not enough. Convinced he could replicate Mix's and Fairbanks's derring-do, Jerry darted in and out of traffic on the narrow roads of his Glenville neighborhood. "Those furious humans driving the cars, who yammered and glared insanely at me," he said, "were mere mortals. But I... I was a leaping, twirling, gleeful phenomenon!" Back at home, with his hip healed after one of those glaring drivers sideswiped him, he climbed onto the roof of the garage holding an umbrella. "I opened the umbrella and leapt. Look out world, here I come!... I did this over and over again. Unexpectedly, the umbrella suddenly turned inside-out as I descended. I banged a knee, when I hit the ground. Just as I had abandoned berserkly dodging in and out between moving automobiles, I gave up jumping off the top of my garage."
As freeing as it felt to mimic his idols, better still was concocting narratives starring Jerry Siegel—not the shunned, tongue-tied adolescent the kids in the schoolyard saw, but the real Jerry, fearless and stalwart. The setting, too, was of his own making, leaving behind Glenville's twenty-five Orthodox shuls and row after row of faded up-and-down duplexes. Crawling into bed at night with pencil and paper, he imagined faraway galaxies full of mad scientists and defiant champions. He loved parody, too, inventing characters like Goober the Mighty, a broken-down knockoff of Tarzan. He went on daydreaming in the classroom, and his writing found its way into the high school newspaper, the _Glenville Torch_ , and onto the pages of his own _Cosmic Stories_ , America's first science fiction magazine produced by and for fans.
Jerry wasn't popular, he wasn't strong, but one thing he knew: He was inventive. Pointing to an empty Coke bottle, he told his cousin, "I could make up a story about that." He even tried an autobiographical novel but flushed it down the toilet after a friend suggested that perhaps not all his experiences were worthy of the label "ecstasy." No theme stuck for long, he confessed in a later-life autobiography. And he still couldn't decide whether good guys or bad made better protagonists.
Clarity came on the wings of his own tragedy. It happened on an overcast evening in June 1932, just after eight o'clock, in a downtrodden strip of Cleveland's black ghetto known as Cedar-Central. Michel Siegel was ready to head home to his family when three men whom police would describe as "colored" entered his secondhand clothing store, one of the few Jewish businesses left in a neighborhood populated by barber shops, billiard parlors, and greasy spoons. One man asked to see a suit, then walked out with it without paying; another blocked the owner's path. Michel, a slight man whose heart muscle was weaker than even he knew, fell to the floor. A month shy of his sixtieth birthday, he stopped breathing before medics could get him to the hospital. His wife, Sarah, was a widow now, on her own with three girls, three boys, and next to no savings. Jerry, her youngest, took the loss of his father the hardest. The boy who had been bullied was bereft. Sitting on his dad's knee and being rocked up and down had been one of Jerry's few safe havens. "Bliss," he called it later. "Supreme rapture." Now his father was gone.
The world of make-believe seemed more alluring than ever to Jerry, who was not quite eighteen. What had been a series of disparate characters with no focus or purpose now merged into a single figure who became a preoccupation. He called him "The Super-Man." Jerry's first story, written shortly after his father's death, envisioned the figure as endowed with exceptional strength, telescopic vision, the capacity to read minds, and a resolve to rule the universe. Over the months that followed, this character would drop "the" and the hyphen, along with his evil inclinations, becoming simply Superman—a bulletproof avenger who beat back bullies, won the hearts of girls, and used his superpowers to help those most in need. And who, in the only artwork that survives from that first imagining, soars to the rescue of a middle-aged man being held up by a robber.
· · ·
SUPERMAN MAY HAVE BEEN A product of the 1930s and Jerry Siegel's teenage imagination, but his DNA traces back twenty-five hundred years to the age of the Tanakh, the Hebrew Bible. The evidence is there in the Book of Judges and the parable of its last and most exalted jurist, Samson. With the Israelites desperate to free themselves from forty years of enslavement by the Philistines, God offered up a strongman who killed a lion with his bare hands and then, using nothing more than the jawbone of an ass, slew a thousand enemy soldiers. The Philistines managed to capture this extraordinary being, gouging out his eyes and bringing him to their shrine in shackles to dance before them, humiliated. But in an act of self-sacrifice and backbone that would set a yardstick for every super-being who came after, Samson brought the enemy's temple crashing down around them as he proclaimed, "Let me die with the Philistines!"
Masterful as the Hebrews were at fashioning powerful and noble warriors, no one outdid the Hellenists. The very word "hero" comes from the Greek _heros_ , meaning "protector" or "defender." The Greek pantheon of demigods began with Perseus, famous for slaying monsters from the sea and the land. There was Jason, who led the heroic Argonauts on a quest for the golden fleece; Euphemus, who could walk on water; Caeneus, who was invulnerable to swords, spears, or any weapon known in his day; and Hermes, speediest and cagiest of the gods. The ultimate exemplar of the Greek ideal of heroism was Herakles, the defender against evil and tamer of beasts, whom the Romans would adopt and rebrand as Hercules. Like Superman, Herakles signaled his special powers in infancy, grabbing by their necks a pair of deadly serpents that had crawled into his cradle and squeezing the life from them. And like Superman, Herakles devoted his days to rescuing ladies in distress, battling a shifting cast of villains, and searing a place in the public imagination as an embodiment of virtue.
Each era that followed produced its own mythic figures that reflected its peculiar dreams and dreads. In 1752, Voltaire anticipated the genre of science fiction and poked fun at contemporary dogmas in his tale of Micromegas, a 120,000-foot-tall super-genius who traveled here from a far-distant planet. Micromegas rendered his verdict on Earth: It's not nearly as special as its inhabitants think. Half a century on, nineteen-year-old Mary Shelley gave us Victor Frankenstein, who tapped his collection of dead body parts to build an eight-foot monster with yellowing skin. More even than Voltaire, Shelley reflected the tremendous leap from Hebrew and Greek legends built on superstition to a more modern reliance on science as the wellspring for fantastic literature. Likewise, her monster foreshadowed Jerry Siegel's early vacillation between Super-Man and Superman. Should his standard-bearer be a contemptible villain, an unwavering hero, or something more ambiguous like Dr. Frankenstein?
History's most infamously ambiguous blueprint for the hero was the German philosopher Friedrich Nietzsche's _Übermensch_ , which translates literally as "overman" and colloquially as "superman." With God dead, Nietzsche argued, man would be tempted to look for salvation in an afterlife or from a society that was naively egalitarian. The real place to look, he said, was among mankind's talented few—its Caesars and Napoleons—who were ready to rule decisively and efficiently. "What is the ape to man?" Nietzsche asked in 1883. "A laughingstock or a painful embarrassment. And man shall be just that for the overman." Some interpreted Nietzsche's answer as a Buddha-like call for humans to reach for an enlightened state; others saw a clearheaded if cold assessment of the unequal allocation of human talents. Adolf Hitler used Nietzsche's argument to bolster not just his theory of a master race of Aryan supermen, but also his obsession with rooting out Jews, Gypsies, gays, and others he saw as subhuman. Whether Hitler appropriated Nietzsche's message or perverted it, the lesson for all hero-framers who followed was clear: Be careful. Whatever your intent, madmen can fuse their nightmares onto your dreams. Fairly or not, history will hold you accountable.
That prehistory was especially resonant in 1932, the year Michel Siegel died and The Super-Man was conceived. America's flirtation with science fiction had, by then, mushroomed into a craze. The only medium that mattered was the written one, with AM radio still in its chaotic early era, FM a year away, and network television but a gleam in its designers' eyes. Action and adventure were still essential, but better still was a story that drew on pseudoscience and a hero endowed with superpowers. Popeye the Sailor Man had both, which let him chase Bluto and Sea Hag all over the planet, popping open a can of spinach whenever he needed to recharge his muscles or fend off bullets or aliens. Buck Rogers's oyster was outer space, where his swashbuckling was such a hit that he spawned an interplanetary imitator: Flash Gordon. Alley Oop started out in the Stone Age, in the kingdom of Moo, and ended up in a time-traveling machine. And when it came to brainwashing there were no rivals: Ask any teenager in the 1930s, "Who knows what evil lurks in the hearts of men?" and they answered as one: "The Shadow knows."
The Shadow, an avenger with the power to cloud men's minds so they couldn't see him, was born on the radio and would catch fire everywhere, from magazines, cartoon strips, and comic books to TV, film, and graphic novels. A more typical launching pad was the funny pages, where tens of millions of readers followed Popeye, Tarzan, and their chums every day in black-and-white, and on Sunday in full color. The adventure strip was taking off in 1932, which was just the right moment given what readers were seeing in the rest of the newspaper.
Who wouldn't want to escape his circumstances, if not his planet, with the world economy in free fall? One in four Americans had no job. The British had just tossed into jail the conscience of the world, Mahatma Gandhi. Millions of Soviets were starving to death. Almost as unsettling was the human-scale drama of a twenty-month-old toddler: Charles Augustus Lindbergh, Jr., son of America's beloved aviator-inventor, was discovered missing from his crib the evening of March 1. The "crime of the century" riveted the nation, as a note from kidnappers told the Lindberghs to "have $50,000 redy" and assured them that "the child is in gut care." Gangster Al Capone promised that if he was let out of jail he would crack the case, while President Herbert Hoover vowed to "move Heaven and Earth" to find the infant. It was truck driver William Allen who actually did, two months after the abduction. Stopping to relieve himself in a grove of trees five miles from the Lindbergh home, he discovered the remains of a baby. The skull was fractured. The left leg was gone, along with both hands, and the torso had been gnawed on by animals. But the overlapping toes of the right foot and a shirt stitched by his nursemaid identified the body as the Lindbergh boy.
Escape indeed. Some kids chose dance marathons—known as "corn and callus carnivals"—to blot out the news and test whether they could keep fox-trotting or waltzing for twelve, twenty-four, or even thirty hours. An easier way to take flight during that decade of despair was through science fiction, and especially through a new trio of mythmakers. Each understood that while Herakles suited the needs of ancient Greece, and Frankenstein was monster enough for the 1800s, the twentieth century's expanding horizons of technology, medicine, and cognition required a paladin who was more expansive, more imaginative, more _today_. Each saw his hero not as a gift from God but as a triumph of fantastic science. All three could claim to be Superman's patron saint if not his progenitor.
First on the scene was Tarzan creator Edgar Rice Burroughs. His protagonist, John Carter of Mars, was actually from Earth, and more precisely Virginia. Carter had served in the Confederate Army and then struck gold in Arizona, but before he could spend it he was killed by Apaches. Instead of in heaven, however, he ended up on the red planet, or rather on its fantastic double that Burroughs dubbed Barsoom, where he stayed forever young and strong enough to defend his new planet from beastly villains. Burroughs introduced Carter to the world in the 1912 novel _A Princess of Mars_ , which foreshadowed Superman in ways substantial and small. John Carter traveled in space and was invulnerable. His strength on Mars came from the planet's having less gravity than Earth, the flip side of what would happen to Superman when he reached Earth from Krypton. And the name Krypton came from the same line of the periodic table of the elements from which Burroughs plucked the name Helium, one of the empires on Barsoom.
In 1930, two years after Burroughs came out with his sixth Mars-Barsoom novel, Philip Wylie published a book called _Gladiator_ with a hero named Hugo Danner. Danner's father, biology professor Abednego Danner, concocted a serum so effective in turning tame animals into ogres that he could not resist trying it on humans. The easiest subject at hand was his pregnant wife. It worked, and she delivered a son with the strength of Samson, the speed of Hermes, and skin, like Caeneus's, that was impervious to injury—a package similar to the one Siegel would unwrap eight years later. Hugo's powers were hinted at in the crib, as Superman's would be; both were cautioned by their fathers to use their gifts judiciously; and the two authors settled on the same superlative to describe their creations: "superhuman" for Wylie, "superman" for Siegel.
The last in the triumvirate of early-twentieth-century science fiction exemplars was Clark Savage, Jr., known to the world as "Doc." His first tale hit the newsstands in February 1933, just as the Depression was reaching its nadir, President Franklin Roosevelt was about to declare a holiday that would close every bank in America, and Jerry Siegel was putting the finishing touches on a superhero he would nickname the Man of Steel. It helped to have as a model Doc Savage, a.k.a. the Man of Bronze. Savage's name rightfully suggested brute strength, but he also was endowed with the deductive skills of Sherlock Holmes, the tree-swinging grace of Tarzan, the scientific-sleuthing acumen of Dick Tracy, and the morals of Abraham Lincoln. Doc crafted a hero's code of conduct that would offer a prototype for Superman and his crime-fighting cohorts: Do not kill your enemy if you can help it. Do not get entangled with women. Do find a remote getaway—he and Superman both picked the Arctic, and both called their getaways the Fortress of Solitude—where you can take a break from saving the world.
On the eve of his birth, then, Superman's world was awash in heroes. So keen was the ferment and the determination to be noticed that the word the Greeks had given us no longer was enough. Now authors and publishers were beating the drums for their champions by inflating their adjectives— _thrilling, marvelous, amazing_ , and soon, the most singular and separating descriptive of them all: _super_ , short for superlative. This new breed of hero started out with the models of strength and courage from the past, then added twists—interplanetary adventure, space-age gadgets, time travel—to spark the imaginations of readers reared in a world of automobiles, airplanes, and skyscrapers. Most had followings measured in the hundreds of thousands or even millions. None—not the buff Doc Savage or the brooding Hugo Danner, the soaring John Carter or the elusive Shadow—had whatever alchemy was needed to separate them from the crowd. As saturated as the market seemed, America was readier than it knew for a hero sized to the age of the metropolis.
· · ·
JEROME THE LONER GOT IT. He may not have been much of a student at Glenville High, where it took him five years to graduate, but Jerry Siegel was a scholar of science fiction and pop culture. He saw the lightning in the air and was determined to bottle it. Being in Cleveland put him far enough away from Oz that the only place to realize his dreams was in his imagination and his writings. Being a loner, with few friends and no girlfriend, let him be single-minded in those pursuits. Being a kid gave him just the right vantage point.
As early as junior high he was poring over the pulps, the ten-cent magazines that were the successors to Britain's aptly titled penny dreadfuls and forerunners to the comic book. Pulps took their name from the coarse paper they were printed on, and they took stories from just about anyone. A few were long-lived gems like _The Maltese Falcon_ and _The Shadow_ , which ran the length of a novel and had heroic narratives; more were trashy tales of sex and mayhem. The earliest pulps had appeared in the 1890s, and by the 1920s some were selling a million copies an issue to working stiffs anxious to escape their vanilla lives or kids with an extra dime and an appetite for wars and westerns. Jerry brought his everywhere, including study hall, which earned him a visit to the principal and a warning to "never, never, never again transgress in this unspeakable manner... or else." But the message he was getting from trade magazines was more persuasive, with announcements that authors nearly as young as he were being published in the pulps, while _Reader's Digest_ trumpeted the big money that comic strip writers were making.
High school was the ideal testing ground for Jerry's fevered imagination. The _Torch_ dubbed him the "master of deduction" and ran stories by him that aped his favorite writer, Edgar Rice Burroughs. He sent handwritten features and letters to the _Sunday Buffalo Times_ and at least one was published, a piece titled "Monsters of the Moon." He corresponded with other wannabe writers. Over time he got bolder, launching The Fantastic Fiction Publishing Company, naming himself president, and putting out a typewritten journal called _Cosmic Stories_. He had but one writer, himself, although in what would become a pattern he took on a pen name, Charles McEvoy. Jerry's explanation for this: "Jerome Siegel did not seem very literary to me." A more likely explanation: A boy who saw himself as a pariah wanted to shield himself from brickbats, and perhaps from anti-Semitism.
_Guests of the Earth_ was his next publication. He printed it after hours on the high school mimeograph machine and published it under an equally Waspy pseudonym: Hugh Langley. When an English teacher saw it, she demanded, "Jerome, why do you write these kinds of stories, when there are so many more worthwhile things you could write about?" His science teacher didn't bother to ask, dismissing Jerry's pamphlets as "junk" and "foolishness." His parents loved their last-born but were equally unimpressed with his scribbling. Michel merely shrugged. Sarah considered Jerry's dream of becoming a writer "a wild, erratic notion and that nothing would really come of it," Jerry remembered later. "She told me that of all her children, she worried the most about me."
No matter. Like his fellow science fiction aficionados, Jerry was not writing for his parents or their peers. His audience was youngsters like himself. So he wrote what he dreamed about—messages from the planet Mars, bullied boys who got even, an ape-man named Goober who was raised by lions. Neither the characters nor the writing was especially persuasive, not with lines like these: " 'Goober!' he shouted, 'Don't you recognize your old pal? It's me—Izzy the Ape! I've changed my name, that's all. Nobody knew the difference in Cleveland, I look so much like the people there.' " But the school paper published it for just that reason: It was not overthought. The language and plots were precisely right for kids like Jerry, kids with outsize imaginations and visions of a world beyond Cleveland, and who had peed in their pants or worried they might.
What did matter to Jerry was money. His dad, a tailor from Lithuania, had squeezed out a living selling and altering suits and other secondhand clothes. Sarah helped out behind the counter. At its best the shop's revenues were barely enough to sustain a family of eight, and during the Depression sales and profits were even lower. After Michel died, the Siegels were destitute. Harry, Jerry's oldest brother, kept them housed and fed with money he earned as a mailman. Others kicked in what they could, even Jerome, Sarah's baby and pet. That meant working after school as a delivery boy for a print shop, where he earned the royal rate of four dollars a week and dreamed of hitting the jackpot with his writing. It also meant sharing a bed with his brother Leo.
Most of the stories he wrote were mailed back by publishers or were gone forever, so Jerry decided to try his hand once more at publishing. His new magazine had a title worthy of his lofty hopes: _Science Fiction: The Advance Guard of Future Civilization_. He was not the only writer this time; fellow Glenville High students chipped in. Jerry was, however, still the sole executive, holding the titles of owner, editor, secretary, treasurer, and office boy. He continued to rely on the school mimeograph machine, but he vowed that would change when ads he placed in more established science fiction publications swelled his magazine's circulation. His rates—fifteen cents a copy, or $1.50 a year—were no bargain, not when thick pulps could be had for a thin dime. And he continued to sign with a name not his own: Herbert S. Fine, which blended the name of his cousin Herbert Schwartz with his mother's maiden name, Fine. Looking back, there are two names that stand out even more in _Science Fiction:_ its primary illustrator, Joe Shuster, and the main character in a story in its third issue, Super-Man.
Jerry's and Joe's names would become as conjoined and revered in the world of comics as those of Rodgers and Hammerstein in song and Tracy and Hepburn in cinema. Jerry and Joe met through Jerry's cousin Jerry Fine, who lived in another part of Cleveland. Fine wrote a column in junior high called "Jerry the Journalist," which Shuster illustrated. When Joe was about to enter ninth grade his family moved to Glenville, and Fine suggested he look up Jerry Siegel. The boys looked almost like brothers even though Jerry was four inches taller and forty pounds heavier, and they seemed fated to become a twosome. Both wore glasses, were petrified of girls, and preferred to stay indoors reading when everyone else their age was in the park playing ball, which made them two-for-one targets for schoolyard toughs. Both were the children of Jewish refugees, and Joe's dad, Julius, was a tailor like Michel Siegel. Both grew up poor, although the Shusters' cramped apartment made the Siegels' two-story home ten blocks away look like a mansion.
Julius Shuster was brilliant with a needle and thread but not with a ledger. He had sunk his meager inheritance into a tailor shop in Toronto's garment district, but the shop failed when he charged too little for clothes that took too long to stitch. His family was accustomed to changing apartments on rent day, but this time they moved all the way to Cleveland, for the promise of a job manufacturing men's suits. Joe had always helped pay the family's bills with the meager wages he could get by peddling newspapers, hawking ice cream cones, and apprenticing with a sign painter. Still, there were winter days and sometimes entire seasons when the Shusters went without coal for the furnace or enough food for three meals. Liabilities like those were not disabling back then, not during the Depression and certainly not to children like Jerry and Joe, whose parents were needy immigrants. The firmest basis for their bond had always been the passion they shared for fantastic stories, which is the way each saw himself getting even with his bulliers and getting out of Glenville. The very day they met, after Joe introduced himself to Jerry at the library, they dashed off to Joe's apartment and got to work on their first project.
Joe Shuster had loved the comics since he was a toddler back in Canada, when his dad would boost him onto his knee and read aloud strips like _The Katzenjammer Kids, Happy Hooligan_ , and Joe's favorite, _Little Nemo in Slumberland_. He also loved drawing, and he used to go on sketching even when the house was so cold he had to wrap up in a pair of sweaters and gloves. Lacking money for a drawing board, he improvised: the wooden slab his mother used to knead her Sabbath challah worked almost as well. Instead of a sketch pad he made do with butcher paper, discarded wallpaper, and the barren walls of their rented apartment. A scholarship let him study at the Cleveland School of Art and he bought himself more instruction, for ten cents a class, at the John Huntington Polytechnic Institute. His artwork was inspired but never visionary. The simple lines and expressive faces got across his message the same way Jerry's craftsmanlike words did—with joy, without flair. They were a well-matched pair.
Joe's limits were a matter of optics as well as talent. Even as a boy he had major eye troubles. "He was in a sight-seeing class," remembers his junior high classmate Jerry Fine. "They had sight-seeing classes for those kids who couldn't see well. Joe was very, very nearsighted. His drawing had to be two to three inches from his face." Rosie Shuster, Joe's niece, describes his eyes as "rheumy and soft-focused. He had Coke-bottle glasses. His family didn't want him to draw in the first place."
Eyesight was only the most obvious way in which Jerry and Joe, for all they shared, were different. Next to most of his peers Jerry seemed reticent, but Joe was so sheepish and sweet he made Jerry look fiery and bossy. Jerry did most of the talking, then and later; Joe trusted that Jerry would represent his interests. Crazy Joey, as one adoring cousin called him, had a broader, deeper inner life than Jerry, with fewer bridges to the real world. That made it easier both to like Joe and to take advantage of him. Joe gave Jerry the two things he most wanted: images to bring his words to life, and the clear understanding that Jerry was in charge.
Their first big collaboration was an illustrated short story called "The Reign of the Super-Man," a twist on the Frankenstein fable that they completed in 1932 and published the next January. The protagonist was Professor Ernest Smalley, a megalomaniacal scientist who tested a mind-bending chemical on a homeless man named Bill Dunn. The experiment yielded a monster with the power not just to read people's thoughts but to control them. Before Smalley could put that power to use, Dunn killed him, then hatched his own plan to manipulate stocks, clean up at racetracks, and generate enough wealth to dominate the planet. At the last instant Dunn lost his powers, returning to the breadline from which Smalley had plucked him. As he did, he had a bout of conscience: "I see, now, how wrong I was. If I had worked for the good of humanity, my name would have gone down in history with a blessing—instead of a curse."
Phew. The world was saved from a monster and Jerry was saved from what he would soon realize was a bad idea. A planet as troubled as his needed a hero, not another villain. At the time, he and Joe were glad to have a full-blown science fiction story, one that filled nine pages of _Science Fiction_. And certain of its themes would stick: an outer-space origin for Smalley's chemicals and Dunn's powers, a newspaper writer as the Super-Man's sidekick, and a sense of whimsy that was pure Jerry. The young writer named his reporter Forrest Ackerman, after the young science fiction fan who would later invent the term _sci-fi_. The Super-Man visited a library where his deskmate was reading _Science Fiction_. He also misbehaved there in much the way Jerry had, with the same result: a reprimand from the librarian. Yet the most striking element in this first take on Superman was its indecision—not just about big matters like whether the central character should be good or evil, but over whether to spell his name with a hyphen (as in the title), as "Superman" (the way it was in the text of the story), or as SUPERMAN (as he contemplated here and switched to in his next rendering). "The Reign of the Super-Man," Jerry explained later, was composed when he was very young, and while the central character was callow, he also "was a giant step forward on: The Road to Superman."
Soon Jerry noticed on newsstands a publication called _Detective Dan_ , a forerunner of the modern comic book that lasted only that single issue but made a mark on at least one of its young readers. He started imagining a comic book that featured him, or rather his Superman. The version he was drafting would again begin with a wild scientist empowering a normal human against his will, but this time the powers would be even more fantastic, and rather than becoming a criminal, the super-being would fight crime "with the fury of an outraged avenger." Jerry and Joe worked up the copy and drawings, scraping together their nickels to pay for the paper, the ink, and the postage to mail everything to the owners of _Detective Dan_. Although the first response was encouraging, the second made it clear that the comic book was so unprofitable that its publishers put on hold any future stories.
Jerry had thought highly enough of Joe's talents in those early years that he gave him the title of art director at _Science Fiction_ and enlisted him as a partner on his cherished Superman project. But now, with that project going nowhere, he had his doubts about their prospects. At first he thought the problem was that since they were just teenagers potential employers might presume their work was just a quick bit of patchwork, so he had Joe re-letter the cover of their mock comic book, backdating its origin from 1933 to 1928 to look as if it had been years in the making. Then he dropped Joe and tried to enlist an older, more established artist.
Hal Foster, who drew the Sunday _Tarzan_ strip, said he was too busy. Another _Tarzan_ illustrator, J. Allen St. John, expressed interest in working with Jerry on a comic strip that Jerry called "Rex Carson of the Ether Patrol," but the strip and the collaboration both died. Next on the list was Leo O'Mealia, who drew the _Fu Manchu_ comic and soon found in his mailbox Jerry's more fully developed script for Superman. This one was set in a future where the Earth was about to explode. Minutes before the blast, a super-powered scientist-adventurer used a time machine to transport himself to the present, where he became a crime-fighter. His character didn't have a Clark Kent secret identity yet, although Jerry later recalled that there probably was a Lois Lane–like character. And he didn't have a publisher, although O'Mealia and the Bell Syndicate, which published _Fu Manchu_ , both showed a brief interest in the sketchy story.
The one measurable result of Jerry's efforts to enlist the help of O'Mealia came from the mild-mannered Joe Shuster. "When I told Joe of this, he unhappily destroyed the drawn-up pages of "THE SUPERMAN," burning them in the furnace of his apartment building," Jerry recalled. "At my request, he gave me as a gift the torn cover." The story of Joe setting fire to the artwork is part of comic book lore, and most tellings say he did it in frustration after publishers rejected "The Superman." In his memoir, Jerry set the record straight: Shuster was angry and depressed not just because publishers weren't interested in their idea but because his partner had been disloyal to him.
Jerry was unbowed by Joe's reaction. His next target for collaboration was Russell Keaton, who drew and wrote the _Skyroads_ strip. They exchanged a series of letters during the summer of 1934, with Keaton going as far as submitting what Jerry thought was inspired artwork for a Superman comic strip. Jerry, meanwhile, was fleshing out his thinking on his hero. He would not just be a man of adventure but would offer "great possibilities for humor," a device that Jerry felt could disarm readers. Superman's backstory also was shifting to what is now familiar ground. The last man on Earth had catapulted his baby rather than himself back in a time machine. The infant was found by passing motorists Sam and Molly Kent, who first turned him over to an orphanage, then adopted him and named him Clark. At the end of the second week of Jerry's scripts, Sam said to his wife, "We've been blind, Molly. The lad's strength is a God-send! I see now that he's destined for wonderful things." The story resonated more this way, and Keaton helped bring it alive with his drawings. The artist was interested enough to set up a meeting with a publisher late in 1934, but then, for reasons Jerry never understood, Keaton told him that "the book is closed." Maybe the publisher wasn't interested. Maybe Keaton was shocked to discover how young and inexperienced Jerry was. Maybe Jerry was simply a pest. Keaton is no longer around to ask, but Denis Kitchen, a comics publisher who represents Keaton's estate, says the illustrator was "a professional in his mid-twenties communicating with a teenager with a wacky idea about a guy with superpowers.... I think Keaton was very surprised by Superman's eventual success, since he didn't have any faith in it."
He may have been betrayed, but Joe Shuster did have faith, still, in Superman and in Jerry Siegel. And while Jerry had no scruples about going back to his abandoned partner, first he needed to retool his superhero. That happened on what he said was a hot summer night of divine-like inspiration whose timing and circumstances swelled slightly each time he revisited them. His most considered version came in his unpublished memoir—a hundred-page document whose exuberance and visual prose give it the feel of a comic book—where he recalled all the telling details of what he had done and thought a half century earlier. Jerry decided his only hope lay in crafting a hero so super that no publisher could resist, one whose story was just unbelievable enough to be credible. He vowed to stay up as late as it took. His newest incarnation of Superman would come from a dying planet called Krypton, not a dying Earth. Clark Kent, the superhero's alter ego, would be a reporter, just the occupation to snoop around for trouble. Lois Lane was here, too—a Lois who "was ga-ga over super-powered Superman" and "had an antipathy toward meek, mild Clark." The premise was easy: His character would have everything Jerry wanted for himself—he would be able to run faster than a train, leap over skyscrapers, and be noticed by a pretty girl. What teenage boy wouldn't love to be able to do all that?
The spine of the story was done late that night before Jerry climbed into his and Leo's bed, leaving paper and pencil nearby. His sleep was fitful. Every time he woke he slipped into the bathroom, flipped on the light, and filled in another piece of the tale. "This went on until the wee hours of the morning," he wrote in his late-life reminiscence. "At dawn, I enthusiastically raced about a dozen blocks to Joe's apartment. I showed Joe the script and asked him if he would be interested in collaborating with me on this newest version of my Superman syndicate[d] comic strip project. He enthusiastically agreed, and got to work at once."
The loyal Joe, hunched barely an inch above the paper as he strained to see, started drawing. Jerry hovered over him. They stopped only to gulp down sandwiches. The reunited partners now saw Superman as their joint property, although they would later disagree on who originated what. "I conceived the character in my mind's eye to have a very, very colorful costume of a cape and, you know, very, very colorful tights and boots and the letter 'S' on his chest," Joe recounted. Jerry begged to differ: it was he who dressed the hero in bold colors and an athlete's tights, and he who came up with the _S_ , along with a cape that "would whip around when the character was in action." They agreed that Superman had to be everything they were not: strapping and dashing, fearless yet composed. As for the superhero's second self—Clark Kent—wasn't it obvious? Like Jerry, Clark wore glasses, wilted at the sight of blood or a pretty girl, and spent his days penning articles for the newspaper. Both lost their fathers and had their childhoods interrupted. When Joe was unsure how Clark should look, Jerry would pose for him. When it came to Superman, Joe often posed himself, in front of a mirror—contorting his face to look enraged, beaming with self-satisfaction, and, most convincingly, making his hero look uncertain about what he was doing but ready to plow ahead.
Lois was harder to picture. Joe and Jerry wanted to get her right, but there was no model at hand. So they hired one, from a Situation Wanted ad in the Cleveland _Plain Dealer_ , at the lavish rate of $1.50 an hour, which was more than either boy made in a day. Jolan Kovacs, a skinny kid whose only training was posing before her bedroom mirror, had advertised herself as an "attractive model" named Joanne Carter. At two in the afternoon on a frigid Saturday in January, the scared high school student showed up at Joe's apartment. "My heart was pounding," she remembered forty years later, by which time she had still another name: Mrs. Jerry Siegel. "I knocked on the door, and a boy my age, wearing glasses, opened the door a crack, and I said, 'I'm the model Mr. Shuster wrote to.' So he opened the door and he motioned me in. We hit it off right away. We started talking about movies, we were talking about everything, and I was thawing out. And a woman stuck her head out of the kitchen and said, 'Hello'—an older woman—and a little girl ran through the living room, chased by a little boy, and out again. And we were talking for the longest time, and finally I said, 'Does Mr. Shuster know I'm here?' And he said, ' _I'm_ Mr. Shuster.'...
"So I posed for him, and his mother would look in, and I was turning blue," Joanne confessed. "My sister's bathing suit was too big, so I pinned it in the back. And he said, 'Never mind. I'll put a little bit more here, a little bit more there.' But he used my face and my hairdo and my poses that just made me look more voluptuous—and older. I had to be older." Jerry remembered that day even more distinctly and fondly: " 'Wow,' I thought. 'She's terrific!' But I was too meek and mild to let Joanne know how great I thought she was. She was a very attractive girl, and I was just a teenaged kid, with a giant sized inferiority complex, who had nothing but grandiose plans that seemed very far from materializing into actuality."
Just how far is clear in hindsight. There they were, two young men who had just turned twenty, with no prospects of any kind nearly a year after graduating from high school. Both were shy of the grades and money needed for college. Neither had a real job, nor a place to live other than where they always had: with their parents. Yet they were paying money they did not have, to a teenage model who was not a model, to play a voluptuous newswoman who existed only in their imaginations. So far those imaginations had come up short, producing four renderings of a superhero who remained a closely held secret, not by design but because Jerry could not find anyone to buy his manuscript or Joe Shuster's drawings. Now even they were having doubts. "I have a feeling of affection for those lost supermen of the comics into whom I tried to breathe life, but who never surfaced onto comics pages," Jerry said. "On paper, they were the mightiest men on Earth. But in real life they were very, very fragile."
· · ·
MAJOR MALCOLM WHEELER-NICHOLSON'S biography reads like a Jerry Siegel adventure strip. The son of an early suffragette, he grew up at the turn of the twentieth century in a household that counted Herbert Hoover's wife and Teddy Roosevelt's mother among its intimates. A horseman from boyhood, Wheeler-Nicholson attended a military academy and then enlisted in the cavalry. He was one of its youngest senior officers and led one of the African American units known as Buffalo Soldiers. In Texas, he operated under the command of General John J. Pershing and chased bandits back across the Mexican border. In the Philippines, his squad set a world speed record for assembling its tripod-mounted machine gun. In Siberia, he was an intelligence officer working out of the Japanese embassy. When he turned on the military, publicly denouncing the outmoded way it trained and promoted troops, he was court-martialed and became the target of what his family has reason to believe was an Army-orchestrated assassination attempt.
His life as a gadabout is just half the story of the man his children knew as the Old Man, his grandchildren called Nick, and the comic book world knows as the Major. One of the most prolific pulp fiction writers of his day, he published more than ninety novels, novellas, short stories, and serials, which turned up on shelves as far away as New Zealand. He was a historian and a visionary. He knew the long and fertile life that comics had enjoyed in U.S. newspapers, beginning in the late 1800s with the launch of strips such as _The Yellow Kid_ and _The Katzenjammer Kids_. The titles made clear the young audience newspaper titans were going after, and Wheeler-Nicholson was fixated on attracting that audience, along with its parents, by blending the magazine form of pulps with the picture presentation of comic strips. What he really longed for was a graphic novel—the perfect way, this literary entrepreneur thought, to bring culture to the masses—but it would be another generation before anyone even imagined that format. Instead he settled for what was at hand, launching National Allied Magazines in 1934 and a year later publishing what many consider the prototype of today's comic magazine. Unlike its progenitors, _New Fun: The Big Comic Magazine_ featured comic strips that hadn't already run in newspapers and it carried moneymaking advertisements. Its title was apt both because of its ground-breaking contents and because it measured a whopping ten by fifteen inches.
For America, _New Fun_ helped kick-start the emerging medium of comic books. For Jerry Siegel and Joe Shuster, it was a career-launcher. The Major needed writers to produce his strips, ideally ones who came cheap. It was his shortage of funds as much as creative zeal that inspired him to include only original material in his books, rather than buying the expensive right to reprint what newspapers already had published. No matter that Jerry and Joe were young and untested, or that their heroes were an unlikely swordsman from seventeenth-century France and an even less likely private detective investigating the supernatural. The Major needed filler. The boys needed money. And so the October 1935 issue of _New Fun_ featured two single-page Siegel and Shuster titles: "Henri Duval of France, Famed Soldier of Fortune" and "Doctor Occult, the Ghost Detective," a story whose protagonist had not just Superman's cape but his face.
It seemed like manna from heaven. For eight months, Jerry and Joe had tried to peddle their revamped superhero but found no takers. The Publishers Syndicate of Chicago turned them down. Owners of the _Famous Funnies_ comic book returned their package unopened, Jerry's knotted strings still intact. The Bell Syndicate wrote that "the drawings are well done and the idea is rather interesting, but we would not care to undertake the syndication." So low were their funds that Joe was hawking ice cream bars on the street, and the only way Jerry could afford a movie was by selling empty milk bottles back to storekeepers. They had even started cannibalizing their sacrosanct Superman, with Doctor Occult getting his face and Henri Duval his penchant for adventure. Now, thanks to the Major, Henri Duval and Doctor Occult were being sold on the streets of Cleveland, with a promise of more work. Wheeler-Nicholson asked Jerry and Joe to prepare a four-page strip about a rip-roaring FBI agent, to be called "Federal Men," which would run in one of the new comic books he was launching. He also had ideas for three new series—"Calling All Cars: Sandy Kean and the Radio Squad," "Spy," and "Slam Bradley," a hard-bitten private eye who seemed like a dry run for Superman. Finally, in a move the young artists had been waiting for forever, Wheeler-Nicholson wrote that their idea for a Superman comic strip "stands a very good chance."
The Major kept his promise about publishing nearly everything Jerry and Joe sent in, but there were red flags. His launches of books like _Detective Comics_ were delayed. Promises of 15 percent of profits and half of syndicate sales remained promises. Some checks that were due never came, and one bounced. Early in 1936, Wheeler-Nicholson sent Jerry a payment with this postscript: "Do not be alarmed over the legal phraseology on the back of the checks. Our lawyers made us put it on after we had a couple of unfortunate experiences with chiselers who tried to hold us up after we'd paid them in full." But the boys were alarmed, less by the request that they release their rights than by the mounting evidence that the Major was running out of money. So while they continued to write and draw for him, and to live off what payments they got, they determined not to trust him with their prize possession. Please, they asked him, give us back our Superman scripts.
The truth is that Wheeler-Nicholson really did believe in Superman. His was the kind of life the Major had lived, a freewheeling and crusading one, and he was convinced that Jerry and Joe were sitting on a gold mine. That is what the Old Man told his children over the dinner table, where, said his son Douglas, Superman "was a major subject of discussion." The truth also is that the Major's approach to money was a lot like Joe's father's: noble intentions, bungled execution. In another business and another era this literary entrepreneur might have squeezed by, but not in the middle of the worst economic downturn in U.S. history, nor in a fledgling industry populated by racketeers and sharks. Two with the sharpest teeth would swallow up Wheeler-Nicholson's publishing houses before he saw what was happening.
The first of that pair, Harry Donenfeld, was a survivor. Born in Romania in 1893 just as it was turning against Jews like him, he came to America with his parents and brother and made his way as a child on the pulsating streets of New York's Lower East Side as a barker—pulling customers into clothing stores or dance halls, or hawking Yiddish and Russian newspapers. Harry neither denied nor embraced his Yiddishkeit roots, but he was hungry to leave his Jewish ghetto and determined never to go back. Money was his way out, and he made his through magazines with titles like _Juicy Tales_ and _Strange Suicides_. Harry couldn't write, edit, or draw. What he did better than anyone was the hard sell. "He could sell ice to the Eskimos," said the now ninety-nine-year-old Jack Adams, who worked for him back then. "Once he got an order from Hearst for nine million inserts for magazines after he entertained their buyer by getting him drunk and laid in Canada.... Harry didn't know from nothing except making money." Some of that money was from printing the publications at his family's shop, but over the years more came from distributing them to drugstores and newsstands. The delivery trucks and drivers were ones bootleggers had used to supply speakeasies with hooch during Prohibition. So were the whatever-it-takes tactics needed to build and sustain regional monopolies, of which Harry had plenty. They were what led him, by the early 1930s, to become a magazine publisher and owner. As long as people were buying, what went inside the publications didn't matter.
The indictments changed all that. New York's new no-nonsense mayor, Fiorello La Guardia, was cracking down on public indecency, and nothing met that era's definition so clearly as pictures Harry ran in _Pep!_ that glimpsed a woman's privates. A grand jury in Kentucky was equally offended. Harry beat the second rap and convinced an underling to take the fall for the first (the favor was returned when Harry gave the patsy, Herbie Siegel, a job for life). One lesson Harry took away was that while smut was profitable enough to sometimes justify the risk, it was best not just to sell it from behind the counter but to be a silent partner to avoid any potential embarrassment or jail time. The other lesson was to diversify.
Jack Liebowitz was everything Harry Donenfeld wasn't. His roots were in the same Jewish ghetto, and both had ties to the rabble-rousing International Ladies' Garment Workers' Union, but while Harry merely printed the guild's brochures and took its money, Jack was an officer and true believer. What Jack remembers most from his boyhood was not really having one. He was born in Ukraine in 1900, the year his father died. In 1904 his mother disappeared to America, leaving him and his older brother behind with their grandparents. By the time she came back his brother had succumbed to diphtheria. Five months before his tenth birthday, Jack, his mother, and his little sister stole across the border in the dead of night into Austria, then made their way to Holland and got tickets in steerage for the journey to America. In New York, Jack slept on the roof of his tenement to escape the summer heat and his stepfather. With other kids, he used trash can covers to fend off rival gangs. He never owned an overcoat, or slept in a bed he didn't share with three brothers.
While Harry came away from his hardscrabble upbringing as a joyful backslapper, Jack emerged hardheaded and glum. Harry always had an expensive cigar in his mouth; Jack smoked cigarettes he bummed from friends, which was his way of disciplining himself not to smoke too much and to save money. Harry was the bluffer who showed his cards only when he had to and generally walked away with the jackpot; Jack was the house, getting a cut of every bet and never trusting to chance. His soul, like his training, was that of an accountant.
Jack went to work for Harry in 1935 after the Depression killed his hopes of making it on his own. Newly unearthed home movie footage of the two from those early years attests to the awkward pair they were—Harry short and thick, Jack six inches taller, thirty pounds lighter, and blanching as a beaming Harry, knowing the reaction he will get, reaches up and presses his face against Jack's in a warm embrace. But their differences made them perfect partners. Jack worked out the details of deals Harry made with a handshake, turned down bribes Harry had trouble resisting, and tidied up Harry's messes. Seven years Harry's junior, Jack acted like his big brother. In the process, Jack Liebowitz made himself so indispensable that Harry Donenfeld promoted him from bookkeeper to second-in-command and then to partner, with Jack bringing little equity to the table and barely anyone noticing. Not even Jack noticed as his flirtation with socialism yielded to a fondness for money.
Neither Harry nor Jack cared about art or storytelling, but they did know more than the Major about how to make a buck, and they liked the idea of branching out from the smooshies and horrors to cleaner kids' stuff. Not only did they print and deliver _New Fun_ and Wheeler-Nicholson's other comic books, they also loaned him the cash to publish them. And while the Major could stay a step ahead of his writers, artists, and process servers, Harry and Jack were pros at spotting a dodger. Harry had defied the Depression trend that was bankrupting other publishers, getting his financing from a quiet partner named Paul Sampliner and Paul's indulgent mother, and buying up other presses as they teetered. In 1937 Jack became an uninvited partner in Wheeler-Nicholson's publishing house; by the next spring, the whole show was Harry and Jack's. The Major walked away with his debts wiped clean, his wallet empty, and his heart broken. Harry and Jack had all three of the Major's books, all published under the Detective Comics banner. They now owned the trucks, the printing presses, and the actual magazines, which gave them a foothold in this new world of kids' literature along with the diversity Harry the pornographer had been seeking.
Were those shifts in ownership aboveboard? Absolutely, according to Jack Liebowitz, who in his unpublished memoir said it was a straightforward matter: Wheeler-Nicholson couldn't pay his artists or writers or pay back his loans, creditors pushed him into bankruptcy, and "I went down to court and bought the two magazines." What Jack doesn't say but his company's records show is that Harry had orchestrated everything. He bought up, for fifty cents on the dollar, debts the Major owed his printer and engraver, thus meeting the three-creditor standard needed to force Wheeler-Nicholson into bankruptcy. The backdrop was Dickensian. Harry and Jack feted the Major by sending him on vacation to Cuba, tried unsuccessfully to buy him out for seventy-five thousand dollars, then took him to debtors' court during Christmas week. Harry made it happen; Jack dotted the _i_ 's and crossed the _t_ 's. The Major, meanwhile, overcame bouts of booze and nerves and moved on. His wife, Elsa, never did. "She hated them," said Douglas Wheeler-Nicholson, referring to Harry and Jack. "Any time any mention of them came up, she would spit fire."
The upheaval in ownership touched Jerry, Joe, and Superman in two ways. It gave Wheeler-Nicholson's old companies new bosses who could pay their bills, and it put the boys on notice that Harry and Jack relished playing hardball. Jerry spent 1936 and 1937 the way he had the two previous years, pitching Superman to anyone he thought might listen. Getting heard was harder than ever. United Feature Syndicate called the whole Superman concept "a rather immature piece of work. It is attractive because of its freshness and naïveté, but this is likely to wear off after the feature runs for a while." The Ledger Syndicate was equally blunt: "We feel that editors and the public have had their fill for the time being of interplanetary and superhuman subjects." By the end of 1937, copies of Jerry and Joe's Superman comic strip could be found in the backs of filing cabinets and the bottoms of wastebaskets across the world of publishing.
One of those who received the strip years before and never forgot it was Maxwell Charles Gaines, a senior executive at the McClure Newspaper Syndicate. Gaines could not interest his bosses so he sent back the drawings. Three years later, in December 1937, Gaines was looking for new material and decided to take another shot at Superman. Jerry and Joe mailed him proposals for five strips—about a cowboy, an adventurer, a detective, a sports star, and a sci-fi scientist, all knockoffs of icons like the Lone Ranger and Jack Armstrong, the All-American Boy—along with their latest rendition of Superman. The sketches still were on Gaines's mind and in his office a few weeks later when he got a call from the new owners of Major Wheeler-Nicholson's publishing business.
Jack and Harry wanted to launch another book. They had a title, _Action Comics_ , but no material. So, as Jack wrote in his memoir, he phoned his friend Charlie Gaines. "I said, do you have any material laying around. The newspaper syndicate usually had stuff laying around. Stuff that had been submitted to them which had been turned down. So he sent me over a pile of stuff. Among that pile of stuff was Superman which had been submitted by Siegel and Shuster to the syndicate which had turned it down like all other syndicates turned it down. It was six strips, daily strips, made for newspapers. Anyway, we liked it." Charlie and Jack called Jerry in Cleveland. Gaines said he had bad news and good: His syndicate was not interested in any of the comic strips, but Jack might be. Would it be okay, Gaines wanted to know, if he turned over to Jack the scripts—including Superman?
It was the question Jerry had been waiting five years to answer. No matter that to the accountant-turned-publisher, the work of the young writer's dreams was "a pile of stuff." Vin Sullivan, an editor who stayed on during the transition from Wheeler-Nicholson to Donenfeld-Liebowitz, called Jerry in January 1938 to make plans. The boys would have to cut and paste their newspaper-style format into material that would fit in a thirteen-page comic book. And they would have to do it fast, Sullivan said, since Superman was destined for the inaugural issue of _Action Comics_. After lying around lifeless for what seemed like half a lifetime, Superman was on the fast track.
The matter of money was settled almost as fast. On March 1, Jack mailed Jerry and Joe a check for $412—$282 for work that had been done for but not yet paid by the Major, and, almost as an afterthought, $130 for Superman. It was double what they were used to and a fair rate—$10 a page—for the era and their experience, so Jerry and Joe cashed it and split it down the middle. It also was a swindle on the order of the Dutch West India Company's 1626 purchase of Manhattan from the natives for $24, for Jack and Harry were buying not merely the thirteen pages of that first Superman comic, but the right to do what they would with the character. They could clip his powers or his hair, bring him to life in new media or kill him outright, or do whatever else they wanted. Harry and Jack were the honchos now, the boys mere hirelings. Jerry and Joe's deal with the publishing house was for five years; Superman's was forever.
That $130 contract signaled the beginning not just of the Superman character but of what would become a multibillion-dollar industry: comic book superheroes. It also was the defining narrative—the original sin—in the relationship between comic book creators and owners. Jerry and Joe may have brought to life their superhero, but that is not what mattered. What counted then and for decades after was who had the money to put that hero on the printed page and deliver those pages to the public. To the publishers went not just the profits but the power. Each writer or artist who took up that cause in the future would hark back to what Harry Donenfeld and Jack Liebowitz did to Jerry Siegel and Joe Shuster. As if to underline the point—to demonstrate that for $130 Jerry and Joe, like all comics creators back then, were giving up everything—their Superman artwork was destroyed soon after it was used. There would be no chance for them to sell it again or save it for posterity.
IT WAS APRIL 1938 and the world was holding its breath. The Führer's storm troopers had just occupied and annexed Austria and were ready to steamroll into Czechoslovakia. Joseph Stalin had shown the West and his countrymen that he was as ruthless as the Nazis by staging a show trial for Nikolai Bukharin, a champion of the revolution, and then liquidating him. Franklin Roosevelt's New Deal was in full motion, but one in three Americans remained ill-housed, ill-clad, and ill-nourished, and 250,000 teenagers had taken to the road to earn money to send home. Never had America so craved a hero, if not a messiah. Never had a publisher so perfectly timed its release of a new title.
The very cover of _Action Comics_ No. 1 signaled how groundbreaking—how uplifting—this Superman would be. There he was, in bold primary colors: blue full-body tights, a yellow chest shield, and candy-apple cape, booties, and briefs worn over his tights. He looked every bit the circus acrobat, only stronger, more agile, ready for action. No mask for this adventurer; he wanted the world to see who he was. While it was left to the imagination just whom he was fuming at, it was clear that no one would want to suffer the rage of a being who could single-handedly lift a car into the air and smash it against a rock. Hopefully he was on our side. The date printed on the top of the page was a standard bit of misdirection: It said June when, in a bid to ensure it would still look fresh if it sat unsold two months later, it actually went on sale in April. There was little doubt that this was a comic book that would justify its ten-cent price and deliver on its name, _Action_.
The first inside page introduced readers to the handsome, brash avenger that Jerry and Joe crafted during that sleepless night of writing and frenetic day of sketching three years earlier. A scientist on a faraway planet placed his infant son in a spaceship headed to Earth just before his planet died of old age. The child was found by a passing motorist, who turned him over to an orphanage. Reaching maturity in just the fourth panel of the comic, Superman was able to leap an eighth of a mile, vault a twenty-story building, and hoist tremendous weight. Nothing less than a bursting shell could penetrate his skin. How was that possible? The "scientific explanation" was there on page 1: He did it the same way a lowly ant supported weights hundreds of times its own, or a grasshopper leaped what to a man would be several city blocks. That was it: the entire birth, growth, and backstory of a breathtaking superhero laid out in a single comic book page measuring 7¾ by 10½ inches.
By page 2 he was a full-grown Superman, racing off on adventures that didn't stop until the story did, eleven pages later. Along the way he saved an innocent woman from electrocution, beat up a wife-beater, rescued Lois Lane from kidnappers, and intercepted a warmonger. No worries here about laws or social niceties. Bursting into the governor's house was the only way to stop an unjust execution? Barrel ahead. Dashing across live electrical wires could make a lobbyist see the evil of his ways? Up we go. This Superman was a hell-raiser and an insurrectionist. Half Huckleberry Finn, half Robin Hood, he had a technique as straightforward and a purpose as pure as those of his teenage truth-and-justice-seeking creators. His story accounted for just thirteen of _Action_ No. 1's sixty-four pages, but those are the only pages the world remembers.
Literature's most gripping love triangle also was there from the first, or at least the hint of it. Clark Kent was smitten with Lois Lane, asking her out on a date on page 6. Lois agreed, but when he pressed to find out why she was avoiding him, she let him have it: "Please Clark! I've been scribbling 'sob stories' all day long. Don't ask me to dish out another." A page later, she walked out on her timid colleague after he let a thug cut in on their dance, explaining, "You asked me earlier in the evening why I avoid you. I'll tell you why now: because you're a spineless, unbearable _coward_!" Lois's time with Superman in this first story was too brief for her to fall in love, or for him to dodge. Not yet. Their newspaper already had a name, _The Daily Star_ , but Clark and Lois's boss was identified only by the nameplate on his desk: EDITOR.
It did not take long for the buzz to begin. Just who was this costumed hero anyway? writers, artists, and publishers wanted to know. And who were Jerry Siegel and Joe Shuster, the uncredited creators who were unknowns in both the old world of comic strips and the new one of comic books? Was their Superman an original or a knockoff? Would he sell? Would he last? Those questions still resonate seventy-five years later.
There was no question that Superman built on what came before. He was as strong as Samson, as fast as Hermes, and as brain-bendingly smart as Micromegas. Douglas Fairbanks, Sr., and Rudolph Valentino were his model swashbucklers. Popeye and Tarzan showed him how to be a strongman. Whom better to look to for guidance on foppish dual identities than the Scarlet Pimpernel and Zorro? The Shadow offered up an alter ego named Kent and a female sidekick named Lane. Jerry and Joe made no secret of any of their inspirations, just as they acknowledged being saps for the endless newspaper comics, dime novels, science fiction tales, and cliff hanger movies they took in as kids, from _Mutt and Jeff_ to the Merriwell brothers.
When does influence became borrowing and borrowing become plagiarism? Doc Savage lent Superman some of his best stuff. In less heroic settings, Doc used his formal first name, Clark, a nod to film star Clark Gable. Superman picked the same name with the same nod to the King of Hollywood. Doc had superhuman strength and a moral compass that compelled him never to kill an enemy unless there was no other way; so would Superman. Doc's nickname was the Man of Bronze; Superman's was the Man of Steel. There was no room for dames in Doc's life, or in Supe's. The borrowing was not confined to general concepts: Gimmicks like putting bad guys to sleep by pressing a nerve in the neck were fair game, as were entire plot lines. Jerry Siegel acknowledged having read Doc "with fascination," but that was as far as he went. Comics historian Will Murray, who has documented the close connection between Doc and Superman, says Jerry may have stopped there for fear of being sued, but future Superman writers borrowed even more from Clark Savage, Jr.
The case for a connection with Philip Wylie's Hugo Danner was even stronger. Hugo hurdled across rivers, bounded into the air, raised a cannon skyward with one hand, and lifted an automobile by its bumper. Like Superman, Hugo was said to be as strong as steel, and both used their strength to take on evildoers ranging from arms merchants to entire armies. How did Hugo get to be so strong? "Did you ever watch an ant carry many times its weight? Or see a grasshopper jump fifty times its length?" Professor Danner asked his son, invoking the precise natural principles—even the very same insects—Jerry would to explain Superman's prowess. Philip Wylie's whole approach in _Gladiator_ , blending science fiction with action lore, seemed state of the art when he published his novel in 1930 and a bit less fresh when Superman came along eight years later.
But was superhuman Hugo Danner actually Superman? Wylie thought so. In their first two years of writing and drawing the character, Jerry and Joe "used dialogue and scenes from GLADIATOR," its author wrote a colleague in 1970. "I even consulted my lawyer to see if I ought not to sue for plagerism [ _sic_ ]. He agreed I'd possibly win but found the 'creators' of 'Superman' were two young kids getting $25 a week apiece, only, and that a corporation owned the strip so recovery of damages would be costly, long, difficult and maybe fail owing to that legal set-up." Wylie was right: He might have won had he sued, much as Superman's publishers did later when they went after his imitators. Doc Savage's creators—Lester Dent, John Nanovic, and H. W. Ralston—might have as well, and even Edgar Rice Burroughs. But as Wylie himself conceded later in his letter, "We all borrow in ways from others, tho. The first Superman wasn't my Gladiator but Hercules or Samson."
That was the point. Jerry and Joe did not cook up Superman from scratch. They built on as well as borrowed from a long line of mythmakers and storytellers, the same way Burroughs borrowed from Homer and Wylie from the ancient Hebrews. "Our concept," Joe said, "would be to combine the best traits of all the heroes of history." He and Jerry sometimes took more than they might have, with too little paraphrasing or crediting. But Doc Savage was an earthling whose hardest job was building his muscles and brainpower; Superman was an alien whose biggest challenge was deciding what to do with the powers he was born with. Danner, too, was decidedly different, a dark presence done in by his worry that mankind could not abide a superhuman such as him. Superman was a creature of light, and it was that very optimism that America loved most. And although Savage and Danner were human and Superman wasn't, his pairing with Clark Kent gave him a groundedness and humanity Doc and Hugo couldn't match.
What the two Cleveland teenagers had done was inspired. They had reached into the melting pot of fantastic characters bubbling up in the 1930s, picking out the choicest features then carefully reformulating them. _Voilà:_ a freshly minted Man of Tomorrow for a world not sure it had one. Superman was an alien shipped to Earth rather than an earthling exploring the universe, like Buck Rogers or Flash Gordon—and he had come to help. Superman was no mortal donning an exotic costume like Zorro or the Lone Ranger; just the opposite, this honest-to-God extraterrestrial walked and worked among mortals like us by disguising himself as a bumbling reporter. No wonder we adored him. Although his enemies included run-of-the-mill rogues tracked down by the likes of Dick Tracy and Sherlock Holmes, Superman also took on the demons of his day, from abusive husbands to war profiteers to a penal system that executed the innocent—all in the very first issue.
By his own admission, Joe's renderings of Superman, Lois, and all the rest of _Action_ No. 1 lacked luster and gloss. But that was their genius. They were straightforward and unprettied, making them as easy to follow as an architect's blueprints. His skyscrapers were impressionistic shafts, his criminals had angular mugs and stiff features. Primitive, yes, but primal and even ethereal. Likewise, Jerry's stories had his superhero racing up the sides of buildings and jerking getaway cars off the road: just the thing for ten-year-olds and for a nation tortured by self-doubt. Their creation was brilliant, whether or not Jerry and Joe themselves were. Superman was the ideal character at the right moment, and the boys sensed it even if they couldn't foresee how long it all would last. Jerry and Joe simply wrote and drew what they knew—which is why Superman embraced the vigilante justice that Jerry longed to mete out to his father's robbers, why Lois and Clark's _Daily Star_ was modeled not on a U.S. newspaper but on the _Toronto Star_ , whose cartoons Joe's father had read to him, and why Superman's alter ego looked and acted so much like the young Messrs. Siegel and Shuster.
# **CHAPTER 2**
# **A Hero for His Times**
IT HAD TAKEN SIX YEARS for Jerry and Joe to bring Superman to life in a comic book, six years that seemed like an eternity. It took six weeks for Jack and Harry to send him hurtling onto the bestseller list, and it seemed like an instant.
Turning the newly minted hero into a blockbuster was not a matter of meticulous plotting or cagey strategizing. Neither Harry nor Jack had much clue what they were doing when they took _Action Comics_ No. 1 out for sale in 1938. The partners, whose slicked-back hair and unctuous smiles made them look like the scoundrels in that first Superman story, were pros in printing and delivering magazines. To them, publishing meant pornography and pulps. What did they know of kids or comics? Comic books themselves were new enough never to have produced a runaway hit. The anxious entrepreneurs did print up twenty thousand promotional posters for the newsstands, pharmacies, corner stores, groceries, and bus and train stations that sold National Allied Magazines' comics. But so doubtful were they that their costumed hero would catch fire that they already had resolved to take Superman off the cover after that inaugural issue. They had also arranged for _Action_ No. 1 to stay on store shelves for six weeks rather than the standard four, hoping that if it sat there long enough someone might notice.
Someone did. There were two ways publishers back then learned how a magazine was selling. Dealers counted the comics gone from the spinner racks and overhead displays, recorded their tally on a penny postcard left by the distributor, and dropped it in the mailbox. They did that fifteen days after the comics were delivered and again near the end of the month. Jack the accountant pored over every card that came through the mail slot, but he and Harry knew that system was hit-or-miss. Some shopkeepers didn't keep a precise count; others didn't even mail back the cards. A more exact tally would come at the start of the following month, when trucks delivering new issues would pick up unsold old ones. Yet by then it would be too late: All returns were fully refundable, and too many could put them out of business.
Each new postcard raised hopes. The publishers dispatched their agents around Manhattan to check demand at kiosks and druggists; sales seemed brisk, but New York was not America. What would Youngstown think? Miami? Jack and Harry had printed up 202,000 copies of _Action_ No. 1 and were worried they had placed too much trust in a hero Harry himself had said was a long shot. Finally, with all the numbers in, the verdict was clear: They had guessed right. Vendors had sold 130,000 comic books, or 64 percent of the print run. Anything over 50 percent constituted a success and guaranteed a profit. To top that by 14 percentage points, on the first issue, barely a month after they had taken over from the Major, was more than they had let themselves dream. This new adventure could work after all. Superman just might fly.
To be sure, Harry and Jack ran a test. They wanted to know that it was Jerry and Joe's caped hero who had driven those sales, not "Zatara Master Magician," "Sticky-Mitt Stimson," or any of the eight other features in that first issue, so in _Action_ No. 4, readers were asked to list in order of preference their five favorite stories. As an incentive to respond, twenty-five one-dollar prizes were offered for the best accompanying essay. The results dispelled any doubts: 404 of 542 respondents named Superman as tops, with 59 more listing him second. It did not take a master magician to figure out who was driving sales. The publishers were beside themselves. Even as he waited for the returns, Jack was looking to drive demand. He printed only 200,000 copies of the second and third issues of _Action_ , knowing that would leave some shopkeepers with fewer than they wanted and hoping that would build interest, not resentment. He was rewarded with a cry from wholesalers to "give us more copies." Sales, meanwhile, continued to climb—to 136,000 for the second issue, 159,000 for the third, 190,000 for the fourth, and 197,000 for the fifth. _Action_ No. 13, released on the first anniversary of the original, offered up 415,000 reasons to celebrate. National printed 725,000 copies of _Action_ No. 16 and sold 625,000—an unheard-of success rate of 86 percent.
Who were the buyers? There were no sophisticated surveys, but the drift was clear. Most were schoolchildren. Boys outnumbered girls, but not by much. And they all loved their Superman. He had quickly become the big brother every kid needed, especially half-pints who were being bullied by playground toughs or babied by teachers and parents. TV wasn't around yet to seduce kids, and radios were oversized consoles shared with the family. The one thing youngsters had that was theirs was what they read, and now they read Superman at night, under the bedcovers, using a flashlight to illuminate the pages. They brought him to school, camouflaged in their Dick and Jane readers. No highfalutin dialogue in these books, or ambiguity over right and wrong, winners and losers. Superman should win and did, which is just how kids would have it. He was unadult enough to be appealing, and he was the right price—ten cents—for consumers whose only income was their modest allowance and the pennies they dug out of the sofa. Nine and ten-year-olds finally had a language and hero who was theirs alone. Libraries got the message, too: Children's rooms began promoting books by saying they had Superman's seal of approval, which carried more clout than a recommendation by a librarian or schoolmistress.
And it wasn't just kids. All-Star pitcher Lefty Gomez recalled walking down the street with his Yankees teammate and roommate Joe DiMaggio when "he suddenly turns to me and says, 'Lefty, you know what day today is?' I say, 'Yeah, Wednesday.' Then he says, 'No, no, today is the day the new _Superman_ comes out'... So now he sees this newspaper stand and looks to see if they got comic books. He points to it and wants me to get it for him. He stands off to the side. Hell, he was Joe DiMaggio and if the newsstand guy saw him buy _Superman_ comics it would be all over the world. I got one of those faces nobody could ever recognize so he wants me to buy it for him. 'Joe, is this what you want, the _Superman_ comics?' He looks around at a couple of people there and he says, 'No, you know I wouldn't buy that.' Then I walk away and he motions again. I finally buy it for him and he stuffs it into his pocket. He spends the night with Superman."
Harry and Jack read the trend lines and responded. Superman was back on the _Action_ cover for issue 7, and again for 10, 13, and 15. Nine months after his debut in comic books he was given his own daily newspaper strip, which was where Jerry and Joe had wanted him in the first place. It was the only time a comic hero had ever jumped from comic books to strips rather than the other way around. The _Houston Chronicle_ was the first paper to sign up, followed by the _Milwaukee Journal_ and _San Antonio Express;_ by the end of 1939, sixty papers were running the daily feature and a Sunday strip was gearing up. That June his publishers had made Superman the first character to have an ongoing comic book named after him, although the plan at the time was for just one issue. The first press run of 500,000 sold out, as did subsequent ones of 250,000 and 150,000. The trial balloon quickly became a regular item, with _Superman_ No. 2 selling all of its 850,000 copies, as well as a second run of 150,000, and _Action_ continuing its record-setting pace.
No one had seen numbers like that since the pulps of the Roaring '20s. Superman was outpacing the girlie magazines, the horror titles, and all his comics challengers. There were a dozen other comic books on the racks, all selling about 200,000 copies. Superman's tally was five times that, a pace so far off the charts that competing publishers presumed it was an anomaly and took nearly a year to gin up their own imitations. The industry may have been at a loss, but Superman's owners weren't. They kept pushing. In October 1939 Jack set up Superman, Inc., to protect the trademark and develop new products. Five months later a new tagline showed up on _Action_ covers, trumpeting its status as the "World's Largest Selling Comic Magazine."
If a pair of teenage scribblers from Cleveland had made all that possible, it was a pair of middle-aged fortune hunters from the Lower East Side who were making it happen. Now they were ready to claim credit. Jack, who generally was neither boastful nor sentimental, made an exception with Superman. "We liked it," he said looking back at Jerry and Joe's first set of Superman scripts. "We put it together and made a 13 page story out of it." Who chose the artwork for that _Action_ No. 1 cover? "I remember picking the first cover," he said. Who harmonized the distribution and marketing? "I just wanted to create a demand." What about adding new comic books? "Donenfeld thought it was too many magazines and he didn't want to put out any more." What did all that effort lead to? "That was the beginning of the comic industry as far as I was concerned."
As far as Harry was concerned, Superman had a different heritage. Jerry and Joe wrote and drew him, and Jack did the marketing, but what mattered most were the brains and the bankroll behind the birthing. Both were Harry's. He was the one who bought the story after every other publisher had turned it down. He had the artwork ready for issues 2 and 3 even before _Action_ 1 hit the stands. Gambler that he was, he bet his stash on a dark horse and it came in. In 1940, Jerry and Harry appeared on Fred Allen's radio show to talk about Superman. This would be the only recorded comment Harry would ever offer about Superman, and while it was comedy, it was revealing. Allen asked Jerry whether he was the man behind the hero and, with atypical modesty, he said, "I'm just _one_ of the men, Fred." Then came Harry, who shared the stage with a character the announcer called Superman. When Superman did not recognize him, Harry chided, "Why, I'm Harry Donenfeld, your boss.... I took you off a drawing board and made a man out of you! I splashed your name from coast to coast... and you've never heard of me!" In later years, as others forgot his role, Harry reminded them by showing off a life-size portrait of Superman that hung in his office lobby. He took to calling himself Harry Superman Donenfeld. And, ever the showman, Harry would wear a Superman T-shirt under his suits and even his tuxedo, waiting for the perfect moment to rip open his jacket and shirt and announce, "This looks like a job for _Superman_!"
IT IS NOT OFTEN that a child gets a name and an identity before its parents. That is the freedom of fiction and the reality of comic books, where stories play out in diminutive dialogue balloons and thin panel drawings that put a premium on each word and image. So it was with Superman: Jerry Siegel and Joe Shuster had trouble enough conjuring up and getting published the basic biography of their superhero. The rest they filled in as they went along.
_Action_ 1 referred to Superman's home only as a "distant planet" that "was destroyed by old age." No name or location. No mention of its culture, religion, history, or why advanced age spelled destruction. Just these small teases: Its "inhabitants' physical structure was millions of years advanced of our own," and "upon reaching maturity, the people of his race became gifted with titanic strength!" That April 1938 backstory got more interesting the following January, with the launch of the _Superman_ newspaper comic strip. Word one on day one gave his home planet a name, Krypton, followed by an elaborated context. The distant world was "so far advanced in evolution that it bears a civilization of supermen—beings which represent the human race at its ultimate peak of perfect development!" On day five we learned that Krypton was dying not of old age but of an implosion caused by "an internal cataclysm." Five installments after that, Krypton was no more, having disintegrated into a million fragments. The first _Superman_ comic book, published in the summer of 1939, added one last fact about Krypton that would help explain Superman's super-strength: It was bigger than Earth, which meant that gravity had greater pull on Superman's home planet than on his adopted one.
Superman's parents fared worse. In _Action_ 1 his Kryptonian birth father went nameless, although we were told he was a scientist who placed his infant son "within a hastily devised space-ship, launching it toward earth!" Not a word about his mother, assuming the baby had one. The first series of comic strips, titled "Superman Comes to Earth," started filling in the blanks. Superman's father got a name, Jor-L, and he was crowned "Krypton's foremost scientist." But while he was smart enough to detect his planet's impending doom, he was not persuasive enough to get the ruling council to evacuate. We also met Superman's mom, Lora, who came up with the notion that safety lay in the distant stars and decided that if only one of them could be saved, it would be her overactive newborn. The last we heard of either parent was as the baby rocketed toward Earth: "An instant after their glorious, self-sacrificing gesture, Jor-L and Lora perish in the earth quake's awful grip!" Just the sort of stilted prose and unequivocal heroism that appealed to Joe DiMaggio and every other red-blooded American.
Baby Superman, like his parents, first got a name and a life on Krypton only in the newspapers. He was Kal-L, and he quickly showed himself to be a chip off his dad's strong-willed block by giving his doctor a black eye and leaping from his mother's arms. His existence on Krypton lasted mere months, but after a perilous journey he made it to Earth, where he was found by a passing motorist and left at an orphanage. That motorist had no further role in either the first comic book or the original comic strip, and Superman had neither parents on Earth nor an explanation for why he was called Clark Kent. In _Superman_ No. 1 he got a father without a first name and a mother named Mary, who adopted him from the orphanage where they had deposited him. They lasted for ten panels, or not quite two pages—enough time to name the boy Clark Kent, watch him grow to manhood, see him discover his powers, and caution him to hide those gifts while using them to "assist humanity." Then the elderly Kent couple died, and Clark was off to keep his promise to them. "And so," the comic announced, "was created—SUPERMAN."
It was not the full-scale world-building that visitors to J.R.R. Tolkien's Middle-earth would come to expect—that detailed lore would begin taking shape for Superman in the mid-1940s—but it did give the caped hero a little more context and depth. The creeping pace was partly a matter of the form: It would have taken nearly thirty years, or 350 issues, for a title like _Action_ to equal the word count of the _Lord of the Rings_ trilogy. It also was a matter of serendipity. There was no master plan for the related but distinct storylines of the _Action_ and _Superman_ books and the _Superman_ strip. No one had known that _Action_ would catch on enough for the McClure Newspaper Syndicate, which had turned the strip down twice before, to come begging to bring Superman to the funny pages. Neither Harry nor Jack had planned for a separate _Superman_ comic book, or for that to be ongoing. Having Superman's story play out across different venues presented a challenge for Jerry and the writers who came after him: Each installment needed to seem original yet part of a whole, stylistically and narratively.
Their solution, at the beginning, was to wing it, which presented its own opportunities. The first _Superman_ book opened with six pages that provided a critical introduction to the character and his world missing from the inaugural issue of _Action_. The comic strip allowed for a different pacing, composed as it was of digestible four-panel dailies whose storylines could run for months when the plot justified it. The newspaper funnies also would have five times as many readers as the books, with stories every day rather than once a month. The multiple offerings meant a lot more money—for Jerry and Joe along with Harry and Jack—from rabid readers who _had_ to follow every turn and twist.
Lois Lane was a fixture from the very start, although at first she was mainly a foil for Superman to rescue and Clark to pine over. _Action_ 1 set the pattern: Kidnapped by three thugs, Lois was quickly whisked to safety by Superman and then laughed at by her editor, who, hearing her recount her unlikely adventure, inquired, "Are you sure it wasn't pink elephants you saw?" The editor had his own problems. It took more than a year for him to get a name (George Taylor), and while it was clear from the beginning that his paper was the _Daily Star_ , in _Action_ No. 2 it was inexplicably called the _Evening News_ and situated in Jerry and Joe's Cleveland. The creators must have had Cleveland on the brain, and lax editing in the office, because the Ohio city turned up again as Superman's home in _Action_ 11. Everyone got their geographic bearings three issues later, in September 1939, when the superhero and his newspaper were situated once and forever in Metropolis. It would take until later that fall, and the second issue of the _Superman_ book, for Metropolis to be situated in New York State.
Superman was a man of the world, perennially on call and needing to dash to wherever Lois and others required his help. Flying would have made that easier, but the most he could manage in 1938 was leaping an eighth of a mile and outracing an express train. Two years later, after what must have been intense training, he could vault into and beyond the stratosphere, outrace an airplane, and run a mile in a scant second. By 1942, he could run at the speed of light and outpace an electric current—but still no take-off. There were hints it was coming in a single frame of a story in May 1943, when his jump looked like he might be taking flight, and he did, finally and irrefutably, that October in the _Action_ story "Million-Dollar Marathon." "Let's see ya fly!" adoring boys at Children's Hospital yelled to Superman, and so he did, telling them, "I'll be back for a real visit pretty soon! Up—up—and away!"
Veteran comic book writer Don Cameron, not Jerry Siegel, described that maiden flight, and Joe Shuster's stand-in, Ed Dobrotka, did the artwork. But flying was something Joe had contemplated early on and Jerry had been dreaming about even before he climbed to the top of his garage roof. "To fly, to fly, to fly! What bliss!" he scribbled in his memoir, thinking back to when, as a boy, he climbed onto his father's leg and was hoisted into the air. While Superman wasn't the first comic book hero to fly—that honor belonged to Namor the Sub-Mariner—flight did become the Man of Steel's most defining and coveted feature. It was a dream made real for millions of earthbound readers.
Superman always had the eyes of an eagle and the hearing of an owl, but over time both got sharper still. Within a year he could see what was going on in a building across the street whether or not it had windows, and it wasn't long before he brought into focus objects millions of miles away on a pitch-black night. Only lead could obstruct his view. His glare alone was hot enough to melt metal, a power that would come in handy for trimming his own hair, which wouldn't yield to scissors or even pruning shears. His ears, meanwhile, became so sensitive that he could eavesdrop on police radio calls without a radio and hear an ant fall thousands of miles away.
Following those twists and turns meant paying attention, and his young fans were transfixed. It was not just their hero's possibilities they were piecing together but their own. They knew that Superman could hold his breath for hours underwater or douse a raging fire just by blowing on it. His million-decibel yell had enough intensity and pitch to topple tall buildings. What if a building fell on him? A tickle at most. His nostrils were super-acute. His typing was super-fast. Superman did age, but super-slowly. No need for the FBI to run a fingerprint search; Superman could find the match. Of all his strengths and skills, the most invaluable was his intellect. He had a photographic memory that let him draw an exact likeness of someone he hadn't seen since childhood. His gaze was intense enough to hypnotize a whole tribe of South American Indians at once. He could converse with a mermaid in her native tongue and beat a checkers expert his first time playing. All that was partly a matter of nature, but he nurtured his intellect by reading, which is easier when you can scan the full contents of a library in under five minutes.
The challenge was to keep him human. The Kryptonian superhero was alien enough with the powers he had when he landed on Earth, and every issue or two the voltage was amped up. Humor helped soften him, as when he hoisted a circus strongman in one hand and iron barbells in the other, wondering, "Which is the greatest dumbbell?" Jerry's words made the point; Joe's drawings of a beaming Superman and a terrified muscleman ensured that no one missed it. Being an orphan twice over—having lost his parents on Krypton, then watched the Kents pass away—made the invulnerable Superman more empathetic. So did mild-mannered Clark Kent, although an alter ego could only go so far in warming up a gladiator who could blast boulders to dust and move mountains. Humility went further. The name "Superman," the hero pointed out, was not his idea. Neither was saving the world. Both conceits came from his parents, were fanned by reporters like Lois Lane, and made him uneasy. He was not just the most manly of superheroes but the most modest, which made his fans hold him even closer.
Themes like those made Siegel and Shuster seem wise if not old, yet other aspects of their writing and drawing reminded the reader how young and green they were. Their very first story, which they had had years to fine-tune, ended nearly every sentence with the most exuberant, least subtle punctuation mark: an exclamation point! Early issues were full of awkward or inappropriate phrasing. Describing how Clark's posture straightened when he changed into Superman, Jerry wrote, "His figure erects." As Superman got ready to burst in on a boyish gang's secret meeting, he said to himself, "The boys don't know it, but they're going to have another attendant at their meeting." And while a twenty-five-cent word like "sardonic" might have impressed Jerry's high school teachers, a ten-cent one like "smart-alecky" would have been an easier sell to Superman fans. The illustrations also were uneven. Sometimes the superhero had facial expressions that were easy to read; just as often, Joe's blocky depiction made it difficult even to make out his face. Thankfully, the audience that ripped through the dialogue and devoured the drawings was even younger than Jerry and Joe, and they were willing to suspend their judgment along with their belief.
Superman was a money machine from the get-go, but he also was a butt-kicking New Dealer. His coming-out party in comic books saw the callow hero tackling a wife beater and liberating an innocent man from death row. In subsequent issues he upended a munitions manufacturer, humiliated the commanders of warring armies, exposed an unscrupulous mine operator, and finished the career of a crooked college football coach. Those no-goodniks didn't deserve a Bill of Rights, and they wouldn't get one with Superman. His messages were simple and direct: Power corrupts. The average Joe deserves a super-powered friend and rich SOBs deserve a boot in the rear. There's a new sheriff in town. These were the very lessons that Depression-era America wanted to hear and that FDR preached in his fireside chats and legislative crusades. But where the president was soothingly patrician, Superman was neither kind nor gentle. A nearer and more apt role model for the new comic book hero was Eliot Ness, who after pursuing Al Capone's mob in Chicago was taking a broom to Cleveland's scandal-ridden police force. So it was that Superman gave the wife beater a thrashing of his own, in the process sounding a warning that applied to all his lowlife targets: "Tough is putting mildly the treatment you're going to get! You're not fighting a woman, now!"
No story better reflected Superman's take on the world's wrongs, and his faith that they could be set right, than _Action_ No. 8's "Superman in the Slums." It opened with a neighborhood tough appearing before a judge and his mother pleading on his behalf. Covering the proceedings for his paper, Clark Kent thought, "The mother's right! But if I know the court of law... her plea hasn't a chance!" This was a cynical and tough Clark paired with a take-no-prisoners Man of Steel. The courts wouldn't deal the kid a fair hand? Superman would. He rescued the delinquents from their double-crossing adult handler and from the police, then gave them a lecture worthy of settlement house pioneer Jane Addams: "It's not entirely your fault that you're delinquent—it's these slums—your poor living conditions—if there was only some way I could remedy it—!" There was. He simulated a cyclone that left the shantytown in shambles. But it did not stay that way for long: "During the next weeks, the wreckage is cleared. Emergency squads commence erecting huge apartment-projects... and in time the slums are replaced by splendid housing conditions."
It was Superman at his do-gooder best. He had no faith that the government would fix things on its own and no patience to wait and see. So he tweaked the system while being careful not to upend it. His means were more those of a rebellious teenager than an anarchist, his ends more FDR reformer than Leninist revolutionary. "In the eyes and mind and heart of Superman," Jerry explained later, "the problems of a penniless tramp or other troubled soul was important. Wealth meant nothing to a super-being who could have acquired all the riches on Earth for himself if he did not have high ideals."
While Jerry and Superman never stopped caring about penniless tramps, they did stop trying to enrich them, or pushing the government to do so. The law became more sacrosanct and Superman less of an outlaw. Robin Hood was giving way to Prince Valiant, or maybe as he saw more of this world Superman was switching from Democratic idealist to Republican realist. Truth, justice, and the status quo. Lost in the transformation was some of his glee. The remake started in the summer of 1939 and picked up steam as the decades turned. The one-year anniversary issue of _Action_ captured the shift. The story started with the same lifelike premise as in earlier issues, with a criminal syndicate using deadly force to drive independent taxi owners out of business. Superman took off his kid gloves and dispatched a gangster to his leaping death. Four pages from the end, however, the story took a science fiction twist: behind the gangsters was not a conniving capitalist or crooked pol, but a dastardly and masterful scientist named the Ultra-Humanite.
It was a new era and he was a new villain—sort of. Jerry and Joe actually were recycling the evil Super-Man from their high school story "The Reign of the Super-Man." Same bald pate, roots in a scientific laboratory, and hyphenated name. Same dream of dominating the world. The Ultra-Humanite's body might be crippled, but his plots to do in Superman were as ingenious as his getaways. He was Superman's first recurring enemy—a worthy successor to Super-Man and an apt precursor to Lex Luthor, Mr. Mxyzptlk, the Puzzler, and other brilliant (and bald) foes to follow. But culling them from the world of fantasy rather than the headlines carried a price: It meant Jerry and Joe were giving a pass to the real-world bad guys.
The timing was right, as it always seemed to be with Superman. America was in a curious interregnum between the summer of 1939 and the end of 1941: Employment and incomes were climbing back, and while the world was at war, we weren't, not yet. When the nation was mired in economic doldrums it had needed a combative hero. Now that its economy and spirit were rebounding, the new attitude was a return to normalcy. For Superman, that meant toning down the violence and adhering to a stricter set of rules. No killing unless he had to, and then only with his bare hands. No destroying private property. No hint of sex. No alienating parents or teachers. Evil geniuses like the Ultra-Humanite were too otherworldly to give kids nightmares. They leveled the playing field for Superman, upped the action, and escalated the escapism. This was exactly what post-Depression, prewar America needed. The results could be measured at newsstands across the nation as Superman, just two years after his entry into the field, had surpassed in popularity three of the marquee names in comics: Little Orphan Annie, Dick Tracy, and Popeye.
SUPERMAN WAS NOT THE only one who was changing. After a prolonged adolescence, Jerry was growing up fast. A year after the first _Action_ came out and just as the _Superman_ book was launching, he married Bella Lifshitz, the daughter of a Russian-Jewish plumber whose home was catty-corner to Jerry's on Kimberly Avenue. She was eighteen and only a week out of high school; he was six years older but still living at home with his mother. They had known each other forever. It was a June wedding arranged by the Siegels and Lifshitzes and paid for by Superman. Three hundred guests came from as far away as Los Angeles, and the already elegant Hotel Sterling was made more so with candles, palms, and flowers. Leo, the brother Jerry had shared a bed with and who made the family swell with pride when he got his dentist's degree, was Jerry's best man. Their mother, Sarah, was beaming, through gritted teeth.
Sarah had never liked Bella. She knew the bride's mother, who could neither read nor write, and her father, who fixed broken toilets and unclogged sinks for a living. Sarah Siegel came from the same impoverished ghettos of Eastern Europe as Sam and Esther Lifshitz, and had experienced enough hardship in the New World to last two lifetimes. But she was a proud American Jew now—the kind who learned the language and how to get by, and who put on rouge, lipstick, and a hat to go to the grocer because you never knew who you'd run into—and she couldn't abide people like the Lifshitzes who hadn't adjusted. Though Sarah liked living with just Jerry after the other children moved out, she was ready to surrender him to the right woman. But now, when his childish dreams about Superman and being a writer finally were coming true, it was the broad-shouldered, buxom Liftshitz girl with whom he chose to share them. Bella looked matronly even at eighteen, with her black hair pinned back and braided in a way that reminded Sarah both of Bella's mother and of her own babushka. It only got worse when she visited the couple at their new apartment and saw for herself how little Bella knew about keeping house. It broke Sarah's heart.
Jerry and Bella were sure they were in love as they headed to the World's Fair in New York for their honeymoon, although she was too young to know what it was supposed to feel like and he was too green. Once they got back to Cleveland, there wasn't much time to learn. He had to deliver a story for the _Superman_ strip every week and monthly ones for the _Action_ and _Superman_ books. And he still was doing "Slam Bradley," "Federal Men," "Spy," and "Radio Squad." The deadlines pumped him full of adrenaline and depleted him of energy and time for Bella. "I wrote, wrote, wrote no matter how I felt," he said looking back. "I found myself writing about terrific super-deeds, even if sometimes I felt like I could barely drag myself to the typewriter."
He still did much of his typing at home, only this time it was his own home, not his mom's, and he typed to the beat of Benny Goodman vinyl. For variety he would head into the office that he and Joe had rented for the bargain price of thirty dollars a month. That was enough for a reception nook where Jerry had his desk, with five more desks squeezed into the main room, but too little money for a telephone or to inscribe their names on the frosted-glass door panels. So what? Comics were Jerry's life then. They were what he talked about in the office and over lunch, which often was just a candy bar, and they were what he dreamed about at night, same as when he was a kid. Another legendary comic book writer, Stanley Martin Lieber, would masquerade as "Stan Lee," looking ahead to the day when he might give up this kiddy medium to write the great American novel. Not Jerry. This was all he had ever aspired to, and now all his Superman comics were written under the name that Michel and Sarah had given him. He invented the vernacular of Superman as he went along, with his hands moving so fast over the keyboard that it felt as if somebody else were dictating the dialogue.
As time went on, somebody was. Initially Harry, Jack, and the managers they hired to oversee their growing editorial empire had let Jerry do as he wished with the character, and what he did was craft a caped avenger who delighted youngsters with his vigilantism and became a poster child for the goo-goos trying to clean up the government. Once Superman became big business, however, plots had to be sent to New York for vetting. Not only did editors tell Jerry to cut out the guns and knives and cut back on social crusading, they started calling the shots on minute details of script and drawing. Superman must be in costume while using his superpowers. His forelocks couldn't be too curly, his arms should be shorter and less "ape-like," and Joe should get rid of his hero's "nice fat bottom." The latter especially made Superman look too "lah-de-dah," 1940s shorthand for shading toward gay. Lois, too, needed a makeover. Nix the "roly-poly hair-do." Stop accentuating her breasts or tummy in a way that made her look pregnant. "Murray suggests that you arrange for her to have an abortion or the baby and get it over with so that her figure can return to something a little more like the tasty dish she is supposed to be," editor Whitney Ellsworth advised in a letter to Jerry early in 1941. "She is much too stocky and much, _much_ too unpleasantly sexy."
In one now-famous case, an entire 1940 storyline was shelved. It was a twenty-six-page tale about a strange substance called K-Metal. Like Superman himself, the metal floated to Earth from the dying planet Krypton, and even brief exposure to it could rob him of his powers. As the story proceeded, Superman was faced with an onerous decision: Should he rescue Lois from a mine disaster when doing so meant revealing his true identity as Superman? True to form, he saved Lois. The two then agreed to become partners in battling crime and villainy. "How foolish you were not to let me in on the secret! You should have known you could trust me!" she chided. Superman: "You're right! There were many times when I could have used the assistance of a confederate. Why didn't I think of it before?"
It was a plot that would have changed everything. K-Metal would become the once-unstoppable superhero's Achilles' heel. Superman would learn for the first time about his origin on Krypton. There would be no more secret identity, at least with Lois, and no love triangle. Jerry might have been in the mood for humanizing his hero by consummating the longtime flirtation, as he was doing in his own life, but it was more than his bosses could stomach. No one knows for sure who pulled the plug, but the reason seems self-evident: Never meddle with a proven success. A metal from Krypton that could melt the Man of Steel did enter the lore in the form of kryptonite, but that took another three years, and while Superman would eventually learn the full story of his interplanetary origins, it was not until 1949. As for the thick K-Metal script, which would have run twice as long as the normal Superman comic, it remained secreted away in the archives of Detective Comics until a curious young staffer stumbled upon a smudged carbon copy forty-eight years later.
The message to Jerry was clear: Superman no longer belonged just to him. And there was more, as his editors spelled out in an onslaught of letters. Hire assistants so you can meet your deadlines, and set aside "Slam Bradley" and other second-tier characters. Stop crying poor mouth and begging for raises. Don't talk to the press. Say goodbye to Cleveland and come to New York, where we can keep a closer eye on you. Grow up. When they really wanted to make their point, his bosses at Detective Comics had their boss, Jack Liebowitz, sign the letter. "Bear in mind," Jack wrote five months after _Action_ was launched, "that we own the feature 'Superman' and that we can at any time replace you." He wrote again the following April: "You have the germ of a great idea in SUPERMAN but you need constant editorial supervision." By January 1940 Jack seemed at the end of his rope, growling, "From your promised five releases a month I'm down to one. For anyone to have fallen that badly, you are certainly a Superman in reverse."
Jerry had moved to New York in 1939, and he reined in his hero to look more like what Jack and his editors wanted. He took the abuse and toed the line not just because Jack had all the power, but because Jerry liked the money Superman was bringing him and loved the prestige. Just how rich he was is a matter of contention. _The Saturday Evening Post_ reported that he and Joe split $75,000 in 1940; Jerry said they were sharing just over $38,000, counting revenues from the comic books and strips. Even at the lower figure he was earning $307,000 a year in today's dollars, or more than thirty times what he had made two years before. A year later, his share had increased to $29,000. That was enough for him and Bella to move back to Cleveland—or rather to the upscale suburb of University Heights—and buy a home with two and a half baths, a paneled rec room with a bar, silk draperies, and air-conditioning, which was a luxury then. He got Bella a mink coat and a diamond bracelet, and bought himself a hip-reducing gizmo in hopes of undoing the damage done by all those candy bars.
What mattered more than money, to a boy who grew up thinking he was a pariah and sharing a bed with his brother, was being a man-about-town—not just in Cleveland, where newspapers and adoring fans sang his praises, but in New York. His editors might have been trying to cut Jerry down to size, but on the streets of the Big Apple there was no denying his hero's rise to iconic stature. It was on full display at the 1940 Macy's Thanksgiving Day Parade, where the biggest balloon was an eighty-foot-high replica of the Man of Tomorrow, and at the World's Fair in Flushing Meadows, which staged a first-of-its-kind Superman Day. Best of all: America's best-read magazine, _The Saturday Evening Post_ , ran a seven-page feature on Superman and his creators that included a picture of Jerry lying in an oversized bed reading a book with hard covers and no graphics.
Joe's picture in that article showed him not in bed but at the head of a bountiful Sabbath table with his mother, Ida; father, Julius; younger brother, Frank; and kid sister, Jean. It seemed designed to underline how differently he and Jerry had responded to Superman's success. Joe acknowledged early on that he could not handle the workload. His eyes were bad and getting worse, and his left hand was a problem, too. He had what he called a spastic condition that had started a year after Superman first appeared in print; it prevented him from drawing for long stretches, made him switch to his right hand for lettering, and eventually forced him to wear a leather brace that completely immobilized his bad hand. So he hired assistants, although their bylines never showed up on the comic book or comic strip, and at first even Jack and Harry were kept in the dark. There was just one to start, Paul Cassidy, who worked from his home in Milwaukee and helped with "Slam Bradley," "Spy," and other early Siegel and Shuster efforts. Once Superman got going, Cassidy moved to Cleveland and Joe hired three more artists, including Wayne Boring, who would keep on drawing and redefining Superman for two decades. By early 1939 Joe was lightly and sparingly sketching the scenes and leaving them to others to fill in and fine-tune; the one thing he insisted on keeping control over was Superman's head, which he felt defined his hero. When he moved to New York later that year he did even less, not showing up at the office for long stretches and frustrating his bosses as much as Jerry did. In Joe's case the complaints were less about his attitude than his health and work ethic.
Joe drew the same salary as Jerry but his assistants had to be paid out of his cut, which in 1941 meant that he earned about $29,000 and kept $14,500. That may not sound like much to live on in New York, but in today's terms it is $220,000. It was enough to let Joe move out of the YMCA and into a small apartment, then into a ten-room house. He needed something that big because his parents and siblings were with him again, and to ensure they got to enjoy their new surroundings he had a car waiting in the driveway and a maid who came in twice a week. Joe was spending more of his weeks holidaying in Pennsylvania's Pocono Mountains. When he was around, he headed to Barney Kofron's gym, hoisting 175-pound weights and trying to ensure that the body that stared back at him in the mirror really was a model for Superman. That, along with a T-bone steak and two quarts of milk a day, inflated his weight from 112 to 128 pounds. To lift his height, which was just five foot two, he wore elevator shoes.
Joe stayed single all through those years, but it was not for lack of interest in girls, or for lack of their reciprocation. The once-shy teenager had more confidence now that he was Superman's personal artist, and he started dating the showgirls he had always dreamed about. Not just anyone would do; they had to have a model's looks and they had to be tall. "He loved _shiksas_ ," recalled his sister, Jean. "They were always tall and slender blondes like his dream girl Lana Turner." He also double-dated a lot, with Jean and with Batman artist Jerry Robinson. Once, Robinson fixed him up with his cousin in Trenton, who was brilliant, pretty, and the same height as Joe. The four young people went out dancing and had what Robinson thought was a terrific time. "Afterwards I said, 'How'd you like Shirley?' He said, 'Oh, well she's great. But she's too short for me.' "
Bob Kane, Batman's creator, loved double-dating with Joe, if only so he could tell the girls they were dating Batman and Superman. One Saturday night in the winter of 1940 he and Joe were due to go out with a couple of girls in Miami Beach, but Joe never showed up. He had stopped on the street near his hotel to ogle an antique car, but with his myopic vision, he had to lean in too close for the comfort of two patrolmen who spotted him. At the station he told them who he was, yet all they could see was a would-be auto thief, which earned him a threat of thirty days in the slammer. Although he drew several sketches of Superman, the police remained skeptical. Luckily Harry Donenfeld was in town and bailed him out, though not before word leaked to the press. SUPERMAN RESCUES HIS CREATOR FROM FLORIDA JAIL, _TheWashington Post_ blared across its front page.
HARRY AND JACK HAD never had things so good. As always they had their pornography profits, and their book and magazine distribution firm was on the way to becoming the biggest in the country. A year after Superman they scored another hit with Batman, and two years after that Wonder Woman gave them a third cash cow. Still, it was Superman who made the most money and generated the most attention for Donenfeld and Liebowitz's Detective Comics, Inc. They acknowledged the debt by amending the company logo to read a SUPERMAN-DC PUBLICATION. The profits from Superman comic books were all theirs. Detective took a cut from the comic strip, too—by 1941 it was up to 10 percent, which was a quarter of what Jerry and Joe were getting but a good deal for middlemen who bore almost no costs. There was yet more money to be made from radio shows, movie serials, merchandising, and a full-length novel, and Jack pushed and pulled for every dollar.
Woe to those who tried to horn in. If Superman was a demon in tracking down his enemies, his accountant and attorneys were even more single-minded in protecting their franchise of super-powered costumed heroes. The only question was whom to sue first. Jack settled on Wonder Man, who had been tapped by a Tibetan yogi to fight evil and was given a ring that endowed him with super powers nearly identical to Superman's. No surprise there: Victor Fox, Wonder Man's owner, was Harry's former partner and had seen Detective's ledger sheets showing how much money Superman made. So the cigar-chomping Fox, a nasty little man who would call himself "king of the comics" but whose kingdom at the time was a single astrology magazine, hired comics wunderkind Will Eisner to beget a hero "with a red, tight-fitting costume, and a red cape." Fox had one more instruction for Eisner: Lie about Wonder Man's parentage when he was called to testify in Jack's lawsuit. Trial records show that Eisner did what he was told, despite his later denials. It didn't matter, not with the evidence there in cartoon panels for the judge to see. Wonder Man became a one-issue wonder and Jack had bragging rights in the first-ever comics copyright lawsuit.
Next up was Captain Marvel, a far more formidable foe who flew onto the comics scene early in 1940 with a chiseled face that looked just like actor Fred MacMurray's. His costume was a stunning red, with a white cape trimmed in yellow. His alter ego was Billy Batson, a radio newsboy who had only to say the magic word "Shazam!" to transform himself into the World's Mightiest Mortal. So compelling were the storylines and illustrations that by mid-1943 Captain Marvel would outsell even Superman. Jack perceived the threat early, and in 1941 he sued Fawcett Publications, alleging that it had stolen Superman's life, looks, and even his shawl. To Jerry it was a slam dunk: "A Mongoloid idiot would have come to the same conclusion, because they both did the same things. They both had super-strength, they both wore costumes, they both had similar identities." A judge agreed that Marvel had infringed on Superman, but in a ruling delivered a full ten years after the suit was filed, he added that Detective had failed to properly copyright its character and therefore had no case. A year later the esteemed jurist Learned Hand overturned the copyright part of the ruling and sent the case back to the lower court. By then Captain Marvel had lost his luster, and rather than pay for another expensive trial, Fawcett gave Jack and Harry four hundred thousand dollars and agreed to retire the Captain.
Jack had made his point. While the superheroes who followed continued to be Superman knockoffs (from Captain America to Plastic Man, Doll Man, and Minute-Man) or purposefully un-Superman-like (Spider-Man), their creators disguised and denied any ties. No one, least of all Jerry, acknowledged the chutzpah in Detective's claims of plagiarism in light of how close Philip Wylie had come to charging Superman with plagiarizing Hugo Danner and the strong case Lester Dent could have made on behalf of Doc Savage. Otto Binder, one of Superman's best scripters in the 1950s, claimed that it was he who in the 1930s had planted the seed with Jerry and Joe for an interplanetary, super-powered orphan, although he never talked about suing. Jerry himself would point an angry finger at surprising targets in 1947, charging that his bosses along with his friend Bob Kane had stolen their Batman brainchild from Superman, and that Wonder Woman was a rip-off of his idea for Superwoman. "It is perfectly clear to the youth of the nation that Batman is really Superman with a mask on," Jerry's lawyer told a judge. And it was not just Batman, but "twenty of these features, all of whom are like Superman." The truth is that Batman was as human and somber as Superman was infallible and uplifting. If the Dark Knight and his second self, Bruce Wayne, stole from anyone, it was from Jerry himself: The death of Bruce's father during a robbery was eerily like what happened to Michel Siegel, and Bruce and Jerry both spent a lifetime trying to get over their early losses.
Shoring up their superhero cartel made it easier for Harry and Jack to live in a manner that their Lower East Side upbringings had not accustomed them to, but had made them crave. How rich was Harry? Jack told _The Saturday Evening Post_ in 1941 that his partner had paid income taxes the year before on "more than $100,000." Harry, according to the _Post_ , "once told a reporter that he netted $500,000 from Superman alone." The low-end projection would be $1.5 million in today's terms, the high end $7.7 million. Both are likely conservative estimates, given all the commerce Harry conducted above the table and under it.
What is clear is that Harry had all the money he needed to keep his wife, Gussie, his son, Irwin, and his daughter, Peachy housed first in a seven-room apartment on Riverside Drive and later in a large duplex on the Upper West Side with a thirty-foot terrace facing Central Park. His and Gussie's last stop was 710 Park Avenue, one of the first apartment houses erected after World War II on America's most prestigious promenade. There was enough money left over to give jobs to nieces, nephews, and others who came looking during the Depression, and to keep his mistress, Sunny Paley, happily ensconced twenty blocks down Park Avenue in a suite at the Waldorf Astoria. He slept in George Pullman's plush hotels on wheels when he traveled for business, and when he arrived he stayed in the best accommodations in town, sometimes having Frank the chauffeur meet him there with his car. The excursions he preferred were purely pleasure—to Miami Beach, Havana, and the circuit of speakeasies and strip clubs of New York City. One hand would lay claim to Sunny's midriff, the other cradled a glass of scotch, but his gaze was planted on the minions gathered round waiting for him to tell a story or buy another round. No one listened harder than his old friend Frank Costello, whom reporters had taken to calling "Prime Minister of the Underworld" and whom soldiers in the Luciano crime family called boss.
Harry savored his contradictions: a mobbed-up publisher whose superhero was the mob's greatest nemesis, a family man whose wife and children accepted that he had a mistress living down the street, a purveyor of adult erotica and kiddy comics who read neither. "He was many things—a heavy drinker, a womanizer, he gambled and he knew Costello—but you're talking to a daughter who absolutely adored him," says Peachy. "He was a very generous, caring, loving man, and he was very progressive."
Holding together his dissolute life was easier with Jack there to mind the businesses and pick him up if he fell apart. Sometimes Jack would have to shout to be heard, telling Harry he couldn't go on spending, or drinking, the way he was. But he never forgot that it was Harry who had brought him into the business when he had nothing. Harry also helped him set up a partnership with Maxwell Gaines to launch All-American Comics, and later helped him buy out Gaines for what Jack said was a million dollars. Jack regularly reminded his kids what it had been like at the beginning, when he and Harry had four or five different companies but "very little cash in the till." They would take whatever there was from one firm to cover the debts from the rest. Now loot was flowing in from everywhere, and especially from Superman, who by 1942 was starring in three comic books with a combined circulation of one and a half million and an estimated readership of four and a half million. Twenty-five million more followed him in 285 newspapers nationwide. For Jack, that cash flow bought him the freedom to launch new companies in the morning and weigh in on how big to make Superman's fanny or Lois's breasts in the afternoon. In the evening, just before the long drive home, he always sat down with the boys from the office for a hand or two of his favorite game, gin rummy. Not a bad life for a boy from the shtetl.
Jack also made investments of his own, in the stock market and in the five-bedroom house he owned on Long Island. Rose and the girls lived the good life in Great Neck—prep schools, their own sitting rooms next to their bedrooms, as many as three servants to tend to their needs—but it hadn't always been that good. They were in the Bronx when Jack was getting started, then moved onto Long Island and up the ladder as he and Superman did—to Laurelton, then Lynbrook, then a smaller home in fashionable Great Neck, and finally to the mansion. Jack was a founder of the Jewish Federation and had a box at the opera. Uncle Harry drove out with Aunt Gussie for family gatherings and games of pinochle, like in the old days; ties between the families were cemented when Harry's only son married Jack's niece. But Jack handled Harry the same way you would any relative you loved but knew wasn't a good example for the kids. "My father was so refined, so un-Harry," recalls his daughter Joan. Jack, Joan adds, was intimidating enough that his family was as afraid of him as his workers were, but when she needed him he always was there. He also always was clear about his expectations for her and her sister, Linda: "A young matron doesn't go out of the house without a hat and gloves. And he wouldn't let us work, it was not what a refined young lady should be doing. If you worked it meant your father wasn't taking care of you."
EVERY KID IN AMERICA knew that Superman could win any war he fought in five minutes, which posed a conundrum when the United States went to war in December 1941. The nation's fleet was in ruins at Pearl Harbor. Germany was holding the line in Russia and pressing ahead in North Africa. Normally bullish Americans were not sure this was a war they belonged in or could win. Adults argued about strategy and tactics at work and over the dinner table, and stayed up nights fretting about their teenage sons who were being drafted to fight. Adolescents had a simpler question: Why not send in Superman?
It made sense. In the space of the first two issues of _Action Comics_ , he had reformed an arms dealer and made peace between warring forces in the fictional South American republic of San Monte. The stories left no doubt that he had clout, or that the conflict in San Monte was a stand-in for the ongoing Spanish Civil War. The vision of all that he could accomplish had become clearer still in the February 27, 1940, issue of _Look_ magazine, in a cartoon story by Jerry and Joe entitled "What If Superman Ended the War?" World War II was at an early stage then and America still was officially neutral, but that did not stop Superman from twisting Nazi cannons into useless metal, intercepting Japanese fighter planes, and hauling Hitler and Stalin before the League of Nations for judgment. "Adolf Hitler and Josef Stalin," the presiding minister announced, "we pronounce you guilty of modern history's greatest crime—unprovoked aggression against defenseless countries."
But once the war heated up and America was forced in, Superman stepped back. It was not for lack of patriotic role models. Hop Harrigan, comic book America's Ace of the Airwaves, took to the skies "to rap the Jap a slap on the yap that'll corrugate his map!" Another comic starred Blackhawk, a crackerjack Polish pilot who had been shot down in 1939 and was back in the cockpit exacting his revenge against the Nazis. Rip Carter and the Boy Commandos did comparable damage on the ground, alongside Captain America, Captain Marvel, the Sub-Mariner, and the Human Torch. So why was Superman, the monarch of American heroes, sitting out the war?
That was a question being asked not just by juvenile fans but by letter writers to _The Washington Post_ and editors at _Time_ magazine. Jerry and Joe were fretting as well, along with Jack and Harry. They knew they had created that expectation by fashioning a hero powerful enough to intervene and righteous enough to recognize that the Allies were the good guys. They also knew that all anyone expected from non-super-powered heroes like Blackhawk was to shoot down an enemy fighter. With Superman the bar would be infinitely higher. Once he entered the fray, readers would demand real-world results that no make-believe character could deliver. Even having him try would make him look like a paper tiger and could damage morale among American soldiers who were fighting for real. "As the mightiest, fightingest American," _Time_ wrote in a story called "Superman's Dilemma," the Man of Steel "ought to join up. But he just can't. In the combat services he would lick the Japs and Nazis in a wink, and the war isn't going to end that soon. On the other hand, he can't afford to lose the respect of millions by failing to do his bit or by letting the war drag on."
The solution: Clark Kent tried to enlist in the Army in 1941, but during his eye exam he inadvertently read a chart in the adjoining room with his X-ray vision. "You're physically superb," the doctor told him, "except that you're obviously blind as a bat.... The Army _doesn't_ want you." It was a remedy attributed to editor Murray Boltinoff, but it may have been cooked up in collaboration with Joe, who failed his own preinduction eye test and was declared unfit for duty. It got Superman off the spot in a way that satisfied everyone from youthful comic book readers to the editors of _Time_. But not Lois. "I might have known the Army would turn you down," she told Clark. "How you summon up enough strength even to peck at your typewriter keys is beyond me!"
He may have been out of the Army, but he was bent on helping. "The United States Army, Navy and Marines are capable of smashing their foes without the aid of a Superman!" he told readers in that same 1942 cartoon strip. "Perhaps I could be of more use to my country working right here at home, battling the saboteurs and fifth columnists who will undoubtedly attempt to wreck our production of vital war materials!" And so it was that his direct confrontations with Hitler, Hirohito, and what he called the Japanazis were confined to comic book covers, which were attention-grabbing but unrelated to the stories inside. The stories had him joining war drills, battling "Japoteurs," and standing up to "Mr. Schickelgruber" and his "so-called master race," making the case without having to say it that the real _Übermensch_ was on our side.
The U.S. military knew that Superman was a flying Uncle Sam—an embodiment of the red, white, and blue virtues for which America was sacrificing its sons—and it used him to solicit blood donations, spur drives for scrap iron, and sell savings bonds. After just one radio appeal from their hero, 250,000 young patriots mailed in pledges to buy war stamps. The British Admiralty named its most powerful oceangoing tug _Superman_. U.S. servicemen had done the same with jeeps, tanks, landing craft, and the planes of the Air Corps Reserve's 33rd Bombardment Squadron. The Navy strove to end illiteracy within its ranks in part by having the dialogue in a Superman comic shaved to single or double syllables, then rolling out to its sailors 15,000 copies a month of the easy-to-read books. After D Day, an anxious infantry officer told war correspondents, "When I saw one of our boys in our landing craft nonchalantly reading a copy of _Superman_ , I knew everything would be all right."
Superman's biggest contribution to the war effort was setting free the imaginations of America's warriors. They got all the blood and guts they needed on the battlefield, and all the run-ins with the Nazis and Japanese. Watching Superman battle with Lex Luthor and other fantastic villains let them escape. It offered them a way to feel like kids again and reminded them of the lives back home that they were fighting to protect. No gift could matter more.
How do we know that? At U.S. military bases, comic books outsold _Reader's Digest, Life_ , and _The Saturday Evening Post_ combined. Estimates said 80 percent of the Army's reading matter was comics, and Superman was tops among the comics. It was a two-way street for the Man of Steel: He helped sell America on the war, and the war pumped up his sales. Kids who had grown up on him were heading overseas, and they brought the knight from Metropolis along as a security blanket. The Navy found Superman so soothing that it tucked his comic books into ration kits bound for the Marine garrison on embattled Midway Island. So concerned was Canada about Superman's influence on its troops that in 1940, when he convinced both sides in a comic strip war between Blitzen and Rutland to temporarily lay down their arms, censors ordered the _Toronto Star_ to leave it out.
U.S. censors didn't step in there, but they were worried four years later when Superman got too close to the sensitive subject of an atom-smashing cyclotron. "The FBI came into the office," Superman editor Jack Schiff recalled years later. "They told us to change the syndicated Superman strip then running in order to eliminate a cyclotron that was featured. For reasons unknown to us then, this was a no-no. We really should have suspected what was happening: the A-Bomb was being developed and this was a possible leak." A 1945 document from the War Department, declassified after the war, downplayed the concern, saying that the very fact that the cyclotron story was playing out in the funny pages would ensure no one took it seriously. The war planners, however, had taken the matter seriously enough to mark their memo SECRET, and Harry's marketing team later used the crackdown to boast that "Superman readers had a comic strip preview of the world's most carefully guarded secret." Two other Superman stories—one with a cover of him watching an atomic bomb test, the other where Lex Luthor tries to use an A-bomb to do away with Superman—saw their publication postponed until after the war.
Superman's owners were good at keeping secrets. They made sure that no one knew that Jerry was not scripting the character at the time of the FBI visit in 1945 and hadn't been for two years, although his was the only writer's name on every story. Jerry had been drafted into the Army in the summer of 1943 and sworn in on July 4. Ghostwriters who had done an occasional story before now were writing nearly everything. Jerry's proxies and Joe's made a motley crew. There was Alvin Schwartz, a Trotskyite who loved debating over lunch with his editor, Jack Schiff, a Stalinist. Leo Nowak, a musician as well as an artist, had been sent Joe's way by a whiskey salesman he met in a bar. Paul Cassidy was a schoolteacher before he drew Superman, and afterward. Nearly everyone who joined up in Cleveland and later in New York backed FDR and the New Deal. Many were in their twenties—only slightly older or more worldly than their readers—and most didn't tell family or friends that they had fallen so low they were writing for the funny pages. They called it Hungry Money; once they had enough, they would find real jobs writing magazine stories, novels, or film scripts. Their tightly packed workspace was a cross between a newsroom and a sweatshop, with an unremitting stream of pages passing from pencillers to inkers, sandwiches sufficing for lunch and dinner, and deadline pressure intense enough to breed ulcers and nervous breakdowns.
Alvin Schwartz had been co-editing a literary magazine that published luminaries like William Carlos Williams and Ezra Pound, but the Depression upended that career. One day he tried to bum a quarter off an old friend; the friend, a comics artist, said there was writing work to be had in his field. Alvin penned his first Superman comic strip in 1944 and his last in 1952. In between he earned enough money to buy a house, even though he was making $250 for a strip that earned Jerry $800. And while he didn't get a byline, he did get a _New York Times_ story revealing that he was one of Superman's writers. He also got to put his stamp on the hero: "I tried to change Superman from being a meathead who simply had a harder punch into something more human and philosophical."
To Jerry, ghostwriters like Alvin were mere seat warmers. After all, Superman was his brainchild, and he kept sending in ideas while he was serving his country in the Army. Although he wore the insignia of the infantry, as a minor celebrity he wasn't about to see real action. His special-assignment company in West Virginia was called DEML, for Detached Enlisted Men's List, but it was known throughout the Army as Damn Easy Military Life. In a 1944 letter, Jerry asked Jack whether he could pull strings to land him an even cushier job in Washington writing for the Army newspaper. And while he was at it, could he send Superman pins and secret codes to all of Jerry's fellow soldiers in DEML?
Jerry ended up not in Washington but in Alabama, then in Hawaii, where he was promoted to corporal and wrote a comic strip for the troops, the first time a member of the military had ever done that. It was called _Super G.I_. and its hero, Private Joe Droop, again was modeled after Jerry—a soldier who was "small, weak... timid." Every human being, Droop told himself, "is two persons. The person he is... and the person he'd like to be. It's all a matter of concentration and psychic conditions." While Jerry loved the attention he was getting in the Army, he was too obsessed to focus just on his military cartoon and his war. He worried that each check coming from Jack was too small, and that Detective was using his old idea for a _Superboy_ comic book without giving him proper control or compensation. He complained less when Joe Shuster—who had been ceding increasing authority to his artistic assistants as his eyesight and work ethic frayed—was stripped of any remaining control over Superman's design.
_Superboy_ wasn't the only new product to emerge from World War II for Detective Comics, and Joe wasn't the only one whose duties changed. Writers and artists were coming and going with the draft and other wartime demands, and Jack Liebowitz found new talent to replace them temporarily or forever. He discovered that a full-length novel worked as well as a thin comic book to tell Superman's story, and that George Lowther was as good a storyteller as Jerry Siegel. And he found all the paper he needed for his books and comics despite a shortage that had forced newspapers to shrink the Sunday strips. No one knew better than Harry Donenfeld and Jack Liebowitz how to dip into black markets and make cross-border deals with Canadian publishers. Beyond that all they had to do was count their money, with monthly sales of comic books doubling to twenty million between 1941 and 1944, dealer return rates plummeting to zero, and war-related prosperity generating a windfall for the whole comics industry. Superman caught the breeze, with thirty million Americans regularly reading his comic books or strips. Soldiers were loyal fans, but truer still were their kid brothers and sisters back home, who bought Superman stories, read and reread them, then traded them in schoolyards and backyards. As for Jerry's complaints that he was being shortchanged while he was in the Army, Jack said that in addition to commissions for stories others were writing, Jerry got a $5,000 bonus at the end of 1945. "You did not see fit to acknowledge [it]," Jack wrote, "though you did deposit it."
The riskiest business decision Jack Liebowitz made during the war—to keep Superman off the battlefield—was rewarded in spades afterward. Hop Harrigan, the Boy Commandos, and nearly all the other comic book heroes who were on the front lines saw their popularity fade, and many disappeared entirely. Finding a role for real warriors in peacetime was hard enough, but it was impossible for fantasy characters meant to provide an escape. Their association with a war America wanted to forget was too painful. Superman's shepherds had bet right, keeping him engaged as a cheerleader rather than a combatant. It was a lesson they would not forget.
An even more valuable lesson was one that Superman himself understood better than anyone. Over the course of World War II, the hero and his country went through a similar process of self-discovery. America learned that it had the world's most vibrant and malleable economy and used it to forge a military complex powerful enough to smash the Axis juggernaut. Superman found he could not just jump high but he could take flight, see through or topple buildings, and defeat any foe. The mightier the superpower and its superhero got, the more both realized that the challenge was to marshal their strength not merely to win battles but to stand for something worthy of the fight. Truth was a good starting point. And of course there was justice. As GIs sacrificed their blood everywhere from the Ardennes Mountains to Iwo Jima, the United States and Superman added another element to their moral code: They were fighting to advance not just universal rights but very particular ones, like life, liberty, and happiness, that they called the American way. The Man of Steel and the nation that loved him were, during the long years of war, growing up together in a way that made him more relevant than ever in postwar America.
# **CHAPTER 3**
# **A Matter of Faith**
HE DIDN'T LOOK JEWISH. Not with his perfect pug nose, electric blue eyes, and boyish spit curl that suggested Anglo as well as Saxon. No hint in his sleek movie-star name, Clark Kent, which could belong only to a gentile, probably one with a lifelong membership at the country club. His social circle didn't give it away either: Lois Lane, George Taylor, and even Lex Luthor were, like him, more Midwest mainstream than East Coast ethnic. The surest sign that Clark was no Semite came when the bespectacled everyman donned royal blue tights and a furling red cape to transform himself into a Superman with rippling muscles and expanding superpowers. Who ever heard of a Jewish strongman?
The evidence of his ethnic origin lay elsewhere, starting with Kal-El, his Kryptonian name. _El_ is a suffix in Judaism's most cherished birthrights, from Isra-el to the prophets Samu-el and Dani-el. It means _God. Kal_ is similar to the Hebrew words for _voice_ and _vessel_. Together they suggest that the alien superbaby was not just a Jew but a very special one. Like Moses. Much as the baby prophet was floated in a reed basket by a mother desperate to spare him from an Egyptian Pharaoh's death warrant, so Kal-El's doomed parents, moments before their planet blew up, tucked him into a spaceship that rocketed him to the safety of Earth. Both babies were rescued by non-Jews and raised in foreign cultures—Moses by Pharaoh's daughter, Kal-El by Kansas farmers named Kent—and the adoptive parents quickly learned how exceptional their foundlings were. The narratives of Krypton's birth and death borrowed the language of Genesis. Kal-El's escape to Earth was the story of Exodus.
Clues mounted from there. The three legs of the Superman myth—truth, justice, and the American way—are straight out of the Mishnah, the codification of Jewish oral traditions. "The world," it reads, "endures on three things: justice, truth, and peace." The explosion of Krypton conjures up images from the mystical Kabbalah, where the divine vessel was shattered and Jews were called on to perform _tikkun ha-olam_ by repairing the vessel and the world. The destruction of Kal-El's planet and people also calls to mind the Nazi Holocaust that was brewing when Jerry and Joe were publishing their first comics, and it summons up as well the effort to save Jewish children through _Kindertransports_. Superman's lingering heartsickness was survivor's guilt. A last rule of thumb: When a name ends in "man," the bearer is Jewish, a superhero, or both.
If most of his admirers did not recognize Superman's Jewish roots, the Third Reich did. A 1940 article in _Das Schwarze Korps_ , the newspaper of the SS, called Jerry Siegel "Siegellack," the "intellectually and physically circumcised chap who has his headquarters in New York." Superman, meanwhile, was a "pleasant guy with an overdeveloped body and underdeveloped mind." Creator and creation were stealthily working together, the Nazis concluded, to sow "hate, suspicion, evil, laziness, and criminality" in the hearts of American youth who "don't even notice the poison they swallow daily."
Superman had even stronger cultural ties to the faith of his founders. He started life as the consummate liberal, championing causes from disarmament to the welfare state. He was the ultimate foreigner, escaping to America from his intergalactic shtetl and shedding his Jewish name for Clark Kent, a pseudonym as transparently WASPish as the ones Jerry had chosen for himself. Clark and Jerry had something else in common: Both were classic nebbishes. Clark and Superman lived the way most newly arrived Jews did, torn between their Old and New World identities and their mild exteriors and rock-solid cores. That split personality was the only way he could survive, yet it gave him perpetual angst. You can't get more Jewish than that.
So compelling were those bonds that decades later TV's Jerry Seinfeld would refer to Superman as his Jewish brother-in-arms, and BBC Radio would air a debate entitled "Is Superman Jewish?" _The Jewish 100_ , a book about the most influential Jews of all time, listed Jerry and Joe alongside Sigmund Freud, Albert Einstein, and Abraham. Jules Feiffer, an authority on cartoons and Jews, said the Last Son of Krypton was born not on Krypton but on "the planet Poland, from Lodz maybe, possibly Crakow, maybe Vilna." The alien superhero was, more than anything, "the striving Jewish boy's goyishe American dream."
Was that what Jerry and Joe had in mind? Neither was an observant Jew or attracted to organized Judaism. Some of Superman's Jewish touches—such as the spelling Kal-El, versus Jerry's more streamlined Kal-L—were added by later writers and editors, the preponderance of whom also were Jewish. But Jerry acknowledged in his memoir that his writing was strongly influenced by the anti-Semitism he saw and felt, and that Samson was a role model for Superman. He also was proud that his anti-Nazi superhero touched a nerve in Berlin. So, undoubtedly, were Harry Donenfeld and Jack Liebowitz, both of whom had experienced Jew-baiting up close in Eastern Europe and the Lower East Side. What Jerry did, as he said repeatedly, was write about his world, which was a Cleveland neighborhood that was 70 percent Jewish, where theaters and newspapers were in Yiddish as well as English, and there were two dozen Orthodox synagogues to choose from but only one place—Weinberger's—to buy your favorite pulp fiction. It was a setting and time where juvenile weaklings and whey-faces—especially Jewish ones, who were more likely to get sand kicked in their faces by Adolf Hitler and the bully down the block—dreamed that someday the world would see them for the superheroes they really were.
THE EVIDENCE THAT HE was a Jew did not stop other faiths from claiming Superman as theirs. Christian enthusiasts saw him as Jesus, the child dispatched to Earth by his omnipotent father to save mankind. No surprise that his Kryptonian name was Hebrew, since Jesus was a Jew, but the _Kal_ in front of _El_ suggested to Christians a presence beyond just God—a son, perhaps. The fact that Clark Kent's adoptive mother originally was called Mary added to their argument, as did Superman's cape, which can look like the wings of an angel. Superman's story reminds Jews of Old Testament heroes from Moses to Aaron and David; Catholics and Protestants find the holy saints when they read between the cartoons' lines.
The case for a Christian Superman was grounded in geography. It was not merely that he came from the heavens—but that he landed in Kansas. It was a state made up mainly of God-fearing Roman Catholics, Baptists, and Methodists. It fell just outside the Bible Belt but deep within the Grain Belt, thanks to its sorghum and wheat, soybeans and corn. It was the American heartland, with plainspoken rules about how to treat others and clear-cut values taken straight from the Ten Commandments. It gave Clark a grounding in right and wrong as he headed to the big Metropolis and gave Middle America a sense that he was one of them.
Symbols mattered, too, as always in religion and popular culture. Baby Kal-El's blanket brought to mind Jesus' swaddling clothes. Clark's Fortress of Solitude rose like a cathedral. His fellow reporters had as much trouble recognizing him as Superman as Jesus' fellow Nazarenes did seeing him as their savior. True believers also have attached meaning to names. _Krypton_ is Greek for _hidden_ , which is one way the New Testament described the kingdom of heaven, and in Kryptonese _Kal-El_ means _Star-Child_ , which could refer to the Star of Bethlehem, signal of the birth of Christ. _Lex Luthor_ sounded and acted like _Lucifer. Clark_ means _cleric_ , in this case one whose middle name—Joseph—is perhaps a wink to the carpenter from Nazareth. These and other connections are laid out in Stephen Skelton's book _The Gospel According to the World's Greatest Superhero_. The dots became even easier to connect in later years. Comic books would kill off Superman, then resurrect him. In the movies, a godlike Marlon Brando would dispense to his son advice straight out of the Book of John to "show the way" to the earthlings who "lack the light" but have the "capacity for good." On stage in _Godspell_ , Jesus would wear a Superman shirt. And a television show about an adolescent Superman would open with an episode showing a young Clark hung on a crucifix by a gang of football players.
While the search for religious meaning has yielded compelling nuggets, sometimes it seems strained. Is Jor-El, Superman's father, a play on Hebrew words for "God teaches"? Or is it, as Jerry insisted, Jor as in Jerome and El as in Siegel—making Jerry Siegel, rather than any deity, the true father of Superman?
The truth is that Superman's most Christ-like features have less to do with how he looks or sounds than how he behaves. He represents our best selves and highest aspirations. He intervenes where he can, as with the abusive husband and death row inmate, but recognizes that man must have free will even when it hurts, as in World War II. That sense of devotion and duty inspired Father John Cush to enter the priesthood, and it resonates with the high school students he teaches in Brooklyn. "Obviously," Cush explains, "as a Roman Catholic priest, I see Superman as a Christ figure." To some clerical leaders, that is apostasy. "The Word became flesh, not steel," said the Reverend Kenneth Reichley of New York's St. Peter's Lutheran Church. "Superman is magic. He manipulates fate and history... Jesus is not magic. He works within history." Not so, said the Reverend Andrew Greeley, Catholic scholar and author: "Superman, I've always thought, is an angel. Probably the angel stories found in all of the world's religions are traces of the work in our world of Superman and his relatives. Who is to say I'm wrong?"
Christians and Jews were not the only ones. Muslims, at least some of them, saw in the Superman creation story a reflection of their own origins, with Jor-El dispatching Kal-El as a messenger to mankind much the way God did Muhammad. Buddhists put in dibs, too, seeing the superhero as the Man of Zen. Superman knew how "to live entirely in the _now_ ," explained Alvin Schwartz, who was born Jewish, was long interested in metaphysics, and wrote Superman's comic strip when Jerry was in the Army. "He's totally fixed on a single point. His one defining act—his rescue mission. That's what he does... and that's why you can't have a Superman without a Clark Kent—because no one can live all the time at that level of experience. There has to be a retreat to ordinariness, to self-recollection."
Nonbelievers had a different take: Superman was a paragon, but he wasn't a Jew or a Christian, a Buddhist or a worshipper of the ancient gods of the sun that gave him his power. He was so strong he could truly move mountains and so pure he would neither litter nor jaywalk. He could crawl away from kryptonite but was undone by moral relativism. He never asked his followers to die in his name or to proclaim themselves the chosen ones. This vision of Superman as a secular messiah tapped into America's cultural myths and oral traditions—into its communal do's and don'ts—which is just the way agnostics, atheists, and spiritualists would have him. It also is just what Geoff Johns calls to mind when he writes Superman stories or edits them. "You can have spirituality and morality without religion," says Johns, today's chief creative officer at the company built by Jack and Harry. "Superman shows that is possible." Mark Waid—who collaborated with Alex Ross on the graphic novel _Kingdom Come_ , the most spiritual Superman story ever—agrees that religion is not the point: "Superman is not a story about faith, it's about inspiration. It's a story about trying to move us into emulating, into being, into doing."
The early Superman did that not by preaching, or even explaining, but by letting his actions speak to his intent. The governor wouldn't pay attention to a prison superintendent who was abusing his inmates? No problem. Superman burst into the governor's mansion just as he had a year before to stop an execution, whisked the state leader to the prison, and let him see for himself the abuse. The superintendent wouldn't confess? Into the sweatbox he went, with the governor looking on and Superman vowing, "I'm going to plug the air-holes... You'll suffocate and die, just like your own victims did!" Superintendent: "No! No! Let me free! I swear I'll never torture the prisoners again!!"
By the time Waid wrote _Kingdom Come_ , more than half a century later, the Man of Tomorrow had found his voice after abandoning—then dramatically reclaiming—his mantle as a superhero. "Years ago, I let those I swore to protect drive me away. We all did," Superman—speaking for Batman, Wonder Woman, and the other crusaders—explained to United Nations leaders. "We saw you as gods," said the UN secretary-general. Superman: "As we saw ourselves. And we both were wrong. But I no longer care about the mistakes of yesterday. I care about coping with tomorrow... together. The problems we face still exist. We're not going to solve them for you.... We're going to solve them with you... not by ruling above you... but by living among you. We will no longer impose our power on humanity. We will earn your trust."
Noble words, but can a comic book character actually move readers to change their behavior and lift their horizons? He can, says Emilio Ramos, Jr., if that character represents not just an abstract notion of good but the good in each of us, the way Superman does. Ramos grew up in the "slums and ghettos" of Holyoke, Massachusetts, with few men to model himself after. He got his first Superman comic at age six and was hooked. "I've never had alcohol, never smoked, never done a drug in my life. People are surprised with that because of the environment I was exposed to growing up. People are even more surprised when they asked how I turned out the way I am and I just say one word: 'Superman,' " says Ramos, who is twenty-nine now and training for a career in law enforcement. "Superman definitely drew me into that field." The Man of Steel taught Peter Lupus how to defend himself and others. "Up until I was fourteen everybody in the neighborhood beat me up for practice," recalls the seventy-nine-year-old actor who was the muscleman on TV's _Mission: Impossible_ and later played Superman in commercials for the U.S. Army. "I started working out to overcome that bullying. Superman was my guy. I could equate with the Clark Kent–to–Superman transformation. I felt maybe I could go through life trying to help the underdog."
Superman changed Tom Maguire's life, too, especially his spiritual side. "I had what I considered to be too much religion in my life as a youngster. I found it confusing. Which was the true god, my grandmother's Armenian Orthodox god, my father's Irish Catholic god, my best friend Bobby Barwald's Jewish god, or my other friend Marty Pushkowicz's Catholic god? Was the true god the Muslim god, the Muslims that my Armenian grandmother escaped from at age 12 to come to America? It was all so confusing to me as a child growing up in the '60s. And then there was Superman, who provided an escape for me as a young reader. Superman didn't ask me to believe in a god," says Maguire, a fifty-six-year-old environmental regulator in Boston. "For 10 cents and later 12 cents an issue, Superman was an escape for me from those who asked me to believe in their god. Superman protected the oppressed and downtrodden and the poor, regardless of their religion or race. And my family was downtrodden at the time. I'm agnostic today, in part because of Superman."
AS WE EXPLORE SUPERMAN'S FAITH, it helps to consider what, if anything, he had in common with his vengeful crime-fighting compatriot Batman. What did Batman share with his archenemy the Joker? Was there anything other than superhero status that bound Superman and Batman to the Spirit, Green Lantern, Captain America, Spider-Man, the Incredible Hulk, the Fantastic Four, the X-Men, the Human Torch, and the Boy Commandos?
All were the products of fertile Jewish imaginations. The midwives to the biggest and boldest comic superheroes and supervillains were young men with names like Lieber, Eisner, Finkelstein, Kurtzberg, Katz, and the dynamic duo of Siegel and Shuster. These aspiring writers and artists went into comics for the same reason bright Jewish doctors in the early 1900s practiced at hospitals like Beth Israel and Mount Sinai: It was the only option open to them. Anti-Semitism barred Jews from advertising agencies, which were lily-white and mainly Protestant, while the lack of a college diploma kept many out of other lucrative careers in publishing. Desperate for an outlet to display their wares and pay their bills, they turned to the nascent comic book industry. Comic books were to the high-end magazines and newspapers of the day what the _shmatte_ , or rag trade, was to high-style clothiers, but that didn't bother these young writers, most of whose parents or grandparents had been in _shmattes_. The comics offered a toehold and a paycheck. Jewish mothers may not have bragged about their son the cartoonist, but it was a way to earn a living and, in a surprising number of cases, fame and fortune.
History equipped them for the task. Jews have been fine-tuning their storytelling skills since the days of Abraham, four thousand years ago. They did it first and best in the Torah, which is the first five books of the Hebrew Bible. Biblical commentaries were collected in the Talmud, rabbis' sermons were published in the Midrash, and mystics' tales found their way into the Zohar. No wonder Muhammad called them the People of the Book. While much of that writing was on esoteric matters of religious practice and legal dictates, it was told with such flair and essence that the books have remained in print for thousands of years, and telling stories still is an esteemed calling for Jews.
Whimsy was another critical attribute for the cartoonist and another rich vein in Judaism. Jews have been cracking jokes for centuries. It was a way to stay sane in the face of repressive Spanish inquisitors, marauding Cossack horsemen, and all the other faces of despotism. Oftentimes the humor was self-mocking, as in the saying that any Jewish holiday can be summed up this way: "They tried to kill us. We won. Let's eat." Sometimes it teased the holy one, Yahweh, the way Moses supposedly did on Mount Sinai: "Let me get this straight. We cut off the tips of our penises and You promise to take care of us until the end of time. You better put that in writing." By one count in the 1970s, more than 80 percent of America's best-paid comics were Jewish, even though Jews made up less than 3 percent of the population.
Telling a good story and making people laugh were good starting points for writers of comic books. But stories about superheroes required prototypes, and there were slim pickings for those in a world of Jewish milquetoasts. Yes, there was Samson, yet it was his very singularity that fueled our interest and his legend. The key, it turned out, was knowing where to look, and a good place to start was with the Golem. A he-man shaped from clay, this mythic character emerged repeatedly throughout history to safeguard Jews from aggressors. In some incarnations he was dim-witted and turned on his creator; at other times he was eloquent and loving. Always, he was big, powerful, and just the thing for a people in trouble. So was Siegmund Breitbart, a flesh-and-blood Polish-Jewish circus performer known to his landsmen as Zishe and to everyone else as the Strongest Man in the World. He could pound nails into wood with his fists and pull a wagon full of people with his teeth. He was becoming famous across America at the very moment that the Jewish creators of Superman, Batman, and other heroes-to-be were coming of age and looking for role models.
Will Eisner was one of those creators. He went to the mainly Jewish, all-boys DeWitt Clinton High School in the Bronx, where his classmates in the 1930s included two teenagers who would invent Batman and two more who would reshape Superman. Eisner was confident that comics could become his vocation based on his writing and drawing for the school newspaper, literary magazine, and yearbook. He also knew from an early age that organized Judaism meant nothing to him and that God was take it or leave it. He was right about the former: He dreamed up the quickly successful crime-busting comic strip _The Spirit_ , inspired generations of comic artists and writers, and, at the ripe age of sixty-one, wrote his first graphic novel and a landmark of the form. As for Judaism, its God and heroes became the centerpiece of his novels and he became a standard-bearer for the American Jewish experience. "I write about the things I know," Eisner explained late in life. "I know about Jews."
Eisner was not the only comic book creator who was ambivalent about his Jewish identity. His fellow writers and artists were routinely reshaping their proud Russian and German Jewish names into monosyllabic, ethnically vanilla ones. So Stanley Martin Lieber became Stan Lee, Max Finkelstein was now Carl Burgos, and Jacob Kurtzberg rebranded himself as Jack Kirby, an appellation that suggested roots in Ireland, not Austria. Green Lantern artist Gil Kane lost not just his Latvian name, Eli Katz, but his arced nose; both changes were conditions his then-girlfriend set for marrying him. Eisner, meanwhile, wrote his early stories under four pen names—Willis B. Rensie, W. Morgan Thomas, Erwin Willis, Wm. Erwin—none of which sounded remotely Jewish. His sensitivity to ethnic appellations, like his compatriots', grew out of boyhood pain. Will's younger brother was named Julian—which neighborhood bullies sounded out as "Jewleen." Will tried fighting for his brother's honor, then settled for convincing Julian to become "Pete," arguing, "That's a better name for around here!"
Their names may not have sounded Semitic, but their writing did. Jack Kirby and his collaborator Joe Simon, also Jewish, created Captain America, the first major comic book hero to declare war on Hitler and the Nazis. Marvel Comics kingpin Stan Lee brought in imaginary characters like Willie Lumpkin and Irving Forbush, whose names were as unmistakably Jewish as Lee's had been. More to the point, he brought to life stars like Spider-Man, whose working-class beginnings, childhood persecution and alienation, and purposeful life mirrored the experiences of Romanian-Jewish immigrants like Celia and Jack Lieber, Stan's parents. Lee, however, resists reading too much into his characters or his own actions. He chose the name "Lee" to preserve "Lieber" for the Pulitzer Prize–winning novel he knew was in him, he says. As for suggestions that his characters came out of his past, "I never consciously added any Jewish qualities or elements."
Conscious or not, those elements were there in the adventures of one superhero after another. Who better to empathize with the fight for the underdog than Jews who had grown up reading signs saying NO NIGGERS, NO JEWS, NO DOGS? And who better to join that battle than the heroes they spawned, such as Lee and Kirby's Mighty Thor and Carl Burgos's Human Torch? The Jewish writers were outsiders by birth. They were conflicted, with one foot in their parents' shtetl and another in their brave new universe of opportunity. They gave life and shape to heroes whose very names, from Batman to Captain America, reflected their creators' reach for the otherworldly and the all-American. Yet the themes and the characters they brought to life grew out of the very past they were trying so hard to escape.
JERRY SIEGEL SHARED THE Old World Jewish heritage of his comic book comrades and he grew up in the same American ethnic melting pot. His father and mother met and married in Kovno, in southern Lithuania, and that is where they had their first two children, Rose and Minnie. The family name then was Sigalowitz; Jerry's dad was Michel, his mom was Sore. Michel took a ship to America in 1900, planting stakes in New York with guidance from his wife's brother, who had arrived earlier. Sore came with the girls two years later. In between, the family name got shortened to Segal; no one knows why, but it didn't happen at Ellis Island, where complicated names often were butchered, and it wasn't to whitewash their religion, since Segal was almost as transparently Jewish as Sigalowitz. The alteration likely arose from a Yiddish-speaking immigrant trying to avoid spelling a long name in a strange tongue. Whatever the motivation, the effect, as her ship's manifest indicates, is that Sore Sigalowitz was met at the dock by her husband M. Segal.
The continuing evolution of their names offers a window into their continuing adjustment to life in a very different land, where they tried their best to fit in and nearly managed to. Sore picked a name that worked better for an American and a Jew—Sarah—and she stuck with it. Michel changed both his names with each new census: from Moses Sigel in 1910 to Michael Siegel in 1920 to Michael Sigel in 1930. Dual death records and dueling obituaries went back and forth between Michel and Michael (although all agreed on Siegel); some stories since then have called him Mitchell. During their brief time in New York, Michel and Sarah had either one or two more children, depending on which census they filled out accurately. Then they headed to the Midwest the way lots of Jewish families were doing, pulled by family or pushed by the German-born Jews who ran the New York community and disdained the poor migrants from Eastern Europe.
The Shusters had more stops and fewer name changes on their way to Cleveland. Joe's father, Julius, was from Rotterdam, in Holland, where Julius's father was a successful hotelier catering to immigrants and where he met his wife, Ida, who was from Russia. The family was doubly blessed: Julius and his brother met Ida and her sister and the outcome was two weddings. All four young newlyweds ended up emigrating to Toronto, where they shared a flat and split their salaries right down the middle. Julius and his family couldn't have made it otherwise, given his difficulty finding work. Later, when his brother and sister-in-law moved with their kids to Niagara Falls, Julius, Ida, and their three kids followed. In May 1924 Julius headed to Cleveland, where a job was waiting. Ida and the children followed in August.
New York and Toronto had been thriving Jewish communities that more than matched in spirit and numbers those the Siegels and Shusters left behind in Europe, but neither family knew what lay ahead in Cleveland. They need not have worried. Cleveland back then had 85,000 Jews, which was more than 10 percent of the city's population, and nearly half of them lived in the Glenville area, where the Siegels started out and the Shusters moved later. Gentiles were a minority in the neighborhood, which was 70 percent Jewish, and they were even rarer at the high school, which had the highest median IQ in the city. Most dads were small businessmen like Julius and Michel. Few moms were as active as Sarah, who volunteered with the Jewish Consumptive Relief Society, the Orthodox Orphan Home, the B'nai B'rith community service organization, the temple sisterhood, and the benevolent society. The Jewish Center, which was on its way to becoming the largest Conservative synagogue in America, had a basketball court and a huge swimming pool; neither Jerry nor Joe had much use for either. When the boys went outdoors it was Yiddish, not English, they heard from the ragman, the bread man, the iceman, and the fruit and vegetable men. When the boys said they didn't grow up very Jewish they meant they didn't go to synagogue much, but Glenville then was like Israel today: Just being there, breathing in the culture and street life, was living a Jewish life.
Just how fully they had absorbed that worldview became apparent in a comic book Jerry and Joe collaborated on in 1948. _Funnyman_ was about, as historians Thomas Andrae and Mel Gordon tell us, "America's first Jewish superhero." Comedian Larry Davis battled mobsters and goons not with super-strength but with wisecracks, wit, and comedic contraptions, the way Jews had been doing for centuries. No carefully sculpted body or small, straight nose on this champion. In costume he wore a Jimmy Durante schnozzola to go with baggy polka-dot pants and blue plaid tails; in street clothes he was drawn to look like the carrot-topped Jewish comedian Danny Kaye. No need for these stories to be scrubbed of their ethnic flavor, which was part of their appeal. One episode centered on a pain-in-the-rump character with the Yiddish handle Noodnik Nogoodnik, while another featured a medieval magician named Schmerlin. In this series the unlucky, awkward schlemiel was no longer the hidden half of our protagonist but the hero himself. It was not just Larry Davis who was showing his true ethnic self, but Jerry Siegel and Joe Shuster.
Jerry says he dreamed up _Funnyman_ when he was in the Army. Joe's sister says it was Joe's idea. The truth is that both had been hams at least since their boyhood days of imagining characters like Goober the Mighty. Jerry had gone so far as to co-write with Joe's brother a 134-page manual entitled _How to Be Funny: A Practical Course of Serious Study in Creative Humor_ and to cook up the Siegel-Shuster School of Humor. By the time they created Funnyman, Joe and Jerry had learned their lesson on giving up control of their character to a corporation and were determined not to do that again. The postwar era, they reasoned, was the right moment for the country to laugh again and forget about being heroic. They were sure they could rebottle the magic of Superman. Unfortunately, they weren't as funny as they thought. Without backing from deep-pocketed publishers like Jack and Harry, the new comic book lasted just eight months, and the syndicated strip stopped the next year. It would be Jerry and Joe's last collaboration, but it was their most joyous.
The Jewish inflections in Funnyman were also there in Superman, although finding them meant looking closer, the way a Torah scholar might. Superman was a refugee who had escaped to America from a world about to explode, just as the Shusters and Siegels did in fleeing Europe before the Holocaust. His parents rocketed him to Earth in hopes he would find a new beginning. He adhered to ethical guideposts as unbending as those of the _tzadik_ , or righteous man, in the Old Testament tradition. Clark Kent was Superman trying to assimilate. Superman was the real thing—as muscle-bound as Siegmund Breitbart, as indestructible as the Golem, and an inspiration to every Jewish schlump who knew there was a super-being inside him. Even kryptonite radiated with symbolism: It showed the influence his homeland still had over its Last Son, threatening to upend his life in the diaspora.
Joe and Jerry were living that immigrant experience alongside their hero. Joe had been hoisting weights for years, building up his body in hopes of snagging a girl. No worries if she wasn't Jewish, so long as she was tall and a dish. Jerry bought a big house, went to restaurants that saved him their best table, and adopted pen names more white-bread than Clark Kent's. Joe and Jerry wanted to fit in, to be all-American big shots. They also wanted it both ways. What they were running away from was not their Judaism but their nebbishness. They hoped people would see that they had been special before they became famous.
One thing that made them special during the war years was the belief that Hitler himself was after them, even if that wasn't entirely true. The German American Bund did send Joe hate mail in the years leading up to the war, and its members picketed outside Detective Comics' headquarters. The SS newspaper did write a story attacking Jerry and Superman after _Look_ ran the Superman story in which he defeated the Führer. That was where the facts ended and the media frenzy began. Each time the story was reported in a newspaper or book it got ratcheted up: Now propaganda chief Joseph Goebbels was angrily attacking Jerry in the middle of a Reichstag meeting, and Adolf Hitler was threatening to exterminate him. Hitler and Goebbels didn't, but their ally Benito Mussolini did ban from Italy Superman, along with every American comic book with the exception of Mickey Mouse.
Those years on either side of World War II were ones in which Jerry and Joe evolved alongside their cartoon characters. Their confidence grew as they were celebrated for their work and sought out by other artists. That made it easier for them to experiment with formats like humor, which had always been there in Superman and took center stage in Funnyman. They did the same with their Jewishness, which became more pronounced when their hero was a comedian instead of a strongman. They had only been willing to reveal so much about Superman's ethnicity, but it was more than anyone else had done with an all-American superhero.
HARRY DONENFELD AND JACK LIEBOWITZ affirmed their Jewish roots in ways that made clear they were children of the shtetl. They believed you should never deny your heritage, and they kept intact names as manifestly Jewish as Liebowitz and Donenfeld. You gave to Jewish causes (both did, generously) and favored Jewish entertainers (Gussie Donenfeld said what she liked most about clarinetist Benny Goodman was "that he never changed his name"). You hired fellow Jews as often as possible, and with the exception of Italian American artists, the roster at Detective Comics and on Superman especially was mainly Jewish. It started with Siegel and Shuster and continued with editors, writers, and illustrators with surnames like Weisinger, Schiff, Joffe, Binder, Dorfman, and a pair of Schwartzes.
There was one more guidepost for children of the ghetto: You didn't let ethnic identity get in the way of making money. That was why Harry overruled himself and opted to call his company not Donenfeld Comics but Detective Comics. "Donenfeld," his daughter Peachy remembers, "sounded too Jewish to him." He and Jack almost certainly would have intervened in the same way if they had felt that Superman, their number one moneymaker, was stepping too far into their Yiddishkeit world.
He never did. Superman's handlers from the beginning planted clues that Superman was Jewish, but that evidence was subtle and ambiguous enough that it convinced many readers that Superman was seriously Christian. Giving him a backstory straight out of the Bible also inoculated Superman against claims of being a false prophet, and those claims would have come, whether from priests, preachers, rabbis, monks, or imams. Presenting him as a moral man in a world of temptation made him compelling to people in search of ethical as well as religious direction. In the end, his appeal was his universality along with his particularity, which ensured his stories would live on the way most parables do. Here was an exemplar, one of the few, who was embraced with equal ardor by Jews and gentiles, believers and agnostics, and anyone else in search of a hero.
# **CHAPTER 4**
# **The Speed of Sound**
IT WAS HOW AMERICANS SPENT their evenings in the era before TV—chairs and sofas pulled around the two-foot-high Philco radio console, the brown Bakelite dial carefully tuned to _Fibber McGee and Molly, Ellery Queen, Amos 'n' Andy_ , or, on sixteen history-making nights in June and July of 1946, _Adventures of Superman_. Listeners came ready to leap out of their corn-fed lives and into their superhero's fantastic one—his slugfests with atom men and mind games with leopard women—which was what had made the Man of Steel such a smash when he debuted on the airwaves two years before the war. What they heard instead during that first summer of peace was a tale of real-life, home-grown fiends who masked their ashen faces with white sheets, twisted their followers' minds with Nazi-like schemes of racial cleansing, and defied Superman or anyone else to try and stop them.
The series was called "Clan of the Fiery Cross" and it was not an easy story to tell. Not then, when professional baseball, public bathrooms, and even the Army and Navy still were divided into white and colored realms. It would be another year before ex–Negro Leaguer Jackie Robinson toppled the color bar when he joined the Brooklyn Dodgers, and another eight before the Supreme Court declared that racially separate schools could never be equal. Jews, Asians, and Roman Catholics still saw signs saying they need not apply. The Ku Klux Klan wasn't as powerful as it once had been, but it didn't have to be. It already had planted doubts about anyone who looked or prayed differently; those who didn't heed its warnings could always be reminded with a flaming cross or lynching noose.
Robert Maxwell didn't care. He detested the Klan and had been given the keys to the Superman radio kingdom by Jack and Harry. The wordsmith turned pitchman turned radio producer knew he had to get to the kids before the haters did. He hired one of America's most trusted education experts to tell him how. They gathered all the intelligence they could on the Klan's passwords and rituals, its ways of corrupting politicians and its means of wrapping itself in the flag. They consulted psychologists, psychiatrists, and propaganda specialists. They tested their approach with five weeks of broadcasts railing against a fictitious organization of anti-Semitic "hate mongers." Now was the time to ratchet up the moralizing and zero in on a real-life hate group. They even had a name for their bold enterprise: "Operation Tolerance." Their secret weapon—the surest way to win over the children and take down the xenophobes—was to sic on them, at the speed of a radio wave, America's most trusted and ferocious do-gooder.
"Clan of the Fiery Cross" ran for sixteen episodes of fifteen minutes each, built around a straightforward storyline. Tommy Lee, who was Chinese, rose to become the star pitcher on his youth baseball team, beating out a hot-headed white hurler named Chuck Riggs. Riggs took his beef to his Uncle Mac, who was secretly the grand scorpion of the Clan. The white-hooded Clansmen terrorized first Tommy and his family, then Jimmy Olsen and Perry White, whose newspaper had taken up Tommy's cause. Superman stepped in just as Mac and his crew were about to finish off Olsen and White.
Every chapter ended with a cliff hanger and most featured dueling sermons from the grand scorpion and Clark Kent. "We're a great society pledged to purify America—American for 100 percent Americans only. One race, one religion, one color," Mac told his nephew. "Are we going to stand idly by and see these scum weasel their way into our neighborhoods and our jobs.... We'll strike back, and the time is now, so get set for action. The fiery cross burns tonight." Not so fast, Kent shot back: "Intolerance is a filthy weed, Jim. I told you before—the only way you can get rid of it is by hunting out the roots and pulling them out of the ground."
Superman's political evolution on the airwaves was the reverse of what had happened in the comics. There, Jerry and Joe molded their avatar into an agitator only to have editors in New York reshape him into something tamer and less likely to offend. Robert Maxwell sprang from the same Eastern European Jewish roots as the boys from Cleveland and he was at least as idealistic, but he was older and wilier, and in those years he had almost complete editorial freedom. His radio Superman carefully picked his enemies: Nazi saboteurs, jewel thieves, witch doctors, and others unlikely to generate sympathy or controversy. He built his audience of kids, stay-at-home moms, and dads who got back home from the office or factory in time to catch the early evening broadcast. It was a full six years into his show when he finally turned Superman loose on the Klan. Even then, Maxwell's venom was directed not at the political corruption or corporate villainy that riled up Jerry and Joe, but rather at narrow-mindedness. The distinction was critical. The new focus might alienate listeners who identified with Mac Riggs, most notably the flesh-and-blood Klansmen who at that very moment were trying to recruit kids in New Jersey, just across the Hudson River from Superman's Manhattan studio. That only helped the cause. Maxwell used the threatening letter he got two days after the series started to stir up publicity. More to the point for his bosses, "Clan of the Fiery Cross" posed little risk of upsetting the advertisers who paid the bills, especially since the focus was prejudice against Asians rather than the more culturally condoned bias against blacks.
Kellogg's, the primary sponsor, was over the moon. Operation Tolerance had given _Superman_ a bump in the ratings. With an audience of 4.5 million listeners it was the number one children's program in America, leaving in its wake old standbys like _Captain Midnight_ and _Hop Harrigan_. The _Superman_ shows also were a boon for Pep Whole Wheat Flakes, the breakfast cereal ballyhooed by the narrator at the beginning, middle, and end of each episode. "Tolerance is rampant in Battle Creek," Maxwell gloated to _Newsweek_ after the airing of the hate mongers series. "Every bit of pep in Rice Krispies is tolerant." The magazine added its own hurrah: "Superman is the first children's program to develop a social consciousness."
The angle that gripped _The New Republic_ was where Superman got his dope on the Klan. Newspaper reporter Stetson Kennedy had gone undercover in the hate group, the liberal journal reported, passing its "code words" to the Anti-Defamation League, which forwarded them to Maxwell. "As a result, Samuel Green, Grand Dragon of the KKK, had to spend part of his afternoon with his ear pressed against the radio. As soon as Superman used a KKK password, Green had to send out urgent orders for a new one. The Grand Dragon is said to have taken this reverse very badly." Kennedy picked up the story in his own book, saying he gave _Superman_ 's producers "the Klan's current password, and promised to keep them informed every time it was changed." The scheme worked so well that kids invented a game they called "Superman Against the Klan," rattling off secret passwords "a mile a minute!" Thanks to his work and the Man of Steel's, Kennedy concluded, "I knew that the millions of kids who had listened to Superman were not likely to grow up to be Klansmen."
It was a seductive backstory, and much of it was true. Kennedy did provide invaluable intelligence on the Klan, although he embellished what he had done and blended his narrative with those of others. Did he pass secret passwords on to Superman? None that he cited were actually broadcast, and the only thing that came close to a code in all sixteen episodes of "Clan" is when "the robed figures solemnly placed their right hands over their hearts, crossing the first two fingers of their left hands," and muttered an "anti-democratic oath." Journalists and authors were so taken with Kennedy's version that no one fact-checked it against the radio script. But compelling though they were, the media accounts missed the point. The wizard hiding behind the studio curtain was not Stetson Kennedy, it was Robert Maxwell.
Siegel and Shuster's comic books and strips already had made Superman a hero on every playground across America; Maxwell's broadcasts made him one in boardrooms, too. The Veterans of Foreign Wars and American Veterans Committee awarded him special citations for leading the battle against bigots. The Mutual Broadcasting System said it was "prideful" to be Superman's station. Sharing that glow were the National Conference of Christians and Jews, the American Newspaper Guild, and the Calvin Newspaper Service, most of whose readers were black. (Apparently none of his progressive boosters minded that a dining-car porter speaking dialect was about the only black face in Superman stories, or that the most prominent mention of Asians was the reminder to wartime readers to "slap a Jap.") In New England, radio stations banded together to get permission to start their broadcasts fifteen minutes late, ensuring that young fans wouldn't miss their hero while their parents listened to coverage of that summer's pennant run by Ted Williams's Boston Red Sox. "We had been getting a lot of complaints about the blood and thunder stuff until we decided to put in these social episodes," said a spokesman for the advertising agency used by Kellogg's, which recognized early on the dividends that could be earned from a crusade for tolerance. "Now all the parents' organizations are congratulating us on the show. The psychologists tell us we're planting a 'thought egg' in the kids' minds."
Whether or not that egg actually hatched, it was clear that Superman and his handlers had staged another triumph. They had inoculated themselves against parents, teachers, and even psychoanalysts who worried about the impact of action heroes on young minds. Thirty-five million American homes had radios in 1946, and Superman was beaming into more of them than ever, entertaining entire families rather than just the kids who read comic books. At a time when nearly all the wartime superheroes were fighting for their lives, the Man of Steel was thriving. To Superman, Inc., crisis meant opportunity, just as it had during the war.
This latest victory was particularly sweet for an old socialist like Jack Liebowitz, who approved Maxwell's hiring and green-lighted Operation Tolerance. Jack had consciously assembled a team of artist-entrepreneurs who were youthful as well as inventive, with the audacity to presume they were shaping not just a fictional character but popular fiction itself. Jack understood that Superman's success in radio helped ensure a growing market for his comics, and vice versa. He also understood that, as he had always said, if you were smart you could do good at the same time you were getting rich.
MAKING SUPERMAN BELIEVABLE ON PAPER was a relative cinch. Good eyes were all that was needed to see him leap and fly, defy bullets along with alien invaders, and metamorphose from buttoned-down Clark Kent in a double-breasted suit to a soaring superhero in a pajama-like uniform. Radio was different. There were no alleyways in which we could witness his makeover, no costumes or hairstyles to show us there had been a switch. Listeners had to visualize for themselves what he looked and acted like to make it work. The fact that it did—from the very first broadcast on February 12, 1940—was a tribute to the skills of the actors and producers and, at least as much, to the supple imaginations of young fans who wanted to believe.
The show's first challenge was finding one performer who could play Superman and another for Clark Kent. The solution was Clayton "Bud" Collyer. The choice seemed obvious to everyone but him. Collyer had trained to be a lawyer, like his dad, but he paid his way through law school by singing and acting on the radio, following in the show business footsteps of his mother and grandfather, sister and brother. After two years as a low-paid law clerk he was back performing—using his mother's maiden name of Collyer so he could preserve his birth name, Clayton J. Heermance, Jr., in case he ever resumed his legal career. By the time _Superman_ was ready to air, Collyer was starring in two radio adventure series— _Renfrew of the Mounted_ and _Terry and the Pirates_ —along with a comedy, several soap operas, and three news features. That was more than enough. The idea of a comic strip on the radio was such a stretch that he made clear he didn't even want to audition. Maxwell tricked him into doing that, twice, but still Collyer tried to get out of it.
That he didn't was a stroke of luck for both Superman and Collyer, since they made a brilliant match for 2,008 radio shows and for thirty years in various media. Collyer drew on his training as a crooner to underscore the difference between Clark and Superman, playing the former in a tenor that oozed milquetoast, then dropping several pitches midsentence to a gravelly baritone that was just right for the world's strongest man, yet making clear that both voices came from the same man. That preserved the essential ego/alter-ego relationship and saved Maxwell from having to hire a second actor. Being the first to impersonate either character meant the only standard Collyer had to meet was the one he was setting. Performing on the radio, where no one could see him, meant he never ran the risk of growing too old for the role. It also reduced the possibility of his being typecast, which was a fear (and reality) that would later plague TV and film actors playing Superman. Even though they couldn't see what he looked like and there were no credits naming him, his voice was rousing enough for female listeners to flood the studio with mash notes addressed to Superman. They might have been disappointed to know that he had a wife and three children and that he taught Sunday school on Long Island. Portraying Superman "was the ultimate in unabashed corn," said Collyer, who would later become known to a generation of baby boomers as host of the TV game show _To Tell the Truth_. "So many people get the least bit embarrassed by fantasy when they're directing it or performing it and it loses all the great charm it could have, but if played honestly and whole-hog all the way, it's great."
Joan Alexander was Collyer's co-star and opposite. She needed the work playing Lois Lane at first to support herself and later to provide for her daughter after she left a troubled marriage. She got the role early on, lost it when Maxwell decided he didn't like her, then disguised herself in a wig and showed up at the audition for her replacement. "The producers hired her!" her daughter recalled sixty years later. "They were astonished to find out they had rehired the woman they'd just let go. This time she kept the part forever."
Jackie Kelk had nailed down the role of Jimmy Olsen by season two. Like Alexander, he needed the part, but like Collyer, he had to fit it around other acting jobs. The solution: Kelk's Jimmy was written into the action four days a week, while on day five—Thursdays, when Kelk was rehearsing with _The Aldrich Family_ —Jimmy was AWOL and a character named Beanie Martin took over as copyboy.
As critical as those and other main characters were, the narrator was more so, especially after Jackson Beck took over the job. He saved Maxwell from needing to have his actors self-consciously stop the action to explain what they were doing or why. Beck set the scene and caught up listeners who had taken a bathroom break or sneaked into the kitchen for a snack. In between his narrations he played as many as four other roles in a single episode, from villains to Beanie the copyboy, which was even more impressive since most of the broadcasts were live. He also was an accomplished huckster, working with his sidekick, Dan McCullough, to weave into high-energy plots messages about Pep cereal, "the eighteen-karat breakfast dish that sparkles with sunny cheerfulness." For Kellogg's, those pitches were what the show was all about, and why they kept Beck around for some 1,600 broadcasts.
The maestro who assembled that extraordinary cast and launched Superman onto the airwaves was Robert Maxwell, the orchestrator of Operation Tolerance. Who better to reconfigure Superman for a new medium and to refresh radio by introducing its first superhero than a thirty-two-year-old artist-entrepreneur who had reinvented himself? Born Robert Maxwell Joffe, this oldest child of Russian-Jewish immigrants had taken the pen name Bob Maxwell to protect himself and his family when, in his early twenties, he wrote stories with titles like "He Had Push" for Harry Donenfeld's bawdy and bloody pulp magazines. The brashness of the tales and their author caught Harry's eye and he drafted Maxwell into Superman, Inc., first to oversee the licensing of toys and other products, then to bring the superhero into the world of broadcast. Before he hired writers or actors, Maxwell sat down with Harry and Jack's masterful press agent, Allen Ducovny, to put together audition discs that would sell the show to prospective sponsors. They couldn't have picked a better moment. Few families had television sets in the early 1940s and almost none had the money, gasoline, or motivation to go out. Radio was the era's hottest medium, with its comedies, harmonies, and mysteries helping to take people's minds off the lingering agony of the Depression.
A new show required another origin story. It had to be familiar enough to loyal comic book and strip readers that they wouldn't see it as tinkering with the legend, and it couldn't presume any preexisting knowledge of Superman lore, since part of the point of a new medium was to attract new fans. So whereas _Action_ No. 1 described Superman's home planet as distant and destroyed by old age and _Superman_ No. 1 simply called it doomed, the inaugural radio show brought the action closer to home and gave more telling details. Krypton was in our own solar system—hidden from us by the sun—and it was that sun's gravitational pull that overpowered the planet and made it "explode like a giant bubble, destroying every living thing on it!" Superman grew up on his way to Earth and by the time he stepped out of his spaceship, in episode two, he was ready to save his adopted planet.
Superman's war against the Nazis also looked different on the radio. In the comics, he stayed in his civvies and fought on the home front. On the airwaves he was commissioned as an undercover Secret Service operative. He still lived by clearly delineated rules, just as he had in print, including doing all he could to spare the lives of his enemies. And he came into America's living rooms with an opening sequence that would become the signature of Superman on the radio and later on TV, even though it has been altered over time and the first paragraph was borrowed from the animated cartoons. So familiar was the refrain that children across the forty-eight states could recite it as readily as the Pledge of Allegiance:
"Faster than a speeding bullet," the narrator intoned. "More powerful than a locomotive. Able to leap tall buildings in a single bound!"
Man: "Look! Up in the Sky!"
Second man: "It's a Bird!"
Woman: "It's a Plane!"
First man: "It's Superman!"
Narrator: "Yes, it's Superman! Strange visitor from another world, who came to Earth with powers and abilities far beyond those of mortal men. Superman, who can change the course of mighty rivers, bend steel in his bare hands! And who, disguised as Clark Kent, mild-mannered reporter for a great metropolitan newspaper, fights a never-ending battle for truth, justice and the American way!"
The last four words, which were added in the summer of 1942 and became part of Superman's motto, were chosen with the help of a child psychologist to ensure they touched the right chords. Superman, of course, had always fought for patriotic principles, but it was only with the nation at war—and Americans thinking more than ever about why their country was worth fighting and dying for—that the idea of a distinctly American way of believing and acting took hold in the public mind and the Superman mythos. Yet again, Superman was reflecting and refracting his era in a way that helped define it.
But words alone wouldn't do. Listeners needed to visualize the action. So as the narrator talked about a speeding bullet, the radio audience heard a burst of machine-gun fire. Locomotive? Let's hear the roar of a passing train. The biggest challenge for the three sound effects men was Superman taking flight. At first a hand-cranked wind machine had to suffice, but the artifice grew more convincing with the addition of several recorded sounds: a wind tunnel playing in reverse, a plane diving with a deafening roar, and a newsreel of an artillery shell whizzing through air during the Spanish Civil War. For Superman's landing, the sound guys slowed by hand those recordings, the way disc jockeys do today. When the record stopped, listeners were assured that Superman was back on solid ground.
A radio writer's mission was different than Jerry Siegel's had been in the comics. Jerry aimed strictly at kids. Radio writers started with young people as the target, but their scripts also had to appeal to grown-ups, who made up more than a third of the listeners. The best way to reach both, advised the show's first director, was to assume the best in each. "Kids can detect the patronizing tone of an adult who tries to reach down to their mental age, and they resent it," said Jack Johnstone. "You've got to be perfectly natural." Another challenge: How to keep the pot boiling when everyone knows that nothing can hurt your hero. "A railroad train runs across his chest. It doesn't hurt him, it hurts the train! Where do you get the suspense from?" asked scriptwriter Edward Langley. "I was a young writer in my twenties. So I asked a guy I knew in his fifties.... He said: 'The one thing Superman can't do is strike a match on a cake of soap!' That was the kind of 'peg' that you used—to try to use what he can't do.... They were wide open to anything, as long as you could make it suspenseful and interesting. Topics didn't make a damn bit of difference."
That challenge got a bit easier in the spring of 1943, when Superman finally got a worthy adversary. It happened while Clark Kent was interviewing Dr. John Whistler at the Metropolis Museum. The scientist showed the journalist an unusual green meteorite that made Kent suddenly feel "as if every ounce of strength had been drained out of me." The narrator explained the game-changing implications: "Superman for the first time in his life faces an enemy against which he is entirely powerless." That enemy, a radioactive fragment of the planet Krypton, prompted Superman to recall his birthplace, his parents, and how and why he had been sent to Earth. It was the first the world heard of kryptonite, although Jerry Siegel's twenty-six-page story on K-Metal was sitting on the library shelf at Detective Comics and could have been read by the radio scriptwriter. It would take another six years for the deadly metal to make its way into the comics and another two before a nation that was still at war, and understandably nervous about the mention of anything involving radiation, would hear much more on the radio about the glowing green element.
When kryptonite returned to the airwaves it became the centerpiece of an epic battle between Superman and the Atom Man. At seventy-seven episodes, this struggle was the longest in all of the Superman radio series. The face-off began in September 1945, less than two months after America dropped its "Little Boy" nuclear bomb on Hiroshima and the world was thrust into the Atomic Age. Perfect timing. The final story aired the following January, two days before the inaugural meeting of the United Nations. In between, the villainous Scarlet Widow stole from the Metropolis Museum the single known sample of kryptonite, only to see a Nazi scientist named Der Teufel (German for "the Devil") steal a chunk of it. Determined to "succeed where Hitler failed," Teufel fed his kryptonite to Heinrich Milch (a.k.a. Henry Miller), turning him into Atom Man, whose radioactive powers were too much even for Superman. Miller seemed unstoppable, as the narrator warned: "Superman is pitting all his strength and speed against the one force on earth which is mightier than he is—the force which twice brought him within the very shadow of death. Can he possibly win this time, when he fights for his very life, and for the lives of those he loves? Monday brings the smashing, dramatic climax of our story, fellows and girls—and a startling surprise, so don't miss it." When Superman triumphed the next evening, he told a grateful Metropolis police inspector, "You don't owe me anything. I'm fighting for the same things you are—the end of tyranny and intolerance—all the things that Miller and the Nazis stood for." Inspector: "Then I'll only say thank heaven that the worst threat America ever faced is over."
Ahh, but it wasn't over. Two more pieces of kryptonite were somewhere in the city, and to track them down, Superman had to enlist the help of Batman and Robin. That meant confessing to them his secret identity, the first time he had ever done that. Showing his full self would pave the way for Superman to forge with Batman his first true friendship, one based on an honesty he couldn't afford with anyone but his foster parents and the sharing that was possible only with a fellow superhero, who understood the anxiety and exhilaration that came with the job. So effective was this radio collaboration in defeating the Atom Man that the Dynamic Duo was back again for more than a dozen other guest appearances. Being able to rely on them as stand-ins proved particularly useful when Superman was taken out of the action by kryptonite—and when Bud Collyer wanted to go on vacation.
It was on the radio, too, that Superman learned to fly. He was aloft by the second episode in 1940, which was three and a half years earlier than his flying debut in the "Million-Dollar Marathon" story in _Action_ , although in the early years of broadcast his power of flight waxed and waned from adventure to adventure. Radio's immediacy and Maxwell's brashness once again let Superman dive into a topic that comics writers would only slowly tease out. Kryptonite seemed like a good idea, so give it a try. Same with turning a high jump into full flight. What didn't change was Superman's boyish delight each time he tested his aerodynamic abilities, as he did in May 1943 when he was battling the Ku Klux Klan. "Up in my arms with you, Chuck," Superman said to the young ballplayer Chuck Riggs. "Are we going to fly?" Riggs asked. Superman: "We are. Hang on now." Riggs: "Oh boy—flying with Superman, I must be dreaming." Superman: "Here we go. Up, up, and away! Is that your house down there, Chuck?"
While some of the radio firsts involved expanding and curtailing his powers, others filled in what was quickly becoming the best supporting cast a superhero ever had. An intrepid copyboy showed up at Clark and Lois's newspaper as early as _Action Comics_ No. 6 in November 1938. He was given a bow tie but not a name or an ongoing role; he would make three more cameo appearances through the end of 1939. Jimmy Olsen was introduced to the world in April 1940, on a radio episode called "Donelli's Protection Racket." He was a red-headed, freckle-faced boy who worked at the paper. His mother had run a candy store since his dad died three years before, and a local mobster named Donelli was trying to extract money from her. Jimmy turned to Clark and Superman for help. Over the next several episodes we learned more about Jimmy: He was a Boy Scout but couldn't find his way out of the woods; he perpetually annoyed his boss by calling him "chief" even as he asked to be promoted to real reporter; he reacted to everything good and bad by gasping "Holy smokes"; and he had a knack for getting into trouble and counting on Superman to bail him out.
Perry White, too, came alive on the radio, fencing with Jimmy from the start and eventually making his way into the comics. The editor was first identified as Paris White, but his first name evolved to one more suited to his gruff demeanor even as his newspaper was changing from the _Daily Flash_ to the _Daily Planet_. Inspector Henderson—first called Charles, then William—was Clark's best source and Superman's closest ally at the Metropolis Police Department. He was made for the radio and was intended to reassure parents that Superman was a friend of the police and not the vigilante he started out as. It took seven episodes on the airwaves for Lois to appear, which was a long time compared to her debut in _Action_ No. 1, but she was the only adult female character on any afternoon action-adventure radio show.
There were actually several Superman radio series, not just one. They ranged from fifteen to thirty minutes, and aired three to five times a week. The shows were meant to be kid-friendly, although for thirteen weeks in 1949 a crime version aimed at adults ran on Saturday evenings. The Mutual Broadcasting System aired the program for most of its run and Kellogg's was by far the longest-lasting sponsor. The end came in March 1951, which was sooner than most fans wanted but more than a year later than for competing shows like _Captain Midnight_ and _Tom Mix_. Because tapes of the old shows became available only recently, the radio _Adventures of Superman_ hasn't gotten the attention from historians and fans that his exploits on TV and film have. But it was radio that lifted the hero from a devoted audience of comic book fans into a broadcast universe that reached nearly every corner of the nation. The radio series came onto the scene two months into 1940 and it wound down fifteen months after the decade did. In between it joined Kilroy and the Slinky, _Citizen Kane_ and Rosie the Riveter, as hallmarks of 1940s America.
HIS SUCCESS IN OTHER MEDIA made it inevitable that Superman would find his way to the big screen. It was equally certain, as the 1930s came to a close, that America someday would produce an animated cartoon star who was not a funny, furry animal. The two trends collided in the Miami studios of a pair of Austrian-Jewish animation geniuses, the brothers Max and Dave Fleischer.
It wasn't the Fleischers' idea. They thought building a "realistic" cartoon around Superman was a lousy idea given the problems they had encountered the only other time they tried to do that, with _Gulliver's Travels_. What they did best was conspicuously unreal characters, like Betty Boop, Popeye, and Koko the Clown. So when they were approached by Paramount Pictures, which owned the screen rights to Superman, Max and Dave had to think about it. They told Paramount that they could produce the ten-minute movies, which theaters craved as lead-ins to feature films—but given the unusual animation requirements and special effects they would have to charge $100,000 per episode, or four times the going rate. "They thought that would be the end of the project—but it wasn't," said Richard Fleischer, Max's son. "Paramount said: 'Okay, go ahead.' "
The brothers were right: It wasn't easy. Special lights were needed to extend shadows and depths and create the right dramatic touch. They tried oblique angles, freeze-framing, double exposures, and other camera techniques that heretofore had been the domain of live-action films and the high-priced Walt Disney Studios. Rotoscoping—a technique the Fleischers had pioneered with Koko in which real-life figures were traced in ink, frame by frame, much as comic book artists sometimes did—made the animated figures believable. Max and Dave's composers knew what Superman, Lois, and the others should look like, thanks to model sheets provided by Joe Shuster. Their voices came from the world of radio, with Bud Collyer playing Superman and Joan Alexander reprising the role of Lois. Composer Sammy Timberg supplied the theme music. The plot lines—thieving robots, rampaging dinosaurs, and jingoistic Japanese saboteurs—were familiar to fans who knew Superman from comic books, comic strips, and the airwaves. So was the sound of an exploding Krypton, which was generated by amplifying the sound of an apple being ripped apart by hand. The films borrowed the expressionism of Orson Welles's _Citizen Kane_ and the futurism of Fritz Lang's _Metropolis_ , blended with the Fleischers' unique feel for scale and vision. To put together a single ten-minute cartoon took a full six months, or twice as long as the normal Fleischer production.
Did it work? _Time_ magazine didn't think so, branding the Fleischer productions "the movie cartoon at its worst. Superman looks and acts like a wooden puppet. So do all his playmates." The _New York Times_ film critic Janet Maslin, writing forty years later, was equally dismissive: "The Fleischers show so little aptitude for—or interest in—realistic animation." Both were right in their own way. There was precious little dialogue and the characters seemed as stiff as Joe's drawings. That was their genius. The Fleischers managed to bring into two dimensions and full motion the same simple strength that Joe had captured on paper, and it was that rendering that led animation historians to deliver a judgment decidedly different from that of newsprint reviewers. "These films are among the best fantasy cartoons ever produced," said Leonard Maltin. "SUPERMAN stands as one of the Fleischer studio's finest achievements." The Academy of Motion Picture Arts and Sciences agreed, nominating the first of the Fleischers' seventeen Superman films for an Academy Award as the Best Short Subject (Cartoon) in 1941. While it lost out to a Disney feature, the nomination sent a message to skeptical critics.
Whatever the experts said, the verdict from fans was boisterous and unanimous. "Some 20,000,000 Supermaniacs can hardly wait for Superman's ten-minute, one-reel cartoon to appear once a month in more than 7,000 U.S. movie houses," _Time_ wrote in July 1942. "Supermania is the only word for their devotion to this irrepressible Citizen Fixit, who smacks death rays back into the cannon, restores toppling skyscrapers to their foundations, knits broken bridges together with his bare hands, and who has brought a new cry into the world: 'It's a bird! It's a plane! It's—SUPERMAN!' " While the films necessarily lacked suspense, with Superman always winning, "his idolators (of all ages) seem satisfied to see him flex his muscles. This vicarious satisfaction has made Superman Paramount's most popular and profitable short.... So popular is the muscular moron that 114 female artists at the Famous studio recently answered a questionnaire asking whether they would prefer Superman for a husband or a boy friend. All said: boy friend. Explained one: 'Trying to live with so super a husband might be awfully fatiguing.' "
Reactions like those earned the Superman cartoons an uncharacteristically prominent spot on theater marquees. But that was not enough to save the series or the Fleischer Studios. The Fleischers were deep in debt on other projects, and in mid-1942 Paramount took over their business and changed its name to Famous Studios. Budgets for the Superman cartoons were cut, quality suffered, and in 1943 Paramount killed the series.
That was not the end of the story. The Fleischer cartoons gave Superman some of his most famous catchphrases—from "faster than a speeding bullet" to "more powerful than a locomotive"—which made their way first to the ongoing radio series and later to TV. These short Superman films were so effective that they even turned up in the comic books: In _Superman_ No. 19, Clark feigned a choking attack and kicked Lois's purse to distract her from a Fleischer cartoon that would have revealed his identity as Superman. Twentieth Century Fox and animators at Terrytoons were paying attention, too, drawing on Superman's cartoon success to create their own super-strong character who could fly and wore a blue costume and red cape. The costume eventually changed to yellow and the new cartoon hero's name went from Super Mouse to Mighty Mouse. One bit of Superman mythology that is never attributed to the Fleischer Studios but should be is the use of a telephone booth as a dressing room. The first time Clark Kent ducked into a phone box to change into Superman was in November 1941 in "The Mechanical Monsters," the second of the animation films. It proved convenient enough that he did it again on the radio, in the newspaper strip and comic books, on Broadway, and, most famously, in the movies. Pulling it off was easy when the booth was the old, heavy wooden box with a small window up front, but privacy would be tougher to come by in the newer, all-glass version.
The Superman animation series' most lasting legacy was in showing that the Man of Steel could conquer yet another medium. He was quickly becoming ubiquitous, succeeding not just in the worlds of comic books and free radio entertainment but on the silver screen. That was a lesson Hollywood would remember.
THE MOVIE SERIALS WERE Hollywood's stepchild, representative not of what the filmmakers could accomplish in their heyday in the 1940s but of what they could get away with. A serial was a short subject that theaters showed alongside the featured movie, with a new chapter each week and a dozen or more chapters in all. The format was a carryover from serialized pulp fiction and a precursor to early TV, where the movie segments were rebroadcast at the rate of one a day. The storylines—westerns and science fiction, crime and espionage—were aimed squarely at youngsters, who never stopped relishing them, from the silent era in 1912 until TV made them obsolete in the early 1950s. The writing was thin, with little love, no sex, and the beginning of each twenty-minute episode wasted recounting the last one. Few came for the acting, either, which was slapdash, with escapes and chases routinely lifted from old films and producers hoping nobody would notice. The draw was the cliffhanger, in which the hero (often literally) hung over a cliff as the villain gloated and fans were reminded to return the following week to see whether what came next was a death (almost never) or a rescue (count on it). Even when they flopped, which was often, studios could dub and resell the shorts in France, Italy, Turkey, China, and, most obliging of all, Spain and Latin America, where episodes were stitched back together for a single five-hour viewing.
"Jungle Sam" Katzman was the king of the serials, for better and worse. The producer and director was Jack Liebowitz's kind of guy, a penny-pincher and autocrat who had never lost money on a film. He started not with a story or idea, but with a wild and colorful title like _Flame of Calcutta_. From that he built a narrative of intrigue set in eighteenth-century India. His biggest earners were the _Jungle Jim_ pictures, or at least they were until the Superman serials he made for Columbia Pictures in the late 1940s. To make sure the new films would be a hit with the adolescent fans who loved Superman in his other media incarnations, Katzman tested them on his fifteen-year-old son, Jerome, and his friends. "If they guess how the guy gets out of the predicament each week," Katzman said, "it goes out immediately and we rewrite until they can't guess." The other key, the producer knew, was a Superman in whom his son and millions of other kids could believe.
Kirk Alyn was an odd choice for the job. He was more a song-and-dance man than an actor, having studied ballet and performed in vaudeville and on Broadway in the 1930s and early forties. That's where he decided to trade in the name he was born with, John Feggo, Jr., for Kirk Alyn, which he felt was better suited to the stage. He appeared in chorus lines and in blackface, modeled for muscle magazines, and performed in TV murder mysteries in the days when only bars had TVs and only dead-end actors performed for the small screen. But he had experience in serials if not in superheroes, so when he got a call from Columbia in 1948 asking if he was interested in trying out for Superman, he jumped into his car and headed to the studio. Told to take off his shirt so the assembled executives could check out his build, the burly performer complied. Then Katzman instructed him to take off his pants. "I said, 'Wait a minute.' They said, 'We want to see if your legs are any good,' " he recalled forty years later. They were good enough, and fifteen minutes after he arrived, Alyn was hired as the first actor to play a Superman whom his fans could see as well as hear.
Alyn and his directors were smart enough not to try to reinvent the character Bud Collyer had introduced so effectively to the airwaves. "I visualized the guy I heard on the radio. That was a guy nothing could stop," Alyn said. "That's why I stood like this, with my chest out, and a look on my face saying, 'Shoot me.' " His demeanor said tough guy, but his wide eyes signaled approachability and mischievousness, just the way Jerry Siegel and Joe Shuster had imagined their Superman a decade before. Alyn understood much as Collyer had that kids like fifteen-year-old Jerome Katzman could spot a phony in an instant. If they didn't think Alyn was having fun—and that he believed in Superman—they wouldn't pay to see his movies. His young audience, after all, didn't just admire the Man of Steel. They loved him. Superman was not merely who they dreamed of becoming but who they were already, if only we could see. The good news for them was that Alyn was having fun, and he did believe in his character in a way that these preteens and teens appreciated even if movie reviewers wouldn't.
Columbia Pictures, too, had learned something from the _Superman_ radio broadcasts: It pays to let children hold on to their fantasies. Sam Katzman announced at his first press briefing that he had despaired of finding an actor capable of portraying the mighty Man of Steel but thankfully had persuaded Superman to appear as himself. The credit lines continued the ruse, billing Kirk Alyn as playing just Clark Kent.
Superman's comic book colors—blue and red—would show little contrast in a black-and-white film, so Alyn wore gray and brown. Being the first live-action Superman meant making a series of adjustments, some of them painful. When it became clear that no stuntman could convincingly stand in for him Alyn performed his own stunts, or so he claimed, including one where he intercepted an electric current the Spider Lady had intended for Lois Lane. No one counted on the sparks catching on the metal of his Superman belt buckle. "I was saved from incineration only by the insulation on my boot soles," he said later, "but it scared the blazes out of me." Producers also promulgated rules about superpowers like X-ray vision. There were two things it shouldn't penetrate: lead, whose X-ray blocking power proved to be Superman's sole defense against the deadly radiation emitted by kryptonite, and clothing, which was Lois's only defense against prying eyes.
Making Superman fly was a more vexing problem. The technical crew strung cables from the studio ceiling to pull Alyn aloft and molded a steel breastplate to hold him there, and for twelve long hours he was filmed dipping, banking, and, yes, flying the way men had dreamed of since Daedalus built wings for Icarus. The flaw this time was the wiring: It was so painfully visible that the crew was fired and Superman was grounded. Filmgoers saw Alyn poised on the window's edge, but what flew away from the building or any other setting was an animated Superman. He always landed behind a bush or wall, from which his human counterpart could dash out and resume the role. The effect was cartoonish.
Parts of the radio Man of Steel were reprised on film, with Alyn self-consciously announcing, "This is a job for Superman!" before each rescue and shouting, "Up, up, and away!" every time his cartoon double took off. That made sense on radio, when listeners needed cues; it seemed like a parody when everyone could see that Superman was on the job and airborne. Equally jarring to observant fans was noticing that Clark went through the identical motions every time he changed into Superman in the _Daily Planet_ storeroom, and that the rough airplane landing in the final episode looked an awful lot like one in the 1947 serial _Jack Armstrong_. Why were they using scavenged film? Surely it wasn't for lack of money: Columbia poured $350,000 into the filming, making _Superman_ one of the most expensive serials ever. In the end, the fifteen-part film that aired in 1948 looked like what it was: a B movie sliced into fifteen disjointed parts.
But kids whose Saturday at the movies was the highlight of their week ate it up. These were the same youngsters who, even before they could read the words, had thumbed through their Superman comic books until the pages grew ragged. In later years the reward for finishing their homework, or the inducement to get started, was the chance to listen to Superman on the radio. Saturdays had meant Superman cartoons at the movie house downtown, while on Sunday he showed up in regal color in the funny pages. Now there was a new treat: their hero, in live action, as part of the weekend matinee. Their parents dropped them at the theater thinking the attraction was Charles Dickens's penniless orphan Oliver Twist, but the real reason they wanted to come was _Hurled to Destruction_ , the Superman short that ran first. That explains not just why _Superman_ played in seven thousand movie houses nationwide but why it took in more than a million dollars, which was three times what Katzman had invested and enough to make it the most successful serial of the time.
In 1950, two years after the first set of shorts, Katzman and Columbia released a fifteen-set sequel called _Atom Man vs. Superman_. The title was borrowed from the radio series that introduced kryptonite, but almost everything else was different. The enemy this time wasn't Der Teufel, the Nazi scientist, but the comic books' Lex Luthor, who slipped in and out of prison and banished Superman to outer space using a secret ray that "breaks down your atoms and reassembles them wherever I desire." Even as the hairless villain was rewriting the laws of physics and inventing flying saucers, his henchmen inexplicably continued pulling off low-tech capers like holding up shoe stores and laundries. _Atom Man_ used a hybrid approach to flying that made it more convincing than in the first serials, though it still left a lot for the fans to wish for. Animated stand-ins were used again, but the transition from human to cartoon was smoothed by filming Alyn with his arms raised above his head, an electric fan blowing from above to simulate whooshing air, in front of blue staging that was supposed to look like sky. Other shots had him straddling an airplane and later a missile. The changes made _Atom Man_ better than its predecessors, but there was no denying that these short Superman movies did not have the taut drama of the radio broadcasts of the same name.
The Superman serials launched the careers of several actors, some of whom, like Alyn, came and went with the short films while others, like Noel Neill, would be back later. Neill, a twenty-seven-year-old "sweater girl," became known to adoring fans as the sweet Lois Lane and to detractors as the saccharine one. Like Superman, Lois had a uniform for the first fifteen shorts: a wide-brimmed white hat, a wool business suit, and wavy black hair bouncing off her shoulders as she walked. In _Atom Man_ her wardrobe expanded to three outfits and her hair was trimmed to well above the shoulder line. Katzman knew Neill from their earlier collaborations and thought she looked enough like the comic book Lois that he didn't require a tryout. The direction he gave her during filming was clipped and pointed: Play yourself.
In an era when studios carefully managed the lives of actors, Columbia was obsessed with keeping intact the illusion of Superman. Every aspect of the making of the serials was to be secret. Katzman banned outsiders from the set when scenes with Superman were being shot. He screened any personal appearances scheduled for Alyn, and told the actor that he "wasn't to appear on the studio lot wearing the 'uniform,' " which is the way his bosses insisted he refer to his costume. But the studio simply couldn't suppress Alyn's swaggering pride at the role he had been given to play. He loved it when Katzman would tease the maître d' at lunch by asking, "Do you know who this is?" then delightedly telling him, "This fellow is Superman!" When he was off the set Alyn refused to brush back his Superman spit curl, which clearly identified him with the character, and when he was on he gleefully told nonstop stories of derring-do. That identification came back to bite him: He was so widely viewed as an alien from outer space that it became difficult for him to get other roles. "Everyplace I'd go," he explained, "they'd say 'Hi ya, Superman!' " This would become a familiar complaint for future actors playing the role.
In the end the serials suffered the same fate as Superman on radio and in animation: they faded out as the spotlight moved elsewhere with changing American tastes and technologies. By 1948 America had four television networks, and in another three years their broadcasts would be beaming across the nation. TV didn't kill the movie business, as Hollywood had feared, but it did change the behavior of the American public, especially young fans like those Superman leaned on. The cabinet radio that had been the focus of family entertainment was replaced by a TV console, and kids who had flocked to Saturday matinees increasingly stayed home and watched for free.
The Man of Steel, though, was far too potent to fade away himself. Kirk Alyn, Noel Neill, and Sam Katzman had proved that their superhero could be wildly popular as live-action entertainment, and Jack and Harry already were lining up the talent and gadgetry to bring Superman into the era of the television.
BACK IN THE COMIC BOOKS, something curious had been happening to Superman: He was maturing and evolving. That had never been a consideration before, not when he went from infant to adult in a single page of _Action Comics_ No. 1 and it was uncertain whether he would last beyond a few issues. No one had given any thought to how or even whether he should continue aging. Joe and his assistants had drawn the character as if he would stay thirty-something forever, even as they went from being young artists to middle-aged ones (editors would later explain his slow aging with the contrivance that he was born on February 29, the leap day, so he added a year only once every four years). Just as important, no one had scoped out which parts of his past readers would want to explore and how his present world should be enlarged. Now, as Superman was looking ahead to his third decade of stardom—and Jerry and Joe were reluctantly relinquishing control over him—a new lineup of artists and writers had begun scoping and enlarging.
Krypton was one of the first elements to take on added dimensions. All we knew to start was that the planet had exploded and Superman had escaped. That seemed like enough, since what mattered was his life with us on Earth. The newspaper strips and the radio show had begun to fill in details about Superman's parents and their world, but the comic books didn't catch up until the summer of 1948, and the first full-length origin story was written not by Superman's creator but by Batman's. Bill Finger's tale opened with this teaser: "Who is Superman? Where did he come from? How did he obtain his miraculous powers? Millions keep asking these and many other questions." The next nine pages took readers back to "the great planet Krypton," populated by "humans of high intelligence and magnificent physical perfection." A handsome, tall Kryptonian scientist wearing a green costume and yellow cape was trying to convince the ruling council that their planet was doomed. The uranium in its core had been quietly churning for ages to the point where "Krypton is one gigantic atomic bomb!" There could be no more hair-raising words for readers, even young ones, who just three years before had lived through the staggering news of what happened to Hiroshima and Nagasaki thanks to the nuclear bombs that ended World War II. Kryptonians' only salvation, Jor-El insisted, would be to build huge rocket ships and flee to Earth, a world with a similar atmosphere and far less gravity. When no one would listen, Jor-El and Lara sent their only child off in a tiny spaceship.
That enlightened readers about Krypton, but Superman himself still didn't know his planet's history or its fate. It took another fifteen months for Finger to bring the hero back to Krypton for the first time since he was an infant and for comic-book-only fans to get their first look at kryptonite. Superman encountered a meteorite infused with the metal in a jewelry store on Earth and, alarmed at how it wilted him, he followed it back through space and time to its source. Arriving on Krypton just as the planet was about to explode, he had a quiet look around. (He "is invisible to these people because he is not of their time and doesn't exist for them," an editor's note explained. "He can only view them as he would a silent movie, but he can read lips.") Following Jor-El and Lara's baby as he rocketed to Earth, Superman did not realize he was following himself until he saw the infant rescued by the Kents, his foster parents.
Going home humanized the Man of Steel. He knew now why the meteor from Krypton had weakened him and why his parents had abandoned him. But that understanding brought with it a loneliness that would never leave. He realized now what Jerry Siegel had confided in us from the beginning: Superman was an orphan and an alien. His planet's sole survivor, he was the last of a long-lived and majestic race. It was not an easy burden to shoulder. "Now I understand," he thought to himself, "why I'm different from Earthmen! I'm not really from Earth at all—I'm from another planet—the planet Jor-El called Krypton!!"
The character named Jimmy Olsen was another instance of comic books catching up with radio as well as with a changing world. While we were introduced to Jimmy and his mother in an April 1940 broadcast, it would take another nineteen months for the comics to give him a personality and a starring role. It took just four panels for the boy, who looked to be ten, to let editor Perry White know his dream: "I—I'd like to become a real reporter—like Clark Kent. And if you'd only give me a chance." White's reply hinted at the repartee the two would continue for decades: "Tell you what I'll do, kid. Come back again in five or ten years.... And I may give you a break." Too impatient to wait, Jimmy stowed away in Lois's car as she chased a story about the villainous Archer, then he helped Lois skedaddle into the woods when the Archer took aim at her. The story ended with Jimmy getting his first byline in the newspaper. It would be four more months before he got a last name, twelve years before he earned a promotion to cub reporter, and seventeen years before he settled in with his trademark blazing red hair after experiments with blond, honey blond, and light red.
As with most of the borrowing back and forth between comics and other media, Jimmy stuck because he tapped a nerve. Being a kid, he could share in Lois and Clark's zany adventures without feeling any of the responsibility that weighed on adults like them. Jimmy was a foil for everyone around him—letting Superman repeatedly sweep to his rescue, Perry snarl at and then warm to him, and Lois display her suppressed mothering instincts. Every ten-year-old who flipped through a Superman comic book and tuned in to the radio show identified with him—which is why, however tired his shtick sounded, Jimmy has lasted through thousands of radio and TV broadcasts and through seventy years and counting of comics.
Perry White was everyone's grandfather or favorite uncle—hard-boiled on the outside but soft as a yolk once you peeled back the shell. And while he, too, got his name and personality on the radio, he quickly came to play a major part in Superman's expanding comic book universe. Whereas Jimmy started out with just a first name, Perry, as befit his age and irascibility, at first had just a last one. Six months later he was humanized with a given name, although it was one that more commonly is a surname. His writers were novices when it came to newspapers, so it shouldn't be surprising that they couldn't decide just what rank he held, moving between editor, editor-in-chief, managing editor, and editor-publisher. The one constant was his championing no-holds-barred journalism.
Perry's most versatile reporter and Superman's most cherished and tormenting friend was Lois Lane, who came onto the scene just five pages after Superman did. Readers could follow her infatuation with Superman, which in 1949 looked as if it might end in marriage, and her exasperation with Clark, which Superman exploited to get out of marrying Lois in the 1949 story with the implausible title "Lois Lane Loves Clark Kent." She lived in unit 1705 at the Ritz Plaza Apartments, which was near Clark and filled with pictures of Superman. At the _Daily Star_ and its successor, the _Daily Planet_ , she held almost every job there was, from sob sister and columnist for the lovelorn to war correspondent, weather editor, question-and-answer editor, and head of the lost and found department. The story she most wanted to write in the early years but could never pin down was "that Clark Kent and Superman are one and the same." We learned that she had a bottomless collection of fashionable hats, a weekly show on radio station WCOD, and a pistol in her purse that she used to defend herself when Superman wasn't there to save her.
What fans couldn't see but did speculate on endlessly was who was the inspiration for America's most famous lady journalist. Was it Margo Lane, girlfriend of one of Jerry Siegel and Joe Shuster's favorite pulp heroes, the Shadow? Perhaps it was Torchy Blaine, the fast-talking, crime-solving newspaper reporter who starred in a series of 1930s films and, in one, was played by actress and singer Lola Lane. That is what Jerry told relatives and friends over the years. Or was it Glenville High's Lois Long, Lois Ingram, Lois Peoples, Lois Donaldson, Bertha Lois Beller, or Lois Amster? Joe repeatedly singled out Amster, a class beauty and National Honor Society member whom he said he had a crush on. Jerry did, too, but he backtracked when that struck a nerve for his new wife, Jolan Kovacs, who had modeled for Joe when he was drawing Lois. A character named Amster also showed up in one of the first issues of Jerry and Joe's early collaboration "Doctor Occult." The real-life Lois Amster, reminiscing at age ninety-three, says she "never spoke to" Joe and Jerry, "never had anything to do with them. They weren't my type.... My type was more sophisticated than they were, more affluent than they were."
Easier to trace were Clark Kent's roots, at least the ones outside the comics. His first name came from Gable, the king of Hollywood and star of _Mutiny on the Bounty_ , and his last name was borrowed from a less well known film star of that era, Kent Taylor. His reporting style was based partly on the kind of journalist Jerry had once fancied becoming himself, when he doubted his comics would sell. A better model was Wilson Hirschfeld, the crusading reporter and managing editor of _The Plain Dealer_ in Cleveland and a high school classmate of Jerry and Joe's. The three had worked together on the _Torch_ , and dreamed up stories together on the front porch of Wilson's home. Hirschfeld, who died in 1974, alternately confirmed and denied that he was Clark, although a warm condolence card from Jerry after Wilson's death helped settle the case for the Hirschfeld family.
Clark's parentage inside the comics remained more ambiguous through the 1930s and 1940s. The couple who found and raised him at first were simply called "the Kents." Ten years on, in a comic book published in the winter of 1948, his foster parents got first names: John and Mary, although those names appeared not in the text but on the Kents' gravestones. A year later Pa Kent would inexplicably become Silas, and it would take until the 1950s for the couple to settle in as Martha and Jonathan. From the first the Kents were big-hearted. They saved the baby boy who rocketed into their world, adopted and raised him as their own, and imbued him with a mission. "No man on Earth has the amazing powers you have. You can use them to become a powerful force for good!" John admonished from his deathbed. "There are evil men in this world... criminals and outlaws who prey on decent folk! You must fight them... in cooperation with the law! To fight those criminals best, you must hide your true identity! They must never know Clark Kent is a... super-man! Remember, because that's what you are... a Superman!"
Inspired though that message was, it was not original. Nearly identical passages had appeared six years earlier in yet another medium, the novel. It was written by George Lowther, a scriptwriter for the Superman radio show, and had the same title as that show: _The Adventures of Superman_. It was the first full-fledged book ever centered on a comic character, and Lowther was the first writer other than Jerry Siegel to get credit for writing a Superman tale. The comic strips' Jor-L and Lora became Jor-el and Lara in Lowther's book, and those spellings stuck although Jor-el became Jor-El. The Kents, too, got new names here: Sarah and Eben. Lowther fleshed out the worlds of Superman's parents on Krypton and his foster parents on Earth. His longest-lasting contribution to the mythos was having Clark slowly discover his powers during his teens, which made him more empathetic and believable. "It was not until his thirteenth year that the incident occurred that was to set him apart from ordinary humans," Lowther wrote. "Clark watched the teacher as she poked about in the desk drawer, and as he did so he became slowly aware that he was also looking at the inside of the desk, that his eyes had pierced the wood.... The simple truth was that he _had looked through the desk_ as though the wood were transparent."
Such narratives about Superman's growing pains were captivating to adolescent readers who imagined themselves in Superman's place. The Superman legend had never waded into any of that. The Kents hid from him his otherworldly origins; he didn't don his Superman identity until he was an adult; and his childhood took up just eight panels in the first _Superman_ comic book and two fewer in _Action_ 1. Jerry Siegel realized what a rich trove he had glossed over and now proposed a comic focusing on the pranks of a noncostumed character he called Superboy. Detective picked up the idea—minus the whimsy and with a costume—while Jerry was in the Army, launching a feature that explored Superman's adventures growing up in the Midwest. It began in 1945 in _More Fun Comics_ , a year later switched to _Adventure Comics_ , and in 1949 Superboy got a comic book of his own. For the first time, readers learned how Martha Kent had stitched her adopted son's playsuit into a red-and-blue cape and tights, and how she used glass from his rocket ship to make Clark's special eyeglasses. The stories were not just about Superboy but about a _Saturday Evening Post_ world of picket fences that needed painting and apple pies warming in brick ovens. Fans went wild—young ones new to the legend and their parents who had grown up with Superman and _The Saturday Evening Post_ —making _Superboy_ the most popular new title of 1949, a time when most superheroes were fading away.
If learning about his roots on Krypton had humanized Superman, learning what he was like as a boy softened him. To his young fans, girls as much as boys, he was more than ever one of them. Yet even as the new stories answered questions that kids had asked since the beginning, they raised concerns that would take years more for Superman's handlers to sort out: How could Superboy know about his origins long before Superman did? How could there even be a Superboy if, as the earliest _Action_ and _Superman_ stories made clear, Clark didn't acquire the Superman costume and identity until he was an adult?
Those questions mattered to readers and writers but not to Harry and Jack. By the mid-1940s they had a better fix on who was buying comic books. The most devoted audience was kids aged six to eleven, with 95 percent of boys and 91 percent of girls reading them regularly. The numbers slowly declined in successive age groups, but even at age thirty half of the men in America and about 40 percent of the women were poring over comic books occasionally, and many were still steady fans. That helped explain the 150 different titles that jockeyed for space on newsstand racks, accounting for record monthly sales of forty million. Detective now faced competition from Fawcett, Timely, Dell, Street & Smith, and a series of other publishers, while Superman was going head-to-head with the likes of Batman, Flash, Green Lantern, and Bulletman. While they wanted to give kids the new Superboy stories they craved, Jack and Harry also wanted to satisfy adults who were devoted to the old Superman tales they'd been raised on. The solution: Reap profits from both old titles and new, contradictions be damned.
The adult Superman was getting a different sort of makeover. Joe Shuster had drawn a sticklike superhero whose skimpy facial expressions were difficult to see, not to mention read. By the late 1940s, Wayne Boring was setting the standard with a more muscular Man of Steel whose face was chiseled, whose jaw jutted onto the page, and who had more stature, bulk, and gravitas than his early incarnation. It was an image that fit with a Superman who had gone from leaping to flying and whose powers were perpetually expanding. Everything around him got bigger, too, from city skyscrapers to the _S_ emblem on his chest. His world was now as outsized as his place in it.
Superman and those he treasured were not the only ones who were evolving. Those he loathed were, too. His first enemy to appear in a costume was the Archer, who, once Superman unmasked him as Quigley the big-game hunter, confessed that "I thought hunting human beings would prove more profitable!" Superman: "Any kid could tell you that crime doesn't pay, Mr. Quigley." The Prankster, the Toyman, the Puzzler, and J. Wilbur Wolfingham, a W. C. Fields lookalike, used tricks and gags instead of a bow and arrows in their bids to conquer Superman. For editors wary of controversy, 1940s villains like those were a way to avoid the sharp edges of the real world. For a nation weary of war, they offered a release. For Superman, the masters of disguise and the tricksters let him demonstrate that his wit was at least as potent as his fists in battling bad guys.
While none of those villains lasted long, Lex Luthor did. When he first turned up in the spring of 1940 he had a full mop of bright red hair. By that summer he was gray, and a year later he was as bald as the evil Super-Man of Jerry and Joe's high school imaginations. What didn't change was Luthor's determination to take down Superman on his way to mastering the universe. In their first encounter Superman confronted him, asking, "What sort of creature are you?" Luthor answered with candor if not modesty: "Just an ordinary man—but with th' brain of a super-genius! With scientific miracles at my fingertips, I'm preparing to make myself supreme master of th' world!"
It is true that you can judge a man by his enemies, and Luthor had a way of bringing out in Superman both his vulnerabilities and his invincibility. From that opening encounter Lex drew on the full mix of villainous tactics—zapping Superman with an all-powerful ray gun, fomenting war as part of his scheme to grab power, and kidnapping Lois Lane. He tested Superman's mettle and exploited his soft spot. But in the end the Man of Steel smashed to bits the ray gun, talked the warring parties into signing an armistice, and rescued Lois. Holding her in his arms, he announced, "And that's th' end of Luthor!" If only it were true. Having created the closest thing Superman would ever get to a nemesis, Jerry and Joe were not about to let him die.
Mr. Mxyztplk, a bald imp who wore a purple suit and derby, was a different sort of adversary: He had superpowers but no interest in world domination. That had been his goal when he arrived on Earth from the fifth dimension, but he decided it would be more fun to discombobulate Superman by playing pranks on him, the way Bugs Bunny did with Elmer Fudd. What he hadn't counted on was that the no-nonsense superhero could be as playful as he was. Superman learned he could send the little man back for at least a month to his home world of Zrfff if he could trick him into saying his name backward—Klptzyxm. So he came up with a different way to do it every time they met. "Let's test your eyesight," the Man of Steel teased his adversary in one such meeting, after convincing him he was losing his sight. "There are three small signs 20 miles from here! If you can read them off, fast..." Before he could finish, Mxyztplk was breezing through sign one, sign two, and finally: "It says, 'Oxygen! Hydrogen! Nitrogen! Klptzyxm—' Oops—I spoke the word that's sending me back to Zrfff!"
Jerry Siegel said he invented the elfin character to give Superman and his readers "a change of pace" after all the battles against deadly adversaries like Luthor. "I think it added something to the feature to show Superman tangling with a magical foe who enjoyed making the idol of millions uncomfortable on his super-pedestal." What made editors at Detective Comics uncomfortable was having to spell the Zrfffian's name. One time when they typed it wrong—Mxyzptlk instead of Mxyztplk—the misspelling somehow stuck.
Names already were an obsession for Detective's writers and artists, most of whom had transformed theirs for reasons of art or assimilation, so it is not surprising that they had fun with the names of their characters. With Superman their obsession was nicknames, the best gift you can give a friend. Man of Steel was the most used, but he also was known then or later as the Last Son of Krypton, Metropolis Marvel, Kryptonian, Citizen Fix-It, Wonder Worker, Man of Tomorrow, Champion of the Underdog, Champion of the Oppressed, Champion of Democracy, Champion of Justice, Colossus of Krypton, World's Mightiest Hero, World's Mightiest Citizen, Man of Might, Big Blue, Big Blue Boy Scout, Big Blue Cheese, Action Ace, Smallville, Strange Visitor, King of Speed, and, the simplest and most intimate, Supes.
The era that stretched from the late 1930s through the end of the 1940s became known as the Golden Age of comic books. It was a time when the comic book became an accepted art form and the superhero played a central role in American culture. Fans would look back longingly as comics hit a rough patch in the 1950s, with fewer heroes and plummeting sales. The dawn of the earlier, more hopeful era was marked by the birth of Superman. Its last important title was the _Superboy_ comic launched in 1949. Who better to bookend comics' gilded age than its reigning monarch? And in Superman's case it wasn't just a Golden Age of comic books but of comic strips, dramatic radio broadcasts, pioneering animation, and wildly successful movie serials. Superman was jumping from medium to medium just as Americans were, responding to society's likes and dislikes and, just as often, shaping them.
# **CHAPTER 5**
# **Superman, Inc**.
THE COMICS HAD NEVER BEHELD a golden goose like him before. Superman was now the marquee attraction in four separate comic books and he shared top billing with Batman in a fifth. Each magazine brought in just ten cents, but a 1940s dime is today's dollar and 3.2 million dimes were rung up every month. True Man of Tomorrow addicts could get a daily dose in the funny pages. They were the newspapers' most fought-over feature, especially Sunday's four-color splash, and every Sunday Superman's strip was delivered to twenty-five million homes, each of which swelled his royalties. Ka-ching.
The cash value of stardom was even easier to measure outside the comics. The radio _Adventures of Superman_ was a runaway hit, with every "Atom Man" or "Clan of the Fiery Cross" adventure bringing a fat check from sponsors such as the snap, crackle, and pop makers of Battle Creek. Superman cartoons and serials were selling out—at forty cents a ticket for a weekend matinee—at theaters from Boston and Baton Rouge to Barcelona, where moviegoers cheered their "El Hombre Supre." Even department stores were mining the gold. Starting in 1942 they bought up and gave away millions of _Superman-Tim_ booklets, featuring cutout puzzles, heroic stories of Superman and his young pal Tim, and a reminder from everyone's favorite strongman not to "Be A Whoo-Shoo! He's The Boy Who Gets This Magazine Every Month, But Never Buys Any Of His Clothes At The Superman-Tim Store! Gee!" Ka-ching.
Then there were the synergies, a newly minted term for the way Harry Donenfeld, Jack Liebowitz, and their apprentices were turning Superman into a product line. As early as 1941, buttons designating the wearer as a paid-up member of the Supermen of America Club were proudly worn by hundreds of thousands of youthful fans, including Mickey Rooney, Our Gang's Spanky, and half a dozen middies at the Naval Academy. Kids across America lathered peanut butter and jelly onto super-flavored Superman bread and, if they ate all the crust, they might get treated to a Superman lollypop or Superman chocolate bar. Their Superman suspenders held up Superman dungarees. They stored their money in Superman billfolds until they had enough to buy Superman bubble gum, squirt guns, lunch boxes, underpants, jammies, moccasins, horseshoes, and a Krypto-Raygun complete with bulb, battery, lenses, and seven strips of film that let them flash onto a wall images of their idol in twenty-eight action-packed poses. Ka-ching.
By 1949 the cash registers were ringing nonstop in Harry and Jack's offices at 480 Lexington Avenue. National Comics Publications, the new name for Detective Comics, had a bullpen of heroes from Batman and Wonder Woman to Hawkman, Flash, and Green Arrow. None could match Superman as a box office headliner or newsstand heavyweight. Nobody was consulted as faithfully by kids across America before they got dressed, decided which game to play, or picked a cereal for breakfast. It was on the Metropolis Marvel's muscled shoulders that Harry and Jack were constructing their empire in the 1940s and that they would withstand the comic book scare of the 1950s. Ka-ching had become the soundtrack for Superman, Inc., much the way "Up, up, and away" was for Superman himself.
The impact went well beyond comics and kids. Disney may have been the guru at building Mickey Mouse and its other four-legged celebrities into long-lasting brands, but Superman, Inc., was becoming master of the fad. It didn't have Disney's patience yet it knew how to bottle the zeitgeist while it was happening. From the five-and-dime to movie houses and corner newsstands, the Man of Steel had been welded into the American consciousness. That model would be applied later, by many of the same marketing whizzes, to clients ranging from James Bond and Dobie Gillis to Major League Baseball. For now, all eyes were on Superman.
There were pitfalls. Push too cavalierly to commercialize your hero and you might threaten his integrity and his moneymaking potential, much the way Aesop's greedy cottagers killed their golden goose. That almost happened with the serial movies, which lacked the radio show's creative scripting and technical wizardry. Thankfully, kids laughed with Superman's cartoonish bid to fly even as the critics laughed at it. Managing Superman across media posed its own challenges: On paper Superman merely leaped even as he was flying across the airwaves, while his super-fueled adolescence in one comic book contradicted his adult-onset power-up in another. Product placement posed a risk, too. It might have been visionary when cellphones, holograms, and even biological weapons turned up in the comics decades before they did in the real world. But what about when writers came up with a three-part series on why Superman needed a Supermobile to promote a toy car the licensing people had dreamed up?
The line between art and merchantry got blurrier still in the case of the Krypto-Raygun. The new toy "looks exactly like the KRYPTO-RAYGUN used by SUPERMAN in his never-ending fight against crime... like the one SUPERMAN had made of KRYPTONITE—that amazing metal from SUPERMAN'S birthplace—the planet KRYPTON!" read an ad from the back cover of _Superman_ 7 in 1940. It would have been an interesting tie-in to the comic book and radio stories except for one thing: Kryptonite wouldn't appear in comic books until 1949, or even on the radio until 1943. It _was_ there on Detective's library shelves, in Jerry's never-used script, but he had called it K-Metal, not kryptonite. Was the ad part of an anticipated marketing campaign for K-Metal or kryptonite on the radio, in the comics, or both? Was it an instance of selling overtaking storytelling?
Jerry and Joe had fantasized from the first about Superman's merchandising potential for everything from box tops and T-shirts to billboards. This was an obvious way to extend his fame and inflate their incomes. But they were torn: Wouldn't commercializing their hero diminish him? They had sounded that alarm just five months after Superman's debut, in the November 1938 issue of _Action_. A fictional con man named Nick Williams claimed to be Superman's personal manager and used his "client" to sell movies and breakfast cereals, gasoline and physical fitness programs. All were items that Harry and Jack had Superman selling, or soon would. And so it was with glee that Jerry and Joe took a shot at the avarice of their bosses by not just exposing Williams and his henchman, but tossing them in a cartoon jail.
JERRY WAS HAVING LESS LUCK in the real world. His home movies offer a lens into his life in the 1940s. He filmed his excursion to the World's Fair in New York, and he filmed the Macy's parade, where he and Superman were treated like royalty. Jack and Harry were there, with Harry riding an elephant and afterward clowning with Jack, who was somber as ever, and both trying to pay attention to the boys from Cleveland. Other surviving tapes show Jerry and Joe in a crowd of kids and a harem of women, with Jerry looking thinner than he ever had and Joe loving being the center of attention. There are movies of Jerry and Bella, too—as newlyweds, then with their son Michael, who was born early in 1944. Jerry was in the Army but got a leave so he could be home for the birth of his baby, who came out weighing a robust nine pounds. He was home again to take movies of Michael's first few birthday parties, each showing the boy getting a bigger gift than he had the year before. There was Michael sitting on his big red truck, riding his scooter, dressed in an aviator's suit climbing aboard his chrome toy airplane, and trying on his dad's Army cap. Mostly it was Michael alone, looking overwhelmed. Jerry seemed equally awkward, more a visitor in Bella's home than a part of his son's life.
Michael never got to meet his grandma Sarah; she died three years before he was born. It happened on a Sunday at the family home on Kimberly Avenue, a place Jerry didn't visit much after his marriage to Bella and his success with Superman. The newspaper listed Sarah Siegel's cause of death as a sudden heart attack, but family members say her descent was more gradual and began the day Jerry announced his wedding plans. Seeing what their married life was like "led to her death," says Jerry Fine, the cousin who introduced Jerry to Joe.
Sarah was right about the marriage: It soured early. Jerry was absorbed in his work, what with all the deadlines for Superman and his other creations, celebrity appearances in New York and elsewhere, and his battles with his bosses. Then came the war, which took him away from Bella when she was four months pregnant—although Jerry's VIP status spared him any combat. The biggest blow came in 1946, when Michael was nearly three and Bella was getting ready to deliver their second child. "Bella went to the hospital a month earlier than expected," Jerry wrote Jack Liebowitz in November 1946. "The boy child that was born lived only eight hours, for somehow the cord had gotten about its throat and cut off its air supply." For Jerry, his son's death was like his father's: He never talked about the pain, not even in his memoir, where he chronicled his many turmoils, and he never got over the feelings of loss and anger.
Jerry's angst was made worse by his inability to bottle lightning again with any of his new comic characters. He had tried in 1940 with the Spectre, a murdered cop whose spirit returned to Earth to battle evil. That flopped, and he was back two years later with Robotman, who had the brain of a murdered scientist inside the body of a robot. In between came the Star-Spangled Kid, who reversed the usual formula by pairing a boy hero with an adult sidekick. Detective Comics rolled the Kid out with a four-page house ad ballyhooing the plotline and the fact that it was the brainchild of the creator of Superman. Instead of becoming Jerry's next big thing, the Kid was yet another ho-hum, spawning whispers that its author was a one-trick pony. _Funnyman_ fueled the doubts. It wasn't nearly as funny as Jerry thought, and Detective wasn't about to grant him the ownership rights he demanded. "I never indicated that we would take your Funnyman feature," Jack wrote stiffly in February 1947, "but as long as your ego tells you that anything you do must be a preordained success, I would be interested, just for the record, in having you name one feature—other than Superman—out of the numerous ones you've developed, which has enjoyed even a modicum of success."
Even Superman was no longer his alone. It wasn't just that other writers were turning out many of the biggest stories now, but that the superhero had left behind his creator as he moved to other media. Jerry played little if any role when it came to writing scripts for the radio drama, the serials, the cartoons, or the endless promotions of the sort Jerry had been spinning out in his head since he dreamed up the character. Whether or not Superman's fans noticed Jerry Siegel's absence, Jerry himself did, and it hurt.
In the meantime, his relations with Joe were turning frosty. He had always resented Joe's wavering work ethic, and it got worse over the years, as Joe went from drawing his own sketches, to filling in with ink his assistants' pencil work, to supervising other artists and rendering just Superman's head, to doing what Jerry said was "practically none of the actual art." It was true that Joe wasn't nearly as diligent as Jerry, but what really ate at Jerry was the suspicion that his childhood buddy was conspiring with another writer on new comics and other creative works. No matter that Jerry himself had been working with other artists since they were kids and still was, collaborating with Hal Sherman on Star-Spangled Kid and with Bernard Baily on the Spectre. "In line with your negotiating increases for yourself alone on 'Superman,' and working on 'Superboy' without first consulting me if it was all right for you to work on my comic, you are now preparing to collaborate on comics with others," Jerry wrote Joe in September 1946. "In the past we've operated under a gentleman's agreement, with mutual trust, but in view of what has occurred since I went into the Army, and your apparent unwillingness to continue our association as it was, I'm afraid that continuing to work with you under just a gentleman's agreement, would be hazardous." To underline his point, Jerry added a postscript: "Since Detective Comics, Inc. is involved in this situation, I am mailing a copy of this letter to Jack Liebowitz."
Jerry likely did not know just how involved Jack and Harry were. They despaired of Jerry's increasingly angry tone with them, too—a rage that, according to Harry's children, saw Jerry and Bella picketing in front of Harry's Long Island home, following the kids around in their car, and writing neighbors to say how unfair Harry had been to Superman's creator. While Harry and Jack knew that Joe was no longer doing the work he was being paid for, they liked him more than they liked Jerry and felt it couldn't hurt to curry favor with him. Harry's son, Irwin, claimed that his father—apparently without Jerry's knowledge—paid for Joe to have an operation to help his failing eyesight, a terrible affliction for an artist. Harry also joined with Joe in an improbable partnership: In December 1946 they filed a certificate to do business as Shuster & Donenfeld in the hamlet of South Fallsburg in New York's Catskill Mountains. While there is no evidence that they actually launched a business, it is fun to wonder what a dreamy artist like Joe Shuster could have cooked up with a wheeler-dealer and pornographer like Harry Donenfeld.
Three months later Jerry and Joe were once again partners, this time for the purpose of suing Harry, Jack, and their Superman empire. Money was a central narrative in the more than one thousand pages of transcripts from the proceedings. Jerry forever regretted that he and Joe hadn't copyrighted Superman for themselves, believing—justifiably—that the oversight had cost them millions of dollars. Now their lawyers made the case that what Harry and Jack had bought for $130 was the ownership rights not to Superman but to the first thirteen-page story alone. Then there was Superboy. "No compensation has ever been paid for this. No permission has ever been secured for this," argued their attorney. "It was purely and simply an act of appropriation of this script." Ditto for Batman and other "union suit characters" who were mere knockoffs of Superman, for comics built around Wonder Woman, for George Lowther's book on Superman, and for the Superman radio show, animation, and accessories, a share of the profits for which had been promised (but never delivered) to Jerry and Joe. "While we are interested in being paid for these past misdeeds," the lawyer said, "the most important thing is this: to satisfy your Honor of the fact that we are entitled to be rid of these people once and for all, and of this contract which keeps these people from sleeping nights and keeps them from earning an honest living." Harry and Jack were the people the boys wanted to be rid of, Jerry and Joe the ones who couldn't sleep. What it would take to make it all better was a round $5 million.
The comic book company had a different take on the law and the numbers. Jerry and Joe weren't the aggrieved schlubs they made themselves out to be but thankless self-seekers. National, the company claimed, had turned over to them everything they were entitled to, which is what enabled them to live the high life. In the ten years from 1938, when the first _Action_ was published, to the filing of the suit in 1947, Jerry and Joe were paid $162,627.08 for their work on comic books, $205,998.21 for comic strips, and $32,569.56 for other uses of the character they had dreamed up, for a total of $401,194.85. That was a king's ransom—more than $5 million in today's terms—even after they split it in half.
Whether or not that was enough wasn't something the Supreme Court of New York ever weighed in on. Its referee affirmed Harry and Jack's ownership of Superman but not Superboy. After further legal wrangling, Jerry and Joe signed an agreement in May 1948 selling the rights to Superboy and related characters to Harry and Jack for $94,013.16. Once the lawyers and broker took their shares, Joe and Jerry each walked away with $29,000—barely one one-hundredth of what they had hoped for. Even then there were onerous terms: The creators had to agree that Superman and Superboy—in all their forms and forever—belonged to Harry and Jack. Gone were Jerry and Joe's jobs writing and drawing the characters. Gone were the bylines indicating that the Man of Steel and Boy of Steel were theirs. Gone even was their right to claim a historic connection if it "may be likely to induce the belief that such past connection still exists."
The lawsuit had always been about Jerry's grievances—the ones he had been carrying around since Superman caught fire a decade earlier, along with the gripes that remained from his forlorn Valentine's Days in grammar school. The suit was never about Joe; although his life would be painfully reshaped by the outcome, he still let Jerry call the shots. In fact, this was the first in a series of lawsuits that would play out over the decades, none of which gave Jerry anything close to what he was after. He could have had more money if that had been all that mattered: Batman founder Bob Kane proved that when he renegotiated his deal with Jack instead of joining Jerry's lawsuit. What Jerry seemed to want was not only for Jack and Harry to give him the money he felt he deserved, but also the homage for having dreamed up the most successful hero in American history. What he couldn't grasp was this: He had already gotten more of everything than any of his peers in an era when comics writers got no bylines, no royalties, and nowhere near the kind of payouts that Jerry and Joe had been enjoying. Jerry, in fact, couldn't enjoy any of it. All that was left was the anger.
Jerry's biggest miscalculation was failing to understand who Jack was—a bean counter so hard-nosed that even his adoring daughter was afraid of him—and failing to deal with him on those terms. In a decade of letters between them, Jerry essentially dismissed his boss as a shyster and Jack branded his scriptsman an ingrate. But there also were heartbreaking moments in the exchanges, like when Jerry confided to Jack—as he might have to his father—news of the wrenching death of his newborn. Jerry's loss in the courtroom was Jack's enormous gain. He walked away with unfettered access to Superman and freedom from a writer who had become a dreadful nag. In the process Jack sent a message to the rest of his writers and artists: Beware. If I can fire the creator of the mighty Superman, any one of you who steps out of line could be next. But in the end Jack lost, too. The narrative had been written: Jerry was the martyr, Jack the bully. What he had done to Siegel and Shuster would remain part of Jack's legacy, as Al Capp, America's most read satirist, made clear in a _Li'l Abner_ strip in 1947. Rockwell P. Squeezeblood, head of the corrupt Squeezeblood Comic Strip Syndicate, was a stand-in for Jack Liebowitz, head of the bare-knuckled National Comics Publications. Squeezeblood published a bestselling feature about a crime-fighting strongman named Jack Jawbreaker, whose creators even looked like Jerry and Joe. "Those boys created 'Jack Jawbreaker' in poverty!!" Squeezeblood told Abner. "Poverty is the greatest inspiration to creative genius!! I won't let all this wealth spoil those innocent boys!!"
Two months after Jerry settled with National Comics, Bella sued him for divorce. During their nine years of marriage, her complaint read, Jerry "absented himself from their home in Cleveland for long periods of time without giving any excuse or reason" and "during the short periods of time he was at home in Cleveland, he displayed a moody, quarrelsome and argumentative attitude toward the plaintiff." Jerry didn't respond. The marriage had been dying for years. Bella had a right to be mad, and he didn't want to fight. He just wanted out. The judge pronounced him guilty of "Gross Neglect of Duty" and said it was "impossible for them [any] longer to live together as husband and wife." Bella got her freedom, custody of their son, all the household furnishings, and 60 percent of their joint assets, which included the house in University Heights, $98,000 in cash and bonds, and a year-old Chrysler. Jerry got the remaining 40 percent, along with his typewriter and Dictaphone. He said he would pay alimony, child support, and 20 percent of his annual earnings once they hit $10,000, which was less likely now that he was jobless. He also said that, with help from Bella, he would pay the Internal Revenue Service $24,000 in back taxes along with whatever extra levy was owed for the settlement money he had just received from Jack.
Jerry would have agreed to anything. He was crazy about a woman he had rediscovered just months before at a Cartoonists Society costume ball at New York's Plaza Hotel. This wasn't just any woman: It was Lois Lane, or at least the young model Jerry and Joe had hired more than a dozen years ago to help them envision what Lois should look like. Joanne Carter was back in his orbit and he was smitten, the more so when she came to the ball dressed as the dreamboat comic strip showgirl Dixie Dugan. In the years since she had posed as Lois, Joanne had lived on both the East and West coasts and gotten married and divorced. It was Joe who fell for her first. He had tumbled more than a decade before, when she spent most of the winter posing for him. He had been corresponding with her ever since then and had invited her to the New York gala, even renting her an elegant gown. No matter. Once they got to the ball, she was drawn to Jerry and he to her. No surprise. With her slim figure, piercing dark eyes, and tightly coiled shoulder-length hair, she was far more attractive than the matronly Bella. Joanne, meanwhile, was eager for a second shot at love and life—not with an artist like Joe, whose type she was weary of after a frustrating career in modeling, but with a celebrity writer. "Jerry and I started dating," she recalled years after, "and a few months later, we were married."
The marriage couldn't happen fast enough. They filed their application on October 13, 1948, asking the state to waive the normal waiting period so they could have a "ceremony at once." Both were already living at the Commodore Hotel in Cleveland, and her marriage had been over for years. His divorce agreement was worked out three months before and accepted by the court on October 7. A justice of the peace married them at the hotel on October 14, and the Cleveland _Plain Dealer_ ran a story the next day. Three days later national gossip columnist Walter Winchell wrote, " 'Superman' creator Jerry Siegel and model Joanne Carter had it sealed in Cleveland." But it wasn't quite sealed. For a reason neither ever talked about, Jerry and Joanne were back with a second marriage application three weeks after the first and were wed again that same day, November 3. Whatever the glitch, it was fixed. Jerry had turned thirty-four in the interval. Joanne told Jerry, _The Plain Dealer_ , and the state of Ohio that she was twenty-five, which would have made her a cherubic twelve or thirteen when she modeled for Joe as a voluptuous Lois Lane. Her birth certificate clears up the confusion: She was, in fact, about to turn thirty-one when she married Jerry, which she must have worried was over-the-hill.
Joanne knew about Jerry and Joe's legal battles with National Comics, which is why she had changed her plan to come to the Plaza costume party as Lois Lane. She hadn't had it easy herself. After high school she had left Cleveland for Chicago, then Boston, when a young man who had spotted her ad in the newspaper invited her to join his skating act. She said she couldn't skate; he said he'd teach her. The act broke up before she had to perform the dangerous stunts he had planned for her, and she stayed for a while in Boston, where she "went to a local bar and began phoning artists listed in the yellow pages. As a result, I became a professional model instead of a professional skater."
Joanne came into Jerry's life just as he was losing his home, his car, his livelihood, and his superhero. His privations became her cause. She had never known high living, so didn't have to make the adjustments he did; having her there made it easier for him. So did being able to lash out at his old bosses in the only forum still open to him, the comics. Less than three weeks after his marriage to Joanne, he published a Sunday _Funnyman_ newspaper strip about a writer (Jerkimer) who was ripped off by a chiseling business executive (Winston Lightfingers of Gypsum Music). The writer was saved through the intervention of Funnyman, the only superhero Jerry still could call his own.
EVEN AS JERRY AND JOE were parting company with Superman, their hero was showing up in places no one expected to find him. There he was in the dentist's office, in an eight-page thriller in which he rescued a U.S. pilot whose aching tooth crippled him in the middle of a dogfight, warning that "smart fellows take good care of their teeth and visit the dentist regularly." Department stores gave out a _Superman's Christmas Adventure_ booklet, as well as _Superman-Tim_ , which was published year-round and was half adventure story, half sales pitch. Superman adorned the backs of cereal boxes, the front of a holiday display at R. H. Macy & Co. that drew a hundred thousand visitors, and ads for Dr Pepper, sandwich spreads, and flour mills. The Man of Steel had gone viral and merchandisers wanted to go with him.
Sometimes the link-ups paid direct dividends to National Comics, as with _Superman-Tim:_ A marketing firm bought the rights to produce the six-by-nine-inch booklets, then sold them to stores. The one million dental brochures that Jack and Harry printed, like the Superman balloon they entered in New York's Thanksgiving Day Parade, earned them nothing directly but built name recognition and trust for their hero. So complete was that trust that a corps of secretaries at National was kept busy answering moms' questions about how to make their kids stop biting their nails, eat egg yolks, and walk the dog. Mothers were learning the lesson librarians had years before: When Superman spoke, kids listened. As for name recognition, popular culture maven Harlan Ellison got it right when he observed that "the urchin in Irkutsk may never have heard of Hamlet; the peon in Pernambuco may not know who Raskolnikov is; the widow in Djakarta may stare blankly at the mention of Don Quixote or Micawber or Jay Gatsby. But every man, woman and child on the planet knows Mickey Mouse, Sherlock Holmes, Tarzan, Robin Hood... and Superman."
That celebrity let National license more Superman products—over one hundred in all—than were commissioned for Sherlock, Tarzan, or Robin Hood, although no one could touch Disney's Mickey. Superman, Inc., started licensing merchandise in 1940 and within months there were more than twenty items, from film viewers to military-style hairbrushes, which netted about one hundred thousand dollars for Harry and Jack. A year later, forty companies were making Superman toys, candies, games, and other items, with profits swelling. Jigsaw puzzles were an early favorite, with more than a dozen being produced in 1940 alone, from intricate five-hundred-piece ones showing him leaping to the rescue to box sets of six puzzles with forty-two pieces each. The first dolls had adjustable wooden joints and squinty eyes and cost just ninety-four cents (although today one in mint condition can fetch eight thousand dollars). For the action-oriented, the Turnover Tank let Superman flip the vehicle all the way over, no mean feat for a toy produced in 1940.
Superman, Inc., offered something for every taste and age. Fathers could drench their cereal in Superman milk, lather up with Superman shaving cream, add Superman hood ornaments to their cars, then drive away using high-octane Superman-certified gasoline. The latter made it especially clear that the only thing needed to make an association with the Man of Tomorrow was a client willing to pay today. Kids could trade seventy-two different Superman picture cards, just as they did baseball cards or comic books. Mothers could take the kids to the Macy's Christmas show, then fit them out in a blue broadcloth shirt, red broadcloth cape, and navy cotton twill pants—all for just ninety-eight cents. Anyone of any age interested in bulking up could try a Superman muscle-building set, with hand grippers, a jump rope, a measuring tape, and a chart to track their progress.
Public relations and advertising were in their adolescence as the 1930s were yielding to the 1940s, and both vocations were hell-bent on making every American a consummate consumer. It was a time before focus groups, when anything went, and Robert Maxwell fit in instantly. Even before his Superman radio show debuted, Maxwell was put in charge of Superman, Inc., where he was anxious to show how much money he could make for Harry and Jack. Some of the promotional deals he negotiated involved products that grew out of the hero and his exploits. Others added Superman's endorsement to items or services already on the market. "Let Superman be your Super-salesman," Maxwell's brochure pitched. "Superman has a tremendous, loyal fan following, a ready-made juvenile market that will respond by boosting your sales volume." It worked from the start and got even better when Maxwell brought it with him to the radio show he was launching. Laundries, dairies, and meat packers signed up as sponsors, then watched their profits shoot up. The Rochester, Minnesota, bottler of Dr Pepper and 7 Up offered an object lesson, its sales doubling after it started sponsoring the show on KROC.
While Superman, Inc., vanished as a corporate entity in the summer of 1946 when it was absorbed into National Comics, the mindset remained that he was a commodity that could be branded, packaged, sold, and incorporated. Neither National nor its predecessors had ever pretended to be a charity. They had always been focused on building a commercial success hand in hand with the Superman story, in whatever form that story might take. Making money was a way of ensuring that Superman would not suffer the fate of the Green Lantern, whose plummeting sales led to his being pulled from publication between 1949 and 1959, or of World War II aviation ace Hop Harrigan, who was grounded forever in 1948. But Maxwell, Liebowitz, and Donenfeld were smart enough to know they could push only so far before they threatened the integrity of their character, and they rarely tested those limits. They took care to associate their all-American hero with all-American products like piggy banks, coloring books, and sliced white bread. Keepers of the legacy from Superman comics would sit in on plotting sessions for his movies, and in-house censors pored over each printed word to ensure the Big Blue Boy Scout stayed true to his name.
That name defined not just one hero but the whole National Comics universe in its early years. Superman, Inc., managed Batman, Wonder Woman, and the rest of National's stable of heroes in the 1940s, none of whom had nearly the product line that Superman did or brought in nearly the revenues. The more success Superman had in one medium, the more others wanted him, with comic books leading to comic strips, radio, cartoons, and film. And the fans who read, listened, and watched couldn't get enough of Superman valentines, timepieces, and the Super-Babe dolls that Macy's introduced just in time for Christmas in 1947. "Superman turned baby by mysterious atomic rays," the ads explained, and for $5.59 ($56 today) kids could have their own fifteen-inch doll with latex rubber skin, moveable arms and legs, and eyes that opened and shut.
Kids were the key to National's strategy, and not just because they were Superman's most avid fans. Hooking them young could mean keeping them forever, and Superman was proving to be a gateway to get young people to try all kinds of other comics. Their parents might have worried about adult conceits like consumerism and commercialism, but kids had their own truth: the Superman they were reading in comic books, listening to on the radio, and watching at the movie house was as pure as ever, and he was theirs. Having Superman toys on the shelf and Superman food in the pantry brought their hero closer and made him more central than ever to their lives.
And it was not just in America. France and Italy imported the Man of Steel barely a year after his debut here, with kids in Paris calling him Yordi and ones in Rome preferring Ciclone, or "hurricane." South American children loved him, as did Germans before the Nazis started railing against his Jewish roots. He was America's most iconic export. Superman is a hero "for the whole universe," explains Vincent Maulandi, a lifelong fan in France. The Last Son of Krypton "had no more homeworld and the Earth would replace that home, not only America." As Superman's comics and films spread around the globe, so did the international flavor of his wares. From France would come a Superman towel rack, from Nepal a can of cooking oil with a large picture on the front of the Man of Steel, and from Mexico a papier-mâché piñata built to look like El Hombre Supre. Ka-ching.
# **CHAPTER 6**
# **The Deadly Truth**
SUPERMAN HAD AN IMAGE PROBLEM. During World War II, the Nazis had denounced him for being a pawn of the Jews and poisoning the minds of America's youth. In the Cold War that followed, a Jewish psychiatrist was accusing him of being a Nazi out to corrupt the adolescents of America. Either way, his detractors were sure that Superman was bad for the kids.
Now they had Melvin Leeland and Billy Becker to prove it. On a cool summer night in 1947, Melvin, a fourteen-year-old from Washington, D.C., was showing a friend how to play Russian roulette. He spun the loaded cylinder, raised the .22-caliber revolver to his right temple, and, as his mate watched in horror, blew a hole in his head. His mother told the police that he had read about the deadly game in a comic book. Two months later Billy, a twelve-year-old from Sewickley, Pennsylvania, tossed a clothesline over a rafter in his basement and hanged himself. Mrs. Becker said he was reenacting a scene from a comic book. "I burned every one I found," she told a coroner's jury, "but Billy always found ways of hiding them."
The common denominator in tragedies like these was "crime" comic books, Dr. Fredric Wertham explained. From Batman to Wonder Woman, Superboy too, all shared the blame for the scourge of juvenile delinquency sweeping the nation. They were "an invitation to illiteracy" and encouraged "criminal or sexually abnormal ideas." Batman and Robin were secret lovers. Wonder Woman was an overt lesbian. Most depraved of all was the Man of Steel, "with the big S on his uniform," Wertham wrote. "We should, I suppose, be thankful that it is not an S.S.... The very children whose unruly behavior I would want to prescribe psychotherapy in an anti-superman direction, have been nourished (or rather poisoned) by the endless repetition of Superman stories."
Looking back, Wertham's warnings read like the ravings of a quack, but he was an esteemed psychiatrist and he wasn't alone. In the spring of 1940, just two years after _Action Comics_ No. 1, the literary editor of the _Chicago Daily News_ leveled the first broadside. "Unless we want a coming generation even more ferocious than the present one, parents and teachers throughout America must band together to break the 'comic' magazine," Sterling North warned. The cure, he added, "can be found in any library or good bookstore. The parent who does not acquire that antidote for his child is guilty of criminal negligence." _Catholic World_ weighed in next, asking, "What's Wrong with the 'Comics'?" Its answer: "The influence of these comics over the popular mind is one of the most striking—and disturbing—phenomena of the century." As for Superman, "in a vulgar way this fantastic character seems to personify the primitive religion expounded by Nietzsche's _Zarathustra_. 'Man alone is and must be our God,' says Zarathustra, very much in the style of a Nazi pamphleteer." The shrillest denunciation came from cultural critic Gershom Legman. "The Superman formula," he wrote in 1949, "is essentially lynching." The Man of Steel "is really peddling a philosophy of 'hooded justice' in no way distinguishable from that of Hitler and the Ku Klux Klan."
Could the cause-and-effect relationship between Melvin and Billy's reading habits and their deaths be that clear-cut? Scores of librarians, teachers, judges, and priests had been saying so for years, but the wider world wasn't buying it. Wertham tipped the debate. He had been taking detailed case histories from troubled youth at his mental health clinics in New York. "Comic-book reading," he concluded in a 1948 _Collier's_ article headlined "Horror in the Nursery," was "a distinct influencing factor in the case of every single delinquent or disturbed child we studied." His findings carried the weight of science and of his résumé. He was the senior psychiatrist for the New York Department of Hospitals and former chief resident psychiatrist at Johns Hopkins University. Born in Nuremberg as Frederic Wertheimer, the Americanized Wertham was a friend of culture critic H. L. Mencken's, a collaborator of renowned criminal attorney Clarence Darrow's and syndicated columnist Walter Lippmann's, and a valued ally of the NAACP as it gathered evidence on the harms of racial segregation. He may not have been trained as a social scientist, but he was a social reformer and his judgment—rendered in repeated press interviews, then in a book with the unnerving title _Seduction of the Innocent_ —was taken as gospel.
His timing couldn't have been better. Postwar America was feeling prosperous in ways it couldn't have imagined when its young men were overseas fighting and its economy was on a full military footing—but that prosperity brought changes that unsettled many Americans, especially ones who had grown up in a more austere and hidebound prewar world. Kids had more money to spend now and they spent more time away from home. Rock and roll was picking up the beat, with Jackie Brenston's Delta Cats and Bill Haley's Comets setting the tempo. Hot rods (cars with tuned-up engines and no hoods or fenders) were the bossest way to get to the passion pit (drive-in theater), and the kookiest films to see there were James Dean's _Rebel Without a Cause_ and _East of Eden_. This new generation even spawned a term for itself—teenagers—and they brought with them a new fear: juvenile delinquency. The FBI said that kids under eighteen were responsible for half of America's car thefts and burglaries and a rising share of robberies and rapes. Gangs of working-class whites were facing off against poor Puerto Ricans on America's urban streets, or so the Broadway hit _West Side Story_ told us. Years later, studies would demonstrate that the rise in youth crime was due to crackdowns by law enforcement and the way family life was disrupted when fathers shipped off to war and mothers took their places in the factories. But in the heat of the moment, parents, newspaper columnists, and crusading scientists like Wertham latched onto an easier target: the mass media, and especially the billion comic books American kids were buying each year.
The finger-pointing was understandable. It was what later generations would do in blaming television, video games, and social media for corrupting the youth of their eras. And it was what Americans in the 1950s were doing on the foreign front, where the Russians had an H-bomb to go with their A-bomb, the Reds in North Korea had invaded the democratic South, and G-man J. Edgar Hoover and Senator Joseph McCarthy were finding communists wherever they looked in Washington, Hollywood, and classrooms across the country. Just as domestic subversives were real and did pose a threat, so comic books were swelling in popularity and challenging more cultured reading habits for youngsters. But both threats also were overtaken by hysteria and overreaction. Borrowing from the term "Red Scare," as the crusade against communists was called, the campaign against Superman and Batman would come to be known as the Great Comic Book Scare. "We are dealing with the mental health of a generation," Wertham told 2.5 million _Collier's_ subscribers, "the care of which we have left too long in the hands of unscrupulous persons whose only interest is greed and financial gain."
Here, finally, was not just a clear diagnosis of what was wrong but a remedy. Five months after the _Collier's_ article, six hundred grade school children in Spencer, West Virginia, gathered to hold last rites for two thousand comic books they had rounded up. "Believing that comic books are mentally, physically and morally injurious to boys and girls, we propose to burn those in our possession," said thirteen-year-old David Mace before he ignited a copy of _Superman_. "We also pledge ourselves to try not to read any more." Additional burnings followed across the country, along with a police raid of a publishing house. More than a dozen states started regulating comic books, with New York dealing the harshest blow by banning any that depicted explicit sex, brutality, or criminal techniques as well as any that used in their titles the words _crime, terror, horror_ , or _sex_. The National Parent Teacher Association pushed for a "national housecleaning" of comic books. The United States Senate held hearings under the leadership of the ever-ambitious Senator Estes Kefauver of Tennessee, who had his eye not just on Superman but on the White House. In Cleveland, which counted Superman among its favorite sons, the city council outlawed comics showing rape, arson, assault, kidnapping, burglary, mayhem, larceny, manslaughter, murder, prostitution, sodomy, or extortion. To show it meant business, the city assigned two policemen to the comic book beat. Comics had always operated around a generational divide between adoring kids and fretting parents. A Gallup poll showed that 70 percent of adults now believed that comic books were at least partly responsible for juvenile delinquency and 26 percent felt they deserved a "great deal" of blame. No one bothered to ask kids how they felt, but the pollster couldn't resist offering his surmise: "Older people are much more inclined to brand both comic books and TV-radio crime programs as factors contributing to juvenile delinquency than are young people."
Wertham and his allies conceded that not all comic books were menacing. It was easy to agree on the increasingly popular and macabre horror books, with titles like _Out of the Night_ and _Weird Thrillers_ and stories about a human head that doubled as a bowling ball and a wife roasting on a barbecue her husband's head, legs, hands, and feet. They had to go. But even ones that seemed harmless might not be, Wertham warned: "You find that what all the little animals are doing involves undue amounts of socking over the head and banging in the jaw, and that the toys that come to life at night sometimes put in the time strangling one another."
Superhero comics were a different animal entirely, focused not on the lurid but on the ennobling, and having aided the nation in its patriotic war against the Axis. To critics, however, that was a bygone era. What mattered now were the crimes depicted in these comic books, their heroes' resort to force, and the very notion of superpowers, which had the stench of fascism. Superman, as king of the superheroes, was especially reviled. He was one of four characters put on trial in 1949 by students at St. Mary's High School in Cape Girardeau, Missouri, no matter that the boy who played him in the mock tribunal had never read a comic book. The Colossus of Krypton was often the first onto the funeral pyre at comic book burnings and was singled out for special scorn at public hearings. He also was Wertham's favorite target. "I would like to point out to you one other crime comic book which we have found to be particularly injurious to the ethical development of children and those are the Superman comic books," the psychiatrist told members of Congress. "They arouse in children fantasies of sadistic joy in seeing other people punished over and over again while you yourself remain immune. We have called it the Superman complex."
THIS WAS NO TIME to launch another Superman experiment. Not in 1951, when the whole comic book world was running scared. Not when the medium into which he was being catapulted, television, was so callow that it was unclear whether it would succeed, and it was perfectly clear that actors with real promise would opt for Hollywood or Broadway. Not when Superman himself was being labeled a sociopath.
But that wasn't the way National Comics or Robert Maxwell thought. They knew that there were more children than ever in America, where soldiers had come home from Europe and made up for lost time by igniting an unprecedented boom in babies. They also knew that radio was dying as a venue for children's adventure and that movie serials wouldn't be far behind. While TV might be new and untested, so were comic books when Superman broke through in that medium. Now was precisely the time when the battered Superman of the static page could use a lift onto the small screens that were turning up in America's dens and playrooms. For Superman's owners, the question wasn't whether but when to push ahead, and whom to sign up as the Man of Steel for this most up-close of media.
Maxwell and his director, Tommy Carr, screened nearly two hundred candidates. Most made their living as actors, although some were full-time musclemen. Nearly all, Carr said, "appeared to have a serious deficiency in their chromosome count." So thorough—and perhaps so frustrating—was their search that the executives stopped by the Mr. America bodybuilding contest in Los Angeles. One candidate they never seriously considered, despite his later claims, was Kirk Alyn, who had done well enough in the serials but had neither the acting skills nor the looks around which to build a Superman TV series. The search ended the day a barrel-chested B-movie actor named George Reeves showed up on the studio lot.
Maxwell's co-producer, Bernard Luber, had recognized Reeves in a Los Angeles restaurant, seeming "rather forlorn," and suggested he come in for a tryout. He did, the next morning, and "from that moment on he was my first choice," said Tommy Carr. "He looked like Superman with that jaw of his. Kirk had the long neck and fine features, but although I like Kirk very much, he never looked the Superman Reeves did." His tough-guy demeanor was no put-on. Standing six foot two and carrying 195 pounds, Reeves had been a light-heavyweight boxing champ in college and could have gone further if he hadn't broken his nose seven times and his mother hadn't made him step out of the ring. It wasn't the first or the last time she would interfere. A headstrong and self-focused girl from Illinois, Helen Lescher had eloped with a pharmacist, Don Brewer, in Iowa in 1914 and within five months they had a son, George Keefer Brewer. The marriage didn't last long and George didn't learn about his real father or his real birthday until he was into his twenties. Helen altered his date of birth to make it look like she was married when he was conceived. She hid his father's fate, telling George he had committed suicide, until Brewer turned up one day. Helen's second marriage wouldn't last, either, nor would the second version of George's name, George Lescher Bessolo.
After giving up boxing George landed a job at the prestigious Pasadena Playhouse, which was more to his mother's liking. It was then that he learned to act and that nearby movie executives got to see what he could do. They liked him enough to give him the modest role of Stuart Tarleton, one of the Tarleton twins and a suitor of Scarlett O'Hara in the 1939 blockbuster _Gone with the Wind_. Even before the film came out, George had been signed by another studio, Warner Bros., where Jack Warner pushed him to change his name to one he felt would look better on movie theater marquees: Reeves. George didn't see his name in lights for anything but lesser films, but he did land minor roles alongside major actors. In the 1949 _Samson and Delilah_ , Victor Mature was Samson and George was a wounded messenger, while that same year, in Bob Hope's _The Great Lover_ , George was a gambler killed in the first three minutes. Between acting jobs, he dug cesspools at the rate of one hundred dollars a hole.
When the offer came in 1951 to play the TV Superman, George was torn. He had barely heard of the Man of Steel, knew that the six hundred dollars a week he was offered was a pittance, and realized that the chance of getting a real acting job would be harder once the movie studios saw him playing a comic book character or any role in a medium that Hollywood disdained. Yet he needed the money, and, as his agent advised, there was a slim chance that the new show would even be broadcast and a slimmer one that anyone in Hollywood would notice. Television, after all, was in its infancy, with the nation just witnessing the first-ever coast-to-coast broadcast in the form of a speech by President Harry Truman. "Take the money and run," George's agent said. Reluctantly, George did. "I've played about every part you can think of. Why not Superman?" he told a friend. To Phyllis Coates, the new Lois Lane, he confided the first time he met her, "Well, babe, this is it: the bottom of the barrel."
Like many in Hollywood and in the growing Superman family, Coates was using a pseudonym. In her case it sounded more like a real name than the one her parents gave her: Gypsie Ann Evarts Stell. The tall, slim brunette had gone from chorus girl on vaudeville to actress in second-tier movies. She was glad to land the role of Lois less because of the professional opportunity—"I'd never read Superman comics, never heard the radio show, never heard of the character"—than because of the paycheck, which she needed to pay doctors' and physical therapists' bills for a daughter born with a displaced hip. Getting the job was easy. Her agent called and told her, "Wear a suit and low-heeled shoes." The next morning at the RKO studio, "I met Bob Maxwell.... I read for him and he said, 'I think you're perfect for the part.' It was that simple: I didn't even get a call-back on it, they just decided that I was it. And there were a _lot_ of good gals up for it."
The first production was a fifty-eight-minute movie called _Superman and the Mole Men_ that was a way to tease as well as finance the TV series. The story was set in the small town of Silsby, where the National Oil Company had just drilled 32,740 feet to create the deepest well in the world. Up came not just petroleum but four small humanoid creatures whose home was in the center of the Earth. The residents of Silsby, stirred to a frenzy by a shotgun-carrying rabble-rouser named Luke Benson, assumed the worst about the subterranean beings and vowed to find and exterminate them. Luckily, Clark Kent was reporting on the oil drilling and did a quick change when he saw the gathering mob. "I'm going to give you all one last chance to stop acting like Nazi storm troopers," Superman lectured the townspeople. When his words were unheeded, he disarmed Benson and the others and helped the underground creatures return down the well. It was exactly the sort of morality tale that Jerry Siegel and Joe Shuster had brought to their early comics and that Maxwell tackled on the radio. Having Superman take this stance was especially brave coming just as Senator McCarthy—striking a pose like Luke Benson's—was beginning his meteoric rise. It also sent a message to Dr. Wertham and his followers: The real threat was the Nazi-like citizens of Silsby, not the superhero roused to violence as a last resort.
_Mole Men_ was a classic Superman enterprise—done on the cheap but at full throttle. The original idea was for children to play the mole men, but Maxwell and crew decided that dwarfs—two of whom had played Munchkins in _The Wizard of Oz_ —would be more believable. Their heads were enlarged to make them scarier, they wore fuzzy jumpsuits to ensure that kids wouldn't be too scared, and their laser weapon was the handiest and cheapest the crew could come up with: a handheld Electrolux vacuum cleaner. Maxwell tried to make the "flying" more realistic than in Kirk Alyn's serials, with George Reeves being hoisted with a harness and piano wire for takeoffs and landings. On-screen, it was believable. Off-screen it was a near-disaster; the wire broke early in the production and George crashed to the ground. Maxwell's first reaction was to cry, "My God—the star!" He knew, given a shoot of just eleven days, what it would mean to lose his Superman. "We went lickety-split," recalled Coates. "I took my money and went home! It was nice working together and everybody liked everybody, but in the final analysis it was a crock of crap!" _Mole Men_ opened in 1951 and for the next year and a half it appeared as a Saturday matinee in movie theaters across America and as far away as Scotland, where a vacationing Coates was shocked to see her "crock of crap" playing at a local movie house. The verdict for National Comics was unequivocal: The movie pilot was a success, so upward and onward to television.
The TV series opened the way every Superman project did, with a creation story. It welcomed back old fans of the comics and radio productions and introduced new ones to the narrative. The opening narration was word-for-word the same as in the radio series, which isn't surprising since Maxwell oversaw both. On Krypton, Jor-El tried and failed to convince the ruling council that its planet was about to be sucked into the sun, then he sent his infant son rocketing to Earth. Here, a young Clark watched his powers slowly surface the way they did in the _Superboy_ comic books, and he heard his mother explain, when he was twelve, why he could see through rocks and do other things that set him apart. His adoptive parents were the Sarah and Eben Kent dreamed up by novelist George Lowther and brought back to life on the radio. The storyline was familiar, but TV added a decidedly new kick to the myth. Here was Superman in real life, and he was sturdier and more steadfast than kids had pictured from the cartoons, imagined on the radio, or seen in the big-screen serials. Here, finally, was a flesh-and-blood Superman worthy of Jerry and Joe's hero.
The pace of filming for TV was even more frenetic than it had been on _Mole Men_ , with just twelve days to complete each batch of five half-hour episodes. That meant working from seven in the morning until dusk six days a week, with no time for retakes. The undertaking was saved by George's photographic mind, which let him memorize the twenty-four pages of dialogue that came his way every day. Scenes were shot in blocks. Monday might be _Daily Planet_ sequences. Tuesday all eyes would be on the gangsters in their boxy suits and rumpled fedoras. It drove the actors mad, reading lines without knowing the context of the story, or even which story it was. The newspaper never had a newsroom—that would have required too many desks and extras—just cramped private offices. Other money-saving precepts: No need for more than two gangsters; limit crowd scenes to the opening, where everyone was looking skyward; and make sure the actors never changed clothes so stock scenes could be spliced in anywhere. Clark stayed in his gray double-breasted suit with padded shoulders. Jimmy wore out his sweater and bow tie. Lois had one hat, one suit, and one set of earrings. On Krypton, Jor-El used Buster Crabbe's old shirt from the _Flash Gordon_ serial while other ruling council members recycled costumes from Captain Marvel and Captain America movies. So what if they were the competition? What mattered to the Superman team, as to most other TV crews back then, was being on budget, which was just $18,500 per episode, or barely enough for a single set in a B picture.
Special effects also were done on the cheap. The bullets that bounced off George were blanks and the revolvers he bent were made of soft lead. With a mere $175 budgeted for each episode's flying sequences, it is not surprising that George took another spill. It was the pulley that gave way this time. "That's enough of that," he announced after he dusted himself off. "Peter Pan can fly with wires, but not Superman!" In another episode, George was set to burst into a room. The cast had rigged a door of balsa wood held up by two-by-fours, but they forgot to take out the extra lumber. "George came running up the stairs right into the frame," recalled Lee Sholem, who directed that show. "The balsa wood barely gave way because George bounced off the heavy wood, and fell to the floor—unconscious." George wasn't the only one taking his knocks. Playing Lois, Phyllis Coates, who prided herself on ad-libbing rather than following a script, moved closer than called for to a thug she was confronting and "he decked me! I was knocked out cold, and they sent me home—that left me a little black-and-blue, but I was back at work the next day." A knockout blow was no reason to stop filming; the director reshot the scene before Lois's face started to swell.
Just getting dressed was a challenge for Superman. George's costume came in two gray-and-brown wool pieces that he dubbed the "monkey suit." It had to be sewn into place on him every day, which meant standing still for an hour and suffering the indignity of having clothespins hold his suit together when the sewing didn't. "What is a man my age doing running around in my underwear?" he would mumble as his personal dresser worked on him. There was a silk cape, too, along with rubber latex padding that he wore under his shirt to lift his sloped shoulders and thrust out his chest. Altogether the outfit weighed twenty pounds, and the materials in it gave him a rash. Imagine effortlessly battling villains with that on, under hot studio lights, with no air-conditioning in the heat of a Los Angeles summer. No wonder he never smiled as Superman.
That also explains why, between takes, George would sit in front of blocks of ice with a wind machine aimed at them and him. The effort of shooting was enormous, so it was a blessed relief when the action came to a temporary halt at precisely four every afternoon and George would pour himself, Phyllis, and sometimes Robert Maxwell what he called "my olive"—a martini or, for a change, a brandy. "This drove the production office crazy, and George would say to them, 'Go shit in your hat!' " said Coates. "George's face was like a baby's butt—he never did show it when he would drink." Another way the actor let off steam was telling stories, which was especially entertaining when his audience included John Hamilton, the gruff white-haired actor who played Perry White. Hamilton liked his olive at least as much as George did, although he did his drinking after work at the Brown Derby. The off-camera scene that first year of filming was worthy of its own shoot: Superman holding court from a director's chair, a cocktail in one hand, a cigarette in a silver-and-black holder in the other, his cape tucked neatly behind his back.
All this was exhilarating to Jack Larson, who was just twenty-three when he signed up to play Jimmy Olsen. Reconnecting with Superman was a boyhood dream come true for Larson, who had adored the character back when he read _Action_ comic books at Campbell's Drug Store in Los Angeles, slurping a cherry phosphate and hoping his parents would keep talking so he could keep reading. His real ambition was performing on the stage in New York. A short stint on the Superman TV show, for which he was offered $250 a week, seemed like the surest route to pay his way to Broadway. But doubts set in for poor Jack on day one, which he spent locked in a safe, sweating while he waited for Superman to rescue him. "This," he explains, "is not what I had been preparing to do in life. I was young and energetic and innocent and eager and dumb as could be. I didn't know that Clark Kent was Superman because he had on a pair of horn-rimmed glasses." Larson was six years older than the radio Jimmy, yet his version was more boyish and innocent. The straight-out comedy didn't come until later. In the first season he was mainly getting in trouble (he did most of his own stunt work, "which meant I sprained a lot of things"), getting wet ("they were always trying to drown me, and the water was cold and dirty"), and getting rescued (in the nick of time). On the radio and in the comics, Jimmy had been one in a series of supporting roles around Superman. Larson's TV Jimmy connected so completely with viewers that the copyboy became a star—such a star that today his trademark bowtie is enshrined at the Smithsonian.
Robert Shayne was another regular in the cast, as Metropolis Police Inspector Bill Henderson. Maxwell had just one condition for hiring the fifty-year-old character actor: that he dye his hair gray so he would look older than George Reeves, who dyed his hair to _hide_ the gray. The true test of Shayne's forbearance, and Maxwell's, came near the end of shooting for the first season, when real-life federal agents came onto the lot with a subpoena ordering Shayne to appear before the House Un-American Activities Committee. He was a self-described "rabid union man," but denied being a communist and surely wasn't a subversive. Reeves and the rest of the cast, along with Maxwell and the honchos at National Comics, stood behind Shayne when he came under fire, first from congressional inquisitors, then from the show's sponsor. They weren't in a position to take on Wertham and McCarthy, but they could stand up for a man they knew and liked.
Impressive as the supporting cast was, the TV show, like the other incarnations of the story, turned around Superman himself. Bud Collyer, the first flesh-and-blood Man of Steel, had set the standard. Collyer lowered and raised the timbre of his voice as he switched between Superman and Clark, making the changeover convincing. Maxwell's wife, Jessica, who was the dialogue director for the TV show, would follow George around the set urging him to do the same—but he just couldn't master the switch, and soon he stopped trying. The result: a Superman who sounded just like his alter ego. They both swallowed their words. They looked and acted alike. There was no attempt here to make Clark Kent into the klutz he was in the comics. No slouching, no shyness. George portrayed the newsman the way that he knew and that Jessica's husband told him to: hard-boiled and rough-edged, Superman in a business suit. As Clark Kent, George would shed his rubber muscles and add thick tortoiseshell glasses with no lenses—that was the sum total of the switch.
But it worked. It worked because fans wanted to be fooled, and because of the way George turned to the camera and made it clear he knew they knew his secret, even if Lois, Jimmy, and Perry didn't. This Superman had a dignity and self-assurance that projected even better on an intimate TV screen than it had in the movies. George just had it, somehow. He called himself Honest George, the People's Friend—the same kind of homespun language Jerry and Joe used for their creation—and he suspended his own doubts just as he wanted viewers to. He looked like a guy who not only could make gangsters cringe but who believed in the righteousness of his hero's cause. His smile could melt an iceberg. His cold stare and puffed-out chest could bring a mob to its knees. Sure, his acting was workmanlike, but it won him generations of fans. Today, when those now grown-up fans call to mind their carefree youth, they think of his TV _Adventures of Superman_ , and when they envision Superman himself, it is George Reeves they see.
In this first season on television, Superman's fans were not just kids but adults, who were now more of a focus for Maxwell than they had been with the radio show. That became especially apparent in the content of the first year's twenty-six episodes. Phyllis Coates's Lois was, as she said, "a ballsy broad" who never let up on Clark or on chasing down a story. The plots were film noir, with enough kidnappings, murders, suicides, and men slugging women to alarm not just parents but Maxwell's bosses. It was dicey to pull the leg braces off a crippled girl, the way heavies did in an episode called "The Birthday Letter," but gassing to death a cocker spaniel? Another episode tested the no-kill decree: Superman deposited two crooks who had learned his secret identity on the top of a snowcapped mountain and, just after he flew away, both fell to their deaths. In "The Evil Three," the most aptly named and nail-biting of the shows, Perry and Jimmy spent the night in the ramshackle Hotel Bayou. Its proprietor had already killed his uncle, and he tried to do the same to the editor and copyboy. That failed, but he did manage to viciously silence a wheelchair-bound old lady named Elsa by shoving her and her chair down a ramp to the basement. Splat.
The melodrama didn't win any fans at the PTA. A self-proclaimed watchdog group, the National Association for Better Radio and Television, listed _Superman_ among seventeen children's series it found "objectionable," and that was before "The Evil Three" had aired. "Monitored programs," the group wrote, "included a demonstration of how to cripple a wrestler, a doctor using drugs to hypnotize patients, torture, and the kidnapping of a child." Wertham would be harsher still in _Seduction of the Innocent:_ "Television has taken the worst out of comic books, from sadism to Superman.... The television Superman, looking like a mixture of an operatic tenor without his armor and an amateur athlete out of a health-magazine advertisement, does not only have 'superhuman powers,' but explicitly belongs to a 'super-race.' "
The brewing controversy didn't help in the search for a company willing to underwrite the broadcast. It took just two and a half months for the production team to shoot twenty-four new episodes, with the _Mole Men_ film cut into two parts and renamed "The Unknown People" to make a total of twenty-six. It took nearly two years to find a sponsor. Flamingo Films, a firm made up of twenty-somethings, bought the distribution rights early on from Harry Donenfeld, paying thirty million dollars for a thirty-one-year deal. Flamingo pitched the production to Kellogg's, which remembered all it had earned from Superman on the radio but, like other would-be sponsors, was afraid of the new medium of television. Higher production expenses meant that it cost ten times more to underwrite a show on TV than on radio. The commercials also were much more expensive to make, since film was needed in addition to audio. Most disquieting of all was the worry that this new medium, with its more explicit format and more intimate entry into people's home, would offend buyers of breakfast cereal and anything else sponsors were peddling.
That last apprehension was especially daunting with _Adventures of Superman_ , which Kellogg's wanted to pitch to children but couldn't with the kind of violence Maxwell was filming. The end result was a painstakingly engineered compromise: Maxwell shortened the mountain death scene, eliminated drive-by shootings, and watered down other elements that were especially offensive to the cereal maker. Elsa and her wheelchair still made their way into the cellar, but it was left to the imagination of viewers how they got there. Ads also had to be shot, including one with George dressed as Clark and telling the children who flanked him about "that favorite new cereal of mine, Kellogg's Sugar Frosted Flakes." Finally, in September 1952, the show aired on its first station, in Chicago. Small cities lined up after that, with Los Angeles coming aboard the next February. By April, the show was airing in New York, the biggest market, along with fifty metropolitan broadcast areas, and it had become the gauge for measuring the success of adventure TV.
The press cheered. "At long last, _Superman_!" wrote the _Brooklyn Eagle_ , with _Variety_ agreeing that "this one's a natural for TV." Kellogg's signed on for two more years. The reaction on the street was even more bullish. Larson recalls that he couldn't walk down Madison Avenue without cabbies yelling, "Hey Jimmy, where's your buddy? Where's Superman?" One day Larson was in a restaurant on Eighty-second Street having lunch with a friend when he noticed that hundreds of kids had gathered outside, peering through the window. "The police came into the restaurant and apologized. They said I was creating a public nuisance and they had to get me out or something bad could happen. They gave me an escort into the Metropolitan Museum around the corner. I was followed by a mob of kids but the museum refused them entrance and I was given refuge. That's the first time I realized I was in a lot of trouble. I was the first TV teen idol."
Phyllis Coates realized her own newfound celebrity less than a week after the show started airing in Los Angeles. "I had to change the color of my hair! From auburn to blonde. And you know why? I couldn't go anywhere without being mobbed. Not only boys and girls but big, grown-up women. They'd spot me in the super-market or just taking a walk with Crinker," her two-and-a-half-year-old daughter, said Coates, who had grown up on a cattle ranch in Odessa, Texas. "The first thing they wanted to know was how it feels to be held in Superman's arms."
THE FIRST SEASON OF THE TV _Adventures of Superman_ was Maxwell's last. Jack Liebowitz, ever the accountant, said the departure was a simple matter of money: Maxwell had promised to spend just fourteen thousand dollars on each episode, which would have ensured a substantial profit, but instead each ended up costing twenty-eight thousand, "so he was out as producer." Maxwell also could be dogmatic, which meant he burned bridges at National as well as on the set, and he was ambitious, to the point that even as he was working on Superman he was negotiating with another company to produce a TV show about Lassie, the heroic Rough Collie. Yet what ultimately did him in, after four years as one of Superman's most influential movers and shakers, was a cultural shift into an era when Americans liked Ike, hated Bolsheviks, and added "under God" to the Pledge of Allegiance. The same way Jerry and Joe couldn't resist their bosses' order to rein in their crusading Superman in the 1930s, so Maxwell was fighting a battle he couldn't win against an image-conscious Kellogg's and a public riled up by Fredric Wertham and his allies. The consensus at National Comics was that it was time for a kinder, gentler Man of Steel on TV and for a loyal company man to handle him.
Whitney Ellsworth brought a fresh perspective and a Superman pedigree even longer than Maxwell's. Ellsworth was the token goy in the National brain trust, having grown up in the Congregational Church, but he shared the Brooklyn background of many comics pioneers and, like Jerry Siegel, he lost his dad as a teenager and it left him bereft. Ellsworth was an editor for Major Wheeler-Nicholson for nearly three years starting in 1935. Harry and Jack brought him back in early 1940 as their first editorial director, a job that included overseeing the Superman books. He picked up some of the writing duties when Jerry was in the Army, penning newspaper strips and the inaugural Superboy story, and helped oversee the Fleischer cartoons, the Kirk Alyn serials, and the radio show. Then his bosses wanted him to ride herd over Jerry and Joe. He did, pressing the creators to "de-sex" Lois, downplay Superman's jockstrap, and generally understand "what sort of censure we are always up against, and how careful we must be."
That background made Ellsworth the perfect choice as Maxwell's successor. He knew Superman. He knew Harry and Jack. He knew what it would take to keep them off his back and Wertham off theirs. He even knew about the televised _Adventures of Superman_ , having written, edited, or otherwise contributed to many of Maxwell's episodes. His budget was nearly 40 percent higher than Maxwell's, and he had the kind of backing from Kellogg's that Maxwell never did. He arrived with so many ideas, and drew up such detailed outlines of how the stories should begin and end, that, as he said, "We never had a writer who had to come in with his own idea." There also was less pressure on the production by the time he moved to Los Angeles. It had proven it could be a blockbuster, the actors knew their roles and their lines, and Superman was the number one reason to turn on the TV for millions of American kids and their parents.
From the start, Ellsworth's shows differed from his predecessor's in ways that were easy to measure. The storylines were ripped more from the comics, which is what Ellsworth knew best. Humor mattered now as much as dark drama. So did science fiction. There were more feel-good plots, like "Around the World with Superman," where, in thirty minutes, the Man of Tomorrow helped a blind girl regain her sight, patched up her parents' shattered marriage, and flew the wide-eyed child around the world. Ellsworth's Superman could make himself invisible and split into two. Crooks had fewer shootouts and wrestling matches with him and were more likely to run into each other or a wall. "Maxwell's first twenty-six shows were a lot more violent than my shows," Ellsworth said looking back. "My concept was that the enemy should be played, for the most part, as semi-comic—and a little bit stupid.... I think the kids liked this better."
More than any episode of his five years, "Panic in the Sky" was Ellsworth at his best. Superman lost his memory trying to stop an asteroid rocketing toward Earth in this show, the last of the 1953 broadcasts. It wasn't the plot that grabbed his audience but the subplots. With Clark not remembering he was Superman, his everyman persona took center stage and the fans loved it. Their hero was manly as always, but also tender and vulnerable. Less alien and more human—more like us. The threat, meanwhile, was not just a bank being robbed but the world being destroyed, an increasingly plausible fear in an era of intercontinental missiles, hydrogen bombs, and flying-saucer sightings. Pint-sized viewers yelled at their TV screens: "I know who you are, Mr. Kent, you're Superman and the world needs you." The show, the most expensive of the 104-episode series, was probably based on a recent comic book and the program's popularity would inspire another comic book forty years later, along with a future TV show and cartoon. Even Superman groupie Jerry Seinfeld would count this episode as his all-time favorite. The key to all that acclaim was simple, says Jackson Gillis, who wrote the script: "Pure fantasy," which was "what I thought the show should be."
Popular, yes, but "Panic" had more than the normal flaws. If you listened closely you could hear the hollow sound of Metropolis residents lumbering down the sidewalk with plywood underfoot, not concrete. Tune your ears when Superman landed on the streaking meteor and you could make out birds chirping. Clark showed himself to Jimmy, Lois, and Perry without his glasses, the centerpiece of his disguise, and without any of these inquiring journalists wondering why Superman disappeared at the very moment Clark had amnesia. Had they been in a mood to question, they might also have asked how viewers were supposed to trust newspeople like them who never carried a notebook. Or how Clark could head for the newspaper storeroom with no hat in sight but be wearing one when he arrived.
One premise that got more credible as the series went on was that a man could fly. After the early mishaps, a mechanical arm was rigged up and Superman lay on an attached Plexiglas pan that had been fitted to his chest and thighs. The tray could be turned and tilted, with twenty stagehands pulling to lift and lower the lumbering arm and the camera following it on a hydraulic dolly. There were no more wires, although there still was a wind machine and compressed air to give the feel of Superman whooshing through the air when really it was the filmed background that was moving. George was skeptical. "I had to get in first, then my helper, and still one other guy to make sure it held," said Thol Simonson, who was brought in to manage the special effects. The aerial action happened now in three phases: George got a running start, jumped on a springboard positioned out of sight below a window, and dove head-first through the opening and onto a pile of wrestling mats. Then came film of the mechanical arm moving him through the air. The last sequence showed him hitting the ground feet-first as if he were landing. This all-human, no-animation approach made more convincing not just the flying but the science fiction itself.
Spicing up the fantasy, upgrading the aeronautics, and toning down the violence weren't the only changes Ellsworth wrought. The action was shot in color starting in 1954, although high costs meant that it would be several years before the color film was aired and it wouldn't be until the 1960s—when color TVs began to proliferate—that most viewers actually saw Superman in his new blue and red costume. The actors were evolving, too, because they felt more comfortable in their roles and because the new producer and his team insisted on it. Perry White was less grizzly and more teddy bear, showing his young journalists how to do their jobs rather than just yelling at them. Inspector Henderson was more relaxed. As for Jimmy, "I loved and admired the first twenty-six shows under Maxwell, but when they lightened up they turned me loose and I got to do comedy," says Larson. "That is exactly what I wanted to do."
The biggest transformation in the cast was a new Lois. Phyllis Coates left after the first season, despite what she said was an offer from Ellsworth to pay her five times what she had been earning. She had the chance to co-star in a new show and was concerned about being typecast as Lois Lane. She also worried about all the drinking on the Superman set, especially by George, and given her family history of alcoholism she felt staying could have a "devastating effect." Looking back, she also had no doubts about what the transition from Maxwell to Ellsworth meant for the show: "I've had people write to me to say that as kids they watched those early shows where the heroes were clear, the bad guys got it. Superman was really Superman.... The new gang came in and turned it all to pudding!"
Her successor, Noel Neill, was a lot like Ellsworth, softer-edged but steeped in the Superman tradition. Daughter of a chorus girl and a journalist, the Minneapolis native had acting in her blood and newspapers in her home. Her résumé included singing for Bing Crosby, playing in westerns and bobby-socks movies, and sharing an agent with Jack Larson. A leg-art poster of her in a bathing suit leaning seductively against a rock ledge was a favorite for GIs. But what really caught Ellsworth's eye was that she had been the first live-action Lois, alongside Kirk Alyn in the serials. She hadn't read Superman growing up—"comic books in those days were a boy thing"—but as soon as she landed the serial role "I rushed out and bought a book to see what Lois Lane looked like. I wanted to see what she wore, how her hair was, this and that."
When Ellsworth called to offer her the role, Neill was ecstatic even though all he was paying was $185 per episode, or half what Coates said she got. The perks were even fewer than in the serials. She had to dress her own hair, bring her own shoes and socks, and shoot scenes in such quick succession that she had no time to learn her lines or to socialize. She got no royalties when the shows were rebroadcast. She was never invited to appear in the commercials for Frosted Flakes, which more than doubled Jack and George's salaries. Kellogg's, she says, worried about how scandalous it would look if she turned up at the breakfast table with Jack or George. Teenager rock and rollers surely would have been delighted, but not their priggish parents.
Neill found it troubling enough to have bosses who were "very, very, very, very cheap," and worse to have a director, Tommy Carr, who gave her a hard time from day one. He didn't like the way she said her hallmark line, "Gee, Superman, am I ever glad to see you!" and had her repeat it until she started to cry. That's when Superman stepped in. "George saw what was happening and immediately asked for a break. He walked over to Tommy and calmly said, 'Why don't you give the kid a break?' " Neill remembered. "After George intervened, he eased up and let me do it my way."
Neill's Lois reflected more than anyone the change from Maxwell to Ellsworth. She was easier on Clark than Coates had been and more dazzled by Superman's rescues. It is no accident that none of Coates's pictures show her smiling, whereas Neill is smiling in every shot. It also is understandable that they didn't like one another. Neill denied ever meeting Coates, whom she called "whatshername." Coates said George introduced them and that Neill told her, "I _hate_ you!" Which Lois was the better fit and whether the show headed downhill after Maxwell's first season are matters still hotly contested. The truth is that the Neill-Ellsworth version had less punch but more warmth than the Coates-Maxwell episodes. Take your pick.
Lost in all those arguments is all that remained the same. Superman still took on third-rail issues like atomic power and political corruption, as he had in the comics, radio shows, and Maxwell TV shows, and he still got skewered by the National Association for Better Radio and Television. Ellsworth had to do the same dance that Maxwell had on the phone every week with Jack Liebowitz. Ellsworth: "Jack, I got no money." Liebowitz: "Well how much do you need?" Ellsworth: "I need $25,000 this week." Liebowitz: "So much?" Ellsworth: "Well yes, so much. Maybe a little more." Kidnappings still were part of _Adventures of Superman_ plots, along with hauntings and rough times for ladies and dogs. There were even killings. What was different with Ellsworth producing the shows is that the dogs survived. In the one Ellsworth episode in which the villains didn't, their deaths happened off-screen. The show knew whom it wanted to reach—kids—and it aimed squarely at them with fare designed to placate their parents and Ellsworth's sponsors.
That same kid-friendly ethic was being applied in the Superman comic books, and for the same reason. Wertham had the entire industry running scared. In just two years, from 1954 to 1956, the number of comic book titles published in America plunged from 650 to 250. It was "like the plague," observed longtime comics artist Carmine Infantino. "If you said you drew comic books, it was like saying you were a child molester." Editor Al Feldstein worried that "the industry's dying, the comics industry. All they need is a spade to bury it." By the fall of 1955, a Comics Code Authority had been set up, and its black-and-white seal appeared in the upper right corner of every book it had approved. Without a seal, most distributors wouldn't carry the comic. Winning the Code Authority's approval meant scrubbing titles and stories of horror and terror, along with anything that could be seen as disrespectful of police, parents, and government. No vampires or werewolves, references to physical afflictions, or exaggerations of the female anatomy. Illicit sex couldn't be hinted at and characters could not be divorced. No ads for liquor, tobacco, fireworks, or weapons. It was the kind of self-policing the motion picture industry had been practicing for twenty-five years, only more so. These restrictions were less severe than the outright bans Wertham and other critics wanted, but they meant much more censorship than printed publications, with their First Amendment protections, usually agreed to. It was a strategy of self-preservation, which made sense given that a dozen publishers had already been forced to shutter their doors and more than eight hundred comic book artists and writers were looking for work.
No publisher pressed the stay-pure strategy on more fronts than Harry and Jack did. They distanced themselves from the "mercenary minority" who were producing most of the violent and vulgar material. They established a "Distinguished Advisory Board" of child behavior experts who gathered the latest evidence of how comic books were helping kids read, helping teachers teach, and, as one panel member told _Time_ , offering "the same type of mental catharsis to their readers that Aristotle claimed was an attribute of the drama." Harry and Jack adopted their own standards of editorial purity, as spelled out in a memo from Ellsworth when he was editorial director, that went even further than the Comics Code. The inclusion of females in Superman stories was discouraged. "Expressions having reference to the Deity" were forbidden, and so were bloodletting and killing, even of killers. "The use of chains, whips, or other such devices is forbidden. Anything having a sexual or sadistic implication is forbidden," the guidelines said. "HEROES SHOULD ACT WITHIN THE LAW, AND _FOR_ THE LAW." Ellsworth had to sanitize his productions even further, banishing killings even offscreen, dumbing down the bad guys, and steering clear of controversial social issues. There was one last move Jack and Harry made in the mid-1950s that Wertham would have found interesting had he known: They quietly acquired the distribution rights to America's sexiest magazine, _Playboy_.
Stories of kids who tried to take wing and fly, as Jerry Siegel had as a child, were always a sore spot for Superman, and never more so than in that era of the inquisition. Newspapers glommed on to the tales. Eight-year-old Larry King of Columbus, Ohio, spread his homemade cape over his back and jumped from his second-floor fire escape, explaining from his hospital bed that "I thought the air would get under my towel and float me down like it does Superman." James Henderson, another eight-year-old second-story boy, took off in his Superman suit and landed with a sprained ankle. "The darned thing wouldn't work," the Des Moines youngster complained of his costume. Twelve-year-old Robert Van Gosig of New York wasn't so lucky: He slipped on a wet ledge while playing Superman on his tenement roof and plunged to his death.
After Jerry read articles like those, his comic book Superman warned readers that only he could perform such feats of derring-do without getting hurt. Superman costumes carried similar cautions. It was one thing for young fans to break their beds as they pretended to fly; it was quite another for them to break their necks. "We were very conscious of that," recalls Jay Emmett, who oversaw the licensing of Superman products. "We couldn't have kids buying costumes if they were going to jump out the window." George Reeves felt a special burden with kids who mimicked him. He gave up smoking and drinking in public. He pushed National Comics to stop selling costumes and capes. And, in a 1955 episode of _Adventures of Superman_ , he warned listeners that "no one, but no one, can do the things Superman does. And that goes especially for flying!"
National's multi-front multimedia campaign worked. Making Superman more squeaky-clean on the printed page—and lightening up on the TV show's noir—eventually inoculated him against the attacks of critics like Wertham. The censorship campaign also backfired by turning Superman stories into the kind of forbidden fruit that made kids more determined to read on and tune in. Jack and Harry's most brilliant stroke was launching the new TV series in the middle of the assault on comic books, when other superheroes were running for cover. The publishing partners had always played better offense than defense. Their radio show, film serials, and animated movies had boosted rather than diminished the popularity of Superman comic books and comic strips, and TV did that in spades. Call it synergy, or just street smarts.
Another measure of success was the way Superman merchandising was taking off. It had always been done well, but in the early 1950s National capitalized on the popularity of its television series and "licensing income soared," says Emmett, who oversaw those sales. Building on his Superman success and his relationship with his uncle Jack Liebowitz, Emmett in 1960 launched a broader-based product-management operation called the Licensing Corporation of America, whose clients would eventually include Pat Boone, Batman, and the National Basketball Association.
TV also spread Superman's fame the rest of the way around the globe. Dubbed into Japanese, _Adventures of Superman_ became that country's most popular TV show, with Emperor Hirohito writing George to say that Superman was his _ninkimono_ , or fave. The show was playing in Paris, too, with simultaneous subtitles in French, Arabic, and Japanese. And all this was happening during the height of the comic book scare. Fredric Wertham had gone to war with the Man of Steel and, like so many Superman foes over the years, the venerable psychiatrist had been humbled.
THERE WAS ONE ADVERSARY SUPERMAN couldn't defeat: himself. Not that anyone imagined the Man of Tomorrow, the world's preeminent Pollyanna, trying to take his own life. The notion was so abhorrent that National Comics had banished any depiction of suicide—by Superman or anyone else—from its pages and airwaves. Yet on the morning of June 17, 1959, America woke to the headline TV SUPERMAN KILLS SELF. Parents tried to hide the paper from their kids. Kids were sure the reports had to be mistaken, since everyone knew bullets bounced off the Man of Steel. But the media spelled out the all too real and gruesome details: Friends had found George Reeves's naked body splayed across his bed, faceup in a pool of blood, with a bullet hole in his temple and a German Luger nestled between his feet.
To his admirers, George had everything to live for. _Adventures of Superman_ had been reaching thirty-five million TV viewers a year, only half of whom were kids. After 104 episodes, there was practically no one in America who did not recognize George Reeves. He delighted moppets with a 1956 appearance on _Romper Room_ and two days afterward he thrilled their parents and grandparents on Tony Bennett's variety show, the first time those with color TVs could see what he looked like in his radiant red-and-blue costume. Later that year he achieved the ultimate measure of celebrity: a guest slot on America's most-watched show, _I Love Lucy_. It was true that _Adventures of Superman_ had stopped production in November of 1957, but reruns were still airing and earning him as much as $1,000 a week, he had been approached about shooting twenty-six new shows, and, having already directed three episodes, he was bullish about directing more.
At the time of his death, George was due to be married in just three days. He was two weeks away from an Australian tour, in which he would play guitar in a band, then wrestle his sidekick Gene LeBell, a judo champion who dressed up as the evil Mr. Kryptonite. He had an offer to star in _Wagon Train_ , a TV western that would go on to run for nearly ten years. "Nothing was bothering him," LeBell says of his friend, who was only forty-five. "He wasn't going to shoot himself."
Other observers offered a less rosy portrait, starting with George's offscreen appearances, which were part of his deal with National. They were the kind of thing Jerry Siegel had loved doing, but George never found them easy. In 1954 a young fan socked him in the eye "to see if he would flinch." Another pointed a loaded Colt .45 at him. The gun story became part of his legend and he played the hero, saying he had convinced the boy to put down the gun because, while Superman could fend off a bullet, a bystander could be hurt by the ricochet. His own variety shows, like the one planned for Australia, were financial disasters, and Noel Neill said he had been staying in his room more on these tours and drinking by himself. The low point was a stop in North Carolina where only three people showed up—a young boy and his parents. George's other investment schemes didn't fare well either, including one for a "Motel of the Stars" run by a con man who defrauded not just George, but Mickey Rooney, Burl Ives, and Debbie Reynolds.
At the time of his death, George hadn't had an acting job since _Adventures of Superman_ went on hiatus nineteen months earlier. "Here I am wasting my life," he told Ben Welden, who played a thug on the Superman series. George had almost died twice in car accidents. The first was in March 1956, when a truck loaded with construction materials reportedly ran a red light and rammed into his sports car, sending him wheeling into another truck and leaving him in the hospital for a week. Three years later he took a curve too fast and skidded into an embankment, leaving him with a concussion and a five-inch gash on the forehead.
The last of his six Superman seasons was aimed more than ever at a young audience and had weak scripts as well as tired execution. He was looking and acting middle-aged now. Clark wore eyeglasses with actual lenses. Lingering pain from the first car crash prompted him to take pills and made it tougher for him and Superman to keep in shape. He strapped on a girdle these days before stepping in front of the camera. His hair was noticeably thinner, whiter, and in need of more frequent touch-ups with food coloring along with hair dye. Karate chops, which were easier to deliver and to fake, had replaced punches. George had always liked his olive, but now he liked it in the morning, too.
His love life was even more unsettled. It had begun promisingly and conventionally when, in 1940, he married a young actress named Ellanora Needles. They were separated first by his enlistment in the Army, then by separate careers and waning interest. The divorce was formalized in 1950, a year before he became Superman. The next year he started an affair with Toni Mannix, a former Ziegfeld Follies beauty who was eight years older than he was and fifteen years younger than her husband, Metro-Goldwyn-Mayer vice president and enforcer Eddie Mannix. She called George "the Boy." To him, she was "Mother" or "Ma," Eddie was "Pop," and the unorthodox relationship among the three churchgoing Catholics was "the Arrangement," which suited everyone perfectly. Eddie got to be with his exotic mistress and, given his weak heart, he had the security of knowing that Toni would have George to look after her. Toni got the man she loved, George, and an exhilarating lifestyle that brought her to the Superman set every day with a picnic basket brimming with gin, vermouth, olives, and hors d'oeuvres that made her a favorite with cast and crew. George got a house in Los Angeles's exclusive Benedict Canyon, an Italian sports car, a Minox camera, a sauna, and an open tab at his favorite restaurant, Scandia—all delivered by Toni and paid for by Eddie. It was a love triangle as tangled as Superman's with Clark and Lois.
The arrangement worked for everyone until George called it off early in 1958. One night at Toots Shor's Restaurant in New York he had met Leonore Lemmon, a barfly infamous for being the only woman ever tossed out of the Stork Club for fistfighting. George couldn't take his eyes off her ample breasts, jet-black hair, and white skin. When she turned up at his hotel room at two the next morning with wine and squab, they became inseparable. She moved to California and into his Benedict Canyon home, alienating his old friends and inviting new ones to what seemed like an around-the-clock party. "She makes me feel like a boy again," George told everyone. Toni called her "the whore" and allegedly had Eddie's yardman make harassing calls to George and Leonore. The two were due to be married three days after his death, or so Leonore said.
She rarely said the same thing twice and never seemed credible to George's friends, especially when it came to what happened the night he died. She told police that George had gone to bed about midnight, after which friends stopped by. George appeared in his robe and got into a squabble with one of the guests, then apologized and went back to bed. As he did, Leonore predicted that "he's probably gonna shoot himself." She could hear him open the dresser drawer and get out the gun. Minutes later, she said, a shot rattled the house. Those basic details remained relatively consistent in her retellings to investigators and friends until thirty years later, when she told a dramatically different version. George never came downstairs that night. She thought he might have planned to kill her, too. And given his difficulty finding work after being typecast as the Man of Steel, "the only thing that killed him is something very simple, very easy: Superman killed him."
The forensic details of the case were sketchy and spilled out haphazardly, fueling suspicion that the crime scene had been tampered with and casting doubt on the suicide declaration by the coroner and police. Why, researchers who have reviewed the evidence wonder, did the embalmer sew together George's head wounds before the coroner thoroughly analyzed them? How did the bullet casing end up under his body? Why were there no fingerprints on the gun or powder burns on his face? What about the extra bullet holes discovered in the floor under a rug in the bedroom? Why did it take forty-five minutes for anyone to call the police and why, in those anxious minutes, did Leonore phone high-powered Washington defense attorney Edward Bennett Williams? He told her to call the cops and keep her mouth shut, and later told friends that "only Lem can turn a suicide into a homicide."
The weeks after the shooting were a circus, with George's mother, Helen, as the ringleader. She came to town in a wheelchair, telling anyone who would listen that her son hadn't killed himself and hiring a prominent criminal attorney, a detective agency, and new pathologists to pore over the evidence. Toni visited the house with Jack Larson and—"to exorcise this evil"—used the heel of her shoe to nail tiny prayer cards over the bullet holes. Leonore went separately, with a friend who ripped the bloody sheets from George's bed and tossed them into the bathtub, then helped Leonore bundle up her things and leave L.A. for good. George's will bequeathed most of his fifty-thousand-dollar estate, including his home and car, to Toni and nothing to his fiancée, leading Leonore to insist there must have been a second will.
The chief coroner personally helped perform an autopsy a week after the death and confirmed it was suicide. George's funeral was held July 1; an open casket revealed his body dressed in a Clark Kent suit, with his head looking as if it had been hastily stitched back together. It would take another seven months for Helen to complete her probes and have her boy cremated. She tried to drown her grief in drink, which was the one thing she shared with Toni and Leonore. Helen died five years after George and her remains lie alongside his in a crypt at the Mountain View Cemetery and Mausoleum in Altadena, California. There are no gravestones, just her simple plaque honoring MY BELOVED SON "SUPERMAN."
Neither his burial nor the deaths of witnesses and investigators have quieted the storm. Was it murder, suicide, or accident? Arguments for all three have been made in fifty years of newspaper and magazine investigations, hours of TV exposés, and a trio of books, with more in the works. George's business manager died before he could complete his tell-all. So did Leonore. A 2006 movie called _Hollywoodland_ , starring Ben Affleck and Adrien Brody, hedged its bets by imagining different death scenarios: Leonore shooting him accidentally, George killing himself, and Eddie Mannix hiring a hit man. Eddie had the contacts to do it but there is no evidence he did—and no motive, since he liked George enough to get him a cherished membership in the movie directors' guild and hoped he would get back together with Toni, who was driving Eddie crazy. Toni was mad enough at George to want him dead, and a young public relations man who was living at her mansion said he heard her confess to a priest that she was part of a murder conspiracy. But that didn't fit with her hope that George would come crawling back to her, or with the shrine she built after he died, where George's picture hung next to Jesus'. Leonore, with her mob friends and gutter diction, would have made a convincing bad gal if this had been a Superman TV treatment: Imagine George calling off their engagement, then Leonore shooting him on purpose or by mistake as they struggled over the gun. But even scriptwriters would have had a difficult time explaining how that happened with three witnesses in the house who vouched for Leonore.
George's young fans advanced their own theory after watching the rerun of _Adventures of Superman_ that aired the day after he died. In "Disappearing Lois," Milton Frome played a thug who fired his pistol at Clark Kent at close range. Since Lois and Jimmy were present Clark was forced to play dead, hitting the deck with a thud. "That night," Frome recalled, "some neighborhood kids came by our house and told my son Michael, 'Hey, your father killed Superman.' "
No one will ever know for sure who or what killed the TV Superman, but good starting points are the typecasting that extinguished his film career and the three unhinged women in his life—Leonore, Toni, and Helen—each of whom wanted him as hers alone. He was aging and slowing, afflictions especially difficult for Superman to handle. On his last night he argued with Leonore and her house-guests, which boiled his blood. None of that would make a self-contained man like George do something he couldn't have imagined otherwise. But he had apparently talked about killing himself as far back as high school and as recently as a year or two before his death, according to one Reeves historian who has spent decades on the case. What might have pushed him to do it this time was the incendiary mixture of alcohol—the coroner said his blood level was .27 percent, three times today's legal limit in California and the equivalent of six martinis—and the painkillers he was taking for the migraines he had suffered since his car crash. His clothes were off because he slept in the buff, and the gun was within reach in the dresser. Whitney Ellsworth, who knew firsthand what dipsomania could do, labeled it temporary insanity. George's admirers then and now call it a tragedy.
There were silver linings even in that calamity. His fans adored him so much that many still remember where they were and what they were doing when they got the news of his death, the same way people do about John F. Kennedy's. Losing George meant being robbed of a touchstone of their youth. But decades of reruns—along with videotapes, DVDs, and an elegant and worshipful fanzine called _The Adventures Continue_ —have let devotees visit with him again and have ensured that George, like President Kennedy, will remain forever young. The actor also seemed to say his own regretful goodbye on the last episode of the TV show. Waking from a dream in which he imagined he had superpowers, Jimmy said, "Golly, Mr. Kent—you'll never know how wonderful it is to be like Superman." After a considered pause, Clark answered: "No, Jimmy, I guess I never will."
As for Superman, the Reeves suicide could have disillusioned fans and tarnished the icon. Jack and Harry were worried enough that they contemplated bringing on another actor to play the part, or centering a series around Jimmy Olsen. Either move would have seemed panicky and neither was needed. While it would take thirty years for the superhero to make his way back to television in a fresh live-action series, he thrived in other media. The 1950s TV show, even more than his radio and film work, had taken Superman beyond the rarefied world of comic books and made him a centerpiece of popular culture. Television was now the medium that mattered in America, and no TV show mesmerized kids of all ages more than _Adventures of Superman_ , which didn't surprise anyone who had watched the Man of Steel conquer comic books, radio, serials, and cartoons.
George's death also spawned a hypothesis called the Superman curse, which argued that misfortune befell anyone who was creatively involved with the Man of Steel. Jerry and Joe had been barred from their creation. Kirk Alyn couldn't land an acting job. Now George was gone. Any news, even the bleakest, added to the legend.
# **CHAPTER 7**
# **Imagine This**
FROM THE BEGINNING SUPERMAN FANS had longed for a story like this: "Mr. and Mrs. Clark (SUPERMAN) Kent!" Now here it was, in the new comic book devoted just to Lois Lane, with a cover sketch of her in an apron and pearls as her humdrum husband hurtles off in cape and tights to save the planet. "Hurry home, dear. Supper will be ready soon!" she chirps, thinking, "Would our neighbors be astounded if they knew my husband Clark is leaving our home through a secret tunnel, as... Superman!"
It started when Clark proposed and Lois said she couldn't, not if there was the faintest hope Superman might someday pop the same question. That did it. To her amazement, he doffed his spectacles, his business suit, and his covert identity. "Why did you wait so long?" she asked as they embraced. Superman: "I feared that if I married you, my enemies would seek to strike at me by harming you! But I've thought of a solution... as far as the world will know, you'll be marrying meek, mild Clark Kent! You alone will share the secret of my real identity!" And so it was, as they said their nuptial vows, then set up a snug household in the suburbs. Even her sister Lucy had to stay in the dark, telling Lois, "I'm glad you and Clark get along so well! Frankly, I thought you'd never get over your crush on Superman."
The tale, published in August 1960, was the first in a series of "imaginary stories" that helped define Superman during America's decade of discontent. This was a bid by National Comics to create fresh and arresting threads for writers who were running out of them. It also was an attempt to tap into the era's capsizing of conventions, even if the comic book outcome was more happily-ever-after than New Frontier. The very notion of tackling heretofore unthinkable topics and offering zany flourishes to timeworn plots was revolutionary, at least to the adolescent keepers of the Superman flame. When Jerry Siegel had proposed a working partnership between Lois and Superman in the K-Metal story in 1940, it was hushed up. Now those two were life partners, upending not just the love triangle and Clark and Superman's bachelorhood but the sacrosanct secret of Clark's alter ego. Romantics cheered, as did all the four-eyed boys who dreamed that a pretty girl would fall for the hero in them. Radical indeed, and vital to keeping Superman atop the sales charts. Yet the powers that be at National knew there were limits. Titillate readers, yes, but not to the point where it toppled the pillars of the Superman biography and canon, which comic book connoisseurs called the "continuity."
So they labeled this story and scores of others imaginary, as if the rest of the comics weren't. In the very first panel they made clear that they were looking into a future "several years hence, on a day that may or may not ever happen." The story ended on a comparably cautionary note, urging readers to "see future issues of this magazine for more stories about the imaginary marriage between Lois and Superman... that may come true... or may not!"
Other titles in the Superman family offered their own counterintuitive narratives. What would Superman have been like if his Kryptonian parents had come to Earth with him? How about if he, like Tarzan, had been raised in the jungle? Or if Ma and Pa Kent had adopted Bruce Wayne as well as Kal-El, making Superman and Batman brothers? We even met Superman's children, whether or not we knew their mother. There had been "imaginary" stories like these since the beginning, but the label was coined and the best of them were published during the Silver Age of comics, from 1956 to 1970. This was the most high-spirited, rules-be-damned Superman storytelling in decades and maybe ever. The World's Mightiest Citizen was letting his hair down.
There was one more unexpected twist to "Mr. and Mrs. Clark (SUPERMAN) Kent!": It was written by Jerry Siegel, or so he claimed, although another writer claimed the same thing. There were no bylines in the Superman comics once Jerry and Joe left, so no one can say for sure. But Jerry was back at Jack and Harry's publishing house, quietly working as a freelancer. It was the first in his series of bids to renew his ties to Superman. Imagining outlandish stories about the Man of Steel is what Jerry had been doing since he was a teenager, and now he needed the work and money more than ever. He brought to the task the insights of the battle-weary adult he was along with those of the lonesome boy who would forever define him. Now he had an opportunity to shape the future of a superhero he had brought to life, even if his legal settlement with National prevented him from whispering it to anyone.
EVERY ERA OF SUPERMAN had a defining medium. In the late 1930s and early 1940s it was comic books and comic strips that introduced the character and generated the buzz. Radio made a splash in the 1940s, with cartoons and serials building the wave. George Reeves and his TV adventures were the center of gravity from the beginning to the end of the 1950s, the decade of McCarthy and conformity. The 1960s were back to the future. The only action on TV was reruns of the Reeves series along with a new batch of cartoons that didn't start until 1966 and lasted just six minutes each. Superman had gone silent on the radio and was off the marquees. The big thing, again, was the comics.
Each medium likewise was defined by one or two creative personalities—from Jerry and Joe in the early comic books and strips, to Bob Maxwell and Bud Collyer on the radio, to Whit Ellsworth and George Reeves for most of the television boom. They were the handlers who tinkered with his biography, shifted his delivery mechanism, and made whatever other changes were needed to keep him alive and compelling. In the 1960s the towering figure was Mort Weisinger, one of the most inspired and influential of Superman's midwives, and hands down the most obnoxious.
This son of Russian-Czech immigrants was born on April 25, 1915—six months after Jerry and nine after Joe—and he matched them both in his passion for space-age fantasy. At sixteen he was being published in science fiction journals, and a year later he helped start the pioneering fan magazine _The Time Traveler_. By eighteen, he and his friend Julius Schwartz had launched Solar Sales Service, the first literary agency to specialize in fantasy and the first to sign up science fiction icons Ray Bradbury and H. P. Lovecraft. Weisinger's sensibilities were formed by the same forces that shaped other early comics leaders: roots in the rough-and-tumble Bronx, the worldview of a Jewish outsider, and teeth cut on the pulp universe of bug-eyed monsters. His parents, who had made a fortune in the shoe business and then lost it when their factory burned, pushed him to be a doctor. He pushed back. In 1941, Mort signed on as an editor with National and Superman—hired by Whit Ellsworth on Jerry Siegel's recommendation—and he stayed for three full decades. That was longer than Jerry, Joe, Whit, or even Harry Donenfeld lasted, and it was time enough for Mort to oversee the production of two thousand stories about Superman and his friends.
Weisinger framed TV and movie scripts and he had edited comics almost from the start, but his watershed came in 1957. That was when Ellsworth, who for years had been focusing on the broadcast world of Los Angeles, formally yielded to Weisinger the Superman realm in New York. Like any good potentate, Weisinger consolidated his authority and pressed ahead with his edicts. Mort's Rule No. 1: Know your readers. Superman's, he thought, were boys aged eight to twelve, a readership that turned over often enough that it was okay to recycle old storylines or reprint old stories. Rule No. 2: Don't let the kids get bored. Imaginary stories with epic sweep were just the beginning. He knew that everyone loves a party, so every six months he gave his youthful followers another reason to celebrate. It might be a new comic book (the two he liked most, even if his bosses didn't, were _Superman's Pal Jimmy Olsen_ and _Superman's Girl Friend Lois Lane_ ), a new character (Supergirl, Superbaby, Krypto the Superdog), or a new explanation for an old conundrum (our yellow sun gave Superman his superpowers). The one enigma he couldn't explain away was why anyone bought Clark's disguise of a battered pair of horn-rimmed glasses. Mort called it the Cinderella Fallacy: "Everyone knows that at midnight all of Cinderella's finery changed back into rags. Yet has anyone ever asked why one of her slippers remained glass?"
Weisinger's last and golden rule was: Listen to your customers. That was why Jimmy got promoted from cub to full-fledged reporter. It also was why Lois got a bouffant hairdo like Jackie Kennedy's and, for her high-altitude flights with Superman, a spacesuit like Alan Shepard's. Not only did Mort publish adolescent fans' letters in all his Superman books, but he printed their home addresses so they could correspond with one another. It helped that he had been an avid science fiction fan as a kid and had never really grown up. He "glowed with an over-the-top enthusiasm for every meal, movie, or book he loved," recalled his daughter, Joyce Kaffel. "He was childlike in his exuberance in that everything he enjoyed was 'the best one' of its kind, even if the 'one' before it had been 'the best.' "
Before Mort came along, Superman's world was ad hoc and seat-of-the-pants, with Jerry and other writers adding elements as they went along without any planning or anyone worrying whether it all hung together. That worked fine when all the books centered around Superman and all the writing was done by a small stable. Now the pool of writers had grown and there were eight different comic books with hundreds of Superman stories a year to worry about. It would take a master choreographer like Weisinger to pull it together. He came up with story ideas and parceled them out to scripters, the way Ellsworth did for the TV shows. He divined a fairy-tale universe with its own laws of nature. Superman got a coherent past. He also got an extended family whose stories were scattered among the various books and collectively accounted for a quarter of National's output. "My greatest contribution to Superman," Mort explained after his retirement, "was to give him a 'mythology' which covered all bases. All this makes Superman credible."
Mort was good for Superman, but he wasn't popular. Everyone had a story about the editor's beastliness. He stole plot proposals from one writer and handed them to another, informing the first that it was a crappy idea and the second that it was Mort's. He took underpaid employees to lunch only when he had a two-for-one coupon, and his tie carried caked-on reminders of mashed potatoes, ketchup, and everything else that happened to be on his plate. He told one protégé, Jim Shooter, to be more like another, Cary Bates; told Bates to be more like Shooter; and told everyone that the people working for him were idiots. He had such trouble telling the truth that Julie Schwartz, his lifelong friend, joked that Mort's gravestone should read, "Here Lies Mort Weisinger—As Usual." Jack Adler, a colleague from the early days, recalled that "I had to bring something to him for approval and he wanted to know whether I was having an affair with someone. He said, 'If you tell me, I'll approve everything you bring to me.' "
Mort couldn't help himself. He was trying to make the important point that Superman, not the writers or artists, was the star. He was born with a heart made for storytelling and a tin ear for dealing with people. He wasn't well, as anyone who met him could see. Weighing three hundred pounds, he had developed a series of maladies: gout, inflamed ulcers, hypertension, insomnia, and what he said was a psychological hang-up that grew out of his being jealous of Superman. Blessedly, he hadn't lost the capacity to laugh at himself. Neal Adams, at the time a young artist, remembered asking Mort why he was so grumpy. "I'll tell you," the editor explained. "Try to imagine that you get up in the morning and you go into the bathroom to shave, and you look into the mirror, and you see _this_ face."
Mort was a lot like his boss and benefactor, Jack Liebowitz, with whom he often commuted to work. Both knew the Man of Tomorrow was in a rut by the late 1950s, with storylines that seemed too yesterday and without the boost the franchise had gotten from George Reeves and _Adventures of Superman_. The newspaper strip was still going, but its nearly thirty-year run would end in 1966 as a growing number of outlets did not renew their contracts for Superman and other superheroes. Dr. Wertham and his watchdogs were making parents reluctant to shell out the money for comics, and television, now soaring in popularity, was competing for kids' time and interest. Superman needed a makeover and Mort delivered it. So what if he wasn't popular? Neither was Jack. The questions that mattered to bottom-line guys like them were _Would it work?_ and _Would it last?_
It did work. In 1960, the first year in which sales data was made public, Superman was selling more comic books than any other title or character, and he stayed on top through much of the decade. The Man of Steel was at the front of a charge that saw superheroes taking over from western and romance-themed comics. Some of that was a dividend from an easing of the comics scare and other, broader forces, but Weisinger's reinventions were key ingredients in Superman's comeback. "Mort kept it alive," says Carmine Infantino, a National Comics artist who would rise to editorial director, then publisher. "He was a damn good editor. Damn good."
You could see Mort's influence in the artwork. Curt Swan, who did his first Superman drawings in 1948 and his last thirty-nine years later, gave the hero a more refined look in the 1960s. Gone was Wayne Boring's muscular virility, replaced by a more dignified and human sensibility. Gorgeous had yielded to handsome. While storylines leaped back and forth between science fiction and reality, Swan's artwork kept Superman grounded and credible. Swan also settled the question of how old the hero was: a clean-cut and youthful twenty-nine, which was how old he'd stay through the next two decades. Weisinger steered all these changes, as Swan and Boring have testified. Swan said battling Mort gave him migraines. Boring said that, after nearly thirty years of drawing Superman, he couldn't believe his ears when Weisinger fired him in 1966, so he asked if he'd understood right. "Do you need a kick in the stomach to know when you're not wanted!?" the editor answered. His biggest nightmare, Boring added, was that "I'd die and go to hell and he'd be in charge!"
Weisinger's writers brought their own flourishes. Some stories, like "Mr. and Mrs. Clark (SUPERMAN) Kent!" used a liberated imagination to take readers places they never thought they would go. Characters also experienced extraordinary things—Clark ending up in a wheelchair, about to marry the magnificent Sally Selwyn—only to have them undone by amnesia. Other oft-used ploys: magic potions, instant aging and even faster rejuvenation, and short-lived superpowers for mortals such as Jimmy and Lois. Mort's favorite trick was making Superman temporarily lose his omnipotence, forcing him to fall back on his wits, à la Sherlock Holmes. Whimsical departures like those would have been out of the question in the 1950s, when the comics were supposed to stay in sync with the TV show, which on its meager budget had trouble convincingly depicting science fiction. Now that the franchise was both flush and confined to print, the only limit was Mort's imagination, and his fanciful devices became so common in the Superman comics of the sixties that covers were often forced to assure fans that what they were about to read was _not_ an imaginary story, a hoax, or a dream.
Reality was a more tender matter. Weisinger stories steered clear of the Vietnam War, the sexual revolution, the black power movement, and other issues that fired the 1960s. There was none of what Mort would have called "touchy-feely" either, much as readers might have liked to know how Clark felt about his split personality, or whether Superman and Lois engaged in the battles between the sexes that were a hallmark of the era. Mort wanted his comics to be a haven for young readers, and he knew his right-leaning politics wouldn't sit well with his leftist writers and many of his Superman fans. That didn't stop Otto Binder from sneaking pro-feminist and anti-homophobic messages into his fantastic Superman adventures. He also got away with insulting Weisinger to his face and surviving at National. "Called him every goddam name I could think of," said Binder, who claimed to have written the Mr. and Mrs. Kent story and to have planted with Jerry and Joe the original idea for a comic about an interplanetary orphan with special powers. After unleashing his invectives against his boss, Binder said, he "walked out, and went to see Julie [Schwartz], and Mort comes down the hall and says, 'Are you going to have lunch with me, Otto?' He didn't believe a word I'd said!"
Another writer Weisinger hired to breathe new life into Superman was the man who had breathed life into him to begin with. Mort called Jerry Siegel "the most competent of all the Superman writers.... What his successors did was just embroidery, including my own contributions. Siegel was the best emotional writer of them all." The feelings were not mutual. While he was glad to have a job, Jerry's correspondence from the early years made clear how he felt about Mort telling him how to write the character he had begotten. "What I find particularly distressing is the editorial attitude, as personified by Mort Weisinger, toward my SUPERMAN magazine scripts," Jerry wrote Jack Liebowitz in 1946. "Mort rejects 'em wholesale, and I find myself in the position of having an editor telling I, who created SUPERMAN, that I don't know how to write it. If this happened only occasionally, I'd shrug and figure every writer is entitled to some rejection slips, and forget it. But he rejects my stuff so consistently, with aggravating comments, that he puts me into a frame of mind where I find it almost impossible to write."
Whatever Jerry thought, it was Mort who now was Superman's boss as well as his mouthpiece. In Russia, Weisinger met Premier Nikita Khrushchev, who he said told him that "the Man of Steel cannot get through the Iron Curtain." In Washington, he sat down for a chat with President Kennedy's press secretary, Pierre Salinger. An MIT class sent Mort a letter from Albert Einstein, who asserted that nothing, not even Superman, could move faster than the speed of light. Mort consulted his "good friend" Isaac Asimov, the science fiction writer, who said that "Professor Einstein's statement is based on theory. Superman's speed is based on fact." Mort knew _everyone_ , or pretended to, and he had no shame promoting himself and his comic book star despite his feigned modesty. "When people asked me what I did for a living, I would suppress the fact that I was editing _Superman_ ," Mort wrote. The truth, his psychologist son Hank said, is that "any time we went anywhere, it only took five minutes for him to let everybody know he was the editor of Superman."
Most helpful to National, Mort took on Dr. Fredric Wertham. In a radio debate, Wertham talked about how all of the two hundred inmates he had interviewed at reform school had read Superman, prompting Weisinger to ask, "Did you get them to confess that they also chew bubble gum, play baseball, eat hot dogs and go to the movies?" There was no joking in an "investigation" he did for _Better Homes and Gardens_ in 1955 entitled "How They're Cleaning Up the Comic Books"—and no mention, as he outlined the "stern self-censorship" publishers were pursuing to purge comics of everything from sex to cannibalism and bad grammar, that Weisinger was not a disinterested investigator but a top editor at America's biggest comic book publisher.
In the end, Weisinger's 1960s remake of Superman was more earth-shifting than any changes the Man of Steel had undergone before, including when Jerry softened him up in the earliest years. Mort was a master plotter. His stories planted the hook with unexpected twists and threats—always by page six of a nine-page story—then reeled in young readers. His formula was to start with a thesis (Lois wants to marry Superman), follow with an antithesis (she is so obsessed she blackmails him into matrimony), and end with a synthesis (she backs out when she realizes an accident has changed her personality and takes a drug that cures her). That became a problem, though: His writers adhered to his rules so faithfully that a refreshing wackiness started looking predictable, and the writers started feeling creatively castrated. Even more limiting was Mort's focus on marketing gimmicks like ultra-powered villains and robotic universes when what kids craved was the hero's humanity and fallibility. Human foibles came naturally to a scarred and human creature like Batman, but not to the infallible Superman. They had come naturally to Jerry and Joe, but not to Mort.
That might not have been such a problem if Superman and National had had the playing field to themselves. Stan Lee and Marvel Comics were not going to give them any such gift. In 1962 Marvel introduced Spider-Man and all his teenage hang-ups. He joined the Hulk and Fantastic Four, and a year later came the ultimate outsiders, the mutant X-Men. Here was a new breed of superhero, insecure, vulnerable, and realistic—a dramatic and intentional challenge to the self-assured, all-powerful brand that was National's specialty. "I wanted to get characters," longtime Marvel boss Lee explains, "whose personal lives were as interesting as their superhero identity." What he really wanted, and found, was the anti-Superman. National now faced a dilemma that was partly of its own making, since it had hounded out of business heroes who were distant facsimiles of Superman and had stressed the Man of Steel's plots over his soul. Even the parent company's name, National Comics Publications, evoked a Wall Street stuffiness, one that fans on Main Street countered by referring to the publisher by its original title of Detective Comics or, simpler still, DC. Now upstart Marvel Comics was unveiling champions who really were different, posing a choice for every comics-reading adolescent who grew up in the 1960s and beyond: Superman or Spider-Man? DC or Marvel?
Spider-Man kids loved romance, Superman disciples were more classicists. It was the choice between the let-it-rip rapture of Little Richard and the self-contained genius of Duke Ellington. Marvel was in your face, the way Robert Maxwell and Phyllis Coates had been. DC was softer-edged, like Whitney Ellsworth and Noel Neill. The competition between the two brands of superhero began slowly, but by the mid-sixties it had become fierce. Looking back, it has the feel of a friendly family squabble; at the time it was a cultural chasm, one built around beliefs so basic some still resonate a half century afterward.
Unused to that sort of challenge, DC second-guessed itself and its standard-bearer. The problem, however, wasn't Superman himself, who "done correctly was the greatest hero of all times. It was his writers who were out of touch, who were all about gimmicks and twists rather than honest emotion," says Jim Shooter, who was about to turn thirteen when he sold his first Superman story to Weisinger and who later became Marvel's top editor. "I was a kid and I knew how kids talked," Shooter explains. Mort realized he was in trouble as Spider-Man sales took off, but the realization came late. And the competition wasn't just Spider-Man: DC's own Sgt. Rock spoke in a way that Superman didn't to young people struggling with issues like the Vietnam War. Though Mort brought in young talent to help restore the youthful energy and relevance that Jerry and Joe had captured by instinct, it wasn't enough and it wasn't in time. The Superman family of comic books stayed the top sellers through the 1960s, but their sales were falling and their lead shrinking. Batman tumbled earlier and deeper, to the point where _Superman's Girl Friend Lois Lane_ was outselling him and National contemplated killing off the Caped Crusader; he was saved by his campy TV show, which started in 1966. Marvel, meanwhile, was in the ascendant. The company not only had heroes in tune with the times, its ongoing storylines made young readers worry about missing a single issue the same way their grandmothers did about missing their TV soaps.
The declining sales and his increasingly grating style would put Mort Weisinger's job in jeopardy when new owners took over in 1968 and his patron Jack Liebowitz could no longer protect him. He admitted later he was losing touch with a new generation of kids and their notions about heroes and villains. He said he had tried for years to leave, repeatedly asking for raises big enough to ensure he'd be turned down—but he never was. Finally, as he reported afterward, he walked away in 1970. Carmine Infantino, the editorial director who had called Weisinger a "damn good editor," says it didn't happen quite that way: "Mort said, 'I'd like to stay.' And I said, 'No, Mort, I think it's time you moved on."
THE SUPERMAN UNIVERSE THAT Mort Weisinger left behind was more expansive than the world of any character in the comics, and perhaps in all of fiction. It began on a reimagined Krypton. In Jerry Siegel's first rendering, back in 1938, the planet lasted a single panel and had a simple, utilitarian purpose: to die so Superman could begin his life on Earth. Now that multiple comic book plots had to be contrived and more material was constantly needed, Otto Binder and other Weisinger writers had the chance to linger on Superman's planet of origin and make it into a jeweled paradise. The Gold Volcano spewed that precious metal instead of lava. Shrinkwater Lake reduced men to the size of ants. Pink leopardlike creatures had horns like unicorns' and, when they got angry, spit fire. Robots did all the hard work, the last war had ended thousands of years ago, and weather towers purified the air and stage-managed the seasons. It was a world that fulfilled science fiction's promise of a planet better than our own, anticipating the kind of intergalactic order that Gene Roddenberry would construct several years later for _Star Trek_. Krypton had gone from a mere launching pad to a wonderland that gave young readers goose bumps as they peered at the stars.
The most emotional of the Krypton tales was written in 1960 by Jerry Siegel, who got a three-part, twenty-six-page story called "Superman's Return to Krypton!" to fill in arcs in the plot he had rushed through more than twenty years before. A fully grown Superman found himself on Krypton before its destruction. He got to know his father and see the sacrifice his mother made by placing him in a rocket meant for her. He fell in love with the Kryptonian movie star Lyla Lerrol and imagined the life he might have lived on this faraway planet. He was getting the chance every child dreams of to see his parents when they were kids, especially boys like Kal-El and Jerry who had been robbed of their fathers, and Jerry was getting the opportunity every writer covets to spruce up an old story. Finally, Superman could purge himself of his survivor's guilt by seeing firsthand that there was nothing he could have done to rescue his planet or his mother and father. "It's impossible for me to save Lyla or my parents!" he told himself as he decided not to perish with them. "Earth needs me!"
The image of Kal-El as Krypton's solitary survivor was a central thesis of Jerry's early narratives. As heartrending as that was, it made for a lonely existence and made it difficult for Superman to learn about his past. Upon revisiting Superman's origins, Jerry discovered that his original thesis was wrong: Others had survived, too. Krypto, baby Superman's dog, had been blasted into space by Jor-El as a test before he dared launch his son. Miraculously, Krypto eventually drifted to Earth and into the arms of Kal-El, who could hardly resist this headstrong canine dressed in a crimson cape. Beppo the Super-Monkey and Titano the Super-Ape took similar paths to Earth. They would be joined by two other super-pets—Comet the Super-Horse and Streaky the Super-Cat—neither of whom came from Krypton but both of whom became members of the Legion of Super-Pets and delighted youthful audiences.
The fellow Kryptonian who gave Superman the greatest joy, and the most sleepless nights, was his cousin Kara Zor-El, known on Earth as Supergirl. While she wasn't launched as a character until 1959, we quickly got the full story. She and all of Argo City had been hurled into the cosmos when the rest of Krypton exploded. Later, when the orbiting Argo itself was threatened, Kara's father launched the child in a spaceship headed for Earth. Save for gender, her story mirrored her famous cousin's: She assumed a secret identity as the pigtailed Linda Lee, she had adoptive parents named Fred and Edna Danvers, she shunned her male admirers, and she had superpowers that she used to help humankind. The Maid of Steel, who would get her own comic book shortly after Mort Weisinger left National, gave Superman a blood relative and fellow outsider with whom he could let down his defenses. If youths of all stripes embraced Superboy, now girls had a heroine made in their own special image. And if H. G. Wells's _The War of the Worlds_ had given aliens a bad name, Supergirl and Superman polished the image of the interplanetary interloper.
Argo wasn't the only city to outlast the cataclysm on Krypton. Its capital, Kandor, had been stolen by the space pirate Brainiac, who shrank it to microscopic size and saved it inside a glass bottle in his spacecraft. Superman eventually recovered the bottle and brought it to Earth. While he was able to shrink himself down to Kandorians' size so he could visit with them, he couldn't enlarge the city or its residents. Kandor, like all of Weisinger's inventions, opened up marketing and storytelling possibilities. It gave readers a dollhouse world where they could see sun lamps that mimicked Krypton's red sun, mental suggestion helmets that aided the mentally deficient, and jet-powered flying belts. Superman's enemies now had a way to get at him—via his defenseless fellow Kryptonians—and they never stopped trying. Kandor also made clear that even Superman couldn't get everything he wanted, since there was nothing he wanted more than to restore the Kandorians to their rightful size.
Superman kept Kandor and everything else that mattered most in his Fortress of Solitude, hidden on the side of a snowbank deep in the Arctic. It was a museum to his life as the Last Son of Krypton. There was a room in honor of Batman and Robin and another dedicated to Supergirl. He kept his atomic-powered robots there, and when he had a free minute he loved challenging them to a tug of war. There were rooms memorializing his birth parents from Krypton and his adopted ones on Earth. There was even a Doghouse of Solitude for the Dog of Steel, although it was situated in outer space rather than at the North Pole. The same way Superman's identity as Clark gave him a break from being Superman, his time at the Fortress let him get away from the pressures of being a hero and a reporter.
Brainiac, Kandor's kidnapper, looked human but was actually a computer. Two giveaways were his green skin and the lightbulbs in his head. Another was his tenth-level intelligence, which made him nearly twice as smart as a clever human. Each of Superman's recurring enemies claimed to be the most menacing and wicked, but none could back up the claim like this emotionless genius with infinite memory who had an unrivaled mastery of science and engineering and was programmed with a single-minded focus on wreaking havoc. He, or it, embodied everything that Space Age America worshipped and feared about technology. At the other end of the evil-genius spectrum was Lex Luthor, Superman's longtime archenemy, who was as smart as any human if not as smart as a computer—and made up for the gap with a kill-Superman obsession no artificial intelligence could match. While he first showed up in 1940, it was not until the 1960s that we learned the source of his hatred. Growing up together in Smallville, he and Superman had been friends, until an accident at Luthor's laboratory made him go bald, an accident he mistakenly blamed on the teenage superhero. Their clashes over the years were made more interesting because each craved what the other had—Lex wanted Superman's powers, Superman envied Lex's humanness.
Bizarro was Superman's twisted mirror image, built from lifeless matter and brought into the world by a scientist showing off his new duplicating ray. With a face the color of chalk and the texture of chiseled stone, the Thing of Steel got everything backward. He used dirt to wash himself instead of water, showed his respect for women by pulling their chairs out from under them, and took as his motto, "Anything Superman kin fix, us kin fix worser!" While he wore Superman's cape and colors, he looked more like Frankenstein. Bizarro, who first surfaced in 1958, offered a compelling device for Superman's writers: He made clear what Superman stood for—from impeccable hygiene to unmatched chivalry—but by doing so in reverse and tongue-in-cheek, he made the hero sound less preachy or self-righteous. Bizarro's monstrous imitation of Superman would have been straight-out funny if he hadn't had Superman's powers, which sometimes made him deadly and always made readers glad the real thing was around to save the day and end the story.
The enemy that had posed the greatest risk to Superman since it first surfaced on the airwaves in 1943 was kryptonite, which in the pre–Mort Weisinger days was a rare element and afterward cropped up all the time. Remnants of the planet Krypton at first were red, then gray, but writers finally settled on green for the metal that in small doses weakened Superman and with prolonged exposure would be fatal. Mort, however, decided that if a little kryptonite made for a good story more would be better, and he tapped the rainbow in coming up with new plots. Gold kryptonite robbed Superman of the powers that made him super. Blue was dangerous only to creatures from the Bizarro world, while the only life that white kryptonite could take was a plant's. Red-green was Brainiac's idea, and the combination was a double-edged sword: It gave Superman an eye he didn't need on the back of his head, but it provided him the extra heat vision he needed to lick the computer genius. Red-gold gave the hero temporary amnesia. Red kryptonite was Mort's favorite threat; it was able to split Superman in two or turn him into an ant. While its effects were unpredictable, as with anything Weisinger-related there were rules: Each scarlet-tinted piece had a unique impact, it worked on Superman just once, and its fallout lasted at most forty-eight hours.
Kara Zor-El, Krypto, and the other super spin-offs made clear how popular Superman was and how determined his handlers were to cash in on his legacy. They said the new additions were to keep Superman from feeling like he was on his own. Mort also used the evolving storyline to unite his universe, bringing old books like _Action_ in line with the stories in new ones like _Superboy_. Mostly the changes were in response to the dip in sales of the Superman comics in the 1950s, and they paid off, at least at first. The Superman family led the rebirth of the costumed hero the same way it had the birth. National made no effort to disguise its devotion to the Man of Steel: In its annual report, the spot typically saved for a picture of the company president was instead used for a drawing of Superman, looking statuesque with his hands on his hips and his cape flowing behind him.
But Weisinger's innovations were taking a quiet toll on the story. Superman's world had become so complicated that readers needed a map or even an encyclopedia to keep track of everyone and everything. (There would eventually be encyclopedias, two in fact, but the first did not appear until 1978.) All the plot complications were beguiling to devoted readers, who loved the challenge of keeping current, but to more casual fans they could be exhausting. Still, Mort pushed ahead, soliciting ideas from neighborhood kids as well as readers and, at their urging, ordering up stories on Superman serving as a fireman (super-breath comes in handy), a postman (easy when you can fly), and even a millionaire (he gave it all away). Superman got a Social Security number (092-09-6616) and an honorary passport to the United Nations, which would have come in handy since he now was appearing in thirty countries. Before executing a new plot twist, Mort would test it on his adolescent son, Hank, the same way "Jungle Sam" Katzman had on his; if Hank guessed the ending too fast, Mort would look for something more demanding.
It was easy to see the direction Mort was taking Superman's closest pals: He was dumbing them down and softening them up. Lois was more of a glamour gal now and less of a shrew or a barrier breaker. She had gone from being a dogged reporter to being fixated on proving that Clark was Superman and on landing Superman for herself. The very name of her comic book— _Superman's Girl Friend Lois Lane_ —suggested the shift in tone, which happened just as the women's movement was firing up. No plotlines here about Lois taking birth control pills or burning her bras. Likewise, Jimmy was less interested in covering the news than making it. He had run-ins with aliens and sorcerers, and he had an ultrasonic watch that he used to signal Superman whenever he needed bailing out, which was usually. What he didn't do was experiment with marijuana, sex, or any of the other forbidden pleasures so tempting to teens of his generation. Even Superman himself was not quite the same. He was less of a firebrand, more of a smoothie. And while it was only in his imagination and comic strip dreams that he or Clark would marry Lois, he was affectionate enough to give little boys the shivers.
One thing neither Mort nor Jerry had trouble with was naming their characters. Both began with the premise that every name had to have two _L_ s, ideally at the start. Jerry had signaled where he was headed with his spelling of Superman's Kryptonian name, Kal-L. The second-most important person in the Superman universe—Lois Lane—was the first to have the Ls take their place up front. Lois's sister was Lucy Lane, while her parents were Ella and Samuel. Lana Lang, Superboy's neighbor and best friend, had a father named Lewis and siblings christened Larry, Alvin, and Ronald. Enemy number one originally had just one name, Luthor, but in 1960 editors decided to soften him up by adding a given name, Lex. His sister was Lena, his parents Lionel and Letitia, and Aunt Lena helped raise him. Over the years Superman, Superboy, or Clark fell in love with Lyla Lerrol, Sally Selwyn, Lyrica Lloyd, Lal Leta, Lahla, and Lori Lemaris, whose sister was Lenora. Superboy would team up with Lightning Lad and Lightning Lass against Lightning Lord.
Why the letter _L_ and the alliterations? Jerry never said, nor did Mort. Not even Superman was talking. When Supergirl told him that she had chosen the name Linda Lee for her disguise, he commented how, "by sheer coincidence," she had picked the same initials as everyone else he held close. The truth was it was just for fun. The echo appealed not just to the ears of his writers and editors but to young readers, just as with fan favorites like Peter Parker, Bruce Banner, Archie Andrews, Mickey Mouse, and Bugs Bunny. Once Jerry got the style rolling, Mort turned it into a game for readers, scores of whom wrote in whenever they uncovered a new one or wondered why someone as close to Superman as Jimmy didn't have the two _L_ s. Mort's answer: He did. Don't you remember the TV episode "The Talkative Dummy," which revealed Jimmy's full name as James Bartholomew Olsen?
SUPERMAN AS A song-and-dance man? It sounded like one of Mort Weisinger's imaginary stories. The Man of Tomorrow had triumphed in so many settings that a crew of theater people decided to give him a tryout in 1966. And this was not just any crew, it was Broadway's finest. Producer Harold Prince was on the way to making _Fiddler on the Roof_ the first musical to run for more than three thousand performances. Music master Charles Strouse and his lyricist partner, Lee Adams, nearly swept the Tony Awards with _Bye Bye Birdie_ , while scriptwriters Robert Benton and David Newman were a year away from their blockbuster movie _Bonnie and Clyde_. No one had ever tried before to build a musical around a comic character, but Superman was used to being first and no one had ever lost money gambling on him.
Jack and Harry gave their blessing in return for a modest share of the profits and—to protect the franchise—a promise that the play couldn't be called _Superman_. So the production was named _It's a Bird... It's a Plane... It's Superman_. Money wasn't a problem either: With Columbia Records on board, Hal Prince was able to raise half a million dollars. Strouse, Benton, and their partners had come up with a doozie of a story, in words and notes: Dr. Abner Sedgwick was convinced he had been cheated out of the Nobel Prize, not once but ten times, and he planned to make the world pay by eliminating its beloved Superman. When force didn't work, the mad scientist turned to psychology. In order to convince the people of Metropolis that Superman couldn't save them, he bombed City Hall while the hero was distracted, then persuaded Superman that he was a Man of Straw for doing nothing. "You're not stopping crime," Sedgwick insisted, "all you're doing is catching criminals after the fact." The logic was compelling. "Could that be true?" Superman asked. "Why must the strongest man in the world be the bluest man—tell me why?" Those were the kinds of questions Spider-Man might have asked himself, but not Superman, or at least not Mort Weisinger's Colossus of Krypton. Before the curtain fell, Benton and Newman's superhero recovered both his confidence and his powers in the nick of time.
The play blended the drama of the _Adventures of Superman_ TV show with the burlesque of the comics for what looked like a winning touch of satire. Even the flying, which had stumped serial and TV producers, worked here by keeping it simple. A flying harness of light leather was strapped onto Superman's chest, upper arms, and back. A wire attached to a wooden clip and pulleys whisked him six feet above the stage. Making this work in front of an audience was a lot to ask, but theatergoers wanted more than anyone to believe. And the producers brought in the same technical staff that had made Mary Martin fly in _Peter Pan_ and Tammy Grimes in _High Spirits_.
Hal Prince and his team knew enough about Superman to realize that everything turned not on stunts or even the story but on finding the right hero. So, like Bob Maxwell and Sam Katzman before them, they launched a far-reaching search. They didn't need a Broadway star to fit this bill; they would have plenty with Jack Cassidy, Linda Lavin, and the rest of the cast. The specs for Superman: He should stand six feet six inches and weigh 190 pounds. A seventeen-inch neck would be ideal, along with biceps of a minimum of eighteen and a half inches. Mid-thirties was the right age. Black hair was a must, along with blue eyes and legs that looked good in tights. And of course he had to know how to act, sing, and fly. Fifty-two actors showed up, including an Olympic pole vaulter and a bass-baritone from the New York City Opera. One fit the bill: Bob Holiday, a thirty-three-year-old singer and comedian on the supper club circuit. He had served overseas in the Army, spinning records for the Armed Services Network, and had been in a Broadway play once, singing the opening song. Most important, he weighed 190 pounds, stood six feet four, and had grown up as an only child in Brooklyn with Superman his favorite comic book friend. When he heard he had the part "a chill went through me. I said, _'Thank you. Thank you.'_ "
To show his gratitude Holiday visited the gym every other day for up to two hours, curling hundred-pound weights and pressing 160 pounds. Breakfast consisted of powdered protein, milk, and wheat germ. Smoking was out, which wasn't easy for a two-pack-a-day guy, as was drinking in public. It all paid off when, after every performance, he would invite hundreds of kids backstage, letting them take their best shot at his midriff. It also helped when he fell from his harness, dropping six feet onto the stage. He bounced back up, turned to the audience as if it were rehearsed, and said, to a standing ovation, "That would have hurt any mortal man."
The first test of what an audience thought of _It's a Bird_ was in February, in an out-of-town run in the historically loving city of Philadelphia. Not this time. "Is it low camp, high camp, medium camp? Is it a musical parody or a cartoon with music?" asked the Philadelphia _Bulletin_. The libretto, chimed in the _Philadelphia Daily News_ , "has a form of humor but no great shining wit," while Strouse and Adams's songbook "only faintly recalls the animation these collaborators brought to _Bye Bye Birdie_." Disappointed but not done in, the creative team went back to work. The lead song was cut and a new showstopper written. Scenes were altered, costumes revised, and when there wasn't time to make the desired changes, outfits were simply turned inside out. Even the pricing at the Alvin Theatre was redone to give the New York opening its best shot. Prince offered the broadest range of rates on Broadway—two dollars at the bargain end, aimed at drawing a new and young audience, with the orchestra split into eight-, ten-, and twelve-dollar options, the last of which was $1.50 more than for any other musical and was meant to subsidize the cheap seats. It was equally novel to offer a third off the high-priced seats for early mail orders, and it worked: Advance sales topped those for _Fiddler_ and Prince's Pulitzer Prize–winning _Fiorello!_
_It's a Bird_ opened on Broadway in mid-March, with Mayor John V. Lindsay and much of the city's establishment on hand. Reviewers were there as well, and their verdicts were up-and-down. _Time_ called it "amiable mediocrity... capable only of inspiring benign indifference." _The Washington Post_ wrote that "on its appointed level of simple-minded casualness it works quite nicely." The New York papers were more generous, with the _Morning Telegraph_ labeling it "a musical show loaded with entertainment" and the _World-Telegram and Sun_ saying, "You leave the theater smiling, and the smile lasts all the way home." The biggest rave came from the highest-minded paper, _The New York Times_ , whose critic Stanley Kauffmann pronounced the play "easily the best musical so far this season, but, because that is so damp a compliment, I add at once that it would be enjoyable in any season."
Prince was convinced he had a smash hit until he called the box office. "They said, 'My God, we haven't sold a single new seat.' " He, like nearly everyone involved with the production, felt like a kid again doing the show, and they all assumed the audience would grow to love it. The timing seemed perfect: Andy Warhol and Roy Lichtenstein were at the height of their popularity, and nobody better defined that pop art craze than the Man of Steel. But the wave had already crested and the crowds at the Alvin started shrinking two months in. Despite an unprecedented four matinees a week and a flurry of ads in the comic books, the curtain fell for the last time on July 17, after just three and a half months and 129 performances.
Everyone had a theory on what went wrong. "It should have had a little more muscle, and some teeth politically. We should have made it about the times," Prince says, looking back to the era when America was shipping young men en masse to fight in Southeast Asia and record numbers of draft dodgers were fleeing to Canada. To Strouse, the trouble was a combination of summer camp and "Capelash." Some kids were away swimming and boating when the show was playing, while others were watching a superhero for free on the new TV show _Batman_ , which was such a hit it aired twice a week. To Benton, the difficulty was the very nature of his Superman story and whom it appealed to: "It was not a children's show and not an adult show. It sort of fell between the two."
Prince went on to produce and direct nearly sixty plays and win a record-setting twenty-one Tony Awards. Strouse wrote the music for twenty-two plays and six movies, including _Bonnie and Clyde_. Benton would make his name with films like _Kramer vs. Kramer_ and _Twilight_. While _It's a Bird_ was a mere footnote in their careers, for Holiday it was the highlight, just as playing Superman had been for Kirk Alyn and George Reeves. It had taken him onto the TV show _I've Got a Secret_ , where he got to joke with Steve Allen and flirt with Miss America. He was a guest on Johnny Carson's _Tonight Show_ and got to recast his Superman role in stage performances in St. Louis and Kansas City. And it had brought him to Broadway, which was a memory that kept him going as he became a home builder in the Pocono Mountains. "I don't think that the supposed 'Superman curse' hit me at all," he says. "It's still a kick to let people know that I was the Man of Steel. My doctor even hangs a picture of me in his office so that all his patients know he fixed Superman right up."
As for Superman himself, he escaped largely unscathed. The critics blamed not him but his handlers. The handlers learned valuable lessons, which they would apply repeatedly. The first was that when people had to pay to see Superman, the target audience should be adults who hopefully would bring along the kids. It also made sense, for a wildly popular character like the Man of Steel, to showcase him in mass media rather than a rarefied venue like a Broadway theater. Strouse and Prince say they wish they'd had a chance to put those lessons to the test. Benton and Newman did. A dozen years after _It's a Bird_ closed, their names were listed in the writing credits for the first major superhero feature film: _Superman: The Movie_.
SUPERMAN WAS EVERYWHERE IN the 1960s. Audiences hooted when the director brought him onstage for the opera _Carmen_ in Bologna, while three hundred thousand subscribers cheered when Superman comics finally appeared in West Germany. Forty-two countries, from Brazil to Lebanon, were translating every issue of the American comic book into their native tongues, which gave the Swedes a hero called Stalmannen, the Mexicans a caped cousin named Superniña, the Dutch an intrepid lady reporter whose byline read Louise Laan, and the Arabic world an undercover male reporter named Nabil Fawzi who worked for the newspaper Al-Kawkab Al Yawmi. Andy Warhol hand-painted Superman into universal pop art fame. Superman and his family were the objects of parody in a comic called _Stupor-Man_ , which also featured Stupor-Snake, Stupor-Rhino, Stupor-Grandpa, and Stupor-Old-Maid-Auntie. He was a star of Jules Feiffer's _The Great Comic Book Heroes_ , which helped lift comic books to the status of high art even as its author prayed they would remain the lowest common denominator of America's fantasy life. "When _Superman_ at last appeared," Feiffer reminisced in 1965, "he brought with him the deep satisfaction of all underground truths: our reaction was less 'How original!' than 'But, of course!' "
That was the reaction kids across America had when a new version of an animated Superman turned up on their TV screens in 1966, two months after Bob Holiday's Man of Steel took his final bows on Broadway. Youngsters who knew the hero from the comics thrilled at seeing him take on new shapes as well as new adventures. Those seeing him for the first time felt the glow of first love that their parents and grandparents had with Max and Dave Fleischer's cartoons a quarter century before. The show was aimed especially at young children but its title, _The New Adventures of Superman_ , was chosen to appeal to older viewers who remembered popular programs of the same name on the radio in the 1940s and on TV in the 1950s. Much of what they heard and saw this time was brand-new: the first appearance in TV cartoons by Jimmy Olsen, Lex Luthor, and Mr. Mxyzptlk; the first time the cartoons came in six-minute features; and the first packaging of Superman shorts with ones starring Superboy and other DC stars, a twist that brought with it name changes to _The Superman/Aquaman Hour of Adventure, The Batman/Superman Hour_ , and finally back to reruns with the familiar if ironic title of _The New Adventures of Superman_.
The _New Adventures_ borrowed more from the past than they changed. Filmation Studios, the producers, used the same low-budget rotoscoping technique the Fleischers had, tracing real-life characters frame by frame. Collyer was back as the voices of Clark and Superman, Joan Alexander reprised her radio role as Lois, Jackson Beck was again the narrator, and Jack Grimes returned in the role of Jimmy, which he had played during the last year of the radio show. Allen Ducovny, Bob Maxwell's partner on the radio show, was executive producer of these cartoons, which may explain why "Up, up, and away" was heard so often. The animated series conjured up one more specter from the past: the censors. This time they took the form of a grassroots group called Action for Children's Television, and their primary target was a worthy one: commercialism aimed at kids. The group, founded in 1968 outside Boston, hated being branded a censor, but that was the effect when it complained that there were too many punches thrown on the Superman cartoons. The result: The series was canceled in 1970, after its third full season.
Given the alarms being sounded by do-gooders, the last place one would have expected to find the Man of Steel was in the schoolhouse. But he had been there for twenty years, helping teach grammar to kids in thousands of classrooms. The number of unique words in a year of Superman comics was twice the vocabulary of the average fourth grader, studies found, and reading his adventures could help adolescents expand their language. Working with a high school teacher from Lynn, Massachusetts, National Comics in the early 1940s had prepared a Superman workbook with lessons on punctuation, grammar, and usage. Teachers across the country jumped aboard. As for the kids, "they loved it," reported _Magazine Digest_. "Children who had been bucking English grammar for years found themselves painlessly answering such questions as 'What punctuation mark ends Superman's speech?' and 'What kind of sentence does he use?' The sugar coating had been found for the pill."
Ron Massengill remembers the first compound word that he learned: Superman. "I couldn't read _bye-bye_ but I could read _Superman_. There was the big _S_ leading to the small _n_. I was sure I had seen that at the drug store or the grocery store," says Massengill, who was born the same year as the Superman comic book and has been a fan since he was a toddler. "Within four months I could read a Superman comic book all the way through. My mom had bought a dictionary, a huge dictionary that weighed like twenty pounds. She explained to me when I saw these groupings of letters in the comics that I could go through and find that grouping in the dictionary." Even more kids might have been using Superman in even more classrooms had it not been for Dr. Wertham, who convinced many parents and teachers in the 1940s and 1950s that it was dangerous to let comic books anywhere near their children.
By the 1960s, as the age of peaceniks and flower children gained steam, Wertham's influence had waned and Superman's had risen to the point that even the White House was laying out the red carpet. The Kennedy administration wanted the hero's help spreading the word about its campaign to close the "muscle gap." Mort Weisinger put two of his best writers on the story, which he called "Superman's Mission for President Kennedy." The Champion of Democracy flew across America pushing young runners to run harder, hurdlers to jump higher, and flabby journalists at the _Daily Planet_ to do fifteen minutes a day of calisthenics. When _The New York Times_ got wind of the preparations it scooped the comic book with an article headlined SUPERMAN MEETS KENNEDY ON VIGOR.
Weisinger's story was all set to run but was pulled back when the president was assassinated in November 1963. Shortly afterward, Weisinger got a call from President Lyndon Johnson saying, "We're waiting for the story. When's it coming out?" Mort explained his worry that running it might be in bad taste, at which point, as he recalled the tale, Johnson interrupted: "Horsefeathers. You can run it with a posthumous foreword, explaining that _I_ ordered it!" Mort did.
It was not the first time President Kennedy had teamed up with Superman. That was in 1962, when Superman was ready to introduce his cousin Supergirl to the world and brought her to the White House to meet the president. High drama, indeed: The Camelot president on the same stage with the Lancelot of comic book heroes. More than a year later Superman took Kennedy into his confidence, sharing his dual identity as Clark Kent. "I'll guard your secret identity as I guard the secrets of our nation!" JFK promised, to which Superman replied, "If I can't trust the President of the United States, who can I trust?" The exchange took on a special poignance when the comic book, which was printed while the president was alive, showed up on newsstands just after he was gunned down in Dallas. There was one other time when the name Jack Kennedy had appeared in Superman's comic books. It was in the very first of the _Superman_ series, in July 1939. A character named Kennedy was murdered and the newly minted Man of Steel saved a wrongly accused man from being executed.
Inside Jack and Harry's business operations, there was a flurry of activity that kept the cash registers ringing in the 1960s, though Harry was less involved than ever. Ivy Leaguer Paul Sampliner, who had supplied Harry with much-appreciated cash back during the Depression, remained the definition of the compliant partner, more interested in being a socialite and civic leader than a business executive. Jack, as always, was looking for ways to boost profits and minimize risks. As early as 1945 he had explored taking the company public, and he would have done so if the stock market had been more bullish. He tried again in 1961, but his broker said the stock exchange would approve the arrangement only if Harry wasn't part of it. "His reputation," Jack wrote, "was not too good. His way of life was quite well known in New York." Jack offered to make Harry, technically still his boss, a millionaire if he resigned from the company. Harry took the bait. Jack became president, with Sampliner still on the board. That night, as Jack recalled, "I guess he [Harry] celebrated, he was drunk and he fell on his head, was in a coma for weeks. He never knew that we went public. Never knew." Doctors operated on Harry but there was nothing they could do. He spent the last four years of his life with round-the-clock nurses and no memory. He learned to recognize his two children, his sister-in-law, and his mistress, but that was the extent of his engagement with the world. Harry died in February 1965, when he was seventy-one and the comic book hero who made him famous was twenty-six.
It was no way to go, and forty years later theories still percolate that Harry's death might not have been an accident. That is what happens when you live life as hard as he did and mix with people who make accidents happen. Harry had spent his adult life in a marriage he wanted out of and he had finally found a way. Gussie, his wife, had just died and he was set to marry Sunny, his mistress. His fall put an end to that. Peachy, his daughter, agrees with Jack that Harry's fall was a drunken accident, but she adds, "We all said that momma came down from heaven and kicked him."
Jack was no sentimentalist but he, too, celebrated the company's listing on the stock exchange, not by getting drunk but by having a chauffeur drive him to his old haunts across Manhattan. There was the Lower East Side tenement where he had slept on the roof and shared a bed with brothers Harry, Lenny, and Mac. He helped out all three over the years with money and advice. (None did anything with his largesse, or even said thanks.) Next stop was the Ladies' Garment Workers' office, or at least the building where it used to be, and where Jack used to be an idealist. On to Greenwich Village and New York University, which taught him that he could be more than a bookkeeper and gave him the skills and worldview of an accountant. It was a _This Is Your Life–_ type revisiting, shared not with a TV audience—that was not Jack's style—but by himself and reconstructed later for his two girls. His was an American success story—a Jewish success story, of a boy from the Ukrainian ghetto making not good but great. All accomplished, as he told older daughter, Linda, with "no help from anybody."
After the nostalgic tour it was back to business. Jack's distribution arm, Independent News, had always been his biggest moneymaker and had helped National stay afloat in the 1950s when other comic book companies were dying. Being a book distributor not only meant that Jack trucked publishers' products from printer to wholesaler, it made him their banker and gatekeeper. Major Wheeler-Nicholson had seen what a vise that could be in the 1930s and now National's biggest competitor, Marvel Comics, was learning the lesson. Independent had taken over as Marvel's distributor and Jack limited the publisher to a dozen titles a month—a third of its peak output and too few to unseat National as king of the comics. "We didn't want the competition," Jack explained in his memoir. He handled _Playboy_ , too, which added thirty-five thousand dollars a month to his bottom line, along with _Mad_ magazine, whose sales he boosted from two hundred thousand an issue to one million. By the 1960s, Independent was America's largest distributor of magazines and paperbacks, and it soon became a player in Europe, too. Jack's nephew Jay Emmett was doing almost as well with his licensing business, cashing in on the success not just of the over-the-top _Batman_ TV show but of James Bond's über-adventure movies. The numbers no longer were in the hundreds of thousands of dollars, or even the millions. "These were large accounts," Jack said of National's diversified holdings. "Tens of millions of dollars."
That money was pouring in to the newly christened National Periodical Publications, Inc., which included both the distribution and publishing operations. Revenues rose still higher in 1961 when National raised the price of its comics from a dime to twelve cents. But the bookkeeper in Jack couldn't keep his eyes off the expense side of the ledger. He knew from his days with the Ladies' Garment Workers how unions could boost employees' wages and benefits, and he wasn't about to let that happen in his shop. So when his workers had talked about a union in the 1950s, he had squashed the effort. When union talk resurfaced in the mid-1960s—talk of veteran freelancers getting health insurance, higher page rates, and partial ownership of the comics characters they created—Jack took a more subtle approach. He turned what became known as the Writers' Rebellion over to his lawyers to study. He said he'd go along with a union if Marvel's publisher would—knowing he wouldn't. Jack gave bonuses to artists who steered clear of the activists, knowing that artists were harder to replace than writers. "[Jack] turned to me at one point during the negotiations to say: 'You don't understand, I'm very sympathetic to the points you're making. When I was a young man, I was a Socialist, too!' " recalled writer Arnold Drake. "The problem was that Liebowitz had a youth of twenty minutes."
The rabble-rousers disappeared, slowly enough that it wasn't until years later that anyone realized how thorough the housecleaning had been. Few old hands were fired outright; rather, they were assigned to lesser comics, given fewer assignments, and supplanted by younger writers who had no idea they were scabs. Mort sided with Jack, as he always had, unaware that he, too, would one day be expendable. Not even Harry had been immune. It wasn't just National that was taking a hard line. Comic book publishers, as Drake said, had always run their businesses like brothels: "They were the madames, and the writers and artists were the girls."
To anyone who accused him of miserliness, Jack could tick off the names of all the employees and relatives he had helped buy a home or pay for a daughter's marriage. What better sign of his generosity could there be than lending yet another hand to the biggest ingrate he had ever hired? It happened in 1959, when Jerry Siegel once again was desperate. He had worked for other publishers, but these jobs never lasted long enough or ended the way he wanted. He and Joanne had been living in a one-bedroom apartment in Great Neck with their baby, Laura. The landlord had threatened eviction, the milk company and diaper service had cut off deliveries, and he couldn't make child-support payments for his and Bella's son, Michael. He went on a hunger strike. He wrote to the media. He touched up friends and neighbors for help. Finally, after years of her pleading on Jerry's behalf, one of Joanne's letters to National got a reply. With Jack's blessing, Harry's son, Irwin Donenfeld, brought Superman's creator back, as a freelancer, at what Jerry said was ten dollars a page, forty dollars less than when he'd left a dozen years before.
Jerry wrote with passion and precision, scripting stories about Superman, Superboy, Supergirl, Lois, Jimmy, and the Legion of Super-Heroes. He invented characters for Mort Weisinger's new universe, from Colossal Boy to Triplicate Girl, Chameleon Boy, Ultra Boy, Shrinking Violet, Sun Boy, and Bouncing Boy. He got along with Mort as well as he could and avoided Jack as much as possible. The arrangement lasted until 1966, when it was clear that Jerry and Joe were planning another lawsuit to try to reclaim the Superman copyright. It took three more years for the suit to be filed and the court didn't rule until 1973. The result was the same "no" as before, only this time Jerry didn't ask for his job back, and if he had Jack wouldn't have given it to him.
Joe never made it back to National as an artist, but he did as a delivery boy. "I was the oldest messenger boy in New York City," he recalled years later about the jobs he had had as a sales clerk, janitor, or gopher. "One day I had to deliver a message to an office located in the same building as the publisher of DC Comics. Someone from their office saw me in the hall, asked me what I was doing there, then told the publisher about it later. He called me that night—very upset—and asked me to come into his office so he could help me out a little. 'How does it look,' he said, 'for the artist/creator of Superman to be running around delivering messages—you're giving us a bad name!' " It looked worse to see Joe milling in front of the Alvin Theatre the night _It's a Bird_ opened, watching ticket holders head in to see his superhero onstage and wishing he had the money to join them. Another time police picked him up in Central Park as a vagrant. After he lost his 1947 lawsuit and his artist's income, he moved in with his invalid mother in Forest Hills, Queens. Later he and his brother shared an apartment in Queens, among broken venetian blinds, sofas with springs poking through, and boxes of yellowing Superman comics.
The best measure of Joe's state of mind, and his finances, was the work he had taken but never breathed a word about: drawings of bare-skinned or nearly naked women being whipped, spanked, and humiliated by men and by other women. Joe didn't sign the illustrations, but they were his. Comics historian Craig Yoe knew it right away when he found the booklets stuffed in an old box at a 1989 antique book sale, and Shuster experts have confirmed it. Most of the women looked like Lois Lane. Some men were dead ringers for Jimmy Olsen and Slam Bradley. The co-publisher was Joe's neighbor. One set of pamphlets, _Nights of Horror_ , was said to have inspired the 1954 rampage by a group of teens known as the Brooklyn Thrill Killers. Dr. Fredric Wertham visited the gang leader in jail, carrying a copy of _Nights of Horror_ but not knowing who had illustrated it. Neither, apparently, did the Supreme Court of New York when it ordered in 1955 the destruction of all copies of the inch-thick pamphlets, which it said were "pornography, unadulterated by plot, moral or writing style.... The many drawings that embellish these stories are obviously intended to arouse unnatural desire and vicious acts."
Why did Joe do it? "Neither he nor Jerry could get work for anything decent, so he had to tender that stuff to make a buck," says his friend and Batman illustrator Jerry Robinson. "I don't think that's the work he would like to be remembered by." Yoe, an author and former creative director for the Muppets, agrees but offers two additional theories: Depicting Superman characters in compromising poses might have been a way to strike back at Jack and Harry for firing him and Jerry. It also could have reflected Joe's fantasy life. "My guess," Yoe concludes, "is that it's probably some of all three of those things."
Jack almost surely never saw the pornographic drawings and he probably never saw Joe after the lawsuit. His preoccupation, starting as early as 1960, was with who would take over the empire when he retired. He was turning sixty and his only heirs were his daughters, whom he didn't want to see working in his business or any other. "I spent all my life accumulating some wealth," he said in his memoir, "but I have nobody to leave it with." So he hired Felix Rohatyn, a high-powered investment banker who would later help save New York City from bankruptcy, to start looking for a takeover partner. It took until 1967 to find one: Kinney National Services, which owned funeral homes, parking lots, rental cars, and an office-cleaning company. It seemed an unlikely match to everyone but Jack, who wrote that "I liked the people, they were hamisha people. Jewish, Jewish oriented. And they had a business that was prosperous." What he didn't say was that Kinney carried the same whiff of not-so-kosher underworld connections that Jack and Harry's businesses had. What mattered in the end was that Kinney paid National $60 million and National gave Kinney the toehold it wanted in the entertainment business. Two years later Kinney bought the Warner Bros. movie studio, and two years after that Kinney renamed itself Warner Communications, Inc.
Jack got a seat on the Kinney board of directors, then on Warner's board. Irwin Donenfeld, Harry's son, got a lot of money, but he lost his job. So did Mort Weisinger. Jay Emmett, Jack's nephew and the whiz kid behind the Licensing Corporation of America, made out best, at least to begin with. He became best friends with and right-hand man to Steve Ross, the ingenious dealmaker behind Kinney and Warner, although Ross later sacrificed his best friend to save himself from a federal racketeering indictment. Emmett said that Kinney's purchase of National was a great move for everyone, but that Ross—an "imaginative genius"—never appreciated that the most important assets he was getting were Superman and Batman: "He had no idea of the worth of those characters, none. They were just two comic characters to him."
# **CHAPTER 8**
# **Believing a Man Can Fly**
BARELY REACHING FIVE FOOT THREE, with a mop of blue-rinsed white hair, Alexander Salkind brought to mind a mole man more than a Superman. His taste ran to white bucks, silk ascots, and jeweled lorgnettes, the elegant spectacles favored by operagoers. His suits were strictly powder blue and Savile Row, with a Légion d'honneur rosette proudly pinned to the wide lapel. He held court amid the faded opulence of luxury hotels and refused to ride an elevator or an airplane. His exotic accent, a thick blend of old-school Romance languages, left no doubt that English was not his native tongue. Indeed, his background was Russian, his homeland Germany, his citizenship Mexican, his ethnicity Jewish, and his passport that of a cultural attaché to Costa Rica. He had bankers in every capital in Europe yet had never paid a bill on time. But this son of Greta Garbo's film producer knew how to make movies—his production credits ranged from Orson Welles's _The Trial_ to the blockbuster _The Three Musketeers_. In the spring of 1974 he was looking for the next big thing.
"Why don't we do Superman?" his son and protégé, Ilya, asked expectantly over dinner at the Café de la Paix in Paris.
"What's Superman?" Alex asked back.
Not an auspicious beginning for the man who was about to define the Last Son of Krypton for a new generation in America and around the globe. But what he lacked in appreciation of popular culture Alex made up for with his instinct that a world disillusioned by Vietnam and Watergate might need a superman. This was the intuition of the Holocaust survivor—an understanding that it wasn't the particular myth that mattered but our aspiring to something bigger. His own life had always been defined half by suspicions and anxieties, half by defying norms and accomplishing the impossible. That fearlessness—what was a tax problem or lawsuit to someone who had been hunted by the Gestapo?—was precisely what was needed to revive Superman more than twenty years after his last radio broadcast and fifteen after his TV show and its star died, when he was again the limited province of adolescent readers of comic books.
"I told my father who Superman was—that he flies, that he's as known as Jesus Christ, that we can't do it tiny—and why it has to be a big movie," Ilya recalls. "He said, 'Sounds very interesting, this Superman. Flies. Powers. Stronger. Known. Ahhh, let me talk a bit with my people."
His people were bankers and other moneymen from Switzerland, the Netherlands, Germany, Britain, and Chicago. Some were reputable while others skated on the edge. It was the same combination that had worked for Harry and Jack over the years. Enough of them approved to give Alex and Ilya confidence. More than enough. Looking back, even his lawyer concedes that Alex sold or traded more than 100 percent of the production, in the style of a Ponzi schemer or of Max Bialystock, Mel Brooks's double-dealing producer. He didn't go to jail only because the film made enough money to pay everyone off handsomely.
Step two was getting Warner Communications, Superman's new owner, to hand over the keys. Warner Bros. executives were busy with their own big films— _Alice Doesn't Live Here Anymore, Night Moves, Dog Day Afternoon_ —and had never imagined Superman as much more than a comic book. A buoyant superhero seemed an especially poor fit at a moment when the nation was reeling from one of the deepest recessions since the 1930s along with the resignation and pardoning of its disgraced president, Richard Milhous Nixon. So whatever they thought of the elfin Alex and his slightly taller son, Warner agreed, turning over twenty-five years of moviemaking rights to the Salkinds in return for $850,000 and the promise of millions more in the unlikely event the producers cashed in. It was a golden chance for Ilya and Alex and a lack of both vision and intestinal fortitude by one of Hollywood's biggest dream factories. "It wasn't one of the studios" that recognized what Superman could be, concedes Terry Semel, Warner's former president. "I'd like to take credit for it, but I think Alex Salkind saw it and he did it."
More than a thousand people would be involved in the production, including six writers and rewriters and three directors. Eleven separate film units shot at three studios in eight countries on three continents. More than a million feet of film were recorded, although just twelve thousand were needed. It took the largest movie budget ever to pull it all off, with more bounced or delayed paychecks than anyone could count. A director, a writer, and the biggest stars all sued the Salkinds afterward, and they all won settlements. Alex had to hijack the film to squeeze the extra money he needed from Warner, and his fear of flying—and of being arrested—kept him from the U.S. premiere.
But what mattered to him, to Ilya, and to studio executives like Semel was that it worked. Five years after that father-son dinner in Paris, the Salkinds released _Superman: The Movie_. It was nominated for three Oscars and took home a Hugo Award for best dramatic presentation and a Grammy for best musical score. The box office results were even more uplifting. It was the second-highest-grossing movie of 1978, bested only by _Grease_ , and the most profitable in Warner Bros.' history. It was the first time a comic book hero had starred in a serious movie and it launched Superman as a film franchise, with three sequels over the next decade. For the Man of Steel, it meant a bold new adventure that would define him for Generation Xers the same way George Reeves's _Adventures of Superman_ had branded him for baby boomers. And it was made possible by one of the few people on the planet who had never heard of Superman.
BEFORE THE SALKINDS COULD MAKE a movie they needed a script, and so, as they would with everything, they opened their checkbooks and went hunting for a big name. Alfred Bester qualified, having written _The Phantom_ comics and award-winning novels like _The Demolished Man_ , and he was hired to produce a treatment. Ilya loved what he wrote; Alex didn't. Bester might be a celebrity in the world of science fiction, father Salkind said, but he wanted _big_. Bester got a generous kill fee and Mario Puzo got a call.
Puzo's _Godfather_ had recently been made into two movies that earned him a pair of Academy Awards for best screenplay. An Oscar was the kind of credential Alex could relate to, and Ilya signed Puzo up for 5 percent of the film's gross sales. His Superman was a TV anchorman at a station where Lois Lane was the weather girl and there was no competition from the _Daily Planet_ , which had folded. Lex was there, too, or rather "Luthor Lux." When Superman went looking for Lux he found a bald Kojak in a trench coat who, sucking a lollipop, asked, "Hey! Superman! Who loves ya, baby?" Puzo thought camp like that gave his movie pizzazz. Everyone who read it, especially the National Periodical people, was sure it would undermine the film's credibility and Superman's. Puzo's opus, which stretched to more than three hundred pages, read more like a novel than a screenplay and would have cost a billion dollars to produce, says Ilya. Yet both sides found silver linings when Puzo walked away at the end of 1975: He eventually got his promised 5 percent, with $300,000 of that up front and an on-screen credit for a largely useless product, while the Salkinds got bragging rights to one of the world's best-known writers, whose legend they used to refill their dwindling coffers.
Next up were Robert Benton and David Newman, who had written the script for the Broadway production _It's a Bird_ , along with Newman's wife and writing partner, Leslie. Ilya offered the new team a million dollars and simple instructions: "Fix it." They spent their first three days tossing out big chunks of Puzo's work, then got their own bead on the hero. "We decided that Superman is our King Arthur, he's our legend," says Leslie. What fascinated Benton was the Clark Kent–Superman split: "Is he Clark Kent until that emergency call happens, or is he Superman? Does he miss going full tilt or does he get used to being this guy who sits in a coffee shop and has a grilled cheese sandwich for lunch?" As for what the Salkinds wanted, "They had no idea and couldn't have cared less," says Benton, although they made clear they wanted screenplays for a film _and_ a sequel. Newman says Alex often asked about what was happening with "Mr. Superman and Mrs. Lois Lane," but "when we would start telling him he would fall asleep in about five minutes. I said to David, 'It's like telling bedtime stories.' " They did get paid—at the end of each day, in cash, with money from whatever country had the best currency exchange rate. They also got ongoing guidance from National's E. Nelson Bridwell, who was a living encyclopedia of everything that Superman had said, done, or imagined.
Heeding Bridwell's advice was less a matter of choice than of law, as spelled out in a fifty-four-page agreement between National and the Salkinds. It prescribed that the films "shall not be satirical or obscene." They had to be G-rated, or at worst PG, and had to be consistent with the way Superman spoke and acted in the comic books. National would get to vet the screenplay and be there during filming. Costumes for Superman and Superboy had to be preapproved, as did the actors who played them and Lois. Just to be sure, the publisher submitted its preferred lists for these parts. Superman and Clark, it suggested, might best be handled by any of twenty-four A-list actors, from Charlton Heston, known for his roles as Moses and Ben-Hur, to tough guy Charles Bronson. Lois's list had twenty-three actresses, from Natalie Wood, who had gotten rave reviews as Maria in _West Side Story_ , to sexpot Raquel Welch.
Before they could worry about stars, the Salkinds needed a director. Tops on their list was _Chinatown_ maestro Roman Polanski, who was still reeling from the murder of his pregnant wife, Sharon Tate, and would soon be accused of sexually abusing a thirteen-year-old girl. "Not exactly my kind of thing, Ilya," Polanski said of Superman. _Jaws_ director Steven Spielberg approached him, Ilya says, but Alex worried that "the shark might go down, let's wait and see how this fish movie does." _Jaws_ was a smash and now Spielberg was out of reach. From there they moved on to a who's who of top Hollywood skippers—from Francis Ford Coppola, who was busy with _Apocalypse Now_ , to John Guillermin, whose hero of the moment was King Kong, to Sam Peckinpah, who pulled a gun on Ilya and said, "You gotta shut up, kid. What do you think you know about movies?" They finally settled on Guy Hamilton, who had made his name with _Goldfinger_. He looked like a gem until the production moved from Rome, where star Marlon Brando had a pending arrest warrant for sexual obscenity, to London, where Hamilton was a tax exile. Moving back to London, Hamilton decided, would cost him too much money.
Richard Donner was a perfect fit. He had grown up in the Bronx as a "comic book man" and his first true love was Lois Lane. He had just finished directing _The Omen_ and was ready for new work. So he listened intently when he got a call on a Sunday morning from a man with what sounded like a Hungarian accent saying, "I am a world famous producer. I am making _Superman_ and I want you to make it." Two hours later Alex Salkind's messenger was at Donner's door with a copy of the Puzo-Benton-Newman script. But the deeper he read, the more alarmed Donner became. "It was a parody on a parody. They were destroying Superman," he recalls. To see whether it could be salvaged, he invited over Tom Mankiewicz, a friend and the screenwriter for some of the James Bond movies. By the time Mankiewicz arrived, Donner had put on a Superman costume and convinced himself that if Mankiewicz agreed to rewrite the script, and Salkind agreed to hire both of them, he would do the movie. "I took the job to protect Superman," he says, "plus the fact that I was being paid a million dollars."
It actually was a million dollars as an advance against 7.5 percent of the film's gross, which made it even more attractive to Donner but still looked like a bargain to the Salkinds. They had already agreed to pay Brando more than any film star had ever received—11.3 percent of domestic gross and 5.6 percent of foreign, with a guarantee of at least $2.7 million—to play Jor-El, who was on-screen for thirteen and a half of the movie's 143 minutes. Two days later they signed up Gene Hackman for $2 million to play Lex Luthor. High-priced talent like that reassured anxious executives at Warner Communications and helped Alex woo his financiers. Still, with no final screenplay in hand, and without a frame of film, the Salkinds had just agreed to hand over a quarter of their profits, or 30 percent including earlier promises to Puzo. They were on the hook for another $10 million in salaries, agents' fees, and bills for gilded suites at hotels like the Plaza in New York and the Beverly Hills in Beverly Hills. And they still had no clue who would be their Superman.
Saturdays were "Superman test day." By the time Donner and Mankiewicz came on board at the end of 1976, a lineup of first-rate stars had refused or been rejected for the part. Alex's first choice was Robert Redford, who said no. So did Paul Newman, although Ilya says Newman "vomited" when he heard later how much Brando was earning. Nearly two hundred other actors were considered, including Sylvester Stallone (too Italian), Arnold Schwarzenegger (too Aryan), Muhammad Ali (too black), James Caan (too greedy), Bruce Jenner (too little talent), and Clint Eastwood (too busy). Ilya's wife had her own favorite—her dentist in Beverly Hills—and he was flown in for a firsthand look (everyone agreed that he "looked terrific" and wasn't worth the risk). Gossip columnists were having a field day and Alex was having a conniption. It was reminiscent of the casting calls that eventually found Kirk Alyn and George Reeves, only worse, with shooting set to start in eight weeks. Ilya says he was all for using an unknown actor who wouldn't overshadow the role but Donner was intent on a big name; Donner says it was just the reverse. They agreed it was time to have a second look at the skinny Juilliard-trained actor whose photo the casting director, Lynn Stalmaster, kept putting back in the in-pile every time they'd toss it out.
Christopher Reeve was an unlikely choice. It wasn't just his honey brown hair, or that his 180 pounds did not come close to filling out his six-foot-four frame. He had asthma and he sweated so profusely that a crew member would have to blow-dry his armpits between takes. He was prep school and Ivy League, with a background in serious theater that made him more comfortable in England's Old Vic theater than in its Pinewood movie lot. He was picked, as he acknowledged, 90 percent because he looked "like the guy in the comic book... the other 10% is acting talent." He also was a brilliant choice. He brought to the part irony and comic timing that harked back to the best of screwball comedy. He had dramatic good looks and an instinct for melding humanism with heroism. "When he walked into a room you could see this wasn't a conventional leading man, there was so much depth he had almost an old movie star feeling," says Stalmaster. Alex loved the price: $250,000, or less than a tenth of what Brando would get. Donner asked Reeve to try on his horn-rimmed glasses. Squinting back at him was Clark Kent. Even his name fit: Christopher Reeve would be assuming the part made famous by George Reeves. "I didn't find him," Donner would say throughout the production. "God sent him to me."
Margot Kidder fell into her part. "She literally tripped into the door when she arrived for her test," says Donner, "and I looked at Lynn and said, 'That's Lois.' " Growing up in Canada's Northwest Territories, Kidder was banned from watching television, reading comic books, or doing anything else that would have put her in touch with Superman. She didn't have to be. To play Lois Lane she just had to be herself: "I'm manic and I'm overambitious and I'm often frantic and disorganized." When Donner told her she had the part that stars like Stockard Channing and Leslie Ann Warren wanted, she thought, " 'Thank God, I really need the money!' Then I went out and to the best lingerie boutique on Beauchamp Place in London and bought six hundred bucks' worth of underwear!" She also went to charm school, courtesy of her director, learning how to wear high heels rather than cowboy boots and to sit in a skirt instead of blue jeans. In the end Kidder was what old fans had always imagined Lois Lane looked like, and what young ones would from 1978 on.
With the big roles filled and the big names signed, Donner and Mankiewicz could zero in on telling their story. Their key was recognizing that, to fans, Superman was not a fantasy character but an embodiment of real hopes and ideals. "It's as simple as that: truth, justice, and the American way. What other comic book hero could say that?" asks Donner. Over the years, Superman's handlers had labored over whether they should aim for kids or parents, longtime fans or new ones. Donner had a less complicated calculus: "I was making it for me.... This picture is the biggest Erector Set given to the biggest kid in the world." This was just what had driven Jerry Siegel to dream up the hero forty years before. To make sure his cast and crew understood his passion and never slipped into parody or pretension, Donner hung on his wall a plastic airborne Superman trailing a banner that read, VERISIMILITUDE.
The movie itself was equally straightforward and came in three acts: the science fiction birth and backstory on Krypton, Clark growing into his down-to-earth values and superhero persona on the wheat fields of Kansas, and nonstop adventures in Metropolis like rescuing Lois from a crashing helicopter and saving the president from a crashing Air Force One, which were what moviegoers had paid to see. Each segment had its own cast, with limited overlap. Each was filmed at its own location. Ground zero was Pinewood Studios, just west of London, where the crew assembled a crystalline version of Krypton along with the world's biggest soundstage. Alberta, Canada, doubled as Smallville, U.S.A. New York was the stand-in for Metropolis the way Metropolis had always been for New York. The settings and stories were truer to the spirit of the comic book Superman than anything filmed before. And it was more than Superman who benefited: Donner, Mankiewicz, and their collaborators were creating a prototype for the new genre of superhero epic, one that held old fans with an elegant rendering of nostalgic origins while it offered neophytes their first bite of the legend. It was a model that everyone from Batman to Spider-Man would follow. The Man of Tomorrow had again shown the way.
What wasn't straightforward was the flying. It never is, but moviegoers in the 1970s were not as forgiving as they had been in the low-tech 1940s and 1950s. Surely a world that had just unveiled videocassette recorders, neutron bombs, and a test-tube baby named Louise Brown could give us a convincing human airliner. Donner tried having his superhero skydive into the action. He hoisted him onto a three-hundred-foot crane behind a miniaturized Golden Gate Bridge. He experimented with flying harnesses and depressurized weightless chambers. Nothing worked. The solution came from an unlikely source: special effects wizard Zoran Perisic, who had read Superman comics growing up in Serbia and had been asked, "Who is Superman?" when U.S. authorities quizzed him for his naturalization papers. He was so convinced he could make Reeve fly that he offered to pay for the tests if his idea didn't work. He put zoom lenses on both the camera and the projector so that the projected image, as seen by the camera, never changed size. Superman, who was in front of that image, appeared to come closer or move farther away—and to be performing aerial maneuvers when the camera/projector rig rotated—when in fact he was standing still. Perisic called the technique "Zoptic." Donner called it a lifesaver. The producers didn't want anyone drawing attention to the invention for fear other filmmakers would use it before they did. The Academy of Motion Picture Arts and Sciences was impressed enough that it gave the film its Special Achievement Award for Visual Effects.
Gimmickry was just half the equation; the other half was Christopher Reeve. He wanted to do more than run and dive the way Kirk Alyn and George Reeves had done. As a licensed aviator, Christopher knew what it felt like to take wing. Even without an airplane or any movement, he banked the turns, rolled, and looped, all with the ease of a stunt pilot. Back on the ground, he studied his predecessors to see what else they did poorly or well. Like them, he performed many of his own capers. George offered critical lessons in how to play the role as if he believed it but none when it came to differentiating Superman from Clark Kent, something he had never managed to do. "How could a thick pair of glasses substitute for a believable characterization?" Christopher asked. "Lois Lane shouldn't have to be blind or dim-witted." His model for Clark was a young Cary Grant—shy, vulnerable, and charmingly klutzy—and his watchword was to underplay the character. By slumping his shoulders and compressing his spine, Christopher's Clark lost a full three inches from his Superman frame. His voice became more nasal and midwestern. He slicked back his hair, flipping the part from left to right and losing the spit curl. His demeanor now suggested a guy who not only couldn't get the girl, he couldn't even get a taxi.
Bulking up for the role was a different kind of challenge for Reeve. "We shoved food down Chris and got him lots of protein drinks, five to six cans a day," recalls Dave Prowse, a bodybuilder and gym owner in London who, having played Superman in a TV commercial, had hoped to land the movie role himself. When he didn't, he agreed to train the man who did. Working out five nights a week with free weights and a trampoline, Prowse and Reeve focused first on Reeve's pectoral muscles, thighs, and back, then on his arms, shoulders, calves, and abdominals. Teatime meant a plate of cakes, and mealtime came four times a day. In just six weeks Reeve put on more than thirty pounds, mostly muscle, adding two inches to his chest, two to his biceps, and enough overall that he could take the muscle padding out of his blue body stocking. Training him was easy at first, but when he had to leave for a week to be with another client, Prowse says, "Chris called me all the names under the sun. He said he was losing weight and strength. Donner called me over and said, 'He really thinks he is Superman.' "
And so he was. Superman himself changed with every artist who filled in his features, every writer who scripted his adventures, and even the marketers and accountants who managed his finances and grew his audience. Each could claim partial ownership. Actors like Christopher Reeve did more molding and framing than anyone and could claim more proprietorship. As each scene was shot it became clearer that he was giving the hero a different face as well as a unique personality. Christopher's Superman would be funnier and more human—if less powerful or intimidating—than any who had preceded him. He was more of a Big Blue Boy Scout now, in contrast to Kirk Alyn's Action Ace and George Reeves's Man of Steel. In the hands of this conservatory-trained actor, Supes was getting increasingly comfortable baring his soul.
As the filming slogged through its second year, the cast and crew were growing temperamental and the media were wondering whether it would ever be done. Ilya and Alex watched Donner spend their money all too freely. Donner says he was the first director ever who never got a budget, so he never knew whether his spending was under or over. Brando had shown up on the set with the flu and what he thought were humorous suggestions—that Jor-El the Kryptonian should look like a bagel, or perhaps a green suitcase—although he left calling the film "a fucking Valentine" to the superhero. Alex's wife, Berta Domínguez D., who called herself the Shakespeare of Mexico, attacked Mankiewicz with a steak knife when he made a joke about Alex's height. Alex apologized for Berta, saying Mexicans shouldn't drink, and he apologized for his perpetual lying, saying, "I can't help it." Jack O'Halloran, an ex-heavyweight boxer playing a Kryptonian supervillain, was so outraged when his paychecks took months to clear that he says he dragged producer Pierre Spengler across his desk, shouting, "This is bullshit. I signed a contract to work. I worked. Now pay me."
Thankfully, the relationship that mattered most off-screen as well as on, Lois and Clark's, was in good shape. The two actors behaved like brother and sister. Christopher was the uptight, ambitious sibling, Margot was loosey-goosey. She reassured him about being typecast as Superman. He pushed her to read the script, not a novel, while they dangled from cranes waiting for the next scene. She couldn't resist pinging his steel codpiece until he'd scream, "For God's sake, stop it!" Their chemistry was most apparent in the movie's most remembered scene, on Lois's balcony. Superman arrived saying, "Good evening, Miss Lane," then cuddled her in his arms for a flight over Metropolis's skyscrapers and bridges. Mankiewicz expanded the scene from two pages to seven and says that when he first heard Chris utter his greeting, "I remember putting my hands together and pleading that he would just keep going like that." He did. When Lois asked, "Who are you?" Superman answered sweetly: "A friend." Margot says they were indeed friends, which made it easy to act that way. What was difficult was summoning the sexual energy the scene demanded. "I had to pretend," she explains, "that Christopher was Harrison Ford." It worked. She asked the man with the X-ray eyes what color underwear she was wearing and, after awkward evasions, he told the truth: "Pink." But then, Reeve's Superman could make even a fib sound guileless, the way he did when he looked into Lois's eyes and promised, "I never lie."
The scene was more a Shakespearean drama—think _Romeo and Juliet_ —than a comic book spoof, and Donner demanded an equally elevated tone for the music. "Superman was the perfect hero to be musicalized in quasi-operatic or balletic fashion," says John Williams, who composed the score and conducted the London Symphony's performance of it. There was a rousing "Superman March" for the opening and closing credits, a mysterious "Krypton crystal" motif to introduce the doomed planet, an all-American melody for Smallville, and a playful "March of the Villains" for Lex and his henchman, Otis. "My challenge and opportunity," Williams says, "was to capture musically Superman's optimism and invincibility and athletics and heroism. The perfect fifth and the perfect octave are heroic intervals that have a strength and a core power to suggest just those qualities of heroism and heroics."
While Donner and his team were working to assemble a movie worthy of their hero, the Salkinds were building an audience that would want to watch. In 1975, before they had a final screenplay, they hired three planes to fly over the Cannes Film Festival every hour with a banner reading, SUPERMAN, SALKIND, PUZO. The next year five planes carried a slightly amended message: SUPERMAN, SALKIND, HAMILTON. By 1977 a blimp was carrying the message, along with a fleet of aircraft worthy of France's Armée de l'Air.
That was just the drumroll. The fully orchestrated rollout was plotted by Warner Bros., which was handling the film's distribution and was finally convinced it had real commercial potential. Super-secrecy was Warner's watchword, with paparazzi kept clear of the studio and street sets, even when the setting was the streets of New York. Pictures of Superman on cranes and wires could undermine the illusion of him flying on his own. There were none of the standard photo handouts, either. That would shatter the intrigue that was building over this new Superman and what he looked like in tights and cape. The secrecy campaign worked so well that someone broke into Pinewood Studios to try to filch shots. The first photographs of Christopher Reeve in uniform and in the air were published just where and when Warner wanted—in the two biggest newsweeklies, just before and after the film's release, with _Newsweek_ 's shot consuming the full cover. As for paid advertising, the Warner team hatched a classic come-on that captured all that was new in the movie and happened to be true: "You'll believe a man can fly!"
But there was a last-minute glitch. Alex Salkind refused to deliver the completed film unless Warner executives agreed to kick in another $15 million. He said it would buy them additional distribution rights for "certain foreign territories." They said it was blackmail. Alex knew that 750 theaters were planning to screen the film, sight unseen, starting December 15, 1978. He also knew that his contract didn't require delivery until December 31. "There was an element of extortion in it," concedes Tom Pollock, Alex's lawyer, "but he was totally legally entitled." So Alex honored his contract, if not his word, and set a price that was $5 million more than Warner had paid for the distribution rights to all of North America and three-quarters of its international markets. The company knew it was over a barrel, and with just two weeks to spare, it agreed to pay.
Finally, five years after that dinner at a Paris café and just ten days before Christmas, the film was ready for viewing. President Jimmy Carter took his daughter, Amy, to see it at a premiere in Washington. Queen Elizabeth brought Prince Andrew to a royal unveiling in London. At the New York bash, Mario Puzo showed up in a blue Superman T-shirt and Norman Mailer wore a blue velvet tuxedo, but Marlon Brando stayed on vacation in Tahiti. A more confounding no-show was Alexander Salkind. He had been arrested by Interpol officers in Switzerland on charges of stealing $20 million from the German company that bankrolled his films and was released only after he displayed the diplomatic credentials he had secured years before courtesy of the president of Costa Rica. Rather than head to the _Superman_ parties in the United States, where he feared another arrest, he overcame his phobia of flying by using heavy sedation and hired a jet to deliver him to the safe haven of Mexico. Had he come to the U.S. gala, Alex could have met Jerry Siegel. As the film ended, Jerry approached National's publisher in tears, saying, "It was exactly how I had imagined it."
Reviewers offered a mixed verdict on Alex's production. _Newsweek_ 's Jack Kroll proclaimed it "a mass entertainment of high class and energy," while Roger Ebert called it "a wondrous combination of all the old-fashioned things we never really get tired of: adventure and romance, heroes and villains, earthshaking special effects, and—you know what else? Wit." Pauline Kael of _The New Yorker_ seemed to be writing about an entirely different movie, saying it was "cheesy looking" and gave "the impression of having been made in a panic," by a director who "can't seem to get the timing right," with a score "that transcends self-parody." Vincent Canby of _The New York Times_ began his review hopefully, writing that _Superman_ offered "good, clean, simple-minded fun." Then he took his shot: "To enjoy this movie as much as one has a right to expect, one has either to be a Superman nut, the sort of trivia expert who has absorbed all there is to know about the planet Krypton, or to check one's wits at the door, which may be more than a lot of people are prepared to do for longer than two hours."
More people qualified than Canby might have expected. The film clocked in as the sixth highest grossing of all time, bringing in just over $300 million worldwide and appearing on screens as far away as Shanghai and Peking. With the average ticket in 1978 selling for $2.50, 120 million people watched Christopher Reeve fly across the screen—one hundred times more than were buying _Superman, Superboy_ , and the rest of their family of comics that year. The movie won twenty-one awards, including best science fiction film of 1978 from the International Society of Science Fiction, Horror and Fantasy. It was an even bigger hit in pharmacies and department stores, where merchants couldn't stock enough thermoses, sneakers, lunch boxes, cereal bowls, cookie jars, and anything else with Christopher Reeve in blue tights. And it wasn't just little boys and their dads who were bewitched by the movie and its star. "I took my 7-year-old son to see the picture, not expecting very much," Penelope Hoover told readers of the _Los Angeles Times_. "When I emerged from the theater afterward, I felt like a 10-year-old kid who had just seen something wonderful.... It made me rediscover the little girl in myself and I'm happy to find her."
Hoover grasped what Warner Communications hadn't. Periodically we all need to recapture our youth and idealism, especially at a moment when America was mired in a malaise that President Jimmy Carter called a "crisis of confidence." Jerry Siegel and Joe Shuster understood that when they introduced their hero in the midst of the Depression and on the eve of a world war. The Salkinds understood it when they bought the rights to Superman and hired two grown-up kids—Donner and Mankiewicz—to make the movie. Superman, the world's biggest optimist, understood it better than anyone, which is why Hoover and her son so adored him.
One group that wasn't sure how to feel about the new film was scriptural literalists. They had plenty to mull over, starting with Marlon Brando as Jor-El. With a long-flowing white robe and a shock of silver hair, he looked as well as acted like God. "They can be a great people, Kal-El," he told his only son, explaining why the boy had been dispatched to Earth. "They wish to be. They only lack the light to show the way. For this reason above all—their capacity for good—I have sent them you. My only son." The Almighty couldn't have said it better. Similarly, Superman's adoptive parents, the Kents, were written into the script as "Christian folk whose morals are as basic as the soil they till." The movie was meant to have religious resonance, says screenwriter Mankiewicz, although the religion could as easily have been Muslim or Jewish as Christian. To many filmgoers, those references made _Superman_ even more compelling, offering grist for editorials, Sunday school discussions, scholarly articles, and more than one book. To some, it was blasphemy. "I got major death threats," remembers Donner. "How dare I symbolize Brando as God and Christopher as Jesus? Studio security brought them to my attention. Some of them were just nuts, fanatics. There was talk of blood running in the streets."
Alex had his own problems. His film, he said, cost $55 million, making it the most expensive ever, although others insisted he was inflating the costs as a bragging right and to downplay his profits to his partners. Marlon Brando sued him for $50 million. Mario Puzo had Ilya served with his legal papers at the Washington premiere of the film. Richard Donner, Christopher Reeve, and Margot Kidder filed their own lawsuits with their own gripes about promises Alex had broken. But no one should have been surprised. Breaking promises had long been Alex's modus operandi. In one of his earlier films, _The Three Musketeers_ , he had made history: He and Ilya paid their actors for one movie but came away with enough footage for a sequel as well. They got away with it that time, but the Screen Actors Guild insisted that all future contracts with them or any other producer have a provision—labeled a "Salkind Clause"—specifying how many films were being made. The lineup of _Superman_ claimants realized too late that they should have included their own clauses to help them sort out which of Alex's movie production figures were real, which hotel and country he was currently calling home, and which of his "people" were real rather than fronts set up to inflate debts and disguise profits. Even Ilya ended up suing his dad, although that wouldn't come until later and it wouldn't get resolved to either's satisfaction.
Nearly all of what people alleged against Alex was true. He had few scruples and no shame. He always had one foot in his Citroën ready to leave town, and he would never say where he was calling from for fear the FBI or Interpol might be listening. He promised shares of _Superman_ to everyone from Brando and Puzo to his German, French, and Swiss lenders. The part of his story that is seldom told is that "everyone got paid off from this movie every dollar they were entitled to," says Tom Pollock, the former MCA/Universal president who was Alex's lawyer when the lawsuits were percolating. "Dick Donner made millions and millions of dollars of profit, as did Marlon Brando, as did Mario Puzo. Warner Bros. made vastly more than anybody. I have no idea what Alex actually kept for himself. He walked away depleted and exhausted but not defeated. Through force of will and money, he put together the team that made a great movie, that generated and spawned other movies, and that created a huge business mostly for other people."
THE 1970S WERE A TIME for rebooting Superman's comic books along with his movies. Gone were Mort Weisinger's imaginary stories, along with Mort himself. Many of the Man of Steel's powers melted away, as did the robots that Mort had inserted in Superman's place to explain his absences when he was pretending to be Clark. The most surprising departure was kryptonite, which had been Superman's most effective adversary. The changes amounted to decluttering an encrusted story. The aim was about marketing as much as storytelling: Bringing Superman closer to Jerry and Joe's Golden Age creation would, his bosses hoped, win back older readers who missed the hero of their youth and educate younger ones on the brilliance of that more streamlined, less gimmicky vision.
Who better to oversee that restoration than an editor who had helped spawn the original, or claimed to have? The son of Romanian-Jewish immigrants, Julie Schwartz grew up in the Bronx—the place that had spawned more comics pioneers than any neighborhood in America. Julie and Mort attended the same high school, shared a passion for science fiction, and teamed up to publish a fan magazine, one of whose first subscribers was Jerry Siegel. Jerry liked what he read and launched his own publication, where he self-published "The Reign of the Super-Man." All of which led Julie to pose, only partly tongue-in-cheek, his Big Bang Theory: "If Mort and I had not created our fanzine, neither would have Jerry Siegel created his—and as a result may never have triggered his creation of the original Living Legend, Superman. No Siegel fanzine, no Siegel Superman!"
Julie had taken his first job in comic books in 1944, as an editor with one of the firms that would be absorbed into National. In the 1950s he was a central force in reviving the Flash and Green Lantern, kicking off a Silver Age of comics that lasted until 1970 and recaptured much of the energy and prosperity of its Golden Age beginnings. In the 1960s, while Mort was managing Superman, Julie was Batman's master. In the 1970s it was Julie's turn. He took over Superman not because he wanted to—he liked Superman but loved Batman—but because he knew the company's preeminent superhero was the comics world's definition of professional success. Now he was in the big time.
It was time for a change. Writers and artists had chafed under Mort's heavy hand. Circulation of the Superman family of comic books had been plummeting since 1966 and by 1970 its most popular title, _Superman_ , was selling barely half what it had five years before. Archrival Marvel was moving up fast; within two years it would, for the first time, wear the mantle of industry leader. That got the attention of the Warner Communications executives who had taken over National and were asking whether they belonged in the comics business. Newly installed publisher Carmine Infantino was the man on the spot, and since his specialty was artwork, not writing, he turned to his friend Julie Schwartz, now in his mid-fifties, to come up with answers for Superman. Julie was the right choice, sharing Mort's deep grounding in comics yet with few of Mort's rough edges or insecurities. The Schwartz empire, however, was not as all-encompassing as Weisinger's, including _Superman_ and _World's Finest_ but not _Action_ or the rest of the Superman-related titles that Mort had overseen.
Julie's impact was apparent from the first issues under his control in 1971. "Superman Breaks Loose" was the aptly named kickoff for a six-part series by lead writer Denny O'Neil in which a freak chain reaction converted all of the Earth's kryptonite into ordinary iron. Fans had complained that the deadly green metal was too omnipresent so, poof, it was gone, along with its gold, red, red-gold, and other rainbow of flavors. KRYPTONITE NEVERMORE! the cover promised. But kryptonite, O'Neil recognized, "was merely a symptom. The disease might have been called elephantiasis of the powers. Superman was just too mighty." Getting rid of kryptonite actually aggravated the illness by making Superman more invulnerable. The remedy, courtesy of Dr. O'Neil, was to have the explosion that rendered kryptonite harmless bring to life a demonic sandman who robbed Superman of critical powers. What was left was a streamlined hero, still super but now requiring both hands rather than the tip of a finger to hold up the world. The goal was to ratchet up the suspense by giving his enemies a better shot at taking him down. It also was to make the Kryptonian more human, more like the heroes that Stan Lee and Marvel were dreaming up.
Clark, too, was different under Julie, although in his case the change had more to do with modernizing than restoring. He moved from being a newspaperman on the _Daily Planet_ to anchoring the news desk at the Galaxy Broadcasting System, which had bought the _Planet_. Young people, Julie explained, "got their news from the television, so therefore it was only natural that Clark Kent should take a job as a television reporter." Not so natural was that whenever Clark needed to change into Superman, the station took a commercial break. His reliable but crusty boss, Perry White, was supplanted by Galaxy president Morgan Edge, who was less steeped in journalism and less trustworthy. Clark's rumpled blue suits were out as well, with a new look snazzy enough to warrant an article in the real _Gentlemen's Quarterly_. More interesting to Marvel readers was Superman's internal struggle over which of his identities—the human reporter or the alien superhero—was the real him. The verdict: Both were indispensable.
Mort's successors took Superman places politically that he hadn't been since Jerry's early days. In "I Am Curious (Black)," which came out in 1970 just as Julie was about to take the reins, Lois was shunned by the black community she was trying to write about because "she's whitey." Superman helped darken her complexion for a day, which she spent exploring the world from an African American perspective. A taxi zoomed past her outstretched arm "as if I don't exist!" Other subway riders stared at her "as if I were a... a... freak?" In the end Lois asked Superman whether her temporarily black skin would stop him from loving her. His answer planted Superman squarely back in his 1930s role as Champion of the Oppressed: "You ask that of me... Superman? An alien from Krypton... another planet? A universal outsider?"
That wasn't the only story in which race was front and center, nor was racial justice the only hot-button issue on which Superman weighed in during the decade that brought us the legalization of abortion, the fall of Saigon, and mood rings. Something important was always at stake now for the hero and his friends. Lois helped recruit Dave Stevens as the _Daily Planet_ 's first black columnist. Superman and Lois promoted Native American rights and she temporarily adopted an Indian baby. Pollution got him even more riled up. He sucked smog out of the air and expelled it into outer space a year after America celebrated its first Earth Day, and he worked to shut a dangerous chemical plant two years before toxins forced the evacuation of the Love Canal section of Niagara Falls, New York. Kal-El already had watched one home, Krypton, disintegrate when its inhabitants failed to acknowledge its impending environmental doom. He was determined to make sure the same thing didn't happen here on Earth.
Julie and his young writers collaborated in ways the scripters never had with Mort, and they answered, more convincingly than Mort's Cinderella Fallacy, the age-old question of why anyone believed Superman's lame masquerade as Clark Kent. Waking from a dream where his secret identity had been exposed, Superman put on his glasses and looked in the mirror, concluding, "That's the dumbest disguise I've ever seen!" By the end of "The Master Mesmerizer of Metropolis!" Superman and all of us had the answer. His power of "super-hypnotism" entranced anyone he met and "automatically projects my subconscious desire to be seen as a weaker and frailer man than I really am!" Not just that, but his glasses—made from the shattered glass of the Kryptonian rocket that sent him to Earth—had "some unknown property" that intensified the hypnotic effect. "Did you realize that the most successful practitioner of mass hypnosis in the world is Superman?" the editors asked as the story closed. "We didn't think so! After all—until today, Superman didn't even know it himself!"
The truth was that real fans didn't need a short-lived gimmick like that—or Christopher Reeve's shifting his hair part—to buy into Superman's disguise as Clark Kent. They loved all that he stood for, from his idealism to his unflinching heroism. Too many of their flesh-and-blood heroes were gone now. Assassins got Jack Kennedy, then Martin Luther King and Jack's brother Bobby. Drugs took Elvis and Marilyn. Baseball great and humanitarian Roberto Clemente died in a plane crash, his body lost at sea. A breakup spelled the end of the Beatles. They were all gone, but Superman endured, seemingly forever, and all those who looked to him as an archetype were grateful. If all he wanted back was for them to play along while he switched in and out of his cape and tights, his fans were ready.
For his part, Julie Schwartz added personal touches that he hoped would make his hero even more appealing and abiding. Julie drew on his Jewish heritage in stories pitting Superman against the galactic golem, Lex Luthor's evil incarnation of the mythic character of clay that watched over Jews. The editor dropped into his comics notes steering readers to old stories or explaining arcane terms, sometimes signing them with an impersonal "editor" and sometimes as "Julie." He tipped his hat, or rather Clark's, to his predecessor: The Kent apartment was furnished with a sculpted bust that looked like Weisinger and when Clark came home he tossed his hat on the statue, saying, "Evening Morty." Julie also tipped his hat to his writers and artists, including their bylines along with his on every story. And, with help from writers like the legendary Jack Kirby, the family of Superman books showed a flourish for the science fiction Julie had been raised on—giving readers a handheld computer decades before it came into use, and delving into genetic research and cloning before they were part of our vocabulary.
Many fans applauded the changes Julie brought as a return to first principles. Others mourned the dulling-down of Weisinger-era tomfoolery. In any case, most of the plot shifts didn't last long and some never made it into _Action_ and other DC titles beyond Julie's control. By 1973 Superman was back with the world on his fingertip and Denny O'Neil was back to Batman. Kryptonite returned as Superman's Achilles' heel in 1977. By the end of 1978, to realign Clark Kent with Christopher Reeve's wildly popular incarnation, Julie's comic book journalist again had print flowing through his veins and a job at the _Daily Planet_. Like reboots of Superman that came before and would follow, Julie Schwartz's arrived with fanfare and fizzled without notice.
That is less a commentary on Julie and Mort than an observation on how Superman shaped his own reality. The same way parents, through a blend of nature and nurture, influence their child's values, politics, and looks, so Superman's handlers animated who he was and what he did. But at some point a child grabs hold of his fate, and so, too, did the superhero. His writers, artists, and editors thought they were in control when it was Superman's personality and legend—what he stood for and what his fans demanded—that set their boundaries. If the DC creative team moved him too close to Clark and away from Superman, or made him more (or less) powerful than he needed to be, he quietly tugged them back toward Jerry and Joe's original vision. Sometimes it took decades, as with Mort's imaginary world; "Kryptonite Nevermore" and Julie's other tinkering unraveled more quickly. Alvin Schwartz was one of the few who saw that it was the fictional hero who was pulling the strings. "Superman directed his own destinies," says Schwartz, who ghostwrote Superman comic strips in the 1940s and 1950s. "All of us were merely his pawns."
By the mid-1970s, even Superman's magic had stopped working. His troubles had less to do with him, his editors, or his writers and more to do with the wider business of comics. Marvel had pulled ahead of National, but both were slumping. Readers continued to age, sales at newsstands were still in free fall, and while specialty comic book stores were catching on, it was not enough to make up the difference. Movies, meanwhile, were making a comeback with special effects blockbusters like _Star Wars;_ TV was attracting young viewers with shows like _All in the Family_ and _Saturday Night Live;_ and video games like Atari's _Pong_ were making a claim on the time and money of bell-bottomed preteens. Even _The New York Times_ wondered whether America's most popular superhero, once a symbol of vitality, had fallen victim to that dreaded affliction: the irrelevancy of middle age. "The famous blue long-john union suit, now faded to the color of old jeans, sags loosely where steely abdominals once stopped speeding locomotives dead on the tracks," humor columnist Russell Baker wrote. "The double chin is nearly a triple. On the back of the skull the hair is sparse, and a bit too blue to be persuasive."
Superman may never have looked like that, but it was a generous take on the cigar-chomping sixty-year-old men who ran National. New blood was needed to spice up the company and refill its coffers, and it arrived in 1976 in the person of Jenette Kahn. At twenty-eight, National's new publisher was the youngest senior executive at Warner Communications and in the world of comics. This daughter of a rabbi also was everything that Jack and Harry hadn't been: college-educated, with a degree from Harvard; an art history major who believed comic books were a form of art; and a neophyte to the industry, although she had grown up reading Superman by flashlight under the bedcovers. Most unsettling, she was a woman in a field where there were almost none. One male colleague later confided that when he heard about her hiring he headed to the men's room and threw up.
But Kahn knew publishing, having launched three successful kids' magazines, and she was willing to try anything to raise her heroes' profiles. When Bill Sarnoff, the head of Warner Publishing, was interviewing her for the job, he actually proposed terminating publication of any new comic books and focusing on licensing and other media, which was where the firm made its money. "Whether he really would have done that I can't say," she says looking back, "but I said that if we were to do that, the characters would have a radioactive half life and all the other revenue would dry up." She sensed she had limited time to make her case, so she started pushing from the day she arrived. One symbolic move was to change the company's name from National Periodical Publications, a colorless title that hid what the company did from would-be censors, to what young readers had always called it: DC Comics. Superman novels were not new, but there were more of them now, including _Superman: Last Son of Krypton_. Kahn's company also was anxious to get more free publicity, as it did when Henry Kissinger showed up on the cover of _Newsweek_ wearing a red cape, blue tights, and the moniker SUPER K, and when a former DC intern began teaching the first accredited college course on comic books. His class was approved only after Michael Uslan convinced the Indiana University dean that Superman and Moses shared an origin story and a teachable moment. Carmine Infantino made history just before he left as publisher with a special-issue comic book teaming Superman with Marvel's Spider-Man; Jenette Kahn did him one better with a book in which Superman partnered with Muhammad Ali to defend the Earth against an alien attack.
She recognized that with so many new forms of entertainment to distract the young, Superman never again would be a million-seller, and that even steep price hikes—comic books began the decade selling for fifteen cents and ended at forty—couldn't make up for the revenue lost with declining circulation. So Jenette and her business-savvy sidekick, Paul Levitz, started viewing comics as creative engines rather than cash cows, able to spin off profitable enterprises in other media. It was a process that Jack Liebowitz had started when comic books themselves were big moneymakers; now those efforts were redoubled.
Superman animated cartoons had come and gone since the Fleischers pioneered them in the 1940s, but by the late 1970s the Colossus of Krypton and Froot Loops were once again a Saturday morning ritual across America. The animator this time was Hanna-Barbera and the lineup of characters came from the _Justice League of America_ comic book. Teamwork was the theme, with Superman collaborating with such heroic friends as Batman, Robin, Wonder Woman, and Aquaman. A not-so-subtle subplot, given the target audience of four- to eight-year-olds, was that violence was verboten the way it had been during the scare of the 1940s and 1950s. The show took on various names, all but one of which included the words _Super Friends_ , and each made friends of sleep-deprived parents, who delighted in the extra hours they got in bed while their kids were mesmerized by Superman.
But the Hanna-Barbera cartoons were more than a distraction. If reach and duration are the measure of a medium's influence, _Super Friends_ gave Superman his biggest stage yet with the small fry. The series ran, with occasional interruptions, from 1973 to 1986. At its height it attracted several million children, most of whom were getting their first look at Superman and many of whom would form a lifelong bond. The show became a paradigm for Kahn's new DC, and it was Superman's most successful venture into animation. _Super Friends_ "drew a humungous audience compared to the comics," Levitz says. "It introduced more kids to our hero than Reeves or Reeve."
Luis Augusto was one of those kids. The forty-year-old architect says the _Super Friends_ cartoons were "totally real to me, then, and Superman was more real than all of them. He could fly! He could bend steel with his bare hands! Nobody could bully him (unless he was pretending to be weak)! Oh, how I dreamed of all these [things]." For Augusto as for so many children, the cartoons were a gateway to other Superman experiences—feasting on Superman comic books, entertaining himself with Superman toys, and cherishing the way Christopher Reeve brought his hero alive in the movies. Superman became a part of Augusto's life, no matter that Metropolis was thousands of miles away from his home in Salvador, Brazil. Superman was "not just some action hero," says Augusto, who today writes and draws his own comic strips, "but a model. A goal to achieve in my life."
Back in the United States, Big Blue had returned to the newspapers with a strip that was launched in 1978 as _The World's Greatest Superheroes_ and the next year was renamed _The World's Greatest Superheroes Presents Superman_. Movies offered even more potential for the synergy that Kahn and Levitz were so keen on. Comics lovers had been gathering for ever-bigger conventions since the mid-1960s, and DC capitalized on these gatherings to get fans geared up for Christopher Reeve's Superman film long before it hit the theaters. Not long after its release in late 1978 they published the first comic book miniseries, _World of Krypton_ , along with a behind-the-scenes book on the movie and a Superman dictionary for kids. The film and its stars also hitched themselves to the Special Olympics, which was good for the charity and for the company's bottom line. Movie-related marketing had become standard fare by then, but it had never been seen on this scale. Two hundred licenses were awarded for more than twelve hundred products, from soap packaged like a telephone booth to velour sweatshirts that sold at Bloomingdale's. Companies paid even more to see their names or merchandise on the screen. It was no accident that we could easily read the name of Lois's Timex watch when she romanced Superman on her balcony, and in the sequel Philip Morris paid forty thousand dollars to get its Marlboro delivery truck into the fight scene with Kryptonian bad guys.
Kahn and Levitz weren't just focused on the present. They were building for a future when comic books would again pay their own way without offshoots like licensing. The turnaround didn't come as soon as they expected—the "DC explosion" of new titles quickly and embarrassingly became the "DC implosion" when many old and new books couldn't pay for themselves—but comics did eventually regain some of their profitability. Advertising helped. Comic books had drawn ads from the beginning, but they took up fewer than two of the sixty-four pages in _Action_ No. 1. Advertising copy quickly grew to 10 percent of the publications and stayed that way through most of the Golden Age. By the 1970s, DC was running up to sixteen pages of ads in books that were down to thirty-six pages, which would prove to be the high-water mark for advertising space in comics, although still not at the 50 percent level of most magazines. The nature of the ads was shifting, too, reflecting comic books' changing readership and society's changing priorities. Gadgets were replaced by beauty aids and muscle manuals. "Sex education" products came and went quickly, thanks to the Comics Code. Breakfast cereals were a perennial advertiser, along with Oreo, Reese's, and other sweets. Pitches for correspondence courses suggested that high school dropouts were a key part of the fan base, just as pitches for older comics made clear how many collectors there were.
The DC brain trust also realized that it paid to treat the creative talent better. When Kahn arrived in 1976, she began giving artists and writers 20 percent of licensing fees for characters they dreamed up, and in 1981, when she became president as well as publisher, she began paying them 5 percent of revenues on comic books that sold more than one hundred thousand copies, a milestone that Superman hit regularly. Freelancers were now getting medical insurance and yearlong contracts, the very benefits that had made Jack Liebowitz blanch. Giving creators a financial stake measurably improved their work. Suddenly a job at DC was a better deal than one at Marvel or at other competitors, which helped lure away the best talent, at least until the competition started matching the benefits. DC Comics, which thanks to Jerry Siegel and Joe Shuster had been a poster child for the old feudal system, now seemed the model of enlightenment.
EVEN JERRY AND JOE would benefit from the new ownership, not that it was easy.
Jerry had spent the first half of the 1970s hoping to win his lawsuit against National and settling into his new life. He was in dire enough straits that he took work first as a writer and then as a proofreader at DC's archrival, Marvel, then he moved his family to California, where there was a healing sun for him and Joanne and inexpensive colleges for their daughter, Laura. To pay for the move, he had to sell off some of his treasured collection of comic books. To make a living, he took a seven-thousand-dollar-a-year job as a clerk-typist with the state of California, while Joanne sold Chevrolets at a car lot in Santa Monica. Jerry earned extra income by writing stories about Mickey Mouse, Goofy, and Donald Duck for Walt Disney's Italian line of comic books. He had fallen so far that he sometimes thought about killing himself, as George Reeves had. A living wage and the California weather helped overcome his depression, but now he worried about his weakening heart and how, if he needed an operation, he would pay for it.
No one found out about any of that until later because Jerry had gone underground, declining to talk to the press and steering clear of most old friends and colleagues. He emerged from the shadows in the fall of 1975, just as Mario Puzo was turning in his movie script and Superman was back on center stage. The creator of Superman knew how to grab the spotlight when he wanted to, and now he did. "Jerry Siegel, the co-originator of SUPERMAN, put a curse on the SUPERMAN movie!" read the press release he tapped out on his manual typewriter and distributed to all the major media. "I hope it super-bombs. I hope loyal SUPERMAN fans stay away from it in droves. I hope the whole world, becoming aware of the stench that surrounds SUPERMAN, will avoid the movie like a plague." For anyone unfamiliar with the stench, Jerry filled them in with a single-sheet summary and a nine-page exposition. National Periodical Publications and Jack Liebowitz especially had "killed my days, murdered my nights, choked my happiness, strangled my career. I consider National's executives economic murderers, money-mad monsters."
It was Jerry at his melodramatic best, showing the same passion and single-mindedness he had tapped to compose and sell his first Superman story forty years before. What led him to cook up Superman? He was inspired by President Roosevelt's fireside chats, by the Nazis' slaughter of fellow Jews, and by a depression that left him and millions of others jobless, which gave him "the great urge to help... help the despairing masses." What would he do if he had the strength of his superhero? "Rip apart the massive buildings in which these greedy people count the immense profits from the misery they have inflicted on Joe and me and our families." What did he want now? A cut of the profits.
Jerry had always been torn as well as tortured, as demonstrated in years of letters to Jack. The angry young man who felt wronged quietly did battle with the lonely one aching to be embraced. His memoir would reflect the latter; his press release bared his mad-as-hell side. No matter that Jack was just a board member now while others ran the company, or that Jerry had promised in his legal settlement never to rehash these issues. It was not just he who was hurting now but his wife and child, and he was out for blood.
The press saw this for the great story it was. A _Washington Star_ reporter visited Joe Shuster in 1975 in the dingy apartment in Queens where he was "slowly going blind, still hoping his Superman would come to his rescue." The next month a _New York Times_ reporter talked to Jerry. "For years," he said, "I've been waiting for Superman to crash in and do something about it all." Their stories played even better on TV, as the _Today_ show, the _Tomorrow_ show, and Howard Cosell appreciated. Orchestrating the publicity was Neal Adams, whose fiery art had brought new life to the Green Lantern, Batman, and Superman. Adams chaperoned the aging creators around New York, persuading the media to pay for their hotel rooms and cartoonist Irwin Hasen to draw his wide-eyed war orphan _Dondi_ with a huge tear on his face and the words, "Is it a plane? Is it a bird? No, it's a pity." Adams remembers that "Joe was like an angel sent from heaven, I never heard him utter an angry word. Jerry was very bitter." Adams drew on both and, with help from _Batman_ artist Jerry Robinson and the Cartoonists Society, he made the case with Warner Communications. "What was my leverage?" Adams asks. "Humanity. Pity. Common sense. I mean, truth, justice and the American way."
Warner's point man in the publicity struggle was Jay Emmett, Superman's longtime marketing whiz and now a man in the middle. On one side, his uncle Jack Liebowitz was adamant that Jerry and Joe had voluntarily signed away their rights and didn't deserve anything more. On the other, morality dictated that Warner, flush with cash from its Superman franchise, help the creators of that golden goose. Business logic bridged the difference. "We were about to put out a movie worth tens of millions and I said, 'Let's not worry about chicken feed,' " recalls Emmett. So after back-and-forth over particulars, Emmett and his company agreed—two days before Christmas—to give Jerry and Joe $20,000 a year for life, an amount that was intended to be fixed but that rose substantially over the years. Their medical expenses, which were enormous, were covered, along with a one-time bonus of $17,500 for each of them. Their bylines were back on the comic books and nearly everywhere else Superman appeared, including a prominent opening credit in _Superman: The Movie_ and its sequels. In return, the creators agreed, again, not to sue for more.
"Joe and I are very happy to be associated with our 'Superman' creation again," Jerry wrote in his memoir, with his joyful tears at the movie premiere making clear he meant it. Joe was happier still. The money let him move to California, where he could be in the sun, in an apartment of his own, back near Jerry and Joanne. It also let him get married for the first time, in December 1976, to Judith Ray Calpini, who seemed to have everything he had been looking for. The attractive blonde was three years younger than he was and five inches taller. Their marriage license listed her as a nurse, while a press photo called her a former showgirl and current writer and artist. It was her fourth marriage, her third having ended nine months before, and it happened almost exactly a year after his settlement with Warner Communications. It lasted eleven months and nineteen days, although their official divorce wasn't granted for another three and a half years. The divorce proceedings listed Judith as a housewife and spelled out what possessions Joe had left: three suits, a topcoat, a color TV, a lounge chair, an eight-year-old Mazda, a few thousand dollars' worth of comic art, no job, and declining health.
Being abandoned was nothing new for Joe Shuster and he didn't let it spoil the fun he was having being reunited with Superman. The settlement with Warner "has meant a tremendous change in our lives," he told _The New York Times_. "We've received marvelous recognition for the Superman movie and our names also appear in the comics. A whole new generation knows us."
# **CHAPTER 9**
# **Back to the Future**
HIS MISSION WAS TO SCRAPE off the barnacles. Take us back to Jerry Siegel and Joe Shuster's primal vision, John Byrne's bosses at DC told him in 1986, but make him a Man of Today. Rewrite Superman's forty-eight-year history, from day one, preserving everything essential and killing all that was timeworn. Don't worry about consistency or the rabid fanboys. Borrow anything you like from the Fleischer brothers' cartoons, Mort Weisinger's imaginary stories, and the Reeves and Reeve Supermen and Clarks. The choices are yours. Just make us remember why we first fell for him.
The moment demanded it. DC Comics had just marked its fiftieth anniversary by blowing up its wider universe, setting the stage for streamlined versions of the heroes who lived on. Who better to point the way than DC's leading man, who was about to celebrate his own golden jubilee? Jenette Kahn had promised bold change and, after a decade of tinkering, it was time to deliver. The new comics-only shops were begging for headliners. Could anyone be more compelling to their baby-boomer patrons than a harder-edged Man of Steel custom-built for the grimmer, edgier Dark Age of comics that kicked off in the mid-1980s?
That didn't mean it would be easy. Recasting the sacred Superman legend was as perilous as trying to jazz up the Bible or formulate a New Coke. It would take somebody with ingenuity, finesse, and a super-sized ego. Byrne had all of those, along with a proven record revitalizing Marvel's cornerstone team of superheroes, the Fantastic Four. No matter that he was born in the British Isles and raised in Canada. Hadn't the Salkinds shown that outsiders could not only be spot-on about an American hero but could even take him to new heights? Byrne said he was scared, but the truth was he was itching to get at it.
Byrne delivered a Clark with flesh and spirit. No more mild-mannered wimp. The new Mr. Kent was a newspaper columnist modeled after the gritty Jimmy Breslin, with stylish round glasses, hair brushed straight back, and the tough-guy demeanor of George Reeves. "More aggressive," explained Byrne. "Not so squeaky clean." Superman changed in reverse, losing his time-traveling powers, freeze breath, and annoying tendency to make any job look easy. He sweated, cursed, and used the toilet. "You can't do interesting stories with a god," Byrne pronounced. "He used to be a _super_ man; now he's a super _man_." The Champion of the Oppressed's politics were moving rightward in an era when Ronald Reagan occupied the White House, Rambo always drew first blood, and "shop till you drop" was the national motto. "If Reagan has done nothing else," Byrne said, "he's gotten us to wave the flag again. Superman practically wears the flag. I'll be shamelessly exploiting that." Byrne's most radical role reversal happened in Superman's head: After half a century of Superman disguising himself as a human—which made him stand out from Batman, Spider-Man, and other humans masked as heroes—now the earthbound Clark would be the real thing and the alien from Krypton the alter ego.
Superman's supporting cast underwent its own retooling. The new Lois was a woman to contend with. Her hair had gone from basic black to in-your-face russet. She could shoot an Uzi and had learned from the Green Berets how to kick a terrorist where he would feel it. When she wasn't winning a Pulitzer Prize for the _Daily Planet_ or signing a lucrative book deal she was fending off advances from Lex Luthor. No time for curling up with Superman or unmasking Clark Kent. Krypton, meanwhile, was more antiseptic and Superman once again was its sole survivor. No more Superdog, Super-Monkey, or Supergirl. Superboy was gone, too, with Clark not emerging as a superhero until he arrived in Metropolis as an adult, which was the way Jerry had written it. The good news for Superman was that his adoptive parents, the Kents, lived on in this romantic retelling and could revel in the man that Clark had become. The bad news was that Lex was even more deadly as a power-grabbing billionaire than he had been as a power-grabbing scientist. Kryptonite was back, but only in the form of a single chunk that had stuck to Superman's spacecraft, and only in green.
This was not the first time Superman had needed a remake. That had come in the early 1940s, with Jerry and Joe doing the work. Mort was the architect of change in the 1960s and Julie Schwartz oversaw 1971's powering down and tossing out of kryptonite. Those revisions were the equivalent of a haircut and nail trim. Byrne performed open-heart surgery, cleaning out arteries that had hardened over the decades and recasting the hero and his universe. The makeover was unveiled in a six-part miniseries called _The Man of Steel_ , released in the summer and fall of 1986. DC then sent Superman on a three-month vacation, suspending his comic book adventures for the first time since 1938. When they resumed in January, the company signaled the milestone and its marketing savvy by launching a new version of the _Superman_ title and starting it off with No. 1. Byrne wrote and drew that and _Action Comics_ , and a year later he was writing the old _Superman_ book, too, which had been renamed _Adventures of Superman_.
There are several ways of judging an overhaul like Byrne's, starting with whether anyone notices. Everyone seemed to, beginning with America's most influential newspaper. _The New York Times_ published four stories—one just before the six-part miniseries, one just after, a third after Superman's hiatus from the regular comic books, and even an editorial, which said to DC Comics, "We write as friends. We like your plan to modernize Superman." _The Washington Post, Los Angeles Times, Time_ , and a lineup of other publications added stories of their own. The last time the comics industry had gotten that kind of coverage was in the 1950s, when it was under siege by Dr. Fredric Wertham and the PTA.
Another measure of change is whether it lasts. Most of Mort's adjustments in the 1960s disappeared when he did in 1970. Julie's remake in 1971 was gone in a year, as was his Superman writer. Byrne's backstory became Superman's defining one for the next eighteen years. It would form the basis for two television series, two TV cartoon shows, and a BBC radio play, and was translated into Chinese. All that meant cash in the DC tills. So did the million copies the first issue of _Man of Steel_ sold, which was the kind of circulation Superman hadn't seen in thirty years. Sales came down to earth once the regular comic books were back on the racks, but at a level above where they were before the remake.
The reaction from longtime readers was harsher and quicker. This was the early days of email, and complaints flooded Byrne's computer. How dare you? they asked, although dare wasn't their four-letter word of choice, and most told rather than asked. "Excoriated" is how Byrne describes the reaction. Thanks to the advance publicity, "anal retentive fanboys" let him have it "even before the work saw print." Paul E. Akers was slightly kinder in his guest column in _The Washington Post_ , arguing that, thanks to Byrne, "the comic book hero, once virtually a deputy of the deity, has fallen to the source of secular superhumanism. Thus confused, comics today try to do almost everything but the one thing they can and should do: tell a simple, imaginative story." Critics drowned out supporters like Russell Hexter, a high school senior from Armonk, New York, who started out skeptical about the need for a reboot but told _The New York Times_ he was delighted to find a Superman who was "more believable and more weak, more like you and me." Over time sentiment has shifted toward the Hexter view, with fans upset by later changes wishing that Byrne would come back, but in 1986 and '87 it was difficult to hear anything but the anger.
Byrne took it to heart. He was equally upset with DC for killing his ideas to keep Superman's Kryptonian mother alive long enough for her to give birth on Earth to Superbaby, bring back a still-learning-the-ropes Superboy, and make other changes he was promised he would be free to undertake. "Double-crossed" is the word he uses, although DC executives say they gave him more money and freedom than anyone had had since Jerry Siegel. And like Jerry, Byrne remained scarred by the wall he had run into. "My time with Superman should have been a dream come true, but it was closer to a nightmare," he says twenty-five years later. "Virtually everything I contributed to the character has been expunged—deliberately—so, in the end, I qualify as little more than a footnote."
BYRNE WALKED AWAY FROM SUPERMAN in 1988, just as the comic book industry was turning around financially. There are two ways to earn money in publishing, as in most enterprises: sell more of what you make, or earn more on each item you sell. DC did both.
Expanding the readership seemed like a lost cause by the 1980s, a decade defined by break dancing, Cabbage Patch dolls, and an _Official Preppy Handbook_. A comics industry that had sold nearly a billion books a year in the 1940s was down to 175 million. _Superman_ 's numbers had been plummeting since 1965, when it reached 823,829 copies per issue. Sales fell to 446,678 in 1970, 296,000 in 1975, 178,946 in 1980, and a paltry 98,767 in 1985. That same year the comic strip, which by then ran in just fifteen newspapers, called it quits for a second time.
Then something happened. In 1987, sales nearly doubled from the year before—from 98,443 to 161,859. Behind-the-scenes initiatives by Kahn and her colleagues finally started to pay off, all at once. Christopher Reeve's movies had introduced Superman to tens of millions of new fans, made millions of aging fans feel young again, and motivated subsets of both to have a look at the hero's comic books. John Byrne's radical remake, and the press attention it was generating, added to the buzz and stirred up collectors. The new look and the newly numbered _Superman_ book made many think the Byrne titles would become classics. They stocked up, keeping the books in their closets wrapped in protective plastic, unread, waiting until the right moment to sell. Not even DC was sure how much each of those factors contributed, and the trends became impossible to monitor after 1987 because the publisher stopped making its circulation figures public. But everyone agreed that, finally, the Superman news was good.
That wasn't enough by itself. Whether the bump lasted a year or several, it was temporary, and sales figures would never rebound to anywhere near their 1950s peak or even the level of the 1970s. The comics business had undergone what _Time_ called an adultification. Older buyers were replacing younger ones, a trend that had begun when Americans replaced their radios with TVs, and kids found CDs, PCs, MTV, and other faster-paced ways to sate their thirst for fantasy. Hard-core fans were the norm now, with fewer casual ones and comics no longer the mass medium they had been for half a century. A survey by Marvel Comics found that its average reader was twenty, which meant he or she was born just after the postwar boom in babies and funnies. On average readers were spending ten dollars a week on comic books, which was beyond the allowance of the average kid. And that ten dollars didn't go very far, since a comic that cost fifteen cents in the 1960s and fifty cents in 1980 was generally a dollar by the end of 1990, with _Superman_ and _Batman_ holding out a bit longer at seventy-five cents.
To DC, predictability was as important as total sales. Historically, comics publishers printed two or more books for every one they sold. Candy stores, five-and-dimes, groceries, and other retailers could return any that were unsold for a full refund, which gave them little incentive to be realistic when ordering and left publishers with garbage bins loaded with untouched comic books. Direct sales changed all that. Kids who had grown up loving comics and still did started opening stores that sold nothing else. Beginning in the 1970s, they made a bargain with publishers: Knowing their customers in a way that general newsstands couldn't, they could predict how many copies they would sell and were willing to give up the right to return unsold ones. In return, publishers gave them lower prices. It worked well enough for both sides that by 1986, when the Byrne reboot was being unveiled, there were four thousand specialty comic shops nationwide accounting for half of all sales. And DC once again was making money from its comics.
OVER THE YEARS SUPERMAN became entangled in a web of inconsistencies. In the 1940s he fought alongside a super-fast hero named Flash who under his scarlet costume was a college student, Jay Garrick, and who lived in the same world we do. In the 1950s he fought beside a Flash named Barry Allen, a police scientist who lived in a parallel Earth where Jay was a comic book character and Barry's inspiration. Superman and Batman were honorary members of the Justice Society of America, a first-of-its-kind team of superheroes formed in 1940, and in 1960 they were founding members of the Justice League of America, which had never heard of the Justice Society. More confounding was how Superman had spent his youth. Did he slowly discover his powers and keep them hidden until he arrived in Metropolis and took on the identity of Superman, as Jerry told us from the first? Or did he start out as an adolescent hero named Superboy until he became a man, as Jerry and others started telling us later? And why, as he carried on decade after decade, didn't Superman ever look a day older than thirty?
Contradictions are a given for comic books, which are fantasies and, in the case of a long-running character like Superman, have been written and drawn by hundreds of different people, each of whom added his own flourish. One way to deal with the incongruities was to have fun, then tell readers it was an imaginary story or a dream. Another was simply to ignore them, the way Superman's handlers did when Pa Kent turned up as John, Jonathan, and Eben, and the Man of Steel went from outrunning a speeding train to flying faster than a wave of light. That worked fine when readers were eight or nine and stayed with the comics only into their teens. It became a problem as Superman's audience shrank, aged, and became at least as versed in everything that had come before as his writers and editors.
So, in time-honored science fiction form, those writers and editors concocted new laws of nature and systems of logic. The Superman adventures from the Golden Age of comic books, from 1938 through the early 1950s, were said to have occurred in a dimension called Earth-2. The Silver Age Superboy and Superman lived in an alternate reality called Earth-1. The parallel dimension concept had been around in science fiction for decades, but DC first introduced it to comic books in 1961, in a story where the Barry Allen Flash met the earlier Jay Garrick version. The two universes occupied the same physical realm but never intersected because they spun at different speeds of vibration. They were given counterintuitive names—Earth-2 for the older reality and Earth-1 for the newer one—in 1963. The changes were made under Julie Schwartz's leadership and they filled in the black holes that had troubled faithful readers since Superboy showed up in 1945. The Justice Society was on Earth-2, where Superman returned to the original spelling of his Kryptonian name, Kal-L, and his adoptive parents were John and Mary Kent. The Justice League was on Earth-1, as were the hero called Superboy and Jonathan and Martha Kent. When the Earth-1 Superman was in his prime, Kal-L started showing his age, which wasn't surprising since he was almost twenty-five years older, and he went into semiretirement. Most supporting characters were tough to tell apart, but not Supergirl, who was Power Girl on Earth-2, where she took over for her cousin when he called it quits.
Over time it got even more muddled. If two Earths were good, why not an Earth-3, Earth-4, Earth-5, Earth-6, Earth-K, and Earth-Prime? Superman was part of several of the new worlds, and each came with its own special heroes and villains. Characters crossed between universes and comics titles at dizzying speed. Superman fans now had to buy other heroes' books to follow his exploits, and vice versa, which was part of the motivation for DC. New readers were left with their heads spinning and many gave up. Even some veterans had trouble keeping track and pined for the simplicity of one Earth and Jerry and Joe's singular Superman.
That is the beauty of comic books: When things get out of control, editors can blow up one reality and create a more reader-friendly one, which is what they did in 1985 for DC's fiftieth anniversary. It came in twelve parts, required years of careful research, and was something that author Marv Wolfman had been aching to try since he was a kid reading comics on his stoop in Brooklyn. They called it _Crisis on Infinite Earths_ , but it could as easily have been named _Let's Start Over_. In a battle between the ever-powerful Monitor and Anti-Monitor, planets were annihilated, heroes died, and archenemies joined forces to save their skins and their worlds. The five worlds that weren't destroyed were condensed into one and no one remembered that there had ever been more. Gone were Luthor 1, Barry Allen's Flash, and Supergirl, among millions. There was but one Superman now, one Lois, one Lex, and one set of stories to keep track of. It was the most cataclysmic event in the history of the Multiverse, rebooting DC's entire line in one swoop, and it set a template both for future comic-cleansings and for impending changes in Superman.
After the _Crisis_ stories and before John Byrne's restart, DC published one of its most moving Superman stories ever, a two-parter called "Whatever Happened to the Man of Tomorrow?" It was Julie Schwartz's goodbye after more than forty years at DC and fifteen years overseeing Superman. It also was the perfect segue into the superhero's new era and new bosses. Julie's first choice to write it was Jerry Siegel, whom he called a "genius," but legal hurdles nixed that idea and he ended up with a more contemporary comic book celebrity, Alan Moore. The story opened by explaining that Superman had died ten years before and a _Daily Planet_ reporter was interviewing Lois (Lane) Elliot to find out what his last days had been like. His bitterest enemies had teamed up against him, she said, and his friends came to his defense. He broke his no-kill rule by slaying Mr. Mxyzptlk. Jimmy Olsen and Lana Lang died, too, and a disheartened Superman had disappeared into the Arctic cold, where he presumably froze to death.
But wait. On the last page, we could see that Lois's husband, Jordan, was in fact Superman and that their son, Jonathan, had inherited his powers. It was the Superman franchise's trademark wink, the kind that George Reeves had perfected in the 1950s and that Julie carried on decades later. It effectively said, "Hold on, readers. Just between us, there's more to the story." No need to spell it out—simply show baby Jonathan playing with a pile of coal, and, just by rubbing it, turning one chunk into a sparkling diamond. In the last panel Jordan looked out at his readers, suggestively closed one eye, and said, "What do _you_ think?" As for the answer to the title's question—Whatever happened to the Man of Tomorrow?—the upshot, as Julie ordained and Moore wrote, was that he, his wife, and his baby were destined "to live happily ever after."
He was actually destined to live the way John Byrne wanted him to, which was more like a human and less like an alien. That approach was why Marvel's heroes were outselling DC's, and that was the kind of Superman that Christopher Reeve had been in the movies. His Kryptonian past, the way Byrne wrote and drew it, was more otherworldly than ever. He came to life not through bodily conception but in a gestation chamber, to parents who looked less human than when Joe Shuster sketched them, and he was rocketed into space not as a cuddly baby but as an unborn child sealed in a futuristic birthing matrix. The idealized Krypton of earlier origin tales was replaced by what Jor-El called "a cold and heartless society, stripped of all human feeling, all human passion and life." No wonder Clark was quick to affirm his Midwestern surroundings and the down-to-earth side of his split personality.
Other Byrne changes were answers to critics who said the superhero had degenerated into a stodgy old man. It was true: Not just Superman but all those around him were showing their age, making him look less like a Man of Tomorrow than Yesterday's Hero. Al Capone had been the right role model for villains in the 1930s, Adolf Hitler in the 1940s, and the Dr. Strangelove–like mad scientist in the 1960s. But in the 1980s the bad guys were corporate raiders who wore Ralph Lauren suits and ID badges to the stock exchange. Enter the new Lex Luthor, Metropolis's wealthiest and slimiest citizen, fashioned in the image of The Donald (Trump), Ivan Boesky, or perhaps Marvel's Kingpin. The new Lois was even more a creature of her times. No more distressed damsel or columnist to the lovelorn—the model now was the sassy Margot Kidder, who had set the standard on-screen by telling Superman, "I've seen how the other half lives. My sister, for instance. Three kids, two cats, and one mortgage. I would go bananas in a week." Byrne's Lois pumped iron and showed thighs and breasts in a way that reflected a far less prudish Comics Code and America. And sex wasn't only for heterosexuals anymore. Policewoman Maggie Sawyer tried a closeted life and marriage but over time became more open to herself and to readers about her attraction to women.
Evolution was inevitable for a character who had lasted as long as Superman. Stasis would have doomed him. New writers and artists picked from a sprawling buffet of ideas and approaches, reflecting what mattered to them and their generation, the same way authors had with folktales like Little Red Riding Hood and with sacred texts like the Torah and the New Testament. "It's a collective work," says Paul Levitz, the former DC publisher who helped steer the 1980s reboot. "It's a long conversation. People come into the room and leave the room and the story keeps getting told. The goal in any process like that is always to preserve the essence of the character."
That is where Byrne got into trouble. He rebuilt the mythology not just around the edges but at its core, changing what not just fans but fellow mythmakers thought should be immutable. "The notion that Clark is the disguise and Superman is the real man was accepted through the 1940s, '50s, '60s, and even the '70s," says Mark Waid, who came to DC just after Byrne did and would get to do his own remake a generation later. "One thing that struck me as off-note with John Byrne's big revamp was the reversal of those roles, with Superman just another disguise. It struck me as taking away one thing that made him unique and it gave short shrift to his alien heritage." What stuck in the craw of Elliot Maggin, another longtime Superman writer, was the idea that Superman had to become less powerful to be more accessible. "You don't define the most powerful character in the world by his limitations. The whole point is that if you have all the power in the world, what do you do with it? How he answered that question is what makes him such an important American character." The most piercing critique came from Len Wein, who, like Byrne, wrote for both Marvel and DC. Byrne's Krypton, Wein says, was so sterile that it "deserved to blow up." And the new Superman "wasn't Superman. He had no heart."
Byrne, many fans agreed, had gone too far. He forgot that the key to Superman is that he is like us even though he isn't us. He isn't human but is a shining example for everyone who is. Sometimes it takes an alien to show us what is special about ourselves, or what could be if we really tried. The Last Kryptonian's greatest powers are his mind and his heart, which are why baby boomers embrace him as much as the greatest generation did.
The Byrne remake and the controversy it spawned came just in time for Superman's fiftieth birthday jubilee, which was perfect timing for DC. Rather than the anniversary conversation being about how calcified he was, it was about what made him a classic. The Smithsonian showed how central it thought Superman was to the American psyche with a yearlong celebration called "Superman: Many Lives, Many Worlds." George Reeves was there alongside Christopher Reeve, in the continuous clips the museum aired from TV, movies, and cartoons. There was a battery-operated Superman wristwatch, a box of Pep cereal with Superman on the back, and, as recognizable as Judy Garland's ruby slippers, Jimmy Olsen's bow tie. _Time_ thought the occasion important enough to justify a cover feature. CBS ran a special assembled by _Saturday Night Live_ creator Lorne Michaels, whose ex-wife was Joe Shuster's niece Rosie. Playwright David Mamet paid tribute, too, writing, "I admire anyone who can make his living in his underwear." And DC ran back-to-back-to-back birthday parties at New York's historic Puck Building, with Mayor Ed Koch toasting the ageless superhero, "May you live to 120!"
The tribute Superman himself might have enjoyed most was a book of essays called _Superman at Fifty! The Persistence of a Legend!_ It explored serious questions such as why Lois was so attracted to Superman ("Because he represents freedom. Freedom from a conventional life. Freedom from the roles women are expected to play. Freedom, in flying, from the gravity that pulls on humans") and fun ones such as is there anything Superman can't do. (A Man of Steel can't get a tattoo, a tan, a vaccination, or a vasectomy.) As to why he persisted for a full half century, the editors concluded it was his "elemental power—a simple grandeur of conception—that sticks in the soul and finds its way to the corner of one's smile."
ALEX AND ILYA SALKIND celebrated their success with _Superman: The Movie_ by doing it again. And again. Then they sold the rights to make a fourth to someone else. Sequels were their specialty—and how they turned red ink black. But by the third, the plot lines were testing the faith of even Superman's fiercest defenders and budgets were plummeting along with box office receipts.
The first of the new movies, _Superman II_ , was the most successful creatively and financially. Relieved of the need to retell the origin story, which the 1978 film had done, this 1980 release could get right into the action, although setting the opening on Krypton served as a reminder of Superman's alien beginnings. The new movie also didn't have to start from scratch with writers and rewriters. Nearly all the story was there in the Puzo-Benton-Newmans-Mankiewicz script, with David and Leslie Newman returning to do touch-ups and write a new ending since the original had been filched for the first movie. Part of the second film had been shot alongside the first, although precisely how much would be hotly contested.
Like its precursor, _Superman II_ was partly a love story. Lois and Clark were dispatched by the _Daily Planet_ to Niagara Falls, where he stuck his hand into a fireplace to rescue her hairbrush—and she realized that he didn't get burned. To her, that proved he was Superman. After half a century of denying it, he inexplicably fessed up. Then he took Lois to his Fortress of Solitude and shared with her the story of his Kryptonian roots. But being with a human like her carried a cost: He must give up his special powers. He did, spending the night of his life with her and intending to spend more. He also experienced what it was like not just to fake physical weakness but to actually be a weakling, as he got throttled by a burly trucker in a seedy diner. Tasting his own blood for the first time, he said to Lois, "Maybe we ought to hire a bodyguard from now on." Lois: "I don't want a bodyguard. I want the man I fell in love with." Clark: "I know that, Lois. And I wish he were here." Humor _and_ romance. What more could a moviegoer want?
What kids wanted was a Superman adventure story, not a yucky love tale. They didn't have to wait long. Three criminals who had been sentenced by Jor-El and his fellow elders to the Hades-like Phantom Zone escaped and came to Earth to exact revenge on Jor-El's son and his adopted planet. They were vicious and single-minded, in stark contrast to the stumblebum villains of the first movie. After subduing the president of the United States they went looking for Superman, who, upon hearing of the havoc they were wreaking, managed to restore his powers. Lex Luthor joined with the murderous trio and they kidnapped Lois. What followed was the most heart-stirring on-screen battle ever for Superman, waged in the streets and into the skies of Metropolis. Weapons were anything they could find—buses, manhole lids, humans ducking for cover—and the action was worthy of Jerry and Joe's early comics. Afraid there would soon be no city left, Superman eventually lured his adversaries to his Arctic fortress. Knowing he couldn't beat three Kryptonians, each with strength comparable to his, he feigned defeat, then stripped them of their powers by exposing them to the same crystalline red light that had weakened him. He had won the battle but was about to lose his love—the toll for getting back his superpowers was forfeiting his dream of human romance. The next day Clark kissed Lois, which he knew would eliminate her memory of his being the Man of Steel, and Superman promised the president that he would never again falter in his primary mission to safeguard humanity.
Director Richard Donner had started filming this story while he was working on the first movie, but he never got to finish it. "My feeling at the time was that if the first picture had been a failure, the Salkinds would have demanded that I come back for the second" just to torture him, Donner recalls thirty years later. "Since it was a success, they figured they could make the second one without me. One day I got a telegram from them saying my services no longer were needed and that my dear close friend Richard Lester would take over. To this day I have never heard from them." Ilya Salkind has a different version: "Dick Donner said, 'I will do the second movie on my terms and without [Pierre] Spengler,' " who was the producer and money manager. "Spengler was my friend since childhood and my father and I were very loyal guys. We said no, and it really boiled down to that."
Much of the cast sided with Donner, including Christopher Reeve, who at the time said, "The mind boggles at the prospect of doing it with someone else, because Dick was so marvelous to work with." A doubling of Reeve's salary to five hundred thousand dollars and a promise that it would be doubled again for a third movie helped unboggle him, as did knowing that enough of the work had been completed to ensure that shooting the sequel would take a fraction of the time he had spent on the first film. Margot Kidder, true to her Lois Lane persona, was less diplomatic and more loyal. She lashed out at the Salkinds, explaining to _People_ magazine that "if I think someone is an amoral asshole I say so." It earned her the admiration of Donner's many fans—and assured that Lois would have fewer than five minutes of screen time in the third movie.
Lester, who had directed two films with the Beatles, had been brought on near the end of the first Superman movie, supposedly to act as an intermediary between Donner and his producers but in fact, he acknowledged later, "to make [Donner] quit and walk out of it and they wouldn't have to pay him any more." Lester was paid twice for shepherding the second movie, first by the Salkinds and again by an anxious Warner Bros., assuring him the highest salary ever for a director. He earned it. First there was the need to substitute Lara for Jor-El in scenes that were supposed to feature the latter. The Salkinds, Lester explained, "decided not to pay Marlon Brando to be in 2, so they had to get rid of all his footage." Gene Hackman already had shot with Donner everything he needed to, and wherever adjustments had to be made, his double stood in for him. As for Kidder, she was in "uncontrollable despair" over problems with her real-world husband and daughter but had to gear up for the scene where Superman would kiss her and make her forget the life they had planned together. "It was the only time that I've ever been quite so manipulative," Lester said. "We shot that scene, and she was so out of it and so emotionally distraught that it was really a lovely performance."
Flying and other special effects continued to be a challenge, especially for the actors. Sarah Douglas, who played the sexy villainess Ursa, started each day by having her long hair tucked into a short-cropped wig. Her freckles needed coating in white makeup—"supervillains don't have freckles"—while her eyebrows were pasted up with glue to make her look haughty and false nails were baked on so they wouldn't keep falling off. The worst part of the job was hanging in the air during flying scenes, held up with just two wires on rings. Her runny nose was repeatedly wiped by a man holding a forty-foot pole with a tissue on the end. All the flying and fighting hurt so much that by the end a nurse had to be on the set to tend to the twenty-year-old Douglas. "I have injuries to that shoulder blade to this day," she says, while fellow villain Jack O'Halloran ruptured a disc and needed an operation. Douglas also earned fans who continue to follow her and her leather-clad, lizardlike Ursa, including "a terrific gay following. Guys say, 'I was struggling with my sexuality and I looked to you.' "
With Brando gone, religion played less of a role in _Superman II_ , although one of its most memorable lines was whispered by an elderly woman as Superman was rescuing a young boy about to topple into Niagara Falls. "What a nice man," she says out of the blue but not out of the script. "Of course he's Jewish." The sets and background, meanwhile, were even more elaborate and expensive for the sequel than for the original. The Fortress of Solitude was completely rebuilt. New York was re-created in miniature in the London studio, with scores of tiny cars with working headlights. Niagara Falls was there, too, along with a look-alike Times Square that would be torn apart by Superman and the trio of villains and showed how quickly costs could mount. Simulating that slice of Manhattan at the British studio required 5,500 tons of sand, cement, and tar; 500,000 feet of scaffolding; 250,000 feet of lumber; 6,000 cubic yards of concrete; and 10,000 square feet of glass. Total cost for the re-creation: $10 million. Total cost for the film: $53 million, which was a record at the time.
To ensure that everyone made back their money, Warner Bros. flipped on its head the conventional marketing strategy used for the first film. Instead of pumping up publicity, the studio did everything it could to avoid attention, at least at first. That was because it had reversed the usual schedule for release, opening the film overseas rather than in America, as was the tradition for blockbuster movies. The goal was to hit every country that mattered during its peak moviegoing season—Christmas in France, South Africa, Spain, Australia, and Italy, and Easter in West Germany and England—and still be ready for a premiere in the United States and Japan during the summer, when kids were out of school and ready for fun. For that to work, Warner's marketing mavens said, it was critical that word not leak back to America about what was happening abroad. They weren't afraid that the film would bomb so much as that it would seem stale once it reached the United States, a concern that had never bothered moviegoers overseas. So while the publicity budget was double what it had been for the first film, and the 1,397 theaters scheduled to air it nearly tripled the first film's bookings, all the ads in newspapers and on radio and TV were held until just three weeks before its June 1981 opening. The all-expenses-paid junket to Niagara Falls for reporters also was last-minute. It was the kind of gamble that Alex Salkind relished but that was anathema to Hollywood studios.
It paid off better than even Alex could have conceived. Its opening day was the highest ever for an opener and for a Friday, at nearly $4.5 million. The next day it smashed the all-time one-day record with $5.6 million, besting _Star Wars_ by more than $2 million. Its $24 million first week was a record, too, $4 million higher than _The Empire Strikes Back_. For the year, _Superman II_ earned $108 million—enough to place third among all releases in 1981, trailing only _Raiders of the Lost Ark_ and _On Golden Pond_ and topping such favorites as _Arthur, Body Heat_ , and James Bond's _For Your Eyes Only_. That jackpot, plus more than $100 million in overseas ticket sales, dug the Salkinds out of their hole from the first film. Still, says Ilya, having spent $120 million to produce the two movies, they "barely broke even."
Critics were split over how good the second movie was and how it compared with the first. "It is that rarity of rarities, a sequel that readily surpasses the original," wrote Richard Schickel of _Time_. "Suffice it to say _Superman II_ is a movie no kid need be ashamed to take his parents to." Janet Maslin of _The New York Times_ was equally effusive: " 'Superman II' is a marvelous toy. It's funny, it's full of tricks and it manages to be royally entertaining, which is really all it aims for." But _The Wall Street Journal_ 's Joy Gould Boyum called it "as loud and distracting as a carnival" and _The Washington Post_ 's Gary Arnold wrote, "What seems to have been lost is the straightforward heroic exuberance of the original film." Most damning of all was Gay S. Gasser, whose _Los Angeles Times_ commentary said, "This Superman is prone to the ugliest of human faults—petulance, envy, vengefulness. He reneges on his commitment to Lois; he throws a temper tantrum when his 'identity' is revealed; he spews out clichés in moments when sincerity would seem vital. What can Lois possibly see in him?"
While fans turned out in numbers unusual for a sequel, loyalists of Richard Donner, whose _Superman: The Movie_ was becoming a classic almost on the scale of the George Reeves _Adventures of Superman_ , were steaming about his having been dumped from _Superman II_. They resented the fact that Donner wasn't listed as director of the second film, given all the work he had done on it. They argued over whether what was left was 25 percent his or 75, and whether the cuts had been made for artistic reasons or so Lester and the Salkinds could claim the work as theirs. The Donner faithful weren't really satisfied until twenty-five years later, when, thanks to their lobbying, Warner Bros. released on DVD a re-edit called _Superman II: The Richard Donner_ _Cut_. While the storyline was largely the same, half the footage had been shot by Donner decades before but never used, including fifteen minutes of Marlon Brando as Jor-El. Donner, not Lester, was credited as the director. Finally Donner fans had the film they longed for and answers to the what-ifs. Ilya took perpetual ribbing when the old Donner crew got together in Los Angeles to watch the new version, but years later he got the last word, saying, "It was mainly because I agreed with Warner's" that the Donner cut came out. "I called Dick after that and I said, 'I love you, man.' And I do."
The Salkinds had made their own special cut of the first Superman movie in 1981. It was aimed at television, and money—not art—was the motive. They added forty-five minutes of footage, knowing that each extra minute meant more money from the ABC network. Gone was a nude baby Kal-El, along with any profanity. The 182-minute film, shown over two nights during the sweeps ratings period in 1982, finished tops in that month's Nielsen survey.
The Salkinds, Lester, Reeve, and the Newmans were back for a third movie in 1983, which, in the spirit of the first sequel, was titled simply _Superman III_. Luckily for him, Donner was long gone, and Margot Kidder was only there long enough for a cameo hello and goodbye. The romance this time was between Superman and Lana Lang, his childhood friend, played by Annette O'Toole in her first appearance in the Superman mythology that she would help redefine a generation later. The villain was an industrialist played by Robert Vaughn, TV's Man from U.N.C.L.E. The real star was Richard Pryor, as an unemployed ne'er-do-well who discovered that his skills with a computer were just what Vaughn needed to dominate the world economy. Pryor had caught the Salkinds' attention when he mentioned on Johnny Carson's _Tonight Show_ that he would love to be in a Superman film, but he ended up distracting the writers and director as they looked for a way to play to the comedian's strengths while staying true to the Superman story. It was futile. The one thing that did work was a literal junkyard brawl between a dark version of the Man of Steel, who'd been corrupted by kryptonite, and his still good-guy alter ego Clark Kent. "The whole good versus evil Superman is something we'd always wanted to fool around with," says Leslie Newman. "Superman is a love triangle with two people in it, it's got multiple personalities built in, and we wanted to take it one step further."
_Superman III_ as a whole indulged in the sort of camp that Donner had consciously avoided in the first movie and Lester only occasionally dabbled in during the second. Its most generous review came from _The New York Times_ , which called it "enjoyable enough." _The Washington Post_ branded it "a high-flying disappointment" and Reeve conceded that it "became more a Richard Pryor comedy vehicle than a proper Superman film." The frankest appraisal came from Pryor. "The movie was a piece of shit," he wrote in his memoir. Why did he do it? "The producers offered me $4 million, more than any black actor had ever been paid. 'For a piece of shit,' I'd told my agent when I finally read the script, 'it smells great.' "
Pryor wasn't the only one the filmmakers were throwing money at. Reeve got more than $1 million this time, along with script approval. Warner added $1 million to what the Salkinds were paying Lester. The film grossed barely half what _Superman II_ did, although the hero still had enough star power to finish eighth at the box office for 1983, ahead of classics like _The Big Chill_ and _Silkwood_. The Salkinds, however, knew that sequels are all about cashing in on old sets, old scripts, and old techniques, which become cheaper each time around. So despite declining ticket sales, Ilya says this was the film where he and his father went from breaking even to "making money," although he won't say how much.
This film stood out in another way: It was the first where Alex found a way around his travel phobias and legal struggles to be at the premiere, which included a reception at Ronald Reagan's White House complete with a picnic, a Beach Boys concert, and a picture with the president. Reagan, like Jack Kennedy before him, was the upbeat kind of leader Superman could relate to, in contrast to the more glum Jimmy Carter and Gerald Ford. But Reagan knew he had no fan in Christopher Reeve, who had publicly branded him a cold-hearted leader who only cared about the rich, so the movie actor turned ruler of the Free World worked his charms on Reeve at the picnic dinner. "I'm just optimist enough to think he might have changed his mind," Reagan wrote in his memoir.
_Superman IV_ could have been Reeve's answer to Reagan. In it, the Man of Steel tried to do what Reeve felt the president ought to be doing but wasn't: work to rid the world of nuclear weapons. The villain this time was Nuclear Man, a radioactive clone reminiscent of the 1940s Atom Man, although Nuclear Man had neither the flair nor the compelling Nazi backstory of his predecessor. The good news for the Salkinds was that they had no part in the 1987 film, having sold their rights to Cannon Films for $5 million. The bad news for Reeve was that he starred in the film and set its theme, which was to wake the world up to the hazards of the escalating U.S.-Soviet atomic arms race. "I thought the character could be used effectively in the real world once again," Christopher wrote later. "Big mistake." He could have saved himself the embarrassment by looking back at how Superman's handlers had dealt with World War II, when they saw that even a superhero couldn't clean up some human messes. Worse still for Reeve, he and the producers were hit with a $45 million lawsuit from two writers who said he stole their idea for the film; he won the legal battle, but at the price of steep legal fees and a tarnished reputation.
Reviewers already had delivered their verdict on _Superman IV_. "One of the cheesiest movies ever made," slammed _The Washington Post. The New York Times_ called the flying sequences "chintzy," the special effects "perfunctory," and the cinematography "so sloppy that Superman's turquoise suit is sometimes green." Critics' reactions to Superman had dimmed with each new film, according to Rotten Tomatoes, the website that compiles and weighs the critiques. Ninety-four percent of reviews for _Superman: The Movie_ were upbeat, a stunning record. The favorable score for _Superman II_ was a still impressive 88 percent. The third film plummeted to 24 percent positive, and _Superman IV_ scored an anemic 10 percent. That movie, which had its world premiere in Cleveland, made just shy of $16 million at the box office. That was less than a third of the disappointing result for _Superman III_ and a signal to Warner and the world that the Reeve Superman saga had run its course.
The Salkinds might have run out of steam for productions about Superman, but not about his youthful offshoots. In 1984, after _Superman III_ and before _IV_ , they released _Supergirl_ , a movie that followed the _Superman_ model by casting an unknown actress, Helen Slater, in the title role and saving the high-priced stars—in this case Faye Dunaway and Peter O'Toole—for the parts of the villain and the Kryptonian elder. Reeve was slated for a cameo but wisely decided against it. The plot saw Supergirl battling Dunaway's evil witch, with a love story buried between the battles much as it was in movies featuring her superhero cousin. The numbers told the real story: Only 8 percent of critics liked _Supergirl_ , an even lower score than for _Superman IV_. The film cost $35 million to make and took in just $14 million at the box office, but Ilya says that the finances were structured in a way that for him and his father, " _Supergirl_ was huge."
With Superboy, the Salkinds opted for the small screen. The half-hour TV series featuring the Boy of Steel ran for one hundred episodes over four seasons, starting in 1988. The timing was paradoxical, since John Byrne had recently killed Superboy in the comic books. Yet many of the show's best scripts were written by Cary Bates, Denny O'Neil, Mike Carlin, and other DC comic book writers who knew the character best. The TV Superboy was a college man, attending the Siegel School of Journalism at Shuster University. He also was pointedly not Christopher Reeve, especially in the first season, when John Haymes Newton played the part. "I just wanted Superboy to be a little introverted and insecure, not bumbling," Newton says looking back. "But I didn't make enough of a distinction [between Superman and Clark].... I went overboard not to do what Chris Reeve did." Newton may have been trying not to think of Reeve, but Stacy Haiduk, the actress who played Lana Lang, couldn't think of anyone else. "He was and always will be the only Superman," she says.
What was on Stan Berkowitz's mind when he wrote scripts for _Superboy_ was how many bosses he was answerable to and the contradictory notes they were sending him. The media conglomerate Viacom was the series' distributor, the Salkinds were the producers, and DC Comics was the keeper of the flame. "DC people were the Jesuits, saying Superboy can't do this and can't do that," recalls Berkowitz. "The single biggest one was killing. He couldn't kill anyone, anywhere, at any time. Otherwise everyone would be afraid of him, even the good people." While _Superboy_ had an audience a hundred times bigger than any comic book, with "the _Superboy_ show on your résumé you are just dirt," the scriptwriter says. "It pretty much destroyed my live-action TV career. They think all you're capable of writing for is children." For producer Julia Pistor it had the opposite effect, especially in England, where she grew up and where "Superman was ubiquitous." _Superboy_ "was the most fun I've ever had in anything I've ever done," she says, "and it did wonders for my career." The show died after its fourth season less because of anything Berkowitz, Pistor, or any of the actors did than because Viacom had the hundred episodes it needed for syndication and amortization and because Superman's owners wanted to reclaim the rights to the character they had turned over to the Salkinds two decades before.
Alex and Ilya, meanwhile, were having a falling-out over money matters involving a movie they were making about another iconic figure, Christopher Columbus. Getting sued for breaching a contract and committing fraud was nothing new for Alex, but this time the plaintiffs were his son and daughter-in-law. She was Jane Chaplin, Charlie's daughter, and she had loaned Alex nearly $7 million to make _Columbus_. When the film bombed, Alex said he had no money to pay her back. "This is a very delicate situation for me because he is my father and I am an only child. I can't even talk to my mother," Ilya said at the time. Alex felt equally aggrieved: "This is all very surprising, very upsetting to see my son come after his father." Looking back, Ilya says he deeply regrets the split and never seeing Alex again before he died in 1997. Their breach also prevented Ilya from moving ahead with plans for a _Superman V_ film, not to mention his dreams of a sixth, seventh, and even an eighth. Alex, he says, "immediately sold the rights for Superman back to Warner's so I couldn't get them."
The Salkinds weren't the only ones whose lives were rocked by the Superman movies. Christopher Reeve grew up over the course of the four films, not always in ways he had wanted. He started out as a twenty-four-year-old actor in serious theater whose role model was Laurence Olivier, not Cary Grant or George Reeves. When the chance arose to play Superman, his mother recalls, he saw it as a "big breakthrough" that could launch his career, but he didn't see himself staying for long in the unfamiliar world of popular culture.
By the time _Superman IV_ came out, Christopher was a thirty-four-year-old millionaire and pop culture icon. Fans' first Superman is the one they hold on to forever, and for millions of young Americans Reeve was it. Typecasting wasn't a problem for him: He had a personality strong enough to resist that, says Gae Exton, his partner at the time and the mother of his first child. "People would say, 'Superman?' and he'd say with a smile, 'No, Christopher.' " The problem, those who knew him best say, is that those ten years had robbed him of energy and at least some of his idealism. He had to count on the makeup department now to keep his hairline youthful and he did painful sit-ups to rein in a bulging gut. He worried that he was offering a negative example to kids, some of whom tried to fly like him and ended up injured or worse, as an earlier generation of kids had when imitating George Reeves's TV Superman. "By the time of _Superman III_ and _IV_ , Christopher didn't see it anymore as his pathway to success and recognition," says his brother, Ben. "It was much more a job and something he did to make money." Christopher himself called _Superman III_ "mostly a misconception," and as far as the next one, "the less said about _Superman IV_ the better." His father, a professor of literature, says that Christopher "asked me not to see 3 and 4. What I saw happening to his career he later agreed with, but at the time he couldn't help himself." He was in a series of films after _Superman_ , including serious ones such as _The Remains of the Day_ , but never as the leading man and never with as much notice as when he was playing the Man of Tomorrow.
One of the few people to play Superman who wasn't transformed by it was Aaron Smolinski, who at age three had the part of baby Clark in _Superman: The Movie_. "When I was cold, Richard Donner would wrap me in a blanket and let me talk on his walkie-talkie," Smolinski recalls. When he first met Christopher Reeve, which wasn't until _Superman III_ , "I thought, what a big man. My hand disappeared in his hand.... It wasn't until I was four or five that it really dawned on me who Superman was and what I had actually played." Is there a curse? "Let's be honest," Smolinksi says. "Having a 'Superman curse' isn't the worst thing to happen. For me, it was and is an incredible experience that I wouldn't trade for anything."
Joanne Siegel had a different take on the curse and a different kind of trade in mind for her husband and his old partner. The three were doing better since they had moved to California, and Warner Communications had hiked Joe's and Jerry's pensions to sixty thousand dollars a year. But even with that, "the quality of our lives is so demoralizing," Joanne wrote to Warner CEO Steve Ross early in 1988. The mailboxes at her and Jerry's apartment building had been broken into, water seepage was rotting the walls, termites were eating their kitchen floor, and they had to haul their soiled laundry to a public Laundromat where "people's dogs and screaming children run up and down the aisles" and "weird looking street people wander in and out." As for Joe, "his rent, like ours has been raised regularly," and if his building were converted to condos he could be evicted.
What did Joanne have in mind? She had worked it out down to the penny and the publicity. "What I'm suggesting is a spectacular gift," she wrote Ross, "presented by you, at a special media event, a gift in the area of 5 million dollars for each, in the form of tax free, gift checks. One million to represent each ten years of Superman's success. In addition, lifetime incomes of $200,000 plus royalties to equal that of top writers and artists working at DC." What if he refused? Joanne didn't threaten the Warner boss, she just reminded him that, in the wake of the triumphant Superman movies and the fiftieth anniversary hoopla, "US, Canadian and Europian [ _sic_ ] media of every kind, has been and currently are trying to get interviews from the three of us." If those journalists were made aware of Jerry's and Joe's plight, Joanne added, it would be "obvious to the media that DC's reform and ethical standards are a sham." In the end the creators didn't get checks for $5 million, but they did get another boost in their annual payouts, to $80,000 a year, with a one-time bonus for each of $15,000 and an agreement that thereafter the annuities would rise with the consumer price index. Ross, the ultimate gamesman, knew when he had met his match.
# **CHAPTER 10**
# **Till Death Do Us Part**
IT STARTED AS A JOKE. Every year Superman's writers and illustrators would hole up in a conference room, sometimes high above Manhattan and other times as far away as Florida, to plot story lines for the next twelve months. Mike Carlin, the editor in charge, taped to the wall a giant chart divided into four columns, one for each monthly book of the interconnected Superman comics. Then the fifteen creators would shout out ideas. "Team Brainiac up with Bizarro," one might say. "What about Lex running for president, and winning?" After a while Jerry Ordway would pipe up with a suggestion he had offered so often that it was a tradition: "Everyone dies—the end!" Ha ha ha.
No one was laughing this time. The team at the 1991 summit wasn't quite desperate, but close to it. The cartoonists had counted on Clark and Lois getting hitched, a story arc that had started six years before with the John Byrne reboot. Lois had fallen out of her infatuation with Superman and into love with Clark. He proposed, she accepted, and the road was paved to their wedding and a year of ready-made storylines. Now that was out. Warner Bros. was developing a television series called _Lois & Clark_ and suddenly the DC honchos were insisting that the marriage wait for TV to catch up, which could take years. The comic book creators would need something new, and fast. So while few took seriously Ordway's tension-breaking refrain, Carlin jotted it down and Ordway, who had been writing and inking Superman for six years, added a refinement. Let's not kill everybody—"Why don't we just kill _him_."
It wouldn't be the first time. Just five years after he came to life, Superman was snuffed out by a second-rate crook, or so it seemed until we learned that it was a ruse to get the thug to confess. The Last Son of Krypton became terminally ill in 1950 in "The Last Days of Superman," which turned out not to be, after all. And so it went, as he appeared to expire in 1957, 1958, 1961, 1962, 1963, 1966, 1968, 1984, and twice in 1987—with each story proving to be his inventive artifice or his writer's imagination. But what if it wasn't? What if Superman really faced off against a villain who was his equal and he couldn't fight or reason his way out? What if he really did die? That would be more riveting even than his getting married, and comic book fans might regret having taken the aging hero for granted. At the very least, it would take the pressure off Carlin and his Super-Team to find a replacement for the wedding stories.
"I'll cop to the fact that I tossed it out there," Ordway says looking back, "but I didn't know how it would work." Writer and artist Dan Jurgens walked into the meeting with two ideas on his scratch pad: "death of Superman" and "bestial foe." Carlin kept the conversation going. "Okay, wise guys," he said, " _if_ we kill him, then what happens?" Next he scribbled on his story board "doomsday for Superman." Others chimed in, trying to imagine a villain omnipotent enough to do in Superman and how a world without him would look. By the time they left the room they had a rough outline for a series of stories. Superman would duke it out with an all-new evil predator who was tearing apart Metropolis. What to call him or it? The answer was there on Carlin's chart: Doomsday. Superman would die as he slew Doomsday. Metropolis would give its savior a royal send-off. Then, after months lying in the grave, he would be brought back to life through an unspecified combination of Kryptonian technology and human faith in resurrection. Good ideas, although all would need amending and none seemed especially radical for a hero who had died and been reborn a dozen times.
Little did they know. Martha Thomases, DC's publicity person, sensed something was brewing over Labor Day weekend in 1992, when editors from TV's _Entertainment Tonight_ tracked her down at the country club in Youngstown, Ohio, where she was celebrating her dad's birthday. What was this about Superman cashing it in? They had read all about it in a front-page story in _Newsday_ titled "The Death of Superman" and now they wanted the full scoop. Cleveland's _Plain Dealer_ provided context in its headline: SUPERMAN TO DIE SAVING METROPOLIS. _The New York Times_ added a touch of whimsy: LOOK! IT'S A BIRD! IT'S A PLANE! IT'S CURTAINS FOR THE MAN OF STEEL. The frenzy had begun, with nearly every paper and station nationwide and many worldwide picking up the story and late-night TV hosts having a feast. It was a distraction for a country mired in a recession and an endless presidential campaign. Having its longest-lived icon die said something about America, even if nobody could agree what. The stories made clear one more thing: Journalists, along with most of their readers and viewers, didn't understand that heroes regularly perished in the comics and almost never stayed dead.
Carlin, who was the only person on Team Superman authorized to talk, played it coy. "Never say we wouldn't kill Superman," he told reporters. "Never say we wouldn't bring him back." The truth was that DC was reeling at all the publicity. The higher-ups at Warner Bros. were caught by surprise when they saw a CNN report on the superhero's impending death. First they were mad as hell. "How dare you kill him without consulting me?" Warner CEO Bob Daly demanded of DC president Jenette Kahn. Then they were embarrassed when a Superman ad for long-running Duracell batteries ("He runs like he's on Duracell") ran in _People_ just before a story about their unflappable mascot meeting his maker. Finally they asked the same question as the people picketing their headquarters: Why would DC liquidate its most cherished character? "Of _course_ he would survive—we weren't stupid!" Carlin would explain later. "None of us wanted to write ourselves out of a job. Or worse, be labeled the people who really did end the Neverending Battle."
The corporate bosses stopped complaining when they saw the consumer response. Readers lined up on the street and around the block outside comics stores. _Superman_ 75, the death issue, tallied the biggest one-day sale ever for a comic book, with more than six million copies printed. A special collector's edition, at $2.50, came in a sealed black polybag and included an obituary from the _Daily Planet_ as well as a black mourning armband. The spin-offs seemed endless, from a wildly successful graphic novel, novel, and young adult book to a beat-'em-up videogame and a tribute ballad from the Crash Test Dummies. T-shirts were flying off the racks, especially the one with blood oozing from a red-and-yellow _S_ shield. It worked so well that DC tried the same thing with Batman, conjuring up a backbreaking assault that temporarily took him out of action. It was the kind of moneymaking lollapalooza worthy of Harry, Jack, and the Salkinds.
For Kahn, this was why she had come to DC sixteen years before. "Our mission was to torture our readers," she explains, "and the best way to torture them is to torture our characters." For Carlin, it was a godsend. He was living out a childhood dream by overseeing Superman but was at a loss as to how to revitalize a character whose comics had resumed their free fall not long after the Byrne retooling. Now he had it. "Okay, world," he told himself, "you want to embrace the antiheroes of the time... and you think Superman is old-school and corny.... Well, how about we take him away?" It was perfect. "To save him," Superman's guardian concluded, "we had to kill him."
KILLING HIM TURNED OUT to be the easiest and most predictable chapter of "The Death of Superman" saga, although it took ten weeks to play out. Superman had confronted a long lineup of evildoers over fifty-four years—from the Ultra-Humanite and Toyman to Lex the mad scientist and Lex the ruthless capitalist—any one of whom would have relished a shot at the ultimate retribution. No prankster or bad guy in a lab coat was man enough for this job. It had to be someone whose very name suggested he was more heinous, more powerful than any evildoer the world had known. His life story would come later, but at first he was a cipher, by design. His unexplained recklessness and unquenchable rage made him ominous in the style of the mindless action comics of the day. All we knew for sure was that he had broken through his burial chamber and cast off most of his chains, that his face and body were masked by a green rubber suit, and that he was a single-minded killing machine.
His first victim was a yellow bird that he crushed in his fist, sending feathers flying. Next came big rigs and other vehicles that happened to be passing on the highway. "HAH... HA HA HAAA," he bellowed, one hand still chained behind his back. The Justice League heroes got word of his rampage and vowed to stop him. Bad idea. The Blue Beetle and Bloodwynd were mere cannon fodder, but the glory-seeking Booster Gold at least put a name to the unrelenting creature when he wondered if it "is biological... or some kind of doomsday machine!"
Then Superman stepped in. It wasn't a fair fight from the start. Doomsday was not distracted by the voices of his victims the way the Man of Steel was. "Superman—you're the only one—help us!" a boy cried as he, his mother, and his baby brother were ensnared. And so their hero did, giving a human dimension to his crusade and making his adversary seem even more maniacal. The stakes quickly became apparent, with Superman promising, "I'll stop Doomsday... if it's the last thing I do!" And so the slugfest proceeded toward Metropolis, with Doomsday turning uglier still as he shed his green covering to reveal stringy white hair and a body rippling with muscles and pocked with bony spurs. Superman's family and friends offered support and concern. "That's our son, Jonathan!" Martha Kent cried to her husband. "He's being beaten to a pulp—and those TV reporters are treating it like entertainment!" In the tenth and last installment of the series, _Superman_ 75, Superman embraced Lois, telling her, "Just remember... no matter what happens... I'll always love you. Always."
That issue was a rarity in comic books, less for its storyline than its storytelling technique. Preceding chapters had scaled back their panels from four per page to three to two, building to the climax. In this last book, each of the first twenty-two pages was a single breathtaking panel documenting another stage of the bone-crushing, beat-to-a-pulp showdown. Fists were clenched, blows were landed, and fists were clenched again, like those of boxers; elbow blows to the body ended with arms interlocked, like wrestlers'. By the end hero and villain had shredded their costumes and battered their bodies. Forget about DC's no-killing rule: This Armageddon was kill or be killed. As the narrator recounted, "In the years to come a few witnesses will tell of the power of these final punches... that they could literally feel the shockwaves." The last two drawings were spread across four foldout pages showing an anguished Lois cradling a dying Superman. "You stopped him!" she said. "You saved us all! Now relax." The narrator had the last words: "But it's too late. For this is the day—that a Superman died."
It was that comic book with the tombstone cover that journalists and fans had been anticipating, and that collectors thought would someday pay for their kids' college education. And it might have, if millions of others hadn't been thinking the same way. _Superman_ 75 went on sale in mid-November 1992, two weeks after Bill Clinton was elected president and two months before its cover date. "Copies in New York sold out the first day as customers ranging from Wall Street business people to Greenwich Village artists swamped stores," _The Wall Street Journal_ reported. Much of the world mourned Superman's passing, including humor columnist Dave Barry, who urged him to "go on the _Larry King Show_ and announce that he would come back to life if people in all 50 states wanted him to." A few fans were angry enough to call or mail in death threats to DC, one of which looked as if it had been written in blood. But _New York Times_ culture critic Frank Rich said: good riddance. "Superman," he wrote, "was a goner long before Doomsday arrived, as was the heroic ideal ('very phallic, glossy, gleamingly hard-edged, hyper-masculine' in the words of the writer Camille Paglia) he symbolized in American culture."
DC writers knew better, and they already had their next series of tales plotted out. It came in an eight-part package called "Funeral for a Friend," with two stories appearing in each of the four Superman titles, followed by an epilogue. Everybody had a chance to weigh in on the lost hero. Lex Luthor II was distraught—not at Superman's demise, but at someone else getting to claim his scalp; not in public, where he was playing the concerned citizen by planning the funeral; and not for real, since Lex II was really Lex masquerading as a nonexistent son. Jimmy Olsen felt guilty for having snapped the last photo of a dying Superman. Lois wondered how she could go on. The Kents had to watch the funeral on TV because the world didn't know that its loss of a champion meant their loss of a son. Bill and Hillary Clinton were there, though, along with more heads of state than had ever been in one place at one time. At least as impressive was the turnout of the DC superheroes, led by Batman and Wonder Woman, all wearing black armbands with the _S_ logo.
Even more than Superman's death, this funeral was the real story. It was a chance for mankind to reflect on what Superman meant to it, and for DC to remind readers why they should buy his comic books. Nobody captured the memories and meaning more movingly than Bibbo Bibbowski, Superman's tough-talking friend and snitch from Suicide Slum. Standing in his darkened bar, the Ace O'Clubs, Bibbo communed with his divinity: "God? 's me... Bibbo... Been awhile since we talked. I know my pal Superman is with ya now... So I guess he don't really need my prayers.... But the rest 'o us sure do... Take good care o' Superman... Okay, God? I miss 'im... I 'spect just about ever'body misses 'im. God? I gotta ask ya... why? Why should Superman die... when a washed-up 'ol roughneck like me goes on livin'? It ain't right, God...... It just ain't right."
In the comics, even stories about last rites and eulogies had to have action. An enigma was even better. Superman's body was stolen from his grave by the head of the Cadmus Project, a super-secret, government-backed experiment in genetic engineering. The corpse was eventually recovered, with Cadmus's vision of cloning Superman presumably having failed. Also falling short was the bid by Supergirl and other heroes to check the spike in crime rates that greeted Superman's death. Jonathan Kent took his adopted son's loss harder than anyone. He suffered a heart attack and, on the verge of death, had a vivid dream in which he rescued Clark from the afterlife. The series ended with a cliff hanger worthy of the 1950s serials, with Superman's remains again disappearing from his grave. The clear message was: Stay tuned.
All the Superman comic books took the next three months off, time that let readers absorb the loss, gave the press a chance to speculate on what would come next, and allowed DC writers and artists to hold an emergency summit to figure that out. "If this many people are caring about Superman's death—we REALLY need to amp up his return!!!" Carlin told himself as he and his crew headed back to a conference room, this time at a hotel north of Manhattan, in Tarrytown. The meeting was held behind a veil of secrecy in the dead of winter, 1992, and the writers of the four Superman books had four very different ideas on who should replace him while he was deceased and on how to bring him back to life. "It was making me nervous that we'd have to pick one of the 4," Carlin recalls, "and have 3 writers participating in a big story that THEY had less invested in." Then Louise Simonson, one of the quartet, shouted out a solution: "What the hell... Why not do all 4?" It fit the bill as brilliantly as Ordway's "kill him" idea had, and turned into a twenty-two-part series called "Reign of the Supermen." The first book in the resurrection sequence hit the stores on April 15, 1993—four days after Easter. Carlin says that timing was just luck, but its appearance on tax day wasn't. "Now," he explained, "only taxes are certain for Superman."
Back in the world of Metropolis, it was just days after Jonathan Kent had dreamed about Superman's soul following him back to the world of the living. Suddenly, four Superman look-alikes were sighted across the city. The Cyborg Superman was one-quarter human, three-quarters machine, and with his wiring and skeleton showing he was 100 percent creepy. Claiming to be the real thing, he announced, "I'm back." The Metropolis Kid was a teenage clone of Superman, constructed by the Cadmus Project using genetic material from Superman's stolen corpse. He loved sunglasses and women and hated being called Superboy. The Last Son of Krypton was a solar-powered alien who had some of Superman's memories but none of his warmth or aversion to killing. Steel, the last new arrival, was an ironworker who used his expertise in weapons design to fashion a suit of armor, mighty sledgehammer, and _S_ shield. Superman had once come to his rescue and told him to "live a life worth saving." He was. He had tried to assist Superman in fending off Doomsday and now he was helping safeguard Metropolis. He had no superpowers, made no claim to be Superman, and was African American, which was still an oddity in the world of comics.
Each would-be hero battled demons and laid out his worldview in a separate comic book—Cyborg in _Superman_ , the Metropolis Kid in _Adventures of Superman_ , the Last Son in _Action Comics_ , and Steel in _Superman: The Man of Steel_. The first issue of each came with a poster of that hero. Readers got to decide for themselves over more than five months which, if any, seemed a worthy successor to the dead superhero, or whether one of the four might actually be that hero in disguise. In the end, succeed though they did in picking up parts of Superman's mantle, the more his replacements tried to take the place of Superman the clearer it became why he had reigned for so long. None had the real hero's blend of empathy and strength, brains and a story so compelling that readers prayed it never would end. The series' name, "Reign of the Supermen," turned out to be more ironic than literal, calling to mind the miscreant a teenage Jerry Siegel gave us in "The Reign of the Super-Man."
The final stories exposed as evil imposters the two Supermen who had made the strongest claim on being the real one. The Cyborg had hatched a plot to conquer the Earth, with help from Superman's old enemy Mongul. The Last Son of Krypton turned out to be the Eradicator, an ancient Kryptonian originally created as a weapon and re-created by Superman's robots after he died. The Eradicator drew his energy from Superman's stolen body, but his wreckful intentions slowly mellowed, and in the end he joined with the Metropolis Kid, Supergirl, and the Green Lantern in standing up to the Cyborg. As the battle unfolded, the real Superman returned to life just in time, via the birthing matrix that had empowered the Eradicator. While the explanations for how that happened came from science fiction, the very fact of his rebirth affirmed writers' and readers' faith in humanity and, for many, divinity. The Christ-like nature of his journey could not have been clearer—from a noble death to the discovery of his empty tomb, the resurrection itself, and his making clear that he was back to redeem mankind.
The narratives dreamed up at the two DC conference summits were paying off in ways that were easy to tally. While they couldn't hope to reach the stratospheric levels of _Superman_ 75, the "Reign" issues sold more than a million copies a week month after month, which was better than Superman had done since the 1940s and better than he has done since. "That became our most successful year in the history of DC," remembers Kahn, "and probably in the history of comics." It was a gold mine not just for the company but for its writers and artists, all of whom were earning royalties that, thanks to publishing sales and licensing spin-offs, were bigger than anyone could have imagined. It also was a reminder that both storytelling and marketing were essential to Superman's success, just as they had been when Jerry Siegel teamed up with Jack Liebowitz.
The nearly yearlong death chronicle left a lingering imprint on Superman, too. His hair had grown to shoulder length during his time away and it stayed that way through most of the 1990s. He had more powers now, including being able to survive in space, which was a partial about-face from his depowering by John Byrne. Another reversal was the return of Superboy, who again got his own comic book and a place in the wider DC universe. Steel did, too, although his didn't last as long, and the Eradicator and Cyborg would eventually be back. How would Jerry Siegel have felt about his successors erasing his hero, even temporarily? Mike Carlin worried about that, so he asked and Jerry gave him a splendid answer. "He said, 'I love what you guys are doing,' " Carlin recalled, "and it didn't matter what anyone else thought."
EVEN AS SUPERMAN WAS being resurrected in the comic books, he was back on TV in 1993 in a series bigger than any he had done since George Reeves went off the air. It was called _Lois & Clark: The New Adventures of Superman_. As the title suggested, the show was more interested in the relationship between the two journalists than in the adventures of the superhero, and at least as interested in Lois as in Clark. Money also played a role in that formulation: Down-to-earth entanglements were cheaper to film than flying sequences and other superheroic special effects.
Teri Hatcher portrayed a Lois who was a lot like the one from Superman's earliest days and the one Margot Kidder played in the movies: feisty and ambitious, sexy, and, at first at least, disdainful of her _Daily Planet_ colleague Clark Kent. Hatcher was a known quantity from the soon-to-be-number-one TV show in America, _Seinfeld_ , or at least her breasts were: Jerry Seinfeld ended his relationship with Hatcher's character after being told her buxom build was the product of implants, then he lamented the decision after he learned they were real. It was just the sort of legend that _Lois & Clark_ creator Deborah Joy LeVine relished. "I didn't want her to be this sweet woman, I wanted to make her more of a businesswoman who was getting that story no matter what," recalls LeVine. ABC "thought she was too bossy, too bitchy, not nice enough. I used to get notes from the network saying she's unlikable, which of course was not true. I used to say, 'Guys, don't you get it? Clark puts on a cape and it doesn't really matter what she says to him.' "
Dean Cain's Clark was less the one Jerry Siegel imagined and more John Byrne's remake. Gone was the good-natured bumbler. In his place was a confident, funny hunk in the person of the six-foot, 190-pound Cain, a former football star at Princeton whose dating résumé included actress Brooke Shields and _Baywatch_ siren Pamela Anderson. Another Byrne carryover: Clark was the dominant personality, Superman his alter ego. Breaking with tradition was easier if, like LeVine, you hadn't grown up with it. In her childhood home, Shakespeare passed for popular culture and comics were verboten. "Just doing a show about a heroic guy who helps people was not that interesting to me," she says. "I was much more interested in what his problems were emotionally, how he falls in love with Lois, who is pretty horrible to him. I told ABC I didn't want to do Superman but I would love to do a show that was a love story between this alien and this Earth woman he knows he probably never will be able to have a real relationship with. I wanted to do it as a romantic comedy."
LeVine had other ideas on how to rewrite the myth. She wanted James Earl Jones to play Perry White, and she says he was "very interested," but "the fact that he was black was anathema to a lot of people" at the network and studio. They rejected her choice for Jimmy, too. "I didn't want him to be so wide-eyed and naïve. I wanted him to be more of a player." As for Perry, the most noticeable change she could manage was having him mumble "Great shades of Elvis" instead of "Great Caesar's ghost." Rather than upending the lore the way she had hoped, LeVine had to settle for tinkering around the edges.
Timing, however, was in her favor. America needed a break and a hero in the fall of 1993. It was still reeling from the Branch Davidian fiasco in Waco, Texas, which left dead close to eighty men, women, and children. A Somali warlord slaughtered twenty-three United Nations peacekeepers, a car bomb at the World Trade Center killed six and injured a thousand, and two Los Angeles police officers were convicted of violating the civil rights of Rodney King, whose vicious beating had been videotaped by a bystander. TV, as always, brought us the gruesome images, then offered viewers who stayed tuned a great escape. The fantasy diet for 1993 included _Murphy Brown, Roseanne, Grace Under Fire_ , and, the most romantic and escapist of all, _Lois & Clark_.
That first season fell short on some expectations but exceeded others. Its time slot, 8 P.M. on Sundays, saw more American households with their TVs on than any other time during the week or weekend, and more of those sets were tuned to _Lois & Clark_ than to Steven Spielberg's ballyhooed science fiction series _seaQuest DSV_. The problem was that far fewer were watching the new Superman series than the old standby _Murder She Wrote_. Yet it was advertising money that mattered most to the networks, and advertisers were more interested in high-spending eighteen- to thirty-four-year-olds who were watching _Lois & Clark_ than in _Murder She Wrote_ 's over-fifty audience. The bottom line: By the end of the second season, ABC was charging advertisers $132,000 for a thirty-second spot on _Lois_ while CBS was getting just $116,000 for ads on the more popular _Murder_.
That was no accident. Warner Bros. and the network purposefully wooed those young, profitable viewers. Erotic innuendo was one key. They tried to craft a Katharine Hepburn and Spencer Tracy–style on-screen electricity, updated for a more modern, less restrained, and less clothed era. Publicity placards showed Cain and Hatcher embracing in their undershirts. AOL promoted pin-up posters of Hatcher that set a record for Internet downloads, making her as much a part of the fantasy life of young men in the 1990s as Noel Neill had been for GIs in the 1940s. Cain was equally appealing to their female counterparts, earning his own reputation as the thinking woman's sex toy. And fans tapped into the exploding World Wide Web to let producers know what they liked, what they hated, and what dialogue and plots they had scripted in case the show's writers were interested. The marketing teams were smart enough not to write off older viewers even as they were trolling for young ones: Phyllis Coates and Jack Larson were back for cameos, to the delight of parents and grandparents who had watched them alongside George Reeves in _Adventures of Superman_.
LeVine got to see Coates, but she was no longer around by the time Larson guest-starred. The final episode of season one was the writer-producer's last, and the full-blown romance she had imagined wouldn't take hold until she was gone. By the end of the first season Lois still was infatuated with Superman but had almost married the ultra-rich, super-evil Lex Luthor. Their wedding was interrupted when Superman burst in, exposing the cunning groom's true persona and watching as Lex seemingly jumped to his death. By the start of season two a male producer was in charge and the show was more action-oriented. The ratings went down, Clark and Lois started dating, and, in the final episode of the year, Clark proposed marriage.
The TV show finally was catching up to the comic books, where Clark and Lois had gotten engaged five years before and where Mike Carlin and his team had been biding their time since 1991. Their wait was not over. At the start of season three, Lois signaled to Clark that she knew his secret identity when—true to her profession as a reporter—she answered his proposal with a question: "Who's asking, Clark or Superman?" Then she turned him down. By the seventh episode she had changed her mind and proposed to him, although that was not the last word either. The wedding was postponed twice—first when television scriptwriters were told to wait for them to get hitched first in the comic books, then to make it even more romantic by delaying it until Valentine's Day. But in a story arc that was part science fiction and part cartoon, Clark ended up marrying a frog-eating clone of Lois while his real fiancée was kidnapped by Lex, who had miraculously survived his fall and been pardoned for his crimes by a cloned president of the United States.
Loyal Superman fans would have known that getting their hero wed wouldn't be any easier than getting him killed. Clark had married Lois for the first time in the _Superman_ comic strip, in a walk down the aisle that started in September 1949 and didn't finish until February 1950. It took another two years for Clark to summon the courage to tell her that he was Superman, and as he did, he awoke to realize that their courtship and nuptials had been a dream. In 1955 the wedding story was in the comic books and it was Lois who was dreaming. They tried and failed again in 1959, and sixteen times in the 1960s. Their record in the 1970s and 1980s was better: six swings, five misses. The one wedding that took, in 1978's fortieth anniversary issue of _Action Comics_ , seemed as if it, too, was going to fizzle, since Clark was under a spell that caused him to forget his heroic side and Lois only discovered it during their seaside honeymoon. She got the spell lifted and offered Superman a way out of his vows, but, shockingly, he and his handlers didn't take the bait. Lois and Clark/Superman actually tied the knot a second time, Krypton style. The only catch was that this was the Earth-2 Superman at a time when most of the comic books were following the adventures of the Earth-1 version.
It was in December 1996 that Clark and Lois got married for real, with no equivocations or caveats. It happened in the same week on TV and in the comic books, with Superman part of the bargain. It was for better as well as worse, since the union of an intrepid reporter and a mettlesome superhero was sure to carry double doses of good and evil.
In the comics, the union occurred in a ninety-six-page _Superman_ issue called "The Wedding Album." DC gave a role to everyone who wanted one and had contributed to the nearly sixty years it took the wedding trio to make it to the altar, which added up to an unheard-of thirty-five writers, pencillers, inkers, letterers, and colorists. The plot was simple, although the scripters had been thinking about it at least since that summit in 1991. Clark and Lois had split up a year before but Lois thought better of it, returning home from her assignment as a foreign correspondent and agreeing to marry a ponytailed Clark, who had temporarily lost his superpowers as part of an attack on the Earth by a sun-eating demon. Time was made for the bridal shower, gown and tuxedo fittings, and apartment hunting, but not much attention was paid to Superman, making clear this was Clark's affair. The entire Superman family was at the ceremony. Jimmy was best man and his on-again, off-again girlfriend, Lucy, Lois's kid sister, was maid of honor. The wedding was held at the Metropolis Chapel of United Faiths, a big tent of a church, and the service was conducted by a cleric who looked a lot like Jerry Siegel. In the pews was the creative talent who had drawn and written Superman's chronicles. The story ended the only way a love triangle could: with Clark kissing Lois and, superimposed on that image at twice the size, Superman kissing her, too.
The TV wedding was staged in season four's third episode, appropriately named "Swear to God, This Time We're Not Kidding." As always when it came to Superman and his bride, the wedding almost didn't come off. The problem this time wasn't at the studio but in the script, where a scary-looking female prison escapee dubbed the Wedding Destroyer was willing to do anything to undermine Lois and Clark's bliss. They managed to stop her and got married on a mountaintop, but the wedding itself was short enough on particulars that real fans had to refer to the comic book version for the full story—and its aftermath spelled the series' demise.
Call it the _Moonlighting_ Effect. That popular 1980s TV show lost viewers once sexual tension gave way to sex. With _Lois & Clark_, interest had started to fade even before domesticity set in, perhaps because of the false starts and stops with the wedding. By the time the nuptials actually happened, only 7.5 million viewers were watching, as opposed to the 12 million who had tuned in the previous season to see Clark marry Lois's clone. With competition stiffening from the rival networks, ABC dropped the show at the end of that season even though it meant having to pay Warner Bros. more than $40 million to cancel the last season of its contract. "Maybe we shouldn't have had them get married," said Robert Singer, the show's executive producer. "But people seemed to be clamoring for that. I guess we bent under the pressure." Jenette Kahn, who had dreamed up the series at her offices at DC Comics and sold Warner on it, says that at the time "marriage was the natural progression. Retrospectively, marrying them off took away some of the excitement of the relationship."
Hatcher, who played Lois, credits the show with making her believe "I was that special," and she went on to a series of movie and TV roles before settling in with the wildly popular _Desperate Housewives_. Cain had less success after _Lois & Clark_, but he said he wasn't disappointed to see the show canceled even though he was earning a reported thirty to sixty thousand dollars an episode, or more than fifty times what George Reeves had. He didn't relish the regimen of living on steamed chicken and vegetables and counting every beer he drank, which is what it took to keep a form that could be shoehorned into his blue-and-red spandex uniform. "There comes a time when you get tired of it," he confessed, adding, "I've been in that costume far, far longer than anyone in history." It was true, as least in one sense. With his show running an hour compared to Reeves's half-hour, Cain had nearly doubled the airtime of his longest-lasting predecessor. Was he afraid of being typecast after Superman? "That's the dumbest thing in the world," he said. "Everybody who's ever been President of the United States—except for four guys—are all dead. So you shouldn't be President of the United States because you might die?"
SUPERMAN WAS ALWAYS a multimedia character, but in the old days that meant one or two formats at a time and those could be tracked with a pencil and ledger, the way Jack Liebowitz did. Now media came in countless segmented forms, and Superman's new boss as of 1990 was the world's largest media conglomerate, Time Warner, Inc. At first, executives at the parent company saw DC as a distant and not especially interesting cousin. What did comic books mean to buttoned-down titans who spent their days fretting about media behemoths like _Time_ magazine and Warner Bros. studios? But the "Death of Superman" bonanza and the success of _Lois & Clark_ had them rethinking the relationship and looking for ways that comic book characters like Superman could contribute across the Time Warner family.
The potential was there to see at Six Flags Magic Mountain, the amusement park near Los Angeles that had the world's tallest and fastest ride, called "Superman: The Escape." Where he was escaping from was unclear, but kids with queasy stomachs had to be thinking getaway when, in just seven seconds, the roller coaster accelerated from zero to one hundred miles per hour—going backward. It also rose 415 feet, which was thrilling heading up and terrifying coming down. Why name it after Superman? Perhaps because anything that powerful conjured up images of the world's fastest, most powerful hero. Or because Time Warner owned Six Flags.
Superman was appearing at Carnegie Hall, too, where the Baltimore Symphony Orchestra performed the five-part _Metropolis Symphony_ , with Krypton, Lex, and even Mr. Mxyzptlk getting movements named for them. Superman was also back in the theater, although not on Broadway. In 1992 the Goodspeed Opera House in Connecticut restaged the 1966 musical _It's a Bird... It's a Plane... It's Superman_ , and a year later the show appeared at Theater Three in Port Jefferson, New York. The Man of Steel even had his own Nintendo video game, _Superman 64_ , although video kids gave it a thumbs-down and _The New York Times_ called it a poor fit: "In Virtual Metropolis, Superman is out of his element. He lacks the reflexive grace of characters born and bred in the video game universe so he seems befuddled and trapped.... Suddenly, you realize that Superman from behind looks just like Al Gore with a cape. After that, it's hard to take him seriously."
It was easier to take him seriously in the cartoon show that inspired the video game. This was produced by Warner Bros. and broadcast on Time Warner's WB Television Network, beginning its run in 1996, just before Lois and Clark got married on competing ABC stations. There wasn't much overlap in viewership, with the cartoons targeting the Saturday-morning Froot Loops audience of preadolescents while _Lois & Clark_ aimed at their older siblings and parents. At first called just _Superman_ , then _Superman: The Animated Series_ , these cartoons may have been the character's best ever. They used animation techniques that the Fleischer brothers couldn't have imagined, borrowed half their story line from Jerry Siegel and the rest from John Byrne, and picked up an Emmy nomination. The series lasted through the end of the century, with the last episode being broadcast in February 2000.
When he wasn't starring in his own TV program Superman was guest starring in someone else's. Hawkeye Pierce referred to him at least a dozen times in the 1970s and 1980s on _M*A*S*H_ , and the prankish Army surgeon spent an entire episode dressed as the Man of Steel. In the 1990s, _Seinfeld_ went _M*A*S*H_ several episodes better. In a 1994 show called "The Visa," George observed that Jerry's "whole life revolves around Superman and cereal." The next season, in "The Marine Biologist," Jerry told Elaine that "when Superman saves someone no one asks if he's trying to hit on her." Elaine: "Well, you're not Superman." Jerry: "Well, you're not Lois Lane." Seinfeld's obsession with Superman, the ultimate man of action, was ironic given that Jerry's show was, by design, "about nothing" and that his career and life were all talk, almost no action. That was precisely Superman's appeal: It would have been difficult for the comedian to dream up a better straight man for contrast. So strong was the bond that in 1998 Jerry teamed up with an animated Superman for an American Express TV commercial in which Superman tried to help Lois when she forgot her wallet at the market, but his uniform had no pockets to carry money in and Jerry had to rescue her with his AmEx card. The ad made Superman as regular a guy as Seinfeld, and it made enough money for him, Time Warner, and American Express that they would do it again six years later in a pair of four-minute web-based commercials directed by film maven Barry Levinson.
Superman's handlers would not let him shill just any product. Jeeps were in; liquor was not. Also in were causes to which DC thought that Superman could be of service. That was the case in 1996 when it collaborated with the Defense Department and UNICEF in publishing a special comic book in which Superman swept down and saved two boys about to step on a landmine. The safety lessons were written in Serbo-Croatian, printed in both the Cyrillic characters used by Serbs and the Roman ones used by Muslims and Croats, and half a million copies of the comic were shipped to Bosnia and Herzegovina. More of the same lessons, in Spanish, would be shipped to war zones in Central America. Why Superman? "He is a citizen of the world," explained Jenette Kahn.
Canada knew that, but Canadians liked to think of Superman as one of their own, since Joe Shuster was born in Toronto, so in 1995 Canada Post issued a Superman stamp in honor of Joe. The U.S. Postal Service followed suit three years later, making clear that whatever else he was, the Big Blue Boy Scout was as all-American as baseball and jazz. The fifteen-stamp set of which Superman's was a part honored icons of the 1930s, from Franklin and Eleanor Roosevelt to Jesse Owens. But rather than choose FDR's hometown of Hyde Park, New York, in which to unveil the series, or Owens's birthplace of Oakville, Alabama, it picked Superman's home, Cleveland, Ohio. It was the Man of Steel's picture and life story that headlined the Postal Service's press release, and he was front and center on the special-edition comics sent to three hundred thousand classrooms nationwide. The first book in that series became DC's largest-circulation title ever, reaching more than 10 million people and helping teach Americans of all ages a little bit about their history.
Comic book collectors had less noble intentions. There had been hobbyists since the beginning, most of whom loved the Superman stories and art and who traded issues with one another. The trend was fueled in the 1970s and 1980s with the opening of specialty stores and publication of limited-edition books. By the early 1990s, the landscape looked decidedly different. Collectors now could cash in and even achieve a certain status by showing or selling a special issue or artifact. Many illustrators were selling their original art while Christie's and Sotheby's were staging "comic art" auctions, with bidders wearing business suits and thinking of vintage _Superman_ books as commodities not unlike pork belly futures. No wonder. A copy of _Action Comics_ No. 1 fetched $54,625 at a Sotheby's sale in 1994, which was more than the auction catalog price and more than what a copy of the first _Batman_ brought in (a surprise only to Batman boosters who had argued for years that Bats had left Supes in the wings).
DC and other companies targeted this new market by manufacturing comics aimed at the new breed of collectors with their get-rich-quick fantasies. Some books had gimmicks like glow-in-the-dark covers; others, like the Superman death special, came hermetically sealed in plastic. The latter paid off in two ways: It cost twice as much as the normal comic book, and anyone who actually wanted to read it had to buy a second copy, since the first would stay in mint condition only if the seal remained unbroken. But ultimately it was a comic book's scarcity that gave it value, which made the mass-marketed collector's edition a contradiction in terms and ensured that the Death of Superman books would, in the end, be worth no more than the cover price. By the end of the 1990s the collector's market was stronger than ever, but only for truly rare comics starring time-tested heroes.
Even as auction houses were soliciting bids for old Superman classics, a stable of DC writers was trying to create new ones. _Kingdom Come_ showed how good the new crew could be. It was a comic book series published in 1996, and two years later a different author turned it into a no-pictures book. In both, Batman's archnemesis, the Joker, attacked the _Daily Planet_ , killing everyone but Lois. Then he finished her off, too. A superhero called Magog killed the Joker, yet was acquitted by a jury that believed he had done the world a service. The only one who objected was Superman, who even after losing Lois maintained his credo that murder couldn't be justified. He was so disillusioned at the jury's verdict and Magog's new celebrity that—in this story, at least—he retired for a decade, coming back only when the world seemed about to self-destruct. After a long battle pitting one set of superheroes against another, and eventually against the United Nations, Superman finally picked up the pieces of his old existence, then, in a confirmation of his faith in the future, conceived a child with Wonder Woman. The Man of Steel imagined his baby as "a battler for truth... justice... and a new American way. I can hardly wait to see it for myself."
The plot was complicated, action-packed, and beside the point. The real aim of the series and book were for Mark Waid and Elliot Maggin, two of Superman's most loyal and skilled disciples, to reaffirm first principles. Superheroes weren't gods, the writers told us, but with their strengths came responsibilities they couldn't walk away from the way Superman had. Rules mattered, too, including ones that seemed quaint, such as murder being wrong even when the murderer meant well and the victim deserved to die. Never forgetting anything was Superman's greatest burden, especially while he was mourning Lois, and his ability to inspire hope in others was more powerful than his X-ray vision. Waid underlined the spirituality of his tale by making his narrator a minister. Maggin called the mantra about truth and justice Superman's "personal Torah." Both writers treated the superhero as if he were real. "You absolutely have to, otherwise you're just writing a cartoon," explains Waid. While he and his bosses originally conceived of _Kingdom Come_ as a story about the broader DC universe, they soon realized that "Superman is such a strong character that any story with Superman in it becomes a Superman story. He is a first among equals. If he retires, if he gives up, if he surrenders, nobody else wants to get out of bed. A world without Superman would be a world in which everybody else who's followed in his footsteps would just throw up their hands and go, 'Why go on?' "
Another landmark from the era was Jeph Loeb's 1998 _Superman for All Seasons_. Each of its four issues was a season of Superman's life, and each was told from the perspective of a new narrator. Jonathan Kent spoke of a father's love for and legacy to his son. Lois Lane described the coming to life of her superhero. Lex Luthor explained the genesis of a vengeful rival. Lana Lang talked about Superman becoming comfortable with his two sides, the human and the heroic. Like Maggin and Waid, Loeb was a devotee and wanted to get back to fundamentals. For him, though, the heart of the character was the Kansas-bred Clark Kent, who "made a choice to put on a costume and realized he had a greater destiny." Loeb thought that having him marry Lois was a mistake, that it made her and Superman more predictable and less interesting. He also wanted to make clear how Superman differed from other characters he had worked on, including Batman and Spider-Man. "Spider-Man tells us that even heroes are human and can be hurt, and that _you_ can be a superhero. Batman tells us this is a dark, terrible thing and you don't want to do it. He says, 'I'm here to scare the hell out of you,' " explains Loeb. "Superman is here to say, 'This is as good as we can be. I'm not going to preach to you. I'm not going to tell you this. I'm just going to show you through my actions that, as in the line from the Superman movie, "There are good people." ' "
This was what every DC partisan had dreamed of since the 1960s, when the world split into DC versus Marvel people, Superman versus Spider-Man. Here, thanks to writers like Waid, Maggin, and Loeb, was a Superman who was not just in touch with his motivations, as Spider-Man was, but with his and our aspirations. Spider-Man had been telling us what he thought and felt in a way that seemed self-indulgent and even narcissistic. Now Superman was showing us in a way that made us want to listen and follow. "I don't want to relate to a superhero," twenty-three-year-old Chris Clow, a political science major at Western Washington University, says in explaining why he prefers Superman to Spider-Man. "Superman continues to inspire me not because I can relate to him, but because I aspire to act as he does. He, and by extension the storytellers that have given him life, have taught me how to live well. Not financially, but socially. Spiritually. Morally. And I am better for it."
In his regular line of comic books, meanwhile, Superman was changing with the times, as he had every decade since his birth. His costume underwent tinkering, as did his haircut and his powers, which now included making himself invisible and teleporting across dimensions. His Reagan-era boosterism and bellicosity were tempered to fit the Clinton era of lowered U.S. rhetoric and America as the single superpower. There also were new characters, many of whom came from the childhood worlds or adult fantasies of their writers. Bibbo Bibbowski, Superman's pal from Suicide Slum, was the reincarnation of Jo Jo Kaminski, a hard-as-nails softie whom Jerry Ordway had adored growing up in Milwaukee. Elliot Maggin managed to insert into his stories names of girls he was dating.
Such stories were more likely than ever to play out over months and even years, which let writers develop intricate plots and subplots that they weaved in and out of _Superman, Adventures of Superman_ , and _Action_. For readers, this meant purchasing every title and issue if they wanted to keep up, which was more than okay with DC. Such ongoing narratives had been around since the 1930s, but they became more frequent in the 1960s, were ramped up again in the 1970s and 1980s, and by 1991 DC had added a numbered triangle to the covers of each primary Superman series to let fans know the order in which they should be read.
More remarkable and counterintuitive was the injection of race into Superman stories and into the staff at DC, which for twenty years had struggled with its reputation as the home of heroes who were both white and white-bread. Now the "Reign of the Supermen" story arc had parachuted a black man, John Henry Irons, into the middle of the most popular comics narrative ever. He was the least egocentric of the four replacement heroes and the easiest to warm to. When the real Superman came back, Irons—known as the Man of Steel, or by Superman as simply Steel—got a comic book of his own. That led to appearances on two cartoon shows, a role on a BBC radio series, and a feature film in which Irons was played by the biggest Superman fan of all, Shaquille O'Neal.
Was the comic book Steel a credible African American character and role model? Louise Simonson thought so when she and artist Jon Bogdanove dreamed him up, seeing him as embodying Superman's spirit if not his powers. She didn't want to make him a racial stereotype or a generic good guy, which would have been the kiss of death in an escapist medium like the comics. "Steel was a character who had made a mistake in inventing weapons, doing what he thought was a good thing until they fell into the wrong hands, and he felt guilty about it," Simonson says. It seemed to work for more than three years—until plans for the _Steel_ movie picked up steam and her bosses at DC started paying more attention. "I was told I was fired because I had sent Steel into space and he should be an earthbound character," Simonson says. "I think I was fired because if there was any publicity related to the movie they didn't want a middle-aged white woman being the face of Steel." Christopher Priest, who took over, is African American, but he says he "wrote John Henry a lot whiter than Louise wrote him. I made him droll." It didn't matter, Priest adds, because few at DC still seemed to be paying attention, and not many readers were, either. As for making Superman more appealing to black readers, Priest says that would have been difficult sixty years into the legend. Superman, he explains, "represents white culture in an intensely megalomaniacal way. To many blacks, he is not Superman so much as he is SuperWhiteMan. There's no sign on the comics shop window that reads WHITE POWER, but the sensibility is implied."
Not to everyone. Growing up in the 1950s, Harvard professor Henry Louis Gates, Jr., "used to watch _Superman_ on television every Monday night, sitting in a galvanized tub in the kitchen while my mother did the laundry." To Gates, the African American literary critic, filmmaker, and scholar, "Superman was America's big brother, getting us out of every scrape. Watching him was as soothing as the warm, soapy water in our tin tub." Celebrity weatherman Al Roker was even more dazzled growing up in Queens, reading about Superman in the comics and watching him on TV: "There was something about this guy, the fact that he's theoretically invulnerable, that he can't be killed, that he's a stranger from another planet." His hero's being white didn't enter into it, says Roker, who is black. "He was, after all, an alien, which was as different as being African American or Jewish." No one was or is more of a fan than Shaquille O'Neal, the fifteen-time NBA All-Star with size 23 shoes, who had more than five hundred framed Superman comic book covers hanging in his home in Orlando, Superman logos engraved in the headlights of his silver Mercedes, and a Superman _S_ tattooed on his left bicep. When he dies, the sports star wants to be entombed in a mausoleum "with Superman logos everywhere." While he relishes being a role model to kids, especially black ones, Shaq has looked to Superman as his own role model since he was seven. He is drawn to the Man of Steel mainly because he is a force for good, but he also identifies with his hero's split personality, which made O'Neal even more anxious to star in the film _Steel_. "Shaquille is corporate, nice-looking, soft-spoken, wears suits, and is very cordial to people, whereas Shaq is the dominant athlete," he explained. "It's kind of like Clark Kent and Superman. During the day, I am Shaquille, and at night I am Shaq."
Another sign of the changing times for Superman was when, in a three-part series in 1998, he traveled back in time to take on the horrors of the Holocaust. It was the kind of story Jerry and Joe probably wished they had done from the start in 1938, although the world knew much less then about what was happening in Germany, and editors would have been reluctant for Superman to step in, just as they were for him to get involved in a war he couldn't stop. While there was a minor flare-up over why the word _Jew_ wasn't used in the 1998 stories, there was no question who the victims were and how much Superman and his writers wanted to help. Before he was transported back to the present, Superman did manage to break up a Nazi rally in America and free some of those trapped in the Warsaw Ghetto. "I'm not a golem and I'm no angel... but it's time Superman got busy," Clark told a community elder and two young boys who were stand-ins for Jerry and Joe. Jon Bogdanove, who drew and co-wrote the series and was so taken with Superman that he named his son Kal-El, said, "I wanted to do a story that didn't just look in the style of Jerry and Joe, but could have been a story that they would have done."
All the comic books and books, TV shows and licensed products, added up to a mixed balance sheet for DC. An optimist would have reveled in the records that the death stories set and the revenues flowing in to Time Warner from licensing and subsidiary products. Realists worried that comic books—the core business built by Jack Liebowitz and Harry Donenfeld—would never again be the moneymakers that they had been. In 1991, DC accounted for slightly more than 28 percent of the nation's comic book sales, with Marvel topping 46 percent. In August 1992, DC for the first time fell to an embarrassing albeit temporary third place, with 17 percent of the market compared to Marvel's 39 percent and 18 percent for the short-lived upstart Malibu Comics. Specialty comic book stores, which in the late 1980s were the industry's savior, were in trouble, in part because they got stuck with so many nonreturnable copies of special series like the ones on Superman's death and Batman's crippling. By the late nineties, two-thirds of the stores operating earlier in the decade had shuttered their doors and direct-distribution sales had plunged from $900 million to $300 million. Even market leader Marvel was slashing production, laying off staff, and, in 1996, filing for bankruptcy protection. Superman's death and resurrection yielded spikes in sales, but the abiding death story was what was happening to comics in an age of video games and high-tech toys.
The silver lining for Superman, if not DC, was that he remained king of whatever hill was left. A Gallup poll taken around the time of the death stories showed that just 25 percent of respondents saw him as passé, compared to 60 percent who wanted him brought back to life. The nationwide survey, which included people age eighteen and older, showed Superman to be more popular than all the other superheroes combined—with 44 percent picking him as their favorite, 8 percent liking Batman, and just 5 percent choosing Spider-Man. Thirty-nine percent knew Superman came from Krypton and 66 percent said Lois Lane was his girlfriend. By comparison, just 13 percent of Americans knew that Delaware was the first state and only 34 percent recognized John Adams as the second president, according to another Gallup survey done a year earlier. "The public," the pollster concluded, "evidently knows more about Superman than it does about American history."
It wasn't just that people recognized the hero's name. He was their main man. He made them soar and moved them to imagine the best in themselves. Bill Necessary of Tyler, Texas, fell in love with Superman when he saw Christopher Reeve's first movie. Today, the forty-seven-year-old cleric is known at his Catholic church as "Superdeacon" because "of my great love for the character and for the fact that I always wear an _S_ t-shirt under my clergy shirt.... I have around 80 different versions of the _S_." Donald Wurzelbacher, a religious studies teacher in Cincinnati, is fifty and the father of four—a son named Kirk Allan, after Kirk (Superman) Alyn; a daughter named Kara, after Kara (Supergirl) Zor-El; another daughter whose middle name is Therese, after Teri (Lois Lane) Hatcher; and his oldest daughter, who was born on Christopher Reeve's birthday. The last is a matter of chance, but the first three are matters of love, as is the basement in Wurzelbacher's new home, which is dedicated to Superman memorabilia from as far back as the 1940s.
Ken Cholette, a corrections officer in Massachusetts, grew up in the 1960s watching reruns of the George Reeves _Adventures of Superman_. When he got married in 2008, his wife surprised him with a wedding cake topped by a statue of Superman carrying Lois Lane, which she had had shipped from Japan, and the couple walked down the aisle to John Williams's theme song from _Superman: The Movie_. His first Father's Day gift was getting the Superman _S_ painted on a stone in their front yard. Best of all, Cholette says, "she bought me a belated wedding gift: The Superman tattoo that I now wear proudly on my right forearm."
What makes grown men feel such connection to and even ownership of a fantasy character from their long-past childhoods? "It's the belief that with all the things that are wrong in the world there is still one thing that can't be corrupted," explains Cholette. "Superman is something that stands for everything that is good and decent." Wurzelbacher finds it sad "that many young people today seem to want a DARK hero." Superman, he adds, not only isn't dark but has shown other superheroes the light. "There would not be other superheroes if it weren't for Superman." Necessary's love is simpler and closer to home: "When I was a freshman in high school, _Superman: The Movie_ came out. I was in the balcony of the Tyler Theater on opening night, Dec. 15, 1978. I think that was when Superman became my favorite hero. I could so relate to Clark Kent. I wore glasses, I stuttered and was clumsy. I was even the manager of my high school football team, just as young Clark was in the film. That film sealed in my heart that, like Clark, I could always do good for others. I may not have his powers, but I could have his heart!"
WITH SUPERMAN ENTERING HIS seventh decade in the 1990s, it isn't surprising that many of those who were with him early on were getting old and some were dying. For those who believed that a curse of Superman brought misfortune to his friends and handlers, there was more evidence.
Joe Shuster was the first to go, with his heart giving out in the summer of 1992, just as DC writers were plotting Superman's death. Joe had been living in a one-bedroom apartment in West Los Angeles, surrounded by the clutter of his life and Superman's. There was sheet music from the Broadway musical, enlarged photocopies of his earliest Superman sketches, and clippings from _Reader's Digest_ that he needed a magnifying glass to read. Just signing his name had become a chore; his right hand trembled and his left was unable to grasp a pencil. His great escape was classical music, which he listened to on his collection of turntables, tape decks, and CD players, the sound pulsating through any or all of his dozens of stereo speakers. Jerry lived nearby and they got together for dinner regularly until Jerry had his heart bypass surgery.
Just what things were like for Joe near the end became clear in a letter his sister, Jean, wrote to Time Warner three weeks after his death. "I was shocked to learn that Joe did not only not have much money in the bank but that he had almost $20,000 in credit card debts and unpaid bills," she wrote. He had three bank accounts, one with $23,773, the others with $167 and $11. He had no life insurance. What he did have was tax returns documenting that Time Warner had, as Steve Ross promised, increased his pension to $80,000 a year. Where did it all go? His closets were stuffed with sports jackets and other clothing. Stereo components were stacked floor to ceiling in every room. Joe, Jean said, apparently had become "a compulsive buyer triggered by years of deprivation." He showered everyone in his orbit with gifts, including his "ladyfriend" and her son. He also made out a $1,200 check every month to Joanne Siegel. "She had been taking 20 percent of his income as an agent's commission," Jean wrote, "for getting pay raises for Siegel and Shuster." Jean asked that Time Warner help her out and it did, agreeing to give her $25,000 a year for the rest of her life. It also arranged Joe's memorial services, as it would for Jerry.
In 1995, the bad news involved the actor millions still thought of as Superman, Christopher Reeve. Eight years after his final Superman film, he was competing in an equestrian event. His horse inexplicably stopped as they approached a three-foot jump and he was thrown, landing on his head and snapping his spine clear through. "This is called a hangman's injury," he would explain in his memoir. "It was as if I'd been hanged, cut down, and then sent to a hospital. I was heard to say, 'I can't breathe,' and that was it." He lost all movement from the neck down. The world was shocked. How could something like this happen? How could it happen to Superman? The children of baby boomers were asking the same questions the boomers themselves had asked thirty-six years before, when George Reeves shot himself.
The difference was that Christopher Reeve not only would survive, he would offer a different and in ways more compelling model of the hero after his accident. His odds of having any function at all were daunting after doctors used wires to essentially reconnect his head to his body. In despair, he thought about suicide. But his wife, Dana, helped him believe in himself and his potential to recover. He exercised beyond exhaustion. He used electric shocks to stir his moribund nerves. He was rewarded with a wild success: He moved an index finger. His doctors were startled; other patients were inspired. Continuing the backbreaking work of rehabilitation, he regained sensation above his neck, around his shoulders, and down his left leg and arm. From a wheelchair, he directed an HBO film that was nominated for five Emmys. He played the lead in a remake of Alfred Hitchcock's _Rear Window_. He wrote an autobiography that spent eleven weeks on the _New York Times_ bestseller list. He lobbied for federal funding of stem cell research and he became the leading advocate for people with spinal cord injuries. " 'What is a hero?' " he wrote. "I remember how easily I'd talk about it, the glib response I repeated so many times. My answer was that a hero is someone who commits a courageous act without considering the consequences.... Now my definition is completely different. I think a hero is an ordinary individual who finds the strength to persevere and endure in spite of overwhelming obstacles."
Reeve was not the only one from the Superman movies who was suffering. In 1990, Margot Kidder had a serious car crash that left her in a wheelchair for about two years. That was just the beginning for the actress whose Lois Lane had seemed almost as invincible as Superman. After several surgeries and unsuccessful appeals to her insurance carrier, she declared bankruptcy. She had been married and divorced three times. In 1996 she was back in the hospital, this time a psychiatric one, after being found in the backyard of a suburban Los Angeles home, bedraggled and disoriented, claiming she had been stalked and attacked. "She was frightened for her life," a police spokesman said. "We do not feel there has been a crime at this time." The explanation was bipolar disorder. Since then she has reengaged in the world and in politics, and she has introduced her grandson to Superman. "It's the first movie that little boys really get," she says. "For little kids, mostly boys, it's their introduction to morality and I think that's a pretty powerful thing."
Reeve and Kidder both confirmed and refuted the chestnut about the Superman curse. Yes, unimaginably bad things happened to them, but what did that have to do with Superman? Yes, they had been typecast, but to themselves as well as to the public, any future role seemed like a footnote. It was their assignments with the Man of Tomorrow that stuck in their minds and America's. They believed in his story and became part of it. For Reeve, Kidder, and most of the artists associated with Superman, he was more of a blessing than a curse.
It was true even for Jerry Siegel. His heart had been giving him trouble for years and it gave way in January 1996, after a short illness. His last years were relatively comfortable. He had moved to Marina del Rey, a seaside enclave of Los Angeles, into a waterfront apartment much nicer than the one Joanne had complained about. He was collecting the same pension Joe was from Time Warner along with reimbursements for steep medical bills, and occasionally he got the acknowledgment he craved from Superman's latest midwives. It had happened during a visit to the _Lois & Clark_ set, when Joanne was introduced to Teri Hatcher, the 1990s model for Lois. It happened again when the Siegels were guests of honor at a dinner of the DC creative team during the Death of Superman run. "The fact that he was so gracious to us at all was amazing, given the past history with legal issues over Superman," says Jerry Ordway, whose shout-out had launched the death project. "None of us would have had jobs without that amazing literary creation." Paul Levitz, DC's second-in-command, was equally touched: "There was an intergenerational blessing going on."
Jerry also made his peace with the new bosses at DC and some of the old ones. "Mort Weisinger had visited him and he and Jerry went out to lunch together, hung out a bit, and became buddies," says Mark Evanier, a comic book writer and historian who visited Jerry six weeks before he died. "Jerry also talked about how much he owed to Paul Levitz of DC, that Paul was responsible for him and Joe getting whatever they had, and I thought to myself, 'Wow, when I first met Jerry if you mentioned anybody at DC Comics to him he'd put a curse on them and he'd turn orange.' It was so pleasing to me to see that he was getting some closure."
Alex Salkind was the next of Superman's intimates to go, in 1997, when he was seventy-five. Since he was ever the enigma, it is fitting that his spokeswoman refused to tell the press what killed him, although he had been hospitalized outside Paris for a stomach ailment. He spent the last years of his life the way he had earlier ones: living in hotels or on his boat, with his wife or mistress, still a fixture in Cannes and still trying to put together movie deals. Two years after Salkind's death, the first on-screen Superman, Kirk Alyn, died at the age of eighty-eight. While he had often said that playing the superhero made it difficult for him to find other movie parts, the truth was that he relished the celebrity the role brought him and, until the last, he would talk to anyone and everyone about his glory days flying over Metropolis.
Jack Liebowitz outlived them all and was proud of it. He survived his loquacious partner Harry Donenfeld by thirty-five years, and his silent partner Paul Sampliner by twenty-five. He mourned the deaths of Bob Maxwell and Mort Weisinger, both of whom went in the 1970s, both at the relatively young age of sixty-three. He never got over the breach with Jerry Siegel and would get livid when anyone mentioned his name. Jack wasn't actively involved with DC when it was shepherding the Christopher Reeve movies and the new TV shows, but he kept a seat on the board with Warner Communications, then with Time Warner, until 1991, and he went into the office every day. For a man whose comic book business and Superman character had helped define twentieth-century America, it was fitting that Jack Liebowitz came into the world in 1900 and left it in 2000, at the age of one hundred.
# **CHAPTER 11**
# **Tights and Fights**
_SMALLVILLE_ WAS _LOIS & CLARK_ in the throes of puberty and raging hormones. The new TV show also was just what the Time Warner doctors ordered to introduce Superman to America's millennial generation, which didn't know George Reeves from Christopher Reeve and in fact barely knew the Man of Steel himself. _Smallville_ had two aims: Let young viewers see why their grandparents and parents were so smitten with Superman, and give them a version of the superhero who was theirs alone.
The show zeroed in on Clark Kent's high school years in the town of Smallville, Kansas, while he was discovering his powers and before he assumed the identity of Superman. It was the Superboy story that Jerry Siegel had imagined nearly sixty years before, but with the focus now on his heart, not his muscles. There would be "no tights, no flights," the producers announced from the first, meaning their reluctant hero wouldn't don the Superman costume, wouldn't fly, and wouldn't stick to the rest of the credo built up over sixty years if that got in the way of exploring his fears and longings. What they wanted was to look deep enough inside this ordinary kid to see how he handled his extraordinary possibilities. Could he have sex? Would the world let him alone if it knew what he could do? What did life have in store for an alien who wanted so badly to be normal? These were the very questions every tortured teen was asking then about his—or her—own life.
To make the point that this was not your grandfather's Superman, _Smallville_ 's pilot episode offered a twist on the standard creation story. Clark's arrival on Earth brought with it a shower of green meteors that struck and transformed the idyllic Smallville, whose previous claim to fame was as Creamed Corn Capital of the World. The damage became clear in everyone close to Clark—from the girl he loved, Lana Lang, whose parents were squashed by the falling kryptonite, to his friend Lex Luthor, who lost his hair and his innocence. A succession of others turned up with strange and evil powers in a story arc that became known as the Freak of the Week. The comic book Superman may have blamed himself for being Krypton's sole survivor, but his TV stand-in was faced with a more proximate and disabling font of guilt: a body count that grew with each new episode. This revised backstory was half Norman Rockwell, half Stephen King. It also was 100 percent suited to kids weaned on the horrors of 9/11 and craving a hero to help them cope. The new show, debuting a month after the 2001 terrorist attacks, didn't allow young Clark an easy coming of age, but it did explain why a fully grown Superman felt so driven to save the world.
Knowing how the story would turn out made it more fun to imagine how it began. But being familiar with all the details of the grown-up champion wasn't a prerequisite for enjoying _Smallville_. "We made no assumptions that anybody knew anything about Superman," explains Al Gough, one of the show's creators. "When we tested it with teenagers, the boys had to tell the girls that they were watching Superman. The girls were completely engaged in the show but they had no idea that Clark was Superman." For old fans, Gough and his colleagues gave Superman something every hero needs to be believable: a past. For new ones, they offered teenage love, duels of good versus evil, and a gateway to the myth.
Religion was infused in _Smallville_ as it had been in other Superman renditions, only more so, as befit an age when public figures felt obliged to pledge allegiance to a deity along with a flag. Divinity was front and center in the first show of the first season. Football players took Clark on what they intended to be a simple hazing ritual, tying him to a scarecrow stake in a cornfield, unaware that kryptonite had made him so weak he couldn't escape. There stood a hero descended from the heavens, stripped nearly naked, on what looked like a crucifix. The image was so powerful it was splashed on billboards and in magazines as well as at the start of each episode in the early seasons. And that wasn't the only insinuation of faith. Clark was filmed next to the statue of an angel, its wings seeming to sprout from his shoulders. He was bathed in halos of light. There were allusions to the Holy Grail, the wise men, and the Romans. "We were very conscious of the religious tones. We also knew it was dangerous, there's a line you don't cross," recalls Ken Horton, a writer and producer. "The most extreme use of religious symbols was in the pilot with the scarecrow. After that we were far more subtle."
Tom Welling had a lot in common with earlier on-screen Supermen. He didn't read comic books or know much about Superman. He was an unknown, having worked in construction, modeled for Tommy Hilfiger and Calvin Klein, and played minor parts on TV. And he was a hunk, standing nearly six foot three with an oh-my-gosh innocence reminiscent of the young Christopher Reeve. That was an asset, since fans were being asked to believe that the twenty-four-year-old Welling was a student at Smallville High. Eighteen-year-old Kristin Kreuk was a closer match as Clark's classmate Lana Lang. This wasn't the feisty Lana we had met in the movies and comics but a subtle beauty made brittle by the loss of her parents in the meteor storm. She and Clark were two sides of the classic love triangle: He longed for her while she befriended him and dated a football star. But a triangle takes three, and without Superman in the picture the producers had to invent a new character. Chloe Sullivan, editor of the school paper, pined for Clark even as she investigated the strange doings in town and their possible connection to her wished-for boyfriend.
No one planned for Michael Rosenbaum to steal the show, but they didn't rein him in when he portrayed the most riveting Lex Luthor ever. His bald pate made him alluring, in a Yul Brynner sort of way, leading fans to brand him "Sexy Lexy" and the media to crown him the "hairless heartthrob." When he went out in public it was with a toque on his head and, when he was feeling playful, with a fake mustache plastered to his lip. The same way Welling's Clark couldn't help but rise to heroism, Rosenbaum's Lex didn't try to be evil but knew that a descent into darkness was his destiny. Rosenbaum had "just the right mix of creepy entitlement and helpless longing" to make him "the most ambiguous character on any prime-time series," Tom Carson wrote in _Esquire_. "You don't even know if he's lovelorn young Clark's rival for the affections of Lana Lang... or her rival for his." That last element—whether Lex's longing gazes at Clark meant he was gay—was fodder for fans and pundits. "I love it," Rosenbaum said. "In fact, if there's a line where I look at Clark and I say, 'If you need me, I'm there,' we laugh our asses off. It takes us ten takes to get it out. Let the audience think what they want to think."
Pete Ross was Clark's best male friend in the comic books and in _Smallville_. He was mad about Chloe, which added another romantic spin to the plot. And he was played by an African American actor, Sam Jones III, in a casting choice that would have been too controversial for _Lois & Clark_ but that _Smallville_ fans liked so much that they set up an online site lobbying for him to get more airtime. He did and was let in on Clark's secret identity, which hadn't been part of the plan. Jones left the show after three seasons, but he was back for a guest appearance in season seven and might have returned again if real-world DEA agents hadn't arrested him in 2009 on drug trafficking charges.
The first season's Freak of the Week aura was toned down by season two, with true crimes, natural disasters, and other stories mixed in with the monster ones. The world apparently had enough real-life monsters to satisfy even the kids. Clark stayed in high school through the first four years, but after that he grew up and the show ventured into more familiar Superman settings like the _Daily Planet_. He explored his origins on Krypton and heard from Jor-El, even if he didn't see him. He battled familiar villains (with new aliases) like Brainiac, General Zod, Doomsday, and a "really hot foreign exchange student" named Mikhail Mxyzptlk. Most of all, he slowly learned to live with each one of his expanding powers, from seeing through tall buildings to being able to leap over them. Lex left seven seasons in, by which time his friendship with Clark had soured, although he came back for a final face-off in season ten. By then the no-tights-and-flights rule had been lifted and Clark was out of high school, living in Metropolis, and ready to take on the full-time role he had been training for.
The new era opened opportunities for new characters. Newspapers across the country had strong-willed women editors, so it was no surprise that Chloe Sullivan was not just popular on _Smallville_ but was written into the mainstream Superman comic books. Lex's father, the diabolical industrialist Lionel Luthor, was made for TV and a poster child for extreme parenting. Perry White was back, this time as a television hack and a drunk. Settings were different, too. Rural Smallville was a bulwark against the industrialization and urbanization pushed by the likes of Lionel Luthor. Violence and sex were okay now in ways that would have been inconceivable in the finger-wagging era of Dr. Fredric Wertham. Computer graphics finally made what little flying there was seem believable and a world removed from Kirk Alyn's cartoons, George Reeves's wires and springboard, and Christopher Reeve's trick photography.
For all the changes in plots and cast, and his own self-doubts, Clark/Superman was the hero he had always been. He still had an instinctive sense of what was right and acted on it. His love interests were as tangled as ever. He remained the handsomest, mightiest, most captivating super-being on TV and in the universe. No wonder a new generation of kids was smitten.
Behind the scenes of _Smallville_ the watchword remained product synergy, but the cooperation was now broader and deeper. Merchandise ranged from two soundtrack albums to T-shirts, hats, posters, and a monthly magazine. "Save Me," the show's theme song, soared on the _Billboard_ charts, and, in a strange twist on product placement, in one scene a CD of _Smallville_ 's songs was being sold. There were two series of _Smallville_ -inspired young adult novels with eighteen titles. The normal pattern whereby a comic book generated spin-offs was turned on its head, with the TV show launching the bimonthly _Smallville: The Comic_ and inspiring cartoonists to redraw Superman to look more like his televised counterpart. There also were webisodes and a tie-in with Verizon that let registered users watch plot updates. True fans could make their own digital comics in a deal worked out between the network and a chewing-gum company.
The most lucrative of the synergies was playing out at the newly created AOL Time Warner, showing the troubled conglomerate what a merger could mean when its moving parts got in sync. The WB network broadcast the show not just across America but around the world, from San Salvador and Berlin to refugee camps in Nairobi. DC Comics safeguarded the legend, which was easier with Superman comic book writer Jeph Loeb also writing and producing for _Smallville_. Warner Bros.' TV studio oversaw the project, and its film division watched for any overlap with the new Superman movie. Expertise was shared. Money was saved. And sometimes limits were set, like keeping Lois Lane off the show until season four, because DC and Warner wanted to save her for what they hoped would be a new film focused on Superman's early years.
Christopher Reeve was a metaphor for that sharing of Superman across divisions and generations. He watched _Smallville_ and loved it, so much so that he agreed to be a guest star. He played Dr. Virgil Swann, a wheelchair-bound scientist who revealed to Clark that Kal-El was his real name, Krypton was his first home, and he had a mission to fulfill on Earth. The aging star who had defined Superman for baby boomers was, in effect, passing the torch to the young one who would define the hero for generation next. The nostalgia was ramped up a notch when Reeve's Lois, Margot Kidder, came back to play Swann's emissary. The producers also brought in Annette O'Toole, who had played Lana in the 1983 Superman movie and now was Clark's adoptive mother, Martha. Making cameo appearances in new roles were _Lois & Clark_'s Lois and Clark, along with the Salkind-era actors who played Jimmy Olsen, General Zod, and Supergirl. "We were winking to the mythology," says Gough, and it wasn't just in casting. Smallville's high school newspaper borrowed its name from Jerry Siegel's high school paper, the _Torch_. Lana wore a kryptonite stone in a necklace, a reminder of her parents and Superman's. In one episode, when Clark dropped a copy of a book by Nietzsche, Lana asked whether he was a "man or superman?" Clark: "I haven't figured it out yet."
But he did figure it out in new ways that drew rave reviews. " 'Smallville' is one of the few new shows this season to have attained breakout status," wrote Hal Hinson of _The New York Times_. " 'Smallville' peddles its own brand of classic all-American corn, which, when served with a pinch of teenage angst, a hint of paranormality and a fresh take on one of the most durable icons in pop culture history, makes an immensely satisfying meal." _Esquire_ 's Carson was even more swept away, writing, "Seeing the Superman myth in terms of innocence lost gives the material a poignancy it's never had before." _Entertainment Weekly_ added that "finally, Clark Kent has an adolescence that actually makes sense," and _Smallville_ "is luring in those who don't know kryptonite from crapola."
It happened at just the right moment. For the WB, the new show cushioned the blow of losing _Buffy the Vampire Slayer_ , a hit that had jumped to a competing network. For the nation, _Smallville_ 's launch in the wake of 9/11 gave America a hero it could believe in when it needed one, the same way Jerry and Joe had more than sixty years earlier.
_Smallville_ drew the biggest-ever audience for a debut on the WB network, 8.4 million viewers, and by its second season it was the WB's most popular program. The show set viewership records for the age groups that AOL Time Warner executives craved: eighteen- to thirty-four year olds. It attracted young men to a network that had catered to teenage girls. Fathers watched with their daughters, the former loving Superman and the latter thinking Clark was hot. While its audience slowly declined over the years, even at the end it was drawing several million viewers, not bad for a cable network.
This wasn't the first Superman TV show to explore the hero's early life. _Superboy_ did that, although he was in college rather than high school, and he wasn't afraid to don his uniform or take flight. The link to _Lois & Clark_ was even clearer, with the title snubbing Superman and the program showing more interest in Clark than in his heroic alter ego. But _Smallville_ was more interesting, maybe because it focused on a more troubled period of Clark's (or anyone's) life: adolescence. It was a story about family, too. Martha and Jonathan were not mere props here but down-to-Earth parents with rock-solid values, anxious to help their son. Lionel Luthor made the same point in reverse, raising Lex to be a chip off his bad-guy block. The upshot was that while _Superboy_ and _Lois & Clark_ died after four seasons, _Smallville_ lived for ten years, making it not just the longest-lasting of the Superman shows but the most enduring of any TV series based on a comic book hero.
RECASTING SUPERMAN FOR A RETURN to the big screen was harder. Warner Bros. had bought back the movie rights from Alex Salkind in 1993 after he had a falling out with his son, Ilya, and the studio trumpeted its acquisition almost as brashly as Alex had his. It should have known better. For starters, it didn't have in hand a Superman story worth telling. Its earliest version began with the hero dying just after immaculately impregnating Lois with a child so super that he grew to adulthood within weeks. It took even less time for Warner to realize it needed a more plausible narrative. Subsequent scripts traded in Superman's blue-and-red costume for an all-black one, sat him down with a psychologist, built him a robot named L-Ron (patterned after Scientology's L. Ron Hubbard), gave him a third persona, pitted him against Brainiac and Batman, resuscitated not just his birth parents but his home planet, and wrote in references to 9/11 then wrote them out for fear the country wasn't ready. Ten writers came and went over eleven years, along with countless producers, directors, and stars ready to don the cape and tights, at a cost to the studio of tens of millions of dollars and a stack of embarrassing news clippings. Even a title for the movie proved elusive, with the options including _Superman Reborn, Superman: Flyboy, Batman vs. Superman, Superman V_ , and the unintentionally ironic _Superman Lives_.
Finally, in the summer of 2004, Warner Bros. hired Bryan Singer to produce and direct his idea for a story he called _Superman Returns_. The title showed that he understood what fans were thinking seventeen years after the Man of Steel's last appearance on the big screen: Give us back our hero. It also made clear that Singer wasn't planning to rewrite the character but wanted to resurrect the Superman millions of Americans had fallen for in the comics and, more to the point, in the two films directed by Richard Donner. Singer had proven his superhero bona fides by bringing to the screen two successful movies about Marvel Comics' X-Men. He had vetted his Superman ideas with his friend Donner before presenting them to Warner, which was desperate to make good on the promises it had made when it reclaimed the movie rights a decade earlier. Just how delighted it was to have Singer on board became clear when the studio gave him a budget of $210 million for the new film, which might have been a record. Just how novel what he was attempting was wouldn't become clear until later, when a movie historian called _Superman Returns_ the first feature film ever to be part of a franchise without being a remake, a prequel, or a sequel. What it was, Wayne Lewellen added, is "a modern blending of the core elements that have endured for nearly 70 years."
Singer's film, released in June 2006, couldn't answer why it had taken nearly twenty years for Superman to return, but it did explain what the hero had been doing for the last five. He was off looking for what astronomers said were the remains of Krypton. While he didn't find any survivors, he did find his earthly world transformed by the time he got back. Lois had had a son, gotten engaged, and won a Pulitzer Prize for her story "Why the World Doesn't Need Superman." Lex had married an old widow whose fortune would finance his latest plot to dominate the world. Stopping his old enemy was as instinctual as flying for Superman, but it was certainly hard getting used to seeing Lois in another man's arms. Harder still for the Man of Steel: not knowing whether her son, Jason, was his son, too.
The first clue came when the boy seemed to be weakened by the kryptonite Lex had stolen from the Metropolis Museum. The evidence became conclusive when Jason saved himself and his mother by using a piano to crush Luthor's thug. In the end, Lois whispered a secret into Superman's ear when she and Jason visited him in the hospital, unsure if he would come out of a kryptonite-induced coma. When Superman woke up, he visited a sleeping Jason and recited to him the parting words he had heard from his father, Jor-El, in the first Salkind film: "You will be different. You will sometimes feel like an outcast. But you will not be alone. You will never be alone." Lois, meanwhile, wrote a corrective to her award-winning story, calling it "Why the World Needs a Superman."
The dialogue wasn't the only part of _Superman Returns_ intended as an homage to Alex and Ilya's first two movies. Its star was, too. Bryan Singer was looking for an unknown, just as Richard Donner had been when he chose Christopher Reeve, and both directors picked actors who needed to add ballast and brawn to convince anyone they were strongmen. Brandon Routh, like Reeve, had acted in a soap opera, and both were twenty-six when they assumed the mantles of Clark and Superman. Routh, who had an angular jaw and six-foot-three-inch frame, had been told all his life he looked like Reeve, especially when he wore his blue-tinted contact lenses. When, as a young boy, he saw Reeve's first film, he got so excited that "I gave myself a migraine. I was puking through half of the movie." Years later, when Routh was picked to succeed Reeve, Christopher's widow, Dana, told him she was struck by the resemblance. She said he was a fitting successor to her husband, who had died just months before. "I can't tell you what that was like to get her blessing," Routh said. "It's frightening trying to fill Christopher Reeve's shoes." Dana herself died of cancer in March 2006, seventeen months after Christopher and three months before the opening of the movie, which was dedicated to both of them.
Singer took another cue from the earlier movies by signing up big names to play Lex Luthor and Jor-El. Kevin Spacey, whom Singer had gotten to know when Spacey delivered an Oscar-winning performance for the director ten years before in _The Usual Suspects_ , gave his audience a villain who knew how to use humor as disarmingly as Gene Hackman had and was even better at anger. As for Jor-El, Singer couldn't have topped Marlon Brando and he didn't try. Rather, he and Warner Bros. negotiated with the estate of Brando, who died in 2004, for the rights to use not just the footage that had already appeared in the first movie but other scenes unearthed from a warehouse in Kansas.
The Superman legend didn't begin with the Salkind movies and Singer didn't stop with them in trying to reel back old fans. He cast eighty-five-year-old Noel Neill, the original live-action Lois Lane, as Lex's dying wife, Gertrude Vanderworth. Jack Larson, whose Jimmy Olsen played alongside Neill's Lois on TV, was back for his own movie cameo as Bo the bartender, wearing his distinctive bow tie and looking younger than his seventy-eight years. While both were delighted to be part of the ongoing legend, "Our George," as Neill explained, "will always be Superman to us."
One way to measure the change in Superman over the decades was to watch him fly. As determined as George Reeves was, and as much as Christopher Reeve lived up to the hype of making us believe a man could fly, Brandon Routh set a new standard. In scenes where animation was used, a cyber scan of the actor duplicated him down to the hair on his ears and the tastebuds on his tongue. For shots of the real Routh airborne, a new digital camera brought intimacy and editing ease, while computers painted out the wires and cranes as they had for Tom Welling in _Smallville_ but couldn't when Reeve was doing his tour of the city with Margot Kidder. Even Routh's skintight supersuits were space-age. At a cost of hundreds of thousands of dollars, the sixty tricolored costumes were fitted not with clothespins or caster molds, but using an electronic mapping device so precise that the actor wasn't allowed to add or shed even an ounce until the film was out.
Michael Dougherty, who with his co-writer managed to craft the movie-ready script that had eluded a lineup of other writers, says the experience was both universal and highly personal. "Writing a story like this is almost biblical, when you hear how the Bible was passed down orally from year to year. The Superman character has been passed from one person to the next and one generation to the next," he says. "My grandmother still loves more than anything the black and white George Reeves show, the Superman that she grew up with. For Bryan and I and Dan it was the Donner films, and today's teenagers gravitate towards _Smallville_ , which I can appreciate simply because I love the character. There's a Superman for every generation." Staying true to that history was just half the process for Dougherty. The other half was pure fun: "It's indescribable—what it's like to stand on the Fortress of Solitude set. To walk into a room and there's Superman standing there. To hang out at the _Daily Planet_. It's really emotional. It's a blast."
That's not the way things started out for John Ottman, who composed the score. "I was practically getting death threats from fans of the Donner version," says Ottman, who had edited and composed for Singer in _The Usual Suspects_. "They worried I wouldn't do the right thing with the music and asked why John Williams wasn't writing the scores the way he had for Donner. I started getting crippled, worrying what fans were thinking. Finally I said to myself that I have to ignore all that and weave in my own sensibilities and style, and of course nod to the Williams theme, which I'd always intended to. The fan reaction was that if they could have sent me flowers, they would have. They all were very happy."
Bryan Singer was under the most intense microscope. It wasn't just that he wasn't Donner, but he was openly gay, and film critics wondered whether his hero would be, too. It was a fair question; the mutant X-Men in Singer's movies concealed their powers, were shunned as outsiders, and had characters in their orbit who were openly lesbian or gay. "Superheroes—let's face it—are totally hot," arts editor Alonso Duralde argued in a cover story titled "How Gay Is Superman?" in _The Advocate_ , a gay-oriented magazine. "Not for nothing does gay director Bryan Singer have an eye for how to make the Superman suit most flattering to Brandon Routh." It wasn't the kind of publicity Warner Bros. had hoped for, and Singer felt compelled to publicly assure fans that _Superman Returns_ was the "most heterosexual movie I've ever made." Any remaining doubts were washed away when Singer made clear that Superman had fathered Lois's child and still was in love with her, no matter that she had a fiancé.
Sex wasn't the only hot button for the producer-director. America-first bloggers were livid about a superhero who they said had been made too global in his outlook and thereby not all-American enough. The flash point came when Perry White told his reporters to find out whether Superman, after being away for five years, still stood for "truth, justice... all that stuff." "Warner Brothers, the studio distributing the movie, doesn't want to tee off any foreign viewers with pro-U.S. sentiments," railed Bill O'Reilly. "It's bad enough Superman was raised in the Midwest; we can't be having the hero actually standing for the American way, now can we? Some jihadist in Pakistan might throw popcorn at the screen." It was left to Erik Lundegaard, who wrote about movies for the liberal MSNBC.com, to come to the defense of Singer and Superman. "There's no reason to be upset," he wrote in a _New York Times_ commentary. "Superman is right back where he began: fighting a never-ending battle for truth and justice. That should be enough to occupy any man. Even a Superman."
Singer had to coordinate not just with editors at DC Comics, who were determined to keep Superman true to his roots, but with the producers of _Smallville_ , who wanted exclusive rights to Superman's adolescence. He had to cope with mishaps—from his producer being mugged, to his editor puncturing a lung when he fell through a window, to a cameraman fracturing his skull in a tumble down a flight of stairs—that added to the legend of a curse. And he had to do everything in record-setting time. It took just two years from his deciding to make the movie, without a script in hand, to its premiere in movie theaters—a process so demanding that "at one point I just stopped shooting," Singer says. "I was physically exhausted and mentally destroyed. I needed to take three weeks off."
Even with those constraints, Singer told the story he wanted to. Like the Death of Superman comic book series a decade earlier, _Superman Returns_ answered the question of whether the hero still mattered. He did, more than ever. The movie revisited the Christ story by looking at whether society still wanted and needed a savior. The answer was yes. Like the previous Superman films, this one was about secrets, but the secret that mattered here wasn't Superman's alter ego but his past intimacy with Lois and the son they had conceived. The only ones who knew were Lois, Superman, and the sixty million people who watched their movie. Millions more saw it in 3-D, and it was the first live-action film Hollywood had produced in the IMAX format.
"They're very important, these comic book movies, because they're our modern myths," said Singer. "What Superman represents is idealism in a physical form." The film also had a personal resonance for its director, who, like Superman, was an only child, an orphan, and an outsider. " _Superman Returns_ ," he says, looking back, "suddenly opened that whole history to me."
Critics understandably compared Singer's film to the Donner and Reeve renditions, which had turned into icons over time. "Earlier versions of Superman stressed the hero's humanity: his attachment to his Earth parents, his country-boy clumsiness around Lois," wrote Richard Corliss of _Time_. "The Singer version emphasizes his divinity. He is not a super man; he is a god (named Kal-El) sent by his heavenly father (Jor-El) to protect Earth. That is a mission that takes more than muscles; it requires sacrifice, perhaps of his own life. So he is no simple comic book hunk. He is Earth's savior: Jesus Christ Superman." But was it a good movie? Corliss asked himself. "You bet," he answered. "Made with precision and vigor, the film never forgets to entertain.... The best Hollywood movies always knew how to sneak a beguiling subtext into a crowd-pleasing story. _Superman Returns_ is in that grand tradition. That's why it's beyond Super. It's superb." Richard Donner agrees, saying the Singer film "stayed very much within the honest traditions. When that little child shoots the piano [across the room] I thought, 'Oh my God.' Bryan did a super job, a really sensational job."
Others felt the film didn't measure up. Routh "offers not so much his personal interpretation of Superman as his best impersonation of Christopher Reeve playing Superman. This feels constrained, to say the least," Anthony Lane wrote in _The New Yorker_. "Singer's casting errs toward the drippy and the dull, and your heart tends to sink, between the rampant set pieces, as the movie pauses listlessly for thought." Mike D'Angelo of _Las Vegas Weekly_ was harsher still, writing, "Fidelity is one thing; slavish imitation another." And _The New York Times'_ Manohla Dargis called _Superman Returns_ "leaden." Rotten Tomatoes gave it a composite critics' rating of 76 percent—a record that would make any director proud unless they were comparing themselves to Donner, who scored 94 percent for his 1978 film and 88 percent for _Superman II: The Richard Donner Cut_.
Results at the box office also were mixed. The film took in $200 million in domestic sales and another $191 million overseas, which sounded like a lot but wasn't stacked up against production costs and what Marvel Comics had raked in for its blockbuster movies. The original Spider-Man grossed $822 million in 2002, and its sequel two years later hit $784 million. The third X-Men movie, out a month before _Superman Returns_ , brought in $459 million. Superman was even losing out to his DC brother Batman, whose masterwork back in 1989 had taken in $411 million—or $668 million in 2006 dollars—while his _Dark Knight_ in 2008 would gross just over $1 billion. _Superman Returns_ "was a very successful movie, but I think it should have done $500 million worldwide," said Warner Bros. president Alan Horn. "We should have had perhaps a little more action to satisfy the young male crowd."
The box office may have been how Horn measured his studio's success, but parent company Time Warner took a broader view. _Superman Returns_ cashed in on a big-time deal with Burger King under which the hamburger chain displayed Superman toys, Superman banners, and Superman fast-food wrappings at 8,500 of its restaurants. Pepsi rolled out more than a billion cans of super-strength Superman soda. Frito-Lay encouraged its customers to test their super hearing at displays that amplified the sound of them munching Lay's and Cheetos snacks. Mattel sold inflatable muscle suits, Cap'n Crunch cereal came with red S-shaped shields that turned milk blue, an Orlando game maker released a _Superman Returns_ video game, and Brandon Routh sported a white mustache in "got milk" ads. A&E aired a Superman documentary just before the film came out, narrated by Kevin Spacey, and the National Geographic Channel countered with a _Science of Superman_ special. In total Warner Bros. collected more than $80 million from _Superman Returns_ licensing deals in America, and substantially more overseas.
Superman had once again demonstrated why he belonged as a summertime Hollywood blockbuster, even if his handlers had not made the same case for themselves. Singer had been hired with the hope of launching a new film franchise for America's signature superhero, just as Routh was intended to replace Christopher Reeve as the defining Superman for his generation. Even the ending of _Superman Returns_ —with the hero passing the mantle to his son and letting Lois know he was back to stay—was tailor-made for a sequel. But a new film called _The Man of Steel_ , intended to air during the summer of 2009, died on the vine. While Warner Bros. is making another Superman movie, neither Singer nor Routh is involved.
JERRY SIEGEL AND JOE SHUSTER'S story read like a movie script from the beginning, and the drama didn't end with their deaths.
To Jerry's widow, Joanne, it was a tragedy, and after decades of Jerry being the victim, she and her daughter, Laura, now assumed that role. Joanne may have been the model for Lois Lane's looks, but it was Laura who lived Lois's life as an award-winning radio newscaster and talk show host, TV reporter and anchor, and news and documentary producer. Now Laura had multiple sclerosis and couldn't work, and Joanne was getting old. They notified DC, Warner Bros., and Time Warner in 1997 that they planned to reclaim the copyright Jerry had signed away in 1938. So much for the promise not to sue that he made back in 1948 and reaffirmed in 1975. Since then federal laws had made it easier to redress perceived wrongs from the past, and the Siegels had had two more decades to stew over how shabbily they had been treated. It was Jerry's dying wish that they set things right, his daughter said. What she and her mother wanted was the nest egg they felt they deserved. So their attorneys and DC's sat down to talk.
They talked and talked some more. Finally, in the fall of 2001, it looked as if the lawyers had agreed on a deal. On October 16, DC set out the general terms of a plan to pay Joanne and Laura nearly $1 million a year each for the rest of their lives. Three days later, their lawyers signed on. Sensing how near they were to a resolution, DC had already given the Siegels a nonrefundable advance of $250,000, and four months later the company sent them a full-blown agreement. That is when things imploded, although why remains a matter of heated dispute.
"We were stabbed in the back with a shocking contract" that included "new, outrageous demands," Joanne wrote to Time Warner boss Richard Parsons. "The document is a heartless attempt to rewrite the history of Superman's creation and to strip Laura and me of the dignity and respect that we deserve.... My disabled daughter still has not received the medical coverage she and her children were promised several years ago," she added, her anger building as the three-page letter proceeded. "Just like the Gestapo, your company wants to strip us naked of our legal rights. Is that moral?" In spite of the letter, negotiations continued for another four months, at which point the Siegels fired their old attorney, hired a new one, and sued Time Warner, Warner Bros., and DC.
While there were changes between DC's preliminary and final proposals, the switch that mattered more was the Siegels' new legal counsel. The Hollywood studios whom he had skewered regarded Malibu's Marc Toberoff as a Svengali who manipulated vulnerable clients. Admirers, like the heirs of Winifred Knight Mewborn, whose short story became the TV classic _Lassie_ , called him a Robin Hood for restoring rights they had given up for next to nothing half a century before. Everyone agreed that the low-budget filmmaker turned high-stakes litigator had mastered an arcane area of copyright law and exploited it to benefit his clients and himself. In the Superman case, DC argued that Toberoff deceptively lured in the Siegels as clients, falsely promising them $15 million in immediate payouts and the chance to make their own Superman movie. His real motive, the company said in a legal filing against Toberoff, was to secure for himself the right to 45 percent of any payout the Siegels would get and the role of kingmaker in future Superman films. Toberoff called DC's allegations a desperate "smear campaign" and part of Warner Bros.' "last-ditch effort" to hang onto its rights to Superman, a property he believes is worth a billion dollars.
Whoever was right, the result was that the Siegels and Jerry's old employers were back in court. Scores of witnesses were deposed and thousands of pages of yellowing documents were unearthed that traced Superman's development from Jerry and Joe's earliest rendering to the latest TV incarnation in _Smallville_ , which Toberoff claimed was a Superboy knockoff and thus belonged to Jerry's heirs. For historians, the legal battle yielded a trove of material—from Jerry and Joe's original $130 contract to stacks of correspondence between the young creators and their editors and publishers. It also produced Jerry's unpublished memoir.
The documents revealed a Jerry Siegel whose personality was at least as split as his superhero's. One side of him was a creative and bereft boy looking to escape his real life by inventing one in fantasy. Less appealing was the angry young man who never recovered from the real and imagined wounds inflicted by the entrepreneurs to whom he had entrusted his sacred Superman. There were two Joannes as well. One was the nurturing beauty who had Jerry and Joe fawning over her. She coaxed the hard-hearted Jack Liebowitz into rehiring her husband after he had repeatedly burned bridges with the publisher, and looked after her man and girl when Jerry was an emotional wreck and there was barely cash enough to keep Laura in milk and diapers. Joanne's other side was that of a lioness protecting her cubs. She was the mouthpiece for Jerry and Joe, writing letters to DC Comics demanding settlements, cost-of-living raises, and other benefits the aging creators lacked the gumption to ask for. If strong language was needed to get a CEO's attention, she'd brand his company as the Gestapo. When an old classmate of Jerry's was written up as the model for Lois Lane, implying that Joanne might not have been, Joanne had her lawyer send a cease-and-desist letter. No matter that the claims weren't the classmate's, but old ones by Joe and Jerry, and that by the time Joanne was posing for Joe, Lois already was part of the story. Superman over time became Joanne's, too, to the point where she told people she planned to write her own memoir on the whole sordid history.
The legal proceedings dragged on long enough that seven different federal judges pored through the evidence. Their preliminary rulings—in 2008, 2009, and 2011—gave the Siegels much but not all of what they wanted. They had the right to sue despite the agreements they had signed with DC Comics. They also had a right to the Superman story in _Action_ 1 but not the cover, the Superman story in _Action_ 4, parts of _Superman_ 1, and the first two weeks of _Superman_ newspaper strips, which Harry and Jack had authorized Jerry and Joe to produce on their own. That gave Jerry's heirs ownership of Superman's blue leotards, red cape, and boots, as well as his early powers to leap tall buildings, repel bullets, and run faster than an express train. Time Warner owned the flying superhero, the _Daily Planet_ , Jimmy and Perry, the Kents, X-ray vision, and kryptonite, along with the overseas rights to everything. In practice the rulings meant that, for the full-fledged Superman to appear on-screen or anywhere else, Jerry's heirs and Jack and Harry's would have to pool their bifurcated holdings and share the profits. Just how that should happen and how much the Siegels already were owed, the court said, would have to be settled in a trial.
While all that was playing out in public, behind the scenes checks continued to arrive each month from DC to Joanne. By the end of 2010 the payments had exceeded $3.8 million, including coverage of Joanne and Jerry's medical bills, which had peaked at $89,000 the year he died. When a Superman TV show or film did especially well, there was a bonus of between $10,000 and $50,000. The agreement signed in 1975 had called for cutting off benefits to Joanne fifteen years earlier, but DC said it would keep them coming in spite of Jerry's death and Joanne's bid to reclaim the ownership of Superman. In 2001, the company said it would continue paying so long as Joanne was working toward a settlement of the copyright dispute; settlement talks broke off in 2002, but again DC kept sending the checks. Annual payouts that had started at $20,000 were up to $126,000 at the end.
Even as Warner Bros. lawyers have been arguing with Toberoff and hoping for a settlement, they are steeling themselves for October 2013, when Joe Shuster's nephew, Warren Peary, will try to restore his uncle's Superman copyright and make the same claim the Siegels have. After Joe died in 1992, DC agreed to clean up his $20,000 in debts and pay his sister, Jean, $25,000 a year for the rest of her life, which so far has yielded her more than $500,000. In return she promised not to sue. But Jean's son, Warren, never gave his word, and he hired Toberoff to sue DC for what he thinks the Shuster heirs will be entitled to under an amended federal copyright law. This time the lawyer and his client would split the rights 50–50, giving Toberoff a total stake of 47.5 percent in the Siegel-Shuster holdings, compared to 27.5 percent for the Siegels and 25 percent for the Shusters.
Is that excessive? Toberoff is entitled to that much, some legal authorities say, since his firm isn't charging a fee and is absorbing the huge costs of the multiyear lawsuit. Others say any contingency share over one-third is excessive and that Toberoff should have persuaded his aging and ill clients to take the tens of millions DC has offered even though he believes they deserve and can get more. "The whole purpose of these termination provisions [in federal law] is to give authors and their heirs a second bite at the apple, to enable them to finally profit from the market value of their creations," says Toberoff. What about the fear—voiced not just by DC and Warner Bros. but by fans—that the lawsuit could impair and even end the Superman franchise itself by clouding the question of who owns the hero? "The notion that this could be the death of Superman is nonsense and studio counter-spin," Toberoff says. "It's clear that this is simply a financial matter. The Siegels are ready and willing to relicense their recaptured copyrights to Warner Bros. at a price that properly reflects the market value."
Toberoff also tried to strike a deal with Jerry's son, Michael Siegel, saying he had an investor who was interested in buying out Michael's interest in the Superman copyright, which would have been 25 percent of any settlement agreed to by Joanne and Laura. But Michael died suddenly in 2006 from complications of knee surgery—without having approved Toberoff's proposal, which was a fraction of what he would have received under DC's settlement offer, and without a will to pass on his share of a future payout. Michael had never gotten the annual payments that Joanne did and he would never see a penny from his father's role in creating Superman, which was par for Michael's course. Jerry was in the Army for much of his only son's early life. After he and Bella got divorced, when Michael was four, Jerry seldom visited, and he stopped paying child support when he hit hard economic times. Michael became a plumber, like Bella's father, and lived with Bella in Cleveland until she died, just four years before he did. Near the end each took care of the other and both felt abandoned by Jerry.
It was a strange way for a son who never got over the loss of his father to treat his own son, but Jerry was wrapped up in his own troubles and his new family. Jerry told everyone who asked how proud he was of Laura, his own Lois Lane and Supergirl, but seldom mentioned Michael. Michael kept a low profile and seldom talked about Superman or his father, who was a cherished native son in Cleveland and would eventually have his birthplace restored, with a plaque dubbing his street Jerry Siegel Lane and the cross street Lois Lane. What little Michael did say about Jerry, to friends and others, showed how torn he was. He didn't want to be angry but couldn't help it. He was crushed every time Jerry was supposed to pick him up at Howard Johnson's for a custody visit but didn't show up. He wished his dad had been around to see what an athlete he was, which was something Jerry had wanted to be but couldn't. It hurt, again, when Michael turned up as an afterthought in Jerry's will. "Even in death," the aggrieved son wrote, his famous father "continues to shun me! Why?" Michael likely died without knowing the high hopes Jerry had had at the beginning for his firstborn, high enough that he named him after his own hallowed dad.
Like Michael, Joanne died without seeing a dime of settlement money. With her passing in February 2011 at the age of ninety-three, the checks from DC were cut off as agreed back in 1975, which presumably gives Laura more incentive to settle her lawsuit. Warner, meanwhile, has two extra motivations to meet its June 2013 target for a new Superman movie: The judge in the Siegel case has made it clear that any delay could be interpreted as holding back on potential earnings for the heirs, and in the fall of 2013 Warren Peary will be asking to reclaim Joe Shuster's copyright. Other comic book creators have been following the suit in hopes it helps them, while other publishers and studios hope it doesn't threaten their ownership rights. Superman fans, too, are keeping a close watch—praying that if the two sides can't settle, the judge shows the wisdom of Solomon by ensuring that the bid by the heirs of Superman's creators to reclaim him does not kill him.
OFFSCREEN, THE FIRST DECADE of the new millennium looked like the worst of times for Superman. Readership continued to sag for comic books generally, and specifically for Superman titles. The bestselling of those, _Superman_ , had fallen from 720,000 copies a month in 1966, to 98,000 in 1986, to just 62,000 in 2006. Circulation was down again early in 2011, to 42,000, with optimists hoping for a rebound and realists noting that _Action_ and other Superman titles were faring even worse. The remaining audience was dedicated to the point of fanaticism, a trend that was self-reinforcing. No longer did casual readers pick up a comic at the drugstore or grocery, both because the books increasingly required an insider's knowledge to follow the action and because they simply weren't being sold anymore at markets, pharmacies, or even the few newsstands that were left. Cost was another constraint: Superman comics that sold for 10 cents in 1938 were $2.99 to $3.99 by 2011, an increase that was about twice the rate of inflation. The core fan now was a worldly-wise twenty-six-year-old who was shelling out a thousand dollars a year for new comics. And it was a he; females made up barely 10 percent of the readers. Comic books had gone from being a cultural emblem to a countercultural refuge.
Superman's fortunes had soared with those of the comic book, but he didn't crumple when the comic book did. That was partly because, after seventy years, he was as recognizable an American trademark as Mickey Mouse or the Playboy bunny and more resilient than either of them. He had his own Graceland in Metropolis, Illinois, which celebrated its native son with summertime festivities that drew thirty thousand people and were as much grassroots love-fests as the ones that Memphis organized for Elvis. When VH1 compiled its list of the two hundred greatest pop culture icons of all times, Superman ranked second—behind Oprah Winfrey and just ahead of Elvis Presley and Lucille Ball. The superhero was back onstage in Dallas, a featured attraction in Warner Bros.' bustling store in Shanghai, a prominent player in movies as eclectic as _Kill Bill_ and _Hollywoodland_ , the focus of college courses on everything from sociology and immigration to gender studies, and a centerpiece of exhibitions at Jewish museums in Berlin, Paris, and Amsterdam. In the Philippines, a thirty-five-year-old man had extensive cosmetic surgery to make himself look more like Superman. Collectors still traded his oldest stories; a copy of _Action_ No. 1 sold for $1 million in February 2010, a record for any comic, and another copy smashed that mark two years later with a sales price of $2.2 million. Even the cheapest of his old trinkets, if they were in good enough condition, could fetch fat prices: $150 for a 1943 Superman-Tim birthday postcard, $750 for a 1945 Pep cereal box with the picture of a Superman button, $3,500 for a 1948 Superman felt beanie, and $20,000 for a brass ring that could be had in 1941 for two bottle caps and ten cents.
Music is a touchstone in any culture, and Superman's omnipresence in the American songbook underlined the chord he had struck. The Crash Test Dummies despaired that "the world will never see another man like him." Donovan boasted that "Superman or Green Lantern ain't got a-nothin' on me." The Kinks wished they "could fly like Superman," while Hank Williams, Jr., said his "friends all call me Superman." Herbie Mann professed his love and ours for the hero. Bluesmen sang about him as wistfully as country boys, rockers, balladeers, and big bands. Rappers, too, although they were less sentimental. "You could be my boyfriend, you surely can, just let me quit my boyfriend called Superman," the Sugarhill Gang recited in "Rapper's Delight," the first hip-hop single to crack the Top 40 Hits. "I said he's a fairy I do suppose, flying through the air in pantyhose, he may be very sexy or even cute, but he looks like a sucker in a blue and red suit."
While his place in American lore was rock-solid, what sustained Superman as a Man of Tomorrow rather than a dusty icon was the same alchemy that had brought him alive three-quarters of a century earlier: the resonance and relevance of his story. That magic was dramatically displayed in _Superman: Red Son_ , a miniseries published by DC in 2003 that challenged not just the premise of the Man of Steel as a symbol of the red, white, and blue, but the outcome of the recently concluded Cold War. What if his rocket ship had taken a slightly different trajectory and, rather than landing in a Kansas cornfield, it had come down on a collective farm in Joseph Stalin's Ukraine? What, writer Mark Millar wondered, would it mean if instead of the bold _S_ on his chest Superman sported a hammer and sickle? The tale was part Mort Weisinger, part George Orwell. Superman managed to virtually eliminate poverty and ignorance behind the Iron Curtain, along with any vestiges of resistance to the Communist Party, which now presided over a global empire with six billion supplicants. America, not Russia, was economically and politically imploding in the brave new order. Its last hope lay with its most brilliant scientist and new president, Lex Luthor. It was a new way of seeing not just Superman and his supporting cast but our world and ourselves.
Millar wasn't the only one reimagining Superman's roots. Mark Waid got to replot Superman's interplanetary beginnings and terrestrial circumstances in _Superman: Birthright_. "Who am I and why am I here?" the hero asked himself. And, in terms more suited to science fiction fans, "Am I an Earthman or a Kryptonian?" His answer, laid out in twelve comic books published in 2003 and 2004, was that embracing his alien heritage wasn't just a possibility, it was his destiny. Along the way Waid gave us a Superman who was less defensive of the status quo and more like the rebel that Jerry Siegel had envisioned. It was a vision compelling enough that it would supplant John Byrne's reboot and become the defining mythos, at least until the next big reboot seven years later.
Superman and Batman teamed up anew in a comic book called _Superman/Batman_ that premiered in 2003 and explored their mutual empathy and antipathy. They still were DC's most glittering stars, and their differences still offered an illuminating lens on what made each special. Superman played Dudley Do-Right to Batman's Dirty Harry. Metropolis was Central Park and 59th Street at ten on a sunshiny morning on the first day of spring; Gotham was the Lower East Side in the middle of a rainy night in November. Now the two worlds and two heroes, who had forged their friendship on the radio in the 1940s, were united in a single book whose kickoff issue was ominously entitled "Public Enemies."
Crackerjack stories like those generally debuted in the comics, but many were collected into graphic novels and novelizations. Some were written by Superman pundits like Jeph Loeb, Mark Waid, and Cary Bates, who were aching for a chance to remind us why their hero mattered. Others came from newer hands like J. Michael Straczynski, who dispatched Superman on a year-long trek across the country to reconnect with the American way, and David S. Goyer, who took Superman in a different direction by having him renounce his American citizenship. Even Stan Lee, Marvel's longtime boss and DC's old-time nemesis, kicked in with a comic book called _Just Imagine: Stan Lee's Superman_. With no facts to rely on, Superman stories had always been about imagination. They also had been about the Metropolis Marvel giving a mouthpiece to the dreams of his starry-eyed writers and to the star-gazer in all of us. "Hold fast to dreams," Harlem Renaissance poet Langston Hughes had admonished, "For if dreams die / Life is a broken-winged bird / That cannot fly."
In the summer of 2011, DC made a bold gambit to expand its narrowing audience of comic book readers by recasting and simplifying not just Superman but its full lineup of superheroes. In the _Superman_ series, the Man of Steel and his alter ego once again were bachelors, Lois was back on the dating scene, and the touchstone red briefs and blue tights were replaced by high-tech ceremonial armor. _Action Comics_ , meanwhile, was relaunched with a new number 1 and a Superman who was younger, sported jeans and a T-shirt along with his red cape and _S_ logo, and was still finding his way in his new world. It was the most daring makeover in years, one aimed at teenagers and adults rather than the adolescents of old, and early returns were bullish: Sales of the reconstituted comics were the highest in twenty years, with both print and digital versions doing brilliantly.
DC is counting on its newest stories to be a gateway not just for new readers but for Hollywood. Warner Bros. has never needed a research and development department when it came to its superheroes; DC Comics has been the best idea mill in the business. Reading a comic is like watching a film frame-by-frame, letting studio executives see how audiences respond to characters and scripts before they commit millions of dollars. It is no accident that one comic book special after another has ended up on the big screen. _Doomsday_ was a 2007 animated adaptation of the 1990s "Death of Superman" story. It also was the first in a series of cartoon movies based on DC heroes that included _Superman/Batman: Public Enemies, Superman/Batman Apocalypse_ , and _All-Star Superman_. What Warner Bros. hopes will be its biggest Superman movie ever is due out in June 2013, with Hollywood heavyweights in charge. Director Zack Snyder's credits include _Watchmen_ , a superhero drama modeled after the comic book of the same name, and _300_ , based on Frank Miller's graphic novel about the Battle of Thermopylae. Creative consultant Christopher Nolan and his wife, Emma Thomas, who is producing the new Superman film, are responsible for rebooting the Batman film franchise and overseeing the science fiction thriller _Inception_. Picking up the red cape and blue tights (or will it be jeans and a T-shirt?) of Christopher Reeve and Brandon Routh will be British actor Henry Cavill, a relatively unknown hunk who played the first Duke of Suffolk on the Showtime series _The Tudors_. As is the tradition in Superman films, more conventional stars will fill in around Cavill, with Russell Crowe playing Superman's Kryptonian father and Kevin Costner his Earthbound dad.
And it was no accident that in 2009 DC got its first chief executive whose background was in movies, not publishing, and that DC Comics was subsumed under DC Entertainment, which also has moviemaking, online, and digital publishing arms. The move came just nine days after the Walt Disney Company announced its purchase of Marvel Comics, with all the promise that held for making Spider-Man and other Marvel characters even bigger screen stars. Diane Nelson, the new DC president, had shepherded Harry Potter from the printed page to movie screens, toy stores, home videos, and theme parks. Now, she told DC's writers and artists, she was determined to do the same for Superman and the rest of DC's heroes. Her announcement made many old-time comic book fans shudder, but it would have been music to the ears of Jack Liebowitz. The technology may have changed from radio waves to the World Wide Web, and kids are more likely to be reading Superman under the covers using an e-reader with a backlit screen than a flashlight and printed page. But the formula for success is the same one that Jack pioneered a half century earlier: Hire the best writers, artists, and actors; stay true to what made Superman resonate with audiences from day one with _Action_ No. 1; and get his story before as many eyeballs as possible to keep the cash register ringing. Ka-ching.
WILL THE TWENTIETH CENTURY'S longest-lasting hero endure deep into the new century and millennium? That is what fans and pundits are asking as Superman approaches the ripe age of seventy-five, just as they did at his first birthday and his tenth, and at his silver and golden jubilees. He has belied every prediction of his demise and defied the life expectancy for cultural icons and literary properties. We saw what happened when his handlers tried to kill him off: America would not have it. Kids want to be like him, and parents like that because they did, too. Many still do. He has proven tougher and more embedded in our DNA than even Jerry Siegel and Joe Shuster dared dream.
Whether he lives on depends in part on those telling his story—in the comic books, on TV, in the movies, and online—and whether they continue cultivating the richness of his character and illuminating his role in a world that never stands still. All signs suggest they will. It depends, too, on the steadfastness of his owners. If he thrived in the hands of a couple of Jewish kids from the ghetto, he should flourish when backed by the muscle of one of the world's biggest media conglomerates, which would be mad to let its billion-dollar franchise languish. In the end, however, it comes down to us, and whether we remain as besotted by Superman as our parents and grandparents were.
Why wouldn't we be? Heroes like Doc Savage, Ty Cobb, and even Teddy Roosevelt can become dated, reduced to interesting reflections of their era but not ours. Others, like Sherlock Holmes, Babe Ruth, and Franklin Roosevelt, still resonate, tapping into something primal. Superman defines that archetype. Part of it is the irresistible allure of taking flight. Part of it is the seduction of the love triangle and his secret identity. Part of it is just being ten years old again. The more that flesh-and-blood role models let us down, the more we turn to fictional ones who stay true. With them, and especially with Superman, it is about the possibility—of getting the girl, saving the world (or at least Lois and Jimmy), and having it our way. Our longest-lasting hero will endure as long as we need a champion, which should be until the end of time.
# **Acknowledgments**
WE KNEW HIS FACE WOULD tell us whether he liked the book idea or hated it, probably in the first fifteen seconds. So my agent, Jill Kneerim, and I decided to make our pitch in person to editor Will Murphy, without even hinting at the topic. We took the train from Boston to New York and marched over to Random House's offices. "Will, I want to tell the story of the longest-lasting American hero of the last century," I announced bluntly. "Who do you think it is?" A shadow of skepticism appeared, but Will played along. He made half a dozen guesses—politicians, sports stars, literary luminaries—all of them wrong. Then Jill carefully positioned on his desk a picture of Christopher Reeve's Superman in the classic red-and-blue uniform.
A smile spread across Will's face well before my fifteen seconds were up. Then he started asking the same exacting questions Jill had. What new could there possibly be to say about the planet's best-known superhero? What credentials did I have to tell his story? Why did the world, which already had two hundred books about the comics and their leading man, need two hundred and one?
Their refusal to accept anything on faith is part of what I like about Jill and Will. The other part is their willingness to listen, then to get as fired up as I do. There are endless books on Superman, I explained, but most are sociological surveys or picture books, or deal exclusively with the comics, TV shows, or some other limited aspect of his expansive, multimedia career. None is a full-fledged account that approaches him as if he were human, which he is to tens of millions of fans who have followed his loves and deaths, reinventions, resurrections, and redemptions. The fact that he is ethereal lets us fill in our image of Superman from our own imaginations. Our longest-lasting champion, I said, offers a singular lens into our deep-rooted fears and our enduring hopes.
They were sold. Jill, who knows as little about Superman as she did about my last subject, Satchel Paige, helped me flesh out my ideas, pored over my manuscript, and held my hand. Will didn't know much about baseball when he did a crackerjack job editing and advocating for _Satchel_ , but he is crazy about comics and has helped make my story worthy of his passion.
The Superman idea came from the same place so many good things do for me, my wife, Lisa, and she was the first to go at my manuscript with a red pencil and sharp intellect. I enlisted two kinds of readers. First were the experts, and I had the best: Paul Levitz, the longtime boss and guiding light at DC Comics; Superman writer Mark Waid, who doesn't just know more than anyone about the superhero but cares more; and Michael Hayde, whose own book demonstrates his nuanced understanding of Superman on the radio and TV. My other readers were old friends: Tom Maguire, whose blend of humor and serious-mindedness gives counterculturalism a good name, and Lou Ureneck, a seasoned newspaper editor and journalism professor who writes inspired memoirs. Even as she was finishing her own book, Sally Jacobs found the time to help me find the words I needed. Claudia Kalb did the same even as she was making a career change.
Two last words on readers: Evan Camfield. Production editors don't come any better. He caught errors of fact and context, fine-tuned prose, and made what often is an exasperating process a pleasure. Two more Random House people to whom I am grateful: designer Chris Zucker, whose creative flair is here for you to see, and publicist David Moench, who is passionate about Superman and selling books.
Every city I visited and every issue I probed turned up questions and gaps. I filled them in with help from hundreds of authors, experts, and friends, all of whom I list in the bibliography and am grateful to. Those I went back to more than I had the right are Cary Bates, Rick Bowers, Nicky Wheeler-Nicholson Brown, Mike Carlin, Richard Donner, Jay Emmett, Danny Fingeroth, Gary Grossman, David Hyde, Jenette Kahn, Jack Larson, Brian McKernan, John Jackson Miller, Will Murray, Denny O'Neil, Jerry Ordway, Tom Pollock, Louise Simonson, Michael Uslan, and, last and most especially, John Wells.
I hired a stream of student researchers, in Boston, Cleveland, Washington, and Los Angeles, to help with library searches, courthouse and schoolhouse searches, and other inquiries. The ones who stayed the longest were Nick Catoni, Michael Goldsmith, Tim Lewis, Chris McElwain, Maryrose Mesa, Elliot Schwartz, and Josh Willis. The ever-deft Katie Donelan was my in-house, go- to person at Random House. I also had two in-home experts on comics and kids, Alec and Marina. Jim Cahill kept my computers running and me online. Thanks, finally, to my parents, Dot and Mauray, for letting Superman into your house and my heart, which was no small thing in the 1950s.
A couple of notes on style: I quote people I interviewed in the present tense, and use the past tense with those whose words came from earlier writings and recordings. My endnotes generally are abridged listings of sources, with the full references in the bibliography.
# **Appendix**
**SUPERMAN (KAL-EL): CURRICULUM VITAE**
• **Permanent:** 344 Clinton Street, Apt. 3-B, Metropolis, USA
• **Getaway:** Fortress of Solitude, North Pole
## **Personal**
• **Parents:** Lara Lor-Van and Jor-El (birth); Martha and Jonathan Kent (adoptive); Jerry Siegel and Joe Shuster (creative); Jack Liebowitz and Harry Donenfeld (mercantile)
• **Hometowns:** Kryptonopolis (Krypton); Smallville, Kansas (USA, Earth)
• **Planetary homes:** Earth-2 (as Kal-L, 1938 on); Earth-1 (as Kal-El, mid-1950s on)
• **Girlfriends/wives:** Lana Lang, Lois Lane, Lori Lemaris, Wonder Woman, Lyla Lerrol, Sally Selwyn, Maxima
• **Best friends:** Pete Ross, Jimmy Olsen, Perry White, Inspector Henderson, Chloe Sullivan, Batman, Krypto
• **Nemeses:** Ultra-Humanite, Lex Luthor, Mr. Mxyzptlk, Brainiac, General Zod, Myrtle Beech (aka the Wedding Destroyer)
• **Aliases:** Clark Kent, Man of Steel, Last Son of Krypton, Big Blue, Supes
• **Age:** Over 29
• **Social Security No.:** 092-09-6616
## **Education**
• Smallville High School (yearbook: "highest grades—boy most likely to become famous")
• Metropolis University (journalism major; cheerleader football team; fraternity pledge; Bachelor of Arts with honors)
## **Work Experience**
• _Daily Star_ and _Daily Planet_ (Metropolis): Police reporter, war correspondent, advice-to-the-lovelorn editor, Bombay correspondent, editor-in-chief
• WGBS-TV (Galaxy Broadcasting): Reporter and news anchor
• _Newstime_ magazine: Publisher
## **Publications**
• _The Golden Throne_
• _The Janus Contract_
• _Under a Yellow Sun_
• _The Confessions of Superman_
• _I Superman_
• _The Krypton Chronicles_
## **Special Skills**
• Flight (like a bird)
• X-ray vision (can see through buildings)
• Super-strength (can squeeze coal into diamonds)
• Immune to aging (no hair loss, graying, wrinkles, or paunch)
• Super-hearing (can hear an ant's footfall)
• Super-breath (can blow out a celestial star)
• Photographic memory (can digest a 300-page book in seconds)
## **Vulnerabilities**
• Kryptonite (green, red, gold, black, red-green, red-gold)
• Virus X
• Magic
• Unbending moral code
## **Other Media Training**
• Comic books (1938–present)
• Comic strips (1939–1966)
• Radio (1940–1951)
• Cartoons (1941–present)
• Novels (1942–present)
• Movie serials (1948–1950)
• Television (1952–2011)
• Feature films (1978–present)
## **Professional organizations**
• Justice Society of America
• Justice League of America
• Atlas Club
• Strong Man Club
• Round Table Club
• Metropolis Press Club
• Club of Heroes
• Legion of Super-Heroes
## **Languages**
• Kryptonese, English, Atlantean, Interlac, Romance languages, Russian
## **Honors**
• Honorary citizen of all United Nations member countries
• Ambassador for Physical Fitness under President John F. Kennedy
## **References**
• Perry White, _Daily Planet_
• Morgan Edge, Galaxy Broadcasting
• Colin Thornton, _Newstime_
• Batman, Gotham City
• Supergirl (Linda Lee, Kara Zor-El)
• Diane Nelson, president, DC Comics
# **Notes**
### PREFACE
**LETTER WRITER:** A. L. Luther, "Vigilantes Not Needed," Cleveland _Plain Dealer_.
**"HOW DID YOU":** Author interview with Aaron Smolinski.
**"THE GODFATHER":** Email to author from Donald Wurzelbacher.
### 1. GIVING BIRTH
**HIS TROUBLE BEGAN:** Siegel, _Creation of a Superhero_ , unpublished memoir, Chapter 1: pages 2–4. This memoir was likely written in stages over the years, with two earlier versions being titled _The Story Behind Superman #1_ and _The Life and Times of Jerry Siegel_. While none of the three were published or made public, Jerry did register _Creation of a Superhero_ with the Copyright Office of the United States in 1978, when he was living in Los Angeles. The application was made in his name, along with those of his wife, Joanne, and daughter, Laura. He wrote his autobiography, he said in the preface, because so many people had asked him to "straighten out some misconceptions" about Superman's creation and "tell the full story."
**ON VALENTINE'S DAY:** Siegel, _Creation of a Superhero_ , 1: 6–7.
**RECESS, TOO:** Siegel, _Creation of a Superhero_ , 1: 7, 18–19.
**WITH THE REAL:** Siegel, _Creation of a Superhero_ , 1: 9–10.
**POINTING TO:** Author interview with Jerry Fine.
**HE EVEN TRIED:** Siegel, _Creation of a Superhero_ , 1: 19.
**IT HAPPENED:** Coroner's report (June 3, 1932), police report (June 3, 1932), and death record (June 4, 1932) on Michel Siegel. The Siegel family, and even the coroner, have raised suspicions that violence was involved in Michel's death, but there was no evidence of that. While the theft probably induced his heart attack, his death was ruled to be the result of natural causes.
**"BLISS":** Siegel, _Creation of a Superhero_ , 1: 2.
**"LET ME DIE":** _Tanakh: Holy Scriptures_ , 407.
**SHELLEY REFLECTED:** Moskowitz, _Emperors of the Infinite_ , 33.
**"WHAT IS THE APE":** Nietzsche, _Thus Spoke Zarathustra_ , 12.
**HIS PROTAGONIST:** Burroughs, _A Princess of Mars_.
**HUGO DANNER:** Wylie, _Gladiator_.
**KNOWN TO THE:** Dent, _Man of Bronze_.
**HAVE AS A MODEL:** Murray, "The Pulp Connection," _Comic Book Marketplace_.
**AMERICA WAS READIER:** Author interview with and emails from Will Murray.
**EARNED HIM A VISIT:** Siegel, _Creation of a Superhero_ , 1: 13.
**"MASTER OF DEDUCTION":** _Glenville Torch_ , "Master Sleuth."
**PEN NAME:** Jerry's other pseudonyms included Joe Carter, Jerry Ess, Herbert S. Fine, Cleve Jerome, Bernard J. Kenton, Hugh Langley, and Leger (Bails, "Who's Who of American Comic Books").
**GREW UP POOR:** Kobler, "Up, Up and Awa-a-y!" _Saturday Evening Post_.
**JOE SHUSTER HAD:** Mietkiewicz, "Great Krypton!" _Toronto Star_. Comic scholars argue over whether Joe Shuster adorned his Canadian roots and their role in shaping the Superman story.
**"HE WAS IN":** Author interview with Jerry Fine.
**"RHEUMY AND SOFT-FOCUSED":** Author interview with Rosie Shuster.
**THEIR FIRST BIG:** Herbert S. Fine, "The Reign of the Super-Man," _Science Fiction: The Advance Guard of Future Civilization_ No. 3.
**"I SEE, NOW":** "The Reign of the Super-Man."
**"WAS A GIANT":** Siegel, _Creation of a Superhero_ , 1: 20.
**"WITH THE FURY":** Siegel, _Creation of a Superhero_ , 3: 4.
**HAL FOSTER:** Siegel, _Creation of a Superhero_ , 3: 6–10.
**AND HE DIDN'T:** There apparently are no surviving copies of that or other early versions of the Superman story.
**"WHEN I TOLD":** Siegel, _Creation of a Superhero_ , 3: 7.
**BEEN DISLOYAL:** Others say they, too, were approached by Jerry. Reuben Schrank remembered Jerry asking him to collaborate when both were at Glenville High. While his college plans prevented that, Schrank told his daughter decades later, he did introduce Jerry to an artist he said was "a better cartoonist than I am." That artist was Joe Shuster, and Schrank was one of several friends who believed he was Jerry and Joe's matchmaker (material provided by the Schrank family).
**THEY EXCHANGED:** Letter from Jerry Siegel to Russell Keaton, July 12, 1934.
**KEATON TOLD HIM:** Siegel, _Creation of a Superhero_ , 3: 18.
**SAYS THE ILLUSTRATOR:** Author interview with Denis Kitchen.
**HIS MOST CONSIDERED:** Siegel, _Creation of a Superhero_ , 3: 18–20. One detail that Jerry could not recall, or at least didn't include in his memoir, was the date of this restless night of writing. Judging from dates he did specify, it did not happen on a hot summer night as he would later dramatically recount, but at the end of 1934 or the early months of 1935, which in Cleveland probably were chilly and wintery.
**"THIS WENT ON":** Siegel, _Creation of a Superhero_ , 1: 18.
**"I CONCEIVED":** Bishoff and Light, " 'Superman' Grew out of Our Personal Feelings About Life," _Alter Ego_ No. 56, 6.
**JERRY BEGGED:** Siegel, _Creation of a Superhero_ , 1: 20.
**LOIS WAS HARDER:** " 'Superman' Grew," _Alter Ego_ No. 56, 8.
**JERRY REMEMBERED:** Siegel, _Creation of a Superhero_ , 1: 21. Jerry's son, Michael, among others, suspected that Joanne made up the modeling story, and that Jerry and Joe played along. Drafts of the earliest comic, with Lois included, already had been drawn by the time the modeling session supposedly happened in 1935, skeptics point out, and Joanne would have been just eighteen then—or twelve, according to the white lie she told about her age on her certificate of marriage to Jerry. And why would she have bothered lying to Jerry about her age if he knew her as early as 1935? But Joanne, Jerry, and Joe insisted until their dying days that the story was true. And it might have been: Joanne could have been hired to help Joe better visualize a character that he already had sketched out and Jerry had written about.
**"I HAVE A FEELING":** Siegel, _Creation of a Superhero_ , 1: 23
**WHEELER-NICHOLSON'S BIOGRAPHY:** Author interviews with and emails from Nicky Wheeler-Nicholson Brown; Brown, "He Was Going to Go for the Big Idea," _Alter Ego_ No. 88, 39–51; and Brown, "Major Malcolm Wheeler-Nicholson, Cartoon Character or Real Life Hero," _International Journal of Comic Art_ 10, No. 2, 242–53.
**UNLIKE ITS PROGENITORS:** Historians point to a long line of "firsts" in the evolution of the modern American comic book, starting with a collection of newspaper strips published in 1897. Over time they shrank to look more like magazines and less like tabloids, soft covers replaced hard and color pages supplanted black-and-white, newsstands began offering the publications for general sale and publishers stopped relying on special orders from companies like Procter & Gamble, and comic books attracted devoted readers who may or may not also have read newspaper comic strips. Benton, _Superhero Comics of the Golden Age_ , 14–20.
**"DOCTOR OCCULT":** For their "Doctor Occult" stories, Jerry and Joe used the pen names Leger and Reuths, which Jerry said were anagrams of their real names.
**THE BELL SYNDICATE:** Siegel, _Creation of a Superhero_ , 4: 5.
**WHEELER-NICHOLSON WROTE:** Letter from Malcolm Wheeler-Nicholson to Jerry Siegel, October 4, 1935.
**THIS POSTSCRIPT:** Letter from Wheeler-Nicholson to Siegel, May 13, 1936.
**BUT THE BOYS:** Jerry explained in his memoir that he and Joe didn't get the 15 percent of profits or 50 percent of chain store sales for the first comics they sold to Wheeler-Nicholson. "Joe and I were not sold on Wheeler-Nicholson," he added, "and hoped to place 'Superman' with what we hoped would be a more responsible organization" (Siegel, _Creation of a Superhero_ , 4: 7–8).
**THAT IS WHAT:** Douglas Wheeler-Nicholson, "His Goal Was the Graphic Novel," _Alter Ego_ No. 88, 29.
**PULSATING STREETS:** Jones, _Men of Tomorrow_ , 8.
**"HE COULD SELL":** Author interview with Jack Adams.
**HERBIE SIEGEL:** Younger workers at DC would speculate over the years on just what Herbie did. Was he Harry's bodyguard or friend, gofer or babysitter? The truth is that he was all those things. As for Harry's misdeeds, he testified in April 1939 that he was never convicted of a crime but "I pleaded guilty in General Sessions for publishing magazines and paid a fine." _Detective Comics Against Bruns Publications_.
**JACK LIEBOWITZ WAS:** _Jack S. Liebowitz_ , 1993, 1–23. This unpublished memoir consisted of his transcribed responses to a series of questions posed by his daughter, Linda Stillman. One effect of having to share a bed when he was growing, Jack said, was that "when I married I refused to have [a] double bed. I wanted my own bed."
**JACK WORKED OUT:** Liebowitz memoir, 25–26, 40.
**PAUL SAMPLINER:** He apparently owned 25 percent of Harry Donenfeld's publishing operation and 75 percent of the distribution company, with Harry owning the remainder of both.
**WALLET EMPTY:** Comics historian Michael Uslan says that in 1973, he saw evidence in the DC archives that the Major received a payment of $19,703, in addition to having his debts canceled ( _Superman: The Action Comics Archives_ , Vol. 3, 6). Neither the Major's family nor DC can find evidence of a payout, although the company acknowledges that many of its files have been lost over the years.
**ABSOLUTELY, ACCORDING:** Liebowitz memoir, 47.
**HARRY HAD ORCHESTRATED:** Malcolm Wheeler-Nicholson, agreements with Donny Press, Inc., World Color Printing Co., and Photochrome, Inc. Nicky Wheeler-Nicholson Brown, the Major's granddaughter, is assembling evidence that she says will raise further questions about the bankruptcy's legitimacy (author interview with Brown).
**"SHE HATED THEM":** Amash, "His Goal Was," _Alter Ego_ No. 88, 33.
**UNITED FEATURE SYNDICATE:** Siegel, _Creation of a Superhero_ , 4: 12.
**LEDGER SYNDICATE:** Siegel, _Creation of a Superhero_ , 4: 13.
**AS JACK WROTE:** Liebowitz memoir, 48.
**CHARLIE AND JACK:** _Detective Comics Against Bruns_ , 131–36.
**IT WAS THE QUESTION:** Siegel, _Creation of a Superhero_ , 4: 21.
**A SWINDLE:** In fairness to Harry and Jack, most investments for them and other publishers didn't pay off and theirs was fair within the dealings of the times. Were Jerry and Joe as naïve as the natives who were handed trinkets by the Dutch West India Company? Not entirely, since they spoke the language and were aware of what they were doing. But they were naïve, and desperate to make a sale and a living. So while Harry and Jack had lawyers covering their backs and knew they could drive as hard a bargain as they chose, Jerry and Joe had barely any clout or clue.
**WERE BUYING NOT:** Letter from Donenfeld to Siegel and Shuster, September 22, 1938. There is disagreement among relatives and others familiar with the case as to whether Jerry and Joe sought legal advice before signing that famous contract. Some say the boys did, at the suggestion of Jerry's mother, and the lawyer said to sign. Others say they got no counsel. In his memoir Jerry writes, with uncharacteristic understatement, "The legal release, which Joe and I signed, caused us much grief later" (Siegel, _Creation of a Superhero_ , 4: 25).
In a comic book world where everything was written concisely, that legal release included one of the longest sentences ever: "In consideration of $130 agreed to be paid me by you, I hereby sell and transfer such work and strip, all goodwill attached thereto and exclusive right to the use of the characters and story, continuity and title of strip contained therein, to you and your assigns to have and hold forever and to be your exclusive property and I agree not to employee said characters or said story in any other strips or sell any like strip or story containing the same characters by their names contained therein or under any other names at any time hereafter to any other person, firm or corporation, or permit the use thereof by said other parties without obtaining your written consent therefor" (release form signed by Siegel and Shuster, March 1, 1938).
**ARTWORK WAS DESTROYED:** While there has been some confusion over what happened to that artwork, Jack Adler says, "Absolutely, it was destroyed." Adler worked for the engraver of _Action_ No. 1 and later for National Allied Magazines, where it was his job to destroy original art like that. Author interview with Jack Adler.
**JOSEPH STALIN:** Stalin was born Iosif Vissarionovich Dzhugashvili. He adopted the name Stalin, a takeoff on the Russian _stal_ , which means "steel." He liked the notion that he was a Man of Steel, but the more fitting reference is to the iron fist with which he ruled.
**"PLEASE CLARK!":** Siegel, _Action Comics_ No. 1.
**SUPERMAN BUILT ON:** Siegel, _Creation of a Superhero_ , 1: 15 and 5: 7.
**DOC SAVAGE LENT:** Author interview with and emails from Murray; and Murray, "The Pulp Connection."
**CASE FOR A CONNECTION:** Murray, "Gladiator of Iron, Man of Steel," _Alter Ego_ No. 37, 3–18.
**"DID YOU EVER":** Wylie, _Gladiator_ , 46.
**"USED DIALOGUE":** Letter from Philip Wylie to J. Randolph Cox, January 28, 1970.
**"OUR CONCEPT":** ' "Superman' Grew," _Alter Ego_ No. 56, 7.
### 2. A HERO FOR HIS TIMES
**THERE WERE TWO:** _Detective Comics Against Bruns_ , 43–46; and author interviews with Adams and Paul Levitz.
**ALL THE NUMBERS:** _Detective Comics Against Bruns_ , 52. Conventional wisdom says that National Comics printed 200,000 copies of _Action_ No. 1, but Jack Liebowitz testified that it was 202,000.
**RAN A TEST:** _Detective Comics Against Bruns_ , 37–38.
**DRIVE DEMAND:** Liebowitz memoir, 48.
**CONTINUED TO CLIMB:** Uslan figures from DC Archives.
**LIBRARIES GOT:** Seldes, "Preliminary Report on Superman," _Esquire;_ and Sheridan, _Classic Comics_ , 234–35. Enoch Pratt Free Library in Baltimore was the first to use Superman to attract kids in 1940, and the technique spread.
**ALL-STAR PITCHER:** Cramer, _The Hero's Life_ , 109.
**FIRST CHARACTER:** _Detective Dan_ appeared in 1933, but as a one-shot deal with the character thereafter appearing as the star of the _Dan Dunn_ comic strip.
**FIRST PRESS RUN:** Uslan figures from DC Archives.
**THERE WERE A DOZEN:** Benton, _Superhero Comics of the Golden Age_ , 23.
**"WE LIKED IT":** Liebowitz memoir, 48.
**FRED ALLEN'S RADIO:** Transcript by author of Allen's radio show on October 9, 1940; and "Up in the Sky! Look!" _Alter Ego_ No. 26, 29–33. When "Superman" pretended to lift Donenfeld into the air, the publisher pleaded, "I've got a weak stomach and any minute I'm going to lose it. Please take me down!"
**HARRY WOULD WEAR:** Jones, _Men of Tomorrow_ , 159.
**_ACTION_** **1 REFERRED:** Siegel, _Action Comics_ No. 1.
**_SUPERMAN_** **NEWSPAPER COMIC:** "The Superman is Born," _Superman: The Dailies_ , 13.
**ON DAY FIVE:** "Krypton Doomed!" _Superman: The Dailies_ , 15.
**FIVE INSTALLMENTS:** "Speeding Towards Earth," _Superman: The Dailies_ , 17.
**FIRST** **_SUPERMAN_** **COMIC:** Siegel, _Superman_ No. 1.
**IN** **_ACTION_** **1:** Siegel, _Action Comics_ No. 1.
**"AN INSTANT AFTER":** "Speeding Towards Earth," 17.
**IN** **_SUPERMAN_** **NO. 1:** Siegel, _Superman_ No. 1.
**FIRST** **_SUPERMAN_** **BOOK:** Historians have assumed that Joe, Jerry, and their editors merely stuck back pages cut from _Action_ 1 to fill out _Superman_ 1. But in his memoir, Jerry said that "the additional pages were specifically created for use in Superman Magazine no. 1." Siegel, _Creation of a Superhero_ , 6: 3.
**"ARE YOU SURE":** Siegel, _Action Comics_ No. 1.
**NO TAKE-OFF:** His first flying actually was in the Fleischer cartoons, which first aired in 1941.
**"MILLION-DOLLAR MARATHON":** Siegel, _Action Comics_ No. 65.
**"TO FLY":** Siegel, _Creation of a Superhero_ , 1: 2.
**WASN'T THE FIRST:** Pinpointing who _was_ the first isn't so simple, historian John Wells says, since the first widely distributed comic book in which the Sub-Mariner took flight, in October 1939, also featured a flying Human Torch. As for Superman, "the transition from leaping to flying was a gradual, organic process with some subsequent backsliding and emphasizing of the leaping along with other stories where Superman HAD to be flying. To my mind, though, Superman was flying (or at least doing a great imitation) by the end of 1939. That's important because 1940 saw an explosion of flying heroes, including (in rough chronological order) Hawkman, the Spectre, Black Condor, Bulletman, Doctor Fate, and Green Lantern. Captain Marvel was definitively flying by the latter half of 1940, too." Email to author from John Wells.
**FOLLOWING THOSE TWISTS:** Helping me follow the twists and turns were Fleisher, _The Great Superman Book;_ Greenberger and Pasko, _The Essential Superman Encyclopedia;_ and Wells emails.
**"WHICH IS THE":** Siegel, "Superman Joins the Circus," _Action Comics_ No. 7.
**EXCLAMATION POINT!:** That punctuation was at least partly a function of the metal plates used to print the comics back then. "Someone in the printing process could accidentally clean out a period, thinking it was a speck of dust, because it was so small," explains Levitz, the former DC publisher. Because exclamation points were bigger they were more likely to survive, which meant that early writers were more likely to use them. "Today, with offset printing," adds Levitz, "that worry is totally irrelevant and there is very little reliance on the exclamation point in comics. It is used more than in the average literature but mostly for melodrama."
**"HIS FIGURE ERECTS":** Siegel, "Superman in the Slums," _Action Comics_ No. 8.
**"THE BOYS DON'T":** Siegel, "Superman in the Slums."
**WORD LIKE "** ** _SARDONIC_** **":** Siegel, "The Million-Dollar Marathon," 172.
**"TOUGH IS PUTTING":** Siegel, "Superman, Champion of the Oppressed," _Action Comics_ No. 1.
**"THE MOTHER'S RIGHT!":** Siegel, "Superman in the Slums," 42, 52, 54.
**"IN THE EYES":** Siegel, _Creation of a Superhero_ , 7: 1.
**SURPASSED IN POPULARITY:** Slater Brown, "The Coming of Superman," _New Republic_.
**"I WROTE, WROTE":** Siegel, _Creation of a Superhero_ , 7: 2.
**HE STILL DID:** Kobler, "Up, Up and Awa-a-y!"
**AS TIME WENT:** The instructions were laid out in a letter from Liebowitz to Siegel on January 23, 1940, and in letters from Whitney Ellsworth to Siegel on January 22, 1940, November 4, 1940, and February 19, 1941. The "Murray" in Ellsworth's 1941 letter almost surely is Murray Boltinoff, although he wasn't believed to have started working at National for another two years.
**"HOW FOOLISH YOU":** Waid, "K-Metal: The 'Lost' Superman Tale," 13.
**NO ONE KNOWS:** Thomas Andrae argues that it was "editorial quibbles" and the story's length that killed it ( _Creators of the Superheroes_ , 54–55). Will Murray says it is more likely that the profound changes the story proposed were too much, too fast, for a superhero as successful as Superman ("The Kryptonite Crisis," _Alter Ego_ No. 37).
**THERE WAS MORE:** Again, there was a series of letters from Liebowitz to Siegel—on September 28, 1938, January 23, 1940, and January 29, 1940.
**IT WAS ENOUGH:** Kobler, "Up, Up and Awa-a-y!"
**"HE LOVED** **_SHIKSAS_** **":** Andrae and Gordon, _Funnyman_ , 79.
**"AFTERWARDS I SAID":** Author interview with Jerry Robinson.
**DATING BATMAN AND SUPERMAN:** There is an old joke that Superman is the guy girls want to marry, but Batman is the one they want to date.
**_WASHINGTON POST:_** "Superman Rescues His Creator," _Washington Post_.
**NO SURPRISE THERE:** _Detective Comics Against Bruns_ , 79–105; and Andelman, _Will Eisner: A Spirited Life_ , 43–45.
**"A MONGOLOID":** " 'Superman' Grew," _Alter Ego_ No. 56, 11.
**RETIRE THE CAPTAIN:** Twenty years after the settlement, DC Comics licensed the rights to Captain Marvel, and in 1973 DC brought him back to life in a comic book called _Shazam!_
**OTTO BINDER:** Emails to author from science fiction writer Richard Lupoff, who was a friend of Binder's.
**"IT IS PERFECTLY CLEAR":** _Jerome Siegel and Joseph Shuster Against National Comics_ , June 5, 1947, 17.
**TOLD A JUDGE:** Kane told Will Murray that if it hadn't been for Siegel and Shuster, "I wouldn't have created Batman nor would there be a comic book industry" ("Mark of the Bat," _Comic Scene Yearbook_ No. 1).
**HOW RICH:** Kobler, "Up, Up and Awa-a-y!"
**WHAT IS CLEAR:** Jones, _Men of Tomorrow_ , 159–64.
**"HE WAS MANY":** Author interview with Sonia "Peachy" Donenfeld.
**JACK REGULARLY REMINDED:** Liebowitz memoir, 47.
**COMBINED CIRCULATION:** "Superman's Dilemma," _Time_. The three comic books he starred in back then were _Action, Superman_ , and _World's Finest_ , which was launched in 1941 and featured Batman along with (and sometimes teaming up with) Superman.
**ROSE AND THE GIRLS:** Author interview with Joan Levy.
**CARTOON STORY:** Siegel, "How Superman Would End the War," _Look_.
**"TO RAP THE":** Uslan, _America at War_ , 27.
**"AS THE MIGHTIEST":** "Superman's Dilemma," _Time_.
**"YOU'RE PHYSICALLY":** Untitled comic strips, February 16–19, 1942, _Superman in the Forties_.
**THE U.S. MILITARY:** "All's Well in Britain Now—Admiralty Enlists Superman," _Washington Post;_ and Kobler, "Up, Up and Awa-a-y!"
**AFTER D DAY:** Weisinger, "Here Comes Superman!" accessed at superman-through-the-ages.nu.
**AT U.S. MILITARY BASES:** "Comic Culture," _Time;_ Robinson, _Zap! Pow! Bam!_ 21; "Superman's Dilemma," _Time;_ and "Superman Stymied," _Time_.
**"THE FBI CAME":** Overstreet, _The Comic Book Price Guide_ No. 13 (1983), A-65. Superman editor Mort Weisinger went a step further, as always, saying, "I'd discovered the bomb two years before it was first exploded" (Peterson, "Superman Goes Mod," _Indianapolis Star Magazine_ ).
**A 1945 DOCUMENT FROM:** "Superman and the Atom Bomb," _Harper's;_ and "Superman vs. Atom Man—the Prequel—and the Sequel!" _Alter Ego_ No. 98, 13.
**ALVIN SCHWARTZ:** Author interview with Alvin Schwartz; and Schwartz, "The Real Secret of Superman's Identity," _Annual of the Modern Language Association_.
**IN A 1944 LETTER:** Letter from Siegel to Liebowitz, January 1, 1944.
**IT WAS CALLED:** "Introducing 'SUPER GI,' " _Midpacifican_.
**GEORGE LOWTHER:** He already had experience writing Superman stories on the radio, and would go on to an eclectic and impressive career in print and writing, producing, directing, and even acting on television and radio.
**MONTHLY SALES:** De Haven, _Our Hero_ , 76; "Escapist Paydirt," _Newsweek;_ Rossen, _Superman vs. Hollywood_ , x; and "Superman Scores," _Business Week_.
**"YOU DID NOT":** Letter from Liebowitz to Siegel, February 3, 1947.
**POPULARITY FADE:** The only three comic book superheroes to survive in the same form and without interruption from the pre–World War II Golden Age until today are Superman, Batman, and Wonder Woman, although Wonder Woman experienced a brief hiatus in 1986. Aquaman and Green Arrow also still are around, but have come and gone as stars of their own series (Wells emails).
### 3. A MATTER OF FAITH
**_KAL_** **IS SIMILAR:** The Hebrew word for voice is _kole_ and for vessel is kol. _Kal_ itself means "light," as in weight.
**KANSAS FARMERS:** Just where the Kents lived wasn't made clear for decades. An early Superman radio show placed them in southeastern Iowa. In the 1950s and 1960s, Metropolis was set on the East Coast, with the Kents seemingly not far away. By the 1970s it had been pinpointed as Maryland. But promotional material for _Superman: The Movie_ talked about Clark having been raised on the plains of Kansas, no matter that the film itself didn't say that, or that the Kansas scenes were shot in Canada. One influential moviegoer, John Byrne, liked the idea of Clark being from Kansas, and it stuck when Byrne led his reboot of the franchise in 1986. Today it is generally accepted that Clark and his adoptive parents lived in the town of Smallville and that Smallville is in Kansas. Or at least the original Earth-2 Smallville was. On Earth-1, it was back in Maryland (Wells emails; and _The Essential Superman Encyclopedia_ ).
**A 1940 ARTICLE:** "Jerry Siegel Attacks!" _Das Schwarze Korps_.
**_THE JEWISH 100:_** That is one in a series of books that explore Superman's Jewish roots. Others include _Disguised as Clark Kent, From Krakow to Krypton, Jews and American Comics_ , and _Up, Up, and Oy Vey!_
**JULES FEIFFER:** Feiffer, _Backing into Forward_ , 73.
**FATHER JOHN CUSH:** Emails to author from John Cush.
**"THE WORD BECAME":** Cornell, "Superman/Jesus Similarities Examined," _Los Angeles Times_.
**"SUPERMAN, I'VE":** Friedrich, Austin, and Simpson, "Up, Up and Awaaay!!!" _Time_.
**SUPERMAN KNEW:** Schwartz, _An Unlikely Prophet_ , 204–5.
**HE COULD CRAWL:** Author interview with and email from Michael Green.
**IT ALSO IS:** Author interview with Geoff Johns.
**"SUPERMAN IS NOT":** Author interview with Mark Waid.
**THE GOVERNOR WOULDN'T:** Siegel, "Superman Goes to Prison," _Action Comics_ No. 10.
**"YEARS AGO":** Waid and Ross, _Kingdom Come_ , 194–95.
**"I'VE NEVER HAD":** Ramos recently got married—to a "huge Supergirl fan." The pastor who married them "made mention during the ceremony how Superman made me the man I am," he says, and "we even had the Superman theme song played at the end of the ceremony (right after they pronounced us husband and wife)" (emails to author from Emilio Ramos, Jr.).
**"UP UNTIL I":** Author interview with Peter Lupus.
**"I HAD WHAT":** Author interview with and emails from Tom Maguire.
**"LET ME GET":** Andrae and Gordon, _Funnyman_ , 17.
**BY ONE COUNT:** Andrae and Gordon, _Funnyman_ , 5.
**"I WRITE ABOUT":** Andelman, _Will Eisner: A Spirited Life_ , 346.
**BOTH CHANGES WERE:** "I Didn't Want to Know," _Alter Ego_ No. 56, 36.
**BOYHOOD PAIN:** Andelman, _Will Eisner: A Spirited Life_ , 113.
**THEIR NAMES:** Fingeroth, _Disguised as Clark Kent_ , 99–100.
**"I NEVER CONSCIOUSLY":** Author interview with Stan Lee.
**FEWER NAME CHANGES:** Julius Shuster and his family were listed in the 1930 U.S. Census as Schuster, which Rosie Shuster says almost certainly was a mistake by the census taker. "People," she adds, "just want to put that 'c' in there. My epitaph will be—'no "c" in Shuster' " (email to author from Rosie Shuster).
**CLEVELAND BACK THEN:** Vincent, _Memoirs of a Life in Community Service;_ and Rubinstein, _Merging Traditions_.
**_HOW TO BE FUNNY:_** The Library of Congress lists Siegel as the sole author, although his address is given as the Siegel-Shuster School of Humor. The book came out in 1938, just after the first Superman story was published in _Action Comics_.
**WASN'T ENTIRELY TRUE:** Emails from Dwight Decker to author, and Decker, "The Reich Strikes Back," _Alter Ego_ No. 79.
**MORE THAN ANYONE:** There were other Jewish superheroes over time, although none with nearly the reach of Superman. Gardner Fox's Sandman, for instance, was half Jewish, but that incarnation of the character, who came to life a year after Superman, faded during the 1940s.
**"DONENFELD," HIS:** Author interview with Peachy Donenfeld.
### 4. THE SPEED OF SOUND
**HE HIRED:** Josette Frank, head of the Child Study Association of America, was Maxwell's primary expert. The strategy was outlined in a May 20, 1946, letter from Mrs. Hugh Grant Straus to the editor of _PM_ magazine (Child Study Association Files, University of Minnesota).
**"CLAN OF THE FIERY":** Author's transcript of the radio broadcasts.
**MAXWELL USED:** Hayde, _Flights of Fantasy_ , 78.
**BUMP IN THE RATINGS:** Whiteside, "Up, Up and Awa-a-y!," _New Republic_.
**"TOLERANCE IS RAMPANT":** "It's Superflight," _Newsweek_.
**THE ANGLE THAT:** Whiteside, "Up, Up and Awa-a-y!"
**KENNEDY PICKED UP:** Kennedy, _The Klan Unmasked_ , 91–94.
**IT WAS A:** Hayde, _Flights of Fantasy_ , 77–78; and Patton, "Investigation of Stephen J. Dubner & Steven D. Levitt Article," _Florida Times-Union_. Dubner and Levitt, in their bestseller _Freakonomics_ , lionized Kennedy, calling him "courageous and resolute and unflappable." After they learned that Kennedy likely had embellished, the _Freakonomics_ authors questioned his credibility in a _New York Times Magazine_ story entitled "Hoodwinked?" The truth is that Kennedy wasn't the hero he was painted, nor the villain. He did help expose the Klan and he did enlarge his role—which Dubner and Levitt could have determined by asking researchers familiar with Kennedy's work or comparing tapes of "Clan of the Fiery Cross" with Kennedy's claims about the Superman broadcasts. Bids to determine what fact-checking the _Freakonomics_ duo performed were unsuccessful: Levitt referred questions to Dubner, who said he would try to answer, then didn't. Kennedy was more forthcoming. He said he sent the Klan passwords to Superman producer Maxwell, who apparently didn't use them, but that syndicated columnist Drew Pearson did. Kennedy, who died in August 2011, was mentioned in at least one Pearson column (May 6, 1947) that talked about leaked Klan passwords.
**THE VETERANS:** Hayde, _Flights of Fantasy_ , 78–79.
**"SLAP A JAP":** In an undated comic strip called "Superman Scores Again," Jerry and Joe showed U.S. troops destroying a Japanese invasion fleet. But they reminded readers that "most Japanese-Americans are loyal citizens. Many are in combat units of our armed forces, and others are working in war factories. According to government statements, not one act of sabotage was perpetrated in Hawaii or [the] territorial U.S. by a Japanese-American."
**"WE HAD BEEN":** Whiteside, "Up, Up and Awa-a-y!" Not everyone was clapping. Dorothy Lewis of the National Association of Broadcasters wrote that "while Superman often tries to crusade in civic affairs, he does so at the expense of the dignity of the community. This leads to confusion and lack of faith" (November 13, 1947, letter from Lewis to Josette Frank, Child Study Association Files).
And while Maxwell led the chorus for more shows on civil rights, he was less broad-minded when it came to portraying America's wartime enemies to his juvenile listeners. "I am, at the moment, teaching this vast audience to hate. If not to hate individuals, to hate that for which they stand," he wrote in an April 12, 1943, letter to George Zachary at the Office of War Information. "A german is a Nazi and a Jap is the little yellow man who 'knifed us in the back at Pearl Harbor.' " Zachary was taken aback, and consulted his colleagues for their reactions to Maxwell's letter. They observed, Zachary wrote, that "the notion that it is necessary to hate our enemies is crude and childish and unreal. It is the invention of frustrated civilians who don't know anything about war" (April 3, 1943, letter from Zachary to Allen Ducovny, Maxwell's partner at Superman, Inc. Both letters are in the Child Study Association Files).
**COLLYER DREW:** Hayde, _Flights of Fantasy_ , 33; and Tollin, _Smithsonian Historical Performances: Superman on Radio, 12–13_.
**EVEN THOUGH THEY:** "It's Superflight," _Newsweek_.
**PORTRAYING SUPERMAN:** Tollin, _Superman on Radio_ , 14.
**"THE PRODUCERS":** Jane Hitchcock's eulogy for her mother, Joan Alexander.
**THE LAST FOUR:** Superman's motto of "Truth, Justice and the American Way" debuted on August 31, 1942, when his live radio serial debuted on the Mutual Broadcasting System. Prior to that, he fought for "Truth and Justice." Olga Druce—a writer, actress, and child psychologist—took credit for the famous phrase. There was only a one-word difference between the Mutual Broadcasting prelude and the one that would later be used on the George Reeves television show. On radio, the narrator says, "strange visitor from another world," while the TV narrator says, "strange visitor from another planet" (email to author from Michael Hayde).
**"KIDS CAN DETECT":** Tollin, _Superman on Radio_ , 17.
**"A RAILROAD TRAIN":** Interview with Edward Langley by Brian McKernan, July 9, 1985.
**SUDDENLY FEEL:** Tollin, _Smithsonian Historical Performances: Superman vs. Atom Man_ , 7.
**"SUCCEED WHERE":** Freeman, _The Superman Radio Scripts_ , 1–43, 199, 203.
**"UP IN MY ARMS":** Author's transcript of "Clan of the Fiery Cross."
**REASSURE PARENTS:** Hayde, _Flights of Fantasy_ , 63.
**THERE WERE ACTUALLY:** Hayde, _Flights of Fantasy_ , 115–16.
**THEY THOUGHT BUILDING:** Fleischer, _Out of the Inkwell_ , 105.
**THE BROTHERS WERE:** Dooley and Engle, _Superman at Fifty!_ 64–65; Daniels, _DC Comics: Sixty Years_ , 68–69; Rossen, _Superman vs. Hollywood_ , 7–9; and Cabarga, _The Fleischer Story_ , 174–77.
**"THE MOVIE CARTOON":** "The New Pictures," _Time_.
**"THE FLEISCHERS SHOW":** Maslin, "Film: Animation Art of the Fleischers," _New York Times_.
**"THESE FILMS":** Maltin, _Of Mice and Magic_ , 122.
**"SOME 20,000,000":** "The New Pictures," _Time_.
**REACTIONS LIKE:** Hayde, _Flights of Fantasy_ , 47, 58.
**ONE BIT OF:** Younis, "Superman and the Phone Booth," www.supermanhomepage.com.
**SHORTS IN FRANCE:** France blew hot and cold when it came to Superman's comic books. Some were published under alternative names and ascribed to French authors during the Nazi occupation. But the postwar French government banned them, either because they were seditious capitalist influences or, as some reports suggest, because it was too much of a stretch to say he could fly. In any case, they were back by the 1960s (Wells emails; and Bart, "Advertising," _New York Times_ ).
**"IF THEY GUESS":** "Jungle Sam," _Time_.
**"I SAID, 'WAIT' ":** Brennan, "Kirk Alyn: Man of Steel," _Washington Post_.
**FIRST ACTOR:** Technically, the very first was Ray Middleton, an actor hired to portray Superman at the 1940 World's Fair in New York.
**"I VISUALIZED":** Tollin, _Smithsonian Historical Performances: Superman with Batman & Robin on Radio_, 22.
**KATZMAN ANNOUNCED:** Alyn, _A Job for Superman_ , 6.
**"I WAS SAVED":** Alyn, _A Job for Superman_ , 18.
**PRODUCERS ALSO:** Grossman, _Superman: Serial to Cereal_ , 43–44.
**PARTS OF THE:** Schoell, _Comic Book Heroes_ , 23–25.
**TWO YEARS:** Schoell, _Comic Book Heroes_ , 25–28.
**HE LOVED IT:** Grossman, _Superman: Serial to Cereal_ , 23, 30.
**BILL FINGER'S TALE:** "The Origin of Superman," _Superman_ No. 53.
**IT TOOK ANOTHER:** Finger, "Superman Returns to Krypton," _Superman_ No. 61.
**COMICS TO GIVE:** Siegel, "The Archer," _Superman_ No. 13.
**WHO WAS THE INSPIRATION:** Joanne Siegel, "The True Inspiration for Lois Lane," supermanhomepage.com; Sherwood, "Superman Still Makes Millions," _Washington Star;_ and author interview with Lois Amster.
**WILSON HIRSCHFELD:** Emails to author from Dan Hirschfeld, Wilson's son, and materials supplied by Dan.
**"NO MAN ON EARTH":** "The Origin of Superman," 55.
**ALTHOUGH JOR-EL BECAME:** The _el_ became _El_ in the comic book letters columns of the 1960s (Wells emails).
**"IT WAS NOT":** Lowther, _The Adventures of Superman_ , 24–28.
**BETTER FIX ON:** Waugh, _The Comics_ , 334–49.
**"I THOUGHT":** "The Archer."
**"WHAT SORT":** Siegel, "Europe at War," _Action Comics_ No. 23.
**"LET'S TEST":** Cameron, "The Mxyztplk-Susie Alliance," _Superman_ No. 40.
**"A CHANGE OF":** Siegel, _Creation of a Superhero_ , 5: 3.
### 5. SUPERMAN, INC.
**"BE A WHOO":** National Comics Publications, _Superman-Tim_ , 1949.
**THE NEW NAME:** The comic book publisher's name changed often enough that it was difficult to keep track. It started in 1935 as National Allied Magazines, became National Comics Publications in 1946, and switched to National Periodical Publications in 1961. National Periodical Publications merged with Kinney in 1967 to become Kinney National Services, and Kinney National was renamed Warner Communications in 1971, although National Periodical Publications continued to be used to describe the publishing operations. There were other iterations in between, including the holding company called Superman, Inc. Only one title was there from the very first, with Major Malcolm Wheeler-Nicholson, and has survived: Detective Comics. It was there on the banner of America's longest continuously published comic book, was sometimes used to refer to the company as a whole, and was the only name most readers recognized. In 1976 the company let its fans have the final word by adopting an abbreviated version—DC Comics—as its official title (www.dccomicsartists.com; and author interview with Levitz).
**"LOOKS EXACTLY":** Murray, "The Kryptonite Crisis."
**"LED TO HER":** Author interview with Jerry Fine.
**"BELLA WENT":** Letter from Siegel to Liebowitz, November 11, 1946.
**"I NEVER":** Andrae and Gordon, _Funnyman_ , 53.
**"PRACTICALLY NONE":** _Siegel and Shuster Against National Comics Publications_ , 1947, 204.
**"IN LINE":** Letter from Siegel to Shuster, September 18, 1946.
**THEY DESPAIRED OF:** Jones, _Men of Tomorrow_ , 215–16, 228; author interview with Peachy Donenfeld; and "Company Formed," _Middletown Times Herald_.
**IN THE TEN YEARS:** _Joanne Siegel and Laura Siegel Larson v. Warner Bros. Entertainment Inc._ , 2004.
**SIGNED AN AGREEMENT:** _Siegel and Shuster Against National Comics Publications_ , Final Judgment, May 21, 1948, 6.
**BOB KANE PROVED:** There has been endless speculation whether Kane got a better deal than Siegel and Shuster, and if so, why. Some say his deal was comparable but looked more lucrative because he didn't have to split it in two the way Jerry and Joe did. Others say Kane's father helped him negotiate better terms, in part by tapping the father's friendship with Liebowitz, and that Kane used Siegel and Shuster's lawsuit to get Liebowitz to settle amicably and lucratively with him.
**_LI'L ABNER:_** Capp, _The World of Li'l Abner_ , 120–26.
**BELLA SUED:** _Bella Siegel vs. Jerome Siegel_ , Petition for Divorce, July 14, 1948.
**CARTOONISTS SOCIETY:** Siegel family lore says that Marlon Brando judged the costumes (Weber, "Joanne Siegel, the Model for Lois Lane, Dies at 93," _New York Times_ ).
**"JERRY AND I":** Andrae, "Of Supermen and Kids with Dreams," _Nemo_ No. 2.
**THE MARRIAGE COULDN'T:** Winchell, " 'Superman' Artist Weds a Model," _Syracuse Herald-Journal;_ Jolan Kovacs and Jerome Siegel's applications for marriage license, October 13, 1948, and November 3, 1948; and Jolan's birth certificate. One possible explanation for the dual marriage licenses was that court fees were not paid in Jerry and Bella's divorce settlement until October 29. Could that mean Jerry was still married to Bella when he married Joanne in October 1948? No, say Ohio matrimonial lawyers, explaining that paying the fees is a technicality and should not have held up his divorce or his remarriage.
**AFTER HIGH SCHOOL:** Handwritten letter from Joanne to unknown recipient, May 25, 1992.
**"THE URCHIN IN":** Ellison, "It Ain't Toontown," _Playboy_.
**THAT CELEBRITY LET:** Kobler, "Up, Up and Awa-a-y!"
**SUPERMAN, INC., STARTED:** Matetsky, _The Adventures of Superman Collecting_.
**"LET SUPERMAN BE":** Daniels, _The Golden Age_ , 48; and Daniels, _DC Comics: Sixty Years_ , 74.
**IT WORKED:** Matetsky, _The Adventures of Superman Collecting_ , 17.
**"SUPERMAN TURNED":** Matetsky, _The Adventures of Superman Collecting_ , 138.
**"FOR THE WHOLE UNIVERSE":** Email to author from Vincent Maulandi.
### 6. THE DEADLY TRUTH
**ON A COOL:** "Boy Kills Self Showing Chum Gun Roulette," _Washington Post_. While his mother said he read about Russian roulette in a comic book, police said they were told he had learned about the dangerous revolver gamble in a movie.
**TWO MONTHS:** " 'Comics' Blamed in Death," _New York Times_.
**THE COMMON:** Wertham, _Seduction of the Innocent_ , 34, 97, 118.
**"UNLESS WE":** North, "A National Disgrace," _Chicago Daily News_.
**"WHAT'S WRONG":** Doyle, "What's Wrong with the 'Comics'?" _Catholic World_.
**"THE SUPERMAN FORMULA":** Legman, _Love & Death_, 39–40. Similar warnings were coming from Moscow. "The word superman, as is known, comes from the ideological inspirer of the German Fascists, Nietzsche," charged Korny Chukovsky, a leading writer of children's books. "Mass fascisization of the children fully corresponds to the perspectives of the present bosses of America" ("Russian Says Comic Books 'Fascisize' U.S. Children," _New York Times_ ).
**"COMIC-BOOK READING":** Crist, "Horror in the Nursery," _Collier's_.
**YEARS LATER:** Nyberg, _Seal of Approval_ , 1–21.
**"WE ARE":** Crist, "Horror in the Nursery."
**"BELIEVING THAT":** "600 Pupils Hold Burial Rites," _Washington Post_.
**A GALLUP POLL:** Hajdu, _The Ten-Cent Plague_ , 294.
**IT WAS EASY:** Hajdu, _The Ten-Cent Plague_ , 190.
**"YOU FIND":** Crist, "Horror in the Nursery."
**HE WAS ONE OF FOUR:** Hajdu, _The Ten-Cent Plague_ , 149, 264.
**MAXWELL AND HIS DIRECTOR:** Grossman, _Superman: Serial to Cereal_ , 80.
**NEVER SERIOUSLY:** Whitney Ellsworth heard Alyn's claim that he was offered the TV role, but said, "It just is not true" (transcript of Grossman interview with Ellsworth). And Noel Neill says, "I found out later that [Alyn] was very, very depressed by not being asked to do Superman on the television show" (author interview with Neill).
**MAXWELL'S CO-PRODUCER:** Grossman, _Superman: Serial to Cereal_ , 80, 316. Studio releases told a more dramatic story about casting George: "Maxwell was on a vacation when he saw a man taking a sun bath on Southern California's Muscle Beach. In his sunglasses the man surprisingly resembled Clark Kent." Glut and Harmon, _The Great Television Heroes_ , 26.
**"TAKE THE MONEY":** Grossman, _Superman: Serial to Cereal_ , 82.
**"WELL, BABE":** Author interview with Phyllis Coates; and Weaver, _Science Fiction Stars and Horror Heroes_ , 22.
**"I'D NEVER":** Author interview with Coates.
**"WEAR A SUIT":** Warren, "Superman's Girl Friday," _TV People_.
**"I MET BOB":** Weaver, _Producers and Writers_ , 20.
**"MY GOD":** Grossman, _Superman: Serial to Cereal_ , 128.
**"WE WENT":** Weaver, _Producers and Writers_ , 21–22.
**OTHER MONEY-SAVING:** Glut and Harmon, _The Great Television Heroes_ , 28–29.
**ON BUDGET:** There were varying recollections of what that budget was. Co-producer Robert Luber said it was $18,500 per episode. Outside producer David Wolper recalled it being $20,000. Jack Liebowitz said the ad agency paid $17,000, Maxwell said he could do it for $14,000, and it ended up costing $28,000. (Grossman, _Superman: Serial to Cereal_ , 316; Wolper, _Producer_ , 18; and Liebowitz memoir, 52.)
**"THAT'S ENOUGH":** Hayde, _Flights of Fantasy_ , 155.
**"GEORGE CAME":** Grossman, _Superman: Serial to Cereal_ , 127.
**"HE DECKED":** Weaver, _Producers and Writers_ , 24; and Bifulco, _Superman on Television_ , 3.
**"WHAT IS A MAN":** Noel Neill in _Biography_ TV show, February 9, 2000.
**THERE WAS A SILK:** Hayde, _Flights of Fantasy_ , 144.
**"THIS DROVE":** Weaver, _Producers and Writers_ , 25–26.
**"THIS," HE:** Author interview with Jack Larson.
**TRADEMARK BOW TIE:** Smithsonian officials say the tie is in their collection and that while it isn't currently on display, it may be soon.
**SHOW'S SPONSOR:** Whitney Ellsworth reportedly said that Kellogg's wanted Shayne fired and that Ellsworth insisted the actor be retained (Will Murray, "The Driving Force That Really Made DC Great," _Alter Ego_ No. 98, 17). But Ellsworth told Gary Grossman that "never did either Kellogg or their agency make any suggestion that we not use Shayne or anybody else, in spite of all the talk about the blacklists and everything else" (Grossman interview with Ellsworth).
**"MONITORED PROGRAMS":** Hayde, _Flights of Fantasy_ , 180.
**"TELEVISION HAS":** Wertham, _Seduction_ , 381.
**WAS AFRAID OF:** William H. Young and Nancy K. Young, _The 1950s_ , 42. In those earliest years of television, sponsors routinely engaged in the kind of content control that Kellogg's did with Superman.
**"THAT FAVORITE":** Hayde, _Flights of Fantasy_ , 165.
**"HEY JIMMY":** Author interview with Larson.
**"I HAD TO":** Warren, "Superman's Girl Friday," _TV People_.
**"SO HE WAS":** Liebowitz memoir, 52.
**WHITNEY ELLSWORTH:** He wrote under several pen names, including Fred Whitby and, when he collaborated with Robert Maxwell, Richard Fielding.
**HE DID:** Ellsworth letters to Siegel on February 21, 1940, November 4, 1940, and February 19, 1941.
**"WE NEVER":** Grossman interview with Ellsworth.
**"MAXWELL'S FIRST":** Grossman interview with Ellsworth.
**A RECENT COMIC:** "The Menace from the Stars" was published in _World's Finest Comics_ No. 68, which hit the newsstands just as "Panic in the Sky" was hitting the airwaves. While it is impossible to say for sure which came first, it's likely that "Panic" was inspired by "Menace." Comic books then were written as long as six months in advance of the cover date, whereas Jackson Gillis, who wrote "Panic," said he often gave an idea to his producer at lunch, then sat down and wrote the screenplay almost immediately and not long before its airing. But Gillis also said that the rip-off process was a two-way street, with comic book writers both borrowing from and offering up ideas to the TV screenwriters (Hagen, "From Lassie to Superman: Jackson Gillis," _Comics Interview_ ).
**"PURE FANTASY":** Author interview with Jackson Gillis.
**LISTENED CLOSELY:** www.tv.com/shows/adventures-of-superman/panic-in-the-sky-90378; and Bifulco, _Superman on Television_ , 75.
**"I HAD TO":** Grossman, _Superman: Serial to Cereal_ , 128.
**"I LOVED":** Author interview with Larson.
**SHE ALSO WORRIED:** Author interview with Coates.
**"I'VE HAD":** Weaver, _Producers and Writers_ , 24–25.
**"COMIC BOOKS IN":** Author interview with Neill.
**NEILL FOUND IT:** Author interview with Neill; and Ward, _Truth, Justice, & the American Way_, 82.
**DIDN'T LIKE ONE:** Ward, _Truth, Justice_ , 73; _The Adventures Continue_ No. 2, 66; and Weaver, _Producers and Writers_ , 27.
**"JACK, I GOT":** Grossman interview with Ellsworth.
**THAT SAME KID-FRIENDLY:** Hajdu, _The Ten-Cent Plague_ , 326, 329, 800.
**"MERCENARY MINORITY":** Weisinger, "How They're Cleaning Up the Comic Books," _Better Homes and Gardens_.
**"THE SAME TYPE":** "Are Comics Fascist?" _Time_.
**"EXPRESSIONS HAVING":** Memo from Ellsworth, "Editorial Policy for Superman-DC Publications."
**NEWSPAPERS GLOMMED:** "Superman Emulation Puts Boy in Hospital," _Washington Post;_ "Miscellany," _Time;_ " 'Death-Defying' Leap Kills Boy," _Los Angeles Times_.
**"WE WERE VERY":** Author interview with Jay Emmett.
**"NO ONE":** Grossman, _Superman: Serial to Cereal_ , 84.
**NATIONAL CAPITALIZED:** Author interview with Emmett.
**EMPEROR HIROHITO:** Grossman interview with Ellsworth; and "Reeves, Superman of TV, Kills Himself," _Los Angeles Times_.
**$1,000 A WEEK:** Grossman interview with Ellsworth.
**"NOTHING WAS BOTHERING":** Author interview with Gene LeBell.
**"TO SEE IF":** Ames, "Superman George Reeves and Producers Disagree," _Los Angeles Times_.
**HIS OWN VARIETY:** Ward, _Truth, Justice_ , 100–101.
**"MOTEL OF THE STARS":** Hayde, _Flights of Fantasy_ , 277.
**"HERE I AM":** Grossman, _Superman: Serial to Cereal_ , 275–76.
**GEORGE HAD ALMOST:** Hayde, _Flights of Fantasy_ , 260.
**THREE YEARS LATER:** "TV Superman Hero Injured in Auto Crash," _Los Angeles Times_.
**HE WAS LOOKING:** Hayde, _Flights of Fantasy_ , 283.
**"HE'S PROBABLY":** Transcript of Lee Saylor interview with Leonore Lemmon.
**FORENSIC DETAILS:** Author interviews with Craig Harvey, chief coroner investigator, Los Angeles County, and Dr. Eugene Mark, pathologist, Massachusetts General Hospital.
**RESEARCHERS WHO:** Author interviews with Jim Beaver, Chuck Harter, Hayde, and Jan Alan Henderson
**"ONLY LEM":** Thomas, _The Man to See_ , 145.
**TONI VISITED:** Author interview with Larson.
**PUBLIC RELATIONS MAN:** Edward Lozzi says Toni kept George's clothes in a bedroom in her mansion that became a shrine to him. "Her deathbed confession was totally the opposite of what she had been telling me," Lozzi says. "She was blaming it all on Leonore Lemmon." Author interview with Lozzi.
**GEORGE'S YOUNG:** Grossman, _Superman: Serial to Cereal_ , 102; and Hayde, _Flights of Fantasy_ , 272.
**NO ONE WILL:** Author interviews with Beaver, Harter, Harvey, Hayde, Henderson, and Eugene Mark.
### 7. IMAGINE THIS
**STORY LIKE THIS:** "Mr. and Mrs. Clark (SUPERMAN) Kent!" _Superman's Girl Friend Lois Lane_.
**ACTION ON TV:** The airwaves were dominated then by Westerns like _Bonanza_ and _Wagon Train_ and personality-driven comedies like Red Skelton's and Andy Griffith's.
**"EVERYONE KNOWS":** Weisinger, "I Flew with Superman," _Parade_.
**HE "GLOWED":** Joyce Kaffel, "Digging up Superman," _Alter Ego_ No. 98.
**EIGHT DIFFERENT:** The eight were _Action Comics, Superman, World's Finest, Superman's Pal Jimmy Olsen, Superman's Girl Friend Lois Lane, Adventure Comics, Justice League of America_ , and _Superboy_.
**"MY GREATEST":** Will Murray, "Superman's Editor Mort Weisinger," _The Krypton Companion_ , 12.
**"HERE LIES":** Jones, _Men of Tomorrow_ , 131.
**"I HAD TO":** Author interview with Adler. Jim Shooter says Weisinger called him a "fucking retard who couldn't spell." But his family was poor, Shooter adds, and the work Weisinger gave him "saved our house and kept us alive. That was the two sides of Mort" (author interview with Shooter). Others say that various DC staffers were so frustrated with Weisinger that they tried to toss him out the window, although with steel mesh surrounding the frame he wouldn't have gone very far.
**"I'LL TELL":** Michael Eury, "Neal Adams Interview," _The Krypton Companion_ , 101.
**"MORT KEPT IT":** Author interview with Carmine Infantino.
**CURT SWAN:** Zeno, _Curt Swan: A Life in Comics_ , 173, 734–75.
**"DO YOU NEED":** Pachter, "A Rare Interview with Superman's Godfather," _Amazing Heroes_ No. 41, 33. After he was fired from DC, Boring worked on several newspaper strips, then took a part-time job as a security guard.
**STEERED CLEAR:** There were exceptions, like the imaginary story in 1963—"The Amazing Story of Superman-Red and Superman-Blue"—in which the hero got Khrushchev to dump all his missiles into the sea and Fidel Castro to free all his political prisoners. And in 1969, Mort sent Clark Kent, Lois Lane, and Superman to Vietnam in a story entitled "The Soldier of Steel!" It was written by DC's war comics whiz, Robert Kanigher, and illustrated by longtime war artist Joe Kubert.
**"CALLED HIM":** Schelly, _Words of Wonder_ , 39, 142; and Lupoff email.
**"THE MOST COMPETENT":** Murray, "Superman's Editor Mort Weisinger," 13.
**"WHAT I FIND":** Letter from Siegel to Liebowitz, July 13, 1946.
**WEISINGER MET:** Murray, "Superman's Editor Mort Weisinger," 14. 166 **"WHEN PEOPLE":** Lillian, "Mort Weisinger: The Man Who Wouldn't Be Superman," _Amazing World of DC Comics_.
**"ANY TIME WE":** Murray, "Superman's Editor Mort Weisinger," 17.
**"DID YOU GET":** Weisinger, "I Flew with Superman."
**AN "INVESTIGATION":** Weisinger, "How They're Cleaning Up the Comic Books," _Better Homes and Gardens_.
**"I WANTED":** Author interview with Lee.
**"DONE CORRECTLY":** Author interview with Shooter.
**BATMAN TUMBLED:** Evanier, Beerbohm, and Schwartz, "There's a Lot of Myth Out There!" _Alter Ego_ No. 26, 23; and author interview with Infantino.
**FINALLY, AS HE REPORTED:** In another version, Mort said he had the economic security to quit DC after Columbia Pictures paid him $250,000 for his novel _The Contest_. How did he feel about leaving Superman after all that time? "I guess my baby has grown up," Mort said, "and doesn't need daddy any more" (Peterson, "Superman Goes Mod").
**CARMINE INFANTINO:** Author interview with Infantino. Julie Schwartz's version is that "every year or so Mort would tell our boss Jack Liebowitz that he wanted to retire, and Jack (he always wanted us to call him Jack) would talk him out of it. Then one day in 1970, (surprise! surprise!) he accepted the resignation since he himself was leaving DC" (Schwartz and Thomsen, _Man of Two Worlds_ , 131).
**THE MOST EMOTIONAL:** Siegel, "Superman's Return to Krypton!" _Superman_ No. 141.
**TENTH-LEVEL:** Lex Luthor would amp up Brainiac's intellect from level ten to twelve.
**"ANYTHING SUPERMAN":** Siegel, "The Bizarro Invasion of Earth!" _Superman_ No. 169.
**THE ENEMY:** Waid, "Red Kryptonite," _Amazing Heroes_ No. 41, 44–45.
**IN ITS ANNUAL:** Bart, "Advertising: Superman Faces New Hurdles," _New York Times_.
**SOCIAL SECURITY:** That number was issued to a real person, Giobatta Baiocchi, who was born in 1887 and whose relatives say they don't know of any connection he might have had to Superman.
**BEFORE EXECUTING:** Eddy Zeno, "A Fond Remembrance of Mort Weisinger by His Son," _The Krypton Companion_ , 17.
**COMIC STRIP DREAMS:** Those dreams lasted from 1949 to 1952, with Clark waking just as he was about to tell Lois he was Superman. The story was dreamed up by Whitney Ellsworth, who started to write it, but then got distracted, and Alvin Schwartz claims he wrote nearly all of it.
**"YOU'RE NOT":** Newman and Benton, _It's a Bird... It's a Plane... It's Superman: The New Musical Comedy_.
**"A CHILL":** Holiday and Harter, _Superman on Broadway_ , 18.
**THE FIRST TEST:** Schier, " 'Superman' Needs a Quick Course in Muscle Building," Philadelphia _Bulletin;_ and Murdock, " 'Superman' Lands in Town," _Philadelphia Daily News_.
**_IT'S A BIRD_** **OPENED:** "Paper Cutups," _Time;_ Coe, "Not Peter Pan, It's 'Superman,' " _Washington Post;_ Nadel, " 'Superman,' Airy, Merry," _New York World-Telegram and Sun;_ and Kauffmann, " 'It's a Bird... It's a Plane...' " _New York Times_.
**"THEY SAID, 'MY' ":** Author interview with Hal Prince.
**EVERYONE HAD:** Author interviews with Prince, Charles Strouse, and Robert Benton.
**"I DON'T THINK":** Author interview with Bob Holiday.
**SUPERMAN WAS:** Shabecoff, "Look! Up in the Air!" _New York Times;_ and Weisinger, "Superman and His Friends Around the World," _Superman's Pal Jimmy Olsen_ No. 113.
**_STUPOR-MAN:_** Not to be confused with _Mad_ magazine's Superduperman.
**"WHEN** **_SUPERMAN_** **":** Feiffer, _The Great Comic Book Heroes_ , 17.
**"THEY LOVED IT":** Thomas, "Superman Teaches School," _Magazine Digest_.
**"I COULDN'T READ":** Author interview with Ron Massengill.
**TWO OF HIS BEST:** The two writers were Bill Finger and E. Nelson Bridwell. "Superman's Mission for President Kennedy," _Superman_ No. 170.
**WHEN** **_THE NEW:_** "Superman Meets Kennedy on Vigor," _New York Times_.
**"WE'RE WAITING":** Weisinger, "I Flew with Superman."
**IT WAS NOT:** _The Essential Superman Encyclopedia_ , 151–52.
**COMPLIANT PARTNER:** Sampliner served on the boards of the Anti-Defamation League, the New York City Anti-Crime Commission, and the New York State Commission Against Discrimination. He stayed on as an owner of DC Comics until 1967, and in 1969 he was named chairman of the board of Independent, the distribution company (Kleefeld, "Paul Sampliner," kleefeldoncomics.blogspot.com).
**AS EARLY AS:** Liebowitz memoir, 55–56.
**HE NEVER KNEW:** Others, including Jack Adams, say the accident happened after the company went public.
**"WE ALL SAID":** Author interview with Peachy Donenfeld.
**HAVING A CHAUFFEUR:** Liebowitz memoir, 56, 62.
**"WE DIDN'T WANT":** Liebowitz memoir, 51.
**"TURNED TO ME":** Barr, "The Madame and the Girls," _Words & Pictures_ No. 5, 5.
**"THEY WERE THE":** Barr, "The Madame and the Girls," 10.
**WAS DESPERATE:** Knutzen, "Man of Steel Splinters an American Dream," _Los Angeles Times_.
**"I WAS THE OLDEST":** Knutzen, "Man of Steel Splinters an American Dream."
**IT LOOKED WORSE:** Author interview with Neal Adams.
**THE BEST MEASURE:** Yoe, _Secret Identity: The Fetish Identity of Superman's Co-Creator Joe Shuster_.
**"PORNOGRAPHY, UNADULTERATED":** _City of New York v. Kingsley Books_ , Supreme Court of New York.
**WHY DID JOE:** Author interviews with Robinson and Craig Yoe.
**"I SPENT ALL":** Liebowitz memoir, 57. He could, of course, have brought his daughters into the business, but that was unthinkable to an old-school father like him.
**SO HE HIRED:** Liebowitz memoir, 57.
**STEVE ROSS:** Bruck, _Master of the Game_ , 129–33.
**EMMETT SAID:** Author interview with Emmett.
### 8. BELIEVING A MAN CAN FLY
**"WHY DON'T WE":** Author interview with Ilya Salkind.
**HIS LAWYER:** Author interview with Tom Pollock.
**OWN BIG FILMS:** Curiously, Warner Bros. was willing to make a movie about _Doc Savage: The Man of Bronze_ , but not about Superman, the caped hero who had borrowed from Doc, then left him in the dust.
**"IT WASN'T":** Author interview with Terry Semel.
**MORE THAN A:** Petrou, _The Making of Superman the Movie_ , 21.
**BOUNCED OR DELAYED:** The Salkinds' producer and money man, Pierre Spengler, concedes, "there were times of extreme cash flow difficulties." But he adds that "everybody got paid in full" (emails to author from Spengler).
**GROSS SALES:** That is money received by the distributor(s) of the film, in this case mainly Warner Bros. It includes about half of box office receipts along with the distributor's share of proceeds from television, VHS/DVD, and other offshoots.
**EVERYONE WHO READ:** Author interviews with Salkind, Tom Mankiewicz, Richard Donner, and Infantino. Infantino says the Puzo script was "the worst thing you ever saw in your life." Transcripts of meetings with Puzo show Infantino and others trying to tone down the script's sexuality and beef up its fealty to Superman and his forty-year history.
**NEXT UP:** Author interviews with Leslie Newman and Robert Benton.
**AGREEMENT BETWEEN NATIONAL:** Agreement with Alex and Ilya Salkind, November 6, 1974.
**NEEDED A DIRECTOR:** Author interview with Ilya Salkind.
**PENDING ARREST:** The warrant had to do with Brando's role in the film _Last Tango in Paris_. His interest in Superman was piqued by an old girlfriend.
**IT ACTUALLY:** Author interviews with and emails from Spengler and Pollock.
**A LINEUP:** Petrou, _The Making of Superman the Movie_ , 36–37.
**PREP SCHOOL:** Christopher said that his poet-professor father, upon hearing his son was playing Superman, assumed he meant George Bernard Shaw's _Man and Superman_. Frank Reeve, the father, says that is "a great line" apparently dreamed up by Christopher and his buddy, comedian Robin Williams. "Later," Frank adds, "it became a line said so often he [Christopher] came to believe it." Frank says he watched all his son's movies except those Christopher asked him not to. "I found _Superman 1_ delightful and enthusiastically said so." (Emails from Frank Reeve to author.)
**"LIKE THE GUY":** Dangaard, "Reeve Flies to the Rescue of 'Superman,' " _Los Angeles Times_.
**"SHE LITERALLY":** Author interview with Donner.
**"I'M MANIC":** Petrou, _The Making of Superman the Movie_ , 48.
**"IT'S AS SIMPLE":** Author interview with Donner. Christopher Reeve, in his 1978 film, was the first Superman to explain for himself that he was fighting for "truth, justice and the American way," to which Lois, with characteristic sarcasm, replies, "You'll wind up fighting almost every elected official in this country."
**"THIS PICTURE IS":** Anderson, "It's a Bird! It's a Plane! It's a Movie!" _New York Times_.
**THE SOLUTION CAME:** Author interviews with Zoran Perisic and Donner.
**HIS OWN CAPERS:** Spengler says Reeve would have liked to do _all_ of his own stunts but "I had to stop him at times for security and/or insurance reasons" (Spengler email).
**"HOW COULD A":** Reeve, _Still Me_ , 192.
**"WE SHOVED":** Author interview with Dave Prowse.
**WAS SO OUTRAGED:** Rossen, _Superman vs. Hollywood_ , 90.
**BEING TYPECAST:** Sean Connery, who played James Bond in seven films, told Reeve not to worry. If the first film isn't good, he said, there won't be more. If you do a low-budget film next, it might hit big by the time _Superman_ airs. And if the producers or studio give you trouble, "get a good lawyer and sue the bastards" (Davis, "Marketing the Man of Steel!" _Maclean's_ ).
**"FOR GOD'S SAKE":** Transcript, _Studio 360_ , "American Icons" series.
**MANKIEWICZ EXPANDED:** Author interview with Mankiewicz.
**"I HAD TO PRETEND":** Author interview with Margot Kidder.
**"SUPERMAN WAS THE":** Author interview with John Williams.
**LAST-MINUTE GLITCH:** Author interviews with Donner, Semel, and Ilya Salkind; author interview with and emails from Pollock; and Blue and Delugach, " 'Superman': Rare Look at Film Finances," _Los Angeles Times_. Semel says that Warner Bros. sent a plane to Europe and managed to get a copy of the negative from Technicolor, the film storage people. "We called Alex in London to say, 'Oh, by the way, the negative is here, in Burbank, we're printing—we started printing last night—maybe we'll see you at the premiere, maybe we won't,' " recalls Semel. Tom Pollock has a different take on what happened more than thirty years ago: "Technicolor denied it, and in any case, had they [Warner Bros.] used it, it would have opened them up to lawsuits, as well as Technicolor. The letters of credit were contingent on DELIVERY by FilmExport, not by theft by Warners."
**CONFOUNDING NO-SHOW:** " 'Superman': Rare Look at Film Finances."
**"IT WAS EXACTLY":** Author interview with Jenette Kahn.
**REVIEWERS OFFERED:** Kroll, "Superman to the Rescue," _Newsweek;_ Ebert, "Superman," _Chicago Sun-Times;_ Kael, "The Current Cinema," _New Yorker;_ and Canby, "Screen: It's a Bird, It's a Plane, It's a Movie," _New York Times_.
**AND PEKING:** The _Peking Evening News_ said that Superman is not really a savior but "a narcotic which the capitalist class gives itself to cast off its serious crises." There were no problems anywhere else in China and all this happened in 1985, when China finally was catching up with old movies from America (Mann, " 'Superman' Shanghaied in Peking Screen Test," _Los Angeles Times_ ).
**"I TOOK MY":** Hoover, "What Women See in Man of Steel," _Los Angeles Times_.
**THE MOVIE WAS MEANT:** Author interview with Mankiewicz.
**"I GOT MAJOR":** Author interview with Donner.
**$55 MILLION:** Spengler says that "the aggregate cost of the first two movies was $109 million. The split would be approximately 75 for the first and 34 for the second" (Spengler email).
**FILED THEIR OWN LAWSUITS:** Tom Pollock says it wasn't just Alex who was targeted in the lawsuits, but "everybody under the sun. Alex, Ilya, Pierre [Spengler], all of Alex's companies, Credit Lyonnais Bank, Warner Brothers, etc etc etc." (email from Pollock).
**PRODUCTION FIGURES:** That matters because the percentage payouts to Puzo, Donner, and the others specified that they were a share of the profits left after Alex recovered his production costs. No postproduction profits, no payouts.
**"EVERYONE GOT PAID":** Author interview with Pollock. He says that Warner made more than $100 million, with Spengler adding that it was "considerably more." Credit Lyonnais, one of Alex's banks, "made the next most," says Pollock, and DC Comics got 5 percent of the gross. Spengler adds in an email that DC got "7.5% domestic or 5% worldwide, which ever is greater."
**BIG BANG THEORY:** Schwartz and Thomsen, _Man of Two Worlds_ , 15. If Julie had asked Jerry for his response to Big Bang, the Superman creator likely would have pointed out that he already had a magazine by and for fans, _Cosmic Stories_ , which predated not only Mort and Julie's fanzine but Jerry's own _Science Fiction_.
**"WAS MERELY":** O'Neil, _Superman: Kryptonite Nevermore_ , 189.
**YOUNG PEOPLE:** Schwartz and Thomsen, _Man of Two Worlds_ , 134.
**"I AM CURIOUS":** Kanigher, "I Am Curious (Black)!" _Superman's Girl Friend Lois Lane_ No. 106.
**"THAT'S THE":** Pasko, "The Master Mesmerizer of Metropolis!" _Superman_ No. 330.
**"SUPERMAN DIRECTED":** Author interview with Alvin Schwartz.
**"THE FAMOUS BLUE":** Baker, "Sad Feet in the Sky," _New York Times_.
**HE ACTUALLY PROPOSED:** Author interview with Kahn.
**MUHAMMAD ALI:** It was bad karma: By the time the book came out, Ali had been dethroned as boxing champ by Leon Spinks.
**PRICE HIKES:** The price rise was even steeper if you start in 1969, when a DC comic book sold for twelve cents, and go until 1981, when it was fifty cents. The number of pages fluctuated, often rising as the price did.
**"DREW A HUMUNGOUS":** Email to author from Levitz.
**"TOTALLY REAL":** Email to author from Luis Augusto.
**MOVIE-RELATED:** Harmetz, "The Marketing of Superman and His Paraphernalia," _New York Times;_ Levin, " 'Protect Children Act' Aims to Ban Cigarette Deals," _Los Angeles Times;_ Scivally, _Superman on Film, Television, Radio and Broadway_ , 95; and author interview with Kidder.
**ADVERTISING HELPED:** Gabilliet, _Of Comics and Men_ , 134–37.
**AS A WRITER:** Much of his writing for Marvel was under the pseudonym Joe Carter.
**"JERRY SIEGEL":** Siegel, "Superman's Originator Puts 'Curse' on Superman Movie," archives.tcj.com.
**THE PRESS:** Sherwood, "Superman Still Makes Millions, but Not His Creators," _Washington Star;_ Breasted, "Superman's Creators, Nearly Destitute, Invoke His Spirit," _New York Times;_ and Vidal, "Mild-Mannered Cartoonists Go to Aid of Superman's Creators," _New York Times_.
**ORCHESTRATING THE PUBLICITY:** Author interviews with Neal Adams and Irwin Hasen.
**"WE WERE ABOUT":** Author interview with Emmett.
**ROSE SUBSTANTIALLY:** In 1979, Jerry and Joe each got a check for $15,000 in recognition of the success of _Superman: The Movie_. The next year their annual payments jumped to $30,000. In 1981, after the release of _Superman II_ , the pensions rose again, to $60,000, and they each got onetime bonuses of $50,000 that year and $25,000 the next. After Joanne Siegel made her first formal request for additional money in 1988, the annual payouts were increased to $80,000, with a cost-of-living inflator.
**"JOE AND I":** Siegel, _Creation of a Superhero_ , 7: 5.
**DECEMBER 1976:** Most references to their marriage say it was in 1975, but public records make clear it was December 24, 1976, in Del Mar, California. Wedding pictures show each with a flower pinned to their breast, him wearing a suit and slightly raised shoes, her in a full-length gown standing before their three-tiered wedding cake. Joe was sixty-two then, Judith fifty-nine. Their handwritten notes invited guests to a reception at the Atlantis Restaurant in Mission Bay, an oceanfront oasis at the mouth of the San Diego River.
**THE ATTRACTIVE:** Certificate of Registry of Marriage, Joseph Michael Shuster and Judith Ray Calpini; and Request and Declaration for Final Judgment of Dissolution of Marriage, _Joseph Michael Shuster vs. Judith Ray Calpini_.
**"HAS MEANT A":** "Follow-up on the News," _New York Times_.
### 9. BACK TO THE FUTURE
**"MORE AGGRESSIVE":** Parrott, "For Clark Kent, Wimpery Is Out," _Los Angeles Times_.
**"YOU CAN'T DO":** Melvin, "Cartoonists Explain Superman's New Image to His Fans," _New York Times_.
**"HE USED TO":** Akers, "Bring Back the REAL Superman," _Washington Post_.
**"IF REAGAN HAS":** Kempley, "Superman: The Ramboization of the Comics' Man of Stale," _Washington Post_.
**SUPERMAN'S HEAD:** Psychologists and psychiatrists have suggested that Superman is the classic schizoid personality, although even they have trouble deciding whether Clark or Superman is the primary identity.
**RENAMED** **_ADVENTURES:_** _Adventures of Superman_ picked up the numbering from the old _Superman_ comic book, and a new _Superman_ was launched with a new number one. _Superman_ and _Adventures of Superman_ were published concurrently from 1986 through the spring of 2006, when _Adventures_ was killed and _Superman_ reclaimed its numbering.
**"WE WRITE AS":** "Dear DC Comics," _New York Times_.
**"EXCORIATED":** Email to author from John Byrne.
**"THE COMIC BOOK HERO":** "Bring Back the REAL Superman."
**"MORE BELIEVABLE":** "Cartoonists Explain Superman's New Image to His Fans."
**"DOUBLE-CROSSED":** Byrne email.
**_SUPERMAN'S_** **NUMBERS:** Miller, "Superman Sales," blog.comichron.com.
**THE PUBLISHER STOPPED:** DC stopped making public its circulation figures when it stopped mailing its comic books second class, a discount privilege that carried with it the reporting requirement.
**ADULTIFICATION:** Friedrich, Austin, and Simpson, "Up, Up and Awaaay!!!" _Time_.
**A SURVEY:** Eichenwald, "Grown-ups Gather at the Comic Book Stand," _New York Times_.
**FULL REFUND:** The black market for comics listed as destroyed was so effective that, in 1974, it was estimated that as few as a quarter of all printed comic books were actually placed for sale at retailers. Many if not most of the rest presumably were sold illicitly, then listed as destroyed so wholesalers and distributors could profit a second time by claiming a credit from publishers (Of _Comics and Men_ , 141).
**COUNTERINTUITIVE NAMES:** Julie Schwartz later acknowledged having made a "horrible mistake" by naming the older planet Earth-2 and the younger one Earth-1. "If we knew 30 years later we'd be asked these questions," he said laughing, "we'd have paid more attention" (Schwartz, "Dawn of the Silver Age," _Comics Scene Spectacular_ No. 6).
**AMONG MILLIONS:** Wolfman said his rule was "not to kill any hero who was created before I was born" (Wolfman, _Crisis on Infinite Earths_ , 7).
**"WHATEVER HAPPENED":** Moore, _Whatever Happened to the Man of Tomorrow_.
**SCHWARTZ'S GOODBYE:** Julie was on the cover of _Whatever Happened_ —with Batman, Wonder Women, and others—waving goodbye to his Man of Tomorrow.
**"GENIUS":** It was safe to call Jerry that in the 1980s, with Jack Liebowitz no longer in charge, but wouldn't have been in the 1960s. Author interview with Mark Evanier.
**KRYPTONIAN PAST:** _Superman: The Man of Steel_ , Vol. 1.
**BIRTHING MATRIX:** One could even argue (and fans did) that, having been hatched in outer space or upon arriving on Earth, Superman wasn't an alien at all but an Earthling, an American, and a Kansan.
**MARGOT KIDDER:** She says her role model for a strong-willed Lois was feminist Gloria Steinem. Author interview with Kidder.
**"IT'S A COLLECTIVE":** Author interview with Levitz.
**GOT INTO TROUBLE:** Author interviews with Waid, Elliot Maggin, and Len Wein.
**"I ADMIRE":** Mamet, _Some Freaks_ , 179.
**BIRTHDAY PARTIES:** www.capedwonder.com/dc-70. Superman's birthday, we were told as far back as 1968, was February 29. That device—having a birthday on a leap day that occurs once in four years—also was used by Orphan Annie's handlers to playfully explain why the cartoon character aged so slowly (Wells emails).
**BOOK OF ESSAYS:** Dooley and Engle, _Superman at Fifty!_ 12, 115, 170.
**CLARK KISSED LOIS:** Leslie Newman says she and her writing partner and husband, David, "snuck in our own love story there. When we wrote the scene we both had tears going down our faces." Author interview with Newman.
**"MY FEELING":** Author interview with Donner.
**"DICK DONNER SAID":** Author interview with Ilya Salkind.
**"THE MIND BOGGLES":** Mann, " 'Superman' Sequel: Flying in the Soup," _Los Angeles Times_.
**"IF I THINK":** "Margot Lois Lane Kidder," _People_.
**"TO MAKE [DONNER]":** Soderbergh and Lester, _Getting Away with It_ , 124–25.
**PAID TWICE:** Yule, _The Man Who "Framed" the Beatles_ , 305. Ilya Salkind says Warner Bros. kicked in only on the third film, but Spengler says the studio boosted Lester's salary for the second and third (author interview with Ilya Salkind and Spengler emails).
**"DECIDED NOT":** Soderbergh and Lester, _Getting Away with It_ , 125. It would have cost them $1 million to use any of the Brando footage for _Superman II_ , a cost no one wanted to pay.
**"UNCONTROLLABLE DESPAIR":** Soderbergh and Lester, _Getting Away with It_ , 130–31.
**VILLAINESS URSA:** Author interview with Sarah Douglas.
**MEMORABLE LINES:** Leslie Newman says, "We never wrote that and there's no way on Earth that line would have gotten by DC Comics."
**COSTS COULD MOUNT:** Schoell, _Comic Book Heroes of the Screen_ , 45.
**MARKETING STRATEGY:** Harmetz, "The Marketing of Superman and His Paraphernalia," _New York Times_.
**"BARELY BROKE":** Author interview with Ilya Salkind.
**"CRITICS WERE SPLIT":** Schickel, "Flying High," _Time;_ Maslin, " 'Superman II' Is Full of Tricks," _New York Times;_ Boyum, "One-Dimensional Flights of Fancy," _Wall Street Journal;_ Arnold, " 'Superman II': The Plot Weakens," _Washington Post;_ and Gasser, "Superman, What Happened to You?" _Los Angeles Times_.
**"IT WAS MAINLY":** Author interview with Ilya Salkind.
**"THE WHOLE GOOD":** Author interview with Newman.
**_SUPERMAN III_** **AS A WHOLE:** Maslin, " 'Superman III'; Reeve Joined by Pryor," _New York Times;_ Kempley, "Number III Is Not So Super, Man," _Washington Post;_ Reeve, _Still Me_ , 192; and Pryor, _Pryor Convictions and Other Life Sentences_ , 205.
**ADDED $1 MILLION:** Author interview with Ilya Salkind.
**"MAKING MONEY":** Author interview with Ilya Salkind. Pollock, Alex Salkind's lawyer, says that "overall, Alex made a lot of money.... None of us know the real numbers. I doubt that anyone but Alex knew the real numbers" (Pollock emails).
**"I'M JUST OPTIMIST":** Reagan, _The Reagan Diaries_ , 158–59. Ilya Salkind says President Reagan told him the movie "was 'very nice.' I don't know if he even saw the film" (author interview with Ilya Salkind).
**NUCLEAR WEAPONS:** Reeve's mother, real-world journalist Barbara Johnson, helped found the Coalition for Nuclear Disarmament.
**"I THOUGHT":** Reeve, _Still Me_ , 218.
**"ONE OF THE":** Kempley, "It's Recurred! It's a Pain!" _Washington Post_.
**"CHINTZY":** Maslin, " 'Superman IV: Quest for Peace,' " _New York Times_.
**"** ** _SUPERGIRL_** **WAS HUGE":** Author interview with Ilya Salkind.
**"I JUST WANTED":** Author interview with John Haymes Newton. Newton was replaced after the first season for what the media said was at least one, and perhaps all, of these reasons: The producers weren't taken with his performance, he asked for a pay raise his bosses didn't think he deserved, and he was facing a charge of drunk driving. Newton says he disputed the drunk driving charge and got it dropped, and he regrets that he left the show.
**"HE WAS":** Author interview with Stacy Haiduk.
**BERKOWITZ'S MIND:** Author interview with Stan Berkowitz.
**FOR PRODUCER JULIA:** Author interview with Julia Pistor.
**"THIS IS A":** Brennan, "A Family Feud," _Los Angeles Times_.
**LOOKING BACK:** Author interview with Ilya Salkind.
**HE HAD A:** Author interview with Gae Exton.
**HAIRLINE YOUTHFUL:** His problem was less age-related and more a result of his alopecia areata, which caused the loss of clumps of otherwise healthy hair. He also suffered from mastocytosis, a skin disease that produces lesions and intense itching (email to author from Benjamin Reeve).
**"BY THE TIME":** Author interview with Benjamin Reeve.
**"MOSTLY A":** Reeve, _Still Me_ , 199.
**HIS FATHER:** Frank Reeve emails.
**"WHEN I WAS":** Author interview with Smolinski.
**"WHAT I'M SUGGESTING":** Letter from Joanne Siegel to Steven J. Ross, February 16, 1988.
### 10. TILL DEATH DO US PART
**IT STARTED:** Recollections of the summit and its aftermath were based on author interviews with and emails from Cary Bates, Jon Bogdanove, Mike Carlin, K. C. Carlson, Chris Duffy, Dan Jurgens, Jenette Kahn, Karl Kesel, Jerry Ordway, Frank Pittarese, Louise Simonson, and Martha Thomases.
**COULD TAKE YEARS:** Some Superman writers worried that, given the quick turnover rate in the comic book business, they would be gone by the time the momentous wedding happened.
**"NEVER SAY":** McTernan, "Superman to Die Saving Metropolis," Cleveland _Plain Dealer_.
**"HOW DARE":** Author interview with Kahn.
**DURACELL BATTERIES:** Elliott, "Always a Place for Superman," _New York Times_.
**KILLING HIM:** Jurgens, Ordway, et al., _The Death of Superman_.
**THE CLIMAX:** The average number of panels per page was six, although Marvel artists had been making a splash with fewer pages and more single panels (Wells emails).
**"COPIES IN":** "Superman Death Issue to Go to Second Printing," _Wall Street Journal_.
**"GO ON THE":** "Look! It's a Bird! It's a Plane! It's Curtains for the Man of Steel," _New York Times_.
**"SUPERMAN," HE:** Rich, "Term Limit for the Man of Steel," _New York Times_.
**MORE HEADS OF STATE:** Stern, _The Death and Life of Superman_ , 174.
**"GOD? 'S ME":** _World Without a Superman_ , 46.
**MONTHS OFF:** To satisfy fans and DC bean counters, the company published a series of other comics during that downtime, including _Legacy of Superman_ No. 1.
**"IF THIS MANY":** Carlin email.
**"NOW," HE:** Zinn, "It's a Bird, It's a Plane—It's a Resurrection," _Business Week_.
**"THAT BECAME":** Author interview with Kahn.
**"HE SAID":** Sangiacomo, "Superman Creator, Siegel, 81, Is Dead," Cleveland _Plain Dealer_.
**INTERESTED IN LOIS:** Jenette Kahn, the power behind the scenes, says her first treatment for the TV show was entitled "Lois Lane's _Daily Planet_." She also says she decided not to have her hero fly because the limited special effects available back then would have made it look "cheesy."
**"I DIDN'T WANT":** Author interview with Deborah Joy LeVine.
**ADVERTISING MONEY:** Gordon, "Superman on the Set," _Quality Popular Television_ , 149.
**NETWORK PURPOSEFULLY:** Gordon, "Superman on the Set," 151.
**EARTH-2:** In his 1985 _Crisis on Infinite Earths_ , Marv Wolfman carried this Earth-2 couple to an alternate dimension, where they could carry on without getting in the way of the Earth-1 couple and the newly simplified DC universe.
**ISSUE CALLED:** Superman writers and artists past and present, "The Wedding Album," _Superman: The Wedding Album_ No. 1.
**_MOONLIGHTING_** **EFFECT:** Gordon, "Superman on the Set," 149–50; Flint and Snierson, " 'Clark' Canned," _Entertainment Weekly;_ and author interview with Kahn.
**"I WAS THAT":** Hatcher, _Burnt Toast and Other Philosophies of Life_ , 191.
**CAIN HAD LESS:** "Dean Cain," _People;_ Perigard, "Raising Cain," _Boston Herald;_ and Jacobs, "Citizen Cain," _Entertainment Weekly_.
**"SUPERMAN: THE ESCAPE":** Wharton, " 'Superman' Ride Still Grounded," _Los Angeles Times_. The Escape tied for the world's fastest ride with Tower of Terror II at Dreamworld Theme Park in Australia.
**"IN VIRTUAL":** Herz, "It's a Bird! It's a Plane! (And It Wobbles?)," _New York Times_.
**GUEST STARRING:** Dooley and Engle, _Superman at Fifty!_ 182; www.seinfeldscripts.com; and "Seinfeld Meets a Really 'Super' Salesman," _New_ _York Times_. The American Express commercials—"A Uniform Used to Mean Something" and "Hindsight Is 20/20"—each lasted four minutes.
**"HE IS A CITIZEN":** "To the Rescue: Superman's Big Mission in Bosnia," _Time for Kids_.
**THE FIRST BOOK:** Levitz, _75 Years of DC Comics_ , 574.
**"COMIC ART" AUCTIONS:** Lyne, "The Executive Life," _New York Times;_ and "2 Comic Books Auctioned for $100,000," _New York Times_.
**"A BATTLER FOR":** _Kingdom Come_ , 210.
**THE REAL AIM:** Author interviews with Waid and Maggin.
**FOR HIM, THOUGH:** Author interview with Jeph Loeb.
**"I DON'T WANT":** Email to author from Chris Clow.
**JO JO KAMINSKI:** Author interview with Ordway.
**NAMES OF GIRLS:** Author interview with Maggin.
**SUCH STORIES WERE:** Wells emails.
**WHITE AND:** Around this time, DC was collaborating with the new Milestone Media to distribute multicultural comic books featuring black superheroes like Hardware and Static.
**"STEEL WAS":** Author interviews with and emails from Simonson, Christopher Priest, and Kahn.
**"USED TO WATCH":** Gates, "A Big Brother from Another Planet," _New York Times_.
**"THERE WAS SOMETHING":** Author interview with Al Roker.
**SHAQUILLE O'NEAL:** Mead, "A Man-Child in Lotusland," _The New Yorker_.
**THREE-PART SERIES:** _Superman: The Man of Steel_ No. 81; and author interview with Bogdanove.
**MIXED BALANCE SHEET:** Jensen, "Dead Superman May Revive DC Comics," _Advertising Age;_ Lev, "Reaching Beyond the Ghouls and Gore for Major Payoffs," _New York Times;_ Rhoades, _A Complete History of American Comic Books_ , 129; Gabilliet, _Of Comics and Men_ , 151; and Chang, "SPLAAAAAAAT!," _Los Angeles Times_.
**GALLUP POLL:** Hugick, "Public to DC Comics: Resurrect Superman!" The Gallup Poll News Service.
**IT WASN'T JUST:** Emails to author from Bill Necessary, Wurzelbacher, and Ken Cholette.
**JOE HAD BEEN:** Mietkiewicz, "Great Krypton!" _Toronto Star_.
**"I WAS SHOCKED":** Letter from Jean Shuster Peavy to Marty Payson, August 21, 1992.
**"THIS IS CALLED":** Reeve, _Still Me_ , 15.
**" 'WHAT IS A HERO?' ":** Reeve, _Still Me_ , 267.
**"SHE WAS FRIGHTENED":** "Margot Kidder Is Hospitalized for Psychiatric Observation," _New York Times_.
**"IT'S THE FIRST":** Author interview with Kidder.
**"THE FACT THAT":** Ordway emails.
**"THERE WAS AN":** Author interview with Levitz.
**"MORT WEISINGER":** Author interview with Evanier.
**KEPT A SEAT:** Nash, "Jack Liebowitz, Comics Publisher, Dies at 100," _New York Times_.
### 11. TIGHTS AND FIGHTS
**"WE MADE NO":** Author interview with Al Gough.
**"WE WERE VERY":** Author interview with Ken Horton.
**"JUST THE RIGHT":** Carson, "Small Comforts," _Esquire_.
**"I LOVE IT":** Hiatt, "Lex-Man," _Entertainment Weekly_.
**JONES LEFT:** He was arrested in 2009 by the Drug Enforcement Administration on trafficking charges, pled guilty in 2010, and the next year was sentenced to a year in prison.
**BROADER AND DEEPER:** Scivally, _Superman on Film_ , 151–52.
**"WE WERE WINKING":** Author interview with Gough.
**" 'SMALLVILLE' IS ONE":** Hinson, "Getting to the Heart of a Hero," _New York Times_.
**"SEEING THE SUPERMAN":** Carson, "Small Comforts."
**"FINALLY, CLARK":** Jensen, "Shows of Strength," _Entertainment Weekly_.
**A RECORD:** The cost would have been even higher if the studio hadn't received nearly $20 million in tax credits in Australia, where most of the filming was done, and if it hadn't canceled plans for a $20 million construction of Metropolis intended to be used afterward as a theme park. It reportedly cost $50 just to grow an ear of corn for the film. Jensen, "Greatest American Hero?" _Entertainment Weekly_.
**"A MODERN BLENDING":** Lewellen expert testimony, _Siegel and Larson v. Warner Bros. Entertainment_.
**REMAINS OF KRYPTON:** The next year, real-life scientists in Serbia found a new mineral whose chemistry matched the material described in the movie (Gustines, "Bad News for Superman," _New York Times_ ).
**"I GAVE MYSELF":** Swanson, "Super Troupers," _Premiere_.
**"I CAN'T TELL":** Bowles, " 'Superman' Torch Is Passed," _USA Today_.
**"OUR GEORGE":** Rhodes, "The Continuing Adventures," _New York Times_.
**WATCH HIM FLY:** Grove, "Singer Was Man of Steel," _Hollywood Reporter_.
**AT A COST:** Swanson, "Super Troupers."
**"WRITING A STORY":** Author interview with Michael Dougherty.
**"MY GRANDMOTHER":** Singer, Dougherty, and Harris, _Superman Returns: The Complete Shooting Script_ , 23.
**"IT'S INDESCRIBABLE":** Author interview with Dougherty.
**"I WAS PRACTICALLY":** Author interview with John Ottman.
**"SUPERHEROES—LET'S":** Duralde, "How Gay Is Superman?" _Advocate_.
**"MOST HETEROSEXUAL":** Jensen, "Greatest American Hero?"
**TOO GLOBAL:** That debate over whether Superman is all-American or all-world has surfaced repeatedly over the decades. Superman himself answered it best in 1961 when, in response to being enrolled as an honorary citizen of all the member countries of the United Nations, he said: "What an honor! But of course my main loyalty will always be to the United States, where I grew up!" ( _Superman_ No. 146).
**"WARNER BROTHERS":** "Superman and the Culture War," billoreilly.com.
**"THERE'S NO REASON":** Lundegaard, "Truth, Justice and (Fill in the Blank)," _New York Times_.
**MISHAPS:** Swanson, "Super Troupers."
**"AT ONE POINT":** Author interview with Bryan Singer.
**"THEY'RE VERY IMPORTANT":** Singer, Dougherty, and Harris, _Superman Returns: The Complete Shooting Script_ , 10, 13.
**"** ** _SUPERMAN RETURNS_** **":** Author interview with Singer.
**"EARLIER VERSIONS":** Corliss, "The Gospel of Superman," _Time_.
**"STAYED VERY MUCH":** Author interview with Donner.
**"OFFERS NOT SO":** Lane, "Kryptology," _The New Yorker_.
**"FIDELITY IS ONE":** D'Angelo, "Man, Yes; Super, Not Really," _Las Vegas Weekly_.
**"LEADEN":** Dargis, "Superman Is Back," _New York Times_.
**PRODUCTION COSTS:** Warner Bros. said it suffered a net loss of $81 million on _Superman Returns_ ( _Siegel and Larson v. Warner Bros. Entertainment_ ).
**"WAS A VERY":** Eller, "Picture This," _Los Angeles Times_.
**CASHED IN:** Johannes, "Superman Soars," _Promo Magazine;_ Rossen, _Superman vs. Hollywood_ , 290; and Holson, "More Than Ever, Hollywood Studios Are Relying on the Foreign Box Office," _New York Times_.
**"WE WERE STABBED":** Joanne Siegel letter to Richard D. Parsons, May 9, 2002.
**LEGAL FILING:** _DC Comics v. Pacific Pictures Corporation_ , 2010 (complaint, response, counter-complaint and response, press release); Cieply and Barnes, "Warner Brothers Sues 'Superman' Lawyer," _New York Times;_ and Cieply, "Lawyer Battles Back Against DC Comics in Superman Dispute," _New York Times_.
**CEASE-AND-DESIST:** The letter, from a Florida attorney claiming he represented Joanne and Laura, told Lois Amster to stop claiming she was the model for Lois Lane. But "she never claimed that," says Amster's son, Paul Rothschild. "After my brother sent him a lawyer's letter, we haven't heard from him since," Rothschild adds. "We don't know what his point was unless it was related to their suit against DC Comics" (author interview with and emails from Rothschild).
**PRELIMINARY RULINGS:** _Siegel and Larson v. Warner Bros. Entertainment_.
**EVEN AS:** _Siegel and Larson v. Warner Bros. Entertainment_.
**"THE WHOLE PURPOSE":** Author interview with Marc Toberoff.
**END THE SUPERMAN:** The concern is that once the Shuster heirs enter the case, DC and Warner Bros. would face such uncertainty over their rights and their profits that they would no longer produce Superman movies, TV shows, or even comic books. Arguing against that is the presumption that Toberoff and his clients, short of trying to produce their own movies or other products—and worrying whether that would violate Warner's remaining rights to the character, scaring off any other studios or publishers that might have been interested—would want to reach a settlement to ensure that the Superman franchise they were getting a share of continues to earn money. The problem would come if both sides presume they can either stare down the other or win in a knock-down, drag-out battle through the courts.
**"THE NOTION THAT":** Author interview with Toberoff.
**TOBEROFF ALSO TRIED:** _Siegel and Larson v. Warner Bros. Entertainment;_ and email from Don Bulson, Michael Siegel's lawyer in Cleveland. "Michael was interested in settlement, but the settlement discussions with DC Comics/Time Warner were controlled by Joanne and Laura as they owned a 75% interest," Bulson wrote. "This remained the same after the settlement discussions with DC Comics/Time Warner stalled."
**STOPPED PAYING:** Bella filed suit in Cleveland for nonpayment of child support, and the judge agreed that Jerry should pay up. Because Jerry was living in New York, the Ohio judge sent the petition on to the authorities in New York State ( _Bella Siegel vs. Jerome Siegel_ , "Judge's Journal Entry"). Michael Siegel said Jerry never did pay the child support he and Bella were due and that Jerry "broke all contact with me" after he and Bella were divorced (Michael Siegel emails to Mark Waid, 2005).
**MICHAEL BECAME:** Michael told Waid that he owned a business, apparently plumbing supply, got a college education, and had lived in places other than greater Cleveland. He sent Waid parts of Jerry's will showing that Jerry left everything to his wife, Joanne, and if she wasn't alive to his daughter, Laura, and her offspring. "If they all die and anything is left," Michael wrote, "I can have that if I am still alive." Michael added that it was ironic that Jerry, who made his living from comic books sold to children, treated his own son so poorly. "Why," Michael asked, "did he ignore me for almost my entire life?" (Michael Siegel emails to Waid).
**NATIVE SON:** Joe also got a commemorative plaque designating the street where he lived Joe Shuster Lane, with Lois Lane connecting Jerry's street to Joe's. Joanne never was able to get the permanent memorial for Jerry that she wanted in Cleveland and to which she promised to donate his typewriter, scripts, glasses, and half of his ashes. Cleveland fans are working on a Superman exhibit for the airport there and hoping to unveil Superman license plates. They and others also raised $110,000 to restore Jerry's house, and they set up a Siegel and Shuster Society (Sangiacomo, "Superman Creator's Widow Seeks Memorial," Cleveland _Plain Dealer;_ Sangiacomo, "Joanne Siegel Dies," Cleveland _Plain Dealer;_ author interview with Jamie Reigle).
**JERRY'S WILL:** The irony of Michael Siegel's death is that his half sister, Laura—someone he never had any relationship with and was jealous of—inherits his estate, because he had no will and she is his closest living relative (Probate Court documents, Cuyahoga County, Ohio, 2006). The estate was valued at $250,000—but that didn't include Michael's share of whatever settlement there is in the lawsuit against DC Comics, which would likely be worth millions.
**WARREN PEARY WILL:** Warren is the executor of Joe Shuster's estate, and it is in that capacity that he is trying to reclaim the copyright to Superman. The sole beneficiary of that estate, however, is Warren's mother, Jean Shuster Peavy, who suffered a severe stroke in recent years and has other health issues.
**CONTINUED TO SAG:** "DC Comics Month-to-Month Sales," www.comicsbeat.com; and " _Superman_ Sales Figures."
**REMAINING AUDIENCE:** Gabilliet, _Of Comics and Men_ , 208, 357. Just how much a part of the culture comic books have been is suggested by the fact that, over the last seventy years, more than 150,000 individual issues have been published in America.
**IN THE PHILIPPINES:** Gayle, "Obsessed Superman Fan Has Cosmetic Surgery to Look Like His Hero," dailymail.co.uk.
**COLLECTORS STILL:** Sanchez, "Superman Comic Saves Family Home from Foreclosure," abcnews.go.com; and "Rare Superman Comic Sells for Record $2.16M US," cbc.ca.
**FAT PRICES:** Hake, _Official Price Guide to Pop Culture Memorabilia_.
**WORLD AND OURSELVES:** It was easier to contemplate Superman as a Russian after we had won the Cold War.
**CENTRAL PARK AND:** This is a paraphrase of Denny O'Neil's quip that Gotham is Manhattan below 14th Street at 3 A.M., November 28, in a cold year. Metropolis is Manhattan between 14th and 110th streets on the brightest, sunniest July day of the year.
Actor Michael Caine says Superman is how America sees itself, Batman is how the rest of the world sees us.
**CARY BATES:** At twenty years, Bates is the longest-serving Superman writer, versus seventeen for Jerry Siegel and sixteen for Alvin Schwartz. Bates actually reached twenty-two years if we count his earliest year selling DC story ideas for Superman, and his most recent Superman story—"The Last Family of Krypton"—in 2010. The longest-lasting artist, hands-down, was Curt Swan, at thirty-eight years.
**RENOUNCE HIS AMERICAN:** _Action_ 900, which unleashed a firestorm of criticism when Superman said he planned to renounce his citizenship, was actually a story about his affirming his global connections, the way he had been almost since the beginning. It also was about his standing shoulder-to-shoulder with human rights demonstrators in a repressive Iran and, by extension, the rest of the Middle East and the planet. And while he might have been distancing himself from the American government, it is unlikely he would ever move away from the American people ("The Incident," _Action Comics_ No. 900). Goyer, the writer of this comic book, also is the screenwriter for the new Superman movie due out in 2013.
**"HOLD FAST TO":** Hughes, _Collected Poems_ , 32.
**COMICS WAS SUBSUMED:** Familiar figures remain in charge of key divisions, with chief creative officer Geoff Johns running the comic-books-to-movies operation, and co-publishers Dan DiDio and Jim Lee in charge of comic books. Here is what today's corporate ladder looks like: Time Warner, Inc., is up top. Warner Bros. Entertainment is one of its divisions, along with Time, Turner Broadcasting, and Home Box Office. Under Warner is DC Entertainment, under that is DC Comics, and helping hold it (and everything else) up is Superman.
# **Bibliography**
### INTERVIEWS, CORRESPONDENCE, AND UNPUBLISHED DOCUMENTS
From 2008 to 2011, the author interviewed or exchanged emails with the following Superman writers, artists, editors, scriptwriters, actors, directors, producers, biographers, bloggers, collectors, marketers, fans, and others: Jack Adams, Neal Adams, Jack Adler, Jim Amash, Lois Amster, Bob Andelman, Murphy Anderson, Luis Augusto, Roger Austin, Dick Ayers, Cary Bates, Bart Beaty, Jim Beaver, Robert Beerbohm, Douglas Belkin, Robert Benton, Ofer Berenstein, Stan Berkowitz, Murray Bishoff, Bill Blackbeard, Jon Bogdanove, Judy Bogdanove, Kal-El Bogdanove, Bobby Bonilla, Celine Bonnin, Rick Bowers, Chris Brockow, Nicky Wheeler-Nicholson Brown, Michael Bryan, Peggy A. Bulger, Don Bulson, Kurt Busiek, John Byrne, K. Callan, Mike Carlin, K.C. Carlson, David Chantler, Roberto Chavarria, Ellen Chesler, Ken Cholette, Nick Cirignano, Joshua Clark, Chris Clow, Phyllis Coates, Neil Cole, Chuck Colleta, Toni Collins, Justin Cousson, Rennie Cowan, John Cush, Bob Daly, Les Daniels, Geoff Darrow, Eric Lief Davin, Dwight Decker, J. M. De-Matteis, Sylvain Despretz, Bruce Dettman, Carole Donenfeld, Sonia "Peachy" Donenfeld, Richard Donner, Michael Dougherty, Robert Dougherty, Sarah Douglas, Chris Duffy, Randy Duncan, Jeff East, Jay Emmett, Mark Evanier, Gae Exton, Rob Falcone, Ruth Fine Farber, Jules Feiffer, Michael Feldman, Al Feldstein, Irv Fine, Jerry Fine, Andy Fogelson, Rob Friedman, Jim Galton, Bert Gibbs, Jackson Gillis, Dick Giordano, Robert Goldstein, Al Gough, Michael Green, Robert Greenberger, Larry Greenfield, Phil Grom, Gary Grossman, Jack Guidera, Kevin Gunn, Stacy Haiduk, David Hajdu, Carole Handler, Chuck Harter, Craig Harvey, Irwin Hasen, Michael Hayde, Dennis Hays, Jan Alan Henderson, Dan Hirschfeld, Jane Hitchcock, Bob Holiday, Ken Horton, Rita Huber, Bob Hughes, Ellen Roney Hughes, Stuart Immonen, Carmine Infantino, Al Jaffee, Geoff Johns, Barbara Johnson, Gerard Jones, Dan Jurgens, Joyce Kaffel, Jenette Kahn, Truman Frederick Keefer, Stetson Kennedy, Karl Kesel, Margot Kidder, Orion Kidder, Rose Kirsh, Denis Kitchen, Sidney Kiwitt, Austin Knill, Lisa Kraemer, Joe Kubert, Eric Landriault, Jack Larson, Gene LeBell, Stan Lee, Deborah Joy LeVine, Paul Levitz, Joan Levy, Jeph Loeb, Edward Lozzi, Janet Luez, Dusty Luker, Richard Lupoff, Peter Lupus, Ernesto Machado, Elliot Maggin, Russ Maheras, Eugene Maletta, Tom Mankiewicz, Joe Mararinno, Mo Marcus, Eugene Mark, Ron Massengill, Harry Matetsky, Vincent Maulandi, Scott McCloud, Marc McClure, Shawn McGaughey, Brian McKernan, Daniel Meerkamper, Brad Meltzer, Todd Michney, John Jackson Miller, Erol Molnar, Kevin Moriarity, Will Murray, Carl Murway, Bill Necessary, Noel Neill, Jerry Newingham, Leslie Newman, John Haymes Newton, Jim Nolt, Paul Oddi, Luke Oldfield, Denny O'Neal, Bob O'Neil, Keith O'Neil, Jerry Ordway, John Ottman, Stephanie Shayne Parkin, Derrick Patry, Mark Warren Peary, Dawn Peavy, Jean Shuster Peavy, Zoran Perisic, Brian Peterson, Julia Pistor, Frank Pittarese, Sophie Plageman, Al Plastino, Jeanette Pollack, Tom Pollock, Christopher Priest, Hal Prince, Dave Prowse, Karen Pryor, Emilio Ramos, Jr., William Raymer, Gene Reed, Benjamin Reeve, Frank Reeve, Jamie Reigle, Trina Robbins, Mike Roberts, Jerry Robinson, Al Roker, John Romita, Conor Rooney, Bud Rosenthal, Paul Rothschild, Bob Rozakis, Jacob Rubinstein, Ilya Salkind, Michael Sangiacomo, Lee Saylor, Bill Schelly, Mary Schrank, Alvin Schwartz, Lew Sayre Schwartz, Terry Semel, Fred Shay, Jim Shooter, Rosie Shuster, Craig Shutt, Dave Siegel, Louise Simonson, Thol Simonson, Walt Simonson, Bryan Singer, Aaron Smolinski, Kelly Souders, Merrill Sparks, Pierre Spengler, Lynn Stallmaster, Benjamin Stevens, Robert Strankman, Sidney Strauss, Charles Strouse, Mike Stumbo, Trina Swuff, Craig Tenney, John Tenuto, Mary Jo Tenuto, Roy Thomas, Martha Thomases, Marc Toberoff, Anthony Tollin, Jeff Trexler, Michael Uslan, Mark Waid, Maya Warburg, Len Wein, John Wells, Ted White, Margery Wieder, Bart Williams, Bill Williams, John Williams, Tony Wilson, Norma Wolcov, Marv Wolfman, Donald Wurzelbacher, Alice Wyner, Craig Yoe, and Steve Younis.
Unless otherwise noted, the documents below are part of the public record of court cases filed by Jerry Siegel and his heirs against National Comics Publications and its successors, or were provided privately to the author.
Donenfeld, Harry. Letter to Jerry Siegel and Joe Shuster, September 22, 1938. Ellsworth, Whitney. Letter to DC Staff. "Editorial Policy for Superman-DC Publications," Undated.
———. Letter to Jerry Siegel, January 22, 1940.
———. Letter to Jerry Siegel, February 21, 1940.
———. Letter to Jerry Siegel, November 4, 1940.
———. Letter to Jerry Siegel, February 19, 1941.
Fingeroth, Danny, Gerard Jones, Paul Levitz, and Nicky Wheeler-Nicholson Brown. "Grandchildren of the Golden Age." Panel discussion at the New York Comic Con. October 9, 2010.
Greenberg, Marc H. "Comics, Courts & Controversy: The Cases of the Comic Book Legal Defense Fund & the Superman Copyright Litigation." Paper presented at the annual Comic Arts Conference Comic-Con, San Diego, California, July 22, 2009.
Grossman, Gary. Transcripts of undated interviews with Whitney Ellsworth, based on recordings provided to author by Gary Grossman.
Hitchcock, Jane. Unpublished eulogy for her mother, Joan Alexander. 2009. Courtesy of Jane Hitchcock.
Kidder, Orion. "Telling Stories About Storytelling: The Metacomics of Alan Moore, Neil Gaiman, and Warren Ellis." Ph.D. diss., University of Alberta, 2010.
Lewis, Dorothy. Letter to Josette Frank. November 13, 1947. Child Study Association of America Files. Elmer L. Andersen Library, University of Minnesota.
Liebowitz, Jack S. Letter to Jerry Siegel, March 1, 1938.
———. Letter to Jerry Siegel, September 28, 1938.
———. Letter to Jerry Siegel, January 23, 1940.
———. Letter to Jerry Siegel, January 29, 1940.
———. Letter to Jerry Siegel, July 13, 1946.
———. Letter to Jerry Siegel, February 3, 1947.
———. Unpublished memoir. 1993. Courtesy of Joan Levy.
Maxwell, Robert. Letter to George Zachary, Office of War Information. April 12, 1943. Child Study Association of America Files. Elmer L. Andersen Library, University of Minnesota.
National Periodical Publications. Agreement with Alex and Ilya Salkind. November 6, 1974.
Peavy, Jean Shuster. Letter to Marty Payson, August 21, 1992.
Puzo, Mario and Carmine Infantino. Script Conference on _Superman: The Movie_. July 16, 1975.
Rosen, Leo. Letter to Gabriel Kaslow, April 15, 1952.
Salkind, Ilya, Mark Jones, and Cary Bates. "Superman Reborn." August 3, 1992. Unpublished screenplay.
Saylor, Lee. Transcript of 1989 interview with Leonore Lemmon, based on recording provided to author by Lee Saylor.
Shuster, Joe. Letter to Neal Adams, November 24, 1975.
———. Letter to Jay Emmett, November 23, 1975.
Siegel, Jerry. _Creation of a Superhero_. Unpublished memoir. 1978.
———. Letter to Harry Donenfeld, March 29, 1946.
———. Letter to Russell Keaton, July 12, 1934.
———. Letter to Jack Liebowitz, December 6, 1937.
———. Letter to Jack Liebowitz, February 1, 1940.
———. Letter to Jack Liebowitz, January 1, 1944.
———. Letter to Jack Liebowitz, July 13, 1946.
———. Letter to Jack Liebowitz, September 18, 1946.
———. Letter to Jack Liebowitz, November 11, 1946.
———. Letter to the President of Heilbrunn Associates. December 2, 1967.
———. Letter to Joe Shuster, September 18, 1946.
———. Letter to Laura Siegel. November 21, 1976.
———. _The Life and Times of Jerry Siegel_. Unpublished memoir.
———. _The Story Behind Superman No. 1_. Unpublished memoir.
Siegel, Jerry, and Joe Shuster. Letter to _Detective Comics_ , March 1, 1938.
———. Letter to Harry Donenfeld, May 8, 1940.
Siegel, Joanne. Letter to Richard D. Parsons, AOL Time Warner, May 9, 2002.
———. Letter to Steven J. Ross, February 16, 1988.
———. Letter to unknown recipient. May 25, 1992.
Siegel, Michael. Correspondence with Mark Waid. 2005. Courtesy of Mark Waid.
Straus, Mrs. Hugh Grant. Letter to Editor. _PM_. May 20, 1946. Child Study Association of American Files. Elmer L. Andersen Library, University of Minnesota.
Wheeler-Nicholson, Malcolm. Agreements with Donny Press, Inc., World Color Printing Co., and Photochrome, Inc. November 15, 1937.
———. Letter to Donny Press, November 15, 1937.
———. Letter to Jerry Siegel, October 4, 1935.
———. Letter to Jerry Siegel, May 13, 1936.
Wylie, Philip. Letter to J. Randolph Cox, January 28, 1970. Courtesy of Truman Frederick Keefer.
Yanes, Nicholas. "Graphic Imagery: Jewish American Comic Book Creators' Depictions of Class, Race, and Patriotism." Master's thesis. Florida State University, 2008.
Zachary, George. Letter to Allen Ducovny, April 3, 1943. Child Study Association of America Files. Elmer L. Andersen Library, University of Minnesota.
### COURT CASES
_City of New York v. Kingsley Books, Inc_. Supreme Court of New York. June 13, 1955.
_In Re. Estate of Michael Siegel_. Probate Court documents, Cuyahoga County (Ohio). March 13, 2006, March 16, 2006, and May 12, 2006.
_DC Comics v. Pacific Pictures Corporation et al_. United States District Court, Central District of California. May 14, 2010.
_Detective Comics, Inc., vs. Bruns Publications, Inc_. United States Court of Appeals for the Second Circuit. November 10, 1939.
_National Comics Publications, Inc., v. Fawcett Publications, Inc._ , U.S. Court of Appeals Second Circuit. September 5, 1952.
_Joseph Michael Shuster vs. Judith Ray Calpini_. Request and Declaration for Final Judgment of Dissolution of Marriage. Superior Court of California, County of Los Angeles. April 15, 1981.
_Bella Siegel vs. Jerome Siegel_. Petition for Divorce, July 14, 1948.
_Bella Siegel vs. Jerome Siegel_. Petition for Support. "Judge's Journal Entry," June 9, 1953.
_Jerome Siegel and Joseph Shuster v. National Comics Publications, Inc., et al_. Supreme Court of New York, Westchester County. Final Judgment. May 21, 1948.
_Jerome Siegel and Joseph Shuster v. National Comics Publications, Inc., et al_. U.S. District Court, Southern District of New York. October 18, 1973.
_Joanne Siegel and Laura Siegel Larson v. Warner Bros. Entertainment, Inc., et. al_. United States District Court, Central District of California. October 8, 2004.
### GOVERNMENT DOCUMENTS
City of Cleveland. Department of Health. "Certificate of Death, Michel Siegel." June 4, 1932.
———. Department of Public Health. "Birth Certificate for Jalon Kovacs." December 5, 1917.
———. Police Department. "Casualty Report for Michael Siegel." June 3, 1932.
Cuyahoga County (Ohio). Application for Marriage License. Jerome Siegel and Jolan Kovacs. October 13, 1948.
———. Application for Marriage License. Jerome Siegel and Jolan Kovacs. November 3, 1948.
———. Coroner's Office. "Coroner's Report for Michael Siegel." June 3, 1932.
City of Los Angeles. Police Department. Investigation Report. June 16, 1959.
County of Los Angeles. Archive Autopsy Report for George Reeves. Case Number 1959—45426.
———. Department of Chief Medical Examiner-Coroner. Medical Report. June 23, 1959.
———. Office of County Coroner. Autopsy Report. June 24, 1959.
State of Ohio. Department of Health. "Certificate of Death for Michael Siegel." June 4, 1932.
County of San Diego. Certificate of Registry of Marriage. "Joseph Michael Shuster and Judith Ray Calpini." December 29, 1976.
United States Bureau of the Census. Julius Schuster (1930).
United States Bureau of the Census. Moses Sigel (1910). Michael Siegel (1920). Michael Sigel (1930).
United States Immigration and Naturalization Service. Naturalization form, Michel Siegel. February 16, 1922.
United States Postal Service. Press release. "The Man of Steel and Other Super Heroes Take Flight as Stamps and Stamped Postcards." San Diego, Calif. July 20, 2006.
### COMICS
Unless otherwise noted, the comic books below were published by National Periodical Publications and its successors.
_Action Comics_ No. 1 (June 1938). Written by Jerry Siegel.
"The Amazing Story of Superman-Red and Superman-Blue," _Superman_ No. 162 (July 1963). Written by Leo Dorfman.
"The Archer." _Superman_ No. 13 (November–December 1941). Written by Jerry Siegel.
_The Battle of the Century: Superman vs. the Amazing Spider-Man_ (1976).
Published by National Periodical Publications and Marvel Comics Group. Written by Gerry Conway.
"The Bizarro Invasion of Earth!" _Superman_ No. 169 (May 1964). Written by Jerry Siegel.
_Detective Comics_ No. 38 (April 1940). Written by Jerry Siegel.
"Doctor Occult." _More Fun Comics_ No. 16 (December 1936). Written by Jerry Siegel.
"Europe at War." _Action Comics_ No. 23 (April 1940). Written by Jerry Siegel.
"Fury of the Energy-Eater." _Superman_ No. 258 (November 1972). Written by Len Wein.
"I Am Curious (Black)!" _Superman's Girl Friend Lois Lane_ No. 106 (November 1970). Written by Robert Kanigher.
"The Incident." _Action Comics_ No. 900 (June 2011). Written by David S. Goyer.
_Just Imagine Stan Lee's Superman_ (2001). Written by Stan Lee.
"King of the Comic Books." _Superman_ No. 25 (November–December 1943). Written by Jerry Siegel.
"The Last Earth-Prime Story." _Superman_ No. 411 (September 1985). Written by Elliot S. Maggin.
"The Master Mesmerizer of Metropolis!" _Superman_ No. 330 (December 1978). Written by Martin Pasko.
"The Million-Dollar Marathon." _Action Comics_ No. 65 (October 1943). Written by Don Cameron.
"Mr. and Mrs. Clark (SUPERMAN) Kent!" _Superman's Girl Friend Lois Lane_ No. 19 (August 1960).
"Murder Plunge." _More Fun Comics_ No. 68 (June 1941). Written by Jerry Siegel.
"The Mxyztplk-Susie Alliance." _Superman_ No. 40 (May–June 1946). Written by Don Cameron.
"The Origin of Stuporman." _Not Brand Echh_ No. 7 (April 1968). Published by Marvel Comics.
"The Origin of Superman." _Superman_ No. 53 (July–August 1948). Written by Bill Finger.
"The Satanic Schemes of S.K.U.L." _Superman's Girl Friend Lois Lane_ No. 63 (February 1966). Written by Leo Dorfman.
"The Soldier of Steel!" _Superman_ No. 216 (May 1969). Written by Robert Kanigher.
"The Super-Family of Steel." _Superman's Girl Friend Lois Lane_ No. 15 (February 1960). Written by Edmond Hamilton.
_Superman_ No. 1 (Summer 1939). Written by Jerry Siegel.
"Superman and His Friends Around the World!" _Superman's Pal Jimmy Olsen_ No. 113 (August/September 1968). Written by Mort Weisinger.
"Superman Breaks the Wall!" _Superman: The Man of Steel_ No. 82 (August 1998). Written by Louise Simonson and Jon Bogdanove.
"Superman, Champion of the Oppressed." _Action Comics_ No. 1 (June 1938). Written by Jerry Siegel.
"Superman Goes to Prison." _Action Comics_ No. 10 (March 1939). Written by Jerry Siegel.
"Superman in the Ghetto!" _Superman: The Man of Steel_ No. 81 (July 1998). Written by Louise Simonson and Jon Bogdanove.
"Superman in the Slums," _Action Comics_ No. 8 (January 1939). Written by Jerry Siegel.
"Superman Joins the Circus." _Action Comics_ No. 7 (December 1938). Written by Jerry Siegel.
"Superman Meets Al Capone." _Superman_ No. 142 (January 1961). Written by Otto Binder.
"Superman Returns to Krypton." _Superman_ No. 61 (November–December 1949). Written by Bill Finger.
"Superman Scores Again." Undated comic strip. Written by Jerry Siegel.
"Superman Takes a Wife." _Action Comics_ No. 484 (June 1978). Written by Cary Bates.
"Superman: The Computers That Saved Metropolis." _Action Comics_ No. 509 (July 1980). Written by Cary Bates.
_Superman: The Wedding Album_ No. 1 (December 1996). By Superman writers and artists past and present.
"The Superman vs. The Atomic Skull!" _Superman: The Man of Steel_ No. 80 (June 1998). Written by Louise Simonson and Jon Bogdanove.
"Superman's Mission for President Kennedy." _Superman_ No. 170 (July 1964). Written by Bill Finger and E. Nelson Bridwell.
"Superman's Return to Krypton!" _Superman_ No. 141 (November 1960). Written by Jerry Siegel.
"War in San Monte." _Action Comics_ No. 2 (July 1938). Written by Jerry Siegel.
### COMIC ANTHOLOGIES AND GRAPHIC NOVELS
Binder, Otto, et al. _Showcase Presents: Superman_ , Vol. 1. New York: DC Comics, 2005.
Byrne, John. _Superman: The Earth Stealers_. New York: DC Comics, 1988.
———. _Superman: The Man of Steel_ , Vol. 1. New York: DC Comics, 1991.
———. _The World of Smallville_. New York: DC Comics, 1987.
_DC's Greatest Imaginary Stories_. New York: DC Comics, 2005.
Gold, Mike. _The Greatest Golden Age Stories Ever Told_. New York: DC Comics, 1990.
_The Golden Age Spectre Archives_ , Vol. 1. New York: DC Comics, 2003.
Johns, Geoff, and Richard Donner. _Superman: Last Son_. New York: DC Comics, 2006.
Jurgens, Dan, Karl Kesel, et al. _World Without a Superman_. New York: DC Comics. 1993.
Jurgens, Dan, Jerry Ordway, et al. _The Death of Superman_. New York: DC Comics, 1993.
———. _The Return of Superman_. New York: DC Comics, 1993.
Jurgens, Dan, Louise Simonson, et al. _Superman: The Death of Clark Kent_. New York: DC Comics, 1997.
Kelly, Joe, Marv Wolfman, Geoff Johns, and Jeph Loeb. _Superman: Infinite Crisis_. New York: DC Comics, 2006.
Loeb, Jeph. _Superman for All Seasons_. New York: DC Comics, 1999.
———. _Superman/Batman, Volume One: Public Enemies_. New York: DC Comics, 2005.
_Lois & Clark: The New Adventures of Superman_. New York: DC Comics, 1994.
Millar, Mark. _Superman: Red Son_. New York: DC Comics, 2003.
Moore, Alan. _Superman: Whatever Happened to the Man of Tomorrow?_ New York: DC Comics, 2009.
O'Neil, Dennis. _Superman: Kryptonite Nevermore_. New York: DC Comics, 2009.
Pasko, Martin. _The DC Vault_. New York: DC Comics, 2008.
Ross, Alex, and Paul Dini. _Superman: Peace on Earth_. New York: DC Comics, 1999.
_The Superman Chronicles_ , Vol. 1. New York: DC Comics, 2006.
_The Superman Chronicles_ , Vol. 6. New York: DC Comics, 2009.
_Superman: From the Thirties to the Seventies_. New York: National Periodical Publications, 1971.
_Superman in the Forties_. New York: DC Comics, 2005.
_Superman in the Fifties_. New York: DC Comics, 2002.
_Superman in the Sixties_. New York: DC Comics, 1999.
_Superman in the Seventies_. New York: DC Comics, 2000.
_Superman in the Eighties_. New York: DC Comics, 2006.
_Superman: The Action Comics Archives_ , Vol. 1. New York: DC Comics, 1997.
_Superman: The Action Comics Archives_ , Vol. 3. New York: DC Comics, 2001.
_Superman: The Action Comics Archives_ , Vol. 4. New York: DC Comics, 2005.
_Superman: The Dailies, 1939–1942_. New York: Sterling Publishing Company, 1998.
_Superman: The Greatest Stories Ever Told_ , Vol. 1. New York: DC Comics, 2004.
_Superman: The Greatest Stories Ever Told_ , Vol. 2. New York: DC Comics, 2006.
_Superman: The Sunday Classics, Strips 1–183, 1939–1943_. New York: DC Comics: 1998.
Uslan, Michael. _America at War: The Best of DC War Comics_. New York: Simon and Schuster, 1979.
Waid, Mark. _Kingdom Come_. New York: DC Comics, 2008.
———. _Superman: Birthright_. New York: DC Comics, 2004.
Wolfman, Marv. _Crisis on Infinite Earths_. New York: DC Comics, 2000.
### FILM, TV, RADIO, AND MUSICAL PRODUCTIONS
_Adventures of Superman_. Seasons 1–2. Warner Home Video Inc., 2005.
Allen, Fred. Transcript prepared by author of radio interview with Jerry Siegel and Harry Donenfeld. October 9, 1940.
"American Icons: Superman." _Studio 360_. WNYC Radio. July 7, 2006.
_Atom Man vs. Superman: The Complete 15-Chapter Adventure!_ Warner Home Video, 1989. Written by George H. Plympton, Joseph F. Poland, and David Mathews.
"Clan of the Fiery Cross." _Adventures of Superman_. June–July 1946. Transcript prepared by author.
_Comic Book Confidential_. Toronto: Sphinx Productions, 1988. Written by Ron Mann.
"The Defeat of Superman." _Adventures of Superman_. Warner Home Video, 1953. Written by Jackson Gillis.
"Drums of Death." _Adventures of Superman_. Warner Home Video, 1953. Written by Dick Hamilton.
_George Reeves Double Feature: Thunder in the Pines and Jungle Goddess_. VCI Entertainment. 2006.
"George Reeves: The Perils of a Superhero." Transcript prepared by author. _Biography_. A&E. February 9, 2000.
_Hollywoodland_. Universal Studios, 2007. Written by Paul Bernbaum.
_Kill Bill: Vol. 2_. Miramax Films, 2004. Written by Quentin Tarantino.
_Lois & Clark: The New Adventures of Superman_. Seasons 1–4. Warner Home Video, 2006.
_It's a Bird... It's a Plane... It's Superman: The New Musical Comedy_. Script. New York: Studio Duplicating Services, 1966. Written by Robert Benton and David Newman.
"The Panic in the Sky." _Adventures of Superman_. Warner Home Video, 1953. Written by Jackson Gillis.
_The Paramount Cartoon Classics of Max & Dave Fleischer_. Bosko Video, 1991.
_Seinfeld_. "The Cheever Letters" (1992). "The Chinese Woman" (1994). "The Face Painter" (1995). "The Heart Attack" (1991). "The Implant" (1993). "The Lip Reader" (1993). "The Marine Biologist" (1993). "The Race" (1994). "The Smelly Car" (1993). "The Stall" (1994). "The Stand-In" (1994). "The Stock Tip" (1990). "The Switch" (1995). "The Tape" (1991). "The Visa" (1993). Transcripts can be found at www.seinfeldscripts.com.
_Smallville: The Complete First Season_. Warner Home Video, 2002.
_Smithsonian Historical Performances: Superman on Radio_. Recordings of first 27 episodes. Schiller Park, Ill.: Radio Spirits, 1997.
_Superman and the Mole Men_. Warner Home Video, 1987. Written by Richard Fielding.
_Superman: The Theatrical Serials Collection_. Warner Home Video, 2006.
_Superman: Ultimate Collector's Edition_. Warner Bros. Entertainment, 2006.
_Torchy Blane Collection_. Warner Home Video, 2010.
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### **NEWSPAPERS, MAGAZINES, AND JOURNALS**
Akers, Paul E. "Bring Back the REAL Superman." _Washington Post_. December 31, 1988.
_Albany Times Union_. "Jerry Siegel, 81: Superman Co-Creator." January 31, 1996.
———. "Man of Steel Drops Cape in Wardrobe Makeover." January 4, 1997.
———. "Superman and Lois Finally Tying the Knot." September 8, 1996.
———. "Superman Fan Has Bond with Real-Life Hero Reeve." April 11, 1997.
———. "Time and Economics Prove Superman Mortal." September 4, 1992.
Albrecht, Brian E. "Superman Meets the Grim Reaper." Cleveland _Plain Dealer_. November 15, 1992.
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Amash, Jim. "His Goal Was the Graphic Novel." _Alter Ego_ No. 88 (August 2009).
———. "I Didn't Want to Know [What Other Companies Were Doing]!"
_Alter Ego_ no. 56 (February 2006).
———. "I've Always Been a Writer: Alvin Schwartz on His Long Career in Comic Books—and Elsewhere." _Alter Ego_ No. 98 (December 2010).
———. "A Real Iconic, Quintessential American Figure." _Alter Ego_ No. 88 (August 2009).
Ames, Walter. "Superman George Reeves and Producers Disagree on New Television Deal." _Los Angeles Times_. September 27, 1954.
———. "Video Actress Can't Convince Daughter; Palm Services Set." _Los Angeles Times_. March 29, 1953.
Anderson, Susan Heller. "It's a Bird! It's a Plane! It's a Movie!" _New York Times_. June 26, 1977.
Andrae, Thomas. "From Menace to Messiah: The Prehistory of the Superman in Science Fiction Literature." _Discourse_ 2 (1980).
———. "Of Supermen and Kids with Dreams: An Interview with Jerry Siegel and Joe Shuster." _Nemo_ No. 2 (1983).
_Anti-Defamation League Bulletin_. "Klan Sleuth Gives 'Superman' Secrets of Hooded Order." February 1947.
Arave, Lynn. "Man of Steel, King of the Seas in for Big Changes." _Deseret News_. January 6, 1997.
———. "Still Leaping Buildings in a Single Bound but in Brand-New Disguise." _Deseret News_. February 17, 1997.
———. "Superman No Longer a Man of Steel." _Deseret News_. June 2, 1997.
Arnold, Gary. "It's a Bird! It's a Pain!" _Washington Post_. June 17, 1983.
———. " 'Superman II': The Plot Weakens." _Washington Post_. June 19, 1981.
Ascher-Walsh, Rebecca. "Cape Fear." _Entertainment Weekly_. May 29, 1998.
Associated Press. "Nicolas Cage and Wife Have Baby Boy." October 3, 2005.
———. "Rare Superman Comic Sells for Record $2.16M US." December 1, 2011. cbc.ca.
_Atlanta Constitution_. "Atlanta Boy Falls 3 Floors, Is Not Injured." June 7, 1941.
Aurthur, Kate. "The Past Catches Up with a Future Superman." _New York Times_. February 23, 2005.
———. "Young Male Viewers Boost 'Smallville.' " _New York Times_. May 20, 2006.
———. "Young-Superman Episode Delivers Clout for WB." _New York Times_. January 28, 2006.
Baker, Russell. "Geezer of Steel." _New York Times_. June 17, 1986.
———. "Sad Feet in the Sky." _New York Times_. September 25, 1973.
———. "The Heart of Superman." _New York Times_. December 17, 1978.
———. "Turn That Dial Back in Time: Superman & Co. Return!" _New York Times_. October 24, 1988.
Ballner-Bear, Lisa. "Golly, Miss Lane, Superman's 50." _Omni_ 10 (March 1988).
Barnes, Brooks, and Michael Cieply. "A Custody Battle, Supersized." _New York Times_. March 21, 2010.
———. "Warner Brothers Sues 'Superman' Lawyer." _New York Times_. May 15, 2010.
Barr, Mike W. "The Madame and the Girls: How DC Got Rid of the Troublemakers." _Words & Pictures_ No. 5 (August 1988).
Barron, James. "Boldface Names." _New York Times_. May 3, 2001.
———. "The Mystery of the Missing Man of Steel." _New York Times_. April 19, 2010.
Bart, Peter. "Advertising: Superman Faces New Hurdles." _New York Times_. September 23, 1962.
Bates, Cary, and Elliot Maggin. "The Men Behind the Super-Typewriter." _Amazing World of DC Comics_ No. 2 (September 1974).
Bear, Greg. " 'Superman': Unable to Leap the Changing Decades?" _Los Angeles Times_. December 24, 1978.
Beaty, Bart. "The Recession and the American Comic Book Industry: From Inelastic Cultural Good to Economic Integration." _Popular Communication_ 8, No. 3 (August 2010).
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Becattini, Alberto. "Jerry Siegel's European Comics." _Alter Ego_ No. 59 (June 2006).
Beck, Marilyn. "$138 Mistake: Superman Creators Gave Him Away Cheaply." Cleveland _Plain Dealer_. January 7, 1979.
Belcher, Jerry. "Comic Characters Revised: At 48, Superman Has Slowed Down Just a Bit." _Los Angeles Times_. June 18, 1986.
Belkin, Douglas. "Superman Birthplace Is Restored." _Wall Street Journal_. July 11, 2009.
Beller, Miles. "Hollywood Is Banking on the Comics." _New York Times_. December 9, 1979.
Belloni, Matthew. "Siegels Win a Round in Epic Battle for Superman." _Hollywood Reporter_. December 21, 2010.
———. "Warners Wins Round in Superman Litigation." _Hollywood Reporter_. December 21, 2010.
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Bianculli, David. "Jeepers! Jimmy to Superman's Rescue." _New York Daily News_. May 31, 1996.
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_Boston Globe_. "Comic Book of Steel." February 23, 2010.
———. "Has Kryptonite Been Discovered?" January 26, 2009.
———. "Superman Comic Sold for $317,200." March 15, 2009.
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Brandon, Craig. "The Man of Steel Comics Fans Yawn as Superman Comes Back, Back, Back, Back." _Albany Times Union_. April 15, 1993.
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———. "This Week's Pics." _Washington Post_. February 28, 1988.
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Brown, Nicky Wheeler-Nicholson. "He Was Going to Go for the Big Idea." _Alter Ego_ No. 88 (August 2009).
———. "Major Malcolm Wheeler-Nicholson, Cartoon Character or Real Life Hero?" _International Journal of Comic Art_ 10, No. 2 (Fall 2008).
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———. "Superman Star Is Back Before the Public." _New York Times_. October 17, 1995.
Buchwald, Art. "The Miracle Worker: Super-K Tries Again." _Washington Post_. October 13, 1974.
———. "Henry, This Is a Job for Superman." _Washington Post_. April 20, 1975.
Buckley, Steve. " 'Original' Superman Story Doesn't Fly." _Boston Herald_. January 10, 1997.
Buckley, Tom. "At the Movies." _New York Times_. May 26, 1978.
———. "The Writing of 'Superman': A Fantastic Story." _New York Times_. December 22, 1978.
Buhle, Paul. "Superbad: Joe Shuster's Seamy Scenes of Lois and Clark." _Forward_. August 7, 2009.
Burns, James H. "Sarah Douglas: The Human-Hating Kryptonian Super-Villainess from 'Superman II.' " _Starlog_ No. 47 (June 1981): 22–24.
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_BusinessWeek_. "Superman Scores." April 18, 1942.
Canadian Press. "Lois Lane Just Fed Up, Dumping Man of Steel." February 9, 1996.
Canby, Vincent. "Nothing 'Went Wrong.' " _New York Times_. December 24, 1978.
———. "Screen: It's a Bird, It's a Plane, It's a Movie." _New York Times_. December 15, 1978.
Carlin, Mike. "Ask the Pros: Describe an Average Story Conference for DC's Superman Titles." _Comic Buyer's Guide_ (September–October 1993).
Carlinsky, Dan. "On Krypton, Superman Might Have Been a Plumber." _New York Times_. December 10, 1978.
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Carson, Tom. "Small Comforts." _Esquire_ (February 2002): 48.
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Chang, Gordon H. "Superman Is About to Visit the Relocation Centers & the Limits of Wartime Liberalism." _Amerasia Journal_ 19, No. 1 (1993).
Chang, Kenneth. "SPLAAAAAAAT! Comic Books No Longer Reaping Big Sales in Single Bound." _Los Angeles Times_. March 2, 1996.
Chernin, Donna. "Pow! Superman's Creators Take a Poke at Him." Cleveland _Plain Dealer_. October 26, 1975.
_Chicago Daily Defender_. " 'Superman,' 11 Saved by Snowbank." December 7, 1964.
_Chicago Tribune_. " 'Superman' Fan Injured." February 3, 1979.
Cieply, Michael. "Lawyer Battles Back Against DC Comics in Superman Dispute." _New York Times_. August 16, 2010.
———. "Ruling Gives Heirs a Share of Superman Copyright." _New York Times_. March 29, 2008.
Cieply, Michael, and Brooks Barnes. "Warner Brothers Sues 'Superman' Lawyer." _New York Times_. May 15, 2010.
_Cinefantastique_. "Superman '... Has Finally Been Brought Down to Earth.' " Summer 1979.
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_Cleveland Call and Post_. "Memo to Parents: If Junior Imitates 'Tarzan' on Furniture, He May Make Better Man." June 25, 1949.
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_Cleveland Jewish Review and Observer_. "Deaths." June 10, 1932.
_Cleveland Magazine_. "The Inspirations for Lois Lane." January 2009.
Cleveland _Plain Dealer_. "A Son Is Born to Superman Author." January 29, 1944.
———. "Death Notices." June 4, 1932.
———. "Half of Superman Drafted; Partner Awaits Army Call." June 30, 1943.
———. "Superman Creator Wed." October 15, 1948.
———. " 'Superman' Is Coming to Plain Dealer Sunday." January 18, 1940.
———. "Superman Leaped Years, Too." February 2, 1988.
———. "Superman's Birthplace Is Now Historic Landmark." January 28, 1987.
———. "Superman's Creator Sued." July 15, 1948.
_Cleveland Press_. "Dies After Robbery." June 3, 1932.
Coe, Richard L. "Not Peter Pan, It's 'Superman.' " _Washington Post_. March 31, 1966.
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Dagnal, Cynthia. "It's a Bird, a Plane, a Movie: A Socially Relevant Superman?" _Los Angeles Times_. September 3, 1976.
Dalton, Joseph. "Death of First Superman Still a Supermystery." _Los Angeles Herald Examiner_. February 28, 1988.
Dangaard, Colin. "Reeve Flies to the Rescue of 'Superman.' " _Los Angeles Times_. July 31, 1977.
D'Angelo, Mike. "Man, Yes; Super, Not Really." _Las Vegas Weekly_. June 29, 2006.
Daniel, Clifton. "The Future of Elliot Richardson." _New York Times_. October 25, 1973.
Dargis, Manohla. "Superman Is Back to Save Mankind from Its Sins." _New York Times_. June 27, 2006.
———. "The Quiet Desperation of Superman." _New York Times_. September 8, 2006.
Davenport, Christian. "The Brother Might Be Made of Steel, but He Sure Ain't Super... Man." _Other Voices_ 1, No. 2 (September 1998).
Davies, Patrick, and Julia Surridge, et al. "Superhero-Related Injuries in Paediatrics: A Case Series." _Archives of Disease in Childhood_ 92 (2007).
Dean, Michael. "An Extraordinarily Marketable Man: The Ongoing Struggle for Ownership of Superman and Superboy." _Comics Journal_ No. 263
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Delugach, Al. "Cannon Bid as Major Studio Is Cliffhanger: Firm's Future at Risk in High-Stakes Gamble." _Los Angeles Times_. August 24, 1986.
———. "Salkinds Settle Brando, Puzo 'Superman' Suits." _Los Angeles Times_. April 6, 1982.
———. " 'Superman' and the Sequel: The Lawsuits Continue." _Los Angeles Times_. September 13, 1981.
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Docker, Philip. Interview with Chuck Harter. _The Adventures Continue_ No. 15 (1998).
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———. "Superman Product to B.&B." _New York Times_. June 4, 1982.
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———. "Seinfeld and Superman Join Forces Again in Spots for American Express, This Time on the Web." _New York Times_. March 30, 2004.
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———. "Zowie! Comic Books Are Back." _Los Angeles Times_. January 23, 1966.
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———. "Hollywood Heist: How a Burglary May Impact the Future of
'Superman.' " _Hollywood Reporter_. May 27, 2011.
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———. "Master Sleuth Solves Very Baffling Enigma." May 21, 1931.
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Schier, Ernest. " 'Superman' Needs a Quick Course in Muscle Building." _Philadelphia Bulletin_. February 16, 1966.
Schwartz, Alvin. "The Real Secret of Superman's Identity." _Annual of the Modern Language Association Group on Children's Literature_ 5 (1976).
Schwartz, Julius. "Dawn of the Silver Age." _Comics Scene Spectacular_ No. 6. July 1992.
Seldes, Gilbert. "Preliminary Report on Superman." _Esquire_. November 1942.
Shabecoff, Phillip. "Look! Up in the Air! To Germans, It Could Be Ubermensch, Saubermann, or Superman." _New York Times_. November 7, 1967.
Shay, Don. "The Director of Steel Bends Producers in His Bare Hands." _Cinefantastique_ 8, No. 4 (Summer 1979).
Sherwood, John. "Superman Still Makes Millions, but Not His Creators." _Washington Star_. October 29, 1975.
Shotz, Israel. "Superman a Sear?" Letters to the Editor. _Washington Post_. November 13, 1941.
Siegel, Jerry. "Detective Tries Composing." _Glenville Torch_. April 30, 1931.
———. "Five Men and a Corpse." _Glenville Torch_. January 14, 1932.
———. "Goober the Mighty Discovers Countless Foes in Wilderness." _Glenville Torch_. May 7, 1931.
———. "How Superman Would End the War." _Look_. February 27, 1940.
———. "Jerry Siegel Tells Why His Superman Won't End the War." _Midpacifican_. August 26, 1944.
———. "The Reign of the Super-Man." _Science Fiction: The Advance Guard of Future Civilization_ No. 3 (January 1933). Written by Herbert S. Fine (Jerry Siegel).
———. "New Entertainment Company Puts on 'Shellsapoppin.' " _Midpacifican_. September 2, 1944.
Sisk, Richard. " 'Superman' Creators Don't Want 'Pension.' " _Cleveland Press_. December 11, 1975.
Sloane, Leonard. "Advertising: Comics Go Up, Up, and Away." _New York Times_. July 20, 1967.
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———. "Ordinary Mortals Not Up to the Superman Dream." Cleveland _Plain Dealer_. July 25, 1988.
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Spacey, Kevin. Interview. _Wizard_ No. 172 (February 2006).
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_Starlog_. " 'Superman II' Movie Magazine." June 1981.
———. "Superman II Ready for Flight." July 1979.
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———. "Super-Postscript." _Alter Ego_ No. 37 (June 2004).
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———. "Bang!" October 6, 1986.
———. "Cease Fire." September 3, 1945.
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———. "Comic Culture." December 18, 1944.
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———. "Good Grief." April 9, 1965.
———. "Here Comes Superman!!!" November 27, 1978.
———. "Jungle Sam." December 1, 1952.
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———. "Miscellany." August 10, 1942.
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———. "The New Pictures." July 6, 1942.
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———. "Paperback Godfather." August 28, 1978.
———. "Paper Cutups." April 8, 1966.
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———. "Boy Kills Self Showing Chum Gun Roulette." July 6, 1947.
———. "Bye-Bye 'Batman'." October 2, 1973.
———. "Children in Wartime." January 19, 1942.
———. "Deaths Elsewhere." March 17, 1999.
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———. "It's a Bird, a Plane—a Happy Ending." December 27, 1975.
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———. "Superman Toils for Musical Role." _New York Times_. November 17, 1965.
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Reeve, Christopher. _Nothing Is Impossible: Reflections on a New Life_. New York: Ballantine Books, 2002.
——— _. Still Me_. New York: Ballantine Books, 1998.
Reynolds, Richard. _Super Heroes: A Modern Mythology_. Jackson: University Press of Mississippi, 1994.
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Robinson, Jerry, Michael Chabon, and Jules Feiffer. _The Zap! Pow! Bam! Superhero: The Golden Age of Comic Books, 1938–1950_. Atlanta: The William Breman Jewish Heritage Museum, 2004.
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Rosenberg, Robin S. _The Psychology of Superheroes: An Unauthorized Exploration_. Dallas: Benbella Books, 2008.
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Sadowski, Greg, ed. Introduction to _Supermen!: The First Wave of Comic Book Heroes 1936–1941_. Seattle: Fantagraphics Books, 2009.
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Schwartz, Alvin. _A Gathering of Selves_. Rochester, Vt.: Destiny Books, 2007.
———. _An Unlikely Prophet: A Metaphysical Memoir by the Legendary Writer of Superman and Batman_. Rochester, Vt.: Destiny Books, 1997.
Schwartz, Julius, and Brian M. Thomsen. _Man of Two Worlds: My Life in Science Fiction and Comics_. New York: HarperCollins, 2000.
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———. _Superman: The Never-Ending Battle_. New York: Pocket Star Books, 2004.
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Weisinger, Mort. _The Contest_. New York: New American Library, 1971.
Wertham, Frederic. _Dark Legend: A Study in Murder_. New York: Bantam Books, 1966.
———. _Seduction of the Innocent_. Laurel, N.Y.: Main Road Books, 1953.
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Wolfman, Marv. _Crisis on Infinite Earths_ (novel). New York: ibooks, 2005.
———. _Superman Returns_. New York: Warner Books, 2006.
Wolper, David L. _Producer: A Memoir_. New York: A Lisa Drew Book, 2003.
Wolverton, Mark. _The Science of Superman_. New York: iBooks, 2002.
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Wylie, Philip. _Gladiator_. New York: Manor Books, 1976.
Wylie, Philip, and Edwin Balmer. _When Worlds Collide_. Lincoln: University of Nebraska Press, 1999.
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Yoe, Craig. _Secret Identity: The Fetish Art of Superman's Co-Creator Joe Shuster_. New York: Abrams ComicArts, 2008.
Young, William H., and Nancy K. Young, _The 1950s_. Westport, Conn.: Greenwood Press, 2004.
Yule, Andrew. _The Man Who "Framed" the Beatles: A Biography of Richard Lester_. New York: Donald I. Fine, 1994.
Zeno, Eddy. _Curt Swan: A Life in Comics_. Lebanon, N.J.: Vanguard Productions, 2002.
# **ABOUT THE AUTHOR**
Larry Tye was an award-winning journalist at _The Boston Globe_ and a Nieman Fellow at Harvard University. Tye, who grew up with Superman on his night table and Clark Kent as a role model, now runs a Boston-based training program for medical journalists. He is the author of _The Father of Spin, Home Lands, Rising from the Rails_ , and _Satchel_ , and co-author, with Kitty Dukakis, of _Shock_. Tye is now writing a biography of Robert F. Kennedy.
Jerry Siegel (top) and Joe Shuster got things going for Superman in 1938, when their first story was published in _Action Comics_ No. 1. The young collaborators from Cleveland had been working on that narrative for nearly four years, and their names eventually would become as conjoined and revered in the world of comics as those of Rodgers and Hammerstein in song and Tracy and Hepburn in cinema. _Photos courtesy of Laura Siegel Larson and Jean Shuster Peavy_.
Jerry Siegel's relationship with Jack Liebowitz would sour over the decades, but here they appeared to be the best of friends.
Bud Collyer was the voice of Superman for 2,008 radio shows, and across thirty years in various media. _Library of American Broadcasting, University of Maryland_.
Around the time Bud Collyer was bringing Superman alive on the radio, brothers Max and Dave Fleischer were taking him to movie theaters in the form of animated cartoons that many critics panned but most fans loved. They featured action and adventure, rescues and thrilling battles, such as the one above, in which Superman takes on mean-spirited robots, in a 1941 Paramount Pictures release called _The Mechanical Monsters_. The animators were Steve Muffati and George Germanetti. _Courtesy of Warner Bros. Entertainment Inc_.
Superman was a favorite of Allied troops during World War II, and they showed their gratitude by naming after him their jeeps, tanks, landing craft, and, pictured here, a B-17 Flying Fortress bomber. _Time & Life Pictures/Getty Images_.
While he was a favorite of parents and even grandparents, Superman's success in the early years resulted from his capturing the imagination of youths like this boy, seen reading a comic book in New York in 1946. Another favorite reading spot: under the bedcovers, at night, where a flashlight illuminated the pages. _Time & Life Pictures/Getty Images_.
The 1950s TV series _Adventures of Superman_ welcomed back old fans of the comics and radio productions and introduced new ones to the Man of Steel narrative. For millions of children who grew up glued to that show, and for others who have watched it in reruns, when they envision Superman they see George Reeves, who is shown here nabbing two thugs. _ABC via Getty Images_.
It was the ultimate measure of celebrity in 1956: a guest slot on America's most-watched TV show, _I Love Lucy_. It is tough to tell here who was having more fun: George Reeves as he flexed his biceps, or Lucille Ball as she felt his super-strong muscle. _Getty Images_.
The pace on the _Adventures of Superman_ set was frenetic, given the measly budgets and tight deadlines, but off the set George Reeves could relax with Noel Neill, who played Lois Lane alongside him for five seasons and earlier had starred next to Kirk Alyn in such film serials as _Atom Man vs. Superman. Colorized photo courtesy of Larry Thomas Ward_.
Jack Larson, who was just twenty-three when he signed up to play Jimmy Olsen, connected so completely with viewers that he took the character from a supporting role to a star. Larson and Neill, shown here in 1956, have sustained their friendship through the decades. _Colorized photo courtesy of Larry Thomas Ward_.
This comic book cover—from _Superman Annual_ No. 1 in 1960—was drawn by Curt Swan and colored by Stan Kaye. Swan, who did his first Superman drawings in 1948 and his last thirty-nine years later, gave the hero a more dignified and human sensibility. _"Superman Annual" No. 1_ © _1960 DC Comics. Used with Permission_.
Two of Superman's most insidious, relentless, and hairless enemies—the evil genius Lex Luthor and the computerized space pirate Brainiac—join forces in this comic book from 1964. _"Superman" #167_ © _1964 DC Comics. Used with Permission_.
Superman spends much of his life fending off Lois Lane's bids to ensnare him, even as she is doing the same with Clark Kent. In this comic from 1966, the Man of Steel offers a novel line of reasoning: He can't marry anyone dumb enough not to see through his lame disguise. _"Superman's Girl Friend, Lois Lane" #63_ © _1966 DC Comics. Used with Permission_.
Superman became a regular at the Macy's Thanksgiving Day Parade in New York as early as 1940, when the biggest balloon was an eighty-foot-high replica of the Man of Tomorrow. This shot is from the 1966 parade. _New York Daily News via Getty Images_.
To sleep-deprived parents in the 1970s, a cartoon like _Super Friends_ was a twofer: Kids were mesmerized by the animation, orchestrated by Hanna-Barbera, and the collaboration between Superman and such heroic friends as Aquaman, Wonder Woman, Robin, and Batman; Mom and Dad, meanwhile, delighted in the extra hours they got in bed. _ABC via Getty Images_.
Many fans of the 1978 movie _Superman_ , the first in a series starring Christopher Reeve, wondered what the sequel would have been like if Richard Donner had been kept on as director. They found out twenty-five years later, when, thanks to their lobbying, Warner Bros. released on DVD a re-edit called _Superman II: The Richard Donner Cut_. Donner is shown here with producer Michael Thau (right) and a cutout of Reeve. _Getty Images_.
The Superman TV show launched in 1993 was called _Lois & Clark: The New Adventures of Superman_. As the title suggested, the show was more interested in the relationship between the two journalists than in the adventures of the superhero, and at least as interested in Lois as in Clark. Teri Hatcher played Lois while Dean Cain doubled as Clark and Superman. _ABC via Getty Images_.
_Smallville_ debuted on the WB network just a month after the 9/11 terrorist attacks. The long-running series let young viewers see why their grandparents and parents were so smitten with Superman, and it gave them a version of the superhero who was theirs alone. Tom Welling (top) played a youthful Clark Kent, and Michael Rosenbaum (above) portrayed what may have been the most riveting Lex Luthor ever. _Warner Bros./Getty Images_.
After nearly seventy-five years in which scores of artists have offered up their unique visions of Superman, it isn't easy to stand out. But Alex Ross's mournful rendering—from the 1999 collection _Peace on Earth_ —does. _From_ Superman: Peace on Earth © _1998 DC Comics. Used with Permission_.
In the early 1990s DC Comics decided to kill off Superman, and while it was for real, it wasn't forever. Readers lined up on the street and around the block outside comic stores to buy _Superman_ No. 75, the death issue, which tallied the biggest one-day sale ever for a comic book, with more than six million copies printed. _"Superman" #75_ © _1993 DC Comics. Used with Permission_.
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/dcds.2008.22.1081","text":"# American Institute of Mathematical Sciences\n\nDecember\u00a0 2008,\u00a022(4):\u00a01081-1090. doi:\u00a010.3934\/dcds.2008.22.1081\n\n## Quasi-$m$-accretivity of Schr\u00f6dinger operators with singular first-order coefficients\n\n 1 Department of Mathematics, Science University of Tokyo, 26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601, Japan, Japan\n\nReceived\u00a0 May 2007 Revised\u00a0 October 2007 Published\u00a0 September 2008\n\nThe Schr\u00f6dinger operator $T = (i\\nabla +b)^2+a \\cdot \\nabla + q$ on $\\mathbb{R}^N$ is considered for $N \\ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \\in L^4(\\mathbb{R}^N)+L^{\\infty}(\\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \\in L^4_{loc}(\\mathbb{R}^N)$.\nCitation: Noboru Okazawa, Tomomi Yokota. Quasi-$m$-accretivity of Schr\u00f6dinger operators with singular first-order coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1081-1090. doi: 10.3934\/dcds.2008.22.1081\n [1] Jean Bourgain. On quasi-periodic lattice Schr\u00f6dinger operators. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 75-88. doi: 10.3934\/dcds.2004.10.75 [2] Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934\/cpaa.2012.11.2239 [3] Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schr\u00f6dinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934\/ipi.2014.8.475 [4] Jaime Cruz-Sampedro. Schr\u00f6dinger-like operators and the eikonal equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934\/cpaa.2014.13.495 [5] Jean Bourgain. On random Schr\u00f6dinger operators on $\\mathbb Z^2$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 1-15. doi: 10.3934\/dcds.2002.8.1 [6] Fengping Yao. Optimal regularity for parabolic Schr\u00f6dinger operators. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934\/cpaa.2013.12.1407 [7] Fritz Gesztesy, Roger Nichols. On absence of threshold resonances for Schr\u00f6dinger and Dirac operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3427-3460. doi: 10.3934\/dcdss.2020243 [8] M. Burak Erdo\u01e7an, William R. Green. Dispersive estimates for matrix Schr\u00f6dinger operators in dimension two. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4473-4495. doi: 10.3934\/dcds.2013.33.4473 [9] Jon Chaika, David Damanik, Helge Kr\u00fcger. Schr\u00f6dinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934\/jmd.2009.3.253 [10] Xinlin Cao, Yi-Hsuan Lin, Hongyu Liu. Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr\u00f6dinger operators. Inverse Problems & Imaging, 2019, 13 (1) : 197-210. doi: 10.3934\/ipi.2019011 [11] Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schr\u00f6dinger operators with certain potentials. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934\/cpaa.2018095 [12] Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schr\u00f6dinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934\/cpaa.2008.7.533 [13] Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schr\u00f6dinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934\/cpaa.2018012 [14] Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schr\u00f6dinger operators with a singular potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5827-5867. doi: 10.3934\/dcds.2015.35.5827 [15] Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schr\u00f6dinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934\/dcdss.2017033 [16] Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schr\u00f6dinger operators in Lp-spaces. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934\/dcdss.2020086 [17] David Damanik, Zheng Gan. Spectral properties of limit-periodic Schr\u00f6dinger operators. Communications on Pure & Applied Analysis, 2011, 10 (3) : 859-871. doi: 10.3934\/cpaa.2011.10.859 [18] Younghun Hong. Strichartz estimates for $N$-body Schr\u00f6dinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934\/dcds.2017233 [19] Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schr\u00f6dinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934\/ipi.2009.3.139 [20] Michael Goldberg. Strichartz estimates for Schr\u00f6dinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934\/dcds.2011.31.109\n\n2019\u00a0Impact Factor:\u00a01.338","date":"2020-09-24 17:47:21","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\": 0.5958542823791504, \"perplexity\": 5587.155455346035}, \"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-2020-40\/segments\/1600400219691.59\/warc\/CC-MAIN-20200924163714-20200924193714-00719.warc.gz\"}"} | null | null |
A tablet interactive and quiz, along with oral histories, help visitors make the most of this small, local, independent museum located in a listed mediaeval building.
Since it opened in 1991, Amersham Museum has thrived as a small, local, independent museum located in a listed mediaeval building. In 2013 it acquired the adjoining building as part of its plan to become a more accessible museum for the local community, representing the town's history "from market town to metro-land", focusing on five key dates.
The museum's space restricts the size of groups – both formal and informal education parties – that can visit. Whilst the acquisition of additional space has enabled the museum to make more of the collection available to visitors, the use of technology helps to make best use of all the space available and allows exploration of the building through stories, stimulating the visitor to explore further. Working with Querceus, we have developed a number of interactive exhibits which do just that.
Visitors can explore a series of topics related to life across five different time periods, through eight questions. They can compare and contrast each question with the other time periods, and at the end of the quiz complete a challenge enabling them to design their own home based on the knowledge they have gained.
This tablet-based interactive enables visitors to explore Amersham Museum's extensive archive of over 4,000 images and is managed using our content management system to allow staff to update the content as they choose. An attractor screen sets the scene, encouraging visitors to explore further. On touching the screen, the visitor then gets to choose from a main menu of themes.
Designed as an old-fashioned sound booth, this exhibit allows visitors to enjoy oral histories from the 1960s. When a visitor approaches and triggers a proximity sensor the exhibit plays a holding audio of music to attract their attention. The visitor can then press different buttons to hear different stories.
An old-fashioned, specially adapted, radio allows visitors to listen to the oral histories of individuals from the 1930s. The tactile nature of the exhibit not only gives a flavour of the time but allows the visitor to 'tune in' to one of the different 'stations' using the dial. To give an authentic experience, when the dial is turned, white noise plays as if the radio is being 'tuned' to a station.
A proximity sensor is triggered when a visitor passes close by, playing a holding audio file which catches their attention and invites them to find a 'station' to listen to.
This exhibit presents "hands-on history", the chance to touch and feel the past.
This exhibit projects a looping slideshow directly onto the fabric of the building in the corridor area. Each slide shows a still taken at Amersham's theatre with a simple deco style filigree border and captions to explain the context.
We provided a turnkey solution on this project including digital interactives and AV development, hardware specification, procurement and installation, and adaption of the original 1930s radio, along with training and manuals to show the correct procedures for start-up, operation, shut-down, maintenance and updates. | {
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Rams QB Jared Goff regains his grip on starting…
Rams QB Jared Goff regains his grip on starting role
John Wolford's neck injury will keep him out of playoff game at Green Bay. Goff says his thumb is sore but improving.
Seattle Seahawks defensive end Carlos Dunlap (43) leaps to try and deflect a pass from Los Angeles Rams quarterback Jared Goff during the second half of an NFL wild-card playoff football game, Saturday, Jan. 9, 2021, in Seattle. (AP Photo/Scott Eklund)
By Kevin Modesti | kmodesti@scng.com | Daily News
THOUSAND OAKS — First Jared Goff's thumb was fractured and dislocated. Then his nose was out of joint.
The emotion came through in Goff's clipped answers to questions in the week before the playoff game against the Seattle Seahawks and his uncharacteristically edgy comments after he came off the bench to help the Rams win, 30-20.
Rams coach Sean McVay didn't mind the edge in the voice of the famously even-keel quarterback.
"I think that's a good edge," McVay said. "It shows his competitive side. I like seeing that."
Goff was back to his familiar unflappable good humor on Thursday after McVay named him the starter for the Rams against the Green Bay Packers in Saturday's second-round playoff game at Lambeau Field.
McVay declared John Wolford out of the Packers game because of the neck injury that knocked Wolford out of the Seattle game, his second start after Goff's Dec. 28 surgery to stabilize his throwing thumb.
Wolford will travel with the team. But Blake Bortles will back up Goff. McVay said practice-squad quarterback Bryce Perkins might be promoted to the active roster as insurance.
Goff said his right thumb continues to get better. He wore gloves during practice and said he might wear them to help grip the ball in cold weather at Green Bay.
"There is slight soreness," Goff said. "But overall it's progressing really well."
A week earlier, Goff gave vague answers to questions about his thumb, trying to hide his knowledge and maybe his hurt that McVay had chosen a then-healthy Wolford to start against Seattle.
After the win, Goff expressed satisfaction about beating the Seahawks two weeks after "we saw them smoking cigars and getting all excited about beating us and winning the division."
But he reserved some of that edge for saying he'd been "not thrilled" with McVay choosing Wolford to start even though Goff insisted he could play.
Thursday, Goff and McVay dismissed hints of a rift.
"We are able to disagree," Goff said. "We are two grown men who disagreed on the status of my thumb. It was not the end of the world, I think. I was able to come in and help us get the win. That is most important to me."
McVay said he and Goff had "great conversations" about who would start.
"As far as being able to work together, figure out how we move forward from whether good things or somewhat of a setback, I thought he handled last week really well," McVay said.
Goff is 2-2 in playoff starts after quarterbacking the Rams to the Super Bowl two years ago. He should need no extra motivation for Saturday's game against the top-seeded Packers and fellow Cal alum Aaron Rodgers. Asked if he has something to prove to McVay and the Rams, Goff replied smoothly, "I think every day you feel that way."
But thinking back to the Seattle game, he said, "I think it's good to play with an edge."
McVay said he learned something new about Goff in seeing him handle the controversy that followed his first serious football injury.
"It's a credit to who he is as a man that he can be able to step in and do what he did," McVay said. "And then this week has represented an opportunity for him to build on last week."
SCOUTING EXEC IS LEAVING
Rams college scouting director Brad Holmes, who always appeared to be going places, is going to the Detroit Lions to be their general manager.
As compensation, the Rams will get draft picks at the end of the third round in 2021 and 2022, becoming the first team to benefit from an NFL rule introduced in November that rewards franchises for developing minority employees who are hired away as coaches or GMs. Holmes is African American.
The Rams reacted to Thursday's announcement with mixed emotions.
"We are all excited for this opportunity for Brad," Rams GM Les Snead said. "He has spent his entire career with the Rams, and he earned this position with the Lions due to his dedication to being an astute evaluator of football talent, dynamic intelligence, unwavering leadership and humility. All of those qualities will ensure he is set up to be successful in this next chapter of his career."
Holmes, 41, started as a PR intern with the St. Louis Rams and worked his way up to his current position in 2013. He oversaw the drafts that brought the Rams Goff, Aaron Donald, Todd Gurley, Cooper Kupp and John Johnson, among other stars. Current rookie starters Cam Akers and Jordan Fuller were taken in the team's second draft in a row without a first-round pick.
Rams GM Les Snead looks to remodel instead of rebuild
Sean McVay tells Rams he will be back for 2023 season
Charles White, USC RB and 1979 Heisman Trophy winner, dies at 64
Game Day. Rams' Sean McVay isn't first to face doubts about coaching
In 2021, in addition to the end-of-third-round compensatory pick for losing Holmes, the Rams have regular picks in rounds 2, 3, 6 and 7, and were projected to have round 3 and 4 compensatory picks for their free-agent losses in 2020.
Defensive tackle Aaron Donald practiced Thursday and was declared good to go for Saturday after missing most of the second half at Seattle with a rib injury. The Rams' injury report listed wide receiver Cooper Kupp (knee) and left guard David Edwards (ankle) as questionable, and linebacker Terrell Lewis (ankle) as well as quarterback John Wolford out. … Wolford said the Rams phoned his family to tell them he was OK after the TV broadcast showed him in an ambulance, leaving the stadium in Seattle for a precautionary exam. "It's unfortunate that picture was taken because it made it look worse than it was," Wolford said. …
Green Bay is forecast to have a high of 35 degrees on Saturday, with gusty wind and a chance of snow and drizzle. It was 83 in Thousand Oaks when the Rams practiced Thursday. … The Houston Texans requested permission to interview Rams defensive coordinator Brandon Staley, who has already talked with the Chargers and New York Jets about head-coach jobs. … McVay said he and Packers coach Matt LaFleur, his friend and former Rams assistant, talked this week. "We didn't talk about the game. That's kind of understood," McVay said. "We'll get back to talking ball after we play one another. We did connect. Just kind of saying it's pretty cool and crazy that we're getting the chance to do this."
Kevin Modesti | reporter
Kevin Modesti is a reporter for the Los Angeles Daily News and the Southern California News Group who hosts Game Day With Kevin Modesti, the papers' sports newsletter. An L.A. native, he has been a sports writer, columnist and editor, an editorial board member, writer and editor in the Opinion section, and a political reporter. He lives in the San Fernando Valley.
kmodesti@scng.com
Follow Kevin Modesti @KevinModesti
More in Los Angeles Rams | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,408 |
It's obvious to state that humans are now completely dependent on their phones and computers for communication, information, and commerce. In fashion, however, despite the fact that many labels are focused on upping their Instagram footprints or selling clothes with the help of a digital avatar, precious handmade things are au courant again. Sure, there are luxury brands like Balenciaga and Maison Margiela pushing the future-is-now agenda, but consumers seem to be drawn to the finer details of a non-machine-made garment these days.
Designer Hanako Maeda has always appreciated and utilized the handcrafted in her work. When she launched her label Adeam in 2013, Maeda set out to design clothes that showcased functionality and Japanese artisanship, and to sell those clothes at a not-so-astronomical price point. Her latest outing is a sort of greatest hits album, one that showcases her love for shirting, convertible clothing, the mix of masculine and feminine silhouettes, and nods to the kawaii (cute in Japanese) aesthetic.
There are pretty tiny bow embellishments on the sleeves of a button-down made with water-resistant gingham, a ruffled obi-style belt that can cinch the waist of a boxy jacket, and sharply tailored bustier dresses. A plaid dress with tied sleeves is reversible, as is a cool denim ruffled top. Every piece is made in Maeda's Tokyo atelier using fabrics like Japanese crepe and a synthetic material that feels as light as linen but won't wrinkle. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,207 |
Q: Looking for something like onlive desktop with Visual Studio for Mac I'm trying to see if there is an online application that would allow me to 'rent' a PC remotely. In much the same way as you could remote connect to a server within Windows (via Remote Connection), I'd like to be able to remote into a PC that is off-site, where the data stays secure, and where I could install new applications.
Yes, this is pretty much asking if there is something that already exists that would be a remote connection to a Windows PC. I'm on a Mac and am a developer, but Bootcamp means I have to keep switching machines, and the VM's/Parallels are just not fast enough. I don't want to be running two machines for the different OS's if I can.
Any advice or pointers in the right direction would be greatly appreciated.
Thanks!
A: I'd use Amazon Web Services EC2. It's a cloud service where you can spin up instances of whatever OS you need. I often use a Windows Server VM on AWS so I can do Windows stuff on my Mac when I'm on the road. I believe micro Windows instances are now available under AWS's free tier (double-check me though) so you could try it out with no risk. Even if you do have to pay, it's like 10 bucks/month to run a Windows server nonstop with moderate bandwidth usage.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,147 |
Josef Schreiber (24 December 1919 – missing as of 1 February 1945) was an Oberfeldwebel in the Wehrmacht during World War II. He was the namesake of the Oberfeldwebel-Schreiber barracks in Immendingen which was closed in March 2016.
Bundeswehr's role model
His actions leading to the presentations of Knight's Cross of the Iron Cross with Oak Leaves () made him a role model in the Bundeswehr. During Operation Citadel, on his own initiative, he led an attack against Soviet trenches, forcing the enemy to retreat. In August 1943, supported by 30 soldiers, he again forced the Soviets to retreat at Karachev. On 27 May 1967, the Bundeswehr named the barracks at Immendingen the Oberfeldwebel-Schreiber barracks.
Awards and decorations
Infantry Assault Badge (16 August 1941)
Iron Cross (1939)
2nd Class (10 September 1941)
1st Class (18 September 1941)
Knight's Cross of the Iron Cross with Oak Leaves
Knight's Cross on 31 March 1943 as Feldwebel and Zugführer (platoon leader) in the 4./Sturm-Regiment 14
309th Oak Leaves on 5 October 1943 as Oberfeldwebel and Zugführer (platoon leader) in the 7./Sturm-Regiment 14
Close Combat Clasp
in Bronze (?)
in Silver (20 October 1943)
References
Citations
Bibliography
1919 births
Recipients of the Knight's Cross of the Iron Cross with Oak Leaves
German Army personnel killed in World War II
1945 deaths
Missing in action of World War II
People from Konstanz (district)
People from the Republic of Baden
Military personnel from Baden-Württemberg
German Army officers of World War II | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,910 |
\section{INTRODUCTION}
\subsection{Uncertainty in Environmental Management}
Environmental risk managers must often make decisions under uncertainty, especially under multiple objectives \citep{NAP12568}. To take this uncertainty into consideration, uncertainty must be characterized, assessed and conveyed \citep{2014:fischhoff}. There are several types and sources of uncertainty to consider in management of environmental systems \citep{2011:maxim}, including ambiguity in the decision maker's objectives \citep{2008:ascough,2005:dewulf,2014:mccarthy}.
Decision theory offers solutions on how to deal with uncertainty. However, these solutions must be transferred to practical applications, and explained in a way that enables managers to identify which solution to use for a particular problem. In addition to uncertainty due to lack of knowledge, eliciting judgments or preferences is sensitive to psychological factors resulting in cognitive biases \citep{OHagan_2006, hemming_burgman_hanea_mcbride_wintle_2017}. For example, value ambiguity can be stronger when uncertainty in outcomes is high. A structured approach to decision making is required to overcome cognitive biases in expert's judgments and values, and to make appropriate use of the relevant decision theory \citep{2012sdm,NAP12568}.
Bayesian decision analysis includes both learning and optimization, and is widely used in environmental management \citep{2012:carriger,2014:mccarthy,2016:carriger}. Bayesian analysis (statistical inference) for learning has been applied in environmental management to quantify uncertainty in management outcomes due to parameter uncertainty \citep{2013:heard, 2011:mcgowan,Hartig2018}, uncertainty in underlying mechanisms \citep{2012:buhle,VANOIJEN2013}, and to integrate expert knowledge \citep{hemming_burgman_hanea_mcbride_wintle_2017,Martin2012, OHagan_2006}.
In standard Bayesian decision analysis,
we start with a prior distribution over the parameters within the assessment model, which
embodies all expert information that is not captured in the data.
Next, we need a model to connect the data to the parameters.
The standard way of doing so goes via the likelihood function,
which models how the data are generated for any known fixed value of the
parameters.
Finally, we need a utility function which encapsulates the
decision maker's preferences over the possible decision outcomes.
The prior, likelihood, and utility function are then combined into
a so-called posterior expected utility.
The optimal decision is found by maximizing this quantity
over all decision alternatives.
In this way, Bayesian decision theory
combines prior knowledge with evidence, allows us to quantify uncertainty in the impact of decisions, and provides a method for selecting the optimal decision alternative.
Standard Bayesian analysis is limited to uncertainty quantified by a single probability distribution.
A feature of Bayesian analysis is that when data are sparse, the
analysis hinges on a correct specification of the prior.
However, when experts cannot express their uncertainty with high confidence, or when they disagree, specifying a full prior distribution may be difficult.
In such cases, we would like to avoid a situation
where the analysis depends on arbitrary choices in our prior
\citep{1854:boole,1921:keynes,2007:troffaes:decision:intro, 2011:sahlinne}.
A second issue is that the decision maker's preferences over outcomes
may only be partially quantified \citep{1962:aumann}, for instance due to the decision
maker's unfamiliarity with the outcomes, or due different interest
groups having conflicting goals (e.g. different relative values of management costs versus negative ecological impacts).
Therefore, some argue for alternative or second order expressions of uncertainty to handle situations where probabilities or utilities are not well known \citep{2014:fischhoff}.
For example, \citet{regan2005} proposed the use of information gap theory to deal with severe uncertainty in environmental management decisions, \citet{todd1998:fuzzy} proposed fuzzy sets to represent severe uncertainty about conservation status of species, and \citet{Lempert2007} addressed uncertainty through scenarios. However, both assessors and decision makers may find it difficult to deal with alternative ways to express uncertainty, especially if it requires different methods for data analysis and modelling.
One way to resolve these issues is to work with sets of
prior distributions and sets of utility functions with respect to those aspects of the problem that cannot be
fully specified with confidence
\citep{1974:levi,1984:berger,1991:walley,1995:seidenfeld,2000:rios:bayesian:sens:anal}.
This allows us to work with weaker model assumptions.
For example, robust Bayesian analysis explores the
influence of the prior and the utility, through sensitivity analysis
on posterior inference \citep{2000:rios:bayesian:sens:anal}.
Specifically, robust Bayesian decision analysis performs a standard Bayesian analysis for each choice of prior and utility function in a given set.
The resulting bounds on the set of posterior expected utility values
can be interpreted as a
quantification of the decision maker's indeterminacy towards the
management decisions themselves resulting from lack of knowledge
\citep{1991:walley,2007:troffaes:decision:intro,2014:troffaes:itip:decision}.
In this paper, we study a way to deal with severe uncertainty both in values and in system knowledge by bounding probability distributions and utility functions. In this, what we refer to as, robust Bayesian decision analysis, we still express uncertainty using probability and utility, however we relax some of the requirements of standard Bayesian decision analysis. We demonstrate methods to learn from data and derive utilities by revisiting a real and typical environment management problem facing both severe uncertainty and value ambiguity.
To incorporate data, standard Bayesian updating is applied on a set of distributions. We allow for value ambiguity when eliciting utilities for different management alternatives, and there is a simultaneous propagation of imprecision in probability and utility in the analysis.
In this approach, we identify management decisions that are consistently bad under the entire range of probability and utility bounds, and that we can therefore clearly exclude.
We also investigate how the performance of the remaining decisions varies as a function of our beliefs about the world.
Thereby, we show how the proposed robust version of Bayesian decision analysis enables a transparent use of information in environmental problems where little data are available and/or where the objectives are ambiguous.
\section{BACKGROUND}
\subsection{An Environmental Risk Management Problem}
In November 2012, specimens of the crayfish marmorkrebs \textit{Procambarus fallax forma virginalis}
were found in Sweden \citep{2013:bohman} from which twelve were instantly removed. The 2012-2013 winter was very cold, which may have reduced the chances of
any remaining individuals to survive.
Marmorkrebs is an non-indigenous invasive species that recently has established in Europe \citep{2012:chucholl}. It reproduces by cloning \citep{2009:jones}, and therefore marmorkrebs constitute a high risk of bringing new disease vectors, and competition with native crayfish which are already threatened.
According to the Swedish Species Protection Ordinance (2007:845), the species is forbidden to import, move and hold in Sweden. However, illegal activities occur and marmorkrebs could be released into the wild.
In spring 2013, environmental managers were concerned that marmorkreb might still be present. A fast response enhances the chances of successful eradication, and therefore, despite uncertainty in the current state, decision making was urgent.
A group of experts were assigned the task to evaluate the probability of crayfish presence and, together with stakeholders, perform a decision analysis to identify an appropriate action \citep{2013:bohman}.
Some disagreement sustained on how to balance costs, environmental impact, and efficiency. Indeed, whilst a radical decision has higher chances of eradication, it typically comes at higher societal and environmental cost. In addition, there was also uncertainty about the possibility for marmorkreb to successfully survive under each of the management options.
Meanwhile, a sampling scheme was set up to reduce uncertainty by collecting evidence for the presence of marmorkreb. No marmorkreb were observed in any of the trials \citep{2013:bohman}. One conclusion might be that no marmorkreb is present, and therefore no action is needed. To do nothing was also the decision taken by managers in this particular case. A more reflective conclusion acknowledges that even though none were observed, the species, or pathogens brought in by the species, could still be there and actions might still be needed, especially when high values are at stake.
Since 2013, there has been no further individuals observed in Sweden and no major outbreak of a disease associated with marmorkreb.
\subsection{Decision Problems with Uncertainty, Value Ambiguity, or Both}
Environmental decision makers responsible for solving the marmorkreb problem described above face uncertainty in their knowledge about the system, as well as ambiguity in their values. System knowledge is the knowledge about the physical system and
the way in which we can interact with it, that is available for the environmental manager at the time for the decision.
In this case, the lack of observed markmorkrebs in the summer following the introduction does not remove the need to evaluate management alternatives. Instead, the decision must be made under severe (or deep as in \citeauthor{NAP12568} (\citeyear{NAP12568})) uncertainty.
Values influence the way a manager perceive and weight the outcomes of the alternative management actions. Combining severe uncertainty and value ambiguity, \citet{2011:sahlinne} identified four types of situations for decision making with respect to the clarity on uncertainty and values:
\begin{itemize}
\item
In Type 1 situations, the decisions maker has extensive knowledge and information, expressed in terms of precise probability estimates. She also has clear and distinct preferences and values.
\item
In Type 2 situations, the quality and quantity of information is poor, and it is difficult to represent the underlying uncertainty in terms of probability. On the other hand, the decision maker still has clear and distinct preferences and values: she knows what she wants and desires.
\item
In Type 3 situations, the quality and quantity of information is good enough to assess precise probabilities. However, the decision maker lacks harmonious, clear and distinct preferences and values.
\item
In Type 4 situations, both information and preferences are unreliable or ambiguous.
\end{itemize}
Type 1 situations can be seen as the standard, where the usual principles for inference and reasoning work well. As long as you feel you have support to characterise uncertainty by subjective probability you are in Type 1 or Type 3. If not, you are in Type 2 or Type 4. Type 4 situations are not that uncommon. They arise for instance when the actions needed to prevent harm create a conflict within us, since we have to make difficult tradeoffs, and we may be unsure if there is a potential harm in the first place. In invasive species management, this could be rapid action to contain a potentially harmful species by killing all possible hosts within a distance from the sight of observation. Such rapid action creates a conflict between the ambition to protect e.g. trees (which can act as hosts to the alien species) and to eradicate the alien species \citep{2015:porth}. We as humans, are not very good in making such tradeoffs, especially when we are uncertain as well.
\subsection{Extending Bayesian Decision Analysis}
Bayesian decision analysis is able to solve Type 1 problems, while Type 2, 3 and 4 problems requires other ways to represent the impact of knowledge-based uncertainty and/or value ambiguity on the decision objectives and rules how to choose between management alternatives under uncertainty or value ambiguity. Robust Bayesian decision analysis is, as described in this paper, one way to simultaneously deal with uncertainty and value ambiguity. In addition, this approach enable a smooth transition of the specification of the decision analysis over all four types.
In the next section we treat the marmorkreb management problem as a Type 4 problem (however the reasoning is the same under Type 2 and 3) and assess the probability of presence taking into account available evidence (i.e. by going from a prior to a posterior probability of presence). Uncertainty coming from limited knowledge about the system and ambiguous values is characterized by lower and upper bounds on the probability that the crayfish is present after management, and by lower and upper bounds on the expected utility.
\section{METHODS}
\subsection{Management Alternatives}
The management problem is to seek the best management decision for eradicating
any alien crayfish possibly still in the water. To do so, the decision maker needs to assess the
probability of eradication across different decisions, as well as
the associated costs and environmental impacts.
In this particular case, the following management decisions were identified \citep{2013:bohman}:
\begin{enumerate}[I,nosep]
\item Do nothing and inform the public about the problem with non-indigenous species and the need to prevent introductions.
\item Mechanical removal of individual specimens found by fishing.
\item Drain the system on water and removal of individuals by hand.
\item Drain the system of water, dredge and sieve the masses to identify and remove individuals.
\item Use a degradable biocide in combination with drainage to increase the biocide concentration.
\item Increase pH in combination with drainage and removal by hand.
\end{enumerate}
The decision problem is specified through a model that links the variables of the system state and the decision maker's values to parameters and data (\cref{fig:bhm}). The variables and dependencies of this probabilistic network (or, more precisely, influence diagram, since it also includes decision and utility nodes) is further explained in the next two sections.
\subsection{Model of the System}
\begin{figure}
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
semithick]
\tikzstyle{const}=[fill=white,shape=circle,double,draw=black,thick,text=black];
\tikzstyle{var}=[fill=white,shape=circle,draw=black,thick,text=black];
\tikzstyle{dec}=[fill=white,shape=rectangle,draw=black,thick,text=black];
\tikzstyle{util}=[fill=white,shape=diamond,draw=black,thick,text=black];
\node[dec] (deci) {$D$};
\node[var] (beta) [left of=deci] {$\beta$};
\node[var] (hyp2) [left of=beta] {$H'$};
\node[var] (hyp) [above of=hyp2] {$H$};
\node[var] (theta) [above of=hyp] {$\theta$};
\node[const] (s) [above left of=theta]{$s$};
\node[const] (t) [above right of=theta] {$t$};
\node[var] (evi) [above of=deci] {$E$};
\node[const] (alpha) [above of=evi] {$\alpha$};
\node[var] (reward3) [below of=beta] {$A_3$};
\node[var] (reward2) [left of=reward3] {$A_2$};
\node[var] (reward1) [left of=reward2] {$A_1$};
\node[var] (reward4) [right of=reward3] {$A_4$};
\node[util] (util) [below of=reward3] {$U$};
\path (s) edge (theta);
\path (t) edge (theta);
\path (theta) edge (hyp);
\path (hyp) edge (evi);
\path (alpha) edge (evi);
\path (evi) edge (deci);
\path (deci) edge (beta);
\path (beta) edge (hyp2);
\path (hyp) edge (hyp2);
\path (hyp2) edge (reward1);
\path (deci) edge (reward1);
\path (reward1) edge (util);
\path (hyp2) edge (reward2);
\path (deci) edge (reward2);
\path (reward2) edge (util);
\path (hyp2) edge (reward3);
\path (deci) edge (reward3);
\path (reward3) edge (util);
\path (hyp2) edge (reward4);
\path (deci) edge (reward4);
\path (reward4) edge (util);
\node [node distance=0.4cm,left of=theta,left] {$\mathrm{Beta}(st,s(1-t))\sim$};
\node [node distance=0.4cm,left of=hyp,left] {$\mathrm{Bin}(1,\theta)\sim$};
\node [node distance=0.4cm,left of=hyp2,left] {$\mathrm{Bin}(H,1-\beta)\sim$};
\node [node distance=0.4cm,right of=evi,right] {$\sim\mathrm{Bin}(H,\alpha)$};
\end{tikzpicture}
\end{center}
\caption{The graphical model for the crayfish management problem is a probabilistic network of nodes constituting of system states ($H$ presence before management, $H'$ presence after management), evidence (data) ($E$), parameters ($\theta$, $\beta$), hyperparameters ($s$, $t$, $\alpha$), decision ($D$), attributes (Biotic impact ($A_1$), Longevity of impacts ($A_2$), Feasibility ($A_3$) and Cost ($A_4$)) and utility ($U$).
\label{fig:bhm}}
\end{figure}
The state variable $H$ describes whether the crayfish is present in the system ($H=1$) or not ($H=0$). As we are uncertain about
the value of $H$ after the winter, we introduce a parameter $\theta$ which embodies the
probability that $H=1$, i.e. that the alien crayfish is still present during spring 2013. Following
the standard Bayesian approach, to allow us to learn about $\theta$
from data, we need to express our initial belief on this probability by a prior distribution on $\theta$. In this case, we
assume that $\theta$ follows a $\mathrm{Beta}(st,s(t-1))$ distribution,
where $s$ and $t$ are hyperparameters satisfying $s>0$ and $0<t<1$.
We use Walley's parametrisation \citep[Sec.~7.7.3]{1991:walley}
to allow for a straightforward
interpretation of the parameters: $t$ is the prior expectation of
$\theta$, and $s$ controls the variance of the prior
(larger $s$ corresponding to smaller variance).
We learn about $\theta$ through the empirical evidence $E$, where $E=1$ if the
crayfish has been observed in the trial fishing during the summer, and $E=0$ otherwise. If
no alien crayfish is present, obviously none will be
observed. However, even if alien crayfish is present, we are only able to detect it with probability $\alpha$. It is crucial to consider such observation errors when learning from data. In this case, the detection probability reflects that the system is only partially observable.
The presence of the crayfish after the management has taken place is expressed in the model by $H'$. Efficacy, measured as the probability of successful eradication ($H'=0$), is captured by the parameter $\beta$. Since different management methods have different chances of success, $\beta$ depends on the management decision $D$. When decisions impact probabilities, we say that we have \emph{act-state dependence}. This will be important later when we perform sensitivity analysis.
Bringing everything together into a probabilistic causal network (\cref{fig:bhm}), the future state of the system is linked to the decision node and the the evidence is linked to the current state of the system. The laws of probability and Bayesian updating allow us to revise the probability of future state $H'$ given evidence $E$ and decision $D$. This is Bayesian learning, prediction and reasoning.
In this problem, the experts (the assessors) were quite uncertain about the probability $\theta$ of the crayfish being present, about the chance $\alpha$ to see it in the trial fishing,
and about the efficacy (the probability $\beta(d)$) of eradication.
To consider these uncertainties, we could choose to could put a
subjective probability distribution over these parameters. When to use probability or not is a matter of choice. In this case, our uncertainty about these parameters is severe and we do not know which probability distribution to use. Instead, we choose to reflect this uncertainty using robust Bayesian analysis.
Since we can learn about $\theta$ via $E$, we choose to model $\theta$ via a set of Beta distributions, with prior mean $t\in[0.1,0.9]$.
We cannot learn about $\alpha$ and $\beta(d)$, so to keep the analysis simple, we simply model these with intervals,
covering the full range of values that we might reasonably expect.
The experts stated that $\alpha\in[0.1,0.5]$ is a plausible range for detection probabilities in trial fishing of crayfish.
Experts were asked to elicit $\beta(d)$, the efficacy
parameter in our model, representing the probability that the management alternative $d$ is
successful in eradication.
The resulting bounds on the efficacy $\beta(d)$ of successful eradication under the different management decisions $d$ are given in \cref{tab:beta}. A biocide in combination with drainage (V) was judged to always result in successful eradication. Increase pH in combination with drainage and removal by hand (VI) was judged as the second most efficient intervention. The option to drain the system of water, dredge and sieve the masses and remove individuals (IV) were judged to potentially be more successful that the option to drain the system on water and removal of individuals (III), but the experts were uncertain and could not clearly say which one was better than the other. Mechanical removal of individuals had the lowest probability of success.
\begin{table}
\caption{Lower and upper bounds on the probability of successful eradication $\beta(d):=P(H'=0|d)$ for different decisions $d$.
\label{tab:beta}}
\begin{center}
\small
\begin{tabular}{l||cccccc}
&\multicolumn{6}{|c}{\textbf{Decision $d$}}\\
\textbf{Probability} &\textbf{I} & \textbf{II}& \textbf{III}& \textbf{IV} & \textbf{V}& \textbf{VI}\\
\hline
$\underline{\beta}(d)$&0&0.05&0.3&0.4&1.0&0.7 \\
$\overline{\beta}(d)$&0&0.25&0.5&0.7&1.0&0.8
\end{tabular}
\end{center}
\end{table}
\subsection{Derivation of Utility}
To elicit our expert's utility representing the value of different management objectives, we use a modified version of the swing weighting procedure that can cope with severe value ambiguity. A simplified version of the marmorkrebs problem was already treated in \citet{2017:troffaes::swing}, which focused on the theoretical results behind the swing weighting method that we will also use here. In this current paper, we treat the modelling of the likelihood in far more detail, we elicit utility from an expert, and we also focus on the simultaneous propagation of imprecision in probability and utility in a much more realistic setting. It is also possible to use judgements from several experts, but that is beyond the scope of this study.
The impact of the overall outcome for the management problem is described by various attributes identified
as relevant by a group of experts and stakeholders. These were biotic effects and longevity of impacts, feasibility and cost of the method.
These attributes, denoted by $A_1$, \dots, $A_4$, are influenced both by the decision ($D$) and by
whether crayfish is still present or not ($H'$) (\cref{fig:bhm}).
The decision is evaluated through a joint utility function $U$
on these attributes.
Each management decision was scored according to the attributes (\cref{tab:scores1}) using a Likert scale ranging from 1 to 4 constructed for each of these attributes, where
1 corresponds to the worst outcome, and 4 corresponds to the best
outcome (detailed descriptions of all attribute levels are in Appendix \ref{app:scores}).
The expert assessed attribute scores in case of successful eradication for each management decision (\cref{tab:scores1}). The expert, with more than 30 years of experience in crayfish management, based these scores on a literature review on techniques to eradicate freshwater crayfish \citep{2013:bohman}.
Here, the expert provided a point score for every management alternative. We note that, in our analysis, it would also have been possible to assess these scores using a range.
In case of failure to eradicate the invasive species ($H'=1$), the scores for biotic impact and longevity of
impacts drop to their worst values (i.e. a score of 1).
\begin{table}
\caption{Scores (Likert scale 1 to 4) for each attribute and each management decision in case of a successful eradication of the crayfish.
\label{tab:scores1}}
\begin{center}
\small
\begin{tabular}{l|l|l|cccccc}
&\textbf{Worst} & \textbf{Best} &\multicolumn{6}{|c}{\textbf{Decision $d$}}\\
\textbf{Attribute} & \textbf{(score 1)} & \textbf{(score 4)}&\textbf{I} & \textbf{II}& \textbf{III}& \textbf{IV} & \textbf{V}& \textbf{VI}\\
\hline
Biotic impact &High&Low&4&4&3&3&2&2\\
Longevity of impacts&Long&Short&4&4&3&3&2&1 \\
Feasibility&Difficult&Easy&4&4&3&2&1&2\\
Cost&High&Low&4&4&3&1&2&3
\end{tabular}
\end{center}
\end{table}
In order to combine the scoring on all attributes into a utility, we first interpret the scores in \cref{tab:scores1} as marginal utilities (i.e. $U_i(a_i)=a_i$) and make a structural assumption that the joint utility function is a weighted sum of the individual marginal utilities.
So, for given scores (below referred to as a joint reward) $r\coloneqq (a_1,\dots,a_n)$, we assume that:
\begin{equation}
U(r)\coloneqq \sum_{i=1}^n k_i U_i(a_i).
\end{equation}
Although this additive form restricts quite substantially the type of preferences that can be expressed,
the attraction of the linearity assumption is that it reduces the elicitation of the joint utility
to just the elicitation of the weights $k_1$, \dots, $k_n$.
Relaxing this additive form is theoretically possible but
unfortunately it makes the multi-attribute elicitation problem far
more complicated, with many more parameters to be identified, and with
the joint utility function becoming a non-linear function of the
marginal utility functions, even under full mutual utility
independence \citep[Theorem~6.1]{1993:keeney::multiattribute}. For
simplicity, here, we will therefore assume an additive form.
Also, since the resulting utility functions are unique up to a positive linear transformation, we may impose all weights to sum to one.
As a consequence, we only need to elicit $n-1$ of the weights. There are many ways to elicit weights, and we choose an indirect method since experts can find it difficult to interpret the weights directly.
\subsection{Modelling Ambiguity in Attribute Weights}
In order to deal with possible unclear objectives (Type 3 and 4 situations), we will use a method for indirect elicitation described in \citet{2017:troffaes::swing} which models ambiguity in attribute weights. The method allows an almost arbitrary set of rewards to be compared to match the expert's experience, which also allows for ambiguity in the way the different attributes are weighed.
For a detailed mathematical description of the method, we refer to Appendix \ref{app:swingmethod}.
The elicitation method is consistent
under fairly relaxed conditions, which are satisfied in the setting that we
shall study here \citep[Sec.~6]{2017:troffaes::swing}.
It also includes the well known swing weighting method \citep{1986:winterfeldt} as a special case.
A downside of swing weighting is that it considers rewards which
are unnatural for our specific problem, because they consider extreme combinations of attributes, with all but one in their worst state. Therefore, experts may find it difficult to express their preferences over these rewards. From an impact assessment perspective it would instead be more natural to compare rewards made up by only small changes from a reference state. These are thus easier to compare
(regardless of any imprecision in preferences). The method we use is a generalization of swing weighting which deals with these problems.
To simplify the elicitation, we developed a graphical user interface (R code in Supplementary material) using shiny R \citep{shinyr} where the expert goes through the steps annotated below. We ran the elicitation procedure with the same expert twice, and what is reported below is the second iteration. The first iteration had slightly different choices for levels, but the overall conclusions remained the same.
\begin{enumerate}
\item The expert is informed of all attributes, along with a detailed description of all attribute levels (see Appendix \ref{app:scores}).
The expert had some input in setting realistic outcomes for these levels. Throughout the interface, short textual descriptions are used for the levels, rather than numbers, to ensure clarity throughout.
\item The expert is informed that they will be asked to compare these attributes at two levels. As a first step, the expert is asked to identify which pairs of levels they find most comfortable with comparing. Note that, at this stage, we excluded the `no impact' outcomes (level 4) for biotic impact and longevity to ensure meaningful joint outcomes are compared for the next steps.
For example, the expert chose levels $\{1,2\}$ for biotic impact, $\{2,3\}$ for longevity, $\{1,3\}$ for feasibility, and $\{1,3\}$ for cost where the highest levels comprise the reference state.
From these levels, we construct the following joint rewards (directly expressed in terms of marginal utilities):
\begin{center}
\begin{tabular}{l}
rewards \\
\hline
$u_0\coloneqq(1, 3, 3, 3)$ \\
$u_1\coloneqq(2, 2, 3, 3)$ \\
$u_2\coloneqq(2, 3, 1, 3)$ \\
$u_3\coloneqq(2, 3, 3, 1)$ \\
$u_4\coloneqq(2, 3, 3, 3)$
\end{tabular}
\end{center}
Here, $u_4$ is the reference state, $u_0$ modifies $u_4$ in the first attribute,
\dots, and $u_3$ modifies $u_4$ in the fourth attribute.
\item The expert is asked which of the above joint rewards is the worst outcome. In our case, the expert chose $u_2$,
so $r_2\preceq r_j\preceq r_4$ for all $j\in\{0,1,3\}$ (the symbol $\preceq$ means `is less or equally preferred to').
\item Next, we introduce uncertainty in the rewards using lotteries. Given two rewards $a$ and $b$, and a number $\alpha\in[0,1]$, the expression
\begin{equation}
\alpha a\oplus (1-\alpha) b
\end{equation}
denotes an uncertain reward where $a$ is obtained with probability $\alpha$ and
$b$ is obtained with probability $1-\alpha$. Comparing a lottery with a known reward is a common way to indirectly elicit someone's probability of a random event, or someone's utility of a reward.
\item
Uncertainty in weights is obtained by asking the expert to compare and set values on $\alpha$ in a range of rewards (as prescribed in Appendix
\ref{app:swingmethod}). Our expert arrived at:
\begin{align}
\label{eq:expertprefs1}
0.60 r_2 \oplus 0.40 r_4\preceq &r_0\preceq 0.35 r_2\oplus 0.65 r_4 \\
0.50 r_2 \oplus 0.50 r_4\preceq &r_1\preceq 0.40 r_2\oplus 0.60 r_4 \\
0.10 r_2 \oplus 0.90 r_4\preceq &r_3\preceq 0.04 r_2\oplus 0.96 r_4
\end{align}
For instance, \cref{eq:expertprefs1} means that the expert prefers the
certain outcome $r_0$ over the uncertain outcome where $r_2$ happens
with 60\% chance and $r_4$ with 40\% chance. However, when the chance
for $r_2$ is reduced to 35\% and the chance for $r_4$ is increased to
65\%, the expert prefers the uncertain outcome instead.
The other preferences have a similar interpretation.
\end{enumerate}
These assessments then lead to a set of linear inequalities
that determine a convex set of attribute weights.
In this decision analysis it is enough to consider extreme points of this set
(see Appendix \ref{app:swingmethod}) to derive bounds on expected utility. The extreme points were here calculated using the double description method
\citep{1996:fukuda} through the rcdd package in R \citep{r} (R code in Supplementary material).
\subsection{Select Decision}
The last step in the decision analysis, after specifying the parameters of a model to express our beliefs and values of alternative outcomes, is to choose the best management alternative. Because the decision affects the probability of successful management (i.e. we have
act-state dependence), we have to treat the problem using interval dominance
\citep{2007:troffaes:decision:intro}. If probabilities and utilities are precise (Type 1 situations), then this is equivalent to the conventional approach of maximizing expected utility.
In interval dominance, we consider the posterior expected utility interval
of every option. The set of decisions whose intervals are undominated are
then considered as optimal. If there is only one such management alternative, then obviously
that is the decision we ought to pick.
With interval dominance, it is always the case that the best worst case option dominates all non-optimal options. To help visualize this, we depicted the best worst case utility as a vertical dashed line on all plots: every option whose interval intersects with this line is optimal.
If there are multiple undominated management alternatives,
then this means that we have insufficient information to say which is the best. If so, we can deselect poor alternatives and arrived at a set of possible alternatives to select from.
One might then try to refine the set by collecting more information and rerun the analysis,
pick the alternative with the best worst outcome,
or decide which alternative to pick based on other concerns.
It is also possible to refine the interval analysis and eliminate further options by performing a sensitivity analysis over parameters that are not affected by the decision. We defer a discussion of this to \cref{sec:results}.
We estimate the posteriors
for each decision alternative $d$,
each extreme value of $t$, $\alpha$ and $\beta(d)$,
and each extreme attribute weight vector $k$ from \cref{tab:extremeweights}.
The utility was then evaluated through
\begin{equation}
U(d,k)\coloneqq H'\sum_{i=1}^5 k_i U_i(a_i(H'=1,d)) + (1-H')\sum_{i=1}^5 k_i U_i(a_i(H'=0,d))
\end{equation}
where the marginal utilities were taken from \cref{tab:scores1},
and the posterior distribution for $H'$ was sampled using the graphical
model depicted in \cref{fig:bhm} using MCMC sampling in the R package rjags calling JAGS \citep{Plummer03jags:a} (R code in Supplementary material).
Because inferences for lower $s$ values lead to tighter inferences, fixing $s$ to any specific value automatically covers all lower values for $s$ as well \citep{1996:walley::idm}. Therefore, we need not consider intervals for $s$, and only need to consider a reasonable upper bound.
In our analysis, the parameter $s$ was set to 2.
This choice ensures that all typical choices of precise Bayesian prior distributions for Bernoulli sampling are covered \citep{1996:walley::idm}.
\section{RESULTS}
\label{sec:results}
Interval dominance evaluated from the full robust Bayesian analysis reveal that the options ``Use a degradable biocide in combination with drainage`` (V) and ``Increase pH in combination with drainage and removal by hand`` (VI) are dominated by the other management alternatives (\cref{fig:results:all}).
The intervals on the probability of presence after management are in this analysis a consequence from considering uncertainty in terms of sets of probability distributions for the parameter $\theta$ (arising from combinations of values for prior probability $t\in[0.1,0.9]$ and detection probability $\alpha\in[0.1,0.5]$) and intervals on the parameter $\beta(d)$ (the efficiency of each management alternative $d$ (\cref{tab:beta})). The expected utility intervals are a consequence from this uncertainty about the probability of presence after management and sets of values for the utilities.
The expected utility intervals on the four non-dominated decision alternatives in \cref{fig:results:all} are wide and partially overlapping. Therefore, they are all reasonable, but highly uncertain.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig-all.pdf}
\end{center}
\caption{Intervals for posterior probability of crayfish being present after management $P(H'=1|d)$ (left) and expected utility $E(U(d))$ (right) for different decision alternatives $d$: I. Do nothing, II. Mechanical removal by fishing, III. Drainage and removal by hand, IV. Drainage, dredging and sieving before removal by hand, V. Use a degradable biocide in combination with drainage, and VI. Increase pH in combination with drainage and removal by hand, given the prior probability $0.1 \leq t \leq 0.9$
, prior equivalent sample size $s=2$, and detection probability
$0.1 \leq \alpha \leq 0.5$. The highest worst expected utility is indicated by a vertical dashed line. In this case decision alternatives V and VI are dominated.
\label{fig:results:all}}
\end{figure}
A refined analysis evaluating interval dominance for different beliefs in the system state and observation error, may reveal if there are any further dominated alternatives. Uncertainty about the efficiency of management and value ambiguity are not refined in this step, because they are specified in a way such that there is no straightforward way on how refine them any further.
We therefore study if different choices of the hyperparameters $t$ and $\alpha$ from the range reflecting our beliefs, result in additional conclusions about dominance (\cref{fig:results:extremes}).
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig-extremes.pdf}
\end{center}
\caption{Intervals for posterior probability of crayfish being present after management $P(H'=1|d)$ (left) and expected utility $E(U(d))$ (right) for different decision alternatives $d$, for each of the extreme points for prior expected probability (solid line) and detection probability a) $t=0.1,\alpha=0.5$, b) $t=0.1,\alpha=0.1$, c) $t=0.9,\alpha=0.5$, d) $t=0.9,\alpha=0.1$, and prior equivalent sample size $s=2$. The highest worst expected utility is indicated by a vertical dashed line. In this case decision alternatives IV, V and VI are dominated. \label{fig:results:extremes}}
\end{figure}
No individuals were observed in trial fishing, and therefore the probability of the crayfish being present is bounded from above by the prior probability of crayfish presence ($t$), as long as the the detection probability is large enough. As seen from the left hand side of \cref{fig:results:extremes}: the prior (solid vertical line) is almost always at least as high as the interval for the posterior probability of presence after management, with the exception of when $t=0.1$ and $\alpha=0.1$.
A higher prior belief of crayfish presence logically results in a high posterior probability, and in lower values for the expected utility. Also, the differences in expected utilities between decision alternatives becomes smaller, since the utility from the loss when the species is present gets a higher weight compared to the specific utility under each decision alternative.
In contrast, a higher detection probability results in lower risk and narrower bounds for the posterior. This is expected since we put more trust to data compared to the prior belief, and since we do not observe any specimens, the posterior probability becomes relatively lower than before.
For each extreme point of the hyperparameters, we find by the refined analysis, that the option ``Drainage and removal by hand`` (IV) is dominated as well (in addition to V and VI) (\cref{fig:results:extremes}). All three options ``inform only`` (I), ``mechanical removal'' (II), and ``drain the system on water and removal of individuals by hand'' (III), are reasonable. We also see that the option II is the best worst case decision since it has the highest value on the worst possible expected utility (\cref{fig:results:extremes}). The management alternative II also dominates all non-optimal options (since its lower bound on the expected utility exceeds the upper bounds on the expected utilities of IV, V and VI).
\section{DISCUSSION}
We demonstrated the use of robust Bayesian decision analysis to solve
a risk management problem under severe uncertainty and value ambiguity.
Bayesian decision theory is applicable in situations when epistemic uncertainty is judged to be reliably characterized by subjective probability and preferences and values are clear and distinct (a Type 1 problem according to \citet{2011:sahlinne}). Decision problems may face uncertainty in knowledge bases (Type 2), ambiguity in values (Type 3) or both (Type 4). Type 3 problems can occur when there is ambiguity in values from multiple actors and different frames \citep{2005:dewulf}. Here we describe the decision to eradicate an invasive alien species to be taken under severe uncertainty and value ambiguity (Type 4).
To handle the limited possibility to verify the presence of the alien species,
the small chances in detecting the species if present,
and difficulties in predicting the probability of successful eradication under different management options,
we opted to model certain quantities through sets of probability distributions. There will always remain some degree of subjectivity in setting
probability bounds. However, this is no different from standard probability elicitation \citep{OHagan_2006, hemming_burgman_hanea_mcbride_wintle_2017}.
Similarly, to handle the severe ambiguity in how decision makers weigh the different attributes
in the outcome (e.g. cost, feasibility, biotic impact),
we used an extension of the standard swing weighting method
to model these ambiguities through sets of utility functions.
We then propagated these entire sets through a robust Bayesian analysis, and then compared
the posterior expected utility intervals to identify the best possible decisions using interval dominance.
Here, robust Bayesian decision analysis is presented as a modification of Bayesian decision analysis, where the ability to incorporate prior beliefs and evidence is ensured, but uncertainty is treated in a more conservative way and ambiguity in values are acknowledged. This decision theory includes learning under severe uncertainty to identify management decisions that are consistently bad under a plausible range of probability and utility bounds, and that decision makers can therefore clearly exclude. Since updating and decision analysis is done in a single process, it is easy to evaluate sensitivity towards the initial choice of our beliefs about the world.
For the marmorkreb problem, we found that three out of six management alternatives were non-dominated. This conclusion was found after a refined analysis, where interval dominance was evaluated within choices of hyperparameters (similar to paired testing). The analysis gives support to the decision that actually was taken, i.e. to do nothing, but it also shows that mechanic removal by fishing would have been a better choice.
The problem to choose if, and how, to eradicate the alien invasive crayfish is a simple one (from a structural point of view). Note that there are several examples of somewhat similar Bayesian decision analyses to support management of invasive species \citep{2017:Russell, 2017:sakamoto, 2014:rout, 2015:Clarke, Regan2011}. These works address the question as to how long we need to monitor after an eradication attempt to be certain the species is gone. In contrast, in this paper, the management decision is if eradication should be done, and if so, which way to do it.
The calculations for the analysis in this paper can be done within seconds (the R code is available as Supplementary material). For larger problems, with more complex structures, the computational challenge of Bayesian updating and optimization under sets of distributions quickly becomes resource demanding. Future applications of robust Bayesian analysis on risk management require efficient algorithms for learning e.g. relying on MCMC sampling for any type of model or approximations for specific types of models \citep{Rue_INLA_2017}. Today, those algorithms exist for standard Bayesian analysis (e.g. \citet{Plummer03jags:a}).
Value ambiguity may be a larger concern than uncertainty in risk management problems. Eliciting the set of prior distributions and set of utility functions poses a practical challenge.
Although the extended swing weighting method was chosen for its strong
consistency properties, it's not clear whether
such elicitation would work in a practical setting.
In our elicitation procedure, we asked the minimal number of questions to elicit weights, similar to the standard swing weighting method. In order to check the internal consistency, one could elicit further preferences and verify that these are compatible. If inconsistencies appear, our suggestion is to communicate these inconsistencies back to the expert, and ask them to reconsider their preferences to achieve consistency.
In the shiny R app, the expert is given some information about what constitutes a lottery, and how to compare lotteries, involving some hypothetical rewards. We found that this substantially helped the expert to conduct the next step, although it still remained a conceptual challenge for someone not experienced in utility elicitation. Therefore, we recommend the procedure to be run with a facilitator who has good conceptual understanding of the procedure.
We note that robust Bayesian decision analysis is not limited to the how utility was derived in this example.
Moving away from an additive utility function is possible at the expense of a more complicated elicitation problem, where the joint utility function becomes a non-linear function of the marginal utility functions \cite{1993:keeney::multiattribute}.
We chose an additive utility function to keep the analysis straightforward and to allow for linear optimisation, which makes it easy to perform the bounding computationally, even though this choice obviously restricts the type of preferences that can be expressed.
Additionally, we note that the analysis could be expanded to also account for uncertainty in the assessed consequences beyond just eradication and detection. Even though in this case study the expert did not reveal any uncertainty in his assessment, one might well imagine a scenario where under some of the decision alternatives, there is uncertainty in the biotic impact, longevity, feasibility, or cost. One could account for this through probability, or through probability bounding if this uncertainty is severe, making for a more advanced model and more complex analysis.
Any serious attempt to deal with uncertainty and value ambiguity ought to find transparent ways to adapt to the type of decision problem at hand (Type 1 to 4). We suggest that relaxing the assumptions behind standard Bayesian decision theory into robust Bayesian decision theory is one way to do this, and goes from one rigorous principle for learning and quantifying epistemic uncertainty into another \citep{1991:walley,2000:rios:bayesian:sens:anal}.
\section{Acknowledgements}
US was supported by the Swedish research council FORMAS through the project ``Scaling up uncertain environmental evidence'' (219-2013-1271)
and the strategic research environment Biodiversity and Ecosystem Services in Changing Climate (BECC). LE was funded by the Swedish Agency for Marine and Water Management.
\section{Authors' contributions}
US and MT conceived the ideas and designed methodology; US and LE provided the case study; LE provided expert judgment; US and MT did the analysis with equal contribution and the writing of the manuscript; LE reviewed the manuscript for clarity; all have given final approval for publication.
\section{Supplementary material}
The R code for performing the analysis is found at \url{https://github.com/mcmtroffaes/r-crayfish-risk-analysis/}.
\bibliographystyle{apa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,458 |
This subproject contains simple express routes and methods for writing files to a git backed folder and a browser client for interacting with those methods.
## Rationale
While SQL DBs are great for our high volume, stable object types, they might be overkill for some small authored item types, where the schemas change at a much faster clip. More importantly, we get to leverage a huge variety of tools and features around collaboration, versioning, auditing, and editing by using Git as our backend.
## Goals
- Make it easier for authors to edit more things that lie between "content" and "code"
- Allow new ways for authors to edit content in a safe way
- Save dev time by just using Git for versioning author editable content
- Increase dev speed by making it faster and less painful to do migrations
- Increase dev speed by enabling faster syncing to dev of certain authored content from prod
- Hopefully lead to some unplanned benefits by making some content more accessible
## Implementation notes
`GitCmsServer.ts` adds some routes to our API for writing, reading, and deleting files in the "GitCMS". The writes and deletes are committed and pushed.
`GitCmsClient.ts` contains methods for the browser to call these API routes.
## todo:
- Cleanup the routes and the unneeded admin vs api split
- Better test coverage and integration tests
- Support saving drafts to private repos
- Better branch handling
- Better syncing with remote repos
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/www.neetprep.com\/question\/34177-body-moves-rest-constant-acceleration--ms-Its-instantaneousspeed-ms-end--sec-----?courseId=18","text":"# NEET Questions Solved\n\nA body moves from rest with a constant acceleration of 5 m\/s2. Its instantaneous speed (in m\/s) at the end of 10 sec is\n\n(1) 50\n\n(2) 5\n\n(3) 2\n\n(4) 0.5\n\n(1) $v=u+at\u21d2v=0+5\u00d710=50\\text{\\hspace{0.17em}}m\/s$\n\nDifficulty Level:\n\n\u2022 80%\n\u2022 9%\n\u2022 7%\n\u2022 6%","date":"2019-06-20 07:24:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.9511367678642273, \"perplexity\": 7723.227445964784}, \"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-2019-26\/segments\/1560627999163.73\/warc\/CC-MAIN-20190620065141-20190620091141-00193.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaC4"
} | 291 |
UofL Football gives back with Operation Christmas Child
UofL football team goes on shopping spree for Operation Christmas Child
By Kaitlin Rust | December 1, 2019 at 11:18 PM EST - Updated December 2 at 12:21 AM
LOUISVILLE, Ky. (WAVE) - The UofL Football Team stepped off the field and into Walmart Sunday with hopes of making children smile over the holidays.
They filled shoe boxes with toys, school supplies and hygiene items for Operation Christmas Child, which sends them to children in need, ages 2 to 14, in over 100 countries.
Three buses filled with players pulled up to the Bashford Manor Walmart. They only had about 30 minutes and $20 to spend each.
It seemed the biggest struggle for the players was keeping it under budget and small enough to fit into a shoe box.
"This means a lot to me because a lot of kids out there aren't privileged, I was growing up," Offensive Lineman Jean-Luc Childs said. "My parents made sure I had everything I had. It's kind of nice to give them things that they might not get on a regular daily basis."
"Just to think you are going to make some little kids day, month, year just by giving them a box of what we take advantage of every day, toothbrushes, combs, hair ties, soap," Long Snapper Thomas Nauret said.
"I think we take a lot of these things for granted, so getting an opportunity to give back is a great thing," Cornerback PJ Mbanasor said.
"God blessed me with so many blessings, so its important for me to give back and bless others," Free Safety Khane Pass said.
You can see they mean it as they pack each box, many bursting at the seams ready to be sent off to 100 different countries with children in need.
Last year, 11 million boxes were shipped out to mission stations, orphanages, hospitals and churches, often in hard to access war-torn areas and places hit by natural disasters.
Kaitlin Rust
Kaitlin Rust joined the WAVE 3 News team in May 2018 as a multimedia journalist and reporter. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,811 |
arrow-left An arrow pointing left. BMW Z4
2012 BMW Z4
View all 2012 BMW Z4 specs .
Decent ride quality (sDrive30i, sDrive35i)
Driving ease
Vastly improved iDrive system
Compelling styling
Trunk volume with top down
High price, with options
Too much brake-pedal travel
Ride comfort in sDrive35is
Some inconsistent interior pieces
sDrive30i
sDrive35is
Our 2012 BMW Z4 trim comparison will help you decide.
Power-folding hardtop
Available turbocharged six-cylinder
Higher-performance sDrive35is trim
No-cost automatic transmission (sDrive30i)
Standard adjustable drivetrain settings
2012 BMW Z4 review: Our expert's take
Editor's note: This review was written in September 2010 about the 2011 BMW Z4 sDrive35is. Little of substance has changed with this year's model. To see what's new for 2012, click here, or check out a side-by-side comparison of the two model years.
The new Z4 sDrive35is has as tongue-twisting a name as you'll find in the U.S. auto market.
Those who speak BMW can decipher what it means, but all you really need to know is that it fully lives up to the automaker's Ultimate Driving Machine slogan.
The top-dog version of the Z4 retractable-hardtop roadster is a pleasure to drive on the street or the track — just don't expect that experience to come cheap.
There are a number of versions of the Z4, all with excessively complicated names. The base model starts at $46,000 and is called the sDrive30i, and the lineup is capped by the $61,050 sDrive35is, which is new for 2011. To see a side-by-side comparison of all three trim levels, click here.
The first generation of the Z4, which replaced the Z3 as a 2003 model, was more unusual than handsome, but the design hit its stride with a 2009 redesign that softened some of the prior model's more controversial lines. The long hood and passenger compartment set nearly over the rear wheels remain, but a more traditional BMW front-end sets the tone for the entire car.
The top-of-the-line sDrive35is exterior is distinguished from other Z4s by its more aggressive front bumper styling, which calls to mind the high-performance M3, and a restyled rear bumper. Its standard 18-inch alloy wheels are an inch larger than the base ones you get on the other two trims, and there are also sDrive35is badges on the front quarter-panels.
Ride & Handling
You know the minute you get behind the wheel of the sDrive35is that it's been designed to deliver responsive performance. Although there are numerous electronic gadgets, they don't come between you and the car.
The position of the cockpit plays a part in this. As mentioned, the cabin is set near the rear wheels, and the driver looks over a long hood. The orientation enhances the sensation of rotation when carving through a corner; there's no waiting for the rest of the car to make it through the turn because you're practically sitting at the back of the roadster.
The sDrive35is' natural rotation and overall balance make it a fun track car. With its technical corners and long, fast straightaways, Road America in Elkhart Lake, Wis., is good at exposing the shortcomings of production cars in a track environment. The sDrive35is is one of the few production cars I've driven there that managed to hold its own. It continually urges the driver on — unlike some cars, whose response tells you it's time to back off. The low-slung Z4 sDrive35is proved to be a cornering champ, exhibiting little body roll and plenty of grip. While its responses may not be as immediate and direct as a Porsche Cayman's, they're not far behind.
The penalty for this sublime handling is ride quality that can be quite rough on patched asphalt roads. BMW has often impressed me with its ability to combine great handling and good ride comfort, but in the sDrive35is, ride comfort has definitely taken a backseat — or in the case of this two-seat roadster, it's been stuffed in the trunk. Sometimes an adaptive suspension can deliver the best of both worlds, but the feature doesn't do enough here.
A Thrilling Drivetrain
Complementing the handling is a wonderful twin-turbo inline-six-cylinder engine that teams with BMW's seven-speed double-clutch transmission. While I'd still like to have the choice of a traditional manual transmission — it's available only on the lower Z4 trim levels — this gearbox is so good you forget about stick shifts altogether.
The transmission always seems to be in the right gear, and it executes incredibly quick shifts that happen in a snap of your fingers. Matched with the 335-horsepower inline-six, the sDrive35is can go from zero to 60 mph in 4.8 seconds, according to BMW. Whether you leave the transmission in Drive or move the gear selector to its Sport mode, which lets the engine rev higher before shifting gears, the sDrive35is has a high-strung quality about it. It's the automotive equivalent of a Super Ball bouncing around a small room.
The engine sounds great, too. The sDrive35is has a specially tuned exhaust system that results in a louder exhaust sound overall, but there's also a lot of crackling and popping noises emanating from the dual tailpipes that make this Z4 sound like it has an aftermarket exhaust.
The sDrive35is gets an EPA-estimated 17/24 mpg city/highway and takes premium gas.
Convertible Commentary
With its 2009 redesign, the Z4 went from soft-top roadster and hardtop coupe body styles to a retractable-hardtop roadster, which theoretically offers the best of both worlds. It takes about 25 seconds from start to finish to lower or raise the fully powered roof, which is operated by switches in front of the console gear selector. The top stows in the upper portion of the trunk, above a movable partition that reserves enough luggage space for a few soft bags when the top is down.
In terms of chassis rigidity, the contrast between the Z4 and BMW's other retractable hardtop, the 3 Series convertible, is unmistakable. While the 3 Series droptop exhibits noticeable body shudder when traveling on bumpy roads with the top down, the Z4 is solid, without a hint of flex in the windshield frame and no squeaks. It goes to show the advantage that dedicated convertibles like the Z4 have over ones that are based on coupes.
Noise, however, can be a problem in the Z4. With the hardtop over your head and the windows up, there's a fair amount of road noise in the cabin at highway speeds — so much, in fact, that you may have to raise your voice when talking to the person sitting next to you. Cruising on the highway with the top down, road noise is replaced by wind noise and general wind buffeting.
Compared with the previous generation's austere cabin, the current Z4's interior is more welcoming. The two-seat cabin remains quite cozy, and that might be a problem for especially tall drivers. I'm 6-foot-1 and had to move the seat nearly all the way back to get comfortable, and I would have reclined the seat a little more had it not been for the rear bulkhead behind me.
What's a little unusual is that the sDrive35is doesn't come with standard power seats (they're optional), which many potential customers will likely expect considering the car's starting price — regardless of any weight savings the manual seats may offer.
The manual seats have a range of adjustments, including seat tilt and thigh support, but the always important height adjustment doesn't work completely when you're sitting down; you can lower the seat cushion, but you have to climb out of the car — or otherwise take your weight off the seat — to raise it. Like the cabin itself, the seats are snug — some might call them restrictive — but the lateral support they provide is appreciated when cornering. The side bolsters are adjustable.
The Z4's trunk measures 8 cubic feet with the top up and the movable partition out of the way. If you plan on traveling with the top down, bring soft luggage; the partition saves enough room for a few overnight bags below the lowered roof. That's especially valuable, as storage space in the cabin is practically nonexistent.
Standard safety features include antilock brakes, side-impact airbags, an electronic stability system and roll bars behind each of the seats. Check out the Standard Equipment & Specs page for a full list of safety features.
Z4 in the Market
The price of a Z4 has jumped considerably in the past few years, as it's gone from a soft-top to a retractable hardtop, and the sDrive35is represents the latest big bump in cost, coming in at nearly $10,000 more than the midlevel sDrive35i. As tested with a few options, our sDrive35is was $64,225. There's no question this version of the Z4 is a star performer on the street and the track, but that kind of money can buy a Porsche Boxster S, and that fact might give some buyers pause.
Send Mike an email
By Retired in Florida from Orlando, Florida
After moving to Florida, I've been looking for a used sport convertible as a 2nd car for having fun on the road. After test driving the Mazda Miata and the Fiat 124 Spider Abarth, I found the 2012 BMW Z4 to be much more comfortable for my size (6ft. 1in). I also like the classic design, additional luxury accessories and the retractable hardtop. Even though a 2012 Z4 is priced higher than later model Mazdas and Fiats, for me it was worth the extra money.
I LOVE THIS CAR! ZIP AROUND TOWN
By NINAO from FORT WORTH
THIS IS MY FAVORITE CAR EVER! IT'S BEAUTIFUL, FAST, AND FUN! I HAVE TRULY ENJOYED THIS CAR. I LOVE THE DESIGN. ZIPPING AROUND TOWN. IT'S STYLISH AND COMFORTABLE.
Roomy,great handling,eye appeal
By CaptJimUSAF from San Jacinto,CA
Sporty, great european styling, roomy, power hardtop, head turner, steady on road, great handling. Turbo but effecient MPG. Built in Germany. No issues.
See all 5 consumer reviews
New car and Certified Pre-Owned programs by BMW
144 months/unlimited distance
Certified Pre-Owned Elite with less than 15,000 miles; Certified Pre-Owned with less than 60,000 miles
1 year/unlimited miles from expiration of 4-year/50,000-mile new car warranty
All model years for the BMW Z4 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,666 |
\section{Introduction}
A classical topic in combinatorial and geometric group theory is \emph{one-relator groups}, that is, groups that can be defined by a presentation with only one relator.
Magnus proved that a one-relator group has solvable word problem, but the algorithmic complexity of the word problem remains unknown.
A geometric measure of this complexity is given by the Dehn function (see \cite{bridson_geom_wp} for a survey).
The Dehn function of a one-relator group can grow very quickly: the group $\gp{a, t}{a^{(a^t)} = a^2}$ has Dehn function $\operatorname{tower}_2 (\log_2(n))$, which is not bounded by any finite tower of exponents, but its word problem is nonetheless solvable in polynomial time \cite{MUW}.
This is conjecturally the largest Dehn function of a one-relator group; Bernasconi proved a weaker uniform upper bound, namely the Ackermann function \cite{bernasconi}.
On the other hand, much less is known about the intricacies of the geometry of one-relator groups satisfying a polynomial isoperimetric inequality, that is, whose Dehn function is bounded by a polynomial.
All previously known examples are hyperbolic or more generally automatic (see \cite{ECHLPT} for background on automatic groups), and thus have linear or quadratic Dehn function.
For example, every one-relator group with torsion is hyperbolic, and Wise has proved they are virtually special \cite{wise_qhc}.
A standard obstruction to a group having desirable geometry is the presence of a subgroup isomorphic to the Baumslag--Solitar group $BS(m, n) = \gp{a, t}{t^{-1} a^m t = a^n}$ for some $m \neq \pm n$: this group has a distorted cyclic subgroup, and its Dehn function is exponential.
A distorted cyclic subgroup rules out being hyperbolic, acting properly cocompactly on a CAT(0) space, or acting freely on a CAT(0) cube complex (of possibly infinite dimension).
A torsion-free one-relator group has geometric dimension 2 \cite{lyndon} so by a theorem of Gersten such a Baumslag--Solitar subgroup gives an exponential lower bound on Dehn function \cite[Theorem C]{gersten_dehn}, further ruling out automaticity which would require a quadratic isoperimetric inequality.
It has been asked whether such Baumslag--Solitar subgroups are the only pathologies for one-relator groups:
\begin{qn}
\label{qn:muw_bs}
Is it true that one-relator groups with no subgroups isomorphic to $BS(m,n)$, for $m \neq \pm n$, are automatic?
\end{qn}
\begin{conj}[Wise, {\cite[1.9]{wise_tubular}}]
\label{conj:wise}
Every [torsion-free] one-relator group with no subgroup isomorphic to $BS(m, n)$, for $m \neq \pm n$, acts freely on a CAT(0) cube complex.
\end{conj}
Question~\ref{qn:muw_bs} was articulated when the theory of automatic groups was first developing \cite[Problem 11 ff.]{gersten_problems} and was posed more recently by Myasnikov--Ushakov--Won \cite[1.5]{MUW}.
If true, it would imply that all polynomial Dehn functions of one-relator groups are linear or quadratic.
The first-named author introduced in his thesis~\cite{gardam} the one-relator groups
\[
R(m, n, k, l) := \gp{x, y, t}{x^m = y^n, \, t^{-1} x^k t = x^l y } \cong \gp{x, t}{x^m (x^{-l} t^{-1} x^k t)^{-n} }
\]
for $\abs{m}, \abs{n} \geq 2$, $k \neq 0$, $l \not \equiv 0 \mod m$.
He then proved that they have no distorted Baumslag--Solitar subgroups, and that they are CAT(0) precisely when $\abs{k} > \abs{l + \frac{m}{n}}$.
In this paper we consider a subfamily of these groups: define \[
R_{p, q} := R(2, 2, 2q, 2p-1) \cong \gp{x, y, t}{x^2 = y^2, \, t^{-1} x^{2q} t = x^{2p-1} y}.
\]
\begin{thm}
\label{thm:main}
Let $p > q$ be positive integers.
The one-relator group $R_{p, q}$ has Dehn function $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p/q)$.
In particular, it has no subgroup isomorphic to a Baumslag--Solitar group $BS(m, n)$ with $m \neq \pm n$, but is not automatic and not CAT(0).
\end{thm}
This answers Question~\ref{qn:muw_bs} negatively.
The key observation is that $R_{p,q}$ is virtually a \emph{tubular group}.
A group is tubular if it splits as a finite graph of groups with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
It is demonstrated in Theorem~\ref{thm:snowflake_subgroup} below that $R_{p,q}$ has an index two subgroup that is isomorphic to the Brady--Bridson snowflake (tubular) group $G_{p,q}$.
These groups are discussed in full in Section~\ref{section:snowflake}, but the salient fact is that $G_{p,q}$ has Dehn function $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p/q) > 1$; as the Dehn function is invariant up to finite index subgroups the first part of the statement holds.
In contast, automatic and CAT(0) groups have at most quadratic Dehn function.
Since $R_{p,q}$ is of geometric dimension 2 we conclude from Gersten's theorem that there are no such Baumslag--Solitar subgroups as their presence would force at least exponential Dehn function.
\end{proof}
\begin{rmk}
Jack Button has shown that for odd $q \geq 3$, the group $R_{1,q}$ is \emph{not} residually finite, giving the first examples of one-relator groups that are CAT(0) (by \cite[Theorem G]{gardam}) but not residually finite~\cite{Button}.
The proof shows that $G_{1,q}$ is non-Hopfian, which implies that $G_{1,q}$ and thus $R_{1,q}$ are not equationally Noetherian, resolving \cite[Problem 1]{baumslag_problems}; in fact one can extend Button's surjective endomorphism of $G_{1,q}$ with non-trivial kernel to show that $R_{1,q}$ itself is non-Hopfian, resolving \cite[Problem 7]{baumslag_problems}.
Button has informed us that he also has completely determined for which~$p$ and $q$ the group $R_{p,q}$ is residually finite.
\end{rmk}
In~\cite{wise_tubular}, Wise classified the tubular groups that act freely on CAT(0) cube complexes.
In Section~\ref{section:cubulation} we apply this classification to disprove Conjecture~\ref{conj:wise}.
The Dehn function is a quasi-isometry invariant, so there are infinitely many quasi-isometry types of counterexamples to Question~\ref{qn:muw_bs} and Conjecture~\ref{conj:wise}.
\section{Virtually snowflake groups}
\label{section:snowflake}
In~\cite{BB} Brady and Bridson studied the groups \[
G_{p,q} := \gp{ a, b, s, t }{ [a, b], \, s^{-1} a^q s = a^p b, \, t^{-1} a^q t = a^p b^{-1} }
\]
defined for positive integers $p$ and $q$.
Due to the suggestive nature of their van Kampen diagrams, these are called \emph{snowflake groups}.
The main theorem of their paper states that for $p \geq q$, the Dehn function of $G_{p,q}$ is $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p/q)$.
This gives the Dehn function of $R_{p,q}$, via the~following:
\begin{thm} \label{thm:snowflake_subgroup}
The snowflake group $G_{p,q}$ is an index $2$ subgroup of the one-relator group~$R_{p,q}$.
\end{thm}
\begin{proof}
First, we re-write the presentation of $R$ to exploit the fact that $\gp{x, y}{x^2 = y^2}$ is the fundamental group of the Klein bottle: we map $x \mapsto a$ and $y \mapsto ab$ to get \[
R_{p,q} \cong \gp{a,b,t}{ a^{-1} b a b, \, t^{-1} a^{2q} t = a^{2p} b}.
\]
Let $X$ be the graph of spaces for $R_{p,q}$ with a vertex space a Klein bottle and edge space a cylinder.
We can assume that the attaching maps are geodesics in the Klein bottle as in Figure~\ref{fig:cover2}.
Let $X' \rightarrow X$ be the index two regular cover corresponding to the map to $\mathbb{Z}/2$ defined by $a \mapsto 1$, $b \mapsto 0$ and $t \mapsto 0$, indicated in Figure~\ref{fig:cover2}; on the Klein bottle subspace this is just the oriented double cover.
\begin{figure}
\centering
\includegraphics[width=0.55\textwidth]{index2cover.pdf}
\caption{$G_{3,1}$ as an index $2$ subgroup of $R_{3,1}$. The two blue arrows in each cylinder are attached along $a$ and the orange arrows along geodesics representing (lifts of) $a^6 b$.}
\label{fig:cover2}
\end{figure}
The fundamental group of $X'$ has the presentation \[
\gp{x, y, s, t}{ [x, y], \, s^{-1} x^q s = x^p y, \, t^{-1} x^q t = x^{p} y^{-1}}
\] where $x$ and $y$ are the generators of the fundamental group of the torus (corresponding to $a^2$ and $b$ respectively).
This group is none other than $G_{p,q}$.
\end{proof}
\begin{cor}
The set of exponents $\rho$ such that $n^\rho$ is the Dehn function of a one-relator group is dense in $[2, \infty)$.
\end{cor}
\begin{rmk}
\cite[Problem 1.4]{MUW} asks whether quadratic Dehn function implies that a one-relator group is automatic.
The snowflake group $G_{1,1}$ is Gersten's non-CAT(0) free-by-cyclic group introduced in \cite{gersten_cat}.
It has been announced that this group is not automatic \cite{BR_free_by_cyclic}, which would settle this remaining problem as well.
\end{rmk}
\section{Non-cubulated examples}
\label{section:cubulation}
In~\cite{wise_tubular}, Wise gave a necessary and sufficient condition for a tubular group to act freely on a CAT(0) cube complex.
The condition is the existence of an \emph{equitable set} which permits the construction of immersed walls in the graph of spaces associated to the group.
A dual cube complex is then obtained from the corresponding wallspace.
\begin{prop}
Let $p$ and $q$ be positive integers.
The snowflake group $G_{p,q}$ acts freely on a CAT(0) cube complex if and only if $p \leq q$.
\end{prop}
\begin{proof}
The existence of a free action of $G_{p,q}$ on a CAT(0) cube complex is equivalent to the existence of an equitable set: $S = \{ (u_1, v_1), \dots, (u_k, v_k) \} \subseteq \mathbb{Z}^2 \setminus \{ (0, 0) \}$ such that $[ \mathbb{Z}^2: \langle S \rangle ] < \infty$ and
\[
\sum_i \# [(q, 0), (u_i, v_i) ] = \sum_i \# [(p, 1), (u_i, v_i) ],
\quad
\sum_i \# [(q, 0), (u_i, v_i) ] = \sum_i \# [(p, -1), (u_i, v_i) ]
\]
where $\#[(a, b), (c, d)]$ denote the ``intersection number'' $\abs{ad - bc}$.
Thus the problem reduces to solving
\begin{equation}
\label{eq:target}
\tag{$\star$}
\sum_i \abs{q v_i} = \sum_i \abs{p v_i - u_i} = \sum_i \abs{p v_i + u_i}.
\end{equation}
If $p \leq q$, then a solution is $\{ (q, 1), (q, -1) \}$.
If $p > q$, there is no solution:
\begin{align*}
\sum_i \abs{p v_i - u_i} + \abs{p v_i + u_i} \geq \sum_i \abs{(p v_i - u_i) + (p v_i + u_i)} = 2 \sum_i \abs{p v_i} \geq 2 \sum_i \abs{q v_i}
\end{align*}
and equality can only hold in this last inequality if all $v_i = 0$, in which case some $u_i \neq 0$ and \eqref{eq:target} clearly cannot hold.
\end{proof}
\begin{cor}
\label{cor:cube}
Let $p > q$ be positive integers.
Then the one-relator group $R_{p,q}$ has no subgroup isomorphic to $BS(m, n)$ for $m \neq \pm n$ but does not act freely on a CAT(0) cube complex.
\end{cor}
\begin{rmk}
One can also deduce that for $p > q$ the group $G_{p,q}$ does not act freely on a CAT(0) cube complex from the fact that $G_{p,q}$ has a cyclic subgroup with distortion $n^\alpha$ \cite[Corollary 2.3]{BB} whereas cyclic subgroups are undistorted in groups admitting such actions by \cite[Theorem 1.5]{haglund_isometries} (which was generalized to finitely generated virtually abelian subgroups in \cite{woodhouse_axis}).
\end{rmk}
\subsection*{Acknowledgements}
We thank the referee for suggestions that improved the exposition of this paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,422 |
South Australia Electoral Commission has released a Timetable of Events for the State Election to be held on Saturday 17 March 2018.
An information session will be held by the Electoral Commissioner between 6:00 pm – 8:00 pm at Level 6, 60 Light Square Adelaide.
The session will provide information on the electoral processes in South Australia and the requirements of candidates.
This session will be beneficial for those persons intending to nominate for the 2018 State Election for either the House of Assembly or the Legislative Council, who are not endorsed by a Registered Political Party. For information about funding and disclosure please see separate briefing sessions on 7 February and 9 February, detailed below.
Postal vote applications are now available online and at Australia Post outlets.
If you are intending to nominate as an independent candidate at the 2018 State Election, then you should be aware that candidates have obligations under the funding and disclosure scheme. An information session will be held at 2 pm at Level 6, 60 Light Square Adelaide.
If you are intending to nominate as an independent candidate or as an independent group of candidates at the 2018 State Election, then you should be aware that candidates and groups have obligations under the funding and disclosure scheme. An information session will be held at 10 am at Level 6, 60 Light Square Adelaide.
Deadline of 5 pm for registered political parties to lodge nominations on behalf of candidates endorsed by them.
Electoral Commission SA will conduct the declaration of candidates and ballot draw for the Legislative Council as soon as practicable after 11 am on 27 February 2018.
Deadline of 5 pm for postal vote applications to be received by Electoral Commission SA.
Polling places open from 8 am-6 pm. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,811 |
source hack/build/config.sh
source hack/build/common.sh
LINTABLE=(staging/src/kubevirt.io/containerized-data-importer-api pkg cmd tests tools)
ec=0
out="$(gofmt -l -s ${SOURCE_DIRS} | grep ".*\.go")"
if [[ ${out} ]]; then
echo "FAIL: Format errors found in the following files:"
echo "${out}"
ec=1
fi
for p in "${LINTABLE[@]}"; do
echo "running golint on directory: ${p}"
out="$(golint ${p}/...)"
if [[ ${out} ]]; then
echo "FAIL: following golint errors found:"
echo "${out}"
ec=1
fi
done
exit ${ec}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,669 |
Surgical considerations in pediatric necrotizing fasciitis
A Pandey, AN Gangopadhyay, SP Sharma, V Kumar, SC Gopal, DK Gupta
Department of Pediatric Surgery, Institute of Medical Sciences, Banaras Hindu University, Varanasi - 221 005, U.P., India
A N Gangopadhyay
Department of Pediatric Surgery, Institute of Medical Sciences, Banaras Hindu University, Varanasi - 221 005, UP
Background: Necrotizing fasciitis (NF) is a serious infection of soft tissues. This paper presents experience with pediatric NF and suitability of conservative surgery in its management. Materials and Methods : In this retrospective study, 70 patients of NF were managed during the study period of eight years. The study was divided into two time periods- first period (June 1998 to June 2001- group 1) and second period (June 2001 to June 2006- group 2). The parameters studied were age, sex, site of involvement and treatment. The treatment included intravenous antibiotics, supportive therapy and either aggressive (group 1) or conservative surgery (group 2). Results: Age of presentation ranged from 10 days to 11 years. Male to female ratio was 1.69:1. Back was the commonest site to be involved. Culture reports were polymicrobial in 70% with predominance of Staphylococcus species. Predisposing factors included malnourishment, boils, scratch injury, intravenous cannulation and injections. Conservative surgery had better outcome in terms of hospital stay, complications and cost of treatment. Conclusion: NF is a serious and disease which requires immediate and all out attention. Early diagnosis, aggressive supportive treatment and conservative surgery improve survival.
Pandey A
Gangopadhyay A N
Sharma S P
Kumar V
Gopal S C
Gupta D K
necrotizing fasciitis | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,707 |
Aloe Vera gives you guaranteed results on thin weak hairs & roots hair loss, dandruff premature baldness & greying also best for baby boys & girls.
Usage: Apply at hairs and roots before sleep and wash in morning or leave for one hour before washing. Aloe Vera is the fastest remedy to enhance hair thickness and mousturize scalp. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,828 |
Q: Grails shell not functional in Grails 3.0.2? Grails now builds using gradle --which is great-- but a side effect seems to be the shell no longer works? When I run
$ grails shell
It doesn't appear to allow me to type any input, and cursor is stuck on grade's building line.
Groovy Shell (2.4.3, JVM: 1.8.0_45)
Type ':help' or ':h' for help.
----------------------------------------------------------------------------
groovy:000>
> Building 83% > :shell
I've had this problem trying to make interactive groovy scripts that build in gradle - is there a workaround?
A: Grails inherited this bug from springboot. It is now fixed and will be packed in grails 3.03 probably. Till then I would recommend using grails console instead. It will open a swing UI and will have same environment as grails shell.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,783 |
Il distretto di Las Lomas è uno dei nove distretti della provincia di Piura, in Perù. Si trova nella regione di Piura e si estende su una superficie di 522,47 chilometri quadrati.
Istituito il 3 aprile 1936, ha per capitale la città di Las Lomas; nel censimento 2005 contava 26.547 abitanti.
Altri progetti
Collegamenti esterni
Sito dell'Istituto nazionale di statistica e informatica del Perù | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,184 |
Q: Is Geralt's Amnesia because I didn't import a Witcher 1 save file? In the opening scene of the game, you meet the Crinfrid Reavers. Dialog with them implies that Geralt met them previously. Geralt claims to have amnesia.
Is this because I did not import a save-file from witcher 1 (and thus the game doesn't know how I "would" have treated them?) or does Geralt truly have amnesia? Did they appear in Witcher 1 at all?
A: Little late to the party, but about the Reavers, its a pretty significant reference if you read the books/short stories. Its the first reference in TW2 to Geralt's relationship to Yennefer (one of the Reavers called her Connifer or some such). The Reavers are in the short story, "Limit of Possibility," which is heavily referenced later in the game. I don't want to spoil it too much but just know that you (Geralt) was a friend of a Golden Dragon (which was also referenced in TW1).
I wouldn't call the Reavers in the short story background characters. They and their leader, Boholt really caused a lot of trouble for Geralt and Yennefer. Theres also kinda a major story and game spoiler related to the Golden Dragon I don't want to ruin.
A: Cinfrid Reavers appeared only in one short story, as far as I can remember, and even there as background characters. Don't know what's their role in the game, but most people familiar with Sapkowski's writing probably wouldn't even remember them. Think of it as a wink to the people 'in the know'.
I don't remember them in the first game.
A: Geralt's amnesia started in the beginning of the first game, it has nothing to do with save files. Geralt doesn't remember most of what happened to him in Sapkovsky's books.
As far as I remember there was no Crinfrid Reavers in the first game.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,881 |
{"url":"http:\/\/www.plant-ecology.com\/EN\/abstract\/abstract9072.shtml","text":"Chin J Plan Ecolo \u203a\u203a 2018, Vol. 42 \u203a\u203a Issue (3): 327-336.\n\n\u2022 Research Articles \u2022\n\nEffects of enclosure on carbon density of plant-soil system in typical steppe and desert steppe in Nei Mongol, China\n\nYAN Bao-Long,WANG Zhong-Wu*(),QU Zhi-Qiang,WANG Jing,HAN Guo-Dong*()\n\n1. College of Grassland, Resources and Environment, Inner Mongolia Agricultural University, Huhhot 010019, China\n\u2022 Online:2018-03-27 Published:2018-03-20\n\u2022 Contact: Zhong-Wu WANG,Guo-Dong HAN E-mail:zhongwuwang1979@163.com;hanguodong@imau.edu.cn\n\u2022 Supported by:\nSupported by the Strategic Priority Research Program of the Chinese Academy of Sciences(XDA05050402-6);the National Key Research and Development Project of China(2016YFC0500504);the Science and Technology Projects in Inner Mongolia Autonomous Region and West Light Foundation of Chinese Academy of Sciences.\n\nAbstract:\n\nAims As an immense carbon (C) stock, grassland ecosystem plays a crucial role in global C cycling. The objective of this research was to reveal the effects of enclosure on C density of the plant-soil system by comparing the aboveground biomass (AGB), belowground biomass (BGB) and soil C density in enclosure plots with those in grazing plots in the typical steppe (TS) and desert steppe (DS) in Nei Mongol, China.\n\nMethods At each of the 19 study sites, we set up a 100 m \u00d7 100 m plot and 5 quadrats (1 m \u00d7 1 m) along the diagonal transect within each plot. At each quadrat, AGB was harvested first and then a soil core (0-100 cm depth, 7 cm inner diameter) was taken for BGB and soil C content measurement. Each soil core was divided into 7 depth increments (0-5 cm, 5-10 cm, 10-20 cm, 20-30 cm, 30-50 cm, 50-70 cm, 70-100 cm).\n\nImportant findings (1) Enclosure significantly increased C density of AGB and BGB in TS. In DS, enclosure significantly increased C density of AGB, but had no significant effect on the C density of BGB. (2) Enclosure significantly increased soil C density in TS, but had no significant impact in DS although there was an increasing trend. (3) For all increments along the soil profile, enclosure significantly increased BGB and soil C density compared to grazing plots in TS, but this effect was not found in DS. (4) Enclosure increased C density of the plant-soil system by 2.2 and 1.6 times in TS and DS, respectively. 65% and 89% C was stored in soil in TS and DS, respectively, and BGB C stock accounted for more than 90% of total biomass C in both TS and DS. Enclosure is an effective approach to improve C sequestration in grassland ecosystems.","date":"2020-06-06 11:11:19","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\": 0.17527486383914948, \"perplexity\": 10001.635621206922}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2020-24\/segments\/1590348513230.90\/warc\/CC-MAIN-20200606093706-20200606123706-00435.warc.gz\"}"} | null | null |
Independent Advisors Looking to Internally Manage Growth, SEI Forum Finds
Growth is both a blessing and a curse, said advisors at a recent forum hosted by SEI (Nasdaq: SEIC), a leading global provider of outsourced asset management, investment processing and investment operations solutions. The SEI Advisor Network's Advisor Growth Forum assembled 25 of the company's fastest-growing advisor clients to discuss current challenges and share best practices as they strive to grow their businesses.
Attendees belonged to a diverse mix of firms, e.g. solo practitioners, niche advisories, ensemble firms, family firms, etc. Despite different characteristics, however, all pointed to a need to invest in firm infrastructure including operations and staffing to enable growth.
"There comes a critical point in the evolution of your firm when you realize your current structure or model just won't accommodate growth," said Paul Salisbury, Insight Group on Salt Lake City, Utah. "The more we grew, the exponentially more difficult things like maintaining client service, overseeing budgets and running day-to-day operations became to manage."
Advisors noted that they had begun or recognized the need to begin focusing on specific areas of their businesses. Some issues discussed included:
* Standardization of processes. As advisors' practices grew, their processes didn't grow with them. Particularly with specific client exceptions, the gaps in service and inefficiencies increased. By implementing standards and systematizing operations and outsourcing functions, however, they were able to improve efficiencies almost instantly.
* Integration of systems. Most advisors use multiple systems for different functions of their business -- financial planning, aggregation, custody, etc. -- resulting in increased cost and time to the advisor. While advisors acknowledged that integrating all systems might not be realistic, they noted that exploring ways to consolidate disparate systems was extremely beneficial.
* Expansion of business management. Many advisors said they were considering hiring a COO or general manager to oversee operational aspects of their business. Yet in doing so, advisors are presented with new challenges -- quality staff is becoming increasingly difficult to recruit and/or afford.
* Growth - Lifestyle dilemma. Though all advisors attending were in "full growth mode," most stressed that not compromising their current lifestyles was a top priority. Time, income and quality of life were all factors they considered in their struggle to manage growth.
* Analyzing cost of client relationships. Because clients' demand for service has increased over the past few years, say advisors, they dedicate more time per client. By comparing time spent to the value of each relationship, advisors can pinpoint opportunities to increase profitability and free up time.
* Better understanding of existing technology. Many advisors felt that gaining a better understanding of the full capabilities of in-house technologies was critical to finding more time to spend with clients and prospects.
* Focused marketing programs. Although the majority of advisors still rely heavily on referrals, many are realizing the need for structured, proactive marketing programs to obtain new clients. They are investing in numerous traditional marketing vehicles, particularly public relations campaigns.
According to another recent survey by SEI, 43 percent of advisors attributed growth to their decision to outsource investment processes, a common theme among growing firms.
Guest facilitator Mark Tibergien, principal of Moss Adams, spoke on barriers to growth advisors face nationwide, as well as leading targeted work sessions. Cautioning advisors about their investment in infrastructure, Tibergien challenged attendees to ask themselves whether or not they're growing at the right pace, are growing with the right clients or are you adding overhead to support the wrong clients.
Steve Onofrio, Senior Managing Director, Sales and Support, SEI Advisor Network, also led sessions aimed at understanding operational challenges.
"The growth-oriented advisor experiences a unique dilemma," said Onofrio. "The more successful they are, the harder their jobs become. That's why it's critical to look to long-term infrastructure investment to correct this imbalance."
Participating firms, all of which had more than $200 million in firm assets, attended the forum because of their expressed interest in organizational, platform and structural issues associated with growth.
About SEI Advisor Network
SEI Advisor Network provides independent advisors with outsourced wealth management platforms that are designed to meet the demands of a new generation of wealthy clients. In an evolving wealth management industry, the group offers an end-to-end process for successfully transforming their clients' businesses in every critical area, including marketing, practice management, investment strategy and client relationship platforms. The SEI Advisor Network is a strategic business unit of SEI. For more information, visit http://www.SEIAdvisorNetwork.com.
SEI (NASDAQ: SEIC) is a leading global provider of outsourced asset management, investment processing and investment operations solutions. The company's innovative solutions help corporations, financial institutions, financial advisors, and affluent families create and manage wealth. As of the period ending December 31, 2006, through its subsidiaries and partnerships in which the company has a significant interest, SEI administers $366.6 billion in mutual fund and pooled assets and manages $181.5 billion in assets. SEI serves clients, conducts or is registered to conduct business and/or operations, from more than 20 offices in over a dozen countries. For more information, visit http://www.seic.com.
Edited by: InvestmentWires Staff, | {
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Black Swan Society: Nassim Taleb's 10 Principles
September 27, 2010 in Global Crisis Blog | Leave a comment
Black Swan Society: Nassim Taleb's 10 Principles, or
"In French socialism 1980's, the government takes over the banks; in the U.S. 2000's, the banks take over the government…." – Nassim Taleb
Nassim Nicholas Taleb is the author of The Black Swan: The Impact of the Highly Improbable, a New York Times bestseller published in 2007 just before the onset of the global crisis. Taleb has now done a second edition with a new section titled "on robustness and fragility". It includes "The 10 Principles for a black-swan-robust society", originally published as an editorial in the Financial Times in 2009. In a footnote, Taleb notes that the FT editor changed his own original title to "Black-Swan-proof", something which he says does not exist.
Here is a brief version of his 10 principles that can help societies cope with the aftermath of crises that are inevitable and always will be:
Let what's fragile break early, while it's small. "Nothing should ever be too big to fail".
Do not socialize losses and privatize gains. "Whatever needs to be bailed out should be nationalized".
Don't let people wearing blindfolds drive buses ever again. "The economics establishment should be ignored [forever]".
Forbid people with 'incentive' bonuses from managing your financial risks. "Odds are they will cut corners to show 'profit' in order to gain the bonus".
Compensate complexity with simplicity. "Complexity is a form of leverage". Avoid it.
Do not give children dynamite sticks. "Ban complex financial procedures that nobody understands".
Governments should never ever need to 'restore confidence'.
Don't give addicts more drugs if they are in withdrawal. "Using leverage to cure excess leverage is pure denial. The debt crisis is not temporary, it is structural and requires rehab".
Citizens should not use financial assets as a repository of value and should not rely on fallible 'expert' advice for their retirement. "Economic life should be definancialized".
Make an omelet with broken eggs. "Remake the system before it remakes itself (through crisis)."
Secrets of the REAL Start-Up Nation: Inside Israeli Innovativeness
September 27, 2010 in Innovation Blog | Leave a comment
Start-up Nation is a best-selling book about Israeli innovativeness by Dan Senor and Saul Singer.[1] This book has aroused enormous interest in the source of Israel's boundless creativity, mainly in the U.S. but in other countries as well. Thousands are reading the book to learn how Israel invented the cell phone, Copaxone, Azilect, a kind of heart pump, drip irrigation, the Given Imaging pill that 'broadcasts' your intestines' condition, the Pentium and Centrino chips, drip irrigation, cherry tomatoes, and a thousand other life-changing inventions, while fending off enemies and squabbling endlessly with one another. Former NBC TV anchor Tom Brokaw wrote, "There is a great deal for America to learn from the very impressive Israeli entrepreneurial model… START-UP NATION is a playbook for every CEO who wants to develop the next generation of corporate leaders."
"What is driving (Israeli inventiveness)," Senor recently told the cable network CNBC, "is a national ethos, resilience, the fight for survival." The book itself stresses two qualities embedded in Israeli culture: Hutzpah (brash impudence, willingness to challenge anything) and leadership experience gained in the Israeli Army.
Start-Up Nation's stories about Israeli entrepreneurs are truly wonderful. But can we dig deeper beneath these stories and find empirical research that illuminates this phenomenon?
With the aid of Shlomo Gradman, founder of the Israeli Semiconductor Club, we asked those who registered for the club's inaugural event to answer a brief questionnaire: ""Invest" 100 shekels in some 13 innovation factors (personal, cultural, contextual) according to their relative importance". We received nearly 50 responses, out of 200 registered participants. They came from a slice of Israeli high-tech industry engaged in innovative chip design. A summary of the responses is shown above in the diagram and below in the Table.
Table 1. 13 Innovation Factors, by Rank and Mean Score.
RANK FACTOR MEAN
1 Resilience 13.75
2 Stubborn Persistence 10.75
3 Role models 9.93
4 Desire to change the world 9.25
5 Lack of fear of risk 8.59
6 Desire for wealth 8.14
7 Willingness to break rules 6.7
8 Desire for independence 6.23
9 Willingness to challenge authority 5.45
10 Hutzpah 5.16
11 Army technology experience 4.93
12 Army leadership experience 4.32
13 Desire for fame 2.48
Three of the top five innovativeness factors are "cultural" in nature, deriving from Israeli culture: Resilience, stubborn persistence and lack of fear of risk. One is contextual (role models – other Israeli entrepreneurs who have done great things), and one is 'personal' but with a strong cultural element (desire to change the world). It is interesting that army experience in both technology and leadership, cited as strong factors by Start-Up Nation, ranked 11th and 12th out of 13. This may in part reflect that respondents are chip designers, a technology not central to IDF R&D efforts.
[1] Dan Senor, Saul Singer, Start-up Nation: The Story of Israel's Economic Miracle: Published by Twelve (Reed Elsevier), 2009.
Drake Equation: The Biggest Question of All
Drake Equation: The Biggest Question of All, or:
Is There Life in the Universe? (Is There Life on Earth?)
The Answer is 10
Frank Drake
In academic research, as in innovation, creative people dare to take huge gambles, by asking very hard, big humongous questions, questions whose very nature discredits their rationality. Here is an example, featured on the wonderful Discovery program on BBC.
Almost 50 years ago, a young astrophysicist named Frank Drake dared to ask the question, how many other civilizations exist in the Universe, apart from life on earth? Until Drake, that question was regarded as whacko by astrophysicists, and those who researched it generally used methods that were, well, also quite whacko. Drake endangered his career by tackling it. But he was driven by curiosity and passion. So he tackled it anyway. And then, the U.S. Science Foundation (which also thought it was a key question – these were Cold War days, what if Aliens contacted the Russians first instead of the Americans???) asked him to convene a conference on the subject, inviting the top experts. So Drake called a conference to be held in Green Bank, West Virginia, attended by a dozen top astrophysicists, including the celebrity Carl Sagan.
Now, a conference has to have an agenda. So Drake sat down, and built an equation that included the seven key factors that would determine how many civilizations there are in the universe that potentially could be contacted, equal to "N".
The Drake equation states that:
where: N = the number of civilizations in our galaxy with which communication might be possible; and R* = the average rate of star formation per year in our galaxy; fp = the fraction of those stars that have planets; ne = the average number of planets that can potentially support life per star that has planets;fℓ = the fraction of the above that actually go on to develop life at some point; fi = the fraction of the above that actually go on to develop intelligent life; fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space; L = the length of time such civilizations release detectable signals into space
Drake's values give N = 10 × 0.5 × 2 × 1 × 0.01 × 0.01 × 10,000 = 10
The universe is very large. And very old. Our sun is 4.5 billion years old. Other suns are as much as 10 billion years old. So these 10 civilizations are likely very very far away. Millions of light years. So messages sent from Earth by the SETI Study of Extra Terrestrial Intelligence are very unlikely to be answered in our lifetime. But SETI's listening program – if other civilizations have sent us messages, or simply use electro-magnetic radiation to communicate – might hear something interesting.
Imagine the scene. SETI radio telescopes get a message. It is from Civilization #9.
It says: "We need to talk".
Do You Know the Difference Between Height and Weight ? A Prayer for Innovators
"G-d grant me courage to try to change what can change, serenity to accept what cannot, and wisdom to know the difference." – Reinhold Niebuhr, 1937
Recently I took part in a program whose front end featured a talk by Joseph Ackerman, CEO of Elbit Industries, a leading Israeli defense contractor. Ackerman spoke about innovation in management, his own philosophy, and illustrated his approach with the H W principle.
On the screen he showed a large "H" and a large "W". "H" stands for height. You can do nothing about your height, he explained. You are born with it, it is determined by your genes. Accept it, live with it. "W" stands for weight. You CAN do something about your weight. If you are underweight, you can fatten up. If you are overweight, you can exercise and slim. It is not easy, but it is possible.
In innovation and in management, he explained, it is hard sometimes to tell the difference between "H" and "W". What can be changed, and should? What cannot be changed and must be accepted? If you waste energy on "H" problems or issues, you will have insufficient resources for "W" problems.
In his brilliant 1993 book[1], psychologist Martin Seligman cites conclusive evidence that "W" (weight), too, needs not courage but serenity. In truth, weight is very hard to change. Studies show obese people in general consume no more calories than the rest of us. And a huge proportion of those who diet and lose weight gain it all back within a couple of years. An enormous diet and diet food industry exists that perpetuates the myth of weight loss. So, Seligman notes, "W" is probably in the "H" category. The title of his chapter is sardonic: A Waist is a Terrible Thing to Mind.
I think the H-W distinction is crucial for innovators. Innovation begins with a deep passion to change the world, by identifying major issues or problems or needs that must be solved or met. The goal of the innovator must be big enough to be meaningful but not so huge as to be unachievable – in other words, "W" rather than "H". And knowing the difference is very very difficult. Jim Collins' BHAG – Big Hairy Audacious Goal – must be not so audacious as to be unachievable. Perhaps it should be called Big WARY Audacious Goal, BWAG, instead of Big HAIRY Audacious Goal (to stress the W rather than the H).
Some 73 years ago, the German Protestant theologist Reinhold Niebuhr wrote the famous Serenity Prayer in one of his newsletters. I would like to propose a different version, the Innovators' Prayer:
Lord, give me courage, passion and creativity to innovate what can and must be innovated; serenity to accept what cannot be innovated; and wisdom to tell the difference.
[1] Martin E.P. Seligman, WHAT YOU CAN CHANGE…AND WHAT YOU CAN'T: The Complete Guide to Successful Self-Improvement, Fawcett Columbine: New York, 1993.
Mommy, Daddy, Tell Me A Story – Forget It, I'll Tell My Own!
"What matters in life is not what happens to you but what you remember and how you remember it." ~ Gabriel Garcia Marquez
Almost a year ago, on Nov. 4, 2009, I wrote a blog with the title "Mommy, Daddy, Tell Me a Story!", that began thus:
Do you want to build a powerful business innovation? I ask my students. If you do — tell me a story. Build a powerful narrative that has real people in it, a plot, conflict, a story line, and above all, a happy end. These are all elements of every great children's book, stories we all grew up on, Good Night, Moon, Where the Wild Things Are, and so on. Children make meaning out of the world through stories. So do we adults, it seems. War and Peace, Anna Karenina — great novels are all great stories
Writing in an Israeli daily newspaper Haaretz, [1] journalist Ron Pressler describes pathbreaking work by Nobel Prize Laureate Daniel Kahneman, a cognitive psychologist who won the Economics Nobel Prize in 2002. Kahneman has for many years been studying how we remember our everyday experiences. He has shown that there are two separate selves, "the experiencing self" and the "remembering self". The second is utterly different from the first. The remembering self remembers key high points, and ignores many dull moments the experiencing self goes through. The remembering self constructs a plot, a key part of which is the end. "…The story we construct is usually influenced by one or several things on which we focus, and on whose importance we tend to exaggerate." A major thing is the end, whether it is happy or sad.
I personally have experienced this dual phenomenon. And I have applied it. Often, in experiencing an activity or event, I ask myself, how will I remember this in 5 years? I try very hard to shape a happy end so that it will be remembered positively instead of traumatically.
Innovator: As you work on your innovation, think about the script you are writing. Think about the high points. Tell yourself the story as you are living it. Make sure it is as positive as you can make it. Even if it fails, you can still shape your story as one to be remembered positively – for example, dramatic all-night efforts to rescue a failing project.
Tell your own story, innovator. Shape it as you are living it. It is quite possible to transform a traumatic failure into a heroic drama while it is ongoing. Bad memories can be forestalled, good ones can be strengthened.
The film-maker and storyteller Shekar Kapur once said, "We are the stories we tell ourselves." I would strengthen that. "We are the stories we tell about ourselves, to ourselves."
Innovator: What stories do you tell about yourself, to yourself? Do you like these stories? Are they good, strong, energizing, positive, inspiring? If not – rewrite the old ones, and reshape the new ones you are living at present. And as you live them, think about the script you are writing and will remember. In 10 years, you will be grateful you did.
[1] "Living the moment, remembering the high points", by Ron Pressler. Haaretz Weekly Magazine, Friday Sept. 17, 2010.
Benchmark Estonia: It's Run Like a Business
Little Estonia, population 1.3 m., will dump its currency and join the 16 countries in the Euro bloc next January. This has become near-certain following the EU announcement that Estonia has met the Masstricht fiscal preconditions for Euro membership, noting that "Estonia stands out… fulfilling the criteria clearly". The European Commission was far less upbeat about several other large Eastern European countries waiting to adopt the euro, such as Poland and Hungary. The last nations to join the euro were Slovenia in 2007 (another smart small country with about 2 m. people), Cyprus and Malta in 2008 and Slovakia in 2009.
Estonia has been hard hit by the global crisis, and by the EU recession; it has 18 per cent unemployment. But despite this, its budget deficit is modest (less than 3 per cent) and it has a Balance of Payments current account surplus. By embracing the euro, Estonia integrates its capital markets with the EU and trashes all the many problems related to having a weak unstable currency.
Some years ago, I brought a group of Israeli managers to Talinn on a benchmarking trip. We were amazed. Estonia has first-rate IT capabilities. It is even one of the first countries to run elections on-line. And you can file your annual income tax report on-line, too, in 20 minutes, and most people do. Cabinet meetings are held electronically, with absent or travelling ministers joining through their webcams. Estonia has closely integrated its economy with Finland, acting as a kind of off-shore "China" for that country.
Recently, NYT columnist Tom Friedman quoted an expert who said that it is hard to compete with China, because that country "is run like a business". Estonia, too, is run like a business, by its clever political leaders. Let Poland, Hungary, Bulgaria, the Czech Republic, Latvia, Lithuania, Romania and Sweden – the remaining countries struggling to fulfill Masstricht requirements – study Estonia carefully. Let them try to run their countries properly, as businesses.
Innovation Management 2007 Plus 3: Still Sub-Par
Innovation Management: Still Sub-Par
A 2007 survey of global executives by McKinsey Global Research on innovation reveals a paradox that most managers keenly understood already: As the importance of innovation grows, the quality of innovation management remains low and perhaps even declining.
Notes the McKinsey survey:
1. Innovation leadership is weak; though leaders seek breakthrough ideas, nonetheless innovation efforts are focused on products and services:
" ….companies often seem to isolate innovation projects within business units, even when they see bigger opportunities. When asked where change would produce the greatest improvement in performance, for example, top managers rank product and service innovations much lower than breakthrough ideas. Yet a majority also say innovation at their organizations is primarily focused on developing products or services and that dedicated teams within business units are the most common way they develop and commercialize new ideas (Exhibit 2). Less than half of top managers say they frequently define themes for breakthrough innovations.
2. Innovation leaders are not an integral part of the innovation process in their organizations.
"… top managers indicate that they are isolated from the innovators within their companies. Most often, top managers get their new ideas from informal, external sources (such as discussions with peers and interactions with consumers), not from the business units or formal teams where innovation tends to occur.
Three years later, in 2010, after a global crisis, the question arises: Has the innovation management process been repaired? In particular, does the organization's leadership truly lead and drive the innovation process? I am skeptical. The paradox remains. Innovation's importance grows day by day. Innovation management remains highly flawed.
Don't Laugh: "US Toughens Tone on Chinese Currency"
That headline is real. It is from the Global New York Times, Sept. 17, p. 15.
The gist is this: America is this time, for real, we're not kidding, better take us seriously, make no mistake, this isn't playing games, we mean business, …. getting tough with China regarding its manipulation of the renminbi.
What manipulation?
Perhaps, the manipulation that involves China buying $1 b. DAILY of dollar-denominated securities, or $365 b. annually, to keep its exchange rate from appreciating, and thus making its exports more expensive. Perhaps, the manipulation that keeps China's exchange rate today at RMB 6.83 per US dollar, compared to RMB 6.95 in 2008, a level that according to The Economist and even The World Bank and IMF, undervalues China's currency by about 50 percent (that is, the true economic exchange rate would be about RMB 3.5 per dollar, if the exchange rate were allowed to be determined by market forces alone, without the Bank of China). Perhaps the manipulation in which China has accumulated way over $2 trillion in reserves through daily purchase of dollar assets.
If China's exchange rate were indeed allowed to rise to RMB 3.5 per dollar, suddenly all its exports would cost double. Perhaps then global rebalancing could begin – with fewer goods flowing out of China, and more goods flowing into it.
For how long has China's Renminbi been undervalued? Would you believe 15 years? The exchange rate was RMB 8.35 per dollar in 1995. It stayed at that level until 2004. The exchange rate was then allowed to appreciate very very slowly and gently, to RMB 8.19 (2005), 7.97 (2006), 7.61 (2007), 6.95 (2008) and 6.83 (2009). At that rate, excruciatingly slow Chinese 'water torture', global rebalancing will be completed by the year …. 6583.
U.S. Treasury Secretary Timothy Geithner is talking really tough. The Treasury "would take China's actions into account as we prepare the next Foreign Exchange Report", he said; the Report is due Oct. 15. America is threatening to declare China a currency manipulator! China? I have no doubt the entire nation is shivering with fear.
Wake up, America. China has been manipulating its currency for 15 years. And you, America, you are an active partner in crime, because you have been living beyond your means with the money China lends you and loving it, and you still are, and there is no real sign whatsoever that your politicians are willing to even begin inflicting the pain (lower living standards) that is required to end this impossible situation.
So, forgive me if I read the headline and chuckle sadly. The last President who talked tough to a trading partner was Ronald Reagan, who read the Riot Act to Japan in the 1980's. Since then, they've all been marshmallows. The result: Global disaster and America's hollow economy that is unable to create new jobs.
Productivity: Good News or Bad News?
September 16, 2010 in Global Crisis Blog, Innovation Blog | Leave a comment
Global Crisis/Innovation Blog
There are endless numbers of awful good news/bad news jokes. Here is one of the worst. Doctor: I have good news and bad news. Patient: What is the good news? Doctor: You have 24 hours to live! Patient: THAT'S THE GOOD NEWS! WHAT IS THE BAD NEWS? Doctor: I forgot to call you yesterday.
In today's global economy, productivity growth is one of those jokes. The good news is that unlike all other global or local recessions, productivity in most countries has continued to grow strongly. The reason: Companies have been very quick to fire or lay off workers, right from the start, and the remaining workers, fearing for their jobs, have worked far harder and far smarter.
This is also very bad news. Why?
It is true by definition that GDP growth is identically equal to a) the rate of growth in GDP per worker (labor productivity) plus the rate of growth in the number of (employed) workers. If labor productivity grows as fast as GDP, then there is no need to hire more workers. And that is precisely what is happening, in the US, Europe and China.
The data? The IMF has revised its 2010 GDP growth forecast upward for the world, to 4.1 percent. Good news. Almost all of that growth will come from productivity growth, rather than from new hires. Bad news.
In China labor productivity growth for 2010 is forecasted to grow at an astonishing rate of 7.7 per cent. GDP will grow by 8 per cent. That means that employment will barely grow at all. The upside of this is that employment will at least remain steady, without massive layoffs. The downside is that there will be few new jobs for migrants coming from the West.
In the US, 2nd Q. 2010, business-sector output grew by 3.7 per cent (good news) and all of that increase came from increased productivity (bad news for workers).
The implication for individuals all over the world, especially young people entering the labor market or planning to, is rather cruel: Darwin's survival of the fittest has come to global labor markets. In future, you will have to navigate your skills flexibly and rapidly, change them often, learn new skills and abandon old ones, and stay a step ahead of the rapidly changing labor and goods markets. Those who fail will be unable to find gainful employment, and if they remain stuck in old patterns and old skills, they may never work again.
Innovating During/After a Recession: A Three-Pronged Plan
September 11, 2010 in Global Crisis Blog, Innovation Blog | 1 comment
Global managers need to come to terms with the fact that the 'recovery' will be weak, perhaps indistinguishable from recession. Whether GDP in the US and EU grows at 1% or 1.5% is immaterial. It still means that for many businesses sales will remain flat – unless the 'three-pronged plan' proposed by Dartmouth's Tuck Business School Professor Vijay Govindarajan is implemented (or at least, a version of it.)[1]
First Govindarajan recommends setting up a Dedicated Team, dedicated solely to the goal of strategizing the post-recession period. Let the team come up with an innovation, and then implement it as they would conduct a scientific experiment. Why? Because until you get that innovation out into the hands of users, you will lack vital information.
Second, he recommends three steps:
• Formalize the experiment. List hypotheses. Define how you will test them. For instance: In a post-recession period, price sensitivity remains high. Hypothesis: "We will test a very low-end product with low price, and check very carefully whether demand is price sensitive, so that volume increases more than price falls, to compensate for the lower prices. Our hypothesis is that it will."
• Break down the hypothesis. Simplify! Identify the most crucial hypothesis, or hypotheses, and focus on them. Is it price? Is it some new feature that is being added?
• Seek the truth. Jim Collins calls this "face the brutal facts". Innovators who invest the company's hard-earned cash have a vested interest in putting a rosy glow on test data. Make sure the company has a pervasive culture of 'telling the truth'. Companies that delude themselves will pay heavily. Remember, if you launch full-scale a bad product, not only will the product fail, but you have lost valuable time that could have been invested in launching a successful product.
Govindarajan again stresses the obvious point that innovation is more about implementation than inspiration. While readers may be utterly tired of hearing this again and again, it is so important that it is worth repeating, in his words:
"There is too much emphasis on ideas and not nearly enough on execution. As a result, most corporations have more ideas than they can possibly move forward. Too many promising ideas on paper never become anything more than … promising ideas on paper."
Screen your ideas, pick one or two, set up a team, give it some money – and execute. Make sure you take seriously a scenario in which there is weak economic growth for several years to come.
[1] Bloomberg Business Week, "How to innovate after a recession", Viewpoint September 7, 2010 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,872 |
from __future__ import unicode_literals
from django.test import TestCase
from httmock import all_requests, HTTMock
from django_seo_js.tests.utils import override_settings
from django_seo_js.backends import PrerenderIO
MOCK_RESPONSE = b"<html><body><h1>Hello, World!</h1></body></html>"
MOCK_RESPONSE_HEADERS = {"foo": "bar"}
MOCK_RECACHE_RESPONSE = "OK"
MOCK_RECACHE_HEADERS = {"ibbity": "ack"}
@all_requests
def mock_prerender_recache_response(url, request):
return {
'status_code': 200,
'content': MOCK_RECACHE_RESPONSE,
'headers': MOCK_RECACHE_HEADERS,
}
@all_requests
def mock_prerender_response(url, request):
return {
'status_code': 200,
'content': MOCK_RESPONSE,
'headers': MOCK_RESPONSE_HEADERS,
}
class PrerenderIOTestToken(TestCase):
@override_settings(PRERENDER_TOKEN=None)
def test_get_token_throws_exception_if_missing(self):
self.assertRaises(ValueError, PrerenderIO)
@override_settings(PRERENDER_TOKEN="123124341adfsaf")
def test_get_token(self):
self.backend = PrerenderIO()
# Test function
self.assertEqual(self.backend._get_token(), "123124341adfsaf")
# Test __init__
self.assertEqual(self.backend.token, "123124341adfsaf")
class PrerenderIOTestMethods(TestCase):
def setUp(self):
self.backend = PrerenderIO()
def test_get_response_for_url_missing_url(self):
self.assertRaises(TypeError, self.backend.get_response_for_url)
self.assertRaises(ValueError, self.backend.get_response_for_url, None)
def test_get_response_for_url_valid(self):
with HTTMock(mock_prerender_response):
resp = self.backend.get_response_for_url("http://www.example.com")
self.assertEqual(MOCK_RESPONSE, resp.content)
for k, v in MOCK_RESPONSE_HEADERS.items():
self.assertEqual(resp[k], v)
def test_update_url_with_url_only(self):
with HTTMock(mock_prerender_recache_response):
resp = self.backend.update_url(url="http://www.example.com")
self.assertEqual(resp, True)
def test_update_url_with_regex_only(self):
with HTTMock(mock_prerender_recache_response):
resp = self.backend.update_url(regex="http://www.example.com/*")
self.assertEqual(resp, True)
def test_update_url_missing_url_and_regex(self):
with HTTMock(mock_prerender_recache_response):
self.assertRaises(ValueError, self.backend.update_url)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,812 |
Q: to generate pdf download using tcpdf i am not able to generate pdf download,my code is as follows,can anyone tell me what is wrong with this code.
include 'tcpdf.php';
$pdf = new TCPDF();
$pdf->AddPage('P', 'A4');
$html = '<html>
<head></head>
<body><table border="1">
<tr><th>name</th>
<th>company</th></tr>
<tr>
<td>hello</td>
<td>xx technologies</td>
</tr>
</table>
</body>
</html>';
$pdf->writeHTML($html, true, false, true, false, '');
$pdf->Output();
?>
A: I have modified your code and it works. [TESTED]
<?php
require_once('tcpdf_include.php');
$pdf = new TCPDF(PDF_PAGE_ORIENTATION, PDF_UNIT, PDF_PAGE_FORMAT, true, 'UTF-8', false);
$pdf->SetDefaultMonospacedFont(PDF_FONT_MONOSPACED);
$pdf->SetAutoPageBreak(TRUE, PDF_MARGIN_BOTTOM);
if (@file_exists(dirname(__FILE__).'/lang/eng.php')) {
require_once(dirname(__FILE__).'/lang/eng.php');
$pdf->setLanguageArray($l);
}
$pdf->SetFont('helvetica', '', 9);
$pdf->AddPage();
$html = '<html>
<head></head>
<body><table border="1">
<tr><th>name</th>
<th>company</th></tr>
<tr>
<td>hello</td>
<td>xx technologies</td>
</tr>
</table>
</body>
</html>';
$pdf->writeHTML($html, true, 0, true, 0);
$pdf->lastPage();
$pdf->Output('htmlout.pdf', 'I');
?>
OUTPUT:
A: If you want to make the file downloads, use PHP function header before your $pdf->Output(); like this :
header('Content-type: application/pdf');
header('Content-Disposition: attachment; filename="file.pdf"');
$pdf->Output(); # terminate your file with TCPDF output
See PHP function header on php.net
A: Add this line $pdf->Output($downlaodname, 'D'); instead of $pdf->Output();. It will force browser to download file.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,566 |
Drosophila linearidentata este o specie de muște din genul Drosophila, familia Drosophilidae, descrisă de Masanori Joseph Toda în anul 1986.
Este endemică în Myanmar. Conform Catalogue of Life specia Drosophila linearidentata nu are subspecii cunoscute.
Referințe
Legături externe
Drosophila | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,873 |
package com.alcedomoreno.sirme.core.dao.common;
import java.io.Serializable;
import java.util.List;
import org.hibernate.Query;
import org.hibernate.Session;
import org.hibernate.SessionFactory;
import org.hibernate.criterion.Projections;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.orm.hibernate3.support.HibernateDaoSupport;
import com.alcedomoreno.sirme.core.dao.ApplicationsDaoImpl;
import com.google.common.base.Preconditions;
/**
* Clase que implementa todos los métodos genéricos
* @param <T> tipo con el que se extenderá la clase
*/
@SuppressWarnings("unchecked")
public class AbstractHibernateDao<T extends Serializable> extends HibernateDaoSupport implements Operations<T> {
///////////////////////////////////////////////////////////////
// Attributes //
///////////////////////////////////////////////////////////////
private static Logger log = LoggerFactory.getLogger( ApplicationsDaoImpl.class );
private static final String CLASS_NAME = "ApplicationsDaoImpl";
private Class<T> clazz;
@Autowired
private SessionFactory sessionFactory;
///////////////////////////////////////////////////////////////
// End of Attributes //
///////////////////////////////////////////////////////////////
@Autowired
public void init(SessionFactory factory) {
setSessionFactory(factory);
}
protected final void setClazz(final Class<T> clazzToSet) {
clazz = Preconditions.checkNotNull(clazzToSet);
}
@Override
public T findOne(Serializable id) {
return (T) getCurrentSession().get(clazz, id);
}
@Override
public List<T> findAll() {
return getCurrentSession().createQuery("from " + clazz.getName()).list();
}
@Override
public void create(T entity) {
Preconditions.checkNotNull(entity);
// getCurrentSession().persist(entity);
getCurrentSession().saveOrUpdate(entity);
}
@Override
public T update(T entity) {
Preconditions.checkNotNull(entity);
return (T) getCurrentSession().merge(entity);
}
@Override
public void delete(T entity) {
Preconditions.checkNotNull(entity);
getCurrentSession().delete(entity);
}
@Override
public void deleteById(Integer entityId) {
final T entity = findOne(entityId);
Preconditions.checkState(entity != null);
delete(entity);
}
protected final Session getCurrentSession() {
return sessionFactory.getCurrentSession();
}
@Override
public void evict(T entity) {
Preconditions.checkNotNull(entity);
getCurrentSession().evict(entity);
}
@Override
public Integer count() {
return (Integer) getCurrentSession().createCriteria( clazz.getName() )
.setProjection( Projections.rowCount() ).uniqueResult();
}
@Override
public List<T> findAll(int rowId, int size) {
Query query = getCurrentSession().createQuery("FROM " + clazz.getName());
query.setFirstResult( rowId );
query.setMaxResults( size );
return query.list();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,078 |
\section{Introduction}
\label{sec:intro}
Matroids are combinatorial structures that model various types of independence, such as linear independence of vectors in a linear space or algebraic independence of elements in a field extension. For an inspiring recent survey, see \cite{Ard18}. There have been several recent breakthroughs proving inequalities on sequences of numbers associated to matroids. While the proofs in this paper are self-contained, we build off several of these ideas to study the following conjecture of Mason \cite{Mas72}.
\begin{conjecture}[Mason's Conjecture]\label{conj:Mason}
For an $n$-element matroid $M$ with ${\mathcal{I}}_k$ independent sets of size $k$,
\begin{forms}
\item\label{mason:i} ${\mathcal{I}}_{k}^{2} \geq {\mathcal{I}}_{k-1} \cdot {\mathcal{I}}_{k+1}$ (log-concavity),
\item\label{mason:ii} ${\mathcal{I}}_{k}^{2} \geq \left(1 + \frac{1}{k}\right) \cdot {\mathcal{I}}_{k-1} \cdot {\mathcal{I}}_{k+1}$,
\item\label{mason:iii} ${\mathcal{I}}_{k}^{2} \geq \left(1 + \frac{1}{k}\right) \cdot \left(1 + \frac{1}{n-k}\right) \cdot {\mathcal{I}}_{k-1} \cdot {\mathcal{I}}_{k+1}$ (ultra log-concavity).
\end{forms}
\end{conjecture}
Note that \cref{mason:i,mason:ii,mason:iii} are written in increasing strength. \Textcite{AHK18} proved \cref{mason:i} using techniques from Hodge theory and algebraic geometry. Building on this, \textcite{HSW18} proved \cref{mason:ii}. Prior to our work, \cref{mason:iii} was only proven to hold when $n \leq 11$ or $k\leq 5$ \cite{KN11}. We refer to \cite{Sey75, Dow80, Mah85, Zha85, HK12, HS89, Len13} for other partial results on Mason's conjecture. Here, we give a self-contained proof of \cref{mason:iii}.
\begin{theorem}\label{thm:mason}
For a matroid $M$ on $n$ elements with ${\mathcal{I}}_k$ independent sets of size $k$,
the sequence ${\mathcal{I}}_{0}, {\mathcal{I}}_{1},\dots,{\mathcal{I}}_{n}$ is ultra log-concave. That is,
for $1<k<n$,
\[ \parens*{\frac{{\mathcal{I}}_{k}}{\binom{n}{k}}}^2 \ \ \geq \ \ \frac{{\mathcal{I}}_{k-1}}{\binom{n}{k-1}} \cdot \frac{{\mathcal{I}}_{k+1}}{\binom{n}{k+1}} . \]
\end{theorem}
We prove \cref{thm:mason} in \cref{sec:mason}.
The main tool we use will be polynomials that are log-concave as functions on the positive orthant.
For $i\in [n]$, let $\partial_i$ or $\partial_{z_i}$ denote the partial derivative operator that maps a polynomial $f$ to its partial derivative with respect to $z_i$. For a vector $v\in {\mathbb{R}}^n$, we let $D_v$ denote the directional derivative operator in direction $v$,
\[ D_v=\sum_{i=1}^n v_i \partial_i. \]
We call a polynomial $f\in {\mathbb{R}}[z_1,\dots,z_n]$ log-concave over ${\mathbb{R}}_{\geq 0}^n$ if $f$ is nonnegative and log-concave
as a function over ${\mathbb{R}}_{\geq 0}^n$, or in other words if for every $u, v\in {\mathbb{R}}_{\geq 0}^n$ and $\lambda\in [0,1]$, we have $f(u), f(v)\geq 0$ and
\[ f(\lambda u+(1-\lambda)v)\geq f(u)^\lambda \cdot f(v)^{1-\lambda}. \]
Note that the zero polynomial is also log-concave.
If $f(v)$ is positive for some $v\in {\mathbb{R}}_{\geq 0}^n$,
then we call $f$ log-concave at $z=v$ if the Hessian of its $\log$ at $v$ is negative semidefinite.
It is easy to see from the definition that for any fixed $d$ and $n$, the set of
polynomials of degree at most $d$ in $n$ variables that are log-concave on ${\mathbb{R}}_{\geq 0}^n$ is closed in the Euclidean topology on ${\mathbb{R}}[x_1, \hdots, x_n]_{\leq d}$. Also, a nonzero polynomial is log-concave over ${\mathbb{R}}_{\geq 0}^n$
if and only if it is log-concave at every point of ${\mathbb{R}}_{>0}^n$.
\begin{definition}\label{def:clc}
A polynomial $f\in {\mathbb{R}}[z_1,\dots,z_n]$ is \textbf{completely log-concave} if for every set of nonnegative vectors $v_1,\dots,v_k\in {\mathbb{R}}_{\geq 0}^n$, the polynomial $D_{v_1}\dots D_{v_k} f$ is nonnegative and log-concave over ${\mathbb{R}}_{\geq 0}^n$.
\end{definition}
Completely log-concave polynomials were introduced in \cite{AOV18} based on similar notions of strongly log-concave and Alexandrov-Fenchel polynomials first studied in \cite{Gur08}.
In this paper, we prove the properties of complete log-concavity necessary for \cref{thm:mason} and defer a more detailed treatment of completely log-concave polynomials to a future article.
The main ingredient of the proof of \cref{thm:mason} is to show that the homogenization of the generating polynomial of all independent sets of $M$ is completely log-concave, namely that the polynomial
$$ g_M(y,z_1,\dots,z_n)=\sum_{I\in{\mathcal{I}}} y^{n-\card{I}}\prod_{i\in I} z_i$$
is completely log-concave. Then, we use this to show that the bivariate restriction $f_M(y,z)=\sum_{k=0}^r {\mathcal{I}}_{k} y^{n-k}z^k$ is completely log-concave.
Finally, we derive \cref{thm:mason} from the latter fact based on an observation of Gurvits \cite{Gur08}
on the coefficients of completely log-concave polynomials.
\subsection{Independent work}
In a related upcoming work, Br\"and\'en and Huh have independently developed methods that overlap with our work. In particular they also prove the strongest version of Mason's conjecture.
\subsection{Spectral negative dependence}
It is well-known that the uniform distribution over all spanning trees of a graph is negatively correlated and more generally negatively associated, see \cite{Pem00} for background. This fact more generally extends to regular matroids. Prior to our work many researchers tried to approach Mason's conjecture through the lens of negative correlation \cite{SW75, Wag08, BBL09, KN10, KN11}. However, for many matroids the uniform distribution on bases is not negatively correlated and furthermore, negative correlation does not necessarily imply log-concavity of its rank sequences \cite{Wag08}.
Consider the polynomial $p_M=\sum_{B} \prod_{i\in B} z_i$, where the sum is over all bases of the matroid $M$. Then the negative correlation property is equivalent to all off-diagonal entries of the Hessian of $\log p_M$ being non-positive when evaluated at the
all-ones vector ${\mathds{1}} = (1,\hdots, 1)$, i.e.
$$ (\nabla^2 \log p_M({\mathds{1}}))_{i,j} = p_M({\mathds{1}})\cdot \partial_i\partial_j p({\mathds{1}})- \partial_i p_M({\mathds{1}})\cdot\partial_j p_M({\mathds{1}}) \leq 0,$$
for all $1\leq i,j\leq n$, $i\neq j$.
This inequality holds for regular matroids but not necessarily for even linear matroids.
In \cite{AOV18} it was observed that for any matroid $M$, the polynomial $p_M$ is completely log-concave. This means that even though $\nabla^2 \log p_M({\mathds{1}})$ can have positive entries, all of its eigenvalues, and eigenvalues of Hessian of the $\log$ of all partials of $p_M$, are non-positive. We call this property, \emph{spectral negative dependence}. In this paper, we show that for any matroid, the homogenization of the generating polynomial of all independent sets, namely $g_M$ also satisfies spectral negative dependence. Furthermore, spectral negative dependence is enough to prove the strong form of log-concavity of rank sequences as conjectured by Mason.
\paragraph{Acknowledgements.}
Part of this work was started while the first and last authors were visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425. Shayan Oveis Gharan and Kuikui Liu are supported by the NSF grant CCF-1552097 and ONR-YIP grant N00014-17-1-2429. Cynthia Vinzant was partially supported by the National Science Foundation grant DMS-1620014.
\section{Preliminaries}\label{sec:prelim}
A polynomial $f\in {\mathbb{R}}[z_1,\dots,z_n]$ is homogeneous of degree $d$ if every monomial of $f$ has degree $d$, or equivalently $f(\lambda\cdot z_1,\dots,\lambda\cdot z_n)=\lambda^d f(z_1,\dots,z_n)$ for all $\lambda \in {\mathbb{R}}$. We will use $\nabla f$ to denote the gradient of $f$ and $\nabla^2 f$ to denote its Hessian matrix.
We use $[n]$ to refer to $\set{1,\dots,n}$. When $n$ is clear from context, for a set $S\subseteq [n]$, we let ${\mathds{1}}_S\in {\mathbb{R}}^n$ denote the indicator vector of $S$. For variables $z_{1},\dots,z_{n}$ and $S \subseteq [n]$, we let $z^{S} $ denote the monomial $\prod_{i \in S} z_{i}$. Similarly, for an integer vector $\alpha = (\alpha_1, \hdots, \alpha_n) \in {\mathbb{Z}}_{\geq 0}^n$ or a subset $S\subseteq [n]$, we denote differential operators
\[\partial^{\alpha}=\prod_{i=1}^n\partial_i^{\alpha_i} \ \ \text{ and } \ \ \partial^{S} =\partial^{{\mathds{1}}_{S}}=\prod_{i\in S} \partial_i. \]
Note that if $f$ is homogeneous of degree $d$, then $\partial^{\alpha}f$ is homogenous of degree $d-\abs{\alpha}$ where $\abs{\alpha} = \sum_{i=1}^n\alpha_i$.
A symmetric matrix $Q\in{\mathbb{R}}^{n\times n}$, alternatively viewed as a quadratic form $z\mapsto z^\intercal Q z$, is positive semidefinite if $v^\intercal Q v\geq 0$ for all $v\in{\mathbb{R}}^n$ and negative semidefinite if $v^\intercal Q v\leq 0$ for all $v\in{\mathbb{R}}^n$. If these inequalities are strict for $v\neq 0$, then $Q$ is positive or negative definite, respectively. There are several equivalent definitions. In particular, a matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative, which occurs if and only if the all its principal minors are nonnegative.
Since $Q$ is negative semidefinite if and only if $-Q$ is positive semidefinite, these translate into analogous characterizations of negative semidefinite-ness.
\subsection{Matroids}
Formally, a \textbf{matroid} $M=([n],{\mathcal{I}})$ consists of a ground set $[n]$ and a nonempty collection ${\mathcal{I}}$ of \emph{independent} subsets of $[n]$ satisfying the following two conditions:
\begin{conds}
\item If $S\subseteq T$ and $T\in {\mathcal{I}}$, then $S\in {\mathcal{I}}$.
\item If $S,T\in {\mathcal{I}}$ and $\card{T}>\card{S}$, then there exists an element $i\in T \setminus S$ such that $S\cup \set{i}\in {\mathcal{I}}$.
\end{conds}
The \textbf{rank}, denoted by $\rank(S)$, of a subset $S\subseteq [n]$ is the size of the largest independent set contained in $S$
and the rank of $M$ is defined as $\rank([n])$. An element $i\in [n]$ is called a \textbf{loop} if $\set{i}\notin {\mathcal{I}}$, and two
elements $i,j\in [n]$ are called \text{parallel} if neither is a loop and $\rank(\set{i,j}) = 1$. One can check that parallelism defines an equivalence relation on the non-loops of $M$, which partitions the set of non-loops into parallelism classes.
For a matroid $M$ and an independent set $S\in {\mathcal{I}}$, the \textbf{contraction}, $M/S$, of $M$ by $S$ is the matroid on ground set $[n]\setminus S$ with independent sets $\set{T \subseteq [n]\setminus S \given S\cup T\in {\mathcal{I}}}$.
In particular, the rank of $M/S$ equals $\rank(M) - \card{S}$.
See \cite{Oxl11} for more details and general reference.
\subsection{Log-concave polynomials}
In \cite{AOV18}, it was shown that a homogeneous polynomial $f \in {\mathbb{R}}[z_1,\dots,z_n]$ with nonnegative coefficients is log-concave at a point $z=a$ if and only if its Hessian $\nabla^2 f$ has at most one positive eigenvalue at $z=a$.
One can relate this directly to the negative semidefinite-ness of the Hessian of $\log(f)$. Indeed, there are several useful equivalent characterizations of this condition:
\begin{lemma}\label{lem:LC_Quad}
Let $f\in {\mathbb{R}}[z_1, \hdots, z_n]$ be homogeneous of degree $d\geq 2$ with nonnegative coefficients. Fix a point $a\in {\mathbb{R}}_{\geq 0}^n$ with $f(a)\neq 0$, and let
$Q = \eval{\nabla^2 f}_{z=a}$.
The following are equivalent:
\begin{conds}
\item\label{cond:1} $f$ is log-concave at $z=a$,
\item\label{cond:2} $z\mapsto z^{\intercal}Qz$ is negative semidefinite on $(Qa)^{\perp}$,
\item\label{cond:3} $z\mapsto z^{\intercal}Qz$ is negative semidefinite on $(Qb)^{\perp}$ for every $b\in{\mathbb{R}}^n_{\geq 0}$ such that $Qb\neq 0$,
\item\label{cond:4} $z\mapsto z^{\intercal}Qz$ is negative semidefinite on some linear space of dimension $n-1$, and
\item\label{cond:5} the matrix $(a^{\intercal}Qa) Q - (Qa)(Qa)^{\intercal}$ is negative semidefinite.
\end{conds}
For $d\geq 3$, these are also equivalent to the condition
\begin{conds}[resume]
\item\label{cond:6} $D_af$ is log-concave at $z=a$.
\end{conds}
\end{lemma}
One can check that this condition is also equivalent to $Q$ having at most one positive eigenvalue, but we do not rely on
this fact and leave its proof to the interested reader.
\begin{proof}
Euler's identity states that for a homogeneous polynomial $g$ of degree $d$, $\sum_{i=1}^nz_i\partial_{i}g$ equals $d\cdot g$.
Using this on $f$ and $\partial_{j}f$ gives that $ Qa=(d-1)\cdot\nabla f(a)$ and $a^\intercal Qa= d(d-1)\cdot f(a) $.
The Hessian of $\log(f)$ at $z=a$ then equals
\[
\eval{\nabla^2(\log(f))}_{z=a} \ = \
\eval{\parens*{\frac{ f \cdot \nabla^2f- \nabla f\nabla f^\intercal}{f^2}}}_{z=a} \
= \
d(d-1)\frac{a^\intercal Qa \cdot Q- \frac{d}{d-1} (Qa)(Qa)^\intercal }{(a^{\intercal}Qa)^2}.
\]
We can also conclude that $a^\intercal Qa = d(d-1)\cdot f(a)>0$ and that the vector $Qa$ is nonzero.
(\cref{cond:1} $\Rightarrow$ \cref{cond:2})
If $f$ is log-concave at $z=a$, then the Hessian of $\log(f(z))$ at $z=a$ is negative semidefinite.
Restricted to the linear space $(Qa)^{\perp} = \set{z\in {\mathbb{R}}^n \given z^{\intercal}Qa=0}$, the formula above simplifies to $\frac{ d(d-1)}{a^\intercal Qa} \cdot Q$, meaning that
$z\mapsto z^{\intercal}Qz$ is negative semidefinite on this linear space.
(\cref{cond:2} $\Rightarrow$ \cref{cond:4}) Since $Qa$ is nonzero, $(Qa)^{\perp}$ has dimension $n-1$.
(\cref{cond:4} $\Rightarrow$ \cref{cond:5}) Suppose that $z\mapsto z^{\intercal}Qz$ is negative semidefinite on an $(n-1)$-dimensional linear space $L$.
Let $b\in {\mathbb{R}}^n$ and consider the $n\times 2$ matrix $P$ with columns $a$ and $b$. Then
\[
P^\intercal Q P=
\begin{bmatrix}
a^\intercal Qa & a^\intercal Qb \\
b^\intercal Qa & b^\intercal Qb
\end{bmatrix}.
\]
If $P$ has rank one, then so does $P^\intercal Q P$, meaning that $\det(P^\intercal Q P)=0$.
Otherwise $P$ has rank two and its column-span intersects $L$ nontrivially.
This means there is a vector $v\in {\mathbb{R}}^2$
for which $Pv \in L$ is nonzero and $(Pv)^{\intercal}Q(Pv) \leq 0$.
From this we see that $P^\intercal Q P$ is not positive definite.
On the other hand, since the diagonal entry $a^\intercal Qa$ is positive,
$P^\intercal Q P$ is not negative definite.
In either case, we then find that
\[ \det(P^\intercal Q P)=(a^\intercal Q a) \cdot(b^\intercal Q b) - (b^\intercal Qa) \cdot (a^\intercal Qb)\leq 0.
\]
Thus $b^\intercal ((a^\intercal Q a) \cdot Q - (Qa)(Qa)^\intercal) b \leq 0$ for all $b\in {\mathbb{R}}^n$.
(\cref{cond:5} $\Rightarrow$ \cref{cond:1}) Suppose $(a^{\intercal}Qa) \cdot Q - (Qa)(Qa)^{\intercal}$ is negative semidefinite.
Further subtracting $\frac{1}{d-1}(Qa)(Qa)^{\intercal}$
and scaling by the positive number $\frac{d(d-1)}{(a^{\intercal}Qa)^2}$ results in $\eval{\nabla^2(\log(f))}_{z=a} $, as above, which must therefore also be negative semidefinite.
(\cref{cond:3} $\Leftrightarrow$ \cref{cond:4}) Both conditions depend only on the matrix $Q$. We can then use the equivalence (\cref{cond:2} $\Leftrightarrow$ \cref{cond:3}) for the point $z=b$ and the quadratic polynomial $f(z) = \frac{1}{2} z^\intercal Q z$, whose Hessian at any point is the matrix $Q$.
(\cref{cond:1} $\Leftrightarrow$ \cref{cond:6}) For $d\geq 3$, $D_af$ is homogeneous of degree $\geq 2$.
Euler's identity applied to $\partial_{i}\partial_{j}f$ shows that
the Hessian of $D_af$ at $z=a$ is a scalar multiple of the Hessian of $f$ at $z=a$, namely $(d-2)\eval{\nabla^2f}_{z=a}$.
Thus by the equivalence (\cref{cond:1} $\Leftrightarrow$ \cref{cond:4}), $D_af$ is log-concave at $a$ if and only if $f$ is.
\end{proof}
\subsection{Completely log-concave polynomials}
One of the basic operations that preserves complete log-concavity is
an affine change of coordinates.
This was first proved in \cite{AOV18}, but for completeness we include the proof here.
\begin{lemma}\label{lem:LinearPreserver}
If $f\in {\mathbb{R}}[z_1,\dots,z_n]$ is completely log-concave and $T:{\mathbb{R}}^m\to {\mathbb{R}}^n$ is an affine transform such that $T({\mathbb{R}}_{\geq 0}^m)\subseteq {\mathbb{R}}_{\geq 0}^n$, then $f(T(y_1,\dots,y_m))\in {\mathbb{R}}[y_1,\dots,y_m]$ is completely log-concave.
\end{lemma}
\begin{proof}
First, we prove that if $f$ is a log-concave polynomial, then $f \circ T = f(T(y_1,\dots,y_m))$ is also log-concave.
By assumption for any $u,v \in {\mathbb{R}}_{\geq0}^{m}$, we have $T(u), T(v)\in {\mathbb{R}}_{\geq 0}^n$. Thus for any $0 \leq \lambda \leq 1$,
\[ f(T(\lambda u+(1-\lambda)v)) \ = \ f(\lambda T(u)+(1-\lambda)T(v)) \ \geq \ f(T(u))^\lambda f(T(v))^{1-\lambda}. \]
Therefore $f \circ T$ is log-concave.
Now suppose that $f$ is completely log-concave and let $v_{1},\dots,v_{k} \in {\mathbb{R}}_{\geq0}^{m}$.
Since $T({\mathbb{R}}_{\geq0}^{m}) \subseteq {\mathbb{R}}_{\geq0}^{n}$ and $T$ is affine,
$T(x) = Ax + b$ for some $A \in {\mathbb{R}}_{\geq0}^{n \times m}$ and $b \in {\mathbb{R}}_{\geq 0}^{n}$.
In particular, $Av_1,\dots,Av_k\in {\mathbb{R}}_{\geq 0}^n$,
which means that $D_{Av_1}\dots D_{Av_k}f$ is log-concave over ${\mathbb{R}}_{\geq 0}^n$.
By the chain rule for differentiation, we have
\[ D_{v_1}\dots D_{v_k}(f\circ T)=(D_{Av_1}\dots D_{Av_k} f)\circ T. \]
Since composition with $T$ preserves log-concavity, this polynomial is log-concave over ${\mathbb{R}}_{\geq 0}^m$.
\end{proof}
\section{Reduction to quadratics}
\label{sec:clc}
As the main result of this section we will show that, under some mild restrictions,
to check whether a homogeneous polynomial is completely log-concave,
it suffices to check the conditions in \cref{def:clc} for $k=d-2$ and $v_1,\dots,v_k\in \set{ {\mathds{1}}_{\set{1}},\dots,{\mathds{1}}_{\set{n}}}$.
Then $D_{v_1}\cdots D_{v_k}f$ has the form $\partial^{\alpha}f$ where $\alpha_j$ is the number of vectors $v_k$ equal to ${\mathds{1}}_{\set{j}}$.
This provides a powerful tool to check complete log-concavity. The mild restriction we impose is \emph{indecomposability} of $f$ and its derivatives.
\begin{definition}\label{def:indecomposable}
A polynomial $f\in {\mathbb{R}}[z_1,\dots,z_n]$ is \textbf{indecomposable} if it cannot be written as $f_1+f_2$, where $f_1,f_2$ are
nonzero polynomials in disjoint sets of variables. Equivalently, if we form a graph with vertices $\set{i\given \partial_i f\neq 0}$ and edges $\set{(i, j)\given \partial_i\partial_j f\neq 0}$, then $f$ is indecomposable if and only if this graph is connected.
\end{definition}
Now we are ready to state the main result of this section.
\begin{theorem}\label{thm:CLCQuad}
Let $f$ be a homogeneous polynomial $f\in {\mathbb{R}}[z_1, \hdots, z_n]$ of degree $d \geq 2$ with nonnegative coefficients.
If the following two conditions hold, then $f$ is completely log-concave:
\begin{forms}
\item For all $\alpha\in {\mathbb{Z}}_{\geq 0}^n$ with $\abs{\alpha}\leq d-2$, the polynomial $\partial^{\alpha}f$ is indecomposable.
\item For all $\alpha\in {\mathbb{Z}}_{\geq 0}^n$ with $\abs{\alpha}= d-2$, the quadratic polynomial $\partial^{\alpha}f$ is log-concave over ${\mathbb{R}}_{\geq 0}^n$.
\end{forms}
\end{theorem}
The converse of the above statement is also true, namely, every completely polynomial is indecomposable, but
we defer the proof of this fact to a future article.
We build up to the proof of this theorem with a series of lemmas.
The first is a criterion for the sum of two log-concave polynomials to be log-concave. We will then use this to prove that if a polynomial $f$ is indecomposable and all of its partial derivatives $\partial_i f$ are log-concave, then it itself must be log-concave. The proof of \cref{thm:CLCQuad} then follows by an induction on the degree.
\begin{lemma}\label{lem:Add}
Let $f,g\in {\mathbb{R}}[z_1,\hdots, z_n]$ be homogenous with nonnegative coefficients satisfying $D_b f = D_cg \neq 0$ for some vectors $b,c\in {\mathbb{R}}_{\geq 0}^n$.
If $f$ and $g$ are log-concave on ${\mathbb{R}}_{\geq 0}^n$ then so is $f+g$.
\end{lemma}
\begin{proof}
The assumption that $D_b f = D_cg \neq 0$ means that $f$ and $g$ have the same degree $d$. We proceed by induction on $d$.
If $d = 1$, then $f+g$ is a linear form with nonnegative coefficients, which is automatically log-concave on ${\mathbb{R}}_{\geq 0}^n$.
Now suppose $d\geq 2$.
Fix $a\in {\mathbb{R}}_{> 0}^n$ and let $Q_1 = \nabla^2f(a)$ and $Q_2 = \nabla^2g(a)$.
Then $D_b f = D_cg$ implies that for each $i=1, \hdots, n$,
\[
(Q_1b)_i \ = \ \eval{( \partial_{i}D_b f)}_{z=a} \ = \ \eval{( \partial_{i}D_c g)}_{z=a} \ = \ (Q_2 c)_i,
\]
showing that $Q_1b = Q_2 c$. Since $D_b f$ has nonnegative coefficients and is not identically zero, we also have
that $D_b f(a) \neq 0$, meaning that $Q_1b\neq 0$. By \cref{lem:LC_Quad} (\cref{cond:1} $\Rightarrow$ \cref{cond:3}) and the log-concavity of $f$ and $g$,
each quadratic form $z\mapsto z^{\intercal}Q_iz^{\intercal}$ is negative semidefinite on $(Q_1b)^{\perp}=(Q_2c)^{\perp}$.
It follows that their sum $z\mapsto z^{\intercal}(Q_1+Q_2)z$
given by the matrix $Q_1 + Q_2 = \eval{\nabla^2(f+g)}_{z=a}$ is also negative semidefinite on
this $(n-1)$-dimensional linear space, so by \cref{lem:LC_Quad} (\cref{cond:4} $\Rightarrow$ \cref{cond:1}), $f+g$ is log-concave at $z=a$.
\end{proof}
\begin{lemma}\label{lem:DerToCLC}
Let $f\in {\mathbb{R}}[z_1, \hdots, z_n]$ be homogeneous of degree $d\geq 3$ and indecomposable with nonnegative coefficients.
If $\partial_i f$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$ for every $i=1, \hdots, n$,
then so is $D_af$ for every $a\in {\mathbb{R}}_{\geq 0}^n$.
\end{lemma}
\begin{proof}
If $\partial_{i}f$ is identically zero for some $i$, then we can consider $f$ as a polynomial in the other variables.
Without loss of generality, we can assume that $\partial_{i}f$ is nonzero for all $i$, and
if necessary relabel $z_1, \hdots, z_n$ so that
that for every $2\leq j \leq n$, there exists $i<j$ for which $\partial_{i} \partial_{j} f$ is non-zero. The latter follows from indecomposability.
Fix $a\in {\mathbb{R}}_{> 0}^n$. We will show that $D_af$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$.
We show by induction on $k$ that for any $1\leq k\leq n$, $\sum_{i=1}^k a_i \partial_{i} f$
is log-concave on ${\mathbb{R}}_{\geq 0}^n$. The case $k=1$ follows by assumption.
For $1\leq k <n$, let $b$ denote the truncation of $a$ to its first $k$ coordinates, $b = (a_1, \hdots, a_k, 0, \hdots, 0)$ and let $c$ denote the vector $a_{k+1}{\mathds{1}}_{\set{k+1}}$.
By induction both $D_b f$ and $D_c f$ are log-concave, and
\[
D_c D_b f \ = \ D_bD_c f \ = \ \sum_{i=1}^k a_i a_{k+1} \partial_{i} \partial_{{k+1}} f.
\]
Since the coefficients of each summand are nonnegative and $\partial_{i} \partial_{{k+1}} f$ is non-zero for some $1\leq i \leq k$, this sum is also non-zero.
Then by \cref{lem:Add}, $D_b f + D_c f = \sum_{i=1}^{k+1} a_i \partial_{i} f$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$.
For $k=n-1$, this is exactly $D_af$.
Taking closures then shows that $D_af$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$ for all $a\in {\mathbb{R}}_{\geq 0}^n$.\end{proof}
\begin{proof}[Proof of \cref{thm:CLCQuad}]
We induct on $d = \deg(f)$. The case $d=2$ is clear, so let $d\geq 3$.
For any positive vector $v\in {\mathbb{R}}_{>0}^n$, $D_vf$ is also indecomposable.
Indeed for any homogeneous polynomial $g$ of degree $\geq 1$
with nonnegative coefficients (such as $\partial_{i}f $ and $\partial_{i}\partial_{j}f $), $D_vg $ is identically zero if and only if $g$ is.
By taking closure, it suffices to show that for vectors $v_1, \hdots, v_k\in {\mathbb{R}}_{>0}^n$, the polynomial
$D_{v_1}\cdots D_{v_k}f$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$.
If $k\geq d-1$, then $D_{v_1}\cdots D_{v_k}f$ is either identically zero or linear with nonnegative coefficients,
in which case it is log-concave on ${\mathbb{R}}_{\geq 0}^n$, so we take $0\leq k\leq d-2$.
If $k=0$, then to show that $f$ is log-concave at a point $a\in {\mathbb{R}}_{\geq 0}^n$, by \cref{lem:LC_Quad} (\cref{cond:6} $\Rightarrow$ \cref{cond:1}),
it suffices to show that $D_af$ is log-concave at $z=a$. This reduces the case $k=0$ to the case $k=1$.
Suppose $1\leq k \leq d-2$. By induction $\partial_{j}f$ is completely log-concave for all $j=1, \hdots, n$,
and hence $D_{v_1}\cdots D_{v_{k-1}}\partial_{j}f = \partial_{j}D_{v_1}\cdots D_{v_{k-1}}f$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$.
Since $D_{v_1}\cdots D_{v_{k-1}}f$ is indecomposable and has degree $d-k+1\geq 3$, it follows from \cref{lem:DerToCLC} that
$D_{v_1}\cdots D_{v_{k}}f$ is log-concave on ${\mathbb{R}}_{\geq 0}^n$.
\end{proof}
\section{Complete log-concavity of independence polynomials}
\label{sec:IndPoly}
In this section, we use \cref{thm:CLCQuad} to prove
that the homogenization of the generating polynomial of the independent sets of a
matroid is completely log-concave. In the following section we use a restriction of this to derive Mason's conjecture.
\begin{theorem}\label{thm:IndPolyCLC}
For any matroid $M = ([n],{\mathcal{I}})$, the polynomial
\[ g_M(y,z_1, \hdots, z_n) \ = \ \sum_{I\in {\mathcal{I}}} y^{n-\card{I}}\prod_{i\in I}z_i \]
in ${\mathbb{R}}[y,z_1, \hdots, z_n]$ is completely log-concave.
\end{theorem}
We prove this by looking at quadratic derivatives of $g_M$.
\begin{lemma}\label{lem:Rank2}
For any matroid $M = ([n],{\mathcal{I}})$, the quadratic polynomial $\partial_{y}^{n-2}g_M$
is log-concave on ${\mathbb{R}}_{\geq 0}^{n+1}$.
\end{lemma}
\begin{proof}
After taking derivatives and rescaling, we see that
\[ q \ = \ \frac{\partial_{y}^{n-2}g_M }{(n-2)!} \ = \ \frac{n(n-1)}{2}\cdot y^2 +(n-1)\cdot \sum_{\set{i}\in {\mathcal{I}}}y z_i+ \sum_{\set{i, j}\in {\mathcal{I}}} z_iz_j. \]
Let $Q$ denote the Hessian $\nabla^2q$ of $q$.
Note that columns and rows of $\nabla^2 q$ corresponding to loops in $M$ are zero, and the log-concavity
of $q$ only depends on the principal submatrix of $Q$ indexed by non-loops.
In this spirit and in a slight abuse of notation, we use ${\mathds{1}}$ within this proof to denote the indicator vector of the non-loops of $M$.
Then we find that
\[ \nabla^2q \ \ = \ \ Q \ \ = \ \
\begin{bmatrix}
n(n-1)& (n-1) {\mathds{1}}^\intercal\\
(n-1) {\mathds{1}}& B
\end{bmatrix},
\]
where $B$ is an $n\times n$ matrix with $B_{ij}=1$ when $\set{i,j}$ has rank two in $M$ and $B_{ij}=0$ otherwise.
Since $q$ is quadratic, its Hessian does not depend on any evaluation, so $q$ is log-concave on ${\mathbb{R}}_{\geq 0}^{n+1}$ if and only if it is
log-concave at the point $a = (1,0, \hdots, 0)$. By \cref{lem:LC_Quad} (\cref{cond:1} $\Leftrightarrow$ \cref{cond:5}), this happens if and only if the matrix
\[
(a^{\intercal}Qa)Q - (Qa)(Qa)^{\intercal}
\ = \ n(n-1)Q - (n-1)^2 \begin{bmatrix} n \\ {\mathds{1}}\end{bmatrix}\begin{bmatrix} n \\ {\mathds{1}}\end{bmatrix}^{\intercal}
\ = \ (n-1)\begin{bmatrix}
0& 0\\
0& n B - (n-1){\mathds{1}}\1^{\intercal}
\end{bmatrix}\]
is negative semidefinite. Thus it suffices to show that $n B - (n-1){\mathds{1}}\1^{\intercal}$ is negative semidefinite.
As $M$ is a matroid, the matroid partition property tells us that the nonloops of $M$ may be partitioned into equivalence classes of parallel elements $P_{1},\dots,P_{c}$.
This lets us rewrite the matrix $B$ as
\[
B \ = \ {\mathds{1}}\1^\intercal-\sum_{i=1}^c {\mathds{1}}^{\ }_{P_i} {\mathds{1}}_{P_i}^\intercal
\ \ \ \ \text{ and }\ \ \ \
n B - (n-1){\mathds{1}}\1^{\intercal} \ = \ {\mathds{1}}\1^\intercal- n\cdot \sum_{i=1}^c {\mathds{1}}^{\ }_{P_i} {\mathds{1}}_{P_i}^\intercal.
\]
We can now check that this matrix is negative semidefinite. Let $x\in {\mathbb{R}}^n$ and consider
\[
x^\intercal(n B - (n-1){\mathds{1}}\1^{\intercal} ) x \ \ = \ \ \left({\mathds{1}}^\intercal x \right)^2 - n\cdot \sum_{i=1}^c \left({\mathds{1}}_{P_i}^\intercal x \right)^2.
\]
Since $P_1, \hdots P_c$ partition the non-loops of $M$, ${\mathds{1}} $ equals $ \sum_{i=1}^c{\mathds{1}}_{P_i}$.
For any real numbers $u_1, \hdots, u_c$, the Cauchy-Schwarz inequality implies that $(\sum_{i=1}^cu_i)^2 \leq c\cdot \sum_{i=1}^c u_i^2$.
This then gives that
\[
\parens*{{\mathds{1}}^\intercal x}^2 \ \ = \ \ \parens*{\sum_{i=1}^c{\mathds{1}}_{P_i}^\intercal x}^2 \ \ \leq \ \ c\cdot \sum_{i=1}^{c} \parens*{{\mathds{1}}_{P_i}^\intercal x }^2 \ \ \leq \ \ n\cdot \sum_{i=1}^c \parens*{{\mathds{1}}_{P_i}^\intercal x}^2.
\]
For the last inequality, we use the fact the number of equivalence classes $c$ of nonloops of $M$ is at most $n$.
It follows that $x^\intercal(n B - (n-1){\mathds{1}}\1^{\intercal} ) x \leq 0$ for all $x$ and by \cref{lem:LC_Quad}, $q$ is log-concave on ${\mathbb{R}}_{\geq 0}^{n+1}$.
\end{proof}
\begin{proof}[Proof of \cref{thm:IndPolyCLC}]
We will use the criterion in \cref{thm:CLCQuad} to show complete log-concavity.
Here we use $\partial_{i}$ to mean $\partial_{z_{i}}$ and for
$\alpha \in {\mathbb{Z}}_{\geq 0}^n$, $\partial^{\alpha}$ to denote $\prod_{i=1}^n\partial_i^{\alpha_i}$.
We need to show that for every $k\in {\mathbb{Z}}_{\geq 0}$ and $\alpha \in {\mathbb{Z}}_{\geq 0}^n$ with $k + \abs{\alpha}\leq n-2$,
the polynomial $\partial_y^{k}\partial^\alpha g_M$ is indecomposable and that for $k + \abs{\alpha} = n-2$ it is log-concave.
Note that if $\alpha_i\geq 2$ for any $i$, then $\partial^{\alpha}g_M$ is zero, so we may consider $\alpha = {\mathds{1}}_J$ for some $J\subseteq [n]$.
Similarly, if $J$ is not an independent set of $M$, then $\partial^{\alpha}g_M = \partial^{J}g_M =0$.
Therefore is suffices to consider $\alpha = {\mathds{1}}_J$ for $J\in {\mathcal{I}}$. In this case,
the derivative $\partial^{J}g_M$ equals the polynomial $g_{M/J}$ of the contraction $M/J$, namely
\[
\partial^{J}g_M \ \ = \ \ \sum_{I\in {\mathcal{I}} : J\subseteq I}y^{n-\card{I}}\prod_{i\in I\setminus J} z_i
\ \ = \ \ \sum_{I\in {\mathcal{I}} : J\subseteq I}y^{n-\card{J}-\card{I\setminus J}}\prod_{i\in I\setminus J} z_i \ \ = \ \ g_{M/J}.
\]
Recall that $M/J$ is a matroid on ground set $[n]\setminus J$ with independent sets $\set{I\setminus J \given J\subseteq I\in {\mathcal{I}}}$.
First we check indecomposability of $\partial_y^{k}\partial^Jg_M =\partial_y^{k}g_{M/J}$.
Note that if $i \in [n]\setminus J$ is a loop of $M/J$, then the variable $z_i$ does not appear in $g_{M/J}$ and $\partial_ig_{M/J} =0$.
Similarly, $\partial_ig_{M/J}$ is zero for all $i\in J$.
Otherwise, the monomial $y^{n-\card{J}-1-k}z_i$ appears in $\partial_y^{k}g_{M/J}$ with non-zero coefficient.
Since $k+\card{J} \leq n-2$, it follows that $\partial_y\partial_i g_{M/J}$ is non-zero.
In particular, the graph formed in \cref{def:indecomposable} is a star centered at the variable $y$, and thus connected.
Therefore $\partial_y^{k}\partial^Jg_M$ is indecomposable.
Now suppose $k + \card{J}=n-2$. Since $M/J$ is a matroid on $n - \card{J}$ elements, \cref{lem:Rank2} imples that
$\partial_y^{n-\card{J}-2}g_{M/J} = \partial_y^{k}\partial^Jg_{M}$ is log-concave on ${\mathbb{R}}_{\geq 0}^{n+1}$. All together with \cref{thm:CLCQuad}, this implies that the polynomial $g_M$ is completely log-concave.
\end{proof}
\begin{corollary}\label{cor:BivariateIndep}
Given a matroid $M=([n], {\mathcal{I}})$ with ${\mathcal{I}}_{k}$ independent sets of size $k$, the bivariate polynomial
\[
f_M(y,z)\ = \ \sum_{k=0}^r \ {\mathcal{I}}_{k} \ y^{n-k}z^k,
\]
is completely log-concave.
\end{corollary}
\begin{proof}
Note that $f_M$ is the restriction of the completely log-concave polynomial $g_M$ to $z_i = z$ for all $i\in [n]$. Since the image of ${\mathbb{R}}_{\geq 0}^2$ under the linear map
$(y,z) \mapsto (y,z,\hdots, z)$ is contained in ${\mathbb{R}}_{\geq 0}^{n+1}$, \cref{lem:LinearPreserver} implies that
$f_M(y,z) = g_M(y,z, \hdots, z)$ is completely log-concave.
\end{proof}
\section{Proof of Mason's conjecture}\label{sec:mason}
We use the following proposition, which was first observed by Gurvits \cite{Gur08}, and give a short proof for the sake of completeness.
\begin{proposition}[Proposition 2.7 from \cite{Gur08}]\label{thm:CLCtoULC}
If $f = \sum_{k=0}^{n} c_{k}y^{n-k}z^{k} \in {\mathbb{R}}[y,z]$ is completely log-concave, then
the sequence $c_{0},\dots,c_{n}$ is ultra log-concave. That is, for every $1<k <n$,
\[\parens*{\frac{c_{k}}{\binom{n}{k}}}^{2} \geq \frac{c_{k-1}}{\binom{n}{k-1}} \cdot \frac{c_{k+1}}{\binom{n}{k+1}}.\]
\end{proposition}
\begin{remark}
In \cite{Gur08}, Gurvits assumes \emph{strong log-concavity} and also shows the
converse. In a future article, we show the equivalence of strong and complete log-concavity for
homogeneous polynomials.
\end{remark}
\begin{proof}
Since $f$ is completely log-concave, for any $1< k<n$, the quadratic
$q(y,z) = \partial_y^{n-k-1}\partial_z^{k-1} f$ is log-concave over ${\mathbb{R}}_{\geq 0}^2$.
Notice that for any $0\leq m \leq n$,
\[
\partial_y^{n-m}\partial_z^{m} f \ = \ (n-m)! \ m! \ c_{m} \ = \ n! \ \frac{c_{m}}{\binom{n}{m}}.
\]
Using this for $m=k-1,k, k+1$, we can write the Hessian of $q$ as
\[
\nabla^2q \ = \
\begin{bmatrix}
\partial_y^2 q & \partial_y\partial_z q \\
\partial_y\partial_z q & \partial_z^2 q
\end{bmatrix}
\ = \ n! \
\begin{bmatrix}
\left. c_{k-1} \middle/\binom{n}{k-1}\right. & \left.c_{k}\middle/ \binom{n}{k}\right. \\
\left.c_{k} \middle/ \binom{n}{k}\right. & \left.c_{k+1} \middle/\binom{n}{k+1}\right.
\end{bmatrix}.
\]
Since $q$ is log-concave on ${\mathbb{R}}_{\geq 0}^2$, by \cref{lem:LC_Quad} its Hessian cannot be positive or negative definite.
Its determinant is therefore non-positive. This gives the desired inequality:
\[
0 \ \geq \ \det(\nabla^2q) \ = \ (n!)^2\parens*{\frac{c_{k-1}}{ \binom{n}{k-1}} \cdot \frac{c_{k+1}}{ \binom{n}{k+1}} - \parens*{ \frac{c_{k}}{ \binom{n}{k}}}^2}.
\]
\end{proof}
The strong version of Mason's conjecture, \cref{thm:mason}, then follows from \cref{cor:BivariateIndep}.
\printbibliography
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,187 |
{"url":"https:\/\/www.vedantu.com\/maths\/area-word-problems","text":"Courses\nCourses for Kids\nFree study material\nFree LIVE classes\nMore\n\n# Area Word Problems\n\n## Introduction to Area Word Problem\n\nLast updated date: 26th Mar 2023\nTotal views: 26.7k\nViews today: 0.09k\n\nWe all are aware of what squares are and how to find the area of a square. But in area word problems on the topic square, we will find the area in which the four sides are given as equal. To find the area of a given square, we must ensure that the sides of the square are in the same unit of length. In case they are given in different units, change them to the same unit. Some area of square examples is given to better understand the topic.\n\n## What is a Square?\n\nA regular quadrilateral in which all four sides are of equal length and all four angles are equal is considered a square. The angles of the square are 90 degrees each. Also, the square's diagonals are equal and bisect each other at 90 degrees.\n\nA Square\n\nThe above figure represents a square in which all the sides are equal, and each angle is 90 degrees.\n\nSimilarly, a parallelogram with all of its two adjacent sides being equal with one right vertex angle is a square.\n\n## Properties of a Square\n\nThe properties of a square are listed below:\n\n\u2022 All the interior angles are equal to 90\u00b0\n\n\u2022 All the sides of the square are equal and congruent with each other\n\n\u2022 The opposite sides of the square are always parallel to each other\n\n\u2022 The diagonal of the square divides it into similar isosceles triangles\n\n\u2022 The length of diagonals is always greater than the sides of the square\n\n\u2022 The square\u2019s diagonals bisect each other at 90\u00b0\n\n\u2022 Both diagonals of the square are equal to each other\n\n\u2022 The square contains 4 vertices and 4 sides\n\n## Area of Square Formula\n\nSince we know that a square is a shape which has four equal sides and every angle is a right angle, i.e. 90\u00b0. And hence, the opposite sides are also parallel. So the area of the square can be found by measuring the space occupied within the square. The formula to calculate the square of its side gives it.\n\nIf \u2018a\u2019 is the side of the square, then its area is given by ${a}^2$.\n\nArea of a Square\n\n## Solved Word Problems on the Area of a Square\n\nSome Solved Word Problems on the Area of a Square are given below:\n\nQ 1. Find the area of a square whose length is 10 cm.\n\nAns: Given the length of a square = 10 cm\n\nSquare of 10 cm\n\nArea of a square $=$ length $\\times$ length\n\n$=10 \\times 10 \\mathrm{~cm}^2$\n\n$=100 \\mathrm{~cm}^2$\n\nThus, the area of the square is $100 \\mathrm{~cm}^2$\n\nQ 2. Find the area of a square whose side measures 45 cm.\n\nAns: Given the length of a square = 45 cm\n\nSquare of 45 cm\n\nArea of a square $=$ length $\\times$ length\n\n$=45 \\times 45 \\mathrm{~cm}^2$\n\n$=2025 \\mathrm{~cm}^2$\n\nThus, the area of the square is $2025 \\mathrm{~cm}^2$.\n\nQ 3. The length of a side of a square field is 200 m. What will be the cost of levelling the field at a rate of 10 rs per square metre?\n\nAns: Length of the square field = 200 m\n\nArea of the field $=$ side $\\times$ side\n\n$=200 \\mathrm{~m} \\times 200 \\mathrm{~m}$\n\n$=40000 \\mathrm{~m}^2$\n\nCost of levelling the field $=40000 \\times 10 \\mathrm{Rs}$\n\n$=400000 \\mathrm{Rs}$\n\nThus, the cost of levelling the field is 400000 Rs.\n\n## Practice Problems on the Area of a Square\n\nTry the given practice problems on the area of a square:\n\nQ 1. Find the area of the square whose sides are given below:\n\n(i) 15 m\n\n(ii) 250 m\n\n(iii) 5 cm\n\n(iv) 40 cm\n\n(v) 10 m\n\nAns: (i) $225 \\mathrm{~m}^2$\n\n(ii) $62500 \\mathrm{~m}^2$\n\n(iii) $25 \\mathrm{~cm}^2$\n\n(iv) $1600 \\mathrm{~cm}^2$\n\n(v) $100 \\mathrm{~m}^2$\n\nQ 2. Find the area of the square of side $16 \\mathrm{~cm}$.\n\nAns: $256 \\mathrm{~cm}^2$\n\nQ 3. Find the length of the square whose area is $100 \\mathrm{~cm}^2$.\n\nAns: 10 cm\n\n## Summary\n\nIn this article, we learned about a quadrilateral square with four equal sides and all the angles as right angles, i.e. 90 degrees. The square's diagonals are equal and bisect each other at 90 degrees. We also learned about the various properties of squares and how we can calculate the area of squares. We also discussed the area of square examples to get to know how to find the area of a square of a given length easily.\n\n## FAQs on Area Word Problems\n\n1. Can a rhombus be considered a square?\n\nA rhombus can also be considered a square if it follows some conditions. Rhombus is a convex quadrilateral with all four sides equal. It is said to be a square if it has a right vertex angle.\n\n2. How is a square different from a rectangle?\n\nA square has all its sides equal in length, whereas a rectangle has only its opposite sides equal in length. The adjacent sides are unequal in rectangles but are equal in squares.\n\n3. Is zero a square number?\n\nSince zero satisfies all the definitions of being a square number, it is considered a perfect square.","date":"2023-03-30 08:43:36","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\": 0.7123635411262512, \"perplexity\": 426.7427877905332}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949107.48\/warc\/CC-MAIN-20230330070451-20230330100451-00667.warc.gz\"}"} | null | null |
Put some cash together and buy him a few hours of exotic car racing in LA.
I'd love to see Bloodworth tear the racetracks apart in a Lamborghini!
He can record and and call the show "Blood on the tracks"
@TheHashtag0nist The Allies know you better than I. Please pitch that to them. Please.
OMG, I just remembered something. You remember a certain episode of mandatory update when Huber explains that someone "stole" a christmas present for his parents? The gift in question was wine and cheese tasting (or something like that), and the prime suspect was Bloodworth. Wink, wink, nudge, nudge.
I know it probably wasn't him, but it's an interesting idea.
I love how Blood brought up this thread at the beginning of the Power Stone tournament... and still didn't provide any real answers as to what would make a great gift for him. C'mon @Bloodworth, give the people what they want... so they can in turn give you what you want.
@SabotageTheTruth Haha, yea man. When I heard him say that I was like, FINALLY IT WILL BE REVEALED.....and then he didn't say anything.
@Bloodworth I might get you pajama pants, a calming tea and a pillow because it seems like you're never asleep, since you're always on twitter, forum, twitch chat and other social media. You should relax a bit, for all the hard work you do.
@Bloodworth Alright! Thank you for posting, Blood! I appreciate it! I understand what you mean, it being a surprise is fun! And it would be weird, my bad! Just wanted to make it a little easier for people to show you some love! Stay classy, Blood! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,864 |
{"url":"https:\/\/codegolf.stackexchange.com\/questions\/98434\/lets-abbreviate-those-numbers-now-reverse","text":"# Let's abbreviate those numbers! Now reverse?\n\nIntroduction:\n\nLike Twitter and Instagram and others, I wanted to display numbers like 1.2K and 3.8 M instead of 1,222 or 3,823,456.\n\nBut that's not all! As we all know, there might be some human beings which undoubtely won't like these abbreviations and will try to reverse them. So, 1.2k will become 1,200 and 3.8 M will become 3,800,000.\n\n\u2022 your task is to write a program or a function that converts a list of numbers (which are given as strings) into their abbreviate pairs and vice-versa.\n\nFor example, if the input list (or any STDIN) was ['1.4k', '1,234,567', '7.99M'], then you should output:\n\n['1,400', '1.2M', '7,990,000']\n\n\nYou can follow the next schema for abbreviations:\n\n\u2022 103 -> one kilo -> K\n\u2022 106 -> one million -> M\n\u2022 109 -> one billion -> B\n\nYour code may assume all lowercase, all uppercase, mixed case or undefined case for input and use any of these for output, but should be consistent.\n\nRules and restrictions:\n\n\u2022 you may write a program or function, taking input via STDIN (or closest alternative), command-line argument or function argument and outputting the result via STDOUT (or closest alternative), function return value or function (out) parameter.\n\u2022 input may be in any convenient list or string format. You may assume that the ai are less than 231 each and that the list contains at least one element.\n\u2022 each abbreviated number will contain only one . while a normal number will contain as many , as necessary (you may assume that this numbers won't be altered).\n\u2022 you MAY NOT enter a number as '123456' but rather 123,456\n\u2022 standard rules apply.\n\nTest cases:\n\nInput: ['1.5M', '77.6k', '123,456,789'] Output: ['1,500,000', '77,600', '123.4M']\nInput: ['3,000,000,000', '581k', '2b'] Output: ['3B', '581,000', '2,000,000,000']\nInput: ['0.1k'] Output: ['100']\nInput: ['888', '33'] Output: ['0.888k', '0.033k']\n\n\nClarifications:\n\n\u2022 for numbers < 1000 after the decimal point in abbreviation output you should have as many digits as required to get the correct result. (e.g: 2 -> will become 0.002k) - that means 3 decimals at most; for numbers > 1000 you can have a maximum of 1 decimal.\n\u2022 the abbreviation may be in both lower or upper case\n\u2022 I removed the built-ins restriction as suggested in the comments\n\nThe shortest code in bytes wins!\n\n\u2022 Requests for clarification: how many digits after the decimal point in abbreviation output? how to abbreviate numbers < 1000? uppercase or lowercase or both in input and output? \u2013\u00a0edc65 Nov 3 '16 at 12:07\n\u2022 Shouldn't '123,456,789' -> '123.4M'? Also, this doesn't clarify how many decimals to use. Surely anything under 1000 shouldn't need to be abbreviated anyways. \u2013\u00a0Kade Nov 3 '16 at 12:48\n\u2022 @anonymous2 read the third rule. \u2013\u00a0Grajdeanu Alex Nov 3 '16 at 13:25\n\u2022 \"you're not allowed to use any built-in module \/ function\" Any built-in function? \u2013\u00a0Alex Howansky Nov 3 '16 at 14:12\n\u2022 Your usage of \"kilo\" suggests SI prefixes and those would be [\"k\", \"M\", \"G\"]. What does \"while a normal number will contain as many , as necessary\" mean, in my country it would be a mistake to use any. \u2013\u00a0Angs Nov 3 '16 at 14:31\n\n# PHP, 234224213201 205 bytes\n\nfor(;$x=$argv[++$n];){$y=str_replace(\",\",\"\",$x)\/1e3;for($i=0;$y>999;$i++)$y=($y|0)\/1e3;echo(A<$c=substr($x,strlen($x)-1))?number_format($x*[k=>1e3,m=>1e6,b=>1e9][$c]):($i?($y*10|0)\/10:$y).kmb[$i],\" \";} 6 bytes saved by insertusernamehere, 4 bytes inspired by that. \u2022 takes input from command line arguments, prints results space-separated with a trailing separator \u2022 expects lower case abbreviation \u2022 run with -r -2 bytes if underscore as separator is ok: Replace \" \" with _. -1 byte if correct rounding is ok: Replace ($y*10|0)\/10 with round($y,1). -17 bytes for PHP 7.1: Replace substr($x,strlen($x)-1) with $x[-1].\n\n80 (63) bytes for expanding one argument only:\n\n<?=number_format(($x=$argv[1])*[K=>1e3,M=>1e6,B=>1e9][substr($x,strlen($x)-1)]);\n\n\nsave to file, then execute (or replace <?= with echo +space and run with -r.\n\n\u2022 Your second example doesn't compile. \u2013\u00a0Alex Howansky Nov 3 '16 at 14:15\n\u2022 You have unbalanced parens. \u2013\u00a0Alex Howansky Nov 3 '16 at 14:17\n\u2022 -4 bytes: for($j=1;$x=$argv[$j++];) \u2013 instead of foreach($argv as$i=>$x)if($i) \u2013\u00a0insertusernamehere Nov 3 '16 at 17:05\n\u2022 -2 bytes: kmb[$i] \u2013 instead of \"kmb\"[$i]. \u2013\u00a0insertusernamehere Nov 3 '16 at 17:20\n\u2022 @insertusernamehere Negative string indexes are coming in PHP 7.1; and that\u00b4s a RC (yet). Thanks for the other bytes! \u2013\u00a0Titus Nov 4 '16 at 10:18\n\n# JavaScript, 545524522518514508504498494 214 bytes\n\nThanks to @ETHproductions for saving 180 bytes!\n\nd=F=>F.map(f=>1\/f.slice(-1)?f=(f=f.replace(\/,\/g,\"\"))[9]?(f\/1e8|0)\/10+\"B\":f[6]?(f\/1e5|0)\/10+\"M\":f\/1e3+\"k\":R(R(f.slice(0,-1)+\"e\"+' kMB'.indexOf(f.substr(-1))*3-0+\"\").match(\/.{1,3}\/g)+\"\"),R=x=>[...x].reverse().join)\n\n\nTo call the function:\n\nd([\"1.5M\",\"1,500,000\"]) \/\/[\"1,500,500\",\"1.5M\"]\n\n\nOutputs as alert, where each alert contains a different element of the input\n\nd = F => F.map(f => 1 \/ f.slice(-1) ? f = (f = f.replace(\/,\/g, \"\"))[9] ? (f \/ 1e8 | 0) \/ 10 + \"B\" : f[6] ? (f \/ 1e5 | 0) \/ 10 + \"M\" : f \/ 1e3 + \"k\" : R(R(f.slice(0, -1) + \"e\" + ' kMB'.indexOf(f.substr(-1)) * 3 - 0 + \"\").match(\/.{1,3}\/g) + \"\"), R = x => [...x].reverse().join )\n\n\nSummary of edits: converted function to an arrow function\n\n\u2022 removed semi-colons ';'\n\u2022 removed var\n\u2022 converted to an arrow function\n\u2022 used map to iterate through the individual elements of the array\n\u2022 used |0 instead of floor\n\u2022 used regex for testing\n\u2022 used ternary operators instead of if-else statements\n\u2022 included a separate function for .reverse().join''","date":"2021-06-16 02:41:34","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\": 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\": 0.17929847538471222, \"perplexity\": 6338.716167420881}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487621699.22\/warc\/CC-MAIN-20210616001810-20210616031810-00288.warc.gz\"}"} | null | null |
UTA men's and women's cross-country teams post top-5 finishes in final meet
By Chris Amaya, The Shorthorn sports editor
The conference championship was UTA's last meet of the fall season, as the NCAA postponed all Division I fall championships.
UTA cross-country sophomore named Sun Belt Runner of the Week
By Adrian Rodriguez, The Shorthorn staff
Jack Myers was one of three runners to average 5:07 per mile and finished 17 places higher than he did at last season's Sun Belt Conference Championship.
UTA men's, women's cross-country teams take third in first meet of the season
Sophomore Jack Myers and senior Madeleine Rowe recorded the top times for their respective teams.
UTA men's and women's cross-country teams picked to finish top five in Sun Belt preseason poll
As of Friday, the teams have one regular season meet scheduled before the conference championship in October.
The day the UTA sports world stood still
By Arianna Vedia, Chris Amaya, Alonzo Olmedo and Julio Vega, The Shorthorn sports desk
The Shorthorn sports desk talks to UTA's coaches about how the COVID-19 outbreak has affected each of the university's sports programs.
Sun Belt Conference suspends all sports indefinitely in wake of coronavirus
By Arianna Vedia, The Shorthorn sports editor
The Sun Belt Conference has suspended all regular-season competitions and conference championships across all sports indefinitely due to coronavirus concerns.
UTA men's track and field scoops 3 out of 4 top Sun Belt Conference postseason awards
By Julio Vega, The Shorthorn staff
Sophomore Vanessa Ugorji earned All-Sun Belt Second Team honors after helping the women's team win the 4x400-meter race at the conference meet.
Men's indoor track and field falls short of repeat championship, women's team places seventh
Arkansas State University secured the conference title for both the men's and women's teams at the Sun Belt Indoor Championships in Birmingham, Alabama. For UTA, the men's team fell short on retaining its conference title, placing second to the Red Wolves while the women placed seventh of 12…
Sophomore Lucas Van Klaveren leads UTA in Charlie Thomas Invitational
By Chris Amaya, The Shorthorn staff
The men's and women's track teams continued their indoor seasons at the Charlie Thomas Invitational on Saturday at Gilliam Indoor Track Stadium in College Station. The men's team finished in fifth place out of 14 teams, while the women's team placed 11th.
UTA indoor track and field team to compete with 14 other universities at Charlie Thomas Invitational
This is the second-to-last competition before the indoor track and field team makes its way to the Sun Belt Conference Indoor Championships.
UTA indoor track and field gears up for Red Raider Invitational
After a solid showing at last weekend's Ted Nelson Invitational, the team will travel to Lubbock for the Red Raider Invitational, which will begin 10 a.m. Saturday.
UTA indoor track and field teams open season with podium finishes at Ted Nelson Invitational
Among seven competing schools, the men's team finished third, and the women's team finished sixth overall.
Justin Domangue makes history at NCAA meet, becomes UTA's first Cross-Country All-American
Domangue earned the honor after placing 36th in a field of 253 runners in the 2019 NCAA Cross Country Championships.
Junior track and field athlete sets Guinness World Record for highest standing jump
By Alonzo Olmedo, The Shorthorn staff
After three years of training and six failed attempts, Brett Williams set the record at 1.651 meters on Sept. 2.
Dynamic duo: basketball senior joins the track and field team
TiAndre Jackson-Young will compete as both a forward for the men's basketball team and as a jumper for the track and field team, increasing his training and versatility.
UTA cross-country attempts to set the pace for conference meet
The cross-country teams are in the tail end of their season with one last meet before the 2019 Sun Belt Conference Cross Country Championships.
Freshman from Norway goes the distance for UTA cross-country team
Mathilde Ruud made the adjustment to cross-country distances when she was recruited to UTA.
Justin Domangue paces UTA in 31st Annual Chile Pepper Cross Country Festival
Domangue was the top finisher for both teams, placing 17th out of 315 total runners.
Women's cross-country team shows promise after start of 2019 season
For five years, the women's cross-country team has seen its conference results and overall output take a fall.
Women's cross-country wins team title while Domangue takes men's crown in Gerald Richey Invitational
DALLAS — The cross-country teams delivered top-two performances in the 5-kilometer race when they competed at the Gerald Richey Invitational on Saturday at the Jesse Owens Memorial Athletic Complex.
Frontrunner Justin Domangue outruns expectations, leads UTA cross-country team
The "it" factor Domangue displays allows the senior runner to push himself to the extreme, head coach John Sauerhage said.
Sun Belt Conference preseason poll names rankings of UTA men's, women's cross-country teams
Both teams kick off their season at the Baylor Bear Twilight Invitational at 7 p.m. Friday in Waco.
UTA alumnus claims gold at 2019 Parapan American Games
Two-time Paralympian Tobi Fawehinmi will next compete in the World Para Athletics Championships in Dubai.
UTA alumnus to represent Team USA in track and field matchup against Europe
Alumnus Cordero Gray will compete as part of the men's 4x100-meter relay pool with three other teammates.
UTA cross-country teams aim to get a leg up on last season's shortcomings
Both the men's and women's teams are hoping new recruits will reinvigorate the season.
Men's, women's cross-country schedule released
The Mavericks will compete in five regular season meets and three postseason meets from August through November.
UTA track and field to compete in 2019 NCAA Outdoor Track & Field Championships
Two individuals and two relay teams will compete at the 2019 NCAA Outdoor Track & Field Championships starting on Wednesday in Austin.
UTA senior sprinter keeps his home close to heart
A long, dark wooden barn and a shed holding about a dozen sheep sit beside the country road on a small family farm just north of Valbo, Sweden.
UTA sprinter named Sun Belt Conference Men's Track Athlete of the Week
Senior sprinter Erik Martinsson was named Sun Belt Conference Men's Track Athlete of the Week on Wednesday.
Sprinting toward success
By Martin Paredes, Jr., The Shorthorn staff
The opportunity to test himself against higher competition paved the way for senior Deondre Wiltshire's arrival to UTA.
Senior hurdler named Sun Belt Men's Track Athlete of the Week
Senior hurdler Victor Fincher was named Men's Track Athlete of the Week by the Sun Belt Conference on Wednesday.
Photos: Lady Mavericks sweep two-game series against University of Louisiana Monroe
Photos: Lady Mavericks kick off new year with loss to University of Arkansas at Little Rock
Photos: Arlington hosts Wrangler National Finals Rodeo at Globe Life Field
Photos: UTA men's, women's basketball teams tip off their seasons at College Park Center
Photos: The bulls start comin' and they don't stop buckin'
Photos: UTA volleyball drops two out of three matches in weekend series to Louisiana-Lafayette
Photos: UTA volleyball falls to in-state rival Texas State University
Photos: UTA volleyball wins first home matches against Arkansas State after COVID-19 postponement
Photos: Spectators attend Globe Life Field for game 1 of the National League Championship Series | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,618 |
<?php
namespace Appoint\Model;
use Zend\InputFilter\InputFilter;
use Zend\InputFilter\InputFilterAwareInterface;
use Zend\InputFilter\InputFilterInterface;
/**
* Appointmets - ORM of ZF2, Defines the relation with Appoints Table in the DB,
* for any kinds of changes or restriction please update this structure.
*
* @package Appoint
* @subpackage Model
* @implements InputFilterAwareInterface
*/
class Appoint implements InputFilterAwareInterface
{
public $id;
public $date;
public $patient_name;
public $reason;
protected $inputFilter;
/*
* This method simply copies the data from the passed in array to our entity's properties
*/
public function exchangeArray($data)
{
$this->id = (!empty($data['id'])) ? $data['id'] : null;
$this->date = (!empty($data['date'])) ? $data['date'] : null;
$this->reason = (!empty($data['reason'])) ? $data['reason'] : null;
$this->patient_name = (!empty($data['patient_name'])) ? $data['patient_name'] : null;
}
//returns InputFilter and mostly help to validatze requests coming from client for DB operations
public function getInputFilter()
{
if (!$this->inputFilter) {
$inputFilter = new InputFilter();
$inputFilter->add(array(
'name' => 'date',
'required' => true,
));
$inputFilter->add(array(
'name' => 'patient_name',
'required' => true,
'filters' => array(
array('name' => 'StripTags'),
array('name' => 'StringTrim'),
),
'validators' => array(
array(
'name' => 'StringLength',
'options' => array(
'encoding' => 'UTF-8',
'min' => 1,
'max' => 100,
),
),
),
));
$inputFilter->add(array(
'name' => 'reason',
'required' => true,
'filters' => array(
array('name' => 'StripTags'),
array('name' => 'StringTrim'),
),
'validators' => array(
array(
'name' => 'StringLength',
'options' => array(
'encoding' => 'UTF-8',
'min' => 1,
'max' => 100,
),
),
),
));
$this->inputFilter = $inputFilter;
}
return $this->inputFilter;
}
public function getArrayCopy()
{
return get_object_vars($this);
}
public function setInputFilter(InputFilterInterface $inputFilter)
{
throw new \Exception("Not used");
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,425 |
{"url":"https:\/\/zenodo.org\/record\/5547019\/export\/schemaorg_jsonld","text":"Journal article Open Access\n\nText Categorization Techniques and Current Trends\n\nAbhisu Jain; Aditya Goyal; Vikrant Singh; Anshul Tripathi; Saravanakumar Kandasamy\n\nJSON-LD (schema.org) Export\n\n{\n\"inLanguage\": {\n\"alternateName\": \"eng\",\n\"@type\": \"Language\",\n\"name\": \"English\"\n},\n{\n\"@id\": \"\",\n\"@type\": \"CreativeWork\"\n},\n{\n\"@id\": \"https:\/\/hdl.handle.net\/E9620069520\/2020\\u00a9BEIESP\",\n\"@type\": \"CreativeWork\"\n}\n],\n\"description\": \"<p>With the development of online data, text categorization has become one of the key procedures for taking care of and sorting out content information. Text categorization strategies are utilized to order reports, to discover fascinating data on the world wide web. Text Categorization is a task for categorizing information based on text and it has been important for effective analysis of textual data frameworks. There are systems which are designed to analyse and make distinctions between meaningful classes of information and text, such system is known as text classification systems. The above-mentioned system is widely accepted and has been used for the purpose of retrieval of information and natural language processing. The archives can be ordered in three different ways unsupervised, supervised and semi supervised techniques. Text categorization alludes to the procedure of dole out a classification or a few classes among predefined ones to each archive, naturally. For the given text data, these words that can be expressed in the correct meaning of a word in different documents are usually considered as good features. In the paper, we have used certain measures to ensure meaningful text categorization. One such method is through feature selection which is the solution proposed in this paper which does not change the physicality of the original features. We have taken into account all meaningful features to distinguish between different text categorization approaches and highlighted the evaluation metrics, advantages and limitations of each approach. We conclusively studied the working of several approaches and drew conclusion of best suited algorithm by performing practical evaluation. We are going to review different papers on the basis of different text categorization sections and a comparative and conclusive analysis is presented in this paper. This paper will present classification on various kinds of ways to deal and compare with text categorization.<\/p>\",\n\"creator\": [\n{\n\"affiliation\": \"Student, Department of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India.\",\n\"@type\": \"Person\",\n\"name\": \"Abhisu Jain\"\n},\n{\n\"affiliation\": \"Student, Department of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India.\",\n\"@type\": \"Person\",\n},\n{\n\"affiliation\": \"Student, Department of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India.\",\n\"@type\": \"Person\",\n\"name\": \"Vikrant Singh\"\n},\n{\n\"affiliation\": \"Student, Department of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India.\",\n\"@type\": \"Person\",\n\"name\": \"Anshul Tripathi\"\n},\n{\n\"affiliation\": \"Assistant Professor, School of Information Technology and Engineering, Vellore Institute of Technology, Vellore campus, Tamil Nadu, India\",\n\"@type\": \"Person\",\n\"name\": \"Saravanakumar Kandasamy\"\n}\n],\n\"headline\": \"Text Categorization Techniques and Current Trends\",\n\"datePublished\": \"2020-06-30\",\n\"keywords\": [\n\"Attention Mechanism, BRCAN, Convolutional Neural Network, Feature Evaluation Function, Few Short\"\n],\n\"url\": \"https:\/\/zenodo.org\/record\/5547019\",\n\"contributor\": [\n{\n\"affiliation\": \"Publisher\",\n\"@type\": \"Person\",\n\"name\": \"Blue Eyes Intelligence Engineering and Sciences Publication(BEIESP)\"\n}\n],\n\"@context\": \"https:\/\/schema.org\/\",\n\"identifier\": \"https:\/\/doi.org\/10.35940\/ijeat.E9620.069520\",\n\"@id\": \"https:\/\/doi.org\/10.35940\/ijeat.E9620.069520\",\n\"@type\": \"ScholarlyArticle\",\n\"name\": \"Text Categorization Techniques and Current Trends\"\n}\n33\n12\nviews","date":"2022-05-24 18:52:15","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\": 0.20496051013469696, \"perplexity\": 4230.308381337621}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662573189.78\/warc\/CC-MAIN-20220524173011-20220524203011-00561.warc.gz\"}"} | null | null |
Phase 1 Projects
Philomathia Social Sciences Research Programme
Programme Director
Project Principal Investigators
Philomathia Post-doctoral Research Associates
Philomathia Programme Co-ordinator
Previous Investigators and Researchers
Phase 1 Projects overview
(In) fertility, Education and Reproductive Health
The Measure of Inequality: Social Knowledge in Historical Perspective
The consequences of the politics of austerity in the European Union
Realising Genomic Medicine: Intellectual Property Issues Beyond the 'Old' DNA Patent Debates
The Law of Energy Transitions
Philomathia Forum
Philomathia Forum overview
Forum 2015 Events
Philomathia Symposium
Philomathia Symposium overview
Symposium 2016 report: 'Body politics: the dilemmas of regulating new technologies'
Symposium 2014 report - 'Social Sciences at Cambridge: Working with the sciences'
Symposium 2014 response - Professor Paul Wright
The Fourth Philomathia Symposium - Where Does Democracy Reside?
Principal Investigators - Professor Jacqueline Scott and Professor Sarah Franklin, Department of Sociology
In the period after the Second World War all industrialized countries underwent a profound fertility change: the average family size has declined, childbearing has been delayed and the incidence of childlessness has risen substantially. The most common used measure to document the transitions in fertility is the period total fertility rate, which indicates the average family size of a synthetic cohort of women. Until 1965 women had on average two or more children in almost all industrialized countries. In contrast by the new millennium, there were only a handful of countries that had a fertility rate larger than two. Does this matter? For some, the decline in Western fertility is considered to be a positive and necessary development, especially by those commentators who warn against the ecological burdens of continuous world population growth, with a world population of 8 billion forecast by 2040. However, the medium-term demographic reality for UK and much of Europe is one of stagnation and even decline which, along with increased longevity, helps drive the aging population and worsening dependency ratios that threaten the sustainability of welfare systems.
Low and late fertility have consequences at both the societal and individual level. At the societal level, pension and health care costs are driven upwards and the changing age composition of the population can transform the political balance, as well as lead to consideration about how far the state should support services for children and families (Brewer et al., 2011). In addition fecundity declines by age, and childbearing at advanced reproductive ages entails a risk for the health of mother and child. This has stimulated a debate about the availability, affordability, efficiency and societal impact of assisted reproductive technologies (ARTs), such as in vitro fertilization (IVF). At the individual level the delay of entering parenthood could lead to having fewer children than desired, as well as to involuntary childlessness.
The transition towards lower and later fertility has occurred during a period when women's roles have changed markedly, particularly in terms of the huge growth in participation in higher education and the labour market. Change in education has rarely been considered in research examining the influence of policy on fertility behaviour, which is usually conceptualized in terms of purposive pro-natal policies. However, rising educational enrollment matters a lot for both the timing of first childbirth and the size of completed families (Smith and Ratcliffe, 2009). In addition, while the literature has paid great attention to declining (albeit
slowly) inequalities between men and women in both education and labour market participation, there has been less attention given to the way inequalities among women have increased markedly in the UK, across recent decades (Scott et al., 2012). Much of this widening inequalities gap has to do with education. Investigating the way the links between education, motherhood and employment are diverging among women (including women of different ethnic minorities) is one of the main gaps that our research will help address.
There is a very pronounced gap in the UK between education and the period total fertility rate of women born 1945-65, with higher education being associated with lower average family sizes, delayed childbearing and higher childlessness compared to women with low education. This is likely to be because educational level is closely related to a multitude of factors which can influence fertility behavior, such as family background; the timing and form of partnerships; occupation and employment trajectories; value orientations; contraceptive use and (in)fertility knowledge. Little attention has been paid to whether the effects of education are gender specific in part because of data limitations. This is a gap that newly available UK longitudinal data which includes information from both partners in forty thousand households will enable us to address.
Three main projects will be undertaken as part of this research.
1) How does education directly and indirectly affect the fertility decisions of men and women?
Event history and fixed effects models will be employed to track fertility intentions and behaviors of individuals over time within the UK, with particular attention to the mechanisms that underlie the fertility differentials by education for men and women. Most studies to date have focused on women. However, excluding partners misses the fact that fertility decisions are usually made in couples and it is important to examine the dyadic dynamics. Much of the focus to date has been on the changing role of women and the conflicts involved in juggling motherhood and paid job. Yet the role of fathers is changing, albeit at a somewhat slower pace, than the 'second shift' expected of mothers. What are the implications of changing maternal and paternal roles for delays in childbearing and family size? Are women more likely to have a second or third child, when male partners do more of the domestic work? What factors exacerbate or mitigate the differential fertility rates of those with higher and lower education?
2). How does the UK compare in fertility intentions and outcomes, with other European countries where policies differ markedly?
This project will utilize European level attitudinal data sets including the World Values Survey and the European Social Survey to carry out a cross-national comparison of fertility intentions and behaviour. We will pay particular attention to the different policy contexts for encouraging and supporting fertility, for example through measures designed to reduce work-family conflict (including parental leave, child care, flexible work hours etc).
3) How are policies on infertility and its treatment changing?
Infertility is estimated to affect approximately 3.5 million couples in the UK (Human Fertilisation and Embryology Authority, 2011). As romantic partnerships and childbearing are increasingly delayed, subfecundity associated with age may gain importance as a public health concern. In the context of changing fertility patterns, how do policies relating to infertility and social parenthood (adoption, parental custody rights) influence family formation? To what extent does public health policy address the diagnosis and treatment of infertility, and are policies relating to treatment equitable? In the UK, women under the age of 40 can undergo three rounds of IVF covered by the National Health Service; however, priority is given to childless couples, and the availability of coverage for the treatment varies substantially across the country (National Health Service, 2013). What is the effect of the uneven availability and distribution of treatment covered by the NHS? What are the characteristics of couples who seek treatment, with particular attention to parity, and is there unmet need for treatment?
Cambridge and the Philomathia Foundation launch second phase of Social Sciences Research Programme
History of Wealth Project
This project, co-funded by the Philomathia Foundation and the Isaac Newton Trust, investigates the broader significance of wealth an inheritance in 19th and early-20th century Britain.
This programme was launched by POLIS in October 2013, and aims to train future policy-makers to value and promote evidence-based policies that can most benefit society.
Research in Humanities and Social Sciences
philomathia@admin.cam.ac.uk | {
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{"url":"http:\/\/www.absoluteastronomy.com\/topics\/Duhamel's_principle","text":"Duhamel's principle\n\n# Duhamel's principle\n\nDiscussion\n\nEncyclopedia\nIn mathematics\nMathematics\nMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...\n\n, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous\nHomogeneous differential equation\nThe term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...\n\nlinear evolution equations like the heat equation\nHeat equation\nThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...\n\n, wave equation\nWave equation\nThe wave equation is an important second-order linear partial differential equation for the description of waves \u2013 as they occur in physics \u2013 such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...\n\n, and vibrating plate\nVibrating plate\nIn solid mechanics, a branch of mathematics and physics, a plate is modeled as a two-dimensional elastic body whose potential energy depends on how it is bent from a planar configuration, rather than how it is stretched . A vibrating plate can be modeled in a manner analogous to a vibrating drum...\n\nequation. It is named for Jean-Marie Duhamel\nJean-Marie Duhamel\nJean-Marie Constant Duhamel was a noted French mathematician and physicist. His studies were affected by troubles of the Napoleonic era. He went on to form his own school \u00c9cole Sainte-Barbe. Duhamel's principle is named for him. He was primarily a mathematician but did studies on the mathematics...\n\nwho first applied the principle to the inhomogeneous heat equation that models, for instance, the distribution of heat in a thin plate heated from beneath. For linear evolution equations without spatial dependency, such as a harmonic oscillator\nHarmonic oscillator\nIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \\vec F = -k \\vec x \\, where k is a positive constant....\n\n, Duhamel's principle reduces to the method of variation of parameters\u00a0technique for solving linear inhomogeneous ordinary differential equations.\n\nThe philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem\nCauchy problem\nA Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...\n\n(or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy u in Rn. The initial value problem is\nwhere g is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation is\ncorresponds to adding an external heat energy \u0192(x,t)dt at each point. Intuitively, one can think of the inhomogeneous problem as set of homogeneous problems each starting afresh at a different time slice t\u00a0=\u00a0t0. By linearity, one can add up (integrate) the resulting solutions through time t0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.\n\n## General considerations\n\nFormally, consider a linear inhomogeneous evolution equation for a function\nwith spatial domain D in Rn, of the form\nwhere L is a linear differential operator that involves no time derivatives.\n\nDuhamel's principle is, formally, that the solution to this problem is\nwhere Ps\u0192 is the solution of the problem\n\nDuhamel's principle also holds for linear systems (with vector-valued functions u), and this in turn furnishes a generalization to higher t derivatives, such as those appearing in the wave equation (see below). Validity of the principle depends on being able to solve the homogeneous problem in an appropriate function space and that the solution should exhibit reasonable dependence on parameters so that the integral is well-defined. Precise analytic conditions on u and f depend on the particular application.\n\n### Wave equation\n\nGiven the inhomogeneous wave equation:\n\nwith initial conditions\n\nA solution is\n\n### Constant-coefficient linear ODE\n\nDuhamel's principle is the result that the solution to an inhomogeneous, linear, partial differential equation can be solved by first finding the solution for a step input, and then superposing using Duhamel's integral.\nSuppose we have a constant coefficient, mth order inhomogeneous ordinary differential equation\nOrdinary differential equation\nIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....\n\n.\n\nwhere\n\nWe can reduce this to the solution of a homogeneous ODE using the following method. All steps are done formally, ignoring necessary requirements for the solution to be well defined.\n\nFirst let G solve\n\nDefine , with being the characteristic function\nCharacteristic function\nIn mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...\n\non the interval . Then we have\n\nin the sense of distributions\nDistribution (mathematics)\nIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...\n\n. Therefore\n\nsolves the ODE.\n\n### Constant-coefficient linear PDE\n\nMore generally, suppose we have a constant coefficient inhomogeneous partial differential equation\nPartial differential equation\nIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...\n\nwhere\n\nWe can reduce this to the solution of a homogeneous ODE using the following method. All steps are done formally, ignoring necessary requirements for the solution to be well defined.\n\nFirst, taking the Fourier transform\nFourier transform\nIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...\n\nin x we have\n\nAssume that is an mth order ODE in t. Let be the coefficient of the highest order term of .\nNow for every let solve\n\nDefine . We then have\n\nin the sense of distributions\nDistribution (mathematics)\nIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...\n\n. Therefore\n\nsolves the PDE (after transforming back to x).","date":"2019-02-19 14:58:15","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8917736411094666, \"perplexity\": 501.71949349751947}, \"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-2019-09\/segments\/1550247490225.49\/warc\/CC-MAIN-20190219142524-20190219164524-00599.warc.gz\"}"} | null | null |
\section{Introduction}
Symmetries play a quintessential role in nature, from being the progenitors of different phases of matter\cite{Sachdev_book,Balents_NatureReview,Savary2016} to manifesting conservation laws in and out of equilibrium.\cite{Brading_book} The group-theoretic notion of symmetries, i.e., defining symmetry as invariance under a group transformation, has become a cornerstone in the development of modern physics.\cite{Zee_book} Symmetries are also used to simplify a physical problem by reducing its effective Hilbert space based on what symmetry sectors the relevant physics takes place in (see glossary in Sec.~\ref{sec:glossary} for our nomenclature). These sectors involve global or local, as well as both continuous or discrete symmetries, and are manifestations of conservation laws. In the case of continuous symmetries, the associated conservation laws are formalized in Noether's theorem, which asserts an equivalence between them and the continuous symmetries in a physical system. As an example, the Fermi-- and Bose--Hubbard models\cite{Tasaki1998,Gersch1963} host a global $\mathrm{U}(1)$ symmetry, which translates into the conservation of particle number. Nevertheless, this equivalence is not restricted to continuous symmetries, as their discrete counterparts can also endow the system with conserved quantities. For example, coordinate inversion (P), charge conjugation (C), and time reversal (T) are discrete symmetries equivalent to the conservation of spatial, charge, and time parities, respectively.\cite{Zee_book,Peskin2016} Whereas (continuous and discrete) global symmetries give rise to the conservation of \textit{global charge}, local symmetries lead to \textit{local charges} being conserved. A prime example is gauge invariance,\cite{Jackson_review,Cheng_book} which is known as Gauss's law in quantum electrodynamics, and forms the principal property of gauge theories. By default, conservation of local charges leads to conservation of the global charge, but the converse is not necessarily true.
Given the richness afforded to physics by symmetries, a fundamental question is the effect of weakly breaking an underlying symmetry of a model on its subsequent dynamics. Even though traditionally studied in systems with weak integrability breaking, prethermalization has recently been generalized to nonintegrable systems with perturbative coherent breaking of global \cite{Mallayya2019,Ray2020} and local symmetries.\cite{Halimeh2020b,Halimeh2020c} Interestingly, in the case of weak breaking of the local gauge symmetry in a lattice gauge theory of $N$ matter sites, a \textit{prethermalization staircase} arises in the dynamics of the gauge violation (see Sec.~\ref{sec:definition} for definition), composed of prethermal plateaus at timescales $\propto\lambda^{-s}$, with $s=0,1,2,\ldots,N/2$ and $\lambda$ the strength of gauge invariance-breaking unitary errors. This rich prethermal behavior has the property of delaying the timescale of full gauge violation exponentially in system size, and it can also be observed in other local observables.\cite{Halimeh2020b,Halimeh2020c}
Also in case of slight breaking of global symmetries, the equilibrating dynamics is strongly affected. This can occur when conservation laws due to, e.g., integrability are perturbatively broken.\cite{Moeckel2008,Eckstein2009,Kollar2011,Tavora2013,Nessi2014,Essler2014,Bertini2016,Fagotti2015,Reimann2019} In this case, the initial equilibration to a generalized Gibbs ensemble\cite{Rigol2007,Rigol2009a,Rigol2009b,Vidmar2016,Essler2016,Cazalilla2016,Caux2016,Mallayya2018,Mallayya2019} (GGE) steady state is replaced by thermal equilibrium at a later timescale $\propto\lambda^{-2}$, with $\lambda$ the strength of the perturbative integrability-breaking term, as can be shown by kinetic Boltzmann-like equations that can be derived through employing time-dependent perturbation theory starting from the GGE steady state.\cite{Stark2013,DAlessio_review} This behavior is not restricted to quenched systems, but can also be seen in weakly interacting driven models.\cite{Lazarides2014,Canovi2016} As these examples illustrate, coherent errors can drastically change the dynamical properties of a quantum many-body system.
Going beyond unitary closed-system dynamics, decoherence has been a central topic of research in quantum many-body physics.\cite{Zeh1970,Schlosshauer2005} Its mitigation is a necessary capability to achieve reliable quantum computers, because these devices rely on the principles of superposition and entanglement, and are therefore particularly sensitive to interactions with the environment. Examples abound such as $1/f$-noise in superconducting quantum interference devices (SQUIDs) that constitutes a dominant adverse effect on superconducting qubits,\cite{Yoshihara2006,Kakuyanagi2007,Bialczak2007,Bylander2011,Wang2015,Kumar2016} CMB-photon noise in superconducting cryogenic detectors,\cite{Day2003} and thermomechanical motion in microwave cavity interferometers.\cite{Regal2008} In open quantum many-body systems, the effects of decoherence have been studied on light-cone dynamics and the spread of correlations,\cite{Poulin2010,Marino2012,Descamps2013,Bernier2013,Halimeh2018,Bernier2018} and in a recent experiment\cite{Maier2019} a ballistic-to-diffusive crossover in quantum transport has been observed due to environmental noise in a $10$-qubit network of interacting spins. Moreover, the effect of decoherence on weakly driven quantum many-body systems have also been studied in the context of long-time steady states in the presence of approximately conserved quantities, where it is shown that a GGE state can also arise.\cite{Lenarcic2018,Lange2018} Therefore, like their coherent counterparts, incoherent symmetry-breaking errors due to decoherence can fundamentally change the properties and behavior of a physical system.
From a technological point of view, in modern ultracold-atom experiments that aim to quantum-simulate a given target model,\cite{Gross_review,Lewenstein_book,Hauke2012} it is of critical importance to reliably implement certain conservation laws due to both local and global symmetries. As global symmetries define the fixed points of renormalization-group flow, they decisively influence quantum phase diagrams. For example, particle-number conservation in the form of a global $\mathrm{U}(1)$ symmetry is crucial in the transition between the superfluid and Mott-insulator phases in the Bose--Hubbard model,\cite{Zhou2006,Maruyama2007,Roy2019,Donatella2020} a paradigm phase transition of cold-atom experiments.\cite{Greiner2002} Similarly, local gauge symmetries have fundamental consequences such as massless photons and a long-ranged Coulomb law,\cite{Melnikov2000,Tu2004} but their realization in quantum simulators requires careful engineering---in contrast to fundamental theories of nature such as quantum electrodynamics or quantum chromodynamics, in quantum devices they are not given by fundamental laws. This has recently generated a surge of research investigating unitary errors in lattice gauge theories that compromise gauge invariance, including ways of protecting against them.\cite{Zohar2011,Zohar2012,Banerjee2012,Zohar2013,Hauke2013,Stannigel2014,Kuehn2014,Kuno2015,Kuno2017,Negretti2017,Barros2019,Schweizer2019,Halimeh2020a,Yang2020,Halimeh2020d,Halimeh2020e,Mathis2020,Lamm2020,Tran2020}
As these examples highlight, the existence of coherent and incoherent symmetry-breaking errors in realistic setups necessitates a rigorous understanding of their influence on the dynamics of quantum many-body systems. Particularly relevant is to see if such errors are \textit{controlled} insomuch that one may extract the ideal theory dynamics despite their presence.
In this paper, we investigate the effect of experimentally motivated incoherent global and local symmety-breaking errors on the dynamics of quantum many-body systems. We demonstrate that the symmetry violation---the expectation value of the symmetry generator or its square, which has been often used in the past to estimate the effects of errors---is a rigorous divergence measure in Hilbert space that quantifies the deviation of the state from the target symmetry sector.
We exploit this insight to show, using exact numerics and rigorous proofs in time-dependent perturbation theory, the existence of a diffusive-to-ballistic crossover in the dynamics of the symmetry violation as a result of competition of these errors with their coherent counterparts. These results extend and generalize upon the findings presented in Ref.~\onlinecite{Halimeh2020f}, which considered quenches from a separable gauge-invariant initial state in a $\mathrm{Z}_2$ gauge theory. By presenting a thorough analysis of various sources of errors and various different model scenarios, our results provide a guideline for quantum-simulation experiments on noisy intermediate-scale quantum (NISQ) devices that aim to realize target models with a given symmetry.
The rest of the paper is organized as follows. In Sec.~\ref{sec:preamble}, we define the symmetry violation, clarify our nomenclature in a short glossary, and provide a summary of our main results. In Sec.~\ref{sec:meaning}, we rigorously explain the physical meaning of the symmetry violation by equating it with a divergence measure in Hilbert space and by relating it to the decrease of the overlap between states that differ only by a symmetry transformation. Our findings and conclusions are then illustrated on three main models: the extended Bose--Hubbard model in Sec.~\ref{sec:eBHM}, the $\mathrm{Z}_2$ lattice gauge theory in Sec.~\ref{sec:Z2LGT}, and the $\mathrm{U}(1)$ quantum link model in Sec.~\ref{sec:U1QLM}. We conclude and discuss possible future directions in Sec.~\ref{sec:conclusion}. Appendix~\ref{sec:TDPT} contains our detailed derivations in time-dependent perturbation theory that explain the various scalings seen in our exact diagonalization results. Appendix~\ref{sec:NumSpec} provides details on our numerical implementation.
\section{Preamble}\label{sec:preamble}
Before entering in the details of our work, we define in this section our main figure of merit, the symmetry violation, we provide a short glossary for nomenclature clarity, and we present a concise summary of our main findings.
\subsection{Definition of symmetry violation}\label{sec:definition}
The motivation behind our work is the assessment of quantum many-body systems in modern experimental settings where realistically coherent and incoherent errors will always be present at least to a perturbative degree. These errors may break target local and global symmetries, which may or may not be desired in the experiment, but where a thorough understanding of their effects on the dynamics is nevertheless advantageous. These effects can be qualitatively and quantitatively studied by calculating dynamics of the symmetry violation and other relevant observables.
The symmetry violation $\varepsilon$ is defined as the expectation value of a (local or global) symmetry generator\cite{footnote} with respect to a target symmetry sector (see glossary in Sec.~\ref{sec:glossary}), or the square of this expectation value, and is often used to estimate the effect of slightly breaking a symmetry.\cite{Zohar2011,Zohar2012,Banerjee2012,Zohar2013,Hauke2013,Stannigel2014,Kuehn2014,Kuno2015,Kuno2017,Negretti2017,Barros2019,Schweizer2019,Halimeh2020a,Yang2020,Halimeh2020d,Halimeh2020e,Mathis2020,Lamm2020,Tran2020} To formalize its definition, let us consider a many-body model described by the Hamiltonian $H_0$ with a local symmetry generated by the operators $G_j$, with $j$ a lattice-site index (for the local gauge symmetries considered below, matter particles reside on the lattice sites and gauge fields on the links in between). The eigenvalues of $G_j$ are the local charges $g_j$, and a given combination of them on the lattice classifies a gauge-invariant sector $\mathbf{g}=\{g_1,g_2,\ldots,g_N\}$. We select a target gauge-invariant sector $\mathbf{g}_\text{tar}=\{g_1^\text{tar},g_2^\text{tar},\ldots,g_N^\text{tar}\}$, and call a given state gauge-invariant or symmetric iff $G_j\rho_0=g_j\rho_0,\,\forall j$. We prepare the system in an initial state $\rho_0$, which may be gauge-invariant or not. Gauge invariance restricts dynamics within a gauge sector, and so in case $\rho_0$ lies in a given gauge-invariant sector $\mathbf{g}$, then the system will remain in this sector for all times if the dynamics is solely due to $H_0$, because due to the gauge invariance of the latter, $[H_0,G_j]=0,\,\forall j$. In the presence of unitary or incoherent gauge-breaking errors, the gauge violation generically will spread across various gauge sectors $\mathbf{g}$, and can in general be quantified as
\begin{align}\label{eq:MeasureGeneral}
\varepsilon(t)=\frac{1}{N}\sum_j\Tr\Big\{\rho(t)\big[G_j-g_j^\text{tar}\big]^2\Big\},
\end{align}
where $\rho(t)$ is the density matrix of the time-evolved system at time $t$. The motivation behind this measure lies in the assumption that the system is desired to reside within the target gauge-invariant sector $\mathbf{g}_\text{tar}$. Thus, any coherent or incoherent errors during the preparation of $\rho_0$ or the subsequent dynamics that take the system away from $\mathbf{g}_\text{tar}$ will make $\varepsilon$ as defined by Eq.~\eqref{eq:MeasureGeneral} nonzero.
Much the same way, this definition can be extended to the case of global-symmetry models, with the only caveat being that there the deviation is across global-symmetry sectors, each of which consists of all states with a given fixed value of the global charge (see glossary in Sec.~\ref{sec:glossary}). We select a target global-symmetry sector defined by the total global charge $g_\text{tar}$. The system is prepared in an initial state $\rho_0$ which may be in the target sector $g_\text{tar}$ or not. The global symmetry is generated by the operator $G$, and we denote the Hamiltonian of the global-symmetry model as $H_0$, i.e., $[H_0,G]=0$. The initial state $\rho_0$ is said to be symmetric iff $G\rho_0=g\rho_0$. Consequently, the symmetry violation becomes
\begin{align}
\label{eq:violationglobalsymmetry}
\varepsilon(t)=\frac{1}{N^2}\Tr\Big\{\rho(t)\big[G-g_\text{tar}\big]^2\Big\}.
\end{align}
The normalization with $N^2$ is chosen since $G$ typically is an extensive quantity such as the total particle number, see, e.g., Eq.~\eqref{eq:viol_eBHM} below.
\subsection{Glossary}\label{sec:glossary}
\textbf{\textit{Symmetry sectors and symmetric states.}} In case of a local symmetry, a state $\rho$ is said to be gauge-invariant or symmetric iff $G_j\rho=\rho G_j=g_j\rho,\,\forall j$ where $G_j$ are the local-symmetry generators of the gauge group at matter sites $j$ with eigenvalues $g_j$ that depend on the gauge symmetry of the model. A given set of values $\mathbf{g}=\{g_1,g_2,\ldots,g_N\}$ constitute a gauge-invariant sector. In the case of a global symmetry, a state $\rho_0$ is symmetric iff $G\rho=\rho G=g\rho$, where $G$ is the generator of the global symmetry, and its eigenvalue $g$ denotes the global charge of the corresponding symmetry sector. As a concrete example, in the Bose--Hubbard model $G$ can be chosen as the total particle number. In this case, a given symmetry sector $g$ would consist of all Fock states $\ket{n_1,n_2,\ldots,n_N}$ where the individual on-site particle numbers $n_j$ sum to $\sum_{j=1}^Nn_j=g$.
\medskip
\textbf{\textit{Target symmetry sector.}} In an experiment, it is often desired to prepare the system and restrict its dynamics within a given symmetry sector in a local- or global-symmetry model. This is called the target symmetry sector. The symmetry violation measures how far off a state $\rho$ is from this target symmetry sector.
\medskip
\textbf{\textit{Gauge-invariant supersector.}} Whereas both local and global symmetries have sectors, in our nomenclature only local symmetries have supersectors. A given supersector $\alpha$ is the set of all gauge-invariant sectors $\mathbf{g}=\{g_1,g_2,\ldots,g_N\}$ that have a total violation $\alpha$ with respect to the target gauge-invariant sector $\mathbf{g}_\text{tar}$. This definition can be formalized as $\sum_{j=1}^N(G_j-g_j^\text{tar})^p\rho=\alpha\rho$, with $p=1$ for the $\mathrm{Z}_2$ LGT and $p=2$ for the $\mathrm{U}(1)$ QLM. In the case of the $\mathrm{Z}_2$ LGT discussed in Sec.~\ref{sec:Z2LGT} and a target gauge-invariant sector $\mathbf{g}_\text{tar}=\mathbf{0}$, a gauge-invariant supersector can be defined by the set of gauge-invariant sectors $\mathbf{g}$ each carrying $M$ violations with respect to $\mathbf{g}_\text{tar}=\mathbf{0}$. For notational brevity, these supersectors are then denoted by $M$ and the projectors onto them by $\mathcal{P}_M$, with each then being a sum of all projectors onto the constituent sectors; cf.~Eq.~\eqref{eq:SupProj}.
\subsection{Summary of main findings}\label{sec:summary}
The main result of our work is the crossover in the short-time dynamics of the symmetry violation from a diffusive spread through symmetry sectors caused by incoherent errors to a ballistic spread caused by coherent errors. Formally, we consider a system ideally described by a Hamiltonian $H_0$, which has either a local or global symmetry generated by the local or global operators $G_j$ or $G$, respectively, with $j$ indicating a site in the associated lattice. In practice, these symmetries will be compromised in an experiment without unrealistic fine-tuning and isolation from the environment. We represent the coherent errors by the Hamiltonian $\lambda H_1$, with $\lambda$ their strength, while we model decoherence (dissipation and dephasing) using a Lindblad master equation with the coupling strength to the environment denoted by $\gamma$ (see further below for the specific terms used).
We prepare the system in an initial state $\rho_0$ and quench at $t=0$ with $H_0+\lambda H_1$ in the presence of decoherence. We demonstrate that the symmetry violation yields a divergence measure for the quantum state across symmetry sectors, enabling us to identify an increase as $t^a$ with diffusive ($a=1$), ballistic ($a=2$), and hyperballistic scaling ($a>2$). As we elaborate in detail below, the leading order of incoherent contributions to the symmetry violation is $\propto\gamma t$. The leading order of the coherent contribution depends on the structure of $H_1$ and $\rho_0$. If $\rho_0$ is symmetric or is a generic eigenstate of $H_0$, then the coherent contribution cannot be of an order lower than $\propto\lambda^2 t^2$. If $\rho_0$ is neither symmetric nor a generic eigenstate of $H_0$, then theoretically the leading order of the coherent contribution can be $\propto\lambda t$, but we find this to happen only in two rather pathological cases and one engineered (artificial) case. The first pathological scenario is when $\rho_0$ is the ground state of $H_0+\lambda_\text{i} H'_1$ with $\lambda_\text{i}\neq0$ and $H'_1$ a highly nonlocal Hamiltonian, and the second is when $H'_1$ is local but $H_1$ is highly nonlocal. If $H_1=H'_1$, the contribution $\propto\lambda t$ vanishes identically to zero, and then the order of the coherent contribution cannot be lower than $\propto\lambda t^2$. We find this also to be the case if $H'_1$ and $H_1$ are both local, or mildly nonlocal, but not equal. One may also get the scaling $\propto\lambda t$ by artificially engineering the initial state in a common eigenbasis of $H_0$ and the symmetry generator to be an arbitrary superposition of eigenstates degenerate with respect to $H_0$ but not with respect to the symmetry generator. Conversely, one can also construct specific combinations of initial state and coherent gauge breaking that result in hyperballistic expansion across symmetry sectors.
Therefore, it is shown that in quantum many-body systems with small coherent and incoherent symmetry breaking, a crossover from diffusive to ballistic spread takes place in the short-time dynamics of the symmetry violation and certain other observables. The crossover time is $t\propto\gamma/\lambda^2$ (or $t\propto\gamma^{\frac{1}{3}}/\lambda^{\frac{2}{3}}$ in case of hyperballistic spread) for initial states that reside in a symmetry sector or are generic (possibly unsymmetric) eigenstates of $H_0$, and $t\propto\gamma/\lambda$ for unsymmetric initial states that are not eigenstates of $H_0$. Our work presents an extension to and generalization of Ref.~\onlinecite{Halimeh2020f}, which has studied this crossover in a $\mathrm{Z}_2$ lattice gauge theory starting in a gauge-invariant initial product state. Qualitatively, we find that our main findings are similar for systems with breakings of local or of global symmetries.
\section{Meaning of the symmetry violation}\label{sec:meaning}
In this section, we imbue the symmetry violation with mathematical and physical meaning, as a divergence measure across symmetry sectors in case of local or global symmetries, as well as the overlap between states that differ only by a symmetry transformation. This section generalizes the discussion of Ref.~\onlinecite{Halimeh2020f}, which considered local gauge symmetries, and provides further details.
\subsection{Symmetry violation as a divergence measure in Hilbert space}
\label{sec:divergencemeasure}
We follow a similar reasoning as in Ref.~\onlinecite{Hauke2015}, where a measure for divergence in Hilbert space has been defined in the context of many-body localization.\cite{Abanin2019} We first consider gauge-symmetry models, and start by defining the mean-square displacement across the gauge sectors,
\begin{align}
D^{(2)}_{\mathbf{g}_\text{tar}}(t)=\sum_{\mathbf{g}} d^2(\mathbf{g},\mathbf{g}_\text{tar})\Tr\big\{ \rho(t)P_\mathbf{g}\big\},
\end{align}
where $P_\mathbf{g}$ is the projector onto gauge-invariant sector $\mathbf{g}$, and $\mathbf{g}_\text{tar}$ is the target gauge-invariant sector (see Sec.~\ref{sec:definition}). Using the definition $d^2(\mathbf{g},\mathbf{g}_\text{tar})=[\mathbf{g}-\mathbf{g}_\text{tar}]^2$, we can rewrite this expression as
\begin{align}
\label{eq:D2vsepsilon}
D^{(2)}_{\mathbf{g}_\text{tar}}(t)&=\mathrm{Tr}\Big\{\rho(t) \sum_{\mathbf{g}} P_\mathbf{g} \big[\mathbf{g}-\mathbf{g}_\text{tar}\big]^2 \Big\}\nonumber \\
&=\sum_j\mathrm{Tr}\Big\{\rho(t)\big [G_j-g_j^\text{tar}\big]^2\Big\}\\
&=\sum_j\langle \big[G_j-g_j^\text{tar}\big]^2\rangle. \nonumber
\end{align}
By comparison with Eq.~\eqref{eq:MeasureGeneral}, we obtain
\begin{align}
\varepsilon(t)=\frac{1}{N}D^{(2)}_{\mathbf{g}_\text{tar}}(t).
\end{align}
Thus, commonly used measures of gauge violation, in $\mathrm{U}(1)$ as well as $\mathrm{Z}_2$ gauge theories,\cite{Stannigel2014,Halimeh2020a,Halimeh2020b,Halimeh2020c} rigorously define a mean-square displacement across gauge sectors as encapsulated in Eq.~\eqref{eq:D2vsepsilon}.
One can immediately apply exactly the same reasoning to global symmetries, where only $\mathbf{g}$ needs to be replaced by the scalar $g$ representing the global-charge sector (and analogously for $\mathbf{g}_\text{tar}$ and $g_\text{tar}$).
In this sense, $\varepsilon(t)\propto \gamma t$ is associated with a diffusive spreading of the wavefunction across gauge (global-charge) sectors in local- (global-) symmetry models, with diffusion constant $\propto \gamma$, while $\varepsilon(t)\propto \lambda^2 t^2$ is indicative of a ballistic spreading, a characteristic property of coherent quantum dynamics.
\subsection{Symmetry violation as overlap between gauge-transformed states}
We can further understand the gauge violation as quantifying how fast the overlap diminishes between states that differ only by a gauge transformation. Let us again begin by considering the case of local symmetry. By definition, a gauge-invariant state should be oblivious to a local gauge transformation generated by a unitary $U(\boldsymbol{\phi}) = \prod_{j=1}^N e^{i\phi_j G_j}$, with $\boldsymbol{\phi}=\{\phi_1,\dots,\phi_N\}$ and $\phi_{j}$ arbitrary and independent angles.
To quantify by how much gauge-breaking errors compromise gauge invariance, one can use the overlap between the state of interest and the transformed state, which for a pure state reads $\mathcal{C}=|\bra{\psi(t)}U(\boldsymbol{\phi})\ket{\psi(t)}|^2$.
The rate with which the overlap reduces under a gauge transformation is
\begin{align}
\eta_j=-\frac{1}{2}\frac{\partial^2 \mathcal{C}}{\partial \phi_j^2} = \langle{G_j^2}\rangle-\langle{G_j}\rangle^2,
\end{align}
which is nothing else but the variance of the local Gauss's-law generator $G_j$.
For a homogeneous system, and assuming without restriction of generality $\mathbf{g}_\text{tar}=\mathbf{0}$, definition \eqref{eq:MeasureGeneral} yields $\eta_j=\varepsilon-\langle{G_j}\rangle^2$. For the $\mathrm{Z}_2$ gauge theory considered below, where $G_j^2=2G_j$, we further have $\eta_j=\varepsilon(2-\varepsilon)$ [upon normalizing $\varepsilon$ by adding an inconsequential factor of $1/2$ in Eq.~\eqref{eq:MeasureGeneral}].
Thus, the sensitivity of a state towards a gauge transformation is related to the gauge violation $\varepsilon$. For small gauge violations in a homogeneous $\mathrm{Z}_2$ gauge theory, both even coincide (apart from an inconsequential factor of 2), as they do in a homogeneous $\mathrm{U}(1)$ gauge theory with $\langle{G_j}\rangle=0$. This insight lends further physical motivation for $\varepsilon$ as a good measure of gauge violation. As a side note, for mixed states, $\eta_j$ is given by the quantum Fisher information of the Gauss's-law generator, which for thermal states, e.g., can be measured through linear-response susceptibilities \cite{Hauke2016} or engineered quench dynamics.\cite{Almeida2020}
Again, these considerations can be immediately adapted to global symmetries. In this case, the unitary transformation is defined as $U({\phi}) = e^{i\phi G}$ and the rate of change is $\eta=-\frac{1}{2}\frac{\partial^2 \mathcal{C}}{\partial \phi^2} = \langle{G^2}\rangle-\langle{G}\rangle^2$. For translationally-invariant systems with $\mathbf{g}_\text{tar}=0$ and $\langle{G}\rangle=0$, using definition \eqref{eq:violationglobalsymmetry} this relation becomes $\eta=N^2\varepsilon$.
\section{Extended Bose--Hubbard model}\label{sec:eBHM}
In this section, we numerically evaluate the interplay of coherent and incoherent breakings of a global symmetry. To this end, we consider the paradigmatic example of the extended Bose--Hubbard model\cite{Kuehner1998,Kuehner2000,Dutta2015} (eBHM).
\subsection{Model and quench protocol}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{eBHM_InitialStates}
\caption{(Color online). Initial product states prepared at half-filling ($g_\text{tar}=N/2$), as used in the dynamics of the extended Bose--Hubbard model in the presence of coherent and incoherent errors. Green (red) arcs exemplify coherent processes preserving (breaking) global $\mathrm{U}(1)$ symmetry. In this work, we use chains of $N=6$ sites with periodic boundary conditions for this model. (a) A staggered product state where odd sites are empty and each even site contains one hard-cose boson. (b) A domain-wall product state where the left half of the lattice has a hard-core boson at each site and the right half of the chain is empty.
}
\label{fig:eBHM_InitialStates}
\end{figure}
The eBHM considered here is defined on a one-dimensional spatial lattice with $N$ sites and assuming periodic boundary conditions, and is described by the Hamiltonian
\begin{align}\nonumber
\label{eq:HeBHM}
H_0=&-\sum_{j=1}^N\big(J_1a_j^\dagger a_{j+1}+J_2a_j^\dagger a_{j+2}+\text{H.c.}\big)\\
&+\frac{U}{2}\sum_{j=1}^Nn_j(n_j-1)+W\sum_{j=1}^Nn_jn_{j+1},
\end{align}
where $a_j$ is the bosonic annihilation operator at site $j$ satisfying the canonical commutation relations $[a_j,a_l]=0$ and $[a_j,a_l^\dagger]=\delta_{j,l}$. The eBHM is nonintegrable at finite nonzero values of $J_1$, $J_2$, $U$, and $W$. Generically, it has two integrable points: the atomic limit of $J_1=J_2=0$ and the free-boson limit of $U=W=0$.\cite{Kollath2010} In our case, we additionally impose a hard-core constraint on the bosons, through which the term $\propto U$ does not play any role in the dynamics, so we can remove it from the Hamiltonian in Eq.~\eqref{eq:HeBHM} (formally, the hard-core constraint amounts to setting $U=\infty$). This leads to another integrable point at $J_2=0$, where the eBHM becomes equivalent to the XXZ model.\cite{Kollath2010} To avoid any effects due to integrability breaking, we therefore set $J_1=1$, $J_2=0.83$, and $W=0.11$ in our numerics, although we have checked that other generic values of these parameters yield the same conclusions. In all our results for the eBHM we use a periodic chain with $N=6$ sites.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{Neel_OneBodyError_gamma0}\\
\includegraphics[width=.48\textwidth]{Neel_TwoBodyError_gamma0}\\
\includegraphics[width=.48\textwidth]{DW_TwoBodyError_gamma0}
\caption{(Color online). Quench dynamics of the symmetry violation in the closed ($\gamma=0$) extended Bose--Hubbard model in the presence of unitary symmetry-breaking errors $\lambda H_1$ and starting in a symmetric initial state. The staggered initial product state shown in Fig.~\ref{fig:eBHM_InitialStates}(a) is used for (a) and (b), while the domain-wall product state in Fig.~\ref{fig:eBHM_InitialStates}(b) is used as the initial state in (c). System size is $N=6$ sites, and periodic boundary conditions are assumed. $H_1$ is composed of single-body terms in (a), while it is a two-body error in (b) and (c). Generically, the initial increase of symmetry violation is $\propto (\lambda t)^2$ (a,c), while specific combinations of initial state and error terms can increase the order of the short-time behavior, to $\lambda^2 t^4$ in the case of panel (b). Within our simulation times, the maximal value of the violation in all cases is $\propto\lambda^2$, which is the same order in $\lambda$ at which the violation grows at short times.}
\label{fig:eBHM_closed}
\end{figure}
The eBHM conserves the total particle number because $\sum_j[H_0,a_j^\dagger a_j]=0$. This translates to the eBHM hosting a global $\mathrm{U}(1)$ symmetry. In a realistic experiment, the implementation of this model will suffer from coherent and incoherent errors that in the best case slightly break this symmetry. The coherent errors are described by a Hamiltonian $\lambda H_1$, where $\lambda$ denotes the strength of these errors, and dynamics in the presence of decoherence is modeled by the Lindblad master equation\cite{Breuer_book,Manzano2020}
\begin{align}\nonumber
\dot{\rho}=&-i[H_0+\lambda H_1,\rho]\\\label{eq:EOM_eBHM}
&+\gamma\sum_j\Big(L_j\rho L_j^\dagger-\frac{1}{2}\big\{L_j^\dagger L_j,\rho\big\}\Big),
\end{align}
where $L_j=a_j$ are the jump operators describing the dissipation of our system with the environment, and $\gamma$ is the environment-coupling strength. Below, we test various examples of initial state and terms for $H_1$.
In the following, we prepare our system in an initial state $\rho_0$ and solve Eq.~\eqref{eq:EOM_eBHM} for the dynamics of the symmetry violation
\begin{align}\label{eq:viol_eBHM}
\varepsilon(t)&=\Tr\big\{\mathcal{G}\rho(t)\big\},\,\,\,\mathcal{G}=\frac{1}{N^2}\bigg[\sum_{j=1}^Na_j^\dagger a_j -g_\text{tar}\bigg]^2,
\end{align}
where $g_\text{tar}=N/2$ indicates the target half-filling sector.
\subsection{Symmetric initial state}
Let us first prepare our system in the staggered initial product state shown in Fig.~\ref{fig:eBHM_InitialStates}(a). This state is symmetric because it lies in the half-filling sector: $\mathcal{G}\rho_0=0$. The subsequent time evolution of the symmetry violation in Eq.~\eqref{eq:viol_eBHM} \textit{without decoherence} under the unitary errors
\begin{align}\label{eq:H1OneBody}
\lambda H_1=\lambda\sum_{j=1}^N\big(a_j+a_j^\dagger\big),
\end{align}
is shown in Fig.~\ref{fig:eBHM_closed}(a). The error term in Eq.~\eqref{eq:H1OneBody} would describe, e.g., the coupling of the system to another internal state (call its associated annihilation operator $b$), which is occupied by a condensate that we assume to be evenly spread across the entire lattice.\cite{Maruyama2007} Then, a Rabi flop between the two states gives $H_1=\sum_j\big(b^\dagger a_j+a_j^\dagger b\big)\approx \sum_j\big(\langle b\rangle^* a_j+\langle b\rangle a_j^\dagger\big)$.
Without restriction of generality, we set the phase of the condensate to $0$. Thus, we get $H_1 = \langle b\rangle \sum_j\big(a_j+a_j^\dagger\big)$, thereby achieving Eq.~\eqref{eq:H1OneBody} by absorbing $\langle b\rangle$ in the definition of the unitary-error strength $\lambda$. The short-time scaling in Fig.~\ref{fig:eBHM_closed}(a) is $\sim(\lambda t)^2$. As we show in Sec.~\ref{sec:TDPT_coherent} through time-dependent perturbation theory (TDPT), this is the lowest order of coherent contributions to the symmetry violation in the case of a symmetric initial state.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{eBHM_OneBodyError_lam1e-2}\\
\includegraphics[width=.48\textwidth]{eBHM_OneBodyError_gamma1e-5}
\caption{(Color online). Quench dynamics of the symmetry violation in the $N=6$-site open eBHM starting in a staggered initial product state and in the presence of the single-body unitary errors of Eq.~\eqref{eq:H1OneBody} at strength $\lambda$. The coupling to the environment is at strength $\gamma$ with the jump operator $L_j=a_j$ at each site. (a) Symmetry violation over time with fixed $\lambda$ at various values of $\gamma$. (b) Symmetry violation over time with fixed $\gamma$ at various values of $\lambda$. Note how decoherence, regardless the value of $\lambda$, takes the symmetry violation to its maximal value $g_\text{tar}^2/N^2=1/4$ at long-enough times. A diffusive-to-ballistic crossover occurs at $t\propto\gamma/\lambda^2$ where the symmetry violation goes from a diffusive spread $\sim\gamma t$ for $t\lesssim\gamma/\lambda^2$ to ballistic spread $\sim\lambda^2t^2$ for $t\gtrsim\gamma/\lambda^2$.}
\label{fig:eBHM_OneBodyError_open}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{eBHM_TwoBodyError_lam1e-2}\\
\includegraphics[width=.48\textwidth]{eBHM_TwoBodyError_gamma1e-6}
\caption{(Color online). Same as Fig.~\ref{fig:eBHM_OneBodyError_open} but for the two-body error of Eq.~\eqref{eq:H1TwoBody}. The initial state can play a nontrivial role in the short-time dynamics of the symmetry violation. In this case, the crossover is from diffusive to \textit{hyperballistic} spread and occurs at $t\propto(\gamma/\lambda^2)^{\frac{1}{3}}$. This behavior is qualitatively different from the case of the single-body unitary errors in Eq.~\eqref{eq:H1OneBody}, and is due to the fact that $H_1$ in Eq.~\eqref{eq:H1TwoBody} annihilates $\rho_0$ when it acts on it: $H_1\rho_0=0$, which leads to the contribution $\propto\lambda^2t^2$ vanishing, and thus the next leading order $\propto\lambda^2t^4$ sets in for $t\gtrsim(\gamma/\lambda^2)^{\frac{1}{3}}$. Decoherence allows the symmetry violation to reach its maximal value of $1/4$ that unitary errors alone cannot achieve.}
\label{fig:eBHM_TwoBodyError_open}
\end{figure}
However, one can engineer the initial state or coherent error to make higher orders the leading contributions. As an example, we consider the same initial state shown in Fig.~\ref{fig:eBHM_InitialStates}(a), but consider the two-body unitary errors
\begin{align}\label{eq:H1TwoBody}
\lambda H_1=\lambda\sum_{j=1}^N\big(a_ja_{j+1}+a_j^\dagger a_{j+1}^\dagger\big).
\end{align}
Similarly to the motivation behind the error term of Eq.~\eqref{eq:H1OneBody}, one can imagine immersing an optical lattice into a condensate of biatomic molecules, where bonding $\propto a_j a_{j+1}$ gives a molecular annihilation operator $m$.\cite{Zhou2006}
Then, a term that drives the coupling to the molecular condensate is $H_1=\sum_j\big(m^\dagger a_ja_{j+1}+a_j^\dagger a_{j+1}^\dagger m\big)
\approx \langle m\rangle \sum_j\big(a_ja_{j+1}+a_j^\dagger a_{j+1}^\dagger\big)$, which by absorbing $\langle m\rangle$ into the definition of $\lambda$, we achieve the error term in Eq.~\eqref{eq:H1TwoBody}.
When $\rho_0$ is the staggered initial state of Fig.~\ref{fig:eBHM_InitialStates}(a), the coherent contribution $\lambda^2t^2\Tr\{\mathcal{G}H_1\rho_0H_1\}$ to the symmetry violation is finite in case of the unitary errors in Eq.~\eqref{eq:H1OneBody}, but vanishes in case of the unitary errors of Eq.~\eqref{eq:H1TwoBody}. The reason is that the staggered initial occupation yields $H_1\rho_0=0$. As shown in Sec.~\ref{sec:TDPT_coherent}, the next leading contribution $\propto\lambda^2t^4\Tr\{\mathcal{G}H_1H_0\rho_0H_0H_1\}$ now dominates, where it does not vanish because $H_0$ induces tunneling processes that can bring bosons on adjacent sites, allowing $H_1$ to act on $\rho_0$ without destroying it. This is exactly what we see in Fig.~\ref{fig:eBHM_closed}(b). Such an increase of the mean-square displacement with a power of $t$ larger than $2$ is a hallmark of hyperballistic expansion, and sets the crossover timescale to $t\propto(\gamma/\lambda^2)^{\frac{1}{3}}$ in the presence of decoherence (see below).
Nevertheless, the short-time scaling $\sim\lambda^2t^4$ is not a generic feature of the unitary errors of Eq.~\eqref{eq:H1TwoBody}, but is rather a combination of such errors and the fact that we start in the staggered initial state. Indeed, if we consider these same \textit{pairing errors} but instead start in, say, the domain-wall initial state shown in Fig.~\ref{fig:eBHM_InitialStates}(b), then the leading coherent contribution to the symmetry violation is again $\propto\lambda^2t^2$, as shown in Fig.~\ref{fig:eBHM_closed}(c). The reason is that a domain-wall initial state already has bosons on adjacent sites, and thus $H_1$ can act on it nontrivially.
In what follows, we take $\rho_0$ as the staggered initial state and include decoherence through jump operators $L_j^\dagger$. We consider first in Fig.~\ref{fig:eBHM_OneBodyError_open} the unitary errors of Eq.~\eqref{eq:H1OneBody}. As can be shown in TDPT (see Appendix~\ref{sec:TDPT_leadingIncoherent}), the leading incoherent contribution to the symmetry violation in case of a symmetric initial state is $\propto\gamma t\sum_j\Tr\{\mathcal{G}L_j \rho_0L_j^\dagger\}$. As such, at times $t\lesssim\gamma/\lambda^2$, the symmetry violation shown in Fig.~\ref{fig:eBHM_OneBodyError_open} scales diffusively $\sim\gamma t$, before being overtaken by the ballistic spread $\sim\lambda^2t^2$ due to the leading coherent contribution. Note that the maximal value $\approx1/4$ reached by the symmetry violation in the steady-state due to decoherence is larger than that due to purely coherent symmetry breaking.
We now consider the same staggered initial state but use the unitary error terms of Eq.~\eqref{eq:H1TwoBody}. The only qualitative difference here is that the diffusive scaling $\sim\gamma t$ due to incoherent errors is overtaken by hyperballistic spread $\sim\lambda^2 t^4$, due to the leading coherent contribution, at a crossover time $t\propto(\gamma/\lambda^2)^{\frac{1}{3}}$. Again here in the presence of decoherence the symmetry violation attains its maximal value of $1/4$, even though in the purely unitary case it does not.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{MQC_eBHM_OneBody}\\
\includegraphics[width=.48\textwidth]{MQC_eBHM_TwoBody}
\caption{(Color online). Multiple quantum coherences $I_{\Delta g}$ (solid blue curves) at fixed coherent- and incoherent-error strengths $\lambda$ and $\gamma$ (see gray boxes for exact values), respectively, in the quench dynamics of the extended Bose-Hubbard model starting in the staggered initial state of Fig.~\ref{fig:eBHM_InitialStates}(a). For reference, we also show the symmetry violation (solid red curve) as well as the MQCs in the purely coherent case ($\gamma=0$; same color but dotted lines). In (a) the coherent errors are single-body terms given by Eq.~\eqref{eq:H1OneBody}. Decoherence compromises all $I_{\Delta g>0}$, causing them to settle into steady-state values lower than those in the purely coherent case. However, $I_0$ does not deviate much from its coherent steady-state value, suggesting that decoherence does not affect quantum coherences within each sector much. In (b) the coherent errors are the two-body terms given by Eq.~\eqref{eq:H1TwoBody}. Even though $I_0$ behaves the same as in the case of single-body errors, $I_{\Delta g>0}$ here are nonzero only in the case of even $\Delta g$ and settle into steady-state values \textit{larger} than those in the purely coherent case. }
\label{fig:eBHM_MQC}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{eBHM_GSlami1e-1Vi1_coherent}\\
\includegraphics[width=.48\textwidth]{eBHM_GSlami1e-1Vi1}
\caption{(Color online). Starting in the unsymmetric ground state $\rho_0$ of $H_0+\lambda_\text{i} H_1$ with $H_1$ given in Eq.~\eqref{eq:H1OneBody}. (a) Quench dynamics of the symmetry-violation change in the case of a closed eBHM chain with $N=6$ matter sites. Since $\rho_0$ is unsymmetric, the leading coherent contribution to the symmetry violation is $\propto\lambda t^2$. Note here that the steady-state value is $\propto\lambda$ rather than $\propto\lambda^2$ as in Fig.~\ref{fig:eBHM_closed} when the initial state is symmetric. (b) Decoherence brings about a diffusive-to-ballistic crossover at an earlier timescale $\gamma/\lambda<\gamma/\lambda^2$ than the case of a symmetric initial state.}
\label{fig:eBHM_unsymmetric}
\end{figure}
\subsection{Multiple quantum coherences}\label{sec:eBHM_MQC}
In order to further quantify the effects of decoherence especially in the late-time dynamics of our quenches, we analyse multiple quantum coherences (MQC). These are experimentally accessible quantities that have been used to study quantum coherences in nuclear magnetic resonance imaging,\cite{Pines1985,Pines1986,Pines1986} as well as decoherence effects on correlated spins\cite{SpinDecoherence2014} and many-body localization.\cite{NMRLocalizationPRL2010,NMRLocalizationScience2015} Moreover, they have also been connected to multipartite entanglement\cite{Gaerttner2018} and out-of-time-ordered correlators,\cite{Gaerttner2018,LewisSwan2020} and they have been measured in trapped-ion experiments.\cite{Garttner2017}
Let us call $\Delta g$ the difference in global charge between two global-symmetry sectors. The associated MQC is then defined as
\begin{subequations}\label{eq:eBHM_MQC}
\begin{align}
I_{\Delta g}&=\Tr\big\{\rho_{\Delta g}^\dagger\rho_{\Delta g}\big\},\\
\rho_{\Delta g}&=\sum_g P_{g+\Delta g}\rho P_g,
\end{align}
\end{subequations}
where $P_g$ is the projector onto the sector of global charge $g$.
In Fig.~\ref{fig:eBHM_MQC}, we show the MQC for various $\Delta g$ for both unitary errors in Eqs.~\eqref{eq:H1OneBody} and~\eqref{eq:H1TwoBody} at fixed $\lambda$ and $\gamma$. For the single-body coherent errors of Eq.~\eqref{eq:H1OneBody}, we see that odd values of $\Delta g$ give rise to a finite MQC as shown in Fig.~\ref{fig:eBHM_MQC}(a). This behavior is plausible, as in this case a first-order process in $H_1$ removes or adds a single boson on a given site $j$. We see that the MQC is dominated by processes within the same sector (corresponding to $\Delta g=0$). Whereas the symmetry violation (red curve) starts at $t=0$ to grow $\sim\gamma t$, the MQCs, being measures of quantum \textit{coherences} between global-symmetry sectors, do not show any scaling related to incoherent processes at early times. Rather, their growth is $\sim(\lambda^2 t^2)^{\Delta g}$. As can be expected in case of decoherence,\cite{Halimeh2020f} the $I_{\Delta g}$ with $\Delta g>0$ show a decrease at $t\approx1/\gamma$ as compared to the steady-state value of the purely coherent case. In contrast, $I_0$ exhibits an increase in its values after an initial decrease, finally settling at roughly the coherent steady-state value. This behavior suggests that even though decoherence compromises coherences between different sectors, those within the same sector are not affected much by the decoherence studied here.
By changing $H_1$ to Eq.~\eqref{eq:H1TwoBody}, the above picture changes, as shown in Fig.~\ref{fig:eBHM_MQC}(b). First, MQCs due to odd $\Delta g$ are identically zero, as there can be no coherent processes between sectors differing by an odd number of bosons---$H_1$ as per Eq.~\eqref{eq:H1TwoBody} removes or adds two neighboring bosons simultaneously. Furthermore, due to the staggered occupation of $\rho_0$, the MQCs scale $\sim(\lambda t^2)^{\Delta g}$.
Whereas $I_0$ behaves the same as in the case of the single-body error, $I_{\Delta g>0}$ behave fundamentally differently in case of the two-body error in Eq.~\eqref{eq:H1TwoBody}. Interestingly, at $t\approx1/\gamma$ they begin to decrease in value, but at later times they increase again and settle at a value larger than that of their purely coherent dynamics, represented by dotted lines in Fig.~\ref{fig:eBHM_MQC}. This suggests that decoherence populates sectors that cannot be populated by $H_1$ alone. Indeed, in the case of purely unitary dynamics with only coherent errors as in Eq.~\eqref{eq:H1TwoBody}, we have checked that $\langle P_g\rangle=0$ identically for odd $g$. In the case of decoherence, however, $\langle P_g\rangle$ acquire nonzero values, and this allows then, through $H_1$, coherence within and between sectors with odd global charges, but separated by an even $\Delta g$. This is quite counterintuitive in that decoherence here seems to help in building up quantum coherences by allowing access to previously inaccessible sectors.
\subsection{Unsymmetric initial state}
We have so far considered only initial states lying in the half-filling sector. Let us now consider an initial state $\rho_0$ that is \textit{unsymmetric}, i.e., $\big[\mathcal{G},\rho_0\big]\neq 0$. In particular, we consider $\rho_0$ to be the ground state of the Hamiltonian $H_0+\lambda_\text{i} H_1$, where $\lambda_\text{i}=0.1$ is the prequench strength of the error term and $H_1$ is given by Eq.~\eqref{eq:H1OneBody}. Such a scenario may arise when aiming at adiabatically preparing the ground state of $H_0$ in the presence of errors. As has been done until now, we quench $\rho_0$ with $H_0+\lambda H_1$ in the presence of decoherence under the jump operators $L_j=a_j$.
The ensuing dynamics of the change in symmetry violation $|\Delta\varepsilon|$ is shown in Fig.~\ref{fig:eBHM_unsymmetric}. In the case of no decoherence ($\gamma=0$), shown in Fig.~\ref{fig:eBHM_unsymmetric}(a), $|\Delta\varepsilon|\sim\lambda t^2$ at short times rather than $\lambda^2t^n$ (even $n\geq2$) as in the case of a symmetric initial state; cf.~Figs.~\ref{fig:eBHM_closed}--\ref{fig:eBHM_TwoBodyError_open}. As explained in Sec.~\ref{sec:TDPT_coherent} through TDPT, the coherent contribution to the symmetry violation $\propto\lambda t^2$, given by Eq.~\eqref{eq:coherent_last} always vanishes when the initial state is symmetric or an eigenstate of $H_0$, but not when $\rho_0$ is unsymmetric yet not an eigenstate of $H_0$ as in the case of Fig.~\ref{fig:eBHM_unsymmetric}. Moreover, the contribution $\propto\lambda t$ in Eq.~\eqref{eq:lamt} completely vanishes in this case, since the errors $H_1$ in preparing $\rho_0$ are the same as those in the subsequent dynamics (see Appendix~\ref{sec:TDPT_coherent}). Note how, consequently, the steady-state value at which $|\Delta\varepsilon|$ settles is $\propto\lambda$. Upon introducing decoherence ($\gamma>0$) in Fig.~\ref{fig:eBHM_unsymmetric}(b), a diffusive-to-ballistic crossover at $t\propto\gamma/\lambda$ occurs taking the spread of the symmetric violation from diffusive $\sim\gamma t$ to ballistic $\sim\lambda t^2$. Note that the diffusive-to-ballistic crossover here occurs at an earlier timescale than that in the case of a symmetric initial state. Finally, as in the case of a symmetric initial state, the symmetry violation reaches its maximal value of $g_\text{tar}^2/N^2=1/4$ also when $\rho_0$ is unsymmetric.
\section{$\mathrm{Z}_2$ lattice gauge theory}\label{sec:Z2LGT}
In this Section, we present results that supplement those on a $\mathrm{Z}_2$ gauge theory presented in Ref.~\onlinecite{Halimeh2020f} in various ways: by studying the effect of decoherence under different Lindblad operators including those for particle loss; by starting in various initial states including gauge-noninvariant ones; by investigating the effect of dissipation and dephasing set at different environment-coupling strengths; by analyzing the addition of energy-penalty terms on the dynamics; and by adding further results on MQCs under decoherence.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{Z2LGT_InitialStates}
\caption{(Color online). Symmetric initial product states used in the dynamics of the $\mathrm{Z}_2$ LGT in the presence of coherent errors. (a) A staggered product state where odd matter sites are empty and each even matter site contains one hard-cose boson, with the electric fields along links pointing from odd to even matter sites, thereby satisfying Gauss's law at each local constraint. (b) A ``domain-wall'' product state where the left half of the lattice has a hard-cose boson at each site and the right half of the chain is empty. The electric fields along the links are oriented such that Gauss's law is satisfied at each local constraint.}
\label{fig:Z2LGT_InitialStates}
\end{figure}
\subsection{Model and quench protocol}
A recent experiment\cite{Schweizer2019} has employed a Floquet setup to successfully implement a building block of the $\mathrm{Z}_2$ LGT described by the Hamiltonian\cite{Zohar2017,Barbiero2019,Borla2019}
\begin{align}\label{eq:H0}
H_0=&\,\sum_{j=1}^N\big[J_a\big(a^\dagger_j\tau^z_{j,j+1}a_{j+1}+\text{H.c.}\big)-J_f\tau^x_{j,j+1}\big],
\end{align}
where $N$ is the number of matter sites, $a_j$ is the annihilation operator of a hard-core boson at site $j$ obeying the canonical commutation relations $[a_j,a_l]=0$ and $[a_j,a^\dagger_l]=\delta_{j,l}(1-2a_j^\dagger a_j)$, and the Pauli matrix $\tau^{x(z)}_{j,j+1}$ represents the electric (gauge) field at the link between matter sites $j$ and $j+1$. The first term of Eq.~\eqref{eq:H0} represents assisted matter tunneling and gauge flipping at strength $J_a$, which, e.g., forms the essence of Gauss's law in quantum electrodynamics. The electric field's energy is given by $J_f$. In this work, we adopt periodic boundary conditions, which means our effective system size is $2N$, and we set $J_a=1$ and $J_f=0.54$ throughout our paper, even though we have checked that our conclusions are not restricted to these values.
Gauge invariance is embodied in local conservation laws, the generators of which are
\begin{align}
G_j=1-(-1)^j\tau^x_{j-1,j}(1-2a_j^\dagger a_j)\tau^x_{j,j+1},
\end{align}
where $[H_0,G_j]=0,\,\forall j$. As discussed above, the eigenvalues $g_j$ of $G_j$ are known as \textit{local charges}, and a set of their values $\mathbf{g}=\{g_1,g_2,\ldots,g_N\}$ defines a gauge-invariant sector (see Sec.~\ref{sec:definition}). A gauge-invariant supersector $M$ is defined as the set of gauge-invariant sector that satisfy $\sum_jg_j=2M$ (see Sec.~\ref{sec:glossary}).
\begin{figure*}[htp]
\centering
\includegraphics[width=.48\textwidth]{gamma0}\quad
\includegraphics[width=.48\textwidth]{gamma1e-6}\\
\includegraphics[width=.48\textwidth]{P2SM}\quad
\includegraphics[width=.48\textwidth]{P4SM}\\
\includegraphics[width=.48\textwidth]{StaggeredBosonNumber}\quad
\includegraphics[width=.48\textwidth]{ElectricField}
\caption{(Color online). (a) Full unitary dynamics of the quench scenario illustrated in Fig.~\ref{fig:Z2LGT_InitialStates}(a). Two prethermal plateaus at timescales $\lambda^{-1}$ and $J_a\lambda^{-2}$ (see insets) appear as explained numerically and analytically through a Magnus expansion in Refs.~\onlinecite{Halimeh2020b} and~\onlinecite{Halimeh2020c}. (b) Complementary results to those of the joint submission Ref.~\onlinecite{Halimeh2020f}, where here we fix $\gamma$ and scan $\lambda$. The conclusion remains the same, with the gauge violation exhibiting diffusive scaling $\varepsilon\sim\gamma t$ at short times for sufficiently small $\lambda$, and ballistic scaling $\sim(\lambda t)^2$ at sufficiently large $\lambda$, with the timescale of the crossover from the former to the latter being $t\propto\gamma/\lambda^2$. Upper inset shows the second (and final in the case of $N=4$ matter sites) prethermal timescale getting compromised at smaller values of $\lambda$ at which $\gamma=10^{-6}$ dominates. Lower inset shows that the first prethermal timescale is more resilient than the second, as its timescale persists for smaller values of $\lambda$. (c,d) Same as (b) but for the supersector projectors $\mathcal{P}_2$ and $\mathcal{P}_4$. As we can see, the crossover from the diffusive to ballistic regime is again at $t\propto\gamma/\lambda^2$, but whereas $\mathcal{P}_2$ shows the same scaling orders as $\varepsilon$, $\mathcal{P}_4\sim\gamma^2t^2$ in the diffusive regime and $\mathcal{P}_4\sim\lambda^4t^4$ in the ballistic regime. (e,f) Influence of gauge violation on local observables. As exemplified by (e) the staggered particle density and (f) the electric field, the dynamics is practically indistinguishable from the decoherence-free model before the timescale $\propto \gamma^{-1}$. Note how the electric field shows no diffusive behavior at the onset as the gauge violation and supersector projectors do. This is because the corresponding correction $\gamma t\mathcal{L}\rho_0$ to the unitary part of the density matrix makes a vanishing contribution to the electric field, as explained in Sec.~\ref{sec:TDPT_leadingIncoherent}.
}
\label{fig:gammaFixed}
\end{figure*}
In the implementation of the $\mathrm{Z}_2$ LGT without unrealistic fine-tuning, coherent error terms emerge with $[H_1,G_j]\neq0$. Here, inspired by the effective coherent errors of the building block of Ref.~\onlinecite{Schweizer2019}, we assume the errors to have the form of unassisted matter tunneling and gauge flipping, which can be formalized as
\begin{align}\nonumber
\lambda H_1=&\,\lambda \sum_{j=1}^N\Big[\big(c_1a_j^\dagger \tau^-_{j,j+1} a_{j+1}+c_2a_j^\dagger\tau^+_{j,j+1} a_{j+1}+\text{H.c.}\big)\\\label{eq:H1}
&+a_j^\dagger a_j\big(c_3\tau^z_{j,j+1}-c_4\tau^z_{j-1,j}\big)\Big].
\end{align}
The strength of these errors is given by $\lambda$, and the coefficients $c_{1\ldots4}$ depend on a dimensionless driving parameter $\chi$ that is tunable in the experiment of Ref.~\onlinecite{Schweizer2019}. The specific expressions for these coefficients can be found in Appendix~\ref{sec:NumSpec}. Just as in the joint submission,\cite{Halimeh2020f} we show here results for $\chi=1.84$, but we have also checked that our results hold for various values of $\chi$ within the range found in the Floquet setup of Ref.~\onlinecite{Schweizer2019}.
As for the case of a global symmetry discussed above, the system is prepared in an initial state $\rho_0$, which at $t=0$ is quenched by $H_0+\lambda H_1$ and decoherence is turned on. The subsequent dynamics is computed using the Lindblad master equation
\begin{align}\nonumber
\dot{\rho}=&-i[H_0+\lambda H_1,\rho]\\\nonumber
&+\gamma\sum_{j=1}^N\Big(L^\text{m}_j\rho L^{\text{m}\dagger}_j+L^\text{g}_{j,j+1}\rho L^{\text{g}\dagger}_{j,j+1}\\\label{eq:EOM}
&-\frac{1}{2}\big\{L^{\text{m}\dagger }_jL^\text{m}_j+L^{\text{g}\dagger }_{j,j+1}L^\text{g}_{j,j+1},\rho\big\}\Big),
\end{align}
where $L^\text{m}_j$ and $L^\text{g}_{j,j+1}$ are the jump operators coupling the matter and gauge fields, respectively, to the environment at strength $\gamma$. We are interested in dynamics of the gauge violation, supersector projector, electric field, and staggered boson number, given respectively by,
\begin{align}\label{eq:Measure}
\varepsilon(t)&=\Tr\big\{\mathcal{G}\rho(t)\big\},\,\,\,\mathcal{G}=\frac{1}{N}\sum_j\big[G_j-g_j^\text{tar}\big],\\\label{eq:SupProj}
\langle\mathcal{P}_M(t)\rangle&=\Tr\big\{\rho(t)\mathcal{P}_M\big\},\,\,\,\mathcal{P}_M=\sum_{\mathbf{g};\,\sum_jg_j=2M}P_\mathbf{g},\\
m_x(t)&=\frac{1}{N}\Big|\Tr\Big\{\rho(t)\sum_j\tau_{j,j+1}^x\Big\}\Big|,\\
n_\text{stag}(t)&=\frac{1}{N}\Big|\Tr\Big\{\rho(t)\sum_j(-1)^ja_j^\dagger a_j\Big\}\Big|.
\end{align}
Note that the form of the gauge violation in Eq.~\eqref{eq:Measure} is specific for the $\mathrm{Z}_2$ LGT. It is a special case of Eq.~\eqref{eq:MeasureGeneral} using $g_j^2=2g_j$ and dropping an irrelevant factor of $2$.
\subsection{Quench dynamics}
As shown in Ref.~\onlinecite{Halimeh2020f}, the coexistence of unitary and incoherent gauge-breaking processes leads to competing timescales due to $\lambda>0$ and $\gamma>0$ in the $\mathrm{Z}_2$ LGT. While incoherent gauge-breaking processes yield a single timescale $1/\gamma$, coherent errors can generate a sequence or \textit{staircase} of prethermal plateaus with timescales $J_a^{s-1}/\lambda^s$ with $s=0,1,2,\ldots,N/2$.\cite{Halimeh2020b,Halimeh2020c} These coherent timescales arise due to unitary dynamics in a gauge theory as a result of resonances between different gauge-invariant sectors coupled through $H_1$, as can be shown through a Magnus expansion.\cite{Halimeh2020c} In the case of the initial states shown in Fig.~\ref{fig:Z2LGT_InitialStates} with $N=4$ matter sites, this means two plateaus at timescales $1/\lambda$ and $J_a/\lambda^2$ (the one at timescale $\propto\lambda^0$ does not appear in this case\cite{Halimeh2020b}), at which maximal violation is attained; see Fig.~\ref{fig:gammaFixed}(a) for the \textit{staggered} initial product state shown in Fig.~\ref{fig:Z2LGT_InitialStates}(a).
The picture changes significantly when $\gamma>0$; see Fig.~\ref{fig:gammaFixed}(b). When $\lambda=0$ in this case, the gauge violation accumulates diffusively as $\varepsilon\sim\gamma t$ until it reaches a maximal value of unity at $t\approx1/\gamma$. This gauge violation due to purely incoherent gauge-breaking processes shows no signatures of prethermalization. The picture starts to change for $\lambda>\gamma$, as then the prethermal plateau at timescale $\propto1/\lambda$ can still appear unaffected by the decoherence, which becomes dominant for $t\gtrsim1/\gamma$; see lower inset of Fig.~\ref{fig:gammaFixed}(b). The effects of decoherence on the second plateau, which in the purely unitary case appears at timescale $\propto J_a/\lambda^2$, are more prominent as can be seen in the upper panel of Fig.~\ref{fig:gammaFixed}(b). This plateau survives only when $\lambda^2/J_a\gtrsim\gamma$ as then the final prethermal timescale $\propto J_a/\lambda^2$ appears earlier than the decoherence timescale of $/\gamma$.
It is interesting to examine again the short-time scaling of the gauge violation for finite $\lambda$ in Fig.~\ref{fig:gammaFixed}(b). The behavior concurs with the conclusions of Ref.~\onlinecite{Halimeh2020f}, where we observe the diffusive scaling $\varepsilon\sim\gamma t$ for $t\lesssim \gamma/\lambda^2$, while for later times $t\gtrsim\gamma/\lambda^2$ the violation is dominated by coherent errors and $\varepsilon\sim(\lambda t)^2$. Intriguingly, we thus find in general two regimes where incoherent errors dominate at finite $\lambda$: the first at evolution times $t\lesssim\gamma/\lambda^2$ and the second for $t\gtrsim1/\gamma$. At intermediate times, the coherent gauge-breaking processes dominate when $\lambda>\gamma$, and both prethermal plateaus appear for $N=4$ matter sites when $\lambda^2>\gamma J_a$.
Let us now again look at the dynamics of the supersector projectors in the presence of decoherence. This is shown for the projectors onto the supersectors $M=2$ and $M=4$ in Fig.~\ref{fig:gammaFixed}(c,d). Congruent to the conclusions of the joint submission,\cite{Halimeh2020f} we get the same short-time scaling for the supersector projectors $\mathcal{P}_2$ and $\mathcal{P}_4$, with the crossover from the diffusive regime where $\langle\mathcal{P}_2\rangle\sim\gamma t$ and $\langle\mathcal{P}_4\rangle\sim\gamma^2 t^2$ at $t\lesssim\gamma/\lambda^2$ to the ballistic regime where $\langle\mathcal{P}_2\rangle\sim\lambda^2 t^2$ and $\langle\mathcal{P}_4\rangle\sim\lambda^4 t^4$ at $t\gtrsim\gamma/\lambda^2$. The deterioration of the first prethermal timescale can also be observed in these quantities, and that of the second prethermal plateau is observed in $\langle\mathcal{P}_4\rangle$. The larger $\lambda$ is, the greater the integrity of the prethermal plateaus, with the first plateau exhibiting greater resilience as it survives smaller values of $\lambda$ than its second counterpart. Interestingly, at long times $t\gtrsim1/\gamma$, both projectors relax to the values $\langle\mathcal{P}_2\rangle\approx0.125$ and $\langle\mathcal{P}_4\rangle=\langle\mathcal{P}_0\rangle\approx0.75$, meaning that $\langle\mathcal{P}_2\rangle/\langle\mathcal{P}_4\rangle=\langle\mathcal{P}_2\rangle/\langle\mathcal{P}_0\rangle=6$, which is equal to the ratio of number of gauge sectors in each supersector. This generally does not happen in the case of no decoherence ($\gamma=0$), but in the presence of decoherence at any $\gamma>0$, the long-time limit will ascribe to this behavior. This is due to the fact that the gauge violation has fully diffused in the space of gauge sectors, occupying an equal distribution among all of them.
We also include the dynamics of the staggered boson number in Fig.~\ref{fig:gammaFixed}(e) [recall that the total boson number is conserved since $H_0$ and $H_1$ both have global $\mathrm{U}(1)$ symmetry, and there is only dephasing on the matter fields] and the electric field in Fig.~\ref{fig:gammaFixed}(f). One cannot discern any diffusive behavior at early times in these observables (even by looking at the deviation from the fully unitary case). A deeper reason may be that these local observables are not related to a divergence measure through the gauge sectors, contrary to the gauge violation (see Sec.~\ref{sec:divergencemeasure}).
The leading-order (in $\gamma$) correction to the unitary part of the density matrix is $\gamma t\mathcal{L}\rho_0$ makes a vanishing contribution to both of these observables as discussed in Sec.~\ref{sec:TDPT_leadingIncoherent}.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{lam0_violation}\\
\includegraphics[width=.48\textwidth]{lam0_StaggeredBosonNumber}
\caption{(Color online). Dynamics of (a) the gauge violation and (b) the staggered matter field for $\lambda=0$ at various values of the environment-coupling strength $\gamma$ (see legend). The behavior is qualitatively similar to that of Figs.~\ref{fig:gammaFixed}(a,e), respectively, albeit here there is no signature of the prethermal plateaus in the gauge violation since $\lambda=0$.}
\label{fig:obs}
\end{figure}
In the quench dynamics of the joint submission\cite{Halimeh2020f} and Fig.~\ref{fig:gammaFixed}, we have focused on $\lambda>0$. For completeness, we show in Fig.~\ref{fig:obs} the effect of $\gamma$ on the dynamics of gauge violation and staggered boson number, considering quench dynamics for the same initial state as in Fig.~\ref{fig:Z2LGT_InitialStates}(a) but for $\lambda=0$. The gauge violation [Fig.~\ref{fig:obs}(a)] spreads diffusively in the gauge sectors scaling as $\varepsilon\sim\gamma t$ at short times before settling into a maximal-violation steady state at $t\approx1/\gamma$. The staggered boson number [Fig.~\ref{fig:obs}(b)] behaves much the same way as in the case of $\lambda>0$ in Fig.~\ref{fig:gammaFixed}(e): it deviates from the purely coherent dynamics at $t\approx1/\gamma$, with its temporal average decaying $\sim(\gamma t)^{-1}$ at late times.
\subsection{Variations of jump operator}
So far, we have included dissipation in the gauge fields as governed by the jump operator $L_{j,j+1}^\mathrm{g}=\tau^z_{j,j+1}$. To corroborate the generality of our results, we study the effect of a different dissipative jump operator $L_{j,j+1}^\mathrm{g}=\tau^-_{j,j+1}$ at various values of $\gamma$ in the presence of coherent gauge-breaking terms at strength $\lambda=10^{-4}J_a$. We show the associated gauge-violation and supersector-projector dynamics in Fig.~\ref{fig:DiffDiss}. The qualitative picture is unchanged, and we see that the diffusive-to-ballistic crossover is also at $t\propto\gamma/\lambda^2$, with scaling $\sim\gamma t$ ($\sim\gamma^2t^2$) in the diffusive regime and scaling $\sim\lambda^2t^2$ ($\sim\lambda^4t^4$) in the ballistic regime in the short-time dynamics of the gauge violation and $\langle\mathcal{P}_2\rangle$ ($\langle\mathcal{P}_4\rangle$).
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{violDiffDiss}\\
\includegraphics[width=.48\textwidth]{P2DiffDiss}\\
\includegraphics[width=.48\textwidth]{P4DiffDiss}
\caption{(Color online). Same as Fig.~1 of Ref.~\onlinecite{Halimeh2020f} but with a different jump operator on the gauge links. Quench dynamics of (a) the gauge violation, (b) supersector projector $\mathcal{P}_2$, and (c) supersector projector $\mathcal{P}_4$ in the open $\mathrm{Z}_2$ LGT starting in the gauge-invariant initial state of Fig.~\ref{fig:eBHM_InitialStates}(a), in the presence of the unitary gauge-breaking errors of Eq.~\eqref{eq:H1} at strength $\lambda=10^{-4}J_a$, and coupling to the environment at strength $\gamma$ with the jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$ on matter sites and gauge links, respectively. The diffusive-to-ballistic crossover is again at $t\propto\gamma/\lambda^2$.}
\label{fig:DiffDiss}
\end{figure}
\subsection{Other initial states}
Furthermore, our conclusions remain unaltered for other initial states. Whereas in the joint submission Ref.~\onlinecite{Halimeh2020f} and hitherto in this paper our initial state has been the staggered product state in Fig.~\ref{fig:Z2LGT_InitialStates}(a), in Fig.~\ref{fig:DiffInitState} we show the gauge-violation and supersector-projector dynamics for the ``domain-wall'' product state in Fig.~\ref{fig:Z2LGT_InitialStates}(b). Again, here we include coherent gauge-breaking terms at strength $\lambda=10^{-4}J_a$ and study the effects of decoherence at various values of the environment coupling $\gamma$, with the jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$. The qualitative picture is identical to that of Fig.~1 in Ref.~\onlinecite{Halimeh2020f} and Fig.~\ref{fig:DiffInitState}.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{violDiffInitState}\\
\includegraphics[width=.48\textwidth]{P2DiffInitState}\\
\includegraphics[width=.48\textwidth]{P4DiffInitState}
\caption{(Color online). Same as Fig.~\ref{fig:DiffDiss} but starting in the gauge-invariant ``domain-wall'' initial product state of Fig.~\ref{fig:Z2LGT_InitialStates}(b) and with $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$.}
\label{fig:DiffInitState}
\end{figure}
\subsection{Multiple quantum coherences}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{MQC_gamma0_a}\\
\includegraphics[width=0.32\linewidth]{MQC_gamma0_b}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_c}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_d}\\
\includegraphics[width=0.32\linewidth]{MQC_gamma0_e}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_f}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_g}\\
\includegraphics[width=0.32\linewidth]{MQC_gamma0_h}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_i}
\includegraphics[width=0.32\linewidth]{MQC_gamma0_j}
\caption{(Color online). Same as Fig.~2 of Ref.~\onlinecite{Halimeh2020f} but with no decoherence, and additionally showing the MQC spectra at different angles. (a) Dominant MQC. (b-j) MQC spectra, with rows from top to bottom $F(\phi_1,0,\phi_3,0)$, $F(\phi_1,\phi_2,0,0)$, and $F(0,\phi_2,\phi_3,0)$, and columns from left to right $tJ_a=10^4$, $tJ_a=10^5$, and $tJ_a=10^6$. Unsurprisingly, in the absence of decoherence, there is no decay of the intensities, and the bandwidth of the MQC spectra increases with time, in contrast to the case of decoherence in Fig.~2 of Ref.~\onlinecite{Halimeh2020f}, where decoherence diminishes the spectrum.}
\label{fig:Fig2SM}
\end{figure}
In the $\mathrm{Z}_2$ LGT, the MQC are a generalization of those given in Eq.~\eqref{eq:eBHM_MQC}, and read
\begin{subequations}
\begin{align}
I_{\Delta \mathbf{g}}&=\Tr\big\{\rho_{\Delta \mathbf{g}}^\dagger\rho_{\Delta \mathbf{g}}\big\},\\
\rho_{\Delta \mathbf{g}}&=\sum_g P_{\mathbf{g}+\Delta \mathbf{g}}\rho P_\mathbf{g},
\end{align}
\end{subequations}
where now they measure quantum coherences between gauge-invariant \textit{sectors} and not just supersectors as in the case of the eBHM (see Sec.~\ref{sec:eBHM_MQC}). Since the deviations between sectors are vectors, the MQC spectra
\begin{align}
F_{\boldsymbol{\phi}}=\sum_{\Delta\mathbf{g}}I_{\Delta\mathbf{g}}e^{-i\sum_j\phi_j\Delta g_j}
\end{align}
depend on $N$ angles $\boldsymbol{\phi}=\{\phi_1,\phi_2,\ldots,\phi_N\}$. In the joint submission Ref.~\onlinecite{Halimeh2020f}, we provide results for the MQC and their spectra, for a given choice of angles, at $\lambda=10^{-4}J_a$ and $\gamma=10^{-6}J_a$. Let us now look at these results but with no decoherence, i.e., $\lambda=10^{-4}J_a$ and $\gamma=10^{-6}J_a$. The corresponding results are shown in Fig.~\ref{fig:Fig2SM}. In the absence of decoherence, the MQC over evolution time and settle at a steady-state value at long times [Fig.~\ref{fig:Fig2SM}(a)], in contrast to the case with decoherence of Ref.~\onlinecite{Halimeh2020f}, where their temporal averages decay $\sim(\gamma t)^{-1}$ at $t>\gamma^{-1}$. The spectrum also behaves fundamentally differently. Whereas in the case with decoherence the spectrum almost vanishes for $t\gtrsim\gamma^{-1}$, in the closed-system case it is maximal in this temporal regime.
\begin{figure}[htp]
\centering
\includegraphics[width=0.32\linewidth]{MQC_1100_t1e4}
\includegraphics[width=0.32\linewidth]{MQC_1100_t1e5}
\includegraphics[width=0.32\linewidth]{MQC_1100_t1e6}\\
\includegraphics[width=0.32\linewidth]{MQC_0110_t1e4}
\includegraphics[width=0.32\linewidth]{MQC_0110_t1e5}
\includegraphics[width=0.32\linewidth]{MQC_0110_t1e6}
\caption{(Color online). Same as Fig.~2(c-e) of Ref.~\onlinecite{Halimeh2020f} but for different angles of the MQC spectrum, where again $\lambda=10^{-4}J_a$ and $\gamma=10^{-6}J_a$. We show $F(\phi_1,\phi_2,0,0)$ at evolution times (a) $t=10^4/J_a$, (b) $t=10^5/J_a$, and (c) $t=10^6/J_a$, and we show $F(0,\phi_2,\phi_3,0)$ at evolution times (d) $t=10^4/J_a$, (e) $t=10^5/J_a$, and (f) $t=10^6/J_a$. The results show that decoherence diminishes the spectrum, in agreement with the conclusion from Fig.~2(c-e) of the joint submission.}
\label{fig:MQC_SM}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{MQC_DiffDiss_a}\\
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_b}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_c}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_d}\\
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_e}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_f}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_g}\\
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_h}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_i}
\includegraphics[width=0.32\linewidth]{MQC_DiffDiss_j}
\caption{(Color online). Same as Fig.~2 of Ref.~\onlinecite{Halimeh2020f} but with $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$ and including additional angles for the MQC spectra. (a) Dominant MQC. (b-j) MQC spectra, with rows from left to right $F(\phi_1,0,\phi_3,0)$, $F(\phi_1,\phi_2,0,0)$, and $F(0,\phi_2,\phi_3,0)$ and columns from left to right $tJ_a=10^4$, $tJ_a=10^5$, and $tJ_a=10^6$. In contrast to the case of $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ as in Fig.~2 of Ref.~\onlinecite{Halimeh2020f}, here the MQC do not decay to zero, but rather saturate at finite steady-state values. This behavior depends on the fixed points of the Liouvillian superoperator.}
\label{fig:MQC_DiffDiss}
\end{figure}
Our choice of the MQC-spectrum angles in the main text is neither special nor unique. Due to symmetry, we have
\begin{subequations}
\begin{align}
&F(\phi_1,0,\phi_3,0)=F(0,\phi_2,0,\phi_4),\\
&F(\phi_1,\phi_2,0,0)=F(0,0,\phi_3,\phi_4),\\
&F(0,\phi_2,\phi_3,0)=F(\phi_1,0,0,\phi_4).
\end{align}
\end{subequations}
For completeness, we show in Fig.~\ref{fig:MQC_SM} for the MQC spectra $F(\phi_1,\phi_2,0,0)$ and $F(0,\phi_2,\phi_3,0)$ in the presence of gauge-breaking coherent and incoherent errors at strengths $\lambda=10^{-4}J_a$ and $\gamma=10^{-6}$, where the initial state is that of Fig.~\ref{fig:Z2LGT_InitialStates}(a) and the dynamics is governed by Eq.~\eqref{eq:EOM}. Similarly to their counterpart $F(\phi_1,0,\phi_3,0)$, both $F(\phi_1,\phi_2,0,0)$ and $F(0,\phi_2,\phi_3,0)$ diminish in the presence of decoherence.
It is interesting to compare these findings to the dynamics of the MQC with a different jump operator. As such, we again start in the staggered initial state of Fig.~\ref{fig:Z2LGT_InitialStates}(a) and set the jump operator $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$. Once again, we set $\lambda=10^{-4}J_a$ and $\gamma=10^{-6}J_a$, with $L^\mathrm{m}_j=a_j^\dagger a_j$. (The dynamics of the gauge violation and supersector projectors for this quench protocol are shown in Fig.~\ref{fig:DiffDiss}.) The corresponding MQC results are shown in Fig.~\ref{fig:MQC_DiffDiss}. Unlike the case of $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ (see Fig.~2 of Ref.~\onlinecite{Halimeh2020f}), the MQC do not decay to zero when $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$, but rather saturate at finite steady-state values. This behavior depends on the fixed point of the Liouvillian superoperator. Indeed, we have checked (not shown) that when the ``domain-wall'' product state shown in Fig.~\ref{fig:Z2LGT_InitialStates}(b) is the initial state, the MQC take on the same steady-state values as those shown in Fig.~\ref{fig:MQC_DiffDiss}.
\subsection{Variations of relative strength of incoherent errors}
Let us now investigate the effect of turning on the dephasing and dissipation at different strengths $\gamma_\mathrm{m}$ and $\gamma_\mathrm{g}$, respectively. The Lindblad master equation generalizes to
\begin{align}\nonumber
\dot{\rho}=&-i[H_0+\lambda H_1,\rho]\\\nonumber
&+\sum_{j=1}^N\Big[\gamma_\mathrm{m}\Big(L^\text{m}_j\rho L^{\text{m}\dagger}_j-\frac{1}{2}\big\{L^{\text{m}\dagger }_jL^\text{m}_j,\rho\big\}\Big)\\\label{eq:EOM_relative}
&+\gamma_\mathrm{g}\Big(L^\text{g}_{j,j+1}\rho L^{\text{g}\dagger}_{j,j+1}-\frac{1}{2}\big\{L^{\text{g}\dagger }_{j,j+1}L^\text{g}_{j,j+1},\rho\big\}\Big)\Big].
\end{align}
Interestingly, the dephasing strength $\gamma_\mathrm{m}$ has little effect on the short-time dynamics of the gauge violation, which at short times $t\lesssim\gamma_\mathrm{g}/\lambda^2$ scales as $\varepsilon\sim\gamma_\mathrm{g}t$, and at intermediate times $\gamma_\mathrm{g}/\lambda^2<t\lesssim1/\lambda$ scales as $\varepsilon\sim(\lambda t)^2$ due to the dominance of coherent gauge-breaking terms. In fact, it can be shown in TDPT (see Sec.~\ref{sec:TDPT_leadingIncoherent}) that the contribution to the gauge violation due to dephasing at short times vanishes. However, we find that both $\gamma_\mathrm{m}$ and $\gamma_\mathrm{g}$ have a significant effect on the later timescale at which decoherence dominates at maximal violation, with this timescale being roughly $1/\max\{\gamma_\mathrm{g},\gamma_\mathrm{m}\}$, as can be seen in the lower insets of Fig.~\ref{fig:relativeGamma}(a,b). In particular, dephasing incurs a nonperturbative effect on the prethermal plateaus, as shown in the lower inset of Fig.~\ref{fig:relativeGamma}(a).
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{fixedGammaGauge}\quad
\includegraphics[width=.48\textwidth]{fixedGammaMatter}
\caption{(Color online). Dynamics of the gauge-invariance violation at different environment-coupling strengths $\gamma_\mathrm{m}$ for the dephasing on matter fields and $\gamma_\mathrm{g}$ for the dissipation on gauge links, with fixed strength $\lambda=10^{-4}J_a$ of coherent gauge-breaking processes. (a) Gauge violation at various values of $\gamma_\mathrm{m}$ for fixed value of $\gamma_\mathrm{g}=10^{-10}J_a$. (b) Gauge violation at various values of $\gamma_\mathrm{g}$ for a fixed value of $\gamma_\mathrm{m}=10^{-10}J_a$. Dissipation clearly shows an effect on the gauge violation at short times, whereas dephasing does not. However, at late times dephasing also has a clear effect on the prethermal plateaus, as shown in the lower inset of (a).
}
\label{fig:relativeGamma}
\end{figure}
\subsection{Dynamics under gauge protection}
Recently, several theoretical works have proposed to use gauge protection to suppress processes driving the system out of its initial gauge-invariant sector,\cite{Zohar2011,Zohar2012,Banerjee2012,Zohar2013,Hauke2013,Kuehn2014,Kuno2015,Kuno2017,Negretti2017,Barros2019,Halimeh2020a,Halimeh2020e,Mathis2020,Lamm2020,Tran2020} and the principle has been demonstrated experimentally for a $\mathrm{U}(1)$ gauge theory.\cite{Yang2020} The basic idea is to introduce a suitable energy-penalty term, which for a $\mathrm{Z}_2$ gauge theory reads
\begin{align}
VH_G=V\sum_jG_j,
\end{align}
where $V$ controls the protection strength. For sufficiently large $V$, the associated gauge violation due to $H_1$ is suppressed by $(\lambda/V)^2$, and the ensuing dynamics is perturbatively close to a renormalized version of the ideal gauge theory.\cite{Halimeh2020a,Halimeh2020e} Using as quench Hamiltonian $H=H_0+\lambda H_1+VH_G$, and numerically solving the respective Lindblad master equation
\begin{align}\nonumber
\dot{\rho}=&-i[H_0+\lambda H_1+VH_G,\rho]\\\nonumber
&+\gamma\sum_{j=1}^N\Big(L^\text{m}_j\rho L^{\text{m}\dagger}_j+L^\text{g}_{j,j+1}\rho L^{\text{g}\dagger}_{j,j+1}\\
&-\frac{1}{2}\big\{L^{\text{m}\dagger }_jL^\text{m}_j+L^{\text{g}\dagger }_{j,j+1}L^\text{g}_{j,j+1},\rho\big\}\Big)
\end{align}
at fixed values of $\gamma$ and $\lambda$, we obtain the gauge-violation dynamics shown in Fig.~\ref{fig:protection}. We find that a finite $V$ suppresses only coherent contributions to the gauge violation, but not incoherent ones. This finding is not surprising as the dissipative errors in our work are modelled by a Markovian Lindblad master equation that couples states regardless of their energy differences, and which is thus oblivious to energy penalties.
Interestingly, for intermediate values of the protection strength ($V=J_a$ at $\lambda=10^{-2}J_a$), we see that after going from diffusive ($\varepsilon\sim\gamma t$) at $t\lesssim\gamma/\lambda^2$ to ballistic ($\varepsilon\sim\lambda^2 t^2$) dynamics at $t\gtrsim\gamma/\lambda^2$, the gauge violation again exhibits diffusive behavior before settling into a maximal-violation steady state at $t\approx1/\gamma$. The larger $V$ is, the shorter is the intermediate ballistic regime. At sufficiently large $V$, coherent errors are almost completely suppressed and the ballistic regime vanishes, with the gauge violation scaling as $\varepsilon\sim\gamma t$ for all times $t\lesssim1/\gamma$.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{protection}\quad
\includegraphics[width=.48\textwidth]{protection2}
\caption{(Color online). Using the setup of Fig.~\ref{fig:Z2LGT_InitialStates}(a) under decoherence, we add a protection term \cite{Halimeh2020a} $V H_G=V\sum_jG_j$ such that the coherent quench is actuated by $H_0+\lambda H_1+VH_G$, which suppresses only the coherent gauge-breaking errors, but has no effect on the gauge violation due to Markovian decoherence. (a) At fixed $\lambda=10^{-2}J_a$ and $\gamma=10^{-6}J_a$ (blue curves), we see that at large enough protection strength $V$, the diffusive scaling $\varepsilon\sim\gamma t$ seen at short times emerges again before the maximal violation is reached, and, in some cases, after the ballistic scaling $\varepsilon\sim(\lambda t)^2$ has appeared at intermediate times. For reference, we also show the gauge violation for $\lambda=0$ and $\gamma=10^{-6}J_a$ (red curve), which exhibits only scaling $\varepsilon\sim\gamma t$ before saturating at its maximal value for $t\gtrsim1/\gamma$. At nonzero $\lambda$ and without energy protection, the gauge violation scales $\varepsilon\sim\gamma t$ only at early times $t\lesssim\gamma/\lambda^2$ before scaling $\varepsilon\sim(\lambda t)^2$ at intermediate times $t\gtrsim\gamma/\lambda^2$, and finally reaching the maximal violation at $t\propto\min\{\lambda^{-N/2},\gamma^{-1}\}$. (b) The same as panel (a) but for $\lambda=10^{-1}J_a$.}
\label{fig:protection}
\end{figure}
\subsection{Dynamics under particle loss}
Until now, within Sec.~\ref{sec:Z2LGT} we have considered only dephasing in the matter fields in order to allow for the conservation of particle number, thereby enabling us to achieve larger system sizes (see Appendix~\ref{sec:NumSpec} for further details). Here, we consider also dissipation in the matter fields. In particular, we will choose $L^\mathrm{m}_j=a_j$ while also using $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ and fixing $\lambda=10^{-4}J_a$. Again, we consider the initial state in Fig.~\ref{fig:Z2LGT_InitialStates}(a) for $N=2$ matter sites (here, $N=4$ matter sites is numerically intractable for the evolution times we need to reach in our calculations), and solve Eq.~\eqref{eq:EOM} for $\lambda=10^{-4}J_a$ using several values of $\gamma$. The corresponding results for the gauge-violation dynamics are shown in Fig.~\ref{fig:ParticleLoss}(a). For ease of comparison, we repeat these results for the case of dephasing on matter fields with jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ in Fig.~\ref{fig:ParticleLoss}(b), just as in all the results before this point. Aside from the absence of a second prethermal plateau due to the halved matter-site number, the results are very similar to those in Fig.~1(b) of Ref.~\onlinecite{Halimeh2020f}, and the qualitative behavior is identical: the gauge violation displays a crossover from diffusive scaling $\varepsilon\sim\gamma t$ to ballistic scaling $\varepsilon\sim\lambda^2t^2$ at $t\propto\gamma/\lambda^2$ when $\gamma<\lambda$. As such, we can conclude that our results in Ref.~\onlinecite{Halimeh2020f} are general, and are not restricted by considering only dephasing in the matter fields.
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{ParticleLoss_violation}\quad
\includegraphics[width=.48\textwidth]{NoParticleLoss_violation}
\caption{(Color online). Dynamics of gauge violation in the $\mathrm{Z}_2$ LGT with $N=2$ matter sites in the presence of dissipation in the gauge fields with jump operators $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ and (a) dissipation in the matter fields with jump operators $L^\mathrm{m}_j=a_j$ and (b) dephasing in the matter fields with jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$, just as in all the results for the $\mathrm{Z}_2$ LGT before now. Coherent gauge breaking is at strength $\lambda=10^{-4}J_a$. The behavior is qualitatively identical whether the matter fields are subjected to dephasing or dissipation. The fact that we have only $N=2$ matter sites here brings about only a single prethermal plateau instead of two as in the case of $N=4$ matter sites.\cite{Halimeh2020b,Halimeh2020c}}
\label{fig:ParticleLoss}
\end{figure}
\subsection{Decoherence starting from equilibrium}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{H0GS_viol}\\
\includegraphics[width=.48\textwidth]{H0GS_proj}
\caption{(Color online). (a) Time evolution of gauge-invariance violation after starting in the ground state of the $\mathrm{Z}_2$ LGT and switching on decoherence with strength $\gamma$ at $t=0$. Note that the ground state of $H_0$ is not gauge-invariant, a consequence of the fact that different gauge-invariant sectors of $H_0$ have energy overlaps. Similarly to the case of quench dynamics, the gauge violation at short times scales $\sim\gamma t$ before plateauing at its maximal value.
(b) Projectors onto the three accessible gauge-invariant supersectors. Their steady-state expectation values are proportional to the number of constituent gauge-invariant sectors.}
\label{fig:Fig3}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{H0GS_Idg}\quad
\includegraphics[width=.48\textwidth]{H0GS_StaggeredBosonNumber}
\caption{(Color online). Same scenario as in Fig.~\ref{fig:Fig3}: we start in the ground state of $H_0$ and switch on decoherence at $t=0$. Running averages of the (a) MQC intesity $I_{\Delta\mathbf{g}=\{2,2,2,2\}}$ and (b) staggered boson number are shown. Unlike the gauge violation and supersector projectors, these observables show no trace of diffusive behavior at short times, but they decay $\sim(\gamma t)^{-1}$ due to decoherence for $t\gtrsim1/\gamma$.
}
\label{fig:staticsObs}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{GSlami0Vi1_coherent}\quad
\includegraphics[width=.48\textwidth]{GSlami0Vi1}
\caption{(Color online). Gauge violation over evolution time after starting in the (gauge-noninvariant) ground state of $H_0$ and quenching with $H_0+\lambda H_1$ with Markovian decoherence through jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$. (a) In the case of $\gamma=0$ (no decoherence), the pre-onset and final prethermal plateaus appear and the violation at early times scales $\sim\lambda^2 t^2$, because $\rho_0$ is the ground state of $H_0$. (b) When decoherence is turned on, a diffusive-to-ballistic crossover appears at timescale $\propto\gamma/\lambda^2$ taking the violation from a diffusive spread $\sim\gamma t$ to a ballistic scaling $\sim\lambda^2 t^2$.
}
\label{fig:lami0}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{GSlami1e-1Vi1_coherent}\quad
\includegraphics[width=.48\textwidth]{GSlami1e-1Vi1}
\caption{(Color online). Same as Fig.~\ref{fig:lami0} but with $\lambda_\text{i}=0.1$, and thus $\rho_0$ is the ground state of $H_0+\lambda_\text{i}H_1$. The initial state $\rho_0$ is again gauge-noninvariant, but not a ground state of $H_0$. This makes the leading order of coherent contribution $\propto\lambda t^2$, which identically vanishes in the case of $\lambda_\text{i}=0$ of Fig.~\ref{fig:lami0}. This therefore leads to a crossover from a diffusive spread $\sim\gamma t$ to a ballistic scaling $\sim\lambda t^2$ at an earlier timescale $t\propto\gamma/\lambda<\gamma/\lambda^2$. Note here in the purely coherent case (a) how there are three prethermal plateaus instead of just two as in the case of Fig.~\ref{fig:lami0}.
}
\label{fig:gaugenoninvariant}
\end{figure}
Aside from quench dynamics, we also study the effect of decoherence through jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ on the ground state of an LGT with $N=4$ matter sites and periodic boundary conditions. For this aim, we prepare our system in the ground state of $H_0$, and switch on decoherence at $t=0$ according to the Lindblad master equation~\eqref{eq:EOM} with $\lambda=0$. Such a scenario may occur, e.g., when variational state preparation\cite{Kokail2019} is used to achieve a good approximation to a gauge-invariant initial ground state, which is then stored in the quantum computer and thus subjected to decoherence.
As we do not project the ground state into the target gauge-invariant sector $\mathbf{g}_\text{tar}$, the gauge violation at $t=0$ starts at a finite nonzero value.
At $t>0$, it grows as $\varepsilon\sim\gamma t$ at short times until the system settles into a maximal-violation steady state at $t\approx1/\gamma$, as shown in Fig.~\ref{fig:Fig3}(a).
It is also instructive to investigate the projectors onto the three relevant gauge-invariant supersectors $\mathcal{P}_{M=0,2,4}$ (the supersector projectors $\mathcal{P}_M$ with odd $M$ are of zero norm in the half-filling global-symmetry sector of the $\mathrm{Z}_2$ LGT). Figure~\ref{fig:Fig3}(b) shows the three projectors that give rise to nonzero expectation values in the case of $N=4$ matter sites.
Interestingly, the steady-state expectation values are proprotional to the number of gauge-invariant sectors within the associated supersector. Indeed, for $N=4$ matter sites, the supersector $\mathcal{P}_2$ contains six different gauge sectors, while the two supersectors represented by $\mathcal{P}_0$ and $\mathcal{P}_4$ each contains a single gauge-invariant sector. As shown in the inset, $\langle\mathcal{P}_2\rangle\approx0.75$, $\langle\mathcal{P}_0\rangle=\langle\mathcal{P}_4\rangle\approx0.125$, and thus $\langle\mathcal{P}_2\rangle/\langle\mathcal{P}_0\rangle=\langle\mathcal{P}_2\rangle/\langle\mathcal{P}_4\rangle=6$. This indicates that with decoherence the system evolves at late times into a steady state where the gauge violation has spread into an equal distribution over all gauge sectors. This is qualitatively and quantitatively identical to the behavior of these projectors at late times in the quench dynamics with decoherence shown in Fig.~1(c,d) of Ref.~\onlinecite{Halimeh2020f}, despite here the initial state and dynamic process being fundamentally different. Hence, in these results decoherence erases the memory of the initial state. For completeness, we additionally present the associated results for the MQC intensity $I_{\{2,2,2,2\}}$ in Fig.~\ref{fig:staticsObs}(a) and staggered particle density in Fig.~\ref{fig:staticsObs}(b). Starting at its ground-state value, each of these observables shows a decay $\sim(\gamma t)^{-1}$ in its temporal average at a time $t\approx1/\gamma$. The electric field (not shown) behaves also qualitatively the same.
A more interesting scenario is starting in ground state of $H_0+\lambda_\text{i} H_1$, which may become relevant in situations where a pre-quench state-preparation protocol is already subject to gauge-breaking errors. Not only is the initial state here gauge-noninvariant, the presence of $H_1$ allows for a competition between coherent errors and their incoherent counterparts due to decoherence.
We first consider the case when $\lambda_\text{i}=0$, i.e., we have managed to prepare the system in the ground state of the ideal gauge theory without any coherent errors, but we shall assume that upon quenching, unitary errors $\lambda H_1$ will be present. This scenario may appear when the preparation follows one protocol, e.g., a variational principle,\cite{Kokail2019} while the quench dynamics is studied with another, e.g., an analog quantum-simulation scheme. The ensuing dynamics of the gauge-violation change is shown in Fig.~\ref{fig:lami0}(a) for the case without decoherence but with finite $\lambda>0$. The gauge violation grows $\sim\lambda^2t^2$ at early times, with all lower-order coherent contributions vanishing identically as rigorously explained in Sec.~\ref{sec:TDPT_coherent} through TDPT. The gauge violation exhibits the pre-onset and final plateaus at timescales $\propto1$ and $\propto J_a/\lambda^2$, respectively, but the onset plateau at timescale $\propto1/\lambda$, prominent in the case of gauge-invariant states, is missing here. Maximal violation occurs at the final prethermal timescale $\propto J_a/\lambda^2$. Upon introducing decoherence through jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ in Fig.~\ref{fig:lami0}(b) at fixed $\lambda$, a crossover emerges at $t\propto\gamma/\lambda^2$ from a diffusive spread $\sim\gamma t$ in the gauge violation to a ballistic spread $\sim\lambda^2t^2$, similarly to the generic behavior we find when starting in a gauge-invariant initial state. Moreover, decoherence compromises the prethermal plateaus, with its effect more apparent on the later plateaus.
On the other hand, when $\lambda_\text{i}\neq0$, i.e., when $\rho_0$ is gauge-noninvariant but also not the ground state of $H_0$, a lower-order coherent contribution $\propto\lambda t^2$ that vanishes in the case of $\lambda_\text{i}=0$, now becomes finite, and thus the gauge-violation change scales $\sim\lambda t^2$ at early times in the case of no decoherence, as shown in Fig.~\ref{fig:gaugenoninvariant}(a). Interestingly, here we find all three prethermal plateaus: pre-onset at timescale $t\propto1$, onset at $t\propto1/\lambda$, and final at $t\propto J_a/\lambda^2$. By switching on decoherence through jump operators $L^\mathrm{m}_j=a_j^\dagger a_j$ and $L^\mathrm{g}_{j,j+1}=\tau^z_{j,j+1}$ at a fixed values of $\lambda$, we see that a crossover appears where the gauge-violation difference goes from a diffusive scaling $\sim\gamma t$ to a ballistic spread $\sim\lambda t^2$ at the timescale $t\propto\gamma/\lambda$. As expected, decoherence also compromises the prethermal plateaus in this case.
\section{$\mathrm{U}(1)$ quantum link model}\label{sec:U1QLM}
\begin{figure}[htp]
\centering
\includegraphics[width=.48\textwidth]{U1QLM_fixedlambda}\\
\includegraphics[width=.48\textwidth]{U1QLM_fixedgamma}
\caption{(Color online). A system with $N=2$ matter sites and periodic boundary conditions, prepared in a gauge-invariant initial state with zero bosons and a N\'eel configuration of the electric field is quenched with $H_0+\lambda H_1$ in the presence of decoherence through jump operators $L^\mathrm{m}_j=\sigma_j^z$ and $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$ both at environment-coupling strength $\gamma$. Decoherence compromises the prethermal plateaus, and in fact leads the violation to a maximal value not attained by the coherent errors is alone. This is due to the absence of resonance between a few of the gauge-invariant sectors in the $\mathrm{U}(1)$ QLM, which does not have an analog in the $\mathrm{Z}_2$ LGT. This maximal value depends on the fixed points of the Liouvillian superoperator. The short-time dynamics is qualitatively the same as in the generic case of the $\mathrm{Z}_2$ LGT when the initial state is gauge-invariant: a diffusive-to-ballistic crossover arises at $t\propto\gamma/\lambda^2$ where the gauge violation goes from a diffusive spread $\sim\gamma t$ to a ballistic spread $\propto\lambda^2t^2$.}
\label{fig:U1QLM}
\end{figure}
To further demonstrate the generality of our findings, we now study the gauge-violation dynamics in the $\mathrm{U}(1)$ quantum link model (QLM) given by\cite{Wiese_review,Hauke2013,Yang2016,Yang2020}
\begin{align}
H_0=\sum_{j=1}^N\Big[-J\big(\sigma^-_j\tau^+_{j,j+1}\sigma^-_{j+1}+\text{H.c.}\big)+\frac{\mu}{2}\sigma^z_j\Big],
\end{align}
where the Pauli matrix $\sigma^+_j$ is the creation operator of a particle on site $j$, while the Pauli matrix $\tau^+_{j,j+1}$ ($\tau^z_{j,j+1}$) on link $j,j+1$ represents the gauge (electric) field. Here, we consider a lattice of $N=2$ matter sites and periodic boundary conditions. The Gauss's-law generator is
\begin{align}
G_j=\frac{(-1)^j}{2}\big(\tau^z_{j-1,j}+\sigma^z_j+\tau^z_{j,j+1}+1\big).
\end{align}
and has eigenvalues $g_j=-1,0,1,2$. This model has been the subject of recent ultracold-atom experiments.\cite{Mil2020,Yang2020} We calculate the quench dynamics of this model in the presence of decoherence through jump operators $L^\mathrm{m}_j=\sigma_j^z$ and $L^\mathrm{g}_{j,j+1}=\tau^-_{j,j+1}$, both at environment-coupling strength $\gamma$, by solving the Lindblad master equation~\eqref{eq:EOM} with
\begin{align}
\lambda H_1=\lambda\sum_j\big(\sigma^-_j\sigma^-_{j+1}+\sigma^+_j\sigma^+_{j+1}+\tau^x_{j,j+1}\big),
\end{align}
which describes unassisted matter tunneling and gauge flipping, and we consider the jump operators $L^\mathrm{m}_j=\sigma^z_j$ and $L^\mathrm{g}_{j,j+1}=\tau^+_{j,j+1}$, although we have checked that other choices of the jump operators yield the same qualitative picture. Moreover, the gauge-invariant initial state $\rho_0$ has zero matter particles and a N\'eel configuration of the electric field. In our numerics, we have set $J=1$ and $\mu=0.05$, though we have checked that our conclusions are independent of this particular choice of parameters.
The ensuing time evolution of the gauge violation is shown in Fig.~\ref{fig:U1QLM}. Focusing first on the long-time dynamics, we again see how decoherence compromises the prethermal plateau, but, more dramatically than in the case of the $\mathrm{Z}_2$ LGT, it drives the gauge violation to a larger value. This is because the $\mathrm{U}(1)$ QLM has a larger number of gauge-invariant sectors (due to an eigenvalue $g_j$ having four possible values) some of which do not have resonances through $H_1$ with one another---unlike the $\mathrm{Z}_2$ LGT, here the value of $g_j$ restricts the possible values that $g_{j-1}$ and $g_{j+1}$ can take. In the presence of decoherence, resonances between these gauge-invariant sectors are facilitated, and this is what leads to a larger long-time violation compared to the purely unitary-error case. Furthermore, as can be seen in Fig.~\ref{fig:U1QLM}, the short-time dynamics and the diffusive-to-ballistic crossover are identical to those for the $\mathrm{Z}_2$ LGT and the eBHM, further validating the generality of our results. Indeed, as we rigorously show in Sec.~\ref{sec:TDPT} through TDPT, our conclusions are valid for any many-body system with local of global symmetries.
\section{Conclusions and outlook}\label{sec:conclusion}
In this work, we have considered the dynamics of quantum systems where a symmetry is slightly broken, with special focus on the interplay between coherent and dissipative errors.
To obtain a general unifying picture, we have performed extensive numerical studies and analytical derivations, considering a broad range of scenarios including dynamics starting from initial product states and from ground states, as well as a variety of models with global symmetries [global $\mathrm{U}(1)$ symmetry in an extended Bose-Hubbard model corresponding to total particle-number conservation] and local symmetries [$\mathrm{Z}_2$ and $\mathrm{U}(1)$ gauge symmetries corresponding to Gauss's law].
From these, several generic features emerge.
First, the symmetry violation---the expectation value of the symmetry generator---generically reveals a short-time crossover from diffusive to ballistic or even hyperballistic mean-square displacement across symmetry sectors. Second, for purely coherent errors interference effects can prevent the symmetry violation to reach its theoretical maximum, even in the long-time limit. Decoherence can lift these interference effects. As a consequence, the dynamics typically is dominated by decoherence at early and late times, while it is dominated by coherent errors in an intermediate time window. Third, the MQC is a powerful tool to reveal this complex interplay by quantifying the coherence between symmetry sectors encapsulated in the quantum state. Counterintuitively, we find situations where the addition of decoherence to coherent errors can increase the MQCs.
Our findings will be highly relevant for quantum simulation experiments on NISQ devices. They illuminate the general behavior with which the dynamics of a quantum many-body system under slight symmetry breaking deteriorates from the ideal model with intact conservation laws. These enable to estimate, e.g., target time scales that experimental technology needs to achieve in order to observe desired phenomena.
\section*{Acknowledgments}
This work is part of and supported by the Interdisciplinary Center Q@TN --- Quantum Science and Technologies at Trento, the DFG Collaborative Research Centre SFB 1225 (ISOQUANT), the Provincia Autonoma di Trento, and the ERC Starting Grant StrEnQTh (Project-ID 804305).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,730 |
Andrew William Pozzi (born 15 May 1992) is a British hurdling athlete. He was the 2018 indoor World Champion at 60 metres hurdles. He was the 2012 UK 60m and 110m Champion and holds the record for the fastest ever time run by a UK junior hurdler. The record time, 13.29 seconds, was set on 3 July 2011 in Mannheim, Germany at the Bauhaus Junior Gala.
Pozzi is coached by Malcolm Arnold, the former coach of Olympic silver medalist and two-time World Championship gold medalist Colin Jackson and 400m Olympic gold medalist John Akii-Bua.
On 26 November 2011, Pozzi won "Outstanding Athlete of the Year" at the UK Athletics Awards.
Background
Pozzi was born in Stratford-upon-Avon, Warwickshire on 15 May 1992. He attended St Gregory's Catholic Primary School in Stratford and later St Benedict's High School in neighbouring town Alcester. He then went on to complete his A-levels at Alcester Grammar School before attending the University of the West of England, Bristol. Since 2018 he has been in a relationship with fellow athlete Katarina Johnson-Thompson.
Junior career
Pozzi begun his career competing for Stratford AC, of which he is still a first-claim member. From 2010-11 he competed as a junior for 'Avon Spa', a joint effort between Stratford AC and Leamington C&AC in the National Junior Athletic League. Pozzi currently holds the record for the fastest ever time run by a UK junior hurdler (13.29s), setting the record at the Bauhaus Junior Gala in Manheim, Germany, on 3 July 2011.
He won a Silver medal at the 2011 European Junior Athletics Championships in Tallinn, Estonia with a time of 13.57s and immediately began focusing on making the transition from junior events to senior competitions.
He ended his junior career on a high, beating his senior rivals in the 110m hurdles to win the McCain UK Challenge Senior Final with a time of 13.84 – his second fastest wind-legal clocking over the senior hurdles, made all the more impressive given the -1.0 m/s headwind.
Senior career
Pozzi started life as a senior athlete well, winning his first 7 races and clocking a new personal best of 7.62 over 60m hurdles at the Birmingham Games. He went on to be crowned UK Indoor Champion on 12 February after winning at the Aviva European Indoor Trials & UK Championships in Sheffield, once again equalling his personal best.
On 18 February he competed at the Aviva Grand Prix against Olympic Gold medalist and World Champion Liu Xiang, and World Indoor Champion Dayron Robles. Pozzi performed well, finishing 7th in the final after automatically qualifying by coming 3rd in his heat with a time of 7.62.
Pozzi was selected to represent Great Britain at the 2012 World Indoor Championships in Istanbul, his first senior International representation. He was the only British male sprint hurdler selected. He surpassed expectations by winning his heat, beating Liu Xiang in the process, with a new personal best of 7.61s which also made him fastest qualifier. Pozzi continued to impress by finishing second in the semi-finals with another personal best of 7.56s making him the second fastest British 19-year-old ever, just 0.01s behind Colin Jackson's record. He finished 4th in the final with a time of 7.58s.
On 7 May 2012, Pozzi won gold at the BUCS championships at the Olympic Stadium, London. In the heats he achieved the Olympic "A" qualifying standard time of 13.52s. It was his first outdoor race of the season. He went on to win in the final with an even better time of 13.35s, making him the fastest European 19-year-old over 110m hurdles of all time.
London 2012 Summer Olympics
Pozzi was selected to compete for Team GB after winning the British Olympic Trials on 24 June 2012. Whilst competing at a Diamond League event in Crystal Palace on 13 July 2012 he became injured in the final but was set to recover in time for the Olympic games. By the date of his race on 7 August it was reported that Pozzi was not fully fit but would compete. After clearing the first hurdle it became clear that he had not recovered and immediately clutched his hamstring. He later said "The last month has been a nightmare. I haven't been able to get my hamstring sorted. Being in the Olympics is all I've been waiting and training so hard for. To leave like that is heart‑breaking."
Birmingham 2018 World Indoor Athletic Championship
Pozzi qualified for finals with a terrific run in both heats and semi-finals. In the finals, he edged Jarret Eaton of the USA by one-hundredth of a second to clinch the gold medal with a time of 7.46 sec.
Personal bests
References
External links
1992 births
Living people
Sportspeople from Warwickshire
People from Stratford-upon-Avon
English male hurdlers
British male hurdlers
Olympic male hurdlers
Olympic athletes of Great Britain
Athletes (track and field) at the 2012 Summer Olympics
Athletes (track and field) at the 2016 Summer Olympics
Commonwealth Games competitors for England
Athletes (track and field) at the 2018 Commonwealth Games
World Athletics Championships athletes for Great Britain
World Athletics Indoor Championships winners
European Athletics Indoor Championships winners
British Athletics Championships winners
Team Bath track and field athletes
People educated at Alcester Grammar School
English people of Italian descent
Athletes (track and field) at the 2020 Summer Olympics
Commonwealth Games bronze medallists for England
Commonwealth Games medallists in athletics
Athletes (track and field) at the 2022 Commonwealth Games
Medallists at the 2022 Commonwealth Games | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,964 |
An especially large giant clam. Local fisher estimates put the age of this wavekin at over five score years.
After overhearing a drunk fisher claim to his equally drunk shipmate that one of the massive mollusks gnawed the leg (or was it head?) off his infant son, a newly instated Yellowjacket sergeant looking to impress his superiors took it upon himself to order his men to locate and destroy as many of the wavekin as possible, leaving the population decimated. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
\section{Introduction}
If string theory is indeed the correct ultra-violet completion of Quantum Field Theory in presence of gravity, it must be possible to derive the observed Standard Model of Particle Physics as one of the consistent four-dimensional vacuum solutions to the string equations of motion.
A very efficient approach to embedding the Standard Model into string theory is via Grand Unified Theories (GUTs) of particle physics.
In this context, exceptional symmetry based on the Lie groups $E_6$, $E_7$ or $E_8$ is known to play a distinguished role. In fact, the success of particle physics model building in heterotic $E_8 \times E_8$ string theory, recent examples being \cite{Bouchard:2005ag,Braun:2005nv,Lebedev:2006kn,Anderson:2011ns} and references therein,
is largely due to the ease with which exceptional symmetry arises here.
In string vacua with branes, exceptional symmetry requires non-perturbative ingredients such as $[p,q]$-strings in strongly coupled Type IIB theory.
In order to extend the GUT paradigm to string theories with branes one therefore has to leave the perturbative regime. Indeed
considerable effort has been invested recently into exploiting such non-perturbative effects in the context of GUT phenomenology in F-theory as initiated in \cite{Donagi:2008ca,oai:arXiv.org:0802.3391,Beasley:2008kw,Donagi:2008kj} (see e.g the reviews \cite{Heckman:2010bq,Weigand:2010wm,Maharana:2012tu} for more references).
On the other hand, in string compactifications with branes a rather different route towards the Standard Model suggests itself which directly yields the observed gauge group $SU(3) \times SU(2) \times U(1)_Y$ without any detour via a GUT group such as $SU(5)$ or $SO(10)$. This approach has traditionally been pursued in a perturbative context with intersecting D-branes in Type IIA or Type IIB orientifold theories \cite{Blumenhagen:2000wh,Aldazabal:2000cn,
Cvetic:2001nr}, where stacks of multiple D-branes in general position give rise to gauge group $U(N)$ and thus the building block of the Standard Model gauge group.
The apparent unification of the gauge couplings, which is particularly well motivated in models with low-energy supersymmetry, translates into a relation on the volume of the D-branes on which the Standard Model is realised \cite{Blumenhagen:2003jy}.
The construction of Standard-like models via such intersecting branes in Type II orientifolds has been pursued in a rich literature reviewed for instance in \cite{Lust:2004ks,Blumenhagen:2006ci,Marchesano:2007de,Cvetic:2011vz,Ibanez:2012zz}.\footnote{For example, most recently
Standard-like vacua have been constructed in toroidal orientifolds in \cite{Honecker:2012qr,Honecker:2012jd} and in RCFT orientifolds in \cite{Dijkstra:2004cc,Anastasopoulos:2006da}. See \cite{Dolan:2011qu,Cicoli:2012vw} and references therein for recent advances in realistic model building with branes at singularities.}
A distinctive organising principle for the couplings between the charged particles of such brane vacua is provided by abelian selection rules.
In perturbative Type II orientifolds, as pointed out already, the gauge group on a stack of $N$ multiple branes is $U(N) = SU(N) \times U(1)/{\mathbb Z_N}$. The diagonal abelian gauge group factor typically receives
a St\"uckelberg mass induced by the coupling to the closed string axions \cite{Ibanez:1998qp}, but remains as a perturbative selection rule (`massive $U(1)$') constraining the structure of perturbatively allowed couplings.
These selection rules can only be broken non-perturbatively by D-brane instantons \cite{Blumenhagen:2006xt,Ibanez:2006da,Haack:2006cy,Florea:2006si,Blumenhagen:2009qh}, oftentimes to an exact discrete $\mathbb Z_k$ symmetry \cite{BerasaluceGonzalez:2011wy,BerasaluceGonzalez:2012vb,Ibanez:2012wg,Anastasopoulos:2012zu,Honecker:2013hda}.
Such selection rules can be a curse or a blessing:
As one of their advantages, they can forbid undesirable, dangerous couplings, e.g. interactions that would induce unacceptably rapid proton decay.
In this sense, for instance the origin of baryon or lepton number in the Standard Model can be traced to perturbatively exact symmetries from a string theory perspective.
Similarly, extra $U(1)$ symmetries may conspire to perturbatively forbid hierarchically suppressed couplings, which in turn are generated only non-perturbatively, thereby explaining their smallness. A systematic analysis of the phenomenology of such effects in perturbative MSSM quivers can be found in \cite{Ibanez:2008my,Cvetic:2009yh,Cvetic:2009ez}.
On the other hand, extra $U(1)$ symmetries can also forbid desirable couplings. Consider a realisation of the Standard Model via
brane stacks of the form $U(3)_a \times U(2)_b \times U(1)_c$ plus possible extra $U(1)$ branes.
The left-handed quark $Q$ resides in representation $({\bf 3},{\bf 2})_{1_a, -1_b}$ (with subscripts denoting the $U(1)$ charges) and the up-type Higgs $H_u$, say, in representation $({\bf 1},{\bf 2})_{1_b,-1_c}$. If the right-handed up-quarks $u_R^c$ are realised as the antisymmetric representation of $U(3)$, i.e. $u_R^c = ({\bf \overline 3}, {\bf 1})_{-2_a}$,
the $U(1)$ charges prohibit a perturbative Yukawa coupling $Q \, u_R^c \, H_u$ even though this coupling is allowed by the Standard Model gauge symmetry itself.
This example is similar to the well-known \cite{Blumenhagen:2001te,Blumenhagen:2007zk} absence of a perturbative coupling ${\bf 10} \, {\bf 10} \, {\bf 5} $ in attempts to realise an $SU(5)$ GUT with perturbative intersecting branes via $U(5)_a \times U(1)_b$.
This problem is beautifully solved in F-theory models, where such a coupling arises at a non-perturbative point of $E_6$ enhancement \cite{Donagi:2008ca,oai:arXiv.org:0802.3391,Beasley:2008kw,Donagi:2008kj}.
Such enhancement points signal that the brane configuration is not smoothly connected in moduli space to a well-defined Type IIB limit \cite{Donagi:2009ra,Krause:2012yh} and one should think of the matter states as arising from multi-pronged $[p,q]$-strings, which are perturbatively absent.
Equivalently, the underlying reason for existence of a ${\bf 10} \, {\bf 10} \, {\bf 5}$ coupling in F-theory is that the structure of $U(1)$ gauge symmetries in F-theory can be much more general than in the subclass of perturbative Type IIB models.
Indeed, in a more general F-theory model without a smooth Type IIB limit, $SU(N)$ gauge symmetries are not necessarily accompanied by extra diagonal $U(1)$ factors.\footnote{See, however, \cite{Grimm:2010ez,Grimm:2011tb,Braun:2014nva,Anderson:2014yva} for the description of such overall massive $U(1)$s in F-theory.} The structure of abelian gauge symmetries in F-theory is a particularly rich topic with beautiful connections to algebraic geometry and has been investigated in great detail in the very recent literature \cite{Grimm:2010ez,Braun:2011zm,Krause:2011xj,Grimm:2011fx,oai:arXiv.org:1202.3138,Morrison:2012ei,oai:arXiv.org:1210.6034,Mayrhofer:2012zy,Braun:2013yti,Borchmann:2013jwa,Cvetic:2013nia,Braun:2013nqa,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta,Cvetic:2013qsa,Morrison:2014era,Martini:2014iza,Bizet:2014uua}, motivated in part by the need for extra $U(1)$ selection rules in the context of F-theory GUT model building \cite{Marsano:2009wr,Hayashi:2010zp,Dolan:2011iu,Dolan:2011aq,Marsano:2011nn,
Maharana:2012tu,Krippendorf:2014xba}.
Given the crucial role of $U(1)$ selection rules for Standard Model building on the one hand and the striking differences between the structure of $U(1)$ selection rules in perturbative and non-perturbative models on the other hand,
it is an obvious question to what extent direct, non-GUT realisations of the Standard Model in F-theory differ from their perturbative, well-studied counter-parts.
Motivated by our interest in extending our knowledge of realistic vacua to new, unexplored regions of the landscape,
we therefore investigate in this work the possibilities of directly constructing string vacua with Standard Model gauge group and matter representations in F-theory.
For phenomenological reasons we focus on string vacua which in addition to the Standard Model gauge group allow for one abelian group factor. This extra $U(1)$ will eventually acquire a St\"uckelberg mass term upon switching on gauge flux (as required to generate a chiral spectrum) and remain as a perturbative selection rule, which will help avoid in principle dangerous dimension-four (and higher) proton decay operators.
Our approach is therefore to construct elliptic fibrations for F-theory compactifications with gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ such that one linear combination of $U(1)_1$ and $U(1)_2$ will play the role of hypercharge $U(1)_Y$ and stay exactly massless even in presence of gauge fluxes.
To this end we start with an elliptic fibration with gauge group $U(1)_1 \times U(1)_2$ and further constrain the complex structure moduli such as to create the non-abelian gauge group factors $SU(3) \times SU(2)$.
Elliptic fibrations with gauge group $U(1)_1 \times U(1)_2$ have been analysed in detail in \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta}. In section \ref{sec_U1U1} we summarise their main properties with special emphasis on the structure of Yukawa couplings among the charged singlet states.
We then analyse the subclass of all possible gauge group enhancements to $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ which can be achieved torically. This amounts to studying the $SU(2)$ and $SU(3)$ tops \cite{Candelas:1996su,Candelas:1997eh} over the $U(1)_1 \times U(1)_2$-fibrations, which generate the corresponding singularities in the fibre over two independent divisors $W_2$ and $W_3$.
The construction of the Standard Model gauge group in F-theory has previously been approached in \cite{Choi:2013hua,Choi:2010su,Choi:2010nf} via a geometric deformation of an $SU(5)$ singularity along a single divisor such that the $SU(3)$ and $SU(2)$ singularities arise over homologous divisors. This is different to our approach, where both divisors are generically unrelated.
For our $U(1)_1 \times U(1)_2$ fibration, there are three tops for $SU(2)$ and $SU(3)$ each \cite{Bouchard:2003bu}. In sections \ref{sec:SU(2)-tops} and \ref{sec:SU(3)-tops} we compute in turn the structure of matter curves and the geometrically realised Yukawa couplings in models with an extra $SU(2)$ and $SU(3)$ gauge symmetry, respectively. We exemplify our computations for one of the two possible tops and relegate the details of the remaining analysis to appendices \ref{SU2DetailsApp} and \ref{app-SU3}.
In section \ref{sec_3211} we combine these results into a single elliptic fibration with gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$. Of the a priori $3 \times 3$ resulting types of fibrations, only five turn out to be inequivalent. In each of the five inequivalent fibrations,
the structure of matter fields charged only under $SU(3)$ or $SU(2)$ (and/or the abelian gauge groups) carries over, but a new field arises at the intersection of the two non-abelian stacks. In generic situations, to which we restrict ourselves, this field transforms in the $({\bf 3}, {\bf 2})$-representation and plays the role of the left-handed quarks in the Standard Model. We analyse all possible Yukawa couplings involving this state.
Interestingly, one of the couplings turns out to correspond to a non-standard Kodaira fibre.
Our approach is to analyse the elliptic fibre in full generality. For a sufficiently generic base space ${\cal B}$, these fibrations define a smooth elliptically fibred Calabi-Yau fourfold suitable for F-theory compactification.
In section \ref{sec_SM}, we match the various matter representations of the five inequivalent types of fibrations with the Standard Model fields, working in the context of the MSSM with extra singlets and an a priori unspecified supersymmetry breaking scale.
Since the fibrations under consideration give rise to five different types of $({\bf 3},{\bf 1})$-fields and three different classes of $({\bf 1},{\bf 2})$-fields, each localised on a different matter curve and with different $U(1)$ charges, a plethora of possible identifications exists, which we list in appendix \ref{appsec:huge_table}.
In principle, different generations of matter can be distributed over different curves. This way some of the generations may enjoy a perturbative Yukawa coupling, while others remain perturbatively massless.
An important distinctive property of the so-obtained MSSM candidate setups is the spectrum of perturbatively allowed dimension-four and -five couplings. We list all possible such couplings.
For a low supersymmetry breaking scale, certain combinations of these couplings lead to unacceptable proton-decay, while in scenarios with intermediate or high-scale supersymmetry breaking the constraints are more relaxed.
With an eye on the possibility of intermediate scale supersymmetry breaking, we do not exclude any models based on such dimension-four and -five couplings. A more detailed phenomenological assessment will appear in future work.
The philosophy behind our classification of `toric Standard Models' is that gauge fluxes ensure the correct spectrum of MSSM matter. In particular, the remaining fundamental and singlet fields which are exotic states from an MSSM perspective are assumed to be absent at the massless level by virtue of a suitable choice of fluxes.
In section \ref{sec_fluxes} we summarise the constraints on these fluxes, especially from the requirement that hypercharge remain massless, leaving a more systematic treatment of gauge fluxes for the future. Such an analysis will be required to determine which of the Standard Model fibrations can also encompass the physically required number of zero-modes for the various fields, in particular in which cases no chiral exotics are forced upon us.
We conclude in section \ref{sec_Conclusions} with an outlook and open questions.
\section[F-theory with \texorpdfstring{\boldmath $U(1) \times U(1)$}{U(1) x U(1)} Gauge Group]{F-theory with \texorpdfstring{\boldmath $U(1) \times U(1)$}{U(1) x U(1)} Gauge Group} \label{sec_U1U1}
We are interested in engineering F-theory vacua with Standard Model gauge group $SU(3) \times SU(2) \times U(1)_Y$ and one additional abelian gauge group factor.
Our starting point are therefore elliptic fibrations which allow for two abelian gauge groups $U(1)_1$ and $U(1)_2$; a further restriction of the complex structure of such fibrations will then induce the non-abelian factors $SU(3) \times SU(2)$. One linear combination of $U(1)_1$ and $U(1)_2$ will correspond to $U(1)_Y$, while the remaining combination must
become massive by a flux-induced St\"uckelberg mechanism and act as an extra selection rule on the couplings of the model.
F-theory compactifications with two abelian gauge groups are based on elliptic fibrations with Mordell-Weil group of rank two. Such elliptic fibrations allow for a description as the vanishing locus of the hypersurface equation \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta}
\begin{align}
\label{eq:hypersurface-equation}
\begin{split}
P_T = &\mathrm{v} \, \mathrm{w} (c_1 \, \mathrm{w} \, s_1 + c_2 \, \mathrm{v} \, s_0) + \mathrm{u} \, (b_0 \, \mathrm{v}^2 \, s_0^2 + b_1 \, \mathrm{v} \, \mathrm{w} \, s_0 \, s_1 + b_2 \, \mathrm{w}^2 \, s_1^2) + \\
&\mathrm{u}^2 (d_0 \, \mathrm{v} \, s_0^2 \, s_1 + d_1 \, \mathrm{w} \, s_0 \, s_1^2 + d_2 \, \mathrm{u} \, s_0^2 \, s_1^2)
\end{split}
\end{align}
inside a $\text{Bl}_2 \mathbb{P}^2$-fibration. Such types of fibration had previously been considered also in \cite{Klemm:1996hh}.
The ambient space $\text{Bl}_2 \mathbb{P}^2$ of the elliptic fibre is a toric space which has the toric diagram represented by polygon 5 in the classification \cite{Bouchard:2003bu} (see also figure \ref{fig:base_polygon} in the appendix).
Here and in the sequel we will stick to the notation of \cite{Borchmann:2013jwa,Borchmann:2013hta} and denote the coordinates of $\text{Bl}_2 \mathbb{P}^2$ by $\mathrm{u},\mathrm{v},\mathrm{w},s_0,s_1$, where $s_0$ and $s_1$ correspond to two blow-up $\mathbb{P}^1$s inside $\mathbb P^2$ with homogeneous coordinates $[\mathrm{u} : \mathrm{w} : \mathrm{v}]$. The Stanley-Reisner ideal is generated by $\{ \mathrm{u}\,\mathrm{v} , \mathrm{u}\,\mathrm{w} , s_0\,\mathrm{w}, s_1\,\mathrm{v}, s_0 \, s_1 \}$ and the divisor classes associated with the fibre ambient space coordinates are given as follows:
\begin{align}\label{tab:divisor-classes-small}
\begin{array}{c|ccccc}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 \\ \hline
{U} & 1 & 1 & 1 & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1
\end{array}
\end{align}
The quantities $b_i, \, c_j, \, d_k$ transform as sections of certain line bundles over the base ${\cal B}$ of the fibration, whose class is determined by the requirement that the fibration is Calabi-Yau. These classes are collected in table \ref{coeff} (taken from \cite{Borchmann:2013hta}; see also \cite{Cvetic:2013nia,Cvetic:2013uta,Cvetic:2013jta}).
\begin{table}[t]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c
$ b_{0}$ & $ b_{1} $ & $b_{2}$ & $ c_{1} $ & $ c_{2} $&$d_{0}$&$d_{1}$&$d_{2}$\\
\hline
$\alpha-\beta+ \bar {\cal K}$&$\bar {\cal K}$&$-\alpha +\beta+\bar {\cal K}$&$-\alpha+\bar {\cal K}$&$-\beta + \bar {\cal K}$&$\alpha + \bar {\cal K}$&$\beta+\bar {\cal K}$&$\alpha+\beta+\bar {\cal K}$\\
\end{tabular}
\caption{Classes of the sections appearing in (\ref{eq:hypersurface-equation}) for $\alpha$ and $\beta$ pullback of `arbitrary' classes of ${\cal B}$ and $\overline{\mathcal{K}} = \pi^{-1} \overline{\mathcal{K}}_{\cal B}$.}\label{coeff}
\end{table}
For suitable 3-dimensional base spaces ${\cal B}$ the hypersurface (\ref{eq:hypersurface-equation}) then describes a smooth elliptically fibred Calabi-Yau 4-fold
\begin{eqnarray}
\pi: Y_4 \rightarrow {\cal B}.
\end{eqnarray}
It exhibits three independent rational sections
\begin{eqnarray}
U = \{u\}, \qquad S_0=\{s_0\}, \qquad S_1= \{s_1\}.
\end{eqnarray}
Throughout the article we use, unless stated otherwise, $\{ \ldots \}$ as a short-hand notation for $\{ \ldots = 0 \}$.
One of these sections, e.g. $S_0$, can be interpreted as the zero-section.\footnote{The fact that all three independent sections are rational (as opposed to holomorphic) is an artefact of the representation of the fibration as a hypersurface. Indeed, the fibration is birationally equivalent to a complete intersection which does exhibit a
holomorphic zero-section \cite{Borchmann:2013hta}. The pre-image of this section under the birational map can be identified with the zero section \cite{Borchmann:2013jwa,Borchmann:2013hta}. Alternatively, one can define an F-theory compactification with a rational zero-section as in \cite{Cvetic:2013nia,Grimm:2013oga,Cvetic:2013uta}.}
The image of the remaining two independent sections under the Shioda map \cite{Shioda:1989,Wazir:2001,Park:2011ji} then identifies the generators of two independent $U(1)$ gauge groups as \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta}
\begin{align}
\label{eq:U(1)-generators}
\begin{split}
\omega_1 &= S_1 - S_0 - \overline{\mathcal{K}}, \\
\omega_2 &= U - S_0 - \overline{\mathcal{K}} - [ c_1 ],
\end{split}
\end{align}
where $\overline{\mathcal{K}} = \pi^{-1} \overline{\mathcal{K}}_{\cal B}$ is the pre-image of the anti-canonical bundle of the base ${\cal B}$. Here and in the sequel our notation does not distinguish between a divisor (class) and its dual 2-form.
In our analysis of the matter representations of F-theory compactified on $Y_4$ we will also need the form of the singular Weierstrass model which is birationally equivalent to the blow-down of the hypersurface (\ref{eq:hypersurface-equation}) (i.e. the hypersurface inside the $\mathbb{P}^2$-fibration over ${\cal B}$ achieved by setting $s_0 = s_1 =1$ in (\ref{eq:hypersurface-equation})).
The explicit map of this blow-down to Weierstrass form
\begin{eqnarray} \label{Weier}
y^2 = x^3 + f \, x \, z^4 + g \, z^6
\end{eqnarray}
has been worked out in the physics literature in \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta}.
In our subsequent analysis we will make use of the expression for the Weierstrass sections $f$ and $g$ in terms of the defining sections $b_i, c_j, d_k$ appearing in (\ref{eq:hypersurface-equation}) as given after equation (2.38) and (2.39) in \cite{Borchmann:2013jwa}, which we recall here for completeness,
\begin{equation} \label{sec-map1}
f=-\tfrac13\T d^2 + \T c\, \T e\qquad\textmd{and}\qquad g=- f\,\left(\tfrac13\T d\right) -\left(\tfrac{1}{3}\T d\right)^3 + \T c^2\, \T k,
\end{equation}
where
\begin{equation}
\begin{split} \label{sec-map2}
\T d &=b_1^2 + 8\,b_0\,b_2 - 4\,c_1\,d_0 - 4\,c_2\,d_1,\\
\T c &=-\frac{4}{c_1}(b_0\,b_2^2 - b_2\,c_1\,d_0 + c_1^2\,d_2),\\
\T e &=\frac{2 c_1 \left( b_0 \left( b_1 c_1 d_1-b_1^2 b_2+2 b_2 c_1 d_0+2 b_2 c_2
d_1-2 c_1^2 d_2\right)\right)}{ b_0 b_2^2+ c_1 ( c_1 d_2- b_2 d_0)}+\\&\qquad\qquad+\frac{2 c_1 \left(-2 b_0^2 b_2^2+ c_2 ( b_1 b_2 d_0+ b_1
c_1 d_2-2 b_2 c_2 d_2-2 c_1 d_0
d_1)\right)}{ b_0 b_2^2+ c_1 ( c_1 d_2- b_2 d_0)},\\
\T k &=\frac{ c_1^2 ( b_0 b_1 b_2- b_0 c_1 d_1- b_2 c_2
d_0+ c_1 c_2 d_2)^2}{\left( b_0 b_2^2+ c_1 ( c_1
d_2- b_2 d_0)\right)^2}.
\end{split}
\end{equation}
\subsection{Charged Singlets and Their Yukawa Couplings}\label{sec:singlets-prime-ideals}
The fibration gives rise to six types of charged singlet states $ \mathbf{1}^{(i)}$ localised on curves on ${\cal B}$, which have already been analysed in \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta}. Here we continue with the analysis of \cite{Borchmann:2013jwa,Borchmann:2013hta} (alternatively, see \cite{Cvetic:2013nia,Cvetic:2013uta}), which derives the curves as loci in the base over which the fibre of the blown-down version of (\ref{eq:hypersurface-equation}) exhibits a conifold singularity. These loci are given as the union of the set of solutions to each of the following three pairs of equations,
\begin{equation}\label{eq:singlet-locus-1}
\begin{split}
0 & = d_0\, c_2^2 + b_0^2 \,c_1 - b_0\, b_1\, c_2\,,\\
0 & = d_1\, b_0\, c_2 - b_0^2\, b_2 - c_2^2\, d_2 \, ,
\end{split}
\end{equation}
and
\begin{equation}\label{eq:singlet-locus-2}
\begin{split}
0 & = d_0\, b_2\, c_1 - b_0\, b_2^2 - c_1^2\, d_2\,,\\
0 & = d_1\, c_1^2 - b_1\, b_2\, c_1 + b_2^2\, c_2\,,
\end{split}
\end{equation}
and
\begin{equation}\label{eq:singlet-locus-3}
\begin{split}
0 & = d_0\, c_1^3\, c_2^2 + b_0^2\, c_1^4 - b_0\, b_1\, c_1^3\, c_2 -
c_2^3 (b_2^2\, c_2 -b_1\, b_2\, c_1 + c_1^2 d_1)\,,\\
0 & = d_2\, c_1^4\, c_2^2 + (b_0\, c_1^2 + c_2\, (-b_1\, c_1 + b_2\, c_2)) (b_0\, b_2\, c_1^2 + c_2 (-b_1\, b_2\, c_1 + b_2^2\, c_2 + c_1^2\, d_1))\,.
\end{split}
\end{equation}
In \cite{Borchmann:2013jwa,Borchmann:2013hta} three singlet curves were identified as complete intersections: $C^{(1)} = \{b_0\} \cap \{c_2\}$ solves both (\ref{eq:singlet-locus-1}) and (\ref{eq:singlet-locus-3}), $C^{(3)} = \{b_2\} \cap \{ c_1\}$ solves (\ref{eq:singlet-locus-2}) and (\ref{eq:singlet-locus-3}), and $C^{(5)} = \{c_1\} \cap \{ c_2\}$ solves (\ref{eq:singlet-locus-3}). If one inserts these equations into the hypersurface equation (\ref{eq:hypersurface-equation}) one confirms that the fibre factorises into two $\mathbb P^1$s and can identify the singlet states as M2-branes wrapping one of the fibre components. The remaining three curves were represented in \cite{Borchmann:2013jwa,Borchmann:2013hta} as (\ref{eq:singlet-locus-1}) with $b_0 \neq 0 \neq c_2$ ($C^{(2)}$), (\ref{eq:singlet-locus-2}) with $b_2 \neq 0 \neq c_1$ ($C^{(4)}$), and (\ref{eq:singlet-locus-3}) with $b_0 \neq 0 \neq b_2$ and $c_1 \neq 0 \neq c_2$ ($C^{(6)}$). Plugging these more lengthy expressions into the hypersurface
equation also leads to a factorisation of the fibre, i.e.~the appearance of charged singlets. Their location and charges are summarised as follows:
\begin{align}
\label{tab:singlets-charges}
\begin{array}{c|c| r @{} l }
\multirow{2}{*}{\text{singlet}} & \multirow{2}{*}{\text{locus}} & \multicolumn{2}{c}{(U(1)_1, U(1)_2) \text{-}} \\
& & \multicolumn{2}{c}{\text{charges}} \\ \hline \rule{0pt}{3ex}
\mathbf{1}^{(1)} / \overline{\mathbf{1}}^{(1)} & \{b_0\} \cap \{c_2 \} & (1,-1) &/ (-1 ,1) \\
\mathbf{1}^{(2)} / \overline{\mathbf{1}}^{(2)} & C^{(2)} & (1, 0) &/ (-1 , 0) \\
\mathbf{1}^{(3)} / \overline{\mathbf{1}}^{(3)} & \{b_2\} \cap \{c_1 \} & (1,2) &/ (-1,-2) \\
\mathbf{1}^{(4)} / \overline{\mathbf{1}}^{(4)} & C^{(4)} & (1, 1) &/ (-1,-1) \\
\mathbf{1}^{(5)} / \overline{\mathbf{1}}^{(5)} & \{c_1\} \cap \{c_2 \} & (0,2) &/ (0,-2) \\
\mathbf{1}^{(6)} / \overline{\mathbf{1}}^{(6)} & C^{(6)} & (0, 1) &/ (0,-1)
\end{array}
\end{align}
From the charge assignment we expect six types of Yukawa couplings. Of these, the couplings $\mathbf{1}^{(1)} \, \overline{\mathbf{1}}^{(4)} \, \mathbf{1}^{(5)}$ over $b_0 = c_1 = c_2 = 0$, $\mathbf{1}^{(2)} \, \overline{\mathbf{1}}^{(3)} \, \mathbf{1}^{(5)}$ over $b_2 = c_1 = c_2 = 0$, and $\mathbf{1}^{(2)} \, \overline{\mathbf{1}}^{(4)} \, \mathbf{1}^{(6)}$ over $C^{(2)} \cap C^{(4)} \cap C^{(6)}$ can in fact be directly seen \cite{Borchmann:2013jwa, Cvetic:2013nia} when plugging in the corresponding equations of the curves into the hypersurface equation. However the charges also allow for couplings $\mathbf{1}^{(1)} \, \overline{\mathbf{1}}^{(2)} \, \mathbf{1}^{(6)}$, $\overline{\mathbf{1}}^{(3)} \, \mathbf{1}^{(4)} \, \mathbf{1}^{(6)}$ and $\overline{\mathbf{1}}^{(5)} \, \mathbf{1}^{(6)} \, \mathbf{1}^{(6)}$, which due to the form of the curves $C^{(2)}$, $C^{(4)}$, $C^{(6)}$ are more complicated to analyse. The difficulty is that the set of solutions to equations (\ref{eq:singlet-locus-1}) to (\ref{eq:singlet-locus-3}) consists of several irreducible components which intersect each other precisely at the interesting Yukawa points. To find an appropriate form of the singlet curves, we apply a classic method in algebraic geometry (e.g.~\cite{hartshorne:alggeo,cox:alggeo,cox:ideals}), which in the context of F-theory has been first presented in \cite{Cvetic:2013uta,Cvetic:2013jta} (`technique using prime ideals', see also \cite{Piragua}).
The general idea is that the expressions on the right-hand sides of equations (\ref{eq:singlet-locus-1}) -- (\ref{eq:singlet-locus-3}) are elements of the polynomial ring $R = \mathbb{C}[b_i,c_j,d_k]$. If we formally treat the sections $b_i,c_j,d_k$ as independent variables of these polynomials, then basic algebraic geometry tells us that the common zero locus $V(\{f_n\})$ of a set of polynomials $f_n \in R$ is the same as the common zero locus of the ideal generated by $\{f_n\}$, $V(\{f_n\}) = V(\langle \{ f_n \} \rangle)$. Intersections of zero loci are described by the formula $V(I_1) \cap V(I_2) = V(I_1 + I_2)$, where $I_1 + I_2$ is the sum of ideals. Each (proper) ideal $I \varsubsetneq R$ has a so-called primary decomposition $I = \bigcap_{i=1}^{r<\infty} J_i$, where $J_i$ are termed primary ideals; by definition, the radical $\sqrt{J_i}$ is a prime ideal (in fact the smallest containing $J_i$), more precisely it is the (minimal) associated prime ideal. Translated into geometry this means that the vanishing locus is
decomposed into components $V(I) = \bigcup_{i=1}^{r<\infty} V(J_i) = \bigcup_{i=1}^{r<\infty} V(\sqrt{J_i})$, where the last equality is a consequence of the famous `Hilbert's Nullstellensatz' (see e.g.~\cite{cox:ideals}). The components $V(\sqrt{J_i})$ are precisely the irreducible components of $V(I)$. Furthermore, one can compute the codimension of $V(I)$ algebraically via the so-called Krull dimension $\dim_K$ of the quotient ring $R/I$: $\text{codim} V(I) = \dim_K R - \dim_K R/I$.
After this short mathematical interlude, the procedure to find the singlet curves and also the Yukawa points becomes clear: The right-hand sides of equations (\ref{eq:singlet-locus-1}), (\ref{eq:singlet-locus-2}) and (\ref{eq:singlet-locus-3}) define three ideals generated by two polynomials. Their associated prime ideals with such Krull dimension that their vanishing locus has codimension two correspond to curves in the three-dimensional base -- the matter curves. Yukawa points arise at intersections of three matter curves; correspondingly we have to form the sum of prime ideals associated to the matter curves involved. In general, when we compute the associated prime ideals of such a sum and calculate their Krull dimension, we will find that the intersection locus has a number of irreducible components with different codimensions. On a generic three-dimensional base, Yukawa points correspond to the codimension-three components, while all higher-codimension components are absent; hence if we compute the
intersection of
certain curves and only find codimension-four or higher components, we conclude that these curves cannot meet in a generic three-dimensional base and form Yukawa couplings.
We have carried out the calculations using the computer algebra system \textsc{Singular}\footnote{We would like to thank Hernan Piragua for introducing us to the `prime ideal technique' in \textsc{Singular}. See also \cite{Piragua}.} \cite{singular} and indeed find the six associated prime ideals listed in \cite{Cvetic:2013uta,Cvetic:2013jta}. Three of them have only two generators, $I^{(1)} = \langle b_0, c_2 \rangle$, $I^{(3)} = \langle b_2 , c_1 \rangle$, $I^{(5)} = \langle c_1 , c_2 \rangle$; their corresponding vanishing loci are the complete intersection curves listed in (\ref{tab:singlets-charges}). The other three associated prime ideals $I^{(2)}$, $I^{(4)}$ and $I^{(6)}$, which are associated prime ideals of (\ref{eq:singlet-locus-1}), (\ref{eq:singlet-locus-2}) and (\ref{eq:singlet-locus-3}), respectively, now correspond to the curves $C^{(2)}$, $C^{(4)}$ and $C^{(6)}$ in our previous notation. They have more generators, which are quite lengthy polynomial expressions in $b_i, c_j, d_k$; our
findings for their explicit form
coincide with the results presented in \cite{Cvetic:2013uta,Cvetic:2013jta}.
With this technique we can now analyse the Yukawa couplings $\mathbf{1}^{(1)} \, \overline{\mathbf{1}}^{(2)} \, \mathbf{1}^{(6)}$, $\overline{\mathbf{1}}^{(3)} \, \mathbf{1}^{(4)} \, \mathbf{1}^{(6)}$ and $\overline{\mathbf{1}}^{(5)} \, \mathbf{1}^{(6)} \, \mathbf{1}^{(6)}$. To this end we first calculate the associated prime ideals of the sum of the ideals corresponding to the curves. In each case we indeed find one prime ideal corresponding to a codimension-three zero locus, confirming the existence of the intersection points of those triplets of singlet curves. All three codimension-three intersection loci are in fact complete intersections,
\begin{align}
\begin{split}\label{eq:intersection-point1}
& V(I^{(1)}) \cap V(I^{(2)}) \cap V(I^{(6)}) = \\
& \{b_0 \} \cap \{c_2\} \cap \{b_2^2\,d_0^2-b_1\,b_2\,d_0\,d_1+c_1\,d_0\,d_1^2+b_1^2\,b_2\,d_2-2\,b_2\,c_1\,d_0\,d_2-b_1\,c_1\,d_1\,d_2+c_1^2\,d_2^2\} \, ,
\end{split} \\
\begin{split}\label{eq:intersection-point2}
& V(I^{(3)}) \cap V(I^{(4)}) \cap V(I^{(6)}) = \\
& \{b_2\} \cap \{c_1\} \cap \{ b_0^2 \, d_1^2 - b_0\,b_1\,d_0\,d_1 + c_2\,d_1\,d_0^2 + b_0\,b_1^2\,d_2 - b_1\,c_2\,d_0\,d_2 - 2\,b_0\,c_2\,d_1\,d_2 + c_2^2\,d_2^2 \} \, ,
\end{split}\\
\begin{split}\label{eq:intersection-point3}
& V(I^{(5)}) \cap V(I^{(6)}) \cap V(I^{(6)}) = \\
& \{c_1 \} \cap \{c_2\} \cap \{b_1\,d_0\,d_1-b_2\,d_0^2-b_0\,d_1^2-b_1^2\,d_2+4\,b_0\,b_2\,d_2\} \, .
\end{split}
\end{align}
Interestingly, the last set of Yukawa points (\ref{eq:intersection-point3}) coincides with the singular locus of $V(I^{(6)})$. Due to the complicated form of $V(I^{(6)})$ (it has 39 generators which themselves are complicated polynomials), we have not determined the type of the singularity, but the form of the Yukawa coupling involving two $\mathbf{1}^{(6)}$-states suggests that it is a point of self-intersection of the $\mathbf{1}^{(6)}$-curve where also the $\mathbf{1}^{(5)}$-curve passes through.
The final proof for the existence of the Yukawa couplings comes by inspecting the fibre over the intersection points. The couplings $\mathbf{1}^{(1)} \, \overline{\mathbf{1}}^{(2)} \, \mathbf{1}^{(6)}$ and $\overline{\mathbf{1}}^{(3)} \, \mathbf{1}^{(4)} \, \mathbf{1}^{(6)}$ have already been argued to exist geometrically in \cite{Cvetic:2013uta} using the `prime ideal technique', and independently in \cite{Borchmann:2013hta} in an indirect manner by exploiting their formal relation to the chiral index of certain $G_4$-fluxes.
Here we therefore focus on the remaining $\overline{\mathbf{1}}^{(5)} \, \mathbf{1}^{(6)} \, \mathbf{1}^{(6)}$ coupling. If we solve the last equation in (\ref{eq:intersection-point3}) for $b_1=(d_0 \, d_1 \pm \sqrt{d_0^2 - 4 \, b_0 \, d_2} \, \sqrt{d_1^2 - 4 \, b_2 \, d_2})/(2 \, d_2)$, we see that the complete intersection locus (\ref{eq:intersection-point3}) really consists of two sets of points defined by each sign. Note that as far as codimension-three loci are concerned the appearance of the square root or of $d_2$ in the denominator does not pose any problems.
Plugging this together with $c_1 = c_2 =0$ into the hypersurface equation (\ref{eq:hypersurface-equation}) yields, after some tedious algebra, the factorisation
\begin{align}\label{eq:factorisation-singlet-yukawa}
\begin{split}
&P_T |_{c_1=c_2=0 , \, b_1=(d_0 \, d_1 \pm \sqrt{d_0^2 - 4 \, b_0 \, d_2} \, \sqrt{d_1^2 - 4 \, b_2 \, d_2})/(2 \, d_2)} \, = \frac{1}{4\,d_2} \, \mathrm{u} \\
& \times \left[ 2 \, d_2 \, s_0 \, s_1 \mathrm{u} + \left( d_0 - \sqrt{d_0^2 - 4\, b_0 \, d_2} \right) s_0 \, \mathrm{v} + \left(d_1 \pm \sqrt{d_1^2 - 4\,b_2\,d_2} \right) s_1 \mathrm{w} \right] \\
& \times \left[ 2 \, d_2 \, s_0 \, s_1 \mathrm{u} + \left( d_0 + \sqrt{d_0^2 - 4\, b_0 \, d_2} \right) s_0 \, \mathrm{v} + \left(d_1 \mp \sqrt{d_1^2 - 4\,b_2\,d_2} \right) s_1 \mathrm{w} \right] \, ,
\end{split}
\end{align}
which is well-defined since no fibre coordinate appears under the square root. The fibre component defined by the factor $\mathrm{u}$ corresponds to the singlet state $\overline{\mathbf{1}}^{(5)}$, as explicit calculation of the intersection numbers with the $U(1)$ generators (\ref{eq:U(1)-generators}) using the Stanley-Reissner ideal and the divisor table (\ref{tab:divisor-classes-small}) quickly shows. The other two components are obviously in the same divisor class of the fibre ambient space and must have the same intersection numbers; indeed their intersection numbers with the $U(1)$ generators reveal that both correspond to $\mathbf{1}^{(6)}$-states. Furthermore, each component intersects the others exactly once, giving rise to an affine $SU(3)$ diagram. Similar calculations also verify the analogous fibre structure enhancement over the other two Yukawa points (\ref{eq:intersection-point1}) and (\ref{eq:intersection-point2}). More details may be found in \cite{philipp:BA}.
\subsection{Introducing Non-Abelian Symmetry}
Non-abelian gauge symmetry arises if the sections $g_m \in \{b_i, c_j, d_k\}$ appearing in (\ref{eq:hypersurface-equation}) take a non-generic form such that the fibration acquires a singularity in the fibre
over one or several divisors in the base.
A special type of such gauge enhancement is realised by restricting every section $g_m \in \{b_i, c_j, d_k\}$ such that it exhibits a certain vanishing order $k$ along a divisor $W = \{w\}$ in the base.
In other words, one restricts
\begin{eqnarray} \label{gmkform}
g_m = g_{m,k} w^k,
\end{eqnarray}
where now $g_{m,k}$ is a generic section of class $[g_m] - k [W]$ in the base; in particular, it does not vanish identically on $W$.\footnote{Other types of non-abelian singularities would involve non-trivial relations between the $g_m$
rather than factorisations of the type (\ref{gmkform}), as studied recently in the context of fibrations with Mordell-Weil group of rank one in \cite{Mayrhofer:2012zy,Kuntzler:2014ila}.}
The complex structure restrictions of type (\ref{gmkform}) compatible with gauge group $G$ along $W$ can be determined with the help of toric geometry and are encoded in the construction of toric tops \cite{Candelas:1996su,Candelas:1997eh}. The possible tops for all sixteen hypersurface realisations of genus-one fibrations have been classified in \cite{Bouchard:2003bu}. The top construction directly gives the resolution of the singularities in the fibre over $w=0$.
The toric resolution of a singularity associated with gauge group $G$ of rank $r$ introduces blow-up coordinates $e_i, i=1, \ldots,r$ together with new scaling relations in the toric ambient space of the fibre.
The hypersurface equation is replaced by a hypersurface in the blown-up ambient space in which each $g_m$ is replaced by $g_{m,k} \, e_0^k \, e_1^{l_1} \, e_2^{l_2} \ldots e_r^{l_r}$ for suitable powers $l_i$. For details on how to read off these $l_i$ from the toric tops we also refer to \cite{Borchmann:2013hta}.
The vanishing set $\{e_0\} \cap \{P_T\}$ is a divisor in the resolved fourfold which can be identified with the fibration of the original (singular) fibre \textit{without} the singular point over $W$, i.e.~the generic fibre has the topology of a $\mathbb{P}^1$.
Each of the sets $\{e_i\} \cap \{P_T\}, i>0$ is a \textit{resolution divisor} $E_i$, which -- over a generic point in $W$ -- introduces one further $\mathbb{P}^1$ to resolve the singularity of the fibre. The intersection diagram of these $\mathbb{P}^1$s \textit{over a generic point} in $W$ is the affine Dynkin diagram of the non-abelian gauge group. The resolution divisors $E_{i,i>0}$ correspond to (minus) the simple roots of the gauge Lie algebra. As in the case of the singlets, the $\mathbb{P}^1$s in the fibre can split over special curves and points (\mbox{codimension-two and -three}, respectively) in the base, leading to the appearance of matter states charged under the non-abelian gauge group and Yukawa couplings involving these states and also the charged singlets. Note that an alternative procedure to detect such matter via a deformation (as opposed to resolution) of the singular fibres has been described recently in \cite{Grassi:2013kha,Grassi:2014sda}.
The complex structure moduli restrictions (\ref{gmkform}) affect the precise location of the charged singlets in the base (but not their charges). In general the loci displayed in (\ref{tab:singlets-charges}) will contain components with non-abelian matter, which we have to disregard when we focus on the singlets. For the singlets $ \mathbf{1}^{(i)}$ with $i = 1, 3, 5$ this is simply done by replacing the coefficients $g_m$ that are supposed to vanish by their factors $g_{m,k}$ that do not vanish identically on $W$. For the other singlets we can again use \textsc{Singular} to determine the prime ideals associated with the curves.
\section[Toric Fibrations with Additional \texorpdfstring{\boldmath $SU(2)$}{SU(2)} Symmetry]{Toric Fibrations with Additional \texorpdfstring{\boldmath $SU(2)$}{SU(2)} Symmetry}\label{sec:SU(2)-tops}
In this section we analyse in detail toric realisations of gauge group $SU(2)$ along a base divisor
\begin{eqnarray}
W_2: \{w_2 = 0\}
\end{eqnarray}
in elliptic fibrations of type (\ref{eq:hypersurface-equation}). As detailed in appendix \ref{app:tops}, the fibres of such toric models are described by the three $A_1$-tops over polygon 5 \cite{Bouchard:2003bu}.
The resolution of an $SU(2)$ singularity over a divisor $W_2$ requires one resolution divisor $E_1$ corresponding to the single (simple) root $-\alpha$. Over a generic point on $W_2$, the fibre splits into two $\mathbb{P}^1$-components described by
\begin{eqnarray}
\mathbb{P}^1_i = \{e_i\} \cap \{P_T\} \cap Y_a \cap Y_b
\end{eqnarray}
for $i=0,1$, where $Y_{a,b}$ are two generic divisors in the base. These two $\mathbb{P}^1$s intersect in the affine $SU(2)$ diagram and will split into further $\mathbb{P}^1$s over matter curves and Yukawa points.
\subsection[\texorpdfstring{$SU(2)$}{SU(2)}-I Top]{\texorpdfstring{{\boldmath $SU(2)$}}{SU(2)}-I Top}
The first $A_1$-top, depicted in figure \ref{fig:tops}
in appendix \ref{app:tops}, corresponds to the following restrictions of the coefficients of the hypersurface equation (\ref{eq:hypersurface-equation}),
\begin{align}\label{eq:SU(2)-I-coeffs}
b_0 = b_{0,1} \, e_0 , \quad b_2 = b_{2,0} \, e_1 , \quad c_1 = c_{1,0} \, e_1 , \quad d_0 = d_{0,1} \, e_0 , \quad d_2 = d_{2,1} \, e_0,
\end{align}
while the other coefficients remain unrestricted. Concretely, the hypersurface describing the resolved elliptic fibration is given by
\begin{align}\label{eq:SU(2)-I-hypersurface-equation}
\begin{split}
P_T = &\mathrm{v} \, \mathrm{w} (c_{1,0}\,e_1 \, \mathrm{w} \, s_1 + c_2 \, \mathrm{v} \, s_0) + \mathrm{u} \, (b_{0,1}\,e_0 \, \mathrm{v}^2 \, s_0^2 + b_1 \, \mathrm{v} \, \mathrm{w} \, s_0 \, s_1 + b_{2,0}\,e_1 \, \mathrm{w}^2 \, s_1^2) + \\
&\mathrm{u}^2 (d_{0,1}\,e_0 \, \mathrm{v} \, s_0^2 \, s_1 + d_1 \, \mathrm{w} \, s_0 \, s_1^2 + d_{2,1}\,e_0 \, \mathrm{u} \, s_0^2 \, s_1^2).
\end{split}
\end{align}
This is the blow-up of a singular fibration with an $A_1$-singular fibre over the base divisor $W_2=\{w_2\}$ with $\pi^{-1} W_2 = E_0 \, E_1$. The singular fibration is obtained by setting $e_1=1$ and identifying $e_0$ with $w_2$.
One can map this blow-down to Weierstrass form (\ref{Weier}) and confirm a Kodaira fibre of (split) type $I_2$ over $\{w_2\}$ from the vanishing orders $(0,0,2)$ of $(f,g,\Delta)$.
The top allows two different triangulations. For definiteness, we choose one of these triangulations, for which the Stanley-Reisner-ideal (SR-ideal) is generated by
\begin{align}\label{eq:SU(2)-I-SR-ideal}
\mathrm{u} \, \mathrm{v} , \mathrm{u} \, \mathrm{w} , \mathrm{w} \, s_0 , \mathrm{v} \, s_1 , s_0 \, s_1 , e_0 \, \mathrm{w} , e_1 \, s_0 , e_1 \, \mathrm{u} .
\end{align}
From the top one can further read off the scaling relations among the coordinates and their corresponding divisor classes in the ambient space, which are summarised in the following table:
\begin{align}\label{tab:SU(2)-I-divisor-classes}
\begin{array}{c|cccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & e_1 & e_0 \\ \hline
{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot \\
E_1 & \cdot & \cdot & -1 & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
In the presence of non-abelian symmetry the $U(1)$ generators (\ref{eq:U(1)-generators}) need to be corrected such that the $SU(2)$ root has zero $U(1)$ charge. The resulting $U(1)$ generators take the form
\begin{align}\label{eq:SU(2)-I-U(1)-generators}
\begin{split}
\omega_1^\text{I} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{1}{2} E_1 ,\\
\omega_2^\text{I} &= U - S_0 - \overline{\mathcal{K}} - [c_{1,0}] .
\end{split}
\end{align}
Note that the charges (\ref{tab:singlets-charges}) of the singlets are not affected as these states are not charged under the $SU(2)$ root.
\subsubsection*{Matter Curves}
The Kodaira type of the resolved fibre changes in codimension-two, i.e.~over curves along the divisor $W_2$ in the base.
These loci can be found by analysing the vanishing order of the discriminant of the singular blow-down of (\ref{eq:SU(2)-I-hypersurface-equation}) along with the Weierstrass sections $f$ and $g$
which define the birationally equivalent Weierstrass model (\ref{Weier}).
One finds
\begin{eqnarray}
\Delta \simeq w_2^2 \Big( c_{2} \, (c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2) \, \ell_3 \, (b_1^2 - 4 c_2 d_1)^2 + {\cal O}(w_2) \Big)
\end{eqnarray}
with $\ell_3$ a complicated expression given in table (\ref{tab:SU(2)-I-matter}).
A straightforward analysis of the Weierstrass sections $f$ and $g$ reveals that the fibre
over the curves $\{w_2 \} \cap \{c_2\}$, $\{w_2 \} \cap \{c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2\} $ and $\{w_2 \} \cap \{\ell_3\}$ is of split Kodaira type $I_3$, corresponding to vanishing orders $(0,0,3)$ for $(f,g,\Delta)$. This indicates an enhancement of the singularity type from $A_1$ to $A_2$ due to the splitting of one of the fibre components such that the fibre over the curves forms the affine Dynkin diagram of $SU(3)$.
The curves therefore host massless matter multiplets in $SU(2)$ representation ${\bf 2}_{(q_1,q_2)}$ plus their conjugates, with the subscripts denoting the $U(1)$ charges.\footnote{
Note that the anti-fundamental representation of $SU(2)$ is equivalent to the fundamental, but in the present context it has the opposite $U(1)$ charges and will therefore be denoted by $\overline{\mathbf{2}}$.
} We will explicitly analyse the fibre and compute the $U(1)$ charges momentarily.
By contrast, along $\{w_2\} \cap \{ b_1^2 - 4 c_2 d_1\} $ the fibre is of Kodaira type $III$, corresponding to vanishing orders $(1,2,3)$ for $(f,g,\Delta)$. Since the singularity type remains $A_1$, no charged matter representations arise over this curve, consistent in particular with the results of \cite{Grassi:2011hq}. The matter curves and $U(1)$ charges are summarised in table \ref{tab:SU(2)-I-matter}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{c|c|c|c|c}
\multirow{2}{*}{matter} & \multirow{2}{*}{locus = $W_2 \cap \ldots$} & splitting of fibre & $U(1)-$ & highest weight\\
& & components & charges & states\\ \hline\hline \rule{0pt}{3ex}
$\mathbf{2}^\mathrm{I}_1$ & $\{c_{2}\}$ & $\mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \! + \mathbb{P}^1_{0A}$ & $(\frac{1}{2}, -1)$ & $\mathbf{2} \! : \mathbb{P}^1_{0 A} , \, \overline{\mathbf{2}} \! : \mathbb{P}^1_{0 s_1}$ \\ [.5ex] \hline \rule{0pt}{3ex}
$\mathbf{2}^\mathrm{I}_2$ & $\{ c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2 \}$ & $\mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}$ & $(\frac{1}{2},1)$ & $\mathbf{2} \! : \mathbb{P}^1_{0 C} , \, \overline{\mathbf{2}} \! : \mathbb{P}^1_{0 B}$\\ [.5ex] \hline \rule{0pt}{3ex}
\multirow{3}{*}{$\mathbf{2}^\mathrm{I}_3$}& $\{\ell_3\} := \{ b_{0,1}^2\,d_1^2$ & & \\
& $+ b_{0,1}\,(b_1^2\,d_{2,1} - b_1\,d_{0,1}\,d_1- 2\,c_2\,d_1\,d_{2,1})$ & $\mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B}$ & $(\frac{1}{2}, 0)$ & $\mathbf{2} \! : \mathbb{P}^1_{1 B} , \, \overline{\mathbf{2}} \! : \mathbb{P}^1_{1 A}$ \\
& $+ c_2\,(d_{0,1}^2\,d_1 - b_1\,d_{0,1}\,d_{2,1} + c_2\,d_{2,1}^2)\}$ & &
\end{tabular}
\end{center}
\caption{Matter states and their charges in the $SU(2)$-I top. Note that for legibility we have omitted the conjugate $\overline{\mathbf{2}}$-states and their charges, which simply come with the opposite sign as the shown charges.}
\label{tab:SU(2)-I-matter}
\end{table}
The splitting process in the fibre is due to the factorisation of the hypersurface equation over the enhancement loci. For the first curve, the factorisation is straightforward to see after setting $c_2 = e_0 = 0$ in (\ref{eq:SU(2)-I-hypersurface-equation}),
\begin{align}\label{eq:SU(2)-I-splitting-first-curve}
{P_T}|_{(e_0=0, \, c_2=0)} = s_1\,\mathrm{w} \left( c_{1,0}\,e_1\,\mathrm{v}\,\mathrm{w} + b_1\,s_0\,\mathrm{u}\,\mathrm{v} + b_{2,0}\,e_1\,s_1\,\mathrm{w}\,\mathrm{u} + d_1\,s_0\,s_1\,\mathrm{u}^2 \right).
\end{align}
Since $e_0 \, \mathrm{w}$ is in the SR-ideal, $\mathrm{w}$ cannot vanish so that the zero locus of (\ref{eq:SU(2)-I-splitting-first-curve}) splits into the zero locus of $s_1$ and of the expression in brackets, defining the components $\mathbb{P}_{0s_1}$ and $\mathbb{P}_{0A}$. One can further calculate the intersections between these components and $\mathbb{P}^1_1$ (which does not split and remains the root of $SU(2)$) and easily verify the structure to be an affine $SU(3)$ diagram. Explicit calculations identify $\mathbb{P}^1_{0 A}$ with the highest weight state of the $\mathbf{2}$-representation with $U(1)$ charges $(\frac{1}{2},-1)$, whose states we denote by $\mathbf{2}^\mathrm{I}_1$. Correspondingly $\mathbb{P}^1_{0 s_1}$ is the highest weight state of the conjugate representation $\overline{\mathbf{2}}^\mathrm{I}_1$, whose states have $U(1)$ charges $(-\frac{1}{2},1)$.
For the second curve defined by $W_2 \cap \{c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2\}$, one could solve the equation for $b_1$, $c_2$ or $d_1$ and plug the expressions into (\ref{eq:SU(2)-I-hypersurface-equation}) to detect a factorisation. However any of these expressions will involve division by $c_{1,0}$ or $b_{2,0}$, which for the analysis of Yukawa points below turns out to be disadvantageous. Instead we factorise
\begin{eqnarray} \label{eq:SU(2)-I-quadratic-curve}
c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2 = \frac{1}{d_1} \, {\mathcal{C}_+} \, {\mathcal{C}_-} \quad {\rm with} \quad {\mathcal{C}_\pm} = c_{1,0}\,d_1 - b_{2,0} \left( \frac{b_1}{2} \pm \sqrt{\frac{b_1^2}{4} - c_2 \, d_1} \right),
\end{eqnarray}
corresponding to a splitting of the curve into two components $W_2 \cap \{\mathcal{C}_\pm = 0\}$. Note that the square root introduces a branch cut in the base along which the two components are interchanged. Therefore the whole locus $W_2 \cap \{ c_{1,0}^2 \, d_1 - b_1 \, b_{2,0} \, c_{1,0} + b_{2,0}^2 \, c_2 \}$ is still one irreducible curve.
Furthermore the above factorisation is valid for generic points for which $d_1 \neq 0$. Since, as it turns out, at $d_1=0$ no Yukawa points are localised, this is sufficient for our purposes.
The factorisation (\ref{eq:SU(2)-I-quadratic-curve}) now allows us to solve $\mathcal{C}_{\pm}=0$ for $c_{1,0}$ and substitute it into the hypersurface equation. With this substitution we can see that over each part of the curve, $\mathbb{P}^1_0$ splits into two components,
\begin{align}\label{eq:SU(2)-I-splitting-quadratic-curve}
\begin{split}
& P_T|_{(e_0=0, \, {\mathcal C}_\pm=0)} \\%c_{1,0} = \frac{b_{2,0}}{d_1} ( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_2 \, d_1} ) )} \\
= \,& \frac{1}{d_1} \underbrace{\left[ d_1\,s_1\,\mathrm{u} + \mathrm{v} \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_2 \, d_1} \right) \! \right]}_{\mathbb{P}^1_{0B}} \underbrace{\left[ b_{2,0}\,e_1\,s_1 + d_1\,s_0\,s_1\,\mathrm{u} + s_0\,\mathrm{v} \left( \frac{b_1}{2} \mp \sqrt{ \frac{b_1^2}{4} - c_2 \, d_1} \right) \! \right]}_{\mathbb{P}^1_{0C}} ,
\end{split}
\end{align}
where we have set $\mathrm{w}=1$ using the SR-ideal. First note that there is no fibre coordinate appearing under the square roots. Therefore the factorisation defines two irreducible fibre components over each part $W_2 \cap {\cal C}_\pm$. At the branch cut the first/second component over one part of the curve is identified with the first/second component over the other part so there is no monodromy acting on the fibre components, making $\mathbb{P}^1_{0B}$ and $\mathbb{P}^1_{0C}$ well-defined on the whole curve. Explicit calculations show that $\mathbb{P}^1_{0B}$, $\mathbb{P}^1_{0C}$ and $\mathbb{P}^1_{1}$ (which again does not split) intersect each other in the affine $SU(3)$ diagram. $\mathbb{P}^1_{0C}$ is the highest weight state of $\mathbf{2}^\mathrm{I}_2$ with charges $(\frac{1}{2},1)$, and $\mathbb{P}^1_{0B}$ is that of $\overline{\mathbf{2}}^\mathrm{I}_2$ with charges $(-\frac{1}{2},-1)$.
For the third curve, we apply a similar factorisation method; the defining equation can be written as
\begin{align}\label{eq:SU(2)-I-complicated-quadratic-curve}
\ell_3 = 1/d_1^2 \, \mathcal{D_+} \, \mathcal{D}_- \quad {\rm with} \quad \mathcal{D}_\pm = b_{0,1}\,d_1^2 - \left[ c_2\,d_1\,d_{2,1} + (d_{0,1}\,d_1-b_1\,d_{2,1}) \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_2\,d_1} \right)\!\right] .
\end{align}
Again, there is a branch cut in the base coming from the square root which identifies the two parts $W_2 \cap \{\mathcal{D}_\pm=0\}$ at the branch locus. The fibre enhancement over each part can be deduced (after some calculation) by solving $\mathcal{D}_\pm=0$ for $b_{0,1}$ and inserting the expression into the hypersurface equation (\ref{eq:SU(2)-I-hypersurface-equation}). We find that $\mathbb{P}^1_1$ splits into two components,
\begin{align}\label{eq:SU(2)-I-splitting-complicated-quadratic-curve}
\begin{split}
& P_T|_{(e_1=0, \, \mathcal{D}_\pm=0)} = \\
& \frac{1}{d_1} \underbrace{\left[ d_1\,s_1 + \left( \frac{b_1}{2} \pm \sqrt{\frac{b_1^2}{4} - c_2\,d_1} \right)\!\mathrm{v} \right]}_{\mathbb{P}^1_{1A}} \\
& \times \underbrace{\left[ d_{2,1}\,e_0\,s_1 + \left[ d_{0,1} - \frac{d_{2,1}}{d_1} \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_2\,d_1 } \right)\!\right]\!e_0\,\mathrm{v} + d_1\,\mathrm{w}\,s_1 + \left( \frac{b_1}{2} \mp \sqrt{\frac{b_1^2}{4} - c_2\,d_1} \right)\!\mathrm{v}\,\mathrm{w} \right]}_{\mathbb{P}^1_{1B}} .
\end{split}
\end{align}
Analogous to the situation over the second curve, the factors are not interchanged by any monodromy when passing the branch locus in the base, making the splitting $\mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B}$ well-defined over the whole curve. The intersection structure together with $\mathbb{P}^1_0$ (which remains irreducible) turns out to be again the affine $SU(3)$ diagram. $\mathbb{P}^1_{1B}$ is the highest weight state of $\mathbf{2}^\mathrm{I}_3$ with charges $(\frac{1}{2},0)$ and $\mathbb{P}^1_{1A}$ is that of $\overline{\mathbf{2}}^\mathrm{I}_3$ with charges $(-\frac{1}{2},0)$.
\subsubsection*{Yukawa Points}
$SU(2)$ matter and singlet curves intersect at codimension-three loci in the base to form gauge invariant Yukawa couplings of the form $\mathbf{2}-\overline{\mathbf{2}}- \mathbf{1}/\overline{\mathbf{1}}$ and $\mathbf{2}-\mathbf{2}-\mathbf{1}/\overline{\mathbf{1}}$. We list all such couplings in table \ref{tab:SU(2)-I-Yukawas}.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus} = W_2 \cap \ldots & \text{splitting of fibre components} \\ \hline\hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_1 - \mathbf{2}^\mathrm{I}_2 - \overline{\mathbf{1}}^{(2)} & \{c_2\} \cap \{c_{1,0}\,d_1 - b_1\,b_{2,0} \} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0 s_1 C} + \mathbb{P}^1_{0AB} + \mathbb{P}^1_{0AC} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_1 - \overline{\mathbf{2}}^\mathrm{I}_2 - \mathbf{1}^{(5)} & \{c_2\} \cap \{c_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1 B} + \mathbb{P}^1_{0AB'} + \mathbb{P}^1_{0AC'} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_1 - \mathbf{2}^\mathrm{I}_3 - \overline{\mathbf{1}}^{(1)} & \{c_2\} \cap \{b_{0,1}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \!+ \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_1 - \overline{\mathbf{2}}^\mathrm{I}_3 - \mathbf{1}^{(6)} & \{c_2\} \cap \{b_1^2\,d_{2,1} - b_1\,d_{0,1}\,d_1 + b_{0,1}\,d_1^2\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \!+ \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_2 - \mathbf{2}^\mathrm{I}_3 - \overline{\mathbf{1}}^{(4)} & \left(\{\mathcal{C}_+\} \cap \{\mathcal{D}_+\}\right) \cup \left(\{\mathcal{C}_-\} \cap \{\mathcal{D}_-\}\right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_2 - \overline{\mathbf{2}}^\mathrm{I}_3 - \overline{\mathbf{1}}^{(6)} & \left(\{\mathcal{C}_+\} \cap \{\mathcal{D}_-\}\right) \cup \left(\{\mathcal{C}_-\} \cap \{\mathcal{D}_+\}\right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_2 - \mathbf{2}^\mathrm{I}_2 - \overline{\mathbf{1}}^{(3)} & \{b_{2,0}\} \cap \{c_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0Cs_0} + \mathbb{P}^1_{0C'} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{I}_3 - \mathbf{2}^\mathrm{I}_3 - \overline{\mathbf{1}}^{(2)} & \{b_{0,1}\,d_1 - c_2\,d_{2,1}\} \cap \{b_1\,d_{2,1} - d_{0,1}\,d_1\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1A} + \mathbb{P}^1_{1B'} + \mathbb{P}^1_{1B''}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(2)$-I top.}\label{tab:SU(2)-I-Yukawas}
\end{table}
To derive these, one first checks explicitly that none of the possible Yukawa couplings lies at $d_1=0$ so that the factorisations (\ref{eq:SU(2)-I-quadratic-curve}) and (\ref{eq:SU(2)-I-complicated-quadratic-curve}) are applicable.
The first two couplings arise over the intersection locus of the $\mathbf{2}^\mathrm{I}_1$- and $\mathbf{2}^\mathrm{I}_2$-curves. This locus splits into two sets of points, which can be identified with the intersection of $\{c_2\}$ with $\{\mathcal{C}_+\}$ and $\{d_1 \mathcal{C}_-\}$, respectively.
The intersection with $\{\mathcal{C}_+\}$ leads to the first set of Yukawa points, over which the fibre of the divisor $E_0$ splits into three components $\mathbb{P}^1_{0 s_1 C}$, $\mathbb{P}^1_{0AB}$ and $\mathbb{P}^1_{0AC}$. The intersection structure of these three components and $\mathbb{P}^1_1$ (which does not split) forms an affine $SU(4)$ diagram.
The splitting of the fibre components (\ref{tab:SU(2)-I-matter}) over the respective $\mathbf{2}$-curves arises as follows:
\begin{itemize}
\item Approaching the Yukawa points along the $\mathbf{2}^\mathrm{I}_1$-curve, $\mathbb{P}^1_{0 s_1} \rightarrow \mathbb{P}^1_{0 s_1C}$ remains irreducible while $\mathbb{P}^1_{0A}$ splits into two components, $\mathbb{P}^1_{0AB} + \mathbb{P}^1_{0AC}$.
\item From the perspective of the $\mathbf{2}^\mathrm{I}_2$-curve the Yukawa points lie on the $\{\mathcal{C}_+\}$-part, where the fibre components $\mathbb{P}^1_{0B}$ and $\mathbb{P}^1_{0C}$ are defined in (\ref{eq:SU(2)-I-splitting-quadratic-curve}) with `+'-sign for $\mathbb{P}^1_{0B}$ and `$-$'-sign for $\mathbb{P}^1_{0C}$. When we approach the Yukawas by setting $c_2= 0$, $\mathbb{P}^1_{0B} \rightarrow \mathbb{P}^1_{0AB}$ remains irreducible while the equation for $\mathbb{P}^1_{0C}$ splits off a factor $s_1$, giving the splitting $\mathbb{P}^1_{0C} \rightarrow \mathbb{P}^1_{0 s_1 C} + \mathbb{P}^1_{0AC}$.
\end{itemize}
The second set of Yukawa points $W_2 \cap \{c_2\} \cap \{c_{1,0}\}$ can be viewed as the intersection of the $\{d_1 \mathcal{C}_-\}$-part of $\mathbf{2}^\mathrm{I}_2$ with $\mathbf{2}^\mathrm{I}_1$.
Again $E_0$ splits into three components, $\mathbb{P}^1_{0 s_1 B}$, $\mathbb{P}^1_{0AB'}$ and $\mathbb{P}^1_{0AC'}$ (the primes denote that these have different charges under the $SU(2)$ root and the $U(1)$ generators, corresponding to the conjugate $\mathbf{2}^\mathrm{I}_2$-state and a different singlet), which together with $\mathbb{P}^1_1$ form the affine $SU(4)$ diagram. The splitting processes are as follows:
\begin{itemize}
\item Along $\mathbf{2}^\mathrm{I}_1$, again the component $\mathbb{P}^1_{0s_1}$ remains irreducible and $\mathbb{P}^1_{0A} \rightarrow \mathbb{P}^1_{0AB'} + \mathbb{P}^1_{0AC'}$ splits. The primes denote that the charge of the components under the $SU(2)$ root and the $U(1)$ generators are different than over the first Yukawa point.
\item Along $\mathbf{2}^\mathrm{I}_2$, we have to look at the fibre components $\mathbb{P}^1_{0B}$ and $\mathbb{P}^1_{0C}$ over $\{\mathcal{C}_-\}$, which are defined through (\ref{eq:SU(2)-I-splitting-quadratic-curve}) with the second sign choice; setting $c_2=0$ now leaves $\mathbb{P}^1_{0C}$ irreducible, while the equation for $\mathbb{P}^1_{0B}$ just becomes $d_1\,s_1\,\mathrm{u}$, defining the components $\mathbb{P}^1_{0s_1 B}$ (where $s_1=0$) and $\mathbb{P}^1_{0AB'}$ (where $\mathrm{u}=0$).
\end{itemize}
The third and fourth sets of Yukawa points are the intersection points of the $\mathbf{2}^\mathrm{I}_1$-curve over $\{c_2\}$ with $\mathbf{2}^\mathrm{I}_3$ over $\{ d_1 \mathcal{D}_-\}$ and $\{\mathcal{D}_+\}$, respectively. The splitting is straightforward and gives the same $\mathbb{P}^1$s with identical charges over both points. What differs, though, is the intersection \textit{pattern}. Over the third set, the pattern is $\mathbb{P}^1_{0s_1} - \mathbb{P}^1_{0A} - \mathbb{P}^1_{1A} - \mathbb{P}^1_{1B} (- \mathbb{P}^1_{0s_1})$; M2-branes can wrap the combination $\mathbb{P}^1_{0s_1}+\mathbb{P}^1_{1B}$ giving rise to $\overline{\mathbf{1}}^{(1)}$-states, but not $\mathbb{P}^1_{0s_1} + \mathbb{P}^1_{1A}$, which is needed for $\mathbf{1}^{(6)}$-states. Accordingly, the intersection pattern over the fourth set is $\mathbb{P}^1_{0s_1} - \mathbb{P}^1_{0A} - \mathbb{P}^1_{1B} - \mathbb{P}^1_{1A} (- \mathbb{P}^1_{0s_1})$, allowing ${\mathbf{1}}^{(6)}$- but not $\
\overline{\mathbf{1}}^{(1)}$-states. Of course, both patterns have the same structure as the affine $SU(4)$ diagram.
The fifth and sixth types of couplings arise over the intersection points between the $\mathbf{2}^\mathrm{I}_2$- and $\mathbf{2}^\mathrm{I}_3$-curve. With the factorisations (\ref{eq:SU(2)-I-quadratic-curve}) and (\ref{eq:SU(2)-I-complicated-quadratic-curve}), these points group into the four intersection loci of $\{\mathcal{C}_\pm\}$ and $\{\mathcal{D}_\pm\}$. Analogously to the situation over the previous two types of Yukawa points, one finds for both the fifth and sixth coupling the same $\mathbb{P}^1$s with the same charges, but different intersection patterns, leading to either $\overline{\mathbf{1}}^{(4)}$-states over $\{\mathcal{C}_\pm\} \cap \{\mathcal{D}_\pm\}$ or $\overline{\mathbf{1}}^{(6)}$-states over $\{\mathcal{C}_\pm\} \cap \{\mathcal{D}_\mp\}$. The intersection structure is an affine $SU(4)$ diagram in both cases.
The last two Yukawa points are self-intersection points of $\mathbf{2}^\mathrm{I}_2$ resp.~$\mathbf{2}^\mathrm{I}_3$. For $\mathbf{2}^\mathrm{I}_2$ it is the intersection point $W_2 \cap \{b_{2,0}\} \cap \{c_{1,0}\}$ of $\{\mathcal{C}_+\}$ with $\{\mathcal{C}_-\}$ which also lies on the singlet curve $\mathbf{1}^{(3)}$.\footnote{$\{\mathcal{C}_+\}$ and $\{\mathcal{C}_-\}$ also have the codimension-three locus $W_2 \cap \{b_1^2-4\,c_2\,d_1\} \cap \{c_1\,d_1-1/2\,b_1\,b_2\}$ in common, which is just the branch locus of the square root. However this set of points does not lie on any singlet curve; consistently there is no further enhancement in the fibre.} From the factorisation of the hypersurface equation (\ref{eq:SU(2)-I-splitting-quadratic-curve}) one easily sees that, irrespective of along which part we approach the point, $\mathbb{P}^1_{0B}$ remains irreducible, while the equation of $\mathbb{P}^1_{0C}$ splits off a factor $s_0$ as we set $b_{2,0}=0$, thus $\mathbb{P}^1_{0C} \
\rightarrow \mathbb{P}^1_{0Cs_0} + \mathbb{P}^1_{0C'}$. The last coupling is over
the point of $\{\mathcal{D}_+\} \cap \{\mathcal{D}_-\}$ which lies on $\mathbf{1}^{(2)}$. The splitting process here is not obvious from (\ref{eq:SU(2)-I-splitting-complicated-quadratic-curve}), but straightforward calculation reveals that $\mathbb{P}^1_{1B}$ splits into two components. Again, the intersection structure is an affine $SU(4)$ diagram over both self-intersection points.
Note that, while for the discussion of the Yukawas above we have only used the loci of the $\mathbf{2}$-curves to determine the Yukawa points, we have used \textsc{Singular} to verify that indeed all the Yukawa points (\ref{tab:SU(2)-I-Yukawas}) also lie on the corresponding singlet curve. This is consistent with the appearance of the associated singlet states in the split fibres as discussed above.
\subsection[\texorpdfstring{$SU(2)$}{SU(2)}-II and -III Tops]{\texorpdfstring{\boldmath $SU(2)$}{SU(2)}-II and -III Tops}\label{subsec:SU(2)-tops-short}
The analysis of the remaining two $A_1$-tops as depicted in figure \ref{fig:tops} is very analogous and is carried out in appendix \ref{SU2DetailsApp}, to which we refer for more details.
Here we merely collect the massless spectrum and the associated Yukawa couplings as these will be needed for our construction of Standard-Model-like F-theory compactifications.
\subsubsection*{\texorpdfstring{\boldmath $SU(2)$}{SU(2)}-II Top}
The second $A_1$-top, called $SU(2)$-II top in appendix \ref{SU2App1},
leads to an $SU(2)$-charged matter spectrum of the following form:
\begin{align}\label{tab:SU(2)-II-matter-short}
\begin{array}{c|c}
\text{matter} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_1 & (\frac{1}{2}, \frac{3}{2}) \\[.5ex]\hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_2 & (\frac{1}{2},-\frac{1}{2}) \\[.5ex]\hline \rule{0pt}{3ex}
\mathbf{2}^\mathrm{II}_3 & (\frac{1}{2}, \frac{1}{2}) \\
\end{array}
\end{align}
All Yukawa couplings allowed by the $U(1)$ charges are realised geometrically. More precisely the set of Yukawas is given by
\begin{align}
\begin{split}
& \mathbf{2}_1^{\mathrm{II}} - \mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(4)}, \qquad
\mathbf{2}^{\mathrm{II}}_1 - \overline{\mathbf{2}}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(5)}, \qquad
\mathbf{2}^{\mathrm{II}}_1 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(3)}, \qquad
\mathbf{2}^{\mathrm{II}}_1 - \overline{\mathbf{2}}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(6)}, \\
& \mathbf{2}^{\mathrm{II}}_2 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(2)}, \qquad
\mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{2}}^{\mathrm{II}}_3 - \mathbf{1}^{(6)}, \qquad
\mathbf{2}^{\mathrm{II}}_2 - \mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(1)}, \qquad
\mathbf{2}^{\mathrm{II}}_3 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(4)}.
\end{split}
\end{align}
In fact the $SU(2)$-II top is equivalent to the $SU(2)$-I top. On way to see this is to notice that upon identifying the $U(1)$ charges in the two tops as
\begin{align}\label{eq:U1-charge-transformation}
\begin{split}
U(1)_1^{\rm II} &= -U(1)_1^{\rm I} \, , \\
U(1)_2^{\rm II} &= U(1)_2^{\rm I} - U(1)_1^{\rm I} \, ,
\end{split}
\end{align}
the spectrum and Yukawa structure is exactly the same if one identifies the states ${\bf 2}^{\rm II}_i \leftrightarrow \overline{\bf 2}^{\rm I}_i, i=1,2,3$ and exchanges the singlets ${\bf 1}^{(1)} \leftrightarrow \overline{\bf 1}^{(3)}, {\bf 1}^{(2)} \leftrightarrow \overline{\bf 1}^{(4)}$. One can also arrive at this identification from the symmetries of the tops. More details can be found in appendix \ref{app:tops}.
Although both models are the same when considering only the gauge group $SU(2) \times U(1)_1 \times U(1)_2$, they will give rise to different models when combining them with an $SU(3)$-top, as we will discuss below in section \ref{sec_3211}.
\subsubsection*{\texorpdfstring{\boldmath $SU(2)$}{SU(2)}-III Top}
The last $A_1$-top is the $SU(2)$-III top with matter content
\begin{align}\label{tab:SU(2)-III-matter-short}
\begin{array}{c|c}
\text{matter} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_1 & (1,0) \\ \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_2 & (1,1)\\ \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_3 & (0,1)
\end{array}
\end{align}
In this top, in addition to the three fundamental matter curves and a notorious type $III$ enhancement locus with no additional matter, one finds a change of fibre type to \emph{non-split $I_3$} type over yet another curve. As explained in appendix \ref{SU2DetailsApp}, the non-split fibre type can either be seen from the Weierstrass data or be explicitly confirmed by analysing the monodromies along the curve in question. As a result of the monodromy, this locus does not carry massless matter.
The geometrically realised Yukawa couplings
\begin{align}
\begin{split}
& \mathbf{2}^{\mathrm{III}}_1 - \overline{\mathbf{2}}^{\mathrm{III}}_2 - \mathbf{1}^{(6)}, \qquad
\mathbf{2}^{\mathrm{III}}_1 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(4)}, \qquad
\mathbf{2}^{\mathrm{III}}_1 - \overline{\mathbf{2}}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(1)}, \\
& \mathbf{2}^{\mathrm{III}}_2 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(3)}, \qquad
\mathbf{2}^{\mathrm{III}}_2 - \overline{\mathbf{2}}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(2)}, \qquad
\mathbf{2}^{\mathrm{III}}_3 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(5)}
\end{split} ´
\end{align}
exhaust again all gauge invariant combinations.
This top is inequivalent to the first two tops. Under the transformation $U(1)'_1 \equiv - U(1)_1, \, U(1)'_2 \equiv U(1)_2 - U(1)_1$ analogous to
(\ref{eq:U1-charge-transformation}) (with the same identification of the singlets), the spectrum and Yukawa structure of the $SU(2)$-III top is mapped to itself as ${\bf 2}^{\rm III}_1 \leftrightarrow \overline{\bf 2}^{\rm III}_2$ and ${\bf 1}^{(1)} \leftrightarrow \overline{\bf 1}^{(3)}, {\bf 1}^{(2)} \leftrightarrow \overline{\bf 1}^{(4)}$.
\section[Toric Fibrations with Additional \texorpdfstring{\boldmath $SU(3)$}{SU(3)} Symmetry]{Toric Fibrations with Additional \texorpdfstring{\boldmath $SU(3)$}{SU(3)} Symmetry}\label{sec:SU(3)-tops}
The construction of $SU(3)$ gauge symmetry via tops is analogous to the $SU(2)$ cases.
The resolution of the $A_2$-singularity over a divisor $W_3: w_3=0$ in the base introduces three toric divisors $F_0, F_1, F_2$ given by the vanishing locus of the coordinates $f_0, f_1, f_2$.
Each $F_i$ is a $\mathbb{P}^1$-fibration over $W_3$, and $\pi^{-1} W_3 = F_0 F_1 F_2$. Over a generic point on $W_3$ the intersection structure of the $\mathbb{P}^1$-fibres reproduces the affine $SU(3)$ diagram. We choose the root assignment $F_1 \leftrightarrow -\alpha_1, \, F_2 \leftrightarrow -\alpha_2, \, F_0 \leftrightarrow \alpha_1+\alpha_2$. There exist three $SU(3)$ tops, which we will now present in detail.
\subsection[\textit{SU}(3)-A Top]{\texorpdfstring{{\boldmath $SU(3)$}}{SU(3)}-A Top}
The first top corresponds to the following restriction of the hypersurface coefficients,
\begin{align}\label{eq:SU(3)-A-coeffs}
\begin{split}
b_0 &= b_{0,1}\,f_0 ,\quad b_2 = b_{2,0}\,f_1\,f_2 ,\quad c_1 = c_{1,0}\,f_2 ,\quad c_2 = c_{2,1}\,f_0\,f_2 , \\
d_0 &= d_{0,1}\,f_0\,f_1 ,\quad d_1 = d_{1,0}\,f_1 ,\quad d_2 = d_{2,1}\,f_0\,f_1^2 ,
\end{split}
\end{align}
where only $b_1$ remains unchanged. Out of the four different triangulations we choose the one whose SR-ideal is generated by
\begin{align}\label{eq:SU(3)-A-SR-ideal}
\mathrm{u}\,\mathrm{v} , \mathrm{u}\,\mathrm{w} , \mathrm{w}\,s_0 , \mathrm{v}\,s_1 , s_0\,s_1 , f_0\,\mathrm{w} , f_0\,s_1 , f_1\,s_0 , f_1\,\mathrm{v} , f_2\,s_0 , f_2\,s_1 , f_2\,\mathrm{u} .
\end{align}
The coordinates and their corresponding divisor classes are summarised in the following table:
\begin{align}\label{tab:SU(3)-A-divisor-classes}
\begin{array}{c|ccccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & f_1 & f_2 & f_0 \\ \hline
\mathrm{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\
F_1 & \cdot & 1 & \cdot & \cdot & \cdot & 1 & \cdot & -1 \\
F_2 & \cdot & \cdot & -1 & \cdot & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
The $U(1)$ generators (\ref{eq:U(1)-generators}) need to be corrected such that the roots of $SU(3)$ have zero $U(1)$ charge. The generators with this property take the form
\begin{align}\label{eq:SU(3)-A-U(1)-generators}
\begin{split}
\omega_1^\text{A} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{2}{3} F_1 + \frac{1}{3} F_2 ,\\
\omega_2^\text{A} &= U - S_0 - \overline{\mathcal{K}} - [c_{1,0}] + \frac{2}{3} F_1 + \frac{1}{3} F_2.
\end{split}
\end{align}
The charges (\ref{tab:singlets-charges}) of the singlets are not affected as they are not charged under the roots of $SU(3)$.
\subsubsection*{Matter curves}
The matter curves are found by analysing the discriminant of the associated Weierstrass model
\begin{eqnarray}
\Delta \simeq w^3 \Big( b_{0,1} \, c_{1,0} \, (b_{0,1}\,c_{1,0} - b_1\,c_{2,1}) \, (b_1\,b_{2,0} - c_{1,0}\,d_{1,0}) \, (b_{0,1}\,d_{1,0}^2 - b_1\,d_{0,1}\,d_{1,0} + b_1^2\,d_{2,1} ) \, b_{1}^3 + {\cal O}(w_3) \Big). \nonumber
\end{eqnarray}
As can be read off from the Weierstrass sections $f$ and $g$, the fibre along the five curves associated with the first five factors in the bracket is of split Kodaira type $I_4$. The fibre components intersect as in the affine $SU(4)$ diagram, corresponding to $SU(3)$ matter in the fundamental representation (plus conjugate). Their $U(1)$ charges together with the geometric data can be found in table \ref{tab:SU(3)-A-matter}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{c|c|c|c|l}
\multirow{2}{*}{matter} & \multirow{2}{*}{$\text{locus} = W_3 \cap \ldots$} & splitting of fibre & $U(1)-$ & \multirow{2}{*}{highest weight states of...}\\
& & components & charges & \\ \hline \rule{0pt}{3ex}
$\mathbf{3}^{\mathrm{A}}_1$ & $\{b_{0,1}\}$ & $\mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w}} + \mathbb{P}^1_{1A}$ & $(\frac{2}{3}, -\frac{1}{3})$ & $\mathbf{3} \! : \mathbb{P}^1_{1 \mathrm{w}} + \mathbb{P}^1_0 + \mathbb{P}^1_2, \, \overline{\mathbf{3}} \! : \mathbb{P}^1_{1A} + \mathbb{P}^1_0 $\\ \rule{0pt}{3ex}
$ \mathbf{3}^{\mathrm{A}}_2$ & $\{c_{1,0}\}$ & $\mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u}} + \mathbb{P}^1_{0A}$ & $(-\frac{1}{3}, -\frac{4}{3})$ & $\mathbf{3} \! : \mathbb{P}^1_{0 \mathrm{u}} , \, \overline{\mathbf{3}} \! : \mathbb{P}^1_{0 A}$\\ \rule{0pt}{3ex}
$\mathbf{3}^{\mathrm{A}}_3$ & $\{b_{0,1}\,c_{1,0} - b_1\,c_{2,1}\}$ & $\mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C}$ & $(-\frac{1}{3}, \frac{2}{3})$ & $\mathbf{3} \! : \mathbb{P}^1_{1C} + \mathbb{P}^1_{0} + \mathbb{P}^1_{2} , \, \overline{\mathbf{3}} \!: \mathbb{P}^1_{1 B} + \mathbb{P}^1_0$\\ \rule{0pt}{3ex}
$\mathbf{3}^{\mathrm{A}}_4$ & $\{b_1\,b_{2,0} - c_{1,0}\,d_{1,0}\}$ & $\mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}$ & $(\frac{2}{3}, \frac{2}{3})$ & $\mathbf{3} \! : \mathbb{P}^1_{0B} , \, \overline{\mathbf{3}} \! : \mathbb{P}^1_{0C}$ \\ \rule{0pt}{3ex}
\multirow{2}{*}{$\mathbf{3}^{\mathrm{A}}_5$} & $\{b_{0,1}\,d_{1,0}^2 - b_1\,d_{0,1}\,d_{1,0}$ & \multirow{2}{*}{$\mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B}$} & \multirow{2}{*}{$(-\frac{1}{3},-\frac{1}{3})$} & \multirow{2}{*}{$\mathbf{3} \! : \mathbb{P}^1_{2A} + \mathbb{P}^1_0 , \, \overline{\mathbf{3}} \! : \mathbb{P}^1_{2B} + \mathbb{P}^1_0 + \mathbb{P}^1_1$}\\ \rule{0pt}{3ex}
& $+ b_1^2\,d_{2,1}\}$ & & &
\end{tabular}
\caption{Matter states in the $SU(3)$-A top.}
\label{tab:SU(3)-A-matter}
\end{center}
\end{table}
The splitting process over the first four curves can be straightforwardly verified. For the fifth curve, we proceed as for the $SU(2)$ tops und use expressions with square roots to split the curve into two parts; because $d_{2,1}\neq0$ for all the Yukawa points below, we factorise the quadratic term such that we can solve for $b_1$ (analogously to e.g.~(\ref{eq:SU(2)-I-quadratic-curve}), where $c_{1,0}$ played the role of $b_1$ here). Inserting the resulting expressions (one for each of the two parts of the curve) for $b_1$ into the hypersurface equation, we find that $\mathbb{P}^1_2$ splits into two components on both parts of the curve; similarly to the $SU(2)$-tops, there is no monodromy interchanging the components when passing from one part of the curve to the other and back.
In addition, over the curve $\{w_3\} \cap \{b_{1}\}$ the Weierstrass data $(f,g,\Delta)$ vanish to orders $(2,2,4)$. This indicates a Kodaira type $IV$ fibre in which no extra $\mathbb P^1$ splits off, but rather the three fibre components intersect in a single point, as can be checked explicitly. Over this curve, no extra matter representation arises.
\subsubsection*{Yukawa Points}
The $SU(3)$ matter and the singlet curves intersect at certain codimension-three loci in the base to form gauge invariant Yukawa couplings $\mathbf{3}-\overline{\mathbf{3}}-\mathbf{1}/\overline{\mathbf{1}}$. In this case the fibre is enhanced to the affine $SU(5)$-diagram, and the realised couplings are in 1-to-1 correspondence with the gauge theoretic selection rules.
In addition one can also form gauge invariant couplings of the type $\mathbf{3}-\mathbf{3}-\mathbf{3}$. As it turns out all such gauge invariant couplings are indeed realised geometrically, and the fibre structure is the affine $SO(8)$-diagram, cf.~table \ref{tab:SU(3)-A-Yukawas}.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus}=W_3 \cap \ldots & \text{splitting of fibre components} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_1 - \overline{\mathbf{3}}^{\mathrm{A}}_2 - \overline{\mathbf{1}}^{(4)} & \{b_{0,1}\} \cap \{c_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u}} + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w}}+\mathbb{P}^1_{1A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_1 - \overline{\mathbf{3}}^{\mathrm{A}}_3 - \mathbf{1}^{(1)} & \{b_{0,1}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w} C} + \mathbb{P}^1_{1AB}\! +\mathbb{P}^1_{1AC} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_1 - \overline{\mathbf{3}}^{\mathrm{A}}_4 - \mathbf{1}^{(6)} & \{b_{0,1}\} \cap \{b_1\,b_{2,0} - c_{1,0}\,d_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w}}+\mathbb{P}^1_{1A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_1 - \overline{\mathbf{3}}^{\mathrm{A}}_5 - \overline{\mathbf{1}}^{(2)} & \{b_{0,1}\} \cap \{b_1\,d_{2,1}-d_{0,1}\,d_{1,0}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w}} + \mathbb{P}^1_{1D}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A}+\mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_2 - \overline{\mathbf{3}}^{\mathrm{A}}_3 - \mathbf{1}^{(5)} & \{c_{1,0}\} \cap \{c_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u}} + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B}+\mathbb{P}^1_{1C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_2 - \overline{\mathbf{3}}^{\mathrm{A}}_4 - \mathbf{1}^{(3)} & \{c_{1,0}\} \cap \{b_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u} A} \! + \mathbb{P}^1_{0AB} +\mathbb{P}^1_{0AC} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_2 - \overline{\mathbf{3}}^{\mathrm{A}}_5 - \mathbf{1}^{(6)} & \{c_{1,0}\} \cap \left(\text{$\mathbf{3}^\mathrm{A}_5$-locus}\right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u}} + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A}+\mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_3 - \overline{\mathbf{3}}^{\mathrm{A}}_4 - \mathbf{1}^{(2)} & \left(\mathbf{3}^\mathrm{A}_3 \right) \cap \left( \mathbf{3}^\mathrm{A}_4 \right) \setminus \left(\{c_{1,0}\} \cap \{b_1\} \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_3 - \overline{\mathbf{3}}^{\mathrm{A}}_5 - \overline{\mathbf{1}}^{(6)} & \left(\mathbf{3}^\mathrm{A}_3\right) \cap \left(\mathbf{3}^\mathrm{A}_5\right) \! \setminus \! \left( \vphantom{\mathbf{3}^\mathrm{A}_5} \{b_{0,1}\} \cap \{b_1\} \right) & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A}+\mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_4 - \overline{\mathbf{3}}^{\mathrm{A}}_5 - \overline{\mathbf{1}}^{(4)} & \left(\mathbf{3}^\mathrm{A}_4\right) \cap \left(\mathbf{3}^\mathrm{A}_5\right) \! \setminus \! \left( \vphantom{\mathbf{3}^\mathrm{A}_5} \{d_{1,0}\} \cap \{b_1\} \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A}+\mathbb{P}^1_{2B} \\ \hline \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_1 - \mathbf{3}^{\mathrm{A}}_3 - \mathbf{3}^{\mathrm{A}}_5 & \{b_{0,1}\} \cap \{b_1\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{w} B} + \mathbb{P}^1_{1AB'} + \mathbb{P}^1_{AC'}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} , \\
& & \mathbb{P}^1_{AB'}=\mathbb{P}^1_{2A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_2 - \mathbf{3}^{\mathrm{A}}_3 - \mathbf{3}^{\mathrm{A}}_4 & \{c_{1,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{u} C} + \mathbb{P}^1_{0AB'} + \mathbb{P}^1_{0AC'}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} , \\
& & \mathbb{P}^1_{0AC} = \mathbb{P}^1_{1C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{A}}_4 - \mathbf{3}^{\mathrm{A}}_5 - \mathbf{3}^{\mathrm{A}}_5 & \{d_{1,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B0} + \mathbb{P}^1_{2B'} , \\
& & \mathbb{P}^1_{0B} = \mathbb{P}^1_{2B0}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(3)$-A top.}
\label{tab:SU(3)-A-Yukawas}
\end{table}
There, in the last column, the subscripts should help to visualise $\mathbb{P}^1$s' splitting process. E.g.~if we approach the second Yukawa point along the $\mathbf{3}^\mathrm{A}_1$-curve, then we find that $\mathbb{P}^1_{1\mathrm{w}}$ remains a single component and $\mathbb{P}^1_{1A}$ splits into two, $\mathbb{P}^1_{1AB}$ and $\mathbb{P}^1_{1AC}$; $\mathbb{P}^1_{1AB}$ is the component $\mathbb{P}^1_{1B}$ from the $\mathbf{3}^\mathrm{A}_3$-curve that does not split over the Yukawa point, while $\mathbb{P}^1_{1C}$ splits into two components, of which one is identified with $\mathbb{P}^1_{1AC}$ and the other one coincides with $\mathbb{P}^1_{1\mathrm{w}}$, hence the notation $\mathbb{P}^1_{1\mathrm{w} C}$.
Over the last three Yukawa points with the coupling type $\mathbf{3}-\mathbf{3}-\mathbf{3}$, two of the three divisors $F_{0,1,2}$ each split off the same $\mathbb{P}^1$-component, which therefore is a multiplicity 2 component; this corresponds to the central node of the affine $SO(8)$-diagram with dual Coxeter label 2.
\subsection[\texorpdfstring{$SU(3)$}{SU(3)}-B and -C Top]{\texorpdfstring{\boldmath $SU(3)$}{SU(3)}-B and -C Top}
Let us briefly list the main results from the analysis of the remaining two $SU(3)$ tops, with more details relegated to appendix \ref{app-SU3}.
\subsubsection*{\texorpdfstring{\boldmath $SU(3)$}{SU(3)}-B Top}
The $SU(3)$-B top gives rise to five ${\bf 3}$-matter curves with the following $U(1)$ charges:
\begin{align}\label{tab:SU(3)-B-matter-short}
\begin{array}{c|c|| c | c}
\text{matter} & U(1)-\text{charges}& \text{matter} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 & (-\frac{2}{3}, -\frac{4}{3}) & \mathbf{3}^{\mathrm{B}}_4 &(\frac{1}{3}, \frac{2}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 & (-\frac{2}{3}, \frac{2}{3}) & \mathbf{3}^{\mathrm{B}}_5 &(\frac{1}{3},-\frac{1}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_3 & (-\frac{2}{3}, -\frac{1}{3}) & & %
\end{array}
\end{align}
These enjoy a rich spectrum of geometrically realised Yukawa couplings with the singlets,
\begin{align}
\begin{split}
& \mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_2 - \mathbf{1}^{(5)}, \qquad
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_3 - \mathbf{1}^{(6)}, \qquad
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(3)}, \qquad
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(4)}, \\
& \mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_3 - \overline{\mathbf{1}}^{(6)}, \qquad
\mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(2)}, \qquad
\mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(1)}, \qquad
\mathbf{3}^{\mathrm{B}}_3 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(4)}, \\
& \mathbf{3}^{\mathrm{B}}_3 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(2)}, \qquad
\mathbf{3}^{\mathrm{B}}_4 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \overline{\mathbf{1}}^{(6)} \qquad
\end{split}
\end{align}
and among one another,
\begin{eqnarray}
\mathbf{3}^{\mathrm{B}}_3 - \mathbf{3}^{\mathrm{B}}_4 - \mathbf{3}^{\mathrm{B}}_5, \qquad
\mathbf{3}^{\mathrm{B}}_1 - \mathbf{3}^{\mathrm{B}}_4 - \mathbf{3}^{\mathrm{B}}_4, \qquad
\mathbf{3}^{\mathrm{B}}_2 - \mathbf{3}^{\mathrm{B}}_5 - \mathbf{3}^{\mathrm{B}}_5,
\end{eqnarray}
which is in 1-to-1 correspondence with the gauge theoretic selection rules.
Under the transformation $U(1)'_1 \equiv -U(1)_1, \, U(1)'_2 \equiv U(1)_2 - U(1)_1$, the spectrum and Yukawa couplings remains invariant with the identification ${\bf 3}^{\rm B}_1 \leftrightarrow \overline{\bf 3}^{\rm B}_2, \, {\bf 3}^{\rm B}_4 \leftrightarrow \overline{\bf 3}^{\rm B}_5$ and ${\bf 1}^{(1)} \leftrightarrow \overline{\bf 1}^{(3)} , \, {\bf 1}^{(2)} \leftrightarrow \overline{\bf 1}^{(4)}$.
\subsubsection*{\texorpdfstring{\boldmath $SU(3)$}{SU(3)}-C Top}
The third top $SU(3)$-C gives rise to $SU(3)$-charged states with the following $U(1)$ charges:
\begin{align}\label{tab:SU(3)-C-matter-short}
\begin{array}{c|c||c|c}
\text{matter} & U(1)-\text{charges} & \text{matter} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 & (-\frac{2}{3}, -1) & \mathbf{3}^{\mathrm{C}}_4 & (-\frac{2}{3}, 0) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 & (\frac{1}{3}, -1)& \mathbf{3}^{\mathrm{C}}_5 & (\frac{1}{3},0) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_3 & (\frac{1}{3}, 1)
\end{array}
\end{align}
The couplings
\begin{align}
\begin{split}
& \mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_2 - \mathbf{1}^{(2)}, \qquad
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_3 - \mathbf{1}^{(3)} , \qquad
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \mathbf{1}^{(6)} , \qquad
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(4)}, \\
& \mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_3 - \mathbf{1}^{(5)}, \qquad
\mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \overline{\mathbf{1}}^{(1)} , \qquad
\mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(6)} , \qquad
\mathbf{3}^{\mathrm{C}}_3 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \overline{\mathbf{1}}^{(4)},\\
& \mathbf{3}^{\mathrm{C}}_3 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \overline{\mathbf{1}}^{(6)}, \qquad
\mathbf{3}^{\mathrm{C}}_4 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(2)}
\end{split}
\end{align}
and
\begin{eqnarray}
\mathbf{3}^{\mathrm{C}}_1 - \mathbf{3}^{\mathrm{C}}_3 - \mathbf{3}^{\mathrm{C}}_5, \qquad
\mathbf{3}^{\mathrm{C}}_2 - \mathbf{3}^{\mathrm{C}}_3 - \mathbf{3}^{\mathrm{C}}_4, \qquad
\mathbf{3}^{\mathrm{C}}_4 - \mathbf{3}^{\mathrm{C}}_5 - \mathbf{3}^{\mathrm{C}}_5
\end{eqnarray}
are in agreement with gauge theoretic expectations.
Similar to the situation with the $SU(2)$ tops, the $SU(3)$-C top is in fact equivalent to the $SU(3)$-A top (cf.~appendix \ref{app:tops}). Analogous to the $U(1)$ charges identification (\ref{eq:U1-charge-transformation}), we find with
\begin{align}\label{eq:U1-charge-transformation-SU3}
\begin{split}
U(1)_1^{\rm C} &= -U(1)_1^{\rm A} \, , \\
U(1)_2^{\rm C} &= U(1)_2^{\rm A} - U(1)_1^{\rm A}
\end{split}
\end{align}
that the spectrum agrees by identifying the states ${\bf 3}^{\rm A}_i \leftrightarrow {\bf 3}^{\rm C}_i, i=1,...,5$ and ${\bf 1}^{(1)} \leftrightarrow \overline{\bf 1}^{(3)}, {\bf 1}^{(2)} \leftrightarrow \overline{\bf 1}^{(4)}$. Again one needs both the $SU(3)$-A and -C top to construct all inequivalent toric $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ models.
\section[Toric \texorpdfstring{\boldmath $SU(3) \times SU(2) \times U(1) \times U(1)$}{SU(3) x SU(2) x U(1) x U(1)} Realisations]{Toric \texorpdfstring{\boldmath $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$}{SU(3)xSU(2)xU(1)xU(1)} Realisations} \label{sec_3211}
We are now ready to construct F-theory compactifications with gauge group $SU(3) \times SU(2)$, together with two abelian factors.
To this end we start with our elliptic fibration realised as the hypersurface (\ref{eq:hypersurface-equation}) within a $\text{Bl}_2 \mathbb{P}^2$-fibration over ${\cal B}$ and realise an $SU(2)$ and an $SU(3)$ singularity over two independent base divisors $W_2$ and $W_3$, respectively. We focus here on the torically realisable singularities and their resolutions enforced by the tops described in the previous sections.
The base sections $g_m \in \{b_i,c_j,d_k\}$ in (\ref{eq:hypersurface-equation}) must now be of the form
\begin{eqnarray}
g_m = g_{m;k,l} \, w_2^k \, w_3^l
\end{eqnarray}
with $\{w_n = 0\} = W_n$ and $g_{m;k,l}$ generic sections in the class $[g_m] - k [w_2] - l [w_3]$. The powers $k$ and $l$ depend on which of the three $SU(2)$ and $SU(3)$ tops are combined. However, as we will discuss shortly, only 5 of the $3 \times 3 = 9$ possible tops leading to the gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ are inequivalent.
The discriminant of the fibration takes the form
\begin{eqnarray}
\Delta = w_2^2 \, w_3^3 \,( P + {\cal O}(w_2) + {\cal O}(w_3))
\end{eqnarray}
with $P$ a section of the base that
does not vanish identically along $W_2$ and $W_3$.
For generic choice of $w_2$ and $w_3$, the fibration over $w_2$ and $w_3$ then looks like the three individual $SU(2)$ and $SU(3)$ fibrations with vanishing orders $k$ and $l$ over $w_2$ and $w_3$, respectively, but with an additional enhancement over the intersection curve $\{w_2\} \cap \{w_3 \}$ of the two non-abelian loci. Here, the generic choice of $w_2$ and $w_3$ in particular means that this intersection locus is assumed not to coincide with any of the matter curves of the individual tops.
In a toric construction, this means that the toric diagram for the complete five-dimensional ambient space consists of two tops over the polygon for the fibre ambient space. The two tops extend in two mutually orthogonal directions of a five-dimensional lattice. If one projects onto the three-dimensional lattice spanned by the polygon-plane and one of the two directions, one either finds an $SU(2)$ or an $SU(3)$ top, each introducing resolution divisors $E_{0/1} = \{e_{0/1} =0\}$ and $F_{0/1/2} = \{f_{0/1/2}=0\}$. If the toric diagram is reflexive (which has to be checked for each base manifold), then toric geometry guarantees the smoothness of the fourfold $\hat{Y}_4$ that is cut out by the hypersurface equation inside the five-dimensional ambient space. A triangulation of the full polytope will in particular give rise to a triangulation of the $SU(2)$ and $SU(3)$ sub-top, so the corresponding full SR-ideal will -- as sub-ideals -- contain an SR-ideal of each the $SU(2)$ and the $SU(3)$ sub-model; in
addition, there will be further generators that involve both $e_i$ and $f_j$. To use the results of sections \ref{sec:SU(2)-tops} and \ref{sec:SU(3)-tops}, we choose a triangulation of the full polytope that leads to an SR-ideal which as sub-ideals contains the SR-ideals we used when studying the corresponding $SU(2)$ and $SU(3)$ tops individually.
Because the $E_i$ and $F_j$ are fibred over different divisors of the base, intersections of the form
\begin{align}
\int_{\hat{Y}_4} \, E_i \wedge F_j \wedge \pi^{-1} D_a \wedge \pi^{-1} D_b \, ,
\end{align}
with $D_{a/b}$ generic divisors of base, will yield zero. This just means that the roots of $SU(2)$ are uncharged under $SU(3)$ and vice versa -- as one would expect from a product structure $SU(3)\times SU(2)$ of the gauge group. Because the enhancement loci over $W_2$ away from $W_3$
are of the same form as in a model with only the $SU(2)$ singularity over $W_2$ (and similarly for the $SU(3)$ singularity over $W_3$ away from $W_2$), one will also find the same spectrum of matter charged only under $SU(2)$ or $SU(3)$. In addition one will find matter charged both under $SU(2)$ and $SU(3)$ at the enhancement loci $W_2 \cap W_3$. As it turns out, this matter transforms in the bifundamental representation $({\bf 3}, {\bf 2})$.
The $U(1)$ generators are now subject to the condition that the $SU(2)$ and $SU(3)$ roots are uncharged under them. However, since the $SU(2)$ and $SU(3)$ roots are mutually uncharged, this condition is met by setting
\begin{eqnarray} \label{ShiodaSu2Su3}
\omega_i^{SU(2) \times SU(3)} = \omega_i + \Sigma_j \, t_j \, E_j + \Sigma_j \, \tilde{t}_j \, F_j,
\end{eqnarray}
where $\omega_i$ are the generators of the form (\ref{eq:U(1)-generators}) and the correction terms $t_j$ and $\tilde t_j$ are the same as for the individual Shioda maps for the $SU(2)$ and $SU(3)$ tops.
Due to the equivalences amongst the $SU(2)$ and $SU(3)$ tops described in the previous sections, some of the combined $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ models are also equivalent. In fact those models whose spectrum and Yukawa couplings can be mapped onto each other with the $U(1)$ transformation $U(1)'_1 = -U(1)_1, \, U(1)'_2 = U(1)_2 - U(1)_1$ are equivalent. One finds four inequivalent pairs of $SU(2) \times SU(3)$ top combinations, ${\rm I} \times {\rm A} \simeq {\rm II} \times {\rm C}$, ${\rm I} \times {\rm B} \simeq {\rm II} \times {\rm B}$, ${\rm I} \times {\rm C} \simeq {\rm II} \times {\rm A}$, ${\rm III} \times {\rm A} \simeq {\rm III} \times {\rm C}$, and the invariant model ${\rm III} \times {\rm B}$. A more detailed explanation based on the tops can be found in appendix \ref{app:tops}.
To summarise, in order to construct toric F-theory models with $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ gauge symmetry we take an $SU(2)$ and an $SU(3)$ top of the form studied in the sections \ref{sec:SU(2)-tops} and \ref{sec:SU(3)-tops} and combine them into one new top. Some of the tops obtained in this way are equivalent. The previous sections directly give us the spectrum of $\mathbf{2}$- and $\mathbf{3}$-matter including their Yukawa couplings. What we need to compute is the bifundamental matter $(\mathbf{3},\mathbf{2})$ as well as all the Yukawa couplings it is involved in.
The result of this analysis is shown in table \ref{tab:SU(2)xSU(3)} for all five mutually inequivalent combinations of tops. In all cases the geometrically realised Yukawa couplings are in 1-to-1 correspondence with the set of gauge theoretically allowed couplings including the $U(1)_i$ selection rules.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{top-combination} & (U(1)_1, U(1)_2)- & \text{additional gauge invariant Yukawas,} \\
SU(2) \times SU(3) & \text{charge of } (\mathbf{3},\mathbf{2}) & (\mathbf{3},\mathbf{2})- \ldots \\ \hline\hline \rule{0pt}{3ex}
\mathrm{I} \times \mathrm{A} & (\frac{1}{6},-\frac{1}{3}) & \overline{\mathbf{3}}^\mathrm{A}_1-\mathbf{2}^\mathrm{I}_3, \, \overline{\mathbf{3}}^\mathrm{A}_2-\overline{\mathbf{2}}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{A}_3-\overline{\mathbf{2}}^\mathrm{I}_1, \, \overline{\mathbf{3}}^\mathrm{A}_4-\mathbf{2}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{A}_5-\overline{\mathbf{2}}^\mathrm{I}_3; \\ \rule{0pt}{3ex}
& & (\mathbf{3},\mathbf{2})-\mathbf{3}^\mathrm{A}_3 \\ \hline \rule{0pt}{3ex}
\mathrm{I} \times \mathrm{B} & (-\frac{1}{6},-\frac{1}{3}) & \overline{\mathbf{3}}^\mathrm{B}_1-\overline{\mathbf{2}}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{B}_2-\overline{\mathbf{2}}^\mathrm{I}_1, \, \overline{\mathbf{3}}^\mathrm{B}_3-\overline{\mathbf{2}}^\mathrm{I}_3, \, \overline{\mathbf{3}}^\mathrm{B}_4-\mathbf{2}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{B}_5-\mathbf{2}^\mathrm{I}_3; \\ \rule{0pt}{3ex}
& & (\mathbf{3},\mathbf{2})-\mathbf{3}^\mathrm{B}_4 \\ \hline \rule{0pt}{3ex}
\mathrm{I} \times \mathrm{C} & (-\frac{1}{6},0) & \overline{\mathbf{3}}^\mathrm{C}_1-\overline{\mathbf{2}}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{C}_2-\mathbf{2}^\mathrm{I}_1, \, \overline{\mathbf{3}}^\mathrm{C}_3-\mathbf{2}^\mathrm{I}_2, \, \overline{\mathbf{3}}^\mathrm{C}_4-\overline{\mathbf{2}}^\mathrm{I}_3, \, \overline{\mathbf{3}}^\mathrm{C}_5-\mathbf{2}^\mathrm{I}_3; \\ \rule{0pt}{3ex}
& & (\mathbf{3},\mathbf{2})-\mathbf{3}^\mathrm{C}_5 \\ \hline \rule{0pt}{3ex}
\mathrm{III} \times \mathrm{A} & (-\frac{1}{3},-\frac{1}{3}) & \overline{\mathbf{3}}^\mathrm{A}_1-\mathbf{2}^\mathrm{III}_1, \, \overline{\mathbf{3}}^\mathrm{A}_2-\overline{\mathbf{2}}^\mathrm{III}_3, \, \overline{\mathbf{3}}^\mathrm{A}_3-\mathbf{2}^\mathrm{III}_3, \, \overline{\mathbf{3}}^\mathrm{A}_4-\mathbf{2}^\mathrm{III}_2; \\ \rule{0pt}{3ex}
& & (\mathbf{3},\mathbf{2})-\mathbf{3}^\mathrm{A}_4 \\ \hline \rule{0pt}{3ex}
\mathrm{III} \times \mathrm{B} & (-\frac{2}{3},-\frac{1}{3}) & \overline{\mathbf{3}}^\mathrm{B}_1-\overline{\mathbf{2}}^\mathrm{III}_3, \, \overline{\mathbf{3}}^\mathrm{B}_2-\mathbf{2}^\mathrm{III}_3, \, \overline{\mathbf{3}}^\mathrm{B}_4-\mathbf{2}^\mathrm{III}_2, \, \overline{\mathbf{3}}^\mathrm{B}_5-\mathbf{2}^\mathrm{III}_1; \\ \rule{0pt}{3ex}
&&(\mathbf{3},\mathbf{2})-\mathbf{3}^\mathrm{B} \, \text{non-existent}
\end{array}
\end{align*}
\caption{$U(1)$ charges of the bifundamental matter and additional Yukawa couplings involving at least one $(\mathbf{3},\mathbf{2})$, as arising in the five inequivalent combinations of the $SU(2)$ and $SU(3)$ tops studied in the previous chapters.}
\label{tab:SU(2)xSU(3)}
\end{table}
We explain the first example in a bit more detail. Combining the $SU(2)$-I and the $SU(3)$-A top amounts to restricting the sections appearing in (\ref{eq:hypersurface-equation})
as
\begin{align}\label{eq:I-A-coeffs}
\begin{split}
b_0 &= b_{0;1,1}\, e_0 \, f_0 ,\quad b_2 = b_{2;0,0}\,e_1 \, f_1\,f_2 ,\quad c_1 = c_{1;0,0}\,e_1 \, f_2 ,\quad c_2 = c_{2;0,1}\,f_0\,f_2 , \\
d_0 &= d_{0;1,1}\,e_0 \, f_0\,f_1,\quad d_1 = d_{1;0,0}\,f_1, \quad d_2 = d_{2;1,1}\,e_0 \, f_0\,f_1^2.
\end{split}
\end{align}
In table \ref{tab:IxA-divisor-classes} we list the divisor classes and the corresponding scaling relations among the fibre coordinates. The last part shows the lattice vectors of the top that describes the ambient space. The vectors $\underline{x}$ and $\underline{y}$ should be linearly independent, but otherwise unspecified for a generic base $\cal B$. For this top there exist 16 different triangulations. We choose a triangulation for which the SR-ideal is the union of the individual ideals (\ref{eq:SU(2)-I-SR-ideal}) and (\ref{eq:SU(3)-A-SR-ideal}) together with the element $\{f_0 \, e_1\}$, i.e. it is generated by
\begin{align}\label{eq:SR-ideal-IxA}
\mathrm{u} \, \mathrm{v} , \mathrm{u} \, \mathrm{w} , \mathrm{w} \, s_0 , \mathrm{v} \, s_1 , s_0 \, s_1 , e_0 \, \mathrm{w} , e_1 \, s_0 , e_1 \, \mathrm{u} , \, f_0\,\mathrm{w} , f_0\,s_1 , f_1\,s_0 , f_1\,\mathrm{v} , f_2\,s_0 , f_2\,s_1 , f_2\,\mathrm{u} , f_0\,e_1.
\end{align}
\begin{table}[ht]
\begin{align*}
\begin{array}{c || c c c c c | c c | c c c}
& \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & e_0 & e_1 & f_0 & f_1 & f_2 \\ \hline \hline
\left[W_2 \right] & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \cdot & \cdot & \cdot & \cdot \\
\left[W_3 \right] & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \cdot & \cdot \\
\alpha & \cdot & \cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\beta & \cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \hline
U & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \hline
E_1 & \cdot & \cdot & -1 & \cdot & \cdot & -1 & 1 & \cdot & \cdot & \cdot \\ \hline
F_1 & \cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & -1 & 1 & \cdot \\
F_2 & \cdot & \cdot & -1 & \cdot & \cdot & \cdot & \cdot & -1 & \cdot & 1 \\ \hline \hline
\multirow{3}{*}{\text{toric data}} & -1 & 0 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 1 \\
& 1 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
& \underline{0} & \underline{0} & \underline{0} & \underline{0} & \underline{0} & \underline{x} & \underline{x} & \underline{y} & \underline{y} & \underline{y}
\end{array}
\end{align*}
\caption{Divisor classes and coordinates of the ambient space for top-combination $\mathrm{I} \times \mathrm{A}$.}
\label{tab:IxA-divisor-classes}
\end{table}
Irrespective of the chosen triangulation,
one recovers from the discriminant the ${\bf 2}$- and ${\bf 3}$-matter curves of the two individual tops, (\ref{tab:SU(2)-I-matter}) and (\ref{tab:SU(3)-A-matter}), where of course the base sections $g_{m,k}$
defining the matter curves are modified in agreement with (\ref{eq:I-A-coeffs}).\footnote{In addition, the same type $III$ and $IV$ enhancement loci arise as before, which do not carry matter representations.}
For example, the representation ${\bf 2}^{\rm I}_1$ is located at the intersection $\{w_2\} \cap \{c_{2;0,1}\}$ (because in the blow-down the locus $\{w_2\} \cap \{c_2\}$ appearing in $(\ref{tab:SU(2)-I-matter})$ splits into $\{w_2\} \cap \{c_{2;0,1}\}$ and $\{w_2\} \cap \{w_3\}$, but the latter contributes to matter charged under both $SU(2)$ and $SU(3)$),
and the curve hosting ${\bf 3}^{\rm A}_3$ is now given by $\{w_3\} \cap \{ b_{0;1,1} \, w_2 \, c_{1;0,0}- b_1 \, c_{2;0,1}\}$.
What is new is that along the curve $\{w_2\} \cap \{w_3\}$ the Kodaira type of the fibre enhances to split type $I_5$, corresponding to vanishing orders $(0,0,5)$ of $(f,g,\Delta)$ in the Weierstrass model.
Indeed, the fibre components straightforwardly split to form the affine Dynkin diagram of $SU(5)$; more precisely the five fibre components are given by
\begin{align}
\begin{split}
\mathbb{P}^1_{00} \equiv P_T|_{\{e_0\} \cap \{f_0\}} &= b_{2;0,0} \, f_1 \, f_2 \, {\rm u} + d_{1;0,0} \, f_1 \, s_0 \, {\rm u}^2
+ c_{1;0,0} \, f_2 {\rm v} + b_1 \, s_0 \, {\rm u} \, {\rm v}, \\
\mathbb{P}^1_{01} \equiv P_T|_{\{e_0\} \cap \{f_1\}} &= c_{2;0,1} \, f_0 \, f_2 + c_{1;0,0}\, e_1 \, f_2 \, s_1 + b_1 \, s_1 \, {\rm u}, \\
\mathbb{P}^1_{02} \equiv P_T|_{\{e_0\} \cap \{f_2\}} &= d_{1;0,0} \, f_1 + b_1 \, {\rm v}, \\
\mathbb{P}^1_{11} \equiv P_T|_{\{e_1\} \cap \{f_1\}} &= b_{0;1,1} \, e_0 + c_{2;0,1} \, f_2 \, {\rm w} + b_1 \, s_1 \, {\rm w}, \\
\mathbb{P}^1_{12} \equiv P_T|_{\{e_1\} \cap \{f_2\}} &= d_{2;1,1} \, e_0 \, f_1^2\, + d_{0;1,1} \, e_0 \, f_1 \, {\rm v} + b_{0;1,1} \, e_0 \, {\rm v}^2 + d_{1;0,0} \, f_1 \, {\rm w} + b_1 \, {\rm v} \, {\rm w},
\end{split}
\label{fiveP1}
\end{align}
where we used the SR-ideal to set as many coordinates to one as possible. Note that $e_1$ and $f_0$ do not intersect due to the SR-ideal relations so that the locus $P_T|_{\{e_1\} \cap \{f_0\}}$ is absent in (\ref{fiveP1}).
The fibre component $\mathbb{P}^1_{00} + \mathbb{P}^1_{02}$ is identified with the highest weight of the bifundamental representation $({\bf 3},{\bf 2})$; the $U(1)_1$ and $U(1)_2$ charges are found to be $(\frac{1}{6}, -\frac{1}{3})$ by computing the intersection product with the generators
\begin{align}\label{eq:I-A-U(1)-generators}
\begin{split}
\omega_1^{{\rm I} \times {\rm A}} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{1}{2} E_1 + \frac{2}{3} F_1 + \frac{1}{3} F_2 ,\\
\omega_2^{{\rm I} \times {\rm A}} &= U - S_0 - \overline{\mathcal{K}} - [c_{1;0,0}] + \frac{2}{3} F_1 + \frac{1}{3} F_2.
\end{split}
\end{align}
Finally, we have analysed the intersection of $\{w_2\} \cap \{w_3\}$ with each of the ${\bf 3}$- and ${\bf 2}$-curves to identify extra fibre enhancements signalling Yukawa couplings involving the new $({\bf 3},{\bf 2})$-state.
The fibres over the Yukawa points ${\bf (3,2)} - {\bf \bar 3} - {\bf 2}/{\bf \bar 2}$ are of split Kodaira type $I_6$, and the fibre components can be explicitly
\begin{wrapfigure}[13]{r}{.2\textwidth}
\vspace{-13pt}
\begin{center}
\def.98\hsize{.9\hsize}
\input{non-standard-fibre.pdf_tex}
\caption{Non-standard fibre structure at the $(\mathbf{3},\mathbf{2}) - (\mathbf{3},\mathbf{2}) - \mathbf{3}$ Yukawa point.}
\label{fig:non-standard}
\end{center}
\end{wrapfigure}
checked to form the affine Dynkin diagram of $SU(6)$.
The base points are found by intersecting $W_3$ and $W_2$ with the five ${\bf 3}$-curves and noting that the intersection points also lie on top of one of the ${\bf 2}$-curves.
A special role is played by the intersection locus $\{w_3\} \cap \{w_2 \} \cap \{ b_{0;1,1} \, w_2 \, c_{1;0,0}- b_1 \, c_{2;0,1}\}$, where the last term is the ${\bf 3}^{\rm A}_3$-curve.
Apart from the intersection $\{w_3\} \cap \{w_2 \} \cap \{c_{2;0,1}\}$, where the $(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_3-\overline{\mathbf{2}}^\mathrm{I}_1 $ Yukawa coupling is localised,
there is also the intersection $\{w_3\} \cap \{w_2 \} \cap \{b_1\}$, which does not lie on any of the ${\bf 2}$-curves. The fibre components almost form an affine Dynkin diagram of $SO(8)$ (including the multiplicity $2$ for the interior node) except for an additional intersection point between two of four the exterior nodes which makes this fibre non-standard (cf.~figure \ref{fig:non-standard}).
The associated Yukawa coupling is $(\mathbf{3},\mathbf{2}) - (\mathbf{3},\mathbf{2}) - \mathbf{3}^\mathrm{A}_3$.
\begin{table}[b]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus} = W_2 \cap W_3 \cap \ldots & \text{fibre type} \\ \hline\hline \rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_1-\mathbf{2}^\mathrm{I}_3 & \{b_{0;1,1}\} & I_6 \\[.5ex] \hline \rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_2-\overline{\mathbf{2}}^\mathrm{I}_2 & \{c_{1;0,0} \} & I_6 \\[.5ex] \hline\rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_3-\overline{\mathbf{2}}^\mathrm{I}_1 & \{c_{2;0,1}\} & I_6 \\[.5ex] \hline \rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_4-{\mathbf{2}}^\mathrm{I}_2 & \{b_1 \, b_{2;0,0} - c_{1;0,0} \, d_{1;0,0} \} & I_6 \\[.5ex] \hline \rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - \overline{\mathbf{3}}^\mathrm{A}_5-\overline{{\mathbf{2}}}^\mathrm{I}_3 & \{b_{0;1,1,} \, d_{1;0,0}^2 - b_1 \, d_{0;1,1} \, d_{1;0,0} + b_1^2 \, d_{2;1,1} \} & I_6 \\[.5ex] \hline\hline \rule{0pt}{3ex}
(\mathbf{3},\mathbf{2}) - (\mathbf{3},\mathbf{2}) - {\mathbf{3}}^\mathrm{A}_3 & \{b_1\} & {\rm non-standard} \\[.5ex]
%
\end{array}
\end{align*}
\caption{Details on the additional Yukawas involving bifundamental matter in the top combination $\mathrm{I} \times \mathrm{A}$.}
\label{tab:I-A-Yukawas}
\end{table}
We have checked the above calculations for a specific fibration over the base ${\cal B} = \mathbb{P}^3$ with $H^{1,1}(\mathbb{P}^3) = \{ n \cdot H | n \in \mathbb{Z}\}$, where $H$ is the hyperplane class, and $\overline{\cal K}_{\cal B} = 4H$. For simplicity we take $w_2$ and $w_3$ to be two of the four homogeneous coordinates $(z_0 : z_1 : z_2 : z_3)$, e.g.~$w_2 = z_0$ and $w_3 = z_1$; then $W_2 = W_3 = H$. Recall that $b_i,c_j,d_k$ must transform as sections of specific line bundles, see table~\ref{coeff}, where there is freedom left in choosing $\alpha$ and $\beta$. They are subject to further constraints as the restricted sections (\ref{eq:I-A-coeffs}) in the presence of the non-abelian symmetry must be effective classes. Over ${\cal B} = \mathbb{P}^3$, these constraints are met with $\alpha = 2H$, $\beta = H$, implying the classes of the restricted sections to be
\begin{align}\label{eq:explicit-base-classes}
\begin{split}
& [b_{0;1,1}] = 3 H, \quad [b_1] = 4 H , \quad [b_{2;0,0}] = 3 H, \quad [c_{1;0,0}] = 2 H, \\
& [c_{2;0,1}] = 2 H, \quad [d_{0;1,1}] = 4 H , \quad [d_{1;0,0}] = 5H , \quad [d_{2;1,1}] = 5 H \, .
\end{split}
\end{align}
From the choice of $\alpha$ and $\beta$ we have to impose the condition $2 \vec{\mathrm{w}} + \vec{\mathrm{v}} + \vec{e_0} + \vec{f_0} + \vec{z_2} + \vec{z_3} =0$, where $\vec{(\cdot)}$ is the $(\cdot)$-coordinate's lattice vector of the toric diagram of the full fibration (`toric data' in table \ref{tab:IxA-divisor-classes}). This condition is met by the toric ambient space $\hat{X}_5$, whose toric diagram has the lattice vectors shown in table \ref{tab:toric-table-IxA}. The resulting polytope is reflexive, guaranteeing that the fourfold cut out by the hypersurface equation inside this toric ambient space is smooth. The Euler characteristic of the fourfold is 1440.
\begin{table}[ht]
\begin{align*}
\begin{array}{c | c | c | c | c | c | c | c | c | c | c | c}
\vec{\mathrm{u}} & \vec{\mathrm{v}} & \vec{\mathrm{w}} & \vec{s_0} & \vec{s_1} & \vec{e_0} & \vec{f_0} & \vec{z_2} & \vec{z_3} & \vec{e_1} & \vec{f_1} & \vec{f_2} \\ [.3ex] \hline
-1 & 0 & 1 & -1 & 0 & 0 & 0 & -2 & 0 & 1 & 0 & 1 \\
1 & -1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0
\end{array}
\end{align*}
\caption{Toric diagram for the ambiente space $\hat{X}_5$ of the model $ \mathrm{I} \times \mathrm{A}$ over the base ${\cal B} = \mathbb{P}^3$.}
\label{tab:toric-table-IxA}
\end{table}
The above analysis can be repeated for the remaining four inequivalent combinations of tops and leads to the couplings listed in (\ref{tab:SU(2)xSU(3)}). Note that for the top combinations ${\rm III} \times {\rm A}$ and ${\rm III} \times {\rm B}$ no gauge invariant coupling ${\bf (3,2)} - {\overline{\bf 3}}^\mathrm{A}_5 - {\bf 2}$ resp.~${\bf (3,2)} - {\overline{\bf 3}}^\mathrm{B}_3 - {\bf 2}$ exists. In both cases, the intersection point of the ${\bf (3,2)}$- and the corresponding ${\overline{\bf 3}}$-curve lies on the curve of non-split $I_3$-enhancement described after (\ref{eq:SU(2)-III-splitting-quadratic-curve-2}) in appendix \ref{SU2DetailsApp}.
Were it not for a monodromy along that curve, an additional ${\bf 2}$-representation would arise, which in fact would have the correct quantum numbers to couple as in ${\bf (3,2)} - {\overline{\bf 3}}^\mathrm{A}_5 - {\bf 2}$ (or ${\bf (3,2)} - {\overline{\bf 3}}^\mathrm{B}_3 - {\bf 2}$). Correspondingly,
the fibre over the triple intersection of these curves does enhance to form an $I_6$ Kodaira fibre, but due to the described monodromy no physical Yukawa couplings result as the {\bf 2}-state in question is projected out.
More drastically, the top combination ${\rm III} \times {\rm B}$ exhibits a non-Kodaira enhancement in the fibre over $\{w_3\} \cap \{w_2 \} \cap \{b_1\}$.\footnote{In the other four top combinations this point corresponds to the $({\bf 3},{\bf 2})-({\bf 3},{\bf 2})-{\bf 3}$ coupling. For ${\rm III} \times {\rm B}$, however, there is no suitable $\bf 3$ state.}
At this set of points, several matter curves coincide and the vanishing orders of $(f,g,\Delta)$ in the Weierstrass model take the values $(4,6,12)$. For such high enhancement no flat crepant resolution can be found. To see how this manifest itself, consider the triangulation of the ${\rm III} \times {\rm B}$ top leading to an SR-ideal generated by (\ref{eq:SU(2)-III-SR-ideal}) (for the $SU(2)$ part), (\ref{eq:SU(3)-B-SR-ideal}) (for the $SU(3)$ part) and the new elements $ \{ s_1 \, e_0 \, f_1 , e_1 \, f_0 \, f_2 \} $: For this SR-ideal the fibration becomes non-flat over $\{w_3\} \cap \{w_2 \} \cap \{b_1\}$ as the hypersurface equation $P_T$ becomes trivial for $b_1 = e_0 = f_1 = 0$.
Therefore, in order for the top combination ${\rm III} \times {\rm B}$ to give rise to a well-defined F-theory compactification, the point set $\{w_3\} \cap \{w_2 \} \cap \{b_1\}$ must be empty.
Since $b_1$ is universally in the class $\overline{\cal K}$, these points cannot be turned off by a suitable choice of classes $\alpha$ and $\beta$ appearing in table (\ref{coeff}).
While it is not excluded that $W_2$ and $W_3$ can be found such that no intersection points $\{w_3\} \cap \{w_2 \} \cap \{b_1\}$ arise, we do currently not have an example of this type.
\section{ Standard Model Embeddings}\label{sec_SM}
\subsection{Criteria for Standard Model Embeddings }\label{subsec_spectrum_search}
The toric fibrations with gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ constructed in the previous section are the starting point of our search for F-theory vacua with Standard Model gauge group and matter. Our discussion will be phrased in the framework of the ${\cal N}=1$ Minimal Supersymmetric Standard Model (MSSM), potentially extended by further singlets, with the understanding that supersymmetry is broken at an priori unknown energy below the compactification scale in agreement with current lower collider bounds.
To fix our conventions we recall the MSSM spectrum plus right-handed neutrinos $\nu^c_R$ (taking all fields to be chiral ${\cal N}=1$ superfields) in table \ref{tab:SM-spectrum}.
We also allow for a generation of the
$\mu$-term in the Higgs sector via the VEV of an MSSM singlet ${\bf 1}_\mu$, as studied extensively in the literature in the framework of the NMSSM (see e.g. \cite{Ellwanger:2009dp} and references therein).\footnote{A detailed and systematic analysis of such singlet extensions of the MSSM in perturbative Type II intersecting brane quivers has been performed in \cite{Cvetic:2010dz}. }
\begin{table}[ht]
\begin{align*}
\begin{array}{cc|c|c}
\text{matter} & & \text{representation} & \text{hypercharge} \\ \hline \hline \rule{0pt}{3ex}
\text{left-handed quarks} & Q & (\mathbf{3},\mathbf{2}) & \hphantom{-}\frac{1}{6} \\ \rule{0pt}{3ex}
\text{right-handed up-quarks} & u^c_R & (\overline{\mathbf{3}},\mathbf{1}) \equiv \overline{\mathbf{3}}_u & -\frac{2}{3} \\ \rule{0pt}{3ex}
\text{right-handed down-quarks} & d^c_R & (\overline{\mathbf{3}},\mathbf{1}) \equiv \overline{\mathbf{3}}_d & \hphantom{-}\frac{1}{3} \\ \rule{0pt}{3ex}
\text{Higgs-up} & H_u & (\mathbf{1},\mathbf{2}) \equiv \mathbf{2}_u & \hphantom{-}\frac{1}{2} \\ \rule{0pt}{3ex}
\text{Higgs-down} & H_d & (\mathbf{1},\mathbf{2}) \equiv \mathbf{2}_d & -\frac{1}{2} \\ \rule{0pt}{3ex}
\text{left-handed leptons} & L & (\mathbf{1},\mathbf{2}) \equiv \mathbf{2}_L & -\frac{1}{2} \\ \rule{0pt}{3ex}
\text{right-handed electrons} & e^c_R & (\mathbf{1},\mathbf{1}) \equiv \mathbf{1}_e & \hphantom{-}1 \\ \rule{0pt}{3ex}
\text{right-handed neutrinos} & \nu^c_R & (\mathbf{1},\mathbf{1}) \equiv \mathbf{1}_\nu & \hphantom{-}0 \\ \rule{0pt}{3ex}
\text{$\mu$-singlet} && (\mathbf{1},\mathbf{1}) \equiv \mathbf{1}_\mu & \hphantom{-}0
\end{array}
\end{align*}
\caption{Matter spectrum of the MSSM.}
\label{tab:SM-spectrum}
\end{table}
At the level of renormalisable couplings, the superpotential of the singlet-extended MSSM takes the form
\begin{eqnarray} \label{W1W2}
W &=& W_1 + W_2 + W_{\rm singlet}, \\
W_1 & =& Y_u \, Q\,H_u\,u^c_R + Y_d \, Q\,H_d\,d^c_R + Y_e \, L \, H_d \, e_R^c + Y_{\nu} \, L\,H_u\,\nu^c_R + \mu \, H_u \, H_d, \\
W_2 & = & \alpha \, Q \, L \, d_R^c + \beta \, u_R^c \, d_R^c \, d_R^c + \gamma \, L \, L \, e_R^c + \kappa \, L \, H_u, \label{eq:couplings_W2}\\
W_{\rm singlet} &=& \delta_{3,0} \, {\bf 1_{\mu}} \, {\bf 1_{\mu}} \, {\bf 1_{\mu}} + \delta_{2,1} \, {\bf 1_{\mu}} \, {\bf 1_{\mu}} \, \nu_R^c + \delta_{1,2} \, {\bf 1_{\mu}} \, \nu_R^c \, \nu_R^c + \delta_{0,3} \, \nu_R^c \, \nu_R^c \, \nu_R^c, \label{eq:couplings_W_singlet}
\end{eqnarray}
where we are suppressing family indices.
Here, $W_1$ contains the Yukawa couplings that give rise to the masses for the up-quarks, down-quarks and the charged leptons as well as potential Dirac masses for the right-handed neutrinos. We also include here the $\mu$-term for the Higgs sector, with the understanding that this term might originate from a Yukawa coupling $Y_{\mu} \, {\bf 1}_{\mu} H_u \, H_d $ if the scalar in the superfield ${\bf 1}_{\mu}$ acquires a non-trivial VEV.
For completeness we have furthermore listed possible dimension-four singlet couplings
$W_{\rm singlet} $. If $\langle {\bf 1}_{\mu} \rangle \neq 0$ the third term in $W_{\rm singlet} $ effectively contributes to the Majorana mass term for the right-handed neutrinos, while the first term would induce an F-term in the vacuum and is therefore of interest in the context of supersymmetry breaking. We have not listed potential tadpole and holomorphic mass terms involving the singlets, which are also allowed by the MSSM gauge group.
The couplings in $W_2$ each violate R-parity $(-1)^{2S + 3 (B-L)}$ with $S$ the spin and $B$, $L$ baryon and lepton number. The second term does in addition not conserve baryon number, while the remaining terms are lepton-number violating.
In particular, some combinations of terms within $W_2$ lead to rapid proton decay and are therefore severely constrained \cite{Allanach:1999ic,Nath:2006ut}.
Proton decay due to dimension-four operators requires both baryon and lepton-number violating contributions. The most severe constraints arise from tree-level induced proton decay, which is generated only if
both $\alpha$ and $\beta$ are non-zero simultaneously.
However, the precise bounds on the couplings depend, amongst other things, on the scale of supersymmetry breaking. In models with intermediate or high-scale supersymmetry breaking some of the constraints on $W_2$ are considerably relaxed compared to TeV-scale supersymmetric scenarios. For more details of the extremely rich phenomenology of R-partiy violating couplings we refer in addition to \cite{Allanach:1999ic,Allanach:2003eb,Nath:2006ut} and references therein.
At mass dimension five, the MSSM allows for the following baryon or lepton number violating operators \cite{Allanach:2003eb},
\begin{align}
\begin{split}\label{W3K}
W_3 &= \lambda_1 \, Q \, Q \, Q \, L + \lambda_2 \, u_R^c \, u_R^c \, d_R^c \, e_R^c + \lambda_3 \, Q \, Q \, Q \, H_d + \lambda_4 \, Q \, u_R^c \, e_R^c \, H_d \\
& + \lambda_5 \, L \, L \, H_u \, H_u + \lambda_6 \, L \, H_d \, H_u \, H_u,
\end{split} \\
K &\supset \lambda _7 \, u_R^c \, (d_R^c)^* \, e_R^c + \lambda_8 \, H_u^* \, H_d \, e_R^c + \lambda_9 \, Q \, u_R^c \, L^* + \lambda_{10} \, Q \, Q \, (d_R^c)^*. \label{eq:couplings_K}
\end{align}
Additional dimension-five terms are possible which involve the singlets $\nu_R^c$ and ${\bf 1}_\mu$. In particular, any of the dimension-four operators present in (\ref{W1W2}) can in principle be dressed with such a singlet.
We do not list these couplings explicitly here.
Our attitude towards lepton and baryon number violating couplings is as follows:
In order to fully explore the parameter space of possible Standard Models within our framework we do not insist on TeV scale supersymmetry a priori, but rather allow for the possibility of intermediate scale supersymmetry breaking. While the viability of such a scenario will ultimately be determined experimentally, a higher supersymmetry breaking scale is in fact a natural option in direct Standard Model constructions. After all, the exact unification of the gauge couplings at a scale around $10^{16}$ GeV, which is one of the predictions of the TeV scale MSSM, is not immediate if the gauge groups $SU(3)$, $SU(2)$ and $U(1)_Y$ are constructed independently.
More importantly perhaps, intermediate scale supersymmetry is well-motivated by a $126$ GeV Higgs
as studied in string theoretic frameworks recently in \cite{Hebecker:2012qp,Ibanez:2012zg,Hebecker:2013lha,Ibanez:2013gf,Hebecker:2014uaa} (see also \cite{Chatzistavrakidis:2012bb,Hall:2013eko,Ibanez:2014zsa,Hall:2014vga} and references therein for other recent examples in the literature motivating an intermediate supersymmetry breaking scale).
Keeping an open mind towards the supersymmetry breaking scale, we do therefore not require absence of all dimension-four and -five lepton and baryon number violating couplings in our search criterion for Standard Model configurations, but will only list which of these couplings are present. A more detailed study of the associated phenomenology, taking into account the details of supersymmetry breaking, is left for future explorations. Having said that, in many cases the $U(1)$ selection rules do prevent potentially dangerous such operators as we will see explicitly.
For each of the five combinations of tops with gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ a plethora of possibilities arises for identifying the massless representations with the MSSM fields.
This identification will in particular determine which linear combination of $U(1)_1$ and $U(1)_2$ corresponds to hypercharge $U(1)_Y$. The orthogonal combination is then massless in absence of gauge fluxes and will remain as a perturbative selection rule after gauge fluxes induce a St\"uckelberg mass for the associated gauge potential. At the same time it must be ensured that the gauge fluxes do not render hypercharge massive, see section \ref{sec_fluxes}.
In the sequel we classify the possible identifications along the following lines:
\begin{itemize}
\item
Since the fibrations under consideration contain only one type of $({\bf 3},{\bf 2})$-curve, all three generations of left-handed quark fields $Q$ must reside on this single $({\bf 3},{\bf 2})$-curve. The $U(1)_1 \times U(1)_2$ charges of $Q$ for the five possible tops are listed in table \ref{tab:SU(2)xSU(3)}.
\item
In a second step we identify the fields $(H_u,H_d)$ with two of the ${\bf 2}_i$-representations or their conjugate representations ${\bf \overline 2}_i$, $i=1,2,3$.
A definite assignment of $(H_u,H_d)$ together with the $U(1)$ charges of $Q$ determines $U(1)_Y$ as a linear combination
\begin{eqnarray} \label{hyper-ass}
U(1)_Y = a\,U(1)_1 + b\,U(1)_2, \quad a,b \in \mathbb{R}.
\end{eqnarray}
We then identify the different possible choices of ${\bf 2}_i$- or ${\bf \overline 2}_i$-states for the left-handed leptons $L$ based on their hypercharge.
The same value of $(a,b)$ and identification of $(H_u,H_d)$ may be compatible with more than one choice for $L$.
In this case, different generations of leptons $L$ may reside on different matter curves and will then be distinguished by their charge under the linear combination of $U(1)_1$ and $U(1)_2$ orthogonal to $U(1)_Y$.
\item
For the specific values of $(a,b)$ in (\ref{hyper-ass}) we next check which of the six singlets ${\bf 1}^{(k)}$, $k=1, \ldots,6$ (and their conjugates) have the correct hypercharge to be identified with the fields $\nu^c_R$ and $e_R^c$,
and similarly which of the ${\bf \overline 3}_j$-representations for $j=1, \ldots, 5$ have the correct hypercharge to be identified with $u_R^c$ and $d_R^c$.
If there is no possible assignment of $e_R^c$, $u_R^c$ or $d_R^c$ we discard this choice of hypercharge. However, to be as general as possible, we do allow for configurations with no right-handed neutrinos $\nu^c_R$.
\item
There are now two types of right-handed leptons and quarks: If the Yukawa couplings $W_1$ in (\ref{W1W2}) are indeed among the geometrically realised couplings as analysed in the previous sections, the fields acquire a perturbative mass term upon electro-weak symmetry breaking. Those generations of MSSM matter for which this is the case will therefore be called `heavy'.\footnote{\label{footcaveat}Note that if two or more families are localised on the same matter curve the rank of the perturbative Yukawa coupling matrix is non-maximal, at least if there exists only one Yukawa coupling point, as studied in the F-theory GUT literature \cite{Heckman:2008qa,Font:2009gq,Cecotti:2009zf,Conlon:2009qq}. In this case some of the `heavy' fields do not receive a perturbative mass after all. Non-perturbative effects can solve this rank-one problem \cite{Marchesano:2009rz,Aparicio:2011jx,Font:2012wq,Font:2013ida}.}
Otherwise, the Yukawas, which are now forbidden by the extra $U(1)$ selection rules, must be generated either by non-perturbative effects \cite{Blumenhagen:2006xt,Ibanez:2006da,Haack:2006cy,Florea:2006si,Blumenhagen:2009qh}, here by M5-brane instantons\footnote{The generation of charged operators via D3/M5-branes in F-theory along the lines of \cite{Blumenhagen:2006xt,Ibanez:2006da,Haack:2006cy,Florea:2006si,Blumenhagen:2009qh} has been studied recently in \cite{Marsano:2008py,Marsano:2008py,Blumenhagen:2010ja,Donagi:2010pd,Cvetic:2011gp,Marsano:2011nn,Grimm:2011dj,Kerstan:2012cy}, and related aspects of such instantons in F-theory appear in \cite{Cvetic:2009ah,Cvetic:2010rq,Bianchi:2011qh,Bianchi:2012kt,Cvetic:2012ts,Martucci:2014ema}.}, or via higher non-renormalisable couplings involving one or more extra singlet states as these acquire a VEV.
In both cases the mass terms will generically be suppressed and the corresponding fields will be called `light'. Again it is understood that different generations can be distributed over the various matter curves.
In particular, if only one of the generations enjoys a perturbative coupling, this could serve as a realisation of the observed mass hierarchies in the MSSM.\footnote{The generation of such mass hierarchies in perturbative Type II MSSM quivers has been studied systematically in \cite{Ibanez:2008my,Cvetic:2009yh,Cvetic:2009ez}.}
Note that while the generation of masses for the `light' generations by M5-instantons depends on the specific geometry of the base ${\cal B}$, the mechanism involving singlet fields could be analysed already at this general level by checking for the existence of singlets with appropriate $U(1)_i$-charges to form a dimension-5 coupling of the required type. We leave such a more advanced analysis for further studies.
\item
The $U(1)_Y$ charges together with the spectrum of perturbative couplings also provide candidates for $\mu$-singlets ${\bf 1}_\mu$ with a Yukawa coupling ${\bf 1}_{\mu} H_u \, H_d $, which we list. As anticipated, if $\langle {\bf 1}_{\mu}
\rangle \neq 0$ this will induce a $\mu$-term in the Higgs sector. In absence of such a VEV the $\mu$-term can in principle be generated via M5-instantons.
Note that sometimes the same type of singlets can also have several interpretations. We furthermore list which other couplings in $W_{\rm singlet}$ are allowed. Since there is only one type of ${\bf 1}_\mu$, the term ${\bf 1}_\mu^3$ term is always forbidden perturbatively, but for $\nu_R^c$ a cubic coupling for the neutrino involving families distributed over different curves can exist. Note that tadpole terms in the superpotential, linear in the singlets, can only be generated non-perturbatively. Holomorphic quadratic terms involving either different families of $\nu_R^c$ or one $\nu_R^c$ and ${\bf 1}_\mu$ are allowed by gauge invariance for vector-like pairs of such fields, even though we do not list this explicitly. Determining their presence amounts to computing the vector-like spectrum of massless states. Otherwise quadratic singlet terms, especially Majorana mass terms for $\nu_R^c$, are only generated non-perturbatively \cite{Blumenhagen:2006xt,Ibanez:2006da} or as effective couplings from the cubic
interactions with $\langle {\bf 1}_{\mu} \rangle \neq 0$.
\item Based on the various assignments of fields we list the perturbative $R$-parity violating dimension-four couplings $W_2$ in (\ref{W1W2}) which are allowed in view of the structure of geometrically realised Yukawa couplings.
More precisely, we list for which of the possible choice of (`heavy' or `light') right-handed quark and lepton fields a coupling of type $\alpha$, $\beta$, $\gamma$ is realised. The coupling $\kappa$ is allowed by the $U(1)$ section rules whenever $L$ and $H_u$ reside on the same matter curve and are thus vector-like with respect to all gauge symmetries. Such terms correspond to effective mass couplings and their presence can be read off from the precise vector-like spectrum of the compactification, which can be computed in F-theory once the gauge background is specified \cite{Bies:2014sra}.
\item Finally we check for which matter identifications the potentially dangerous dimension-five couplings (\ref{W3K}) are perturbatively allowed, based on the $U(1)_i$ charges of the involved fields.
Note that in principle, this does not necessarily imply that the couplings are actually non-zero; to check this one would have to analyse in more detail how precisely the non-renormalisable couplings arise by exchange of heavy intermediate states. Depending on the details of the setup the resulting couplings can be negligibly small. This is left for a more in-depth analysis, and we take the results based purely on $U(1)_i$ selection rules merely as a first indication.
Let us also note that some of the non-perturbative effects required to induce the Yukawa couplings for the `light' generations may at the same time induce other baryon or lepton-number violating or other undesirable operators \cite{Ibanez:2008my,Kiritsis:2009sf,Cvetic:2009yh,Cvetic:2009ez}. We do not check for this possibility here.
Furthermore we reiterate that the constraints on both dimension-four and -five couplings are relaxed in scenarios with intermediate or even high-scale supersymmetry breaking.
\end{itemize}
The results of this scan over possible Standard Model-like embeddings is presented in appendix \ref{appsec:huge_table}.
In configurations where the matter states can be localised at different curves, we do not list all possible combinations separately. In particular our analysis so far does not make any statements about whether it is possible to realise precisely the Standard Model context by inclusion of fluxes.
Irrespective of our relaxed attitude towards baryon and lepton number violation, a number of configurations exists in which all dangerous dimension-four and in particular the dimensions-five operators $\lambda_1$ and $\lambda_2$ in (\ref{W3K}) are absent as a result of the $U(1)$ selection rules.
\subsection{A Specific Example}
As an example consider model number 5 in the top combination ${\rm I} \times {\rm A}$ listed in table \ref{tab:match_IxA} with $U(1)_Y = U(1)_1$.
In perhaps the simplest scenario, the $H_u$, $H_d$ and all families of left-handed leptons $L$ are realised as the states ${\bf \overline 2}^{\rm I}_1$, ${\bf \overline 2}^{\rm I}_2$ and ${\bf \overline 2}^{\rm I}_3$ respectively.
In particular, $U(1)_2$ therefore distinguishes these states. Perturbative lepton masses arise if we identify $\nu_R^c = { \bf \overline 1}^{(6)} $ and $e_R^c = { \bf 1}^{(1)} $. For the choice $(u_R^c, d_R^c) = ({\bf \overline 3}^{\rm A}_4, {\bf \overline 3}^{\rm A}_3)$ the quark masses are also realised perturbatively with the caveat noted in footnote \ref{footcaveat}.
For the described assignment of matter all R-parity violating dimension-four couplings are perturbatively forbidden, as are the potentially problematic dimension-five couplings $\lambda_1 \, Q \, Q \, Q \, L$ and $ \lambda_2 \, u_R^c \, u_R^c \, d_R^c \, e_R^c $.
However, more complicated assignments are possible. For instance, if one or more families of leptons $L$ are instead identified with the state ${\bf \overline 2}_2$, then in this family the right-handed neutrino, which could be any of the states ${\bf 1}^{(5)}$, ${\bf 1}^{(6)}$ or their conjugates, does not have a perturbative Dirac mass. In this case, the Dirac mass would have to be generated directly by non-perturbative effects as proposed in \cite{Cvetic:2008hi}, naturally explaining the smallness of the neutrino masses via the non-perturbative suppression. If both types of matter identifications are combined for different families, a lepton-number violating dimension-four term ${\bf 2}_3 \, {\bf 2}_3 \, {\bf 1}_2$ arises, where ${\bf 1}^{(2)}$ is now the `massive' $e_R^c$ which couples perturbatively to the $L$-family ${\bf \overline 2}_2$. This coupling is innocuous for the proton as no baryon-lepton number violating terms are created.
\section{Fluxes and Chiral Spectrum} \label{sec_fluxes}
A chiral charged matter spectrum, ideally with three generations of MSSM matter and no chiral exotics, requires the introduction of suitable gauge fluxes.
In F/M-theory, gauge fluxes are described by 4-form fluxes $G_4 \in H^{2,2}(\hat Y_4)$ subject to a number of consistency conditions.
Apart from obeying transversality \cite{oai:arXiv.org:hep-th/9908088},
\begin{eqnarray}
\int_{\hat Y_4} G_4 \wedge Z \wedge \pi^{-1} D_a = \int_{\hat Y_4} G_4 \wedge \pi^{-1} D_a \wedge \pi^{-1} D_b = 0 \qquad\quad \forall D_a, D_b \in H^{1,1}(\cal B)
\end{eqnarray}
with $Z$ the zero-section, and the quantisation condition \cite{oai:arXiv.org:hep-th/9609122,oai:arXiv.org:1011.6388,oai:arXiv.org:1203.4542}
\begin{eqnarray} \label{quantisation}
G_4 + \frac{1}{2} c_2({\hat Y_4}) \in H^4(\hat Y_4, \mathbb Z),
\end{eqnarray}
the gauge flux $G_4$ must not break the $SU(3)$ and $SU(2)$ gauge group,
\begin{eqnarray}
\int_{\hat Y_4} G_4 \wedge E_i \wedge \pi^{-1} D_a =0 \qquad \quad \forall D_a \in H^{1,1}(\cal B),
\end{eqnarray}
satisfy the D-term supersymmetry conditions for both $U(1)$ gauge groups with generating 2-forms $\omega_i$ \cite{Grimm:2010ks,Grimm:2011tb},
\begin{eqnarray}
\int_{\hat Y_4} G_4 \wedge \omega_i \wedge J, \qquad i=1,2
\end{eqnarray}
with K\"ahler form $J$ inside the K\"ahler cone, as well as the D3-tadpole cancellation condition
$
\int_{\hat Y_4} G_4 \wedge G_4 + N_{D_3} = \frac{\chi(\hat Y_4)}{24}$.
In presence of such fluxes, the chiral index of matter in representation $R$ localised on matter curve $C_R \subset {\cal B}$ is given by the topological intersection number \cite{oai:arXiv.org:0802.2969, oai:arXiv.org:0904.1218, Braun:2011zm,oai:arXiv.org:1108.1794,Krause:2011xj, oai:arXiv.org:1111.1232,oai:arXiv.org:1202.3138,oai:arXiv.org:1203.6662}
\begin{eqnarray}
\chi(R) = \int_{{\cal C}_R} G_4 .
\end{eqnarray}
Here ${\cal C}_R$ denotes the matter surface given by the fibration over $C_R$ of the linear combination of fibre $\mathbb P^1$s associated with the highest weight of representation $R$.
Eventually we will need to know not only the chiral index, but the exact massless vector-like spectrum.
A proposal for the computation of the charged localised vector-like matter based on a description of the 3-form potential underlying $G_4$ in terms of Chow groups has been given recently in \cite{Bies:2014sra}.
An important restriction arises from the requirement that the MSSM hypercharge
\begin{eqnarray}
U(1)_Y = a \, U(1)_1 + b \, U(1)_2
\end{eqnarray}
must not receive a St\"uckelberg mass. This is guaranteed precisely if
\begin{eqnarray}
\int_{\hat Y_4} G_4 \wedge \omega_Y \wedge \pi^{-1} D_a = 0 \qquad \quad \forall D_a \in H^{1,1}({\cal B}), \quad {\rm where} \quad \omega_Y = a \, \omega_1 + b \, \omega_2
\end{eqnarray}
is the hypercharge generator defined in terms of the generators of the two $U(1)_i$, c.f. (\ref{ShiodaSu2Su3}). This condition ensures that no $U(1)_Y$-D-term is induced by the flux, which is equivalent to stating that the fluxes do not lead to a $U(1)_Y$-dependent gauging of the axions as would be the case if the $U(1)_Y$ boson received a St\"uckelberg mass.
To explicitly analyse these conditions, recall that the group $H^{2,2}(\hat Y_4)$ splits into a vertical part $H^{2,2}_{\rm vert}(\hat Y_4)$ and a horizontal part $H^{2,2}_{\rm hor}(\hat Y_4)$, the first of which is generated by products of elements in $H^{1,1}(\hat Y_4)$ \cite{Greene:1993vm,oai:arXiv.org:1203.6662}. Elements in $H^{2,2}_{\rm vert}(\hat Y_4)$ are comparatively straightforward to describe, and can in principle be classified completely, see e.g.~\cite{oai:arXiv.org:1202.3138} in the context of so-called $U(1)$-restricted $SU(N)\times U(1)$ models (for $N \leq 5$) and \cite{Cvetic:2013uta,Cvetic:2013jta} for $SU(5) \times U(1)_1 \times U(1)_2$ fibrations. While we leave a more systematic analysis of the possible vertical fluxes for the nine combinations of $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ for future work, we here exemplify the general procedure.
First, the gauge fluxes associated with the $U(1)_1$ and $U(1)_2$ gauge symmetries are guaranteed to satisfy the transversality condition as an immediate consequence of the properties of the Shioda map that leads to the definition of the $U(1)_i$ generators $\omega_i$. In our situation therefore
\begin{eqnarray}
G^{(i)}_4 = \pi^{-1} F_i \wedge \omega_i, \qquad F_i \in H^{1,1}({\cal B}), \qquad\quad i=1,2
\end{eqnarray}
represent viable gauge fluxes subject to the remaining conditions listed above.
One more independent vertical flux exists already for the $U(1)_1 \times U(1)_2$-fibration without further non-abelian gauge group \cite{Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta}.
We will make use of the representation of this extra flux as the 4-form dual to one of the two singlet matter surfaces associated with the singlet ${\bf 1}^{(3)}$ appearing in table \ref{tab:singlets-charges}. More precisely, the extra flux can be described, in the notation of \cite{Borchmann:2013hta}, as
\begin{eqnarray} \label{G4gamma}
G_4^{\gamma} = [\gamma] = \pi^{-1} [b_2] \wedge \pi^{-1}[c_1] - [b_2 \cap c_1 \cap s_0],
\end{eqnarray}
where the matter surface $\gamma$ is given as a complete intersection inside the ambient 5-fold of the fibration $\hat Y_4$,
\begin{eqnarray}
\gamma = b_2 \cap c_1 \cap \tilde P, \qquad\quad {P_T}|_{b_2 = c_1=0} = s_0 \tilde P.
\end{eqnarray}
In the second expression in (\ref{G4gamma}) it is used that the fibre over the curve $\{b_2\} \cap \{c_1\}$ splits into two $\mathbb P^1$s.
Additional vertical fluxes may exist in presence of extra non-abelian gauge groups. In fact, each of the 4-cycles associated with the charged matter surfaces defines a transverse cycle and thus defines a valid type of gauge flux subject to the remaining constraints. However, more work is required to check which of these lead to fluxes independent of the fluxes $G_4^{(i)}$ and $G_4^\gamma$, see \cite{Cvetic:2013uta,Cvetic:2013jta} for this analysis with extra non-abelian gauge group $SU(5)$.
Evaluating the described constraints depends on the details of the tops under consideration.
Let us consider the combination of tops ${\rm I} \times {\rm A}$.
We first need to evaluate the constraints on the fluxes for $U(1)_Y$ to remain massless.
With the help in particular of the intersection numbers (2.18) - (2.22) in \cite{Borchmann:2013hta} and using the explicit the form (\ref{eq:I-A-U(1)-generators}) of the $U(1)_i$ generators, one computes
\begin{eqnarray}
\int_{\hat Y_4} \omega_1 \wedge \omega_1 \wedge \pi^{-1} D_a \wedge \pi^{-1} D_b &=& \int_{\cal B} (-2 \overline{\cal K} + \frac{1}{2} W_2 + \frac{2}{3} W_3) \wedge D_a \wedge D_b, \\
\int_{\hat Y_4} \omega_2 \wedge \omega_2 \wedge \pi^{-1} D_a \wedge \pi^{-1} D_b &=& \int_{\cal B} (-2 \overline{\cal K} - 2 [c_{1;0,0}] + \frac{2}{3} W_3) \wedge D_a \wedge D_b, \\
\int_{\hat Y_4} \omega_1 \wedge \omega_2 \wedge \pi^{-1} D_a \wedge \pi^{-1} D_b &=& \int_{\cal B} (- \overline{\cal K} + [c_{2;0,1}] - [c_{1;0,0}] + \frac{2}{3} W_3) \wedge D_a \wedge D_b,
\end{eqnarray}
where $D_a, D_b \in H^{1,1}({\cal B})$ and $W_2 = \{w_2 \}$ and $W_3=\{w_3 \}$ represent the $SU(2)$ and $SU(3)$ divisors on the base. The base sections $c_{1;0,0}$ and $c_{2;0,1}$ are defined in (\ref{eq:I-A-coeffs}).
To compute the intersection numbers $\int G_4^\gamma \wedge \omega_i \wedge \pi^{-1}D_a$, we follow the procedure in \cite{Borchmann:2013hta} and exploit that the 4-cycle $\gamma$ dual to the flux $G_4^\gamma$ represents the weight of the singlet state ${\bf \overline 1}^{(3)}$ with charges $(-1,-2)$. With $\int G_4^\gamma \wedge \omega_i \wedge \pi^{-1}D_a = \int_\gamma \omega_i \wedge \pi^{-1}D_a$ therefore
\begin{eqnarray}
\int G_4^\gamma \wedge \omega_1 \wedge \pi^{-1} D_a &=& (-1) \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \wedge D_a, \\
\int G_4^\gamma \wedge \omega_2 \wedge \pi^{-1} D_a &=& (-2) \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \wedge D_a,
\end{eqnarray}
because integration of the $U(1)_i$ generators $\omega_i$ over the highest-weight $\mathbb P^1$ in the fibre gives the $U(1)_i$ charge.
Altogether, if we only switch on flux $G_4 = G_4^{(1)} + G_4^{(2)} + \alpha \, G^\gamma_4$, the parameters $F_i \in H^{1,1}(\cal B)$ and $\alpha$ are constrained by masslessness of $U(1)_Y$ such that for all $D_a \in H^{1,1}({\cal B})$
\begin{align}
\begin{split}
\int_{\cal B} \, \Big[ & F_1 \wedge \Big(a (-2 \overline{\cal K} + \frac{1}{2} W_2 + \frac{2}{3} W_3) + b (-2 \overline{\cal K} - 2 [c_{1;0,0}] + \frac{2}{3} W_3) \Big) - \alpha (a + 2b) \, [b_{2;0,0}] \wedge [c_{1;0,0}] + \\
& F_2 \wedge \Big(a (-2 \overline{\cal K} - 2 [c_{1;0,0}] + \frac{2}{3} W_3) + b (- \overline{\cal K} + [c_{2;0,1}] - [c_{1;0,0}] + \frac{2}{3} W_3) \Big) \Big] \wedge D_a = 0 .
\end{split}
\end{align}
For a given base ${\cal B}$ one then expands all forms into a basis of $H^{1,1}_{\mathbb Z}(\cal B)$ and makes an ansatz for the $U(1)_i$ fluxes $F_i$ with suitably quantised coefficients (and same for $\alpha$)
in agreement with the flux quantisation condition (\ref{quantisation}).
The chiral index of the charged matter in representation $R$ with respect to the $U(1)_i$ fluxes is simply given by
\begin{eqnarray}
\int_{{\cal C}_R} G_4^{(i)} = q_i \int_{C_R} F_i,
\end{eqnarray}
where $q_i$ denotes the $U(1)_i$ charge.
To compute the index $\int_{{\cal C}_R} G_4^\gamma$ we analyse the geometric intersection of ${\cal C}_R$ with the 4-cycle ${\gamma}$.
For top ${\rm I} \times {\rm A}$ the only intersections occur with the surfaces associated with the states ${\bf 2}_2$ as well as ${\bf \overline 3}_2$ and ${\bf \overline 3}_4$. The intersection product of $\gamma$ with the various matter surfaces is implicitly contained in tables \ref{tab:SU(2)-I-Yukawas} and \ref{tab:SU(3)-A-Yukawas}, which contain the Yukawa couplings with the singlet ${\bf 1}^{(3)}$.
More precisely, explicit analysis of the intersection shows that the topological intersection number of 4-cycle $\gamma$ with the fibration of $\mathbb P^1_{0B}$ over the ${\bf 2}_2$-curve - see table \ref{tab:SU(2)-I-matter} - is given by
$ - \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \wedge W_2$. Since $\mathbb P^1_{0B}$ is the highest weight of the representation ${\bf \overline 2}_2$, this equals the chiral index for this state.
Similarly for $\mathbb P^1_{0u}$ over the ${\bf 3}_2$-curve listed in table \ref{tab:SU(3)-A-matter} one gets $- \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \ \wedge W_3$.
Altogether thus $G_4^\gamma$ contributes
\begin{eqnarray}
\chi^\gamma({\bf \overline 2}_2) = - \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \wedge W_2, \qquad
\chi^\gamma({\bf \overline 3}_2) = \int_{\cal B} [b_{2;0,0}] \wedge [c_{1;0,0}] \ \wedge W_3 = - \chi^\gamma({\bf \overline 3}_4).
\end{eqnarray}
This merely exemplifies the use of fluxes and the constraints that have to be met in constructing vacua with a realistic particle spectrum. An explicit analysis of the set of possible fluxes is clearly beyond the scope of this work and left for future investigations.
\section{Summary and Outlook}\label{sec_Conclusions}
F-theory is a unifying framework for the description of Type IIB compactifications with 7-branes which extends to an intrinsically non-perturbative regime.
While recent phenomenological studies of F-theory have exploited its non-perturbative nature in the context of GUT models of particle physics,
it is an equally exciting question to what extent direct, non-GUT realisations of the Standard Model within F-theory go beyond the known possibilities of perturbative models.
In view of the very different structure of Yukawa couplings in perturbative and non-perturbative brane vacua, F-theory is expected to encompass realisations of the Standard Model which cannot be studied in purely perturbative approaches.
In this work we have taken some first steps towards a direct embedding of the Standard Model gauge group and matter fields into F-theory.
To this end we have constructed elliptic fibrations for F-theory compactifications with gauge group $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$.
The fibrations considered arise as specialisations of the class of ${\rm Bl}_2 \mathbb P^3$-fibrations constructed in \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta} (see also \cite{Klemm:1996hh} for earlier work) with gauge group $U(1)_1 \times U(1)_2$. We have focused on the class of toric singularity
enhancements leading to extra gauge group $SU(3) \times SU(2)$ along two in principle unrelated divisors $W_3$ and $W_2$. In this sense our construction differs from the earlier approaches to F-theory Standard Models reported in \cite{Choi:2013hua,Choi:2010su,Choi:2010nf}, which geometrically deform an underlying $SU(5)$ theory to the Standard Model. The structure of singularities in the class of toric $SU(3) \times SU(2)\times U(1)_1 \times U(1)_2$ models and their resolution is described by the combination of the $3 \times 3$ possible tops with gauge group $SU(3) \times SU(2)$ over polygon 5 in the classification of \cite{Bouchard:2003bu}; of the nine combinations only five are mutually inequivalent.
For generic choice of $SU(3)$ and $SU(2)$ divisors on the base, the spectrum of charged matter representations consists, in absence of fluxes, of a state $({\bf 3},{\bf 2})$, three types of $({\bf 1},{\bf 2})$-states and five types of $({\bf 3},{\bf 1})$-states plus conjugates, whose $U(1)_i$ charges we have computed. We have analysed the perturbative Yukawa interactions among these states including the sector of $U(1)_i$ charged singlets \cite{Borchmann:2013jwa,Cvetic:2013nia,Cvetic:2013uta,Borchmann:2013hta,Cvetic:2013jta}.
This analysis is independent of the specific base space ${\cal B}$ of the fibration. Given a concrete base space which allows for all the sections defining the fibration one can then construct explicit elliptically fibred Calabi-Yau fourfolds, whose smoothness can be checked torically. We have exemplified this for a toy model over ${\cal B} = \mathbb P^3$; more complicated explicit realisations, ideally with rigid divisors for the non-abelian gauge groups along the lines of \cite{Blumenhagen:2009yv,Grimm:2009yu,Chen:2010ts,Knapp:2011wk}, are left for future work.
Based on these investigations, we have classified the possible identifications of the MSSM matter fields with the various representations present in each of the five inequivalent types of fibrations.
We allow for an extension of the MSSM by right-handed neutrinos and possibly by an extra singlet whose non-vanishing VEV generates a $\mu$-term in the Higgs sector.
One linear combination of $U(1)_1$ and $U(1)_2$ describes the MSSM hypercharge, which must remain massless upon inclusion of gauge fluxes, while the orthogonal linear combination acts as a $U(1)$ selection and must acquire a flux-induced St\"uckelberg mass.
For suitable matter assignments in our list of possibilities, this extra $U(1)$ selection rule indeed forbids all dimension-four R-parity violating and the most dangerous dimension-five lepton- and baryon-number violating effective couplings. As stressed and investigated in \cite{Ibanez:2008my,Kiritsis:2009sf,Cvetic:2009yh,Cvetic:2009ez}, such couplings can be introduced non-perturbatively if some of the required Yukawas are generated by instantons.
A study of this effect is left for future work. In fact, in our classification of possible Standard Model identifications we
do a priori not insist on absence of all lepton- and baryon-number violating dimension-four and -five couplings even at the perturbative level, but merely list which of these couplings are geometrically realised. While most of them would certainly have to be excluded for a TeV-scale supersymmetry breaking scale -- see \cite{Cvetic:2009yh,Cvetic:2009ez,Anastasopoulos:2012zu} for this approach in the context of perturbative MSSM quivers and \cite{Cvetic:2010dz} in the context of MSSM quivers with singlet extensions -- the constraints on some of the couplings are more relaxed in intermediate or high-scale supersymmetry breaking scenarios.
Based on our classification of potential Standard Model identifications, an interesting task for future work will be an in-depth analysis of the effects of such couplings depending on the spectrum of superpartners.
In view of our original motivation as stated at the beginning of this section, it will furthermore be interesting to investigate which of the configurations admit a well-defined weak-coupling limit. This will in turn identify those potential Standard Model realisations which truly go beyond perturbative models, and it will be illuminating to distill their characteristic physical properties in contradistinction to perturbative D-brane vacua.
Another important question to study will be the constraints on the moduli of the compactification for the running of the gauge couplings to be consistent with their observed value at the weak scale.
The gauge couplings depend on the K\"ahler moduli of the compactification, which therefore must obey certain relations in order to reproduce the approximate unification of gauge couplings at a higher scale, the detailed form of which depends of course on the precise spectrum of intermediate states. Such relations have been studied in the perturbative Type IIA framework in \cite{Blumenhagen:2003jy}. The need for the moduli to obey such relations might seem ad hoc to GUT model builders; on the other hand even in GUT realisations of string theory important corrections, in the example of F-theory due to hypercharge flux or Kaluza-Klein states \cite{Donagi:2008kj,Blumenhagen:2008aw,Conlon:2009qa,Mayrhofer:2013ara,Hebecker:2014uaa}, render unification less non-trivial than one might think.
The construction of phenomenologically viable F-theory vacua requires the inclusion of gauge fluxes in such a way as to reproduce the chiral spectrum of the Standard Model.
As we have discussed, the fluxes must be chosen such as to respect masslessness of the specific linear combination of $U(1)_1$ and $U(1)_2$ corresponding to hypercharge.
In future work we plan to systematically analyse large classes of consistent gauge fluxes in order to determine which of the potential Standard Model identifications are actually compatible with three families of MSSM spectrum and absence of chiral exotics. Indeed, consistency of the compactification including the gauge flux data is known to restrict the allowed spectrum, sometimes even beyond a purely 4-dimensional field theory analysis as studied in the perturbative framework in \cite{Cvetic:2011iq} (see also \cite{Halverson:2013ska}).
Apart from the inclusion of suitable fluxes, there are various modifications of the geometry conceivable in order to avoid unwanted exotic states:
Some of the matter curves are inexistent for suitable choices of classes $\alpha$ and $\beta$ which determine the fibration as summarised in table \ref{coeff}. It will be interesting to analyse in detail which special choices are possible such as to `switch off' a maximal number of exotic matter curves without affecting the Standard Model sector. Furthermore, the spectrum of massless states changes in the presence of vacuum expectation values for some of the singlets. This corresponds either to a recombination process or to a gluing \cite{Donagi:2011jy,Donagi:2011dv} of branes and can turn out instrumental in concrete model building.
Finally we stress that the analysis of the massless spectrum has to go beyond the chiral index and include also a computation of the the vector-like spectrum of states.
This implies that the $C_3$-gauge field background must be specified more accurately than merely in terms of the gauge flux $G_4$. In \cite{Bies:2014sra} it was shown how to specify the gauge data via rational equivalence classes of 4-cycles, and a natural candidate was proposed for the cohomology groups counting the exact vector-like matter spectrum individually.
We look forward to applying this technology in our search for realistic F-theory non-GUTs.
\subsubsection*{Acknowledgements}
\noindent We thank Philipp Arras, Arthur Hebecker, Luis Ibanez, Christoph Mayrhofer, Dave Morrison, Eran Palti, Hernan Piragua and Oskar Till for important discussions.
This work was supported in part by Deutsche Forschungsgeimeinschaft under TR 33 `The Dark Universe' and by Studienstiftung des Deutschen Volkes.
\section[Details on the Toric Diagrams]{Details on the Toric Diagrams}\label{app:tops}
In this appendix we present the toric diagrams of the $SU(2)$ and $SU(3)$ tops. Special emphasis will be put on the symmetries which identify some of the models as equivalent pairs. The relationship between toric geometry and the geometry of Calabi-Yau hypersurfaces is described in e.g.~\cite{Bouchard:2003bu}.
The ambient space ${\rm Bl}_2 {\mathbb P}^2$ has the toric diagram depicted on the left in figure \ref{fig:base_polygon}. The lattice points of the dual diagram on the right correspond to the terms of the hypersurface equation (\ref{eq:hypersurface-equation}). Clearly the diagrams share a common reflection symmetry along the dotted diagonal axis.
\begin{figure}[ht]
\begin{center}
\def.98\hsize{.98\hsize}
\input{base_polygon.pdf_tex}
\end{center}
\caption{Polygon 5 (in the classification of \cite{Bouchard:2003bu}) describing the fibre ambient space ${\rm Bl}_2 {\mathbb P}^2$; every lattice point of the dual polygon (right) gives an individual term of the hypersurface equation. The reflection symmetry along the dotted diagonal is manifest.}
\label{fig:base_polygon}
\end{figure}
The symmetry exchanges fibre coordinates $s_0 \leftrightarrow s_1, \, \mathrm{v} \leftrightarrow \mathrm{w}$ and coefficients $b_0 \leftrightarrow b_2, \, c_1 \leftrightarrow c_2 , \, d_0 \leftrightarrow d_1$ of the hypersurface equation. Consequently, the $U(1)$ generators (\ref{eq:U(1)-generators}) are also transformed, namely as
\begin{align}\label{eq:omega-transformation}
\begin{split}
\omega_1 &= S_1 - S_0 - \overline{\cal K} \longrightarrow S_0 - S_1 - \overline{\cal K} = - \omega_1 + 2\overline{\cal K}\, , \\
\omega_2 &= U - S_0 - \overline{\cal K} - [c_1] \longrightarrow U -S_1 - \overline{\cal K} - [c_2] = \omega_2 - \omega_1 - \overline{\cal K}+[c_1]-[c_2] \, .
\end{split}
\end{align}
These forms do not satisfy the verticality condition, i.e.~they are not in the image of the Shioda map. However they only differ from such by the pullback of divisors of the base. Since such pullbacks never contribute to the $U(1)$ charges of any states, the $U(1)$ charges indeed transform as
\begin{align}
\begin{split}
U(1)'_1 &= - U(1)_1 \, , \\
U(1)'_2 &= U(1)_2 - U(1)_1 \, .
\end{split}
\end{align}
The symmetry also exchanges the singlets (as the coefficients $b_i, c_j, d_k$ are exchanged), namely ${\bf 1}^{(1)} \leftrightarrow \overline{\bf 1}^{(3)}, {\bf 1}^{(2)} \leftrightarrow \overline{\bf 1}^{(4)}$, while ${\bf 1}^{(5)}$ and ${\bf 1}^{(6)}$ are invariant.
\begin{figure}[ht]
\begin{center}
\def.98\hsize{.98\hsize}
\input{tops.pdf_tex}
\end{center}
\caption{Possible $SU(2)$ (upper) and $SU(3)$ (lower) tops. The coloured lines and vertices are the `top layer' of the three-dimensional toric diagram, projected down onto the layer containing the base polygon representing the fibre ambient space.}
\label{fig:tops}
\end{figure}
The same symmetry relates the $SU(2)$ and $SU(3)$ tops as well as combinations of those. As shown in figure \ref{fig:tops}, the $SU(2)$-I and -II tops are precisely matched onto each other, however only if one also exchanges the resolution coordinates $e_0 \leftrightarrow e_1$. The $U(1)$ generators on both sides can be similarly matched. The first generator transforms as $\omega^{\rm I}_1 \longrightarrow S^{\rm II}_0 - S^{\rm II}_1 - \overline{\cal K} + \frac{1}{2} E^{\rm II}_0 = -(S^{\rm II}_1 - S^{\rm II}_0 - \overline{\cal K} + \frac{1}{2} E^{\rm II}_1) + 2\overline{\cal K} + \frac{1}{2} \pi^* W_2 = - \omega^{\rm II}_1 + 2\overline{\cal K} + \frac{1}{2} \pi^* W_2$, where the first equality exploits the relation $E_0 + E_1 = \pi^* W_2$ for the resolution divisors of an $SU(2)$ singularity. For the second generator one now needs to take into account that $c^{\rm I}_1 = c^{\rm I}_{1,0} \, e^{\rm I}_1$ is mapped onto $c^{\rm II}_2 = c^{\rm II}_{2,1} \, e^{\rm II}_0$, from which one can easily verify the transformation $\omega^{\rm I}_2 \longrightarrow \omega^{\rm II}_2 - \omega^{\
\rm II}_1 + [c^{\rm II}_1] - [c^{\rm II}_{2,1}] - \overline{\cal K}$. As mentioned after (\ref{eq:U1-charge-transformation}), the spectrum of $SU(2)$-charged states is also exchanged as ${\bf 2}^{\rm I}_i \leftrightarrow \overline{\bf 2}^{\rm II}_i, i=1,2,3$. Obviously the $SU(2)$-III top is invariant under the reflection symmetry. The spectrum and $U(1)$ charges transform as stated in subsection \ref{subsec:SU(2)-tops-short}. Similarly one can see that the $SU(3)$-A and -C tops are equivalent to each other, while the -B top is invariant under reflection. Analogous calculations as above show that the $U(1)$ generators transform accordingly.
When we combine the $SU(2)$ and $SU(3)$ tops, we see that among the nine possibilities there are in fact five inequivalent models with $SU(2) \times SU(3)$ gauge group. The redundancy comes again from reflecting along the symmetry axis of the base polygon, which identifies four pairs of models to be equivalent (cf.~figure \ref{fig:combined_tops}). The combination ${\rm III} \times {\rm B}$ is invariant under the reflection transformation. The equivalence of the pairs of models can also be checked in a similar fashion as above, by inspecting the transformation of the $U(1)$ generators and the matter states.
\begin{figure}[ht]
\begin{center}
\def.98\hsize{.98\hsize}
\input{combined_tops.pdf_tex}
\end{center}
\caption{The combination of $SU(2)$ and $SU(3)$ tops gives rise to five inequivalent models. Their toric diagram lies in a four-dimensional lattice, where the `top layers' corresponding to $SU(2)$ and $SU(3)$ resolution divisors extend into two linearly independent directions that do not lie in the plane spanned by the base polygon. For this figure we have projected the tops down into said plane. The four pairs of tops that are equivalent are related to each other by reflection along the diagonal in the plane of the base polygon.}
\label{fig:combined_tops}
\end{figure}
\section[Details on \texorpdfstring{\boldmath $SU(2)$}{SU(2)}-II and -III Tops]{Details on \texorpdfstring{\boldmath $SU(2)$}{SU(2)}-II and -III Tops} \label{SU2DetailsApp}
In this part we provide more details on the matter and Yukawa couplings of the $SU(2)$-II and -III top which have not been discussed at length in the corpus of this paper.
\subsection[\texorpdfstring{$SU(2)$}{SU(2)}-II Top]{\texorpdfstring{{\boldmath $SU(2)$}}{SU(2)}-II Top} \label{SU2App1}
The top corresponds to restricting the hypersurface coefficients as
\begin{align}\label{eq:SU(2)-II-coeffs}
b_0 = b_{0,1} \, e_0, \quad b_2 = b_{2,0} \, e_1, \quad c_2 = c_{2,1} \, e_0 , \quad d_1 = d_{1,0} \, e_1 , \quad d_2 = d_{2,0} \, e_1.
\end{align}
There are two possible SR-ideals, of which we choose
\begin{align}\label{eq:SU(2)-II-SR-ideal}
\mathrm{u} \, \mathrm{v}, \mathrm{u} \, \mathrm{w} , \mathrm{w} \, s_0, \mathrm{v} \, s_1 , s_0 \, s_1 , e_0 \, s_1 , e_0 \, \mathrm{u} , e_1 \, \mathrm{v} .
\end{align}
The scaling relations and divisor classes for this top are as follows:
\begin{align}\label{tab:SU(2)-II-divisor-classes}
\begin{array}{c|cccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & e_1 & e_0 \\ \hline
\mathrm{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot \\
E_1 & \cdot & 1 & \cdot & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
The $U(1)$ generators, normalised such that the $SU(2)$ root remains uncharged, are given by
\begin{align}\label{eq:SU(2)-II-U(1)-generators}
\begin{split}
\omega_1^\text{II} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{1}{2} E_1, \\
\omega_2^\text{II} &= U - S_0 - \overline{\mathcal{K}} - [c_1] + \frac{1}{2} E_1.
\end{split}
\end{align}
The discriminant locus now takes the form
\begin{eqnarray} \label{Dis2II}
\Delta \simeq w_2^2 \Big( c_{1} \, ( c_{2,1}^2 \, d_0 - b_{0,1} \, b_1 \, c_{2,1} + b_{0,1}^2 \, c_1 ) \, \ell_3 \, (b_1^2 - 4 c_1 d_0)^2 \, + {\cal O}(w_2) \, \Big).
\end{eqnarray}
The fibres over the intersection of the first three factors inside the big bracket in (\ref{Dis2II}) with $\{w_2\}$ are indeed of split Kodaira type $I_3$.
A similar analysis as for the $SU(2)$-I top confirms matter in the $\mathbf{2}$ representation (together with their charge conjugates $\overline{\mathbf{2}}$) on the matter curves displayed in table \ref{tab:SU(2)-II-matter}.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c|c}
\text{matter} & \text{locus} = W_2 \cap \ldots & \text{splitting of fibre components} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_1 & \{c_{1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0}\!+ \mathbb{P}^1_{1A} & (\frac{1}{2}, \frac{3}{2}) \\[.5ex]\hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_2 & \{ c_{2,1}^2 \, d_0 - b_{0,1} \, b_1 \, c_{2,1} + b_{0,1}^2 \, c_1 \} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} & (\frac{1}{2},-\frac{1}{2}) \\[.5ex]\hline \rule{0pt}{3ex}
& \{\ell_3\} := \{ b_{2,0}^2\,d_0^2 & & \\
\mathbf{2}^\mathrm{II}_3 & + b_{2,0}\,( b_1^2\,d_{2,0} - 2\,c_1\,d_0\,d_{2,0} - b_1\,d_0\,d_{1,0}) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} & (\frac{1}{2}, \frac{1}{2}) \\
& + c_1 (d_0\,d_{1,0}^2 + d_{2,0}(c_1\,d_{2,0} - b_1\,d_{1,0} ) ) \} & &
\end{array}
\end{align*}
\caption{Matter states in the $SU(2)$-II top.}
\label{tab:SU(2)-II-matter}
\end{table}
The splitting of the fibre components over the first curve and the resulting enhancement of the intersection structure to that of an affine $SU(3)$ diagram is straightforward to see. For the second curve, we factorise
\begin{eqnarray}\label{eq:SU(2)-II-quadratic-curve}
c_{2,1}^2 \, d_0 - b_{0,1} \, b_1 \, c_{2,1} + b_{0,1}^2 \, c_1 =\frac{1}{d_0} \, {\mathcal{C}_+} \, {\mathcal{C}_-} \quad {\rm with} \quad {\mathcal{C}_\pm} = c_{2,1}\,d_0 - b_{0,1} \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_1\,d_0} \right),
\end{eqnarray}
which splits the curve into two parts $W_2 \cap \{C_\pm=0\}$ that are connected at a branch cut. Similar to the results (\ref{eq:SU(2)-I-splitting-quadratic-curve}) of the first $SU(2)$ top, we find that over these two parts the fibre of the divisor $E_1$ splits into two components, $\mathbb{P}^1_{1B}$ and $\mathbb{P}^1_{1C}$, that can be extended over the whole curve without being interchanged by any monodromy. The intersection structure is again that of the affine $SU(3)$ diagram.
Analogously, the third curve can be written as $ W_2 \cap \ell_3$ with
\begin{align}\label{eq:SU(2)-II-complicated-quadratic-curve}
\ell_3 = 1/d_0^2 \, \mathcal{D}_+ \, \mathcal{D}_-, \qquad \quad \mathcal{D}_\pm = b_{2,0}\,d_0^2 - \left[ c_1\,d_0\,d_{2,0} + (d_0\,d_{1,0} - b_1\,d_{2,0}) \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - c_1\,d_0 } \right)\!\right] .
\end{align}
A similar calculation as (\ref{eq:SU(2)-I-splitting-complicated-quadratic-curve}) shows that the divisor $E_0$ splits into $\mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}$, with both components well-defined over the whole curve. As expected one finds the intersection structure to be an affine $SU(3)$ diagram.
Note that apart from the three matter curves discussed above, the vanishing order of the discriminant increases from $2$ to $3$ also along the curve $\{ w_2 \} \cap \{b_1^2 - 4 c_1 d_0 \}$; however, the fibre is of Kodaira type $III$ since the Weierstrass sections $f$ and $g$
vanish to order $1$ and $2$ respectively. Thus no extra charged matter representations arise here, in agreement with the formalism of \cite{Grassi:2011hq}.
The Yukawa couplings involving $SU(2)$ matter are summarised in table \ref{tab:SU(2)-II-Yukawas}. The fibre structure enhancement for each Yukawa point can be read off from the last column in an analogous fashion as with the first $SU(2)$ top (cf.~table \ref{tab:SU(2)-I-Yukawas}). We find the affine $SU(4)$ diagram as the intersection structure over all Yukawa points. Note that for the second and third pair of Yukawa points, the same split products of each pair arrange themselves into different intersection \textit{patterns}, realising either $\mathbf{2}_i-\mathbf{2}_j-\mathbf{1}/\overline{\mathbf{1}}$ or $\mathbf{2}_i-\overline{\mathbf{2}}_j-\mathbf{1}/\overline{\mathbf{1}}$ couplings.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus} = W_2 \cap \ldots & \text{splitting of fibre components} \\ \hline\hline \rule{0pt}{3ex}
\mathbf{2}_1^{\mathrm{II}} - \mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(4)} & \{c_1\} \cap \{c_{2,1}\,d_0 - b_{0,1}\,b_1 \} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0 C} \! + \mathbb{P}^1_{1AB} + \mathbb{P}^1_{1AC} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_1 - \overline{\mathbf{2}}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(5)} & \{c_1\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0 B} \! + \mathbb{P}^1_{1AB'} + \mathbb{P}^1_{1AC'} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_1 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(3)} & \{c_1\} \cap \{b_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0}\!+ \mathbb{P}^1_{1A} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_1 - \overline{\mathbf{2}}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(6)} & \{c_1\} \cap \{b_{2,0}\,d_0^2 + b_1\,(d_0\,d_{1,0} - b_1\,d_{2,0}) \} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0}\!+ \mathbb{P}^1_{1A} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_2 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(2)} & \left( \{\mathcal{C}_+\} \cap \{\mathcal{D}_+\} \right) \cup \left( \{\mathcal{C}_-\} \cap \{\mathcal{D}_-\} \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{2}}^{\mathrm{II}}_3 - \mathbf{1}^{(6)} & \left( \{\mathcal{C}_+\} \cap \{\mathcal{D}_-\} \right) \cup \left( \{\mathcal{C}_-\} \cap \{\mathcal{D}_+\} \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\[.5ex] \hline\hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_2 - \mathbf{2}^{\mathrm{II}}_2 - \overline{\mathbf{1}}^{(1)} & \{b_{0,1}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} \! + \mathbb{P}^1_{1s_1 C} + \mathbb{P}^1_{1C'} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{II}}_3 - \mathbf{2}^{\mathrm{II}}_3 - \overline{\mathbf{1}}^{(4)} & \{b_1\,d_{2,0} - d_0\,d_{1,0}\} \cap \{c_1\,d_{2,0} - b_{2,0}\,d_0\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} \! + \mathbb{P}^1_{0B'} + \mathbb{P}^1_{0B''}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(2)$-II top.}
\label{tab:SU(2)-II-Yukawas}
\end{table}
\subsection[\texorpdfstring{{$SU(2)$}}{SU(2)}-III Top]{\texorpdfstring{{\boldmath $SU(2)$}}{SU(2)}-III Top}\label{SU2App2}
This top restricts the hypersurface coefficients as
\begin{align}\label{eq:SU(2)-III-coeffs}
c_1 = c_{1,1} \, e_0, \quad c_2 = c_{2,1} \, e_0, \quad d_0 = d_{0,0} \, e_1 , \quad d_1 = d_{1,0} \, e_1 , \quad d_2 = d_{2,0} \, e_1^2.
\end{align}
For this top there are four possible SR-ideals, of which we choose
\begin{align}\label{eq:SU(2)-III-SR-ideal}
\mathrm{u} \, \mathrm{v}, \mathrm{u} \, \mathrm{w} , \mathrm{w} \, s_0, \mathrm{v} \, s_1 , s_0 \, s_1 , e_0 \, \mathrm{u} , e_0 \, s_0 , e_1 \, \mathrm{w} .
\end{align}
The scaling relations and divisor classes for this top are
\begin{align}\label{tab:SU(2)-III-divisor-classes}
\begin{array}{c|cccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & e_1 & e_0 \\ \hline
\mathrm{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot \\
E_1 & \cdot & 1 & 1 & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
The $SU(2)$ root in this top is uncharged under the generators (\ref{eq:U(1)-generators}) so that no correction term is needed,
\begin{align}\label{eq:SU(2)-III-U(1)-generators}
\begin{split}
\omega_1^\text{III} &= S_1 - S_0 - \overline{\mathcal{K}}, \\
\omega_2^\text{III} &= U - S_0 - \overline{\mathcal{K}} - [c_{1,1}].
\end{split}
\end{align}
This time the discriminant of the singular blow-down takes the form
\begin{eqnarray}
\Delta &=& w_2^2 \Big( b_0 \, b_2 \, (b_0 \, c_{1,1}^2 - b_1\,c_{1,1}\,c_{2,1} + b_2\,c_{2,1}^2) (b_1^2 - 4 b_0 b_2)^2 \, \nonumber \\
& & \phantom{w_2^2 \Big( } ( -b_2 d_0^2 + b_1 d_{0,0} d_{1,0} - b_0 d_{1,0}^2 - b_1^2 d_{2,0} + 4 b_0 b_2 d_{2,0}) + {\cal O}(w_2) \Big). \label{DisSu2III}
\end{eqnarray}
Over the intersection of $\{w_2\}$ with the first three factors in the bracket in (\ref{DisSu2III}), the fibre type enhances to split Kodaira type $I_3$. This gives rise to matter states in the $\mathbf{2}$ representation (together with their charge conjugate $\overline{\mathbf{2}}$ states) summarised in tabe \ref{tab:SU(2)-III-matter}.
By setting $b_0 / b_2=0$ in the hypersurface equation, it is again straightforward to see the splitting process and the enhancement of the intersection structure to an affine $SU(3)$ diagram over the first/second $\mathbf{2}$-curve.
The quadratic equation defining the third curve can be again factorised analogously to (\ref{eq:SU(2)-I-quadratic-curve}). However, because of the Yukawa points that are present in this top (see below), we need two different factorisations,
\begin{eqnarray}
b_0 \, c_{1,1}^2 - b_1\,c_{1,1}\,c_{2,1} + b_2\,c_{2,1}^2 = \frac{1}{b_0} \, {\cal C}_+ \, {\cal C}_- = \frac{1}{b_2} \, {\cal D}_+ \, {\cal D}_-
\end{eqnarray}
with
\begin{eqnarray} \label{eq:SU(2)-III-quadratic-curve-2}
{\cal C}_\pm = c_{1,1}\,b_0 - c_{2,1} \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - b_0\,b_2 } \right), \qquad
{\cal D}_{\pm} = c_{2,1}\,b_2 - c_{1,1}\left( \frac{b_1}{2} \mp \sqrt{ \frac{b_1^2}{4} - b_0\,b_2 } \right).
\end{eqnarray}
The two factorisations describe the same splittings of the curve into two parts which are connected at the branch cut of the square root.
With these two factorisations, one can analyse the splitting of the fibre when either $b_0$ or $b_2$ is non-zero.\footnote{On a generic base of complex dimension 3, $b_0$ and $b_2$ cannot both vanish on the codimension 2 curve.} When $b_0 \neq 0$, we can solve $\mathcal{C}_\pm$ for $c_{1,1}$ and plug the result into the hypersurface equation; if $b_2 \neq 0$, we solve ${\cal D}_\pm$ for $c_{2,1}$. Doing so, we find no splitting for $\mathbb{P}^1_0$, but $\mathbb{P}^1_1$
splits as follows:
\begin{align}
& P_T (e_1=0 , \, \mathcal{C}_\pm/\mathcal{D}_\pm=0) \notag \\
\overset{b_0 \neq 0}{=} \, & \frac{1}{b_0} \underbrace{\left[\!\left(\frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - b_0\,b_2} \right)\!s_1 + b_0\,s_0\,\mathrm{v} \right]}_{\mathbb{P}^1_{1B}} \! \underbrace{\left[\!\left( \frac{b_1}{2} \mp \sqrt{ \frac{b_1^2}{4} - b_0\,b_2} \right) \! s_1\,\mathrm{u} + c_{2,1}\,e_0\,\mathrm{v} + b_0\,s_0\,\mathrm{u}\,\mathrm{v} \right]}_{\mathbb{P}^1_{1C}} \label{eq:SU(2)-III-splitting-quadratic-curve-1} \\[-.6ex]
\overset{b_2 \neq 0}{=} \, & \frac{1}{b_2} \overbrace{\left[\!b_2\,s_1 + \left( \frac{b_1}{2} \mp \sqrt{ \frac{b_1^2}{4} - b_0\,b_2} \right)\!s_0\,\mathrm{v} \right]} \! \overbrace{\left[\! b_2\,s_1\,\mathrm{u} + c_{1,1}\,e_0\,\mathrm{v} + \left( \frac{b_1}{2} \pm \sqrt{ \frac{b_1^2}{4} - b_0\,b_2} \right)\!s_0\,\mathrm{u}\,\mathrm{v} \right]} \label{eq:SU(2)-III-splitting-quadratic-curve-2}
\end{align}
The fibre over $\{w_2\} \cap \{ b_1^2 - 4 b_0 b_2 \}$ is of Kodaira type $III$ and thus no massless matter arises.
Interestingly, over the remaining locus $\{w_2\} \cap \{ -b_2 d_0^2 + b_1 d_{0,0} d_{1,0} - b_0 d_{1,0}^2 - b_1^2 d_{2,0} + 4 b_0 b_2 d_{2,0} \} $, $(f,g,\Delta)$ vanish to order $(0,0,3)$, but the fibre is of {\it non-split } Kodaira type $I_3$. This can be read off from the specifics of the Weierstrass sections $f$ and $g$ following Tate's algorithm. Moreover, an explicit analysis of the resolved fibre confirms that it locally factors into three $\mathbb P^1$s, two of which are however exchanged by a monodromy
along the curve in the base. Since the corresponding singularity type is merely ${\rm Sp}(1)$ (as opposed to $SU(3)$) no massless matter arises here. Note that this conclusion is not in contradiction with the results of \cite{Grassi:2011hq}, especially table 9, which would naively indicate fundamental matter along this curve. However, the analysis of \cite{Grassi:2011hq} holds on Calabi-Yau 3-folds and therefore does not account for potential monodromies along the matter loci.
The possible Yukawa couplings are summarised in table \ref{tab:SU(2)-III-Yukawas}. The splitting process over the first type of Yukawa points is straightforward to see when one evaluates the hypersurface equation on the locus.
\begin{table}[t]
\begin{align*}
\begin{array}{c|c|c|c}
\text{matter} & \text{locus} = W_2 \cap \ldots & \text{splitting of fibre components} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_1 & \{b_0\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1}\!+ \mathbb{P}^1_{0A} & (1,0) \\ \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_2 & \{b_2 \} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{v}} + \mathbb{P}^1_{1A} & (1,1)\\ \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_3 & \{b_0 \, c_{1,1}^2 - b_1\,c_{1,1}\,c_{2,1} + b_2\,c_{2,1}^2 \} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} & (0,1)
\end{array}
\end{align*}
\caption{Matter states in the $SU(2)$-III top.}
\label{tab:SU(2)-III-matter}
\end{table}
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus} = W_2 \cap \ldots & \text{splitting of fibre components} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_1 - \overline{\mathbf{2}}^{\mathrm{III}}_2 - \mathbf{1}^{(6)} & \{b_0\} \cap \{b_2\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \! + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{v}} + \mathbb{P}^1_{1A} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_1 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(4)} & \{b_0\} \cap \{c_{2,1}\,b_2 - c_{1,1}\,b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \! + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\[.5ex] \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_1 - \overline{\mathbf{2}}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(1)} & \{b_0\} \cap \{c_{2,1}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0s_1} \! + \mathbb{P}^1_{0A}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B'} + \mathbb{P}^1_{1C} \\ \hline \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_2 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(3)} & \{b_2\} \cap \{c_{1,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{v} B} + \mathbb{P}^1_{1AB} + \mathbb{P}^1_{1AC} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_2 - \overline{\mathbf{2}}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(2)} & \{b_2\} \cap \{c_{1,1}\,b_0 - c_{2,1}\,b_1 \} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{v} C} + \mathbb{P}^1_{1AB'} + \mathbb{P}^1_{1AC'} \\ \hline \rule{0pt}{3ex}
\mathbf{2}^{\mathrm{III}}_3 - \mathbf{2}^{\mathrm{III}}_3 - \overline{\mathbf{1}}^{(5)} & \{c_{1,1}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1\mathrm{u} C} + \mathbb{P}^1_{1C'}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(2)$-III top.}
\label{tab:SU(2)-III-Yukawas}
\end{table}
The second and third groups of couplings arise over the intersection of the $\mathbf{2}^\mathrm{III}_3$-curve with $b_0=0$ and hence require the factorisation (\ref{eq:SU(2)-III-splitting-quadratic-curve-2}). The second Yukawa point lies over $\mathcal{D}_-=0$, corresponding to the downstairs signs in (\ref{eq:SU(2)-III-splitting-quadratic-curve-2}); the third point lies over $\mathcal{D}_+=0$, corresponding to upstairs signs. In both cases, we see that there is no further splitting of $\mathbb{P}^1_{1B}$ and $\mathbb{P}^1_{1C}$ when we set $b_0=0$. Rather, $\mathbb{P}^1_0$ splits, coming from the already present enhancement over the $\mathbf{2}^\mathrm{III}_1$ curve.
The fourth and fifth Yukawa point are the intersection points of $\mathbf{2}^\mathrm{III}_3$ and $b_2=0$, hence we make use of the factorisation (\ref{eq:SU(2)-III-splitting-quadratic-curve-1}). The fourth/fifth point lies on $\mathcal{C}_-/\mathcal{C}_+ =0$, correspondingly we take the downstairs/upstairs signs in (\ref{eq:SU(2)-III-splitting-quadratic-curve-1}). Setting $b_2=0$, we see that for the fourth coupling, $\mathbb{P}^1_{1B}$ splits off a factor $\mathrm{v}$, while $\mathbb{P}^1_{1C}$ remains irreducible; for the fifth coupling, it is $\mathbb{P}^1_{1C}$ that splits.
Finally, over the last Yukawa point, which is a self-intersection point, neither $b_0$ nor $b_2$ are 0, and so both factorisations (\ref{eq:SU(2)-III-splitting-quadratic-curve-1}) and (\ref{eq:SU(2)-III-splitting-quadratic-curve-2}) should give the same splitting process in the fibre. Indeed, setting $c_{2,1}=0$ in (\ref{eq:SU(2)-III-splitting-quadratic-curve-1}) and $c_{1,1}=0$ in (\ref{eq:SU(2)-III-splitting-quadratic-curve-2}) shows that $\mathbb{P}^1_{1B}$ remains irreducible while $\mathbb{P}^1_{1C}$ splits off a factor $\mathrm{u}$.
Over all Yukawa points we find that the intersection structure of the $\mathbb{P}^1$ components is the affine $SU(4)$ diagram.
\section[Details on \texorpdfstring{{\boldmath $SU(3)$}}{SU(3)}-B and -C Tops]{Details on \texorpdfstring{{\boldmath $SU(3)$}}{SU(3)}-B and -C Tops} \label{app-SU3}
Here we go through the remaining $SU(3)$ tops in more detail.
\subsection[\texorpdfstring{{$SU(3)$}}{SU(3)}-B Top]{\texorpdfstring{{\boldmath $SU(3)$}}{SU(3)}-B Top} \label{app-SU3-B}
The $SU(3)$-B top leads to the restrictions of the following coefficients
\begin{align}\label{eq:SU(3)-B-coeffs}
\begin{split}
b_0 &= b_{0,2}\,f_0^2\,f_1 ,\quad b_2 = b_{2,0}\,f_1\,f_2^2 ,\quad c_1 = c_{1,0}\,f_2 ,\quad c_2 = c_{2,1}\,f_0 , \\
d_0 &= d_{0,1}\,f_0\,f_1 ,\quad d_1 = d_{1,0}\,f_1\,f_2 ,\quad d_2 = d_{2,0}\,f_1 ,
\end{split}
\end{align}
while $b_1$ remain unrestricted. There are eight different triangulations. For definiteness, we choose the one leading to the following SR-ideal:
\begin{align}\label{eq:SU(3)-B-SR-ideal}
\mathrm{u}\,\mathrm{v} , \mathrm{u}\,\mathrm{w} , \mathrm{w}\,s_0 , \mathrm{v}\,s_1 , s_0\,s_1 , f_0\,\mathrm{u} , f_0\,s_1 , f_1\,\mathrm{v} , f_1\,\mathrm{w} , f_2\,\mathrm{u} , f_2\,s_0 , f_2\,\mathrm{v} .
\end{align}
The coordinates and their corresponding divisor classes are summarised in the following table:
\begin{align}\label{tab:SU(3)-B-divisor-classes}
\begin{array}{c|ccccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & f_1 & f_2 & f_0 \\ \hline
\mathrm{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\
F_1 & \cdot & 1 & \cdot & \cdot & \cdot & 1 & \cdot & -1 \\
F_2 & \cdot & 1 & -1 & \cdot & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
For $SU(3)$ roots to have zero $U(1)$ charge, the generators (\ref{eq:U(1)-generators}) receive the following correction:
\begin{align}\label{eq:SU(3)-B-U(1)-generators}
\begin{split}
\omega_1^\text{B} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{1}{3} F_1 + \frac{2}{3} F_2 ,\\
\omega_2^\text{B} &= U - S_0 - \overline{\mathcal{K}} - [c_{1,0}] + \frac{2}{3} F_1 + \frac{1}{3} F_2.
\end{split}
\end{align}
We find codimension 2 enhancement with $\mathbf{3}$ and $\overline{\mathbf{3}}$ matter over loci and with charges as presented in table \ref{tab:SU(3)-B-matter}. Over the curve $\{ w_3 \} \cap \{ b_1 \}$ the fibre type changes to Kodaira type $IV$, but such fibres do not give rise to additional charged matter.
The gauge invariant Yukawa coupling appearing are listed in table \ref{tab:SU(3)-B-Yukawas}.
\begin{table}[h!]
\begin{align*}
\begin{array}{c|c|c|c}
\text{matter} & \text{locus} = W_3 \cap \ldots & \text{splitting of fibre components} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 & \{c_{1,0}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0} \!+ \mathbb{P}^1_{1A} & (-\frac{2}{3}, -\frac{4}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 & \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_1} \!+ \mathbb{P}^1_{1B} & (-\frac{2}{3}, \frac{2}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_3 & \{d_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{w}} + \mathbb{P}^1_{0A} & (-\frac{2}{3}, -\frac{1}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_4 & \{b_1^2\,b_{2,0} - b_1\,c_{1,0}\,d_{1,0} + c_{1,0}^2\,d_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C} & (\frac{1}{3}, \frac{2}{3}) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_5 & \{b_{0,2}\,b_1^2 + c_{2,1}^2\,d_{2.0} - b_1\,c_{2,1}\,d_{0,1}\} & \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} & (\frac{1}{3},-\frac{1}{3})
\end{array}
\end{align*}
\caption{Matter states in the $SU(3)$-B top.}
\label{tab:SU(3)-B-matter}
\end{table}
\begin{table}[h!]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus}=W_3 \cap \ldots & \text{splitting of fibre components} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_2 - \mathbf{1}^{(5)} & \{c_{1,0}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0 B} + \mathbb{P}^1_{1 s_1 A} + \mathbb{P}^1_{1AB} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_3 - \mathbf{1}^{(6)} & \{c_{1,0}\} \cap \{d_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0 \mathrm{w}} + \mathbb{P}^1_{0A} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0} \!+ \mathbb{P}^1_{1A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(3)} & \{c_{1,0}\} \cap \{b_{2,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C}, \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1 s_0} \!+ \mathbb{P}^1_{1A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(4)} & \{c_{1,0}\} \cap \left(\mathbf{3}^\mathrm{B}_5\right) & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_0} \!+ \mathbb{P}^1_{1A} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_3 - \overline{\mathbf{1}}^{(6)} & \{c_{2,1}\} \cap \{d_{2,0}\} & \mathbb{P}^1_0 + \mathbb{P}^1_{0\mathrm{w}} + \mathbb{P}^1_{0A} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1 s_1} \!+ \mathbb{P}^1_{1B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(2)} & \{c_{2,1}\} \cap \left( \mathbf{3}^\mathrm{B}_4 \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_1} \! + \mathbb{P}^1_{1B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(1)} & \{b_{0,2}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1s_1} \! + \mathbb{P}^1_{1B} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_3 - \overline{\mathbf{3}}^{\mathrm{B}}_4 - \mathbf{1}^{(4)} & \{d_{2,0}\} \cap \{b_1\,b_{2,0} - c_{1,0}\,d_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{w} C} + \mathbb{P}^1_{0 AC} + \mathbb{P}^1_{0AB} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_3 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \mathbf{1}^{(2)} & \{d_{2,0}\} \cap \{b_{0,2}\,b_1 - c_{2,1}\,d_{0,1}\} & \mathbb{P}^1_0 + \mathbb{P}^1_{0\mathrm{w}} + \mathbb{P}^1_{0A} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_4 - \overline{\mathbf{3}}^{\mathrm{B}}_5 - \overline{\mathbf{1}}^{(6)} & \left(\mathbf{3}^\mathrm{B}_4 \right) \cap \left( \mathbf{3}^\mathrm{B}_5 \right) \setminus (\{d_{2,0}\} \cap \{b_1\}) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_3 - \mathbf{3}^{\mathrm{B}}_4 - \mathbf{3}^{\mathrm{B}}_5 & \{d_{2,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0\mathrm{w} B} + \mathbb{P}^1_{0AB'} + \mathbb{P}^1_{0AC'} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{0AB'} = \mathbb{P}^1_{2A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_1 - \mathbf{3}^{\mathrm{B}}_4 - \mathbf{3}^{\mathrm{B}}_4 & \{c_{1,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C'} + \mathbb{P}^1_{0C1} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1 s_0} \! + \mathbb{P}^1_{1A} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{0C1} = \mathbb{P}^1_{1A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{B}}_2 - \mathbf{3}^{\mathrm{B}}_5 - \mathbf{3}^{\mathrm{B}}_5 & \{c_{2,1}\} \cap \{b_1\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1 s_1} \!+ \mathbb{P}^1_{1B'}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2A} + \mathbb{P}^1_{2B'} + \mathbb{P}^1_{2B1} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{1B} = \mathbb{P}^1_{2B1}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(3)$-B top.}
\label{tab:SU(3)-B-Yukawas}
\end{table}
\subsection[\texorpdfstring{{$SU(3)$}}{SU(3)}-C Top]{\texorpdfstring{{\boldmath $SU(3)$}}{SU(3)}-C Top} \label{app-SU3-C}
The third top leads to the restrictions of the following coefficients
\begin{align}\label{eq:SU(3)-C-coeffs}
\begin{split}
b_0 &= b_{0,1}\,f_0\,f_2 ,\quad b_2 = b_{2,0}\,f_1 ,\quad c_1 = c_{1,1}\,f_0\,f_1 ,\quad c_2 = c_{2,1}\,f_0 , \\
d_0 &= d_{0,0}\,f_2 ,\quad d_1 = d_{1,0}\,f_1\,f_2 ,\quad d_2 = d_{2,0}\,f_1\,f_2^2 ,
\end{split}
\end{align}
while $b_1$ remain unrestricted. The top allows 4 different triangulations. For definiteness, we choose the one leading to the following SR-ideal:
\begin{align}\label{eq:SU(3)-C-SR-ideal}
\mathrm{u}\,\mathrm{v} , \mathrm{u}\,\mathrm{w} , \mathrm{w}\,s_0 , \mathrm{v}\,s_1 , s_0\,s_1 , f_0\,\mathrm{u} , f_0\,s_0 , f_0\,s_1 , f_1\,s_0 , f_1\,\mathrm{v} , f_2\,\mathrm{w} , f_2\,s_1 .
\end{align}
The coordinates and their corresponding divisor classes are summarised in the following table:
\begin{align}\label{tab:SU(3)-C-divisor-classes}
\begin{array}{c|ccccccc|c}
\hphantom{U} & \mathrm{u} & \mathrm{v} & \mathrm{w} & s_0 & s_1 & f_1 & f_2 & f_0 \\ \hline
\mathrm{U} & 1 & 1 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
S_0 & \cdot & \cdot & 1 & 1& \cdot & \cdot & \cdot & \cdot \\
S_1 & \cdot & 1 & \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\
F_1 & \cdot & 1 & \cdot & \cdot & \cdot & 1 & \cdot & -1 \\
F_2 & \cdot & 1 & 1 & \cdot & \cdot & \cdot & 1 & -1
\end{array}
\end{align}
For $SU(3)$ roots to have zero $U(1)$ charge, the generators (\ref{eq:U(1)-generators}) receive the following correction:
\begin{align}\label{eq:SU(3)-C-U(1)-generators}
\begin{split}
\omega_1^\text{C} &= S_1 - S_0 - \overline{\mathcal{K}} + \frac{1}{3} F_1 - \frac{1}{3} F_2 ,\\
\omega_2^\text{C} &= U - S_0 - \overline{\mathcal{K}} - [c_{1,1}] .
\end{split}
\end{align}
The Kodaira type of the fibre enhances from $I_3$ to $I_4$ (split) over the codimension-2 loci displayed in table \ref{tab:SU(3)-C-matter}, which therefore give rise to $\mathbf{3}$ and $\overline{\mathbf{3}}$ matter. In addition, over the curve $\{ w_3 \} \cap \{ b_1 \}$ the fibre type changes to Kodaira type $IV$, but no matter representation arises over this locus.
The gauge invariant Yukawa coupling appearing are summarised in table \ref{tab:SU(3)-C-Yukawas}.
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c|c}
\text{matter} & \text{locus} = W_3 \cap \ldots & \text{splitting of fibre components} & U(1)-\text{charges} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 & \{b_{2,0}\} & \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v}} + \mathbb{P}^1_{2A} & (-\frac{2}{3}, -1) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 & \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{u}} + \mathbb{P}^1_{1A} & (\frac{1}{3}, -1) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_3 & \{b_1\,c_{1,1} - b_{2,0}\,c_{2,1}\} & \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2B} + \mathbb{P}^1_{2C} & (\frac{1}{3}, 1) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_4 & \{b_{0,1}\,b_1 - c_{2,1}\,d_{0,0}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} & (-\frac{2}{3}, 0) \\ \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_5 & \{b_{2,0}\,d_{0,0}^2 + b_1^2\,d_{2,0} - b_1\,d_{0,0}\,d_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} & (\frac{1}{3},0)
\end{array}
\end{align*}
\caption{Matter states in the $SU(3)$-C top.}
\label{tab:SU(3)-C-matter}
\end{table}
\begin{table}[ht]
\begin{align*}
\begin{array}{c|c|c}
\text{coupling} & \text{locus}= W_3 \cap \ldots & \text{splitting of fibre components} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_2 - \mathbf{1}^{(2)} & \{b_{2,0}\} \cap \{c_{2,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{u}} + \mathbb{P}^1_{1A} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v}} + \mathbb{P}^1_{2A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_3 - \mathbf{1}^{(3)} & \{b_{2,0}\} \cap \{c_{1,1}\} & \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v} C} + \mathbb{P}^1_{2AB} + \mathbb{P}^1_{2C'} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \mathbf{1}^{(6)} & \{b_{2,0}\} \cap \{b_{0,1}\,b_1 - c_{2,1}\,d_{0,0}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v}} + \mathbb{P}^1_{2A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(4)} & \{b_{2,0}\} \cap \{b_1\,d_{2,0} - d_{0,0}\,d_{1,0}\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v}} + \mathbb{P}^1_{2A} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_3 - \mathbf{1}^{(5)} & \{c_{2,1}\} \cap \{c_{1,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{u}} + \mathbb{P}^1_{1A} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2B} + \mathbb{P}^1_{2C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \overline{\mathbf{1}}^{(1)} & \{c_{2,1}\} \cap \{b_{0,1}\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{u} B} + \mathbb{P}^1_{1B'} + \mathbb{P}^1_{1AC} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(6)} & \{c_{2,1}\} \cap \left( \mathbf{3}^\mathrm{C}_5 \right) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1\mathrm{u}} + \mathbb{P}^1_{1A}\\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_3 - \overline{\mathbf{3}}^{\mathrm{C}}_4 - \overline{\mathbf{1}}^{(4)} & \left( \mathbf{3}^\mathrm{C}_3 \right) \cap \left( \mathbf{3}^\mathrm{C}_4 \right) \setminus (\{c_{2,1}\} \cap \{b_1\}) & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2B} + \mathbb{P}^1_{2C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_3 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \overline{\mathbf{1}}^{(6)} & \left( \mathbf{3}^\mathrm{C}_3 \right) \cap \left( \mathbf{3}^\mathrm{C}_5 \right) \setminus ( \{b_{2,0}\} \cap \{b_1\} ) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2B} + \mathbb{P}^1_{2C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_4 - \overline{\mathbf{3}}^{\mathrm{C}}_5 - \mathbf{1}^{(2)} & \left( \mathbf{3}^\mathrm{C}_4 \right) \cap \left( \mathbf{3}^\mathrm{C}_5 \right) \setminus ( \{d_{0,0}\} \cap \{b_1\} ) & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_1 - \mathbf{3}^{\mathrm{C}}_3 - \mathbf{3}^{\mathrm{C}}_5 & \{b_{2,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0A} + \mathbb{P}^1_{0B}, \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2\mathrm{v} B} + \mathbb{P}^1_{2 B'} + \mathbb{P}^1_{2AC} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{0A} = \mathbb{P}^1_{2B'} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_2 - \mathbf{3}^{\mathrm{C}}_3 - \mathbf{3}^{\mathrm{C}}_4 & \{c_{2,1}\} \cap \{b_1\} & \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1 \mathrm{u} C} + \mathbb{P}^1_{1C'} + \mathbb{P}^1_{1AB} , \, \mathbb{P}^1_2 \rightarrow \mathbb{P}^1_{2B} + \mathbb{P}^1_{2C} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{1C'} = \mathbb{P}^1_{2C} \\ \hline \rule{0pt}{3ex}
\mathbf{3}^{\mathrm{C}}_4 - \mathbf{3}^{\mathrm{C}}_5 - \mathbf{3}^{\mathrm{C}}_5 & \{d_{0,0}\} \cap \{b_1\} & \mathbb{P}^1_0 \rightarrow \mathbb{P}^1_{0B} + \mathbb{P}^1_{0C'} + \mathbb{P}^1_{0C1} , \, \mathbb{P}^1_1 \rightarrow \mathbb{P}^1_{1B} + \mathbb{P}^1_{1C} \\ \rule{0pt}{3ex}
& & \mathbb{P}^1_{0C'} = \mathbb{P}^1_{1C}
\end{array}
\end{align*}
\caption{Yukawa couplings in the $SU(3)$-C top.}
\label{tab:SU(3)-C-Yukawas}
\end{table}
\section{Matching the MSSM-Spectrum}\label{appsec:huge_table}
With the search criteria and algorithm presented in section \ref{subsec_spectrum_search}, we find, for each of the toric $SU(3) \times SU(2) \times U(1)_1 \times U(1)_2$ models described in section \ref{sec_3211}, a significant number of possibilities to match the geometric spectrum with matter states of the MSSM including right-handed neutrinos und $\mu$-singlets. The results are listed in the left column of the tables below, in the notation introduced in section \ref{subsec_spectrum_search}. In the right column, we have listed which of the baryon and lepton number violating couplings (cf.~(\ref{eq:couplings_W2}) to (\ref{eq:couplings_K})) are allowed by the $U(1)$ selection rules, although -- for space-saving reasons -- in a slightly altered order. Furthermore we do not explicitly write down the states associated with $H_u$, $H_d$ and $Q$ in each coupling that appears, as these states are fixed for each possible match. For example the $\alpha$-term comes from a coupling $Q\,L\,d^c_R$, where there can be,
depending on the matching, several different states for $L$ and $d^c_R$, while $Q$ is given by the unique $({\bf 3}, {\bf 2})$-state. In our table we list such an existing coupling as $({\bf 2} , \overline{\bf 3})$, where the $\bf 2$-state is the lepton $L$ and $\overline{\bf 3}$ the down-quark, i.e.~the states appear in the same order as in the corresponding term in equations (\ref{eq:couplings_W2}) -- (\ref{eq:couplings_K}). When there is no $Q$ and $H$ involved in a coupling we give all the states involved, again in the order as they appear in the corresponding term; e.g.~for the $\beta$-term $u_R^c \, u_R^c \, d_R^c$ the corresponding entry in the table looks like $\overline{\bf 3}_i \, \overline{\bf 3}_j \, \overline{\bf 3}_k$, with the first up-quark involved being the state $\overline{\bf 3}_i$, the second one being $\overline{\bf 3}_j$, and the down-quark being $\overline{\bf 3}_k$. Note that we have summarised all possible terms of $W_{\text{singlet}}$ (\ref{eq:couplings_W_singlet}) in the entry $\delta$. Another special entry is the $\lambda_3$-term in (\ref{W3K}) of the form $Q\,Q\,Q\,H_d$; since there is no ambiguity in this term from the matching of the states, we simply list whether the coupling is allowed by the selection rules ($\checkmark$) or not ($-$).
We need to point out one case where the search algorithm does not completely fix the identification of the states with the MSSM fields. This happens when the choice of the Higgs states $H_{u/d}$ together with the charges of $Q = ({\bf 3},{\bf 2})$ does not completely fix the coefficients $a$ and $b$ of the hypercharge in terms of $U(1)_{1/2}$. In fact this is the case whenever there is a linear combination of $U(1)_{1/2}$ under which $Q$ and $H_{u/d}$ are all uncharged, which is the orthogonal linear combination to $U(1)_Y$. In such a case there might be some other states that are also uncharged under this particular $U(1)$ charge combination and can be identified with some Standard Model states. As these states are uncharged under the orthogonal $U(1)$, they are not subject to any selection rules, so that the dimension-four and -five operators in (\ref{eq:couplings_W2}) -- (\ref{eq:couplings_K}) will be present if all states involved are present. In this case there may be more possibilities to match the
spectrum, which we do not work out explicitly here. For every top combination (except for ${\rm III} \times {\rm B}$, where there is no such case) we have listed the corresponding case at the end of the tables below. In the ${\rm I} \times {\rm A}$-model for example, this happens when $H_u = {\bf 2}^{\rm I}_1$, $H_d = \overline{\bf 2}^{\rm I}_1$, and any assignment $U(1)_Y = (2b + 1) \, U(1)_1 + b \, U(1)_2$ for arbitrary $b$ gives the correct hypercharge for the Higgs and the left-handed quarks, because they are uncharged under the linear combination $2 \, U(1)_1 + U(1)_2$. To match states charged under this $U(1)$ one needs to specify the value of $b$. It would be interesting to see if, after Higgsing the particular linear combination of $U(1)_{1/2}$ under which $H_{u/d}$ and $Q$ are uncharged, the geometric spectrum can be embedded into the most general F-theory compactification with only one abelian factor \cite{Morrison:2012ei}.
Finally, recall from section
\ref{sec_3211} that the top-combination $ {\rm III} \times {\rm B}$ generically suffers from a non-Kodaira point, which, at least as far as our current understanding of F-theory is concerned, must be absent in order for the fibration to describe a well-defined vacuum.
\newpage
\footnotesize
\input{outIxA.txt}
\input{outIxB.txt}
\input{outIxC.txt}
\newpage
\input{outIIIxA.txt}
\newpage
\input{outIIIxB.txt}
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,152 |
import React from 'react'
import PropTypes from 'prop-types'
import { Button } from '@material-ui/core'
const BankButton = ({ bank, selectedBank, onClick }) => (
<Button
mini
variant="contained"
color={bank === selectedBank ? 'primary' : 'secondary'}
onClick={() => onClick(bank)}
>
Bank {String.fromCharCode(65 + bank)}
</Button>
)
BankButton.propTypes = {
bank: PropTypes.number.isRequired,
selectedBank: PropTypes.number.isRequired,
onClick: PropTypes.func.isRequired,
}
export default BankButton
| {
"redpajama_set_name": "RedPajamaGithub"
} | 592 |
\section{Introduction}
For an integer $n\geq 1$, let $\tau(n)=\sum_{k|n}1$ be the divisor function, which is the number of positive divisors of $n$. Dirichlet showed that
\begin{align}\label{eq: asymptotic formula for the sum of tau(n)}
\sum_{n\leq x}\tau(n)=x(\log{x}+2\gamma-1)+\Delta(x)
\end{align}
where is $\gamma$ is the Euler's constant and $\Delta(x)\ll x^{1/2}$. For the estimate of $\Delta(x)$, Voronoi \cite{Voronoi}, Van der Corput \cite{Van der Corput1,Van der Corput2}, Kolesnik \cite{Kolesnik}, Iwanice and Mozzcchi \cite{Iwanice}, Huxley \cite{Huxley1,Huxley2} and others made continuous improvement.
The most recent result is due to Bourgain and Watt \cite{BW}, who showed that
\begin{align}\label{eq: Bourgain and Watt's result}
\Delta(x)\ll x^{\frac{517}{1648}+\varepsilon}
\end{align}
for any $\varepsilon>0$.
By the definition of $\tau(n)$, we note that
$$
\sum_{n\leq x}\tau(n)=\sum\limits_{kl\leq x}1=\sum_{k\leq x}\sum_{l\leq \frac{x}{k}}1=\sum_{n\leq x}\lfloor\frac{x}{n}\rfloor,
$$
where $\lfloor{x}\rfloor$ is the largest integer not exceeding $t$.
Thus we can consider \eqref{eq: asymptotic formula for the sum of tau(n)} as an asymptotic formula for the fractional sum $\sum_{n\leq x}\lfloor\frac{x}{n}\rfloor$. With this viewpoint, Bordell\`{e}s, Dai, Heyman, Pan and Shparlinski \cite{BDHPS} investigated a more general case. For an arithmetic function $f$ satisfying some certain conditions, they proved an asymptotic formula for the sum
$$
S_f(x):=\sum_{n\leq x}f(\lfloor\frac{x}{n}\rfloor).
$$
Zhai \cite{Z} and Wu \cite{W} independently showed that if
$$
f(n)\ll n^{\alpha}\left(\log{n}\right)^{\theta}
$$
for some $\alpha\in[0,1)$ and $\theta\geq 0$, then we have
\begin{align}\label{eq: S_f(x) asymptotic formula of Zhai and Wu}
S_{f}(x)=x\sum_{n=1}^{\infty}\frac{f(n)}{n(n+1)}+O\left(x^{(1+\alpha)/2}\left(\log{x}\right)^{\theta}\right).
\end{align}
It is possible to improve \eqref{eq: S_f(x) asymptotic formula of Zhai and Wu} for a certain function $f$. For examples, if $f=\Lambda$, which is the von Mangoldt function, Ma and Wu \cite{MW} proved that
\begin{align*}
\sum_{n\leq x}\Lambda(\lfloor\frac{x}{n}\rfloor)=x\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n(n+1)}+O\left(x^{\frac{35}{71}+\varepsilon}\right)
\end{align*}
for any $\varepsilon>0$.
The exponent $35/71$ was then improved to $9/19$ by Liu, Wu and Yang \cite{LWZ}. If $f=\tau$, Ma and Sun \cite{MS} showed that
\begin{align*}
S_{\tau}(x)=\sum_{n\leq x}\tau(\lfloor\frac{x}{n}\rfloor)=x\sum_{n=1}^{\infty}\frac{\tau(n)}{n(n+1)}+O\left(x^{\frac{11}{23}+\varepsilon}\right).
\end{align*}
The exponent $11/23$ was then improved to $19/40$ and $9/19$, respectively, by Bordell\`{e}s \cite{Bordell} and Liu, Wu and Yang \cite{LWZ2}. Next in \cite{Stucky} Stucky improved on the previous result and proved that $$ S_{\tau}(x)=x\sum_{n=1}^{\infty}\frac{\tau(n)}{n(n+1)}+O\left(x^{\frac{5}{11}+\varepsilon}\right).$$
Motivated by recent results, it is important to study the sum of the form
$$T_f(x)=\sum_{n_1n_2...n_r\leq x}f(\lfloor\frac{x}{n_1n_2...n_r}\rfloor),$$ taken over the hyperbolic region $$\lbrace(n_1,n_2,...,n_r)\in\mathbb{N}^{r}:n_1n_2...n_r\leq{x}\rbrace$$ for some arithmetic function $f$, where $r$ is a fixed positive integer.\\
In this paper, we consider the following "hyperbolic" sum in the case $f=\tau$ and $r=2$. We put
$$
T(x)=\sum_{n_1n_2\leq x}\tau(\lfloor\frac{x}{n_1n_2}\rfloor).
$$
Using Bourgain and Watt's bound \eqref{eq: Bourgain and Watt's result}, it is not difficult to obtain (see Section \ref{sec: Proof of (1.4)}) that
\begin{align}\label{eq: (1.4)}
T(x)=C_1x\log{x}+C_2x+O\left(x^{\frac{1648}{2779}+\varepsilon}\right),
\end{align}
where
\begin{align}\label{def: C1}
C_1=\sum_{d\geq 1}\frac{\tau(d)}{d(d+1)},
\end{align}
$C_2=C_1(2\gamma -1)+C_3$
with $\gamma$ being the Euler's constant and
\begin{align}\label{def: c3}
C_3=\sum_{d\geq 1}\tau(d)\left(\frac{\log{d}}{d}-\frac{\log(d+1)}{d+1}\right).
\end{align}
We further improve \eqref{eq: (1.4)} and obtain a better error term.
\begin{theorem}\label{theorem:1}
For any $x\geq 2$ and $\varepsilon >0$, we have
\begin{align}\label{eq: asymptotic for T(x)}
T(x)=C_1x\log{x}+C_2x+O\left(x^{\frac{10}{17}+\varepsilon}\right),
\end{align}
where constants $C_1,C_2$ are the same as above.
\end{theorem}
For comparison, we note that $1648/2779\approx 0.593$ and $10/17\approx0.588$.
\bigskip
{\bf{Notations:}} Through out this paper, we use $\lfloor{x}\rfloor$ to denote largest integer not exceeding $x$, $e(x)$ to denote $e^{2\pi ix}$ and $d\sim D$ to denote $D<d \leq 2D$. Moreover, for functions $f,g$, symbol $f\ll g$ or $f=O(g)$ means $|f|<C|g|$ for some unspecified constant $C>0$, and
$f\asymp g$ means $f\ll g$ and $g\ll f$.
\section{Preliminaries}
In this section, we state some lemmas, which will be needed in the proof of Theorem \ref{theorem:1}.
For the error term $\Delta(x)$ in \eqref{eq: asymptotic formula for the sum of tau(n)}, we have the following expression.
\begin{lemma}\cite[Theorem 4.5]{GK}\label{lemma:1}
For any $x\in \mathbb{R}$, we have
$$
\Delta(x)=-2\sum_{n\leq \sqrt{x}}\psi\left(\frac{x}{n}\right)+O(1),
$$
where $\psi(x)=x- \lfloor{x}\rfloor-1/2$.
\end{lemma}
The function $\psi(x)$ is periodic with period $1$. It can be expanded into a Fourier series. We need the following truncated version.
\begin{lemma}\cite[Lemma 4.1]{BDHPS}\label{lemma:2}
For $x\geq 1$ and $H\geq 1$, we have
$$\psi(x)=-\sum_{1\leq |h|\leq H}\Phi\left(\frac{h}{H+1}\right)\frac{e(hx)}{2\pi ih}+R_{H}(x),$$
where $\Phi(t):=\pi t(1-|t|)\cos(\pi t)+|t|$, and the error term $R_{H}(x)$ satisfies
$$\left|R_{H}(x)\right|\leq \frac{1}{2H+2}\sum_{|h|\leq H}\left(1-\frac{|h|}{H+1}\right)e(hx).$$
\end{lemma}
The following lemma is the B-process in the Van der Corput's method.
\begin{lemma} \cite[Lemma 3.6]{GK}\label{lemma:3}
Suppose that $f$ has four continuous derivatives on $[a,b]$, and that $f^{''}<0$ on this interval. Suppose further that $[a,b]\subseteq[N,2N]$ and that $\alpha=f^{'}(b)$ and $\beta=f^{'}(a)$. Assume that there is some $F>0$ such that
$$
f^{(2)}(x)\asymp FN^{-2}, f^{(3)}(x)\ll FN^{-3} \; and \; f^{(4)}(x)\ll FN^{-4}
$$
for $x$ in $[a,b]$. Let $x_v$ be defined by the relation $f^{'}(x_v)=v$, and let
$$
\phi(v)=-f(x_v)+vx_v.
$$
Then we have
$$\sum_{n\in I}e(f(n))=\sum_{\alpha\leq v \leq \beta}\frac{e(-\phi(v)-1/8)}{|f^{''}(x_v)|^{1/2}}+O\left(\log(FN^{-1}+2)+F^{-1/2}N\right).$$
\end{lemma}
\section{Applying Bourgain and Watt's bound} \label{sec: Proof of (1.4)}
In this section, we prove
\eqref{eq: (1.4)}. According to the definition of divisor function $\tau(n)$, we have
\begin{eqnarray*}
T(x)=\sum_{n\leq x}\tau(\lfloor\frac{x}{n}\rfloor)\sum_{n_1n_2=n}1
=\sum_{n\leq x}\tau(n)(\lfloor\frac{x}{n}\rfloor).
\end{eqnarray*}%
Let $N\in [1,x)$ be a parameter that will be chosen, the sum $T(x)$ can be written as
\begin{eqnarray}\label{eq: T(x)=T1(x)+T2(x)}
T(x)=T_1(x)+T_2(x),
\end{eqnarray}%
where$$T_1(x):=\sum_{n\leq N}\tau(\lfloor\frac{x}{n}\rfloor)\tau(n),\ T_2(x):=\sum_{N<n\leq x}\tau(\lfloor\frac{x}{n}\rfloor)\tau(n).$$
For $T_1(x)$, we have
\begin{eqnarray}\label{equation:S1}
T_1(x)=\sum_{n\leq N}\tau(\lfloor\frac{x}{n}\rfloor)\tau(n)\leq Nx^{\varepsilon},
\end{eqnarray}
where we have used the bound $\tau(n)\ll n^{\varepsilon}$ for $n\geq 1$ and any $\varepsilon>0$.
For $T_2(x)$,
let $d=\lfloor{x/n}\rfloor$, then
$$\frac{x}{n}-1< d \leq \frac{x}{n}, \quad \frac{x}{d+1}<n \leq \frac{x}{d}.$$
Thus we have
\begin{eqnarray*}
T_2(x)
=\sum_{N<n\leq x}\sum_{\lfloor{x/n}\rfloor=d}\tau(d)\tau(n)
=\sum_{d\leq \frac{x}{N}}\tau(d)\sum_{\frac{x}{d+1}<n\leq \frac{x}{d}}\tau(n).
\end{eqnarray*}%
By \eqref{eq: asymptotic formula for the sum of tau(n)}, we obtain
\begin{eqnarray}\label{equation:S2}
T_2(x)=\sum_{d\leq \frac{x}{N}}\tau(d)\left(\sum_{n\leq \frac{x}{d}}\tau(n)-\sum_{n\leq \frac{x}{d+1}}\tau(n)\right)=T_{21}(x)+T_{22}(x)+T_{\Delta}(x),
\end{eqnarray}%
where
\begin{eqnarray*}
T_{21}(x):&=&x(\log{x}+2\gamma-1)\sum_{d\leq \frac{x}{N}}\frac{\tau(d)}{d(d+1)},\\
T_{22}(x):&=&x\sum_{d\leq \frac{x}{N}}\tau(d)\left(\frac{\log{d}}{d}-\frac{\log{(d+1)}}{d+1}\right),\\
T_{\Delta}(x):&=&\sum_{d\leq \frac{x}{N}}\tau(d)\left(\Delta\left(\frac{x}{d}\right)-\Delta\left(\frac{x}{d+1}\right)\right).
\end{eqnarray*}%
For $T_{21}(x)$, we write
\begin{eqnarray*}\label{equation:S21}
T_{21}(x)
=x(\log{x}+2\gamma-1)\left(\sum_{d> 1}\frac{\tau(d)}{d(d+1)}-\sum_{d>\frac{x}{N}}\frac{\tau(d)}{d(d+1)}\right).
\end{eqnarray*}
Note that
\begin{align*}
\sum_{d>\frac{x}{N}}\frac{\tau(d)}{d(d+1)}\ll \sum_{d>\frac{x}{N}}d^{-2+\varepsilon}\ll\left(\frac{x}{N}\right)^{-1+\varepsilon}.
\end{align*}
We then obtain
\begin{eqnarray}\label{bound of T_21}
T_{21}(x)
=C_1x(\log{x}+2\gamma-1)+O\left(N^{1+\varepsilon}x^{2\varepsilon}\right),
\end{eqnarray}
where $C_1$ is given by \eqref{def: C1}.
For $T_{22}(x)$, similarly as above, we have
$$
T_{22}(x)=C_3x+O\left(\sum_{d>\frac{x}{N}}\tau(d)\left(\frac{\log{d}}{d}-\frac{\log{(d+1)}}{d+1}\right)\right),
$$
where $C_3$ is given by \eqref{def: c3}.
Note that
$$
\frac{\log{d}}{d}-\frac{\log{(d+1)}}{d+1}=\frac{\log{d}}{d}-\frac{\log d+O(1/d)}{d+1}\ll \frac{\log d}{d^2}.
$$
It follows that
\begin{align}\label{bound of T_22}
T_{22}(x)=C_3x+O\left(N^{1+\varepsilon}x^{2\varepsilon}\right).
\end{align}
Combining \eqref{bound of T_21} and \eqref{bound of T_22} with \eqref{equation:S2}, we have
\begin{align}\label{T_2(x)=}
T_2(x)=C_1x(\log x+2\gamma+1)+C_3x+T_{\Delta}(x)+O(N^{1+\varepsilon}x^{2\varepsilon}).
\end{align}
Inserting
\eqref{equation:S1} and \eqref{T_2(x)=} into \eqref{eq: T(x)=T1(x)+T2(x)}, we obtain
\begin{align}\label{T(x)=T+T(delta)}
T(x)=C_1x\log{x}+C_2x+T_{\Delta}(x)+O\left(N^{1+\varepsilon}x^{2\varepsilon}\right),
\end{align}
where $C_2=C_1(2\gamma-1)+C_3$.
For $T_{\Delta}(x)$, using Bourgain and Watt's bound \eqref{eq: Bourgain and Watt's result}, we have
\begin{eqnarray}\label{equation:T(delta)}
T_{\Delta}(x)\ll\sum_{d\leq \frac{x}{N}}\tau(d)\left(\frac{x}{d}\right)^{\frac{517}{1648}+\varepsilon}
\ll x^{1+\varepsilon}N^{-\frac{1131}{1648}+\varepsilon}.
\end{eqnarray}%
Inserting \eqref{equation:T(delta)} into \eqref{T(x)=T+T(delta)} gives
\begin{eqnarray*}
T(x)=C_x\log{x}+C_2x+O\left(N^{1+\varepsilon}x^{2\varepsilon}+x^{1+\varepsilon}N^{-\frac{1131}{1648}+\varepsilon}\right).
\end{eqnarray*}
Now \eqref{eq: (1.4)} follows from taking $N=x^{\frac{1648}{2779}}$.
\section{An three dimensional exponential sum}\label{an exponetial sum}
In the next two sections, we show how to improve \eqref{eq: (1.4)} and prove Theorem \eqref{theorem:1}. To this aim, we need to estimate a three dimensional exponential sum first.
For $x\geq 2$, $h\geq 1$, $1\ll D$ and $1\ll L$, define
$$
\mathfrak{S}_{\delta}(x,h,D,L):=\sum_{d_1d_2\sim D}\sum_{l\sim L}e\left(\frac{hx}{l(d_1d_2+\delta)}\right)
$$
with $\delta=\{0,1\}$. We have the following bound for $\mathfrak{S}_{\delta}(x,h,D,L)$.
\begin{proposition}\label{prop: bound for G_delta(x,H,D,L)}
We have
\begin{align*}
\mathfrak{S}_{\delta}(x,h,D,L)\ll x^{\frac{2\kappa+1}{2}}h^{\frac{2\kappa-1}{2}}D^{\frac{\lambda-3\kappa}{2}}L^{-\kappa-\frac{1}{2}}+x^{-\frac{1}{2}}h^{-\frac{3}{2}}(D^{\frac{3}{2}}L^{\frac{1}{2}}+D^{\frac{1}{2}}L^{\frac{3}{2}})+D(xhL)^{\varepsilon},
\end{align*}
where $(\kappa,\lambda)$ is an exponent pair.
\end{proposition}
The proof of Proposition \ref{prop: bound for G_delta(x,H,D,L)} is following.
Note that
\begin{align*}
\left|\sum_{d_1d_2\sim D}\sum_{l\sim L}e\left(\frac{hx}{l(d_1d_2+\delta)}\right)\right|=\left|\sum_{d_1d_2\sim D}\sum_{l\sim L}e\left(-\frac{hx}{l(d_1d_2+\delta)}\right)\right|.
\end{align*}
Applying Lemma \ref{lemma:3}, the inner sum over $l$ is equal to
\begin{eqnarray*}
&&\frac{1}{\sqrt{2}}e\left(-\frac{1}{8}\right)\sum_{\frac{hx}{4(d_1d_2+\delta)L^2}\leq v \leq \frac{hx}{(d_1d_2+\delta)L^2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)\left(\frac{hx}{(d_1d_2+\delta)v^3}\right)^{1/4}\\
&&+O\left(\log \left(\frac{hx}{(d_1d_2+\delta)L}+2\right)+\left(\frac{hx}{(d_1d_2+\delta)L}\right)^{-1/2}L\right).
\end{eqnarray*}
By trivial estimate, we see that the contribution of the $O$-term to $\mathfrak{S}_{\delta}(x,h,D,L)$ is
\begin{eqnarray*}
\ll D(xhL)^{\varepsilon}+x^{-\frac{1}{2}}h^{-\frac{3}{2}}D^{\frac{1}{2}}L^{\frac{3}{2}}.
\end{eqnarray*}
Hence we have
\begin{align}\label{eq: hS(x)}
\mathfrak{S}_{\delta}(x,h,D,L)\ll \mathfrak{S}_{\delta}^{'}(x,h,D,L)+D(xhL)^{\varepsilon}+x^{-\frac{1}{2}}h^{-\frac{3}{2}}D^{\frac{1}{2}}L^{\frac{3}{2}},
\end{align}
where
$$
\mathfrak{S}_{\delta}^{'}(x,h,D,L)= \left|\sum_{d_1d_2\sim D}\sum_{\frac{hx}{4(d_1d_2+\delta)L^2}\leq v \leq \frac{hx}{(d_1d_2+\delta)L^2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)\left(\frac{hx}{(d_1d_2+\delta)v^3}\right)^{1/4}\right|.
$$
In the following, we deal with $\mathfrak{S}_{\delta}^{'}(x,h,D,L)$.
Applying partial summation to the sum over $v$ and changing the order of summations, we derive that
\begin{align}\label{eq: hS1(x)}
\mathfrak{S}_{\delta}^{'}(x,h,D,L)
\ll\left(\frac{L^3}{hx}\right)^{\frac{1}{2}}\max_{\frac{hx}{4(d_1d_2+\delta)L^2}\leq \theta_1\leq \frac{hx}{(D+\delta)L^2}}\sum_{\frac{hx}{4(d_1d_2+\delta)L^2}\leq v \leq \theta_1}|\mathfrak{D}_{\delta}(v,h,x)|,
\end{align}
where
$$
\mathfrak{D}_{\delta}(v,h,x)=\sum_{d_1d_2\sim D}\left(d_1d_2+\delta\right)^{\frac{1}{2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right).
$$
For $\mathfrak{D}_{\delta}(v,h,x)$, we will show that
\begin{align}\label{eq: bound for D_delta(v,h,x)}
\mathfrak{D}_{\delta}(v,h,x)\ll x^{\frac{\kappa}{2}}h^{\frac{\kappa}{2}}D^{\frac{\kappa}{6}+\frac{\lambda}{2}+1}v^{\frac{\kappa}{2}}+x^{-\frac{1}{2}}h^{-\frac{1}{2}}D^{2+\varepsilon}v^{-\frac{1}{2}}+D^{\frac{1}{2}}.
\end{align}
Inserting this bound to \eqref{eq: hS1(x)}, we obtain Proposition \ref{prop: bound for G_delta(x,H,D,L)} by trivial estimating.
Now we prove \eqref{eq: bound for D_delta(v,h,x)}. Since
$$
\left(d_1d_2+\delta\right)^{\frac{1}{2}}=\left(d_1d_2\right)^{\frac{1}{2}}\left(1+O\left(\frac{1}{d_1d_2}\right)\right),
$$
then we have
$$
\mathfrak{D}_{\delta}(v,h,x)=\sum_{d_1d_2\sim D}\left(d_1d_2\right)^{\frac{1}{2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)+O(D^{\frac{1}{2}}).
$$
Divide the sum over $d_1$ into two sum according to $d_1\leq \sqrt{2D}$ or not. We get
\begin{align}\label{eq:D_delta(v,h,x)}
\mathfrak{D}_{\delta}(v,h,x)=\mathfrak{D}_{\delta}^{'}(v,h,x)+\mathfrak{D}_{\delta}^{''}(v,h,x)+O(D^{1/2})
\end{align}
where
$$
\mathfrak{D}_{\delta}^{'}(v,h,x)=\sum\limits_{d_1\leq \sqrt{2D}}\sum\limits_{\frac{D}{d_1}<d_2\leq\frac{2D}{d_1}}(d_1d_2)^{\frac{1}{2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)
$$
and
$$
\mathfrak{D}_{\delta}^{''}(v,h,x)=\sum\limits_{d_2\leq\sqrt{2D}}\sum\limits_{\max\left\{\frac{D}{d_2},\sqrt{2D}\right\}<d_1\leq \frac{2D}{d_2}}(d_1d_2)^{\frac{1}{2}}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right).
$$
For $\mathfrak{D}_{\delta}^{'}(v,h,x)$, applying partial summation to the sum over $d_2$, we obtain
\begin{align}\label{eq: D'_delta(v,h,x)<<}
\mathfrak{D}_{\delta}^{'}(v,h,x)\ll D^{\frac{1}{2}}\sum_{1\leq d_1\leq \sqrt{2D}}\max\limits_{\frac{D}{d_1}\leq\theta_2\leq\frac{2D}{d_1}}\left|\sum_{\frac{D}{d_1}\leq d_2\leq\theta_2}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)\right|.
\end{align}
Applying the exponent pair $(\kappa,\lambda)$ to the sum over $d_2$, we get
\begin{eqnarray*}\label{exponent pair}
\sum_{\frac{D}{d_1}\leq d_2\leq\theta_2}e\left(2\sqrt{\frac{hxv}{d_1d_2+\delta}}\right)&\ll & \left(\left(\frac{hxv}{d_1}\right)^{\frac{1}{2}}\left(\frac{D}{d_1}\right)^{-\frac{3}{2}}\right)^{\kappa}\left(\frac{D}{d_1}\right)^{\lambda}+\left(\frac{hxv}{d_1}\right)^{-\frac{1}{2}}\left(\frac{D}{d_1}\right)^{\frac{3}{2}}\nonumber\\
&=& x^{\frac{\kappa}{2}}h^{\frac{\kappa}{2}}d_1^{\kappa-\lambda}v^{\frac{\kappa}{2}}D^{\lambda-\frac{\kappa}{3}}+x^{-\frac{1}{2}}h^{-\frac{1}{2}}d_1^{-1}v^{-\frac{1}{2}}D^{\frac{3}{2}}.
\end{eqnarray*}
Insert this to \eqref{eq: D'_delta(v,h,x)<<}, it comes
\begin{align}\label{eq:D_delta^{'}}
\mathfrak{D}_{\delta}^{'}(v,h,x)\ll x^{\frac{\kappa}{2}}h^{\frac{\kappa}{2}}D^{\frac{\kappa}{6}+\frac{\lambda}{2}+1}v^{\frac{\kappa}{2}}+x^{-\frac{1}{2}}h^{-\frac{1}{2}}D^{2+\varepsilon}v^{-\frac{1}{2}}.
\end{align}
For $\mathfrak{D}_{\delta}^{''}(v,h,x)$, by similar argument as above, we also have
\begin{align}\label{eq:D_delta^{''}}
\mathfrak{D}_{\delta}^{''}(v,h,x)\ll x^{\frac{\kappa}{2}}h^{\frac{\kappa}{2}}D^{\frac{\kappa}{6}+\frac{\lambda}{2}+1}v^{\frac{\kappa}{2}}+x^{-\frac{1}{2}}h^{-\frac{1}{2}}D^{2+\varepsilon}v^{-\frac{1}{2}}.
\end{align}
Now inserting \eqref{eq:D_delta^{'}} and \eqref{eq:D_delta^{''}} to \eqref{eq:D_delta(v,h,x)}, we obtain \eqref{eq: bound for D_delta(v,h,x)}. This completes the proof of Proposition \ref{prop: bound for G_delta(x,H,D,L)}.
\section{Proof of Theorem \ref{theorem:1}} \label{sec: proof of our theorem}
In this section, we use Proposition \ref{prop: bound for G_delta(x,H,D,L)} to give a better upper bound for $T_{\Delta}(x)$, which implies Theorem \ref{theorem:1}. In view of the definition $T_{\Delta}(x)$, we only need to bound the sum
$$
S_{\delta}(x,N):=\sum_{d\leq \frac{x}{N}}\tau(d)\Delta\left(\frac{x}{d+\delta}\right)
$$
with $\delta=0,1$.
By Lemma \ref{lemma:1}, we have
\begin{align}\label{eq: def of S_delta(s,N)}
S_{\delta}(x,N)=-2\sum_{d\leq \frac{x}{N}}\tau(d)\sum_{l\leq \sqrt{\frac{x}{d}}}\psi\left(\frac{x}{l(d+\delta)}\right)+O\left(\frac{x}{N}\right).
\end{align}
By the dyadic argument and the definition of $\tau(d)$, we have
\begin{eqnarray}\label{equation:Smax}
\sum_{d\leq \frac{x}{N}}\tau(d)\sum_{l\leq \sqrt{\frac{x}{d}}}\psi\left(\frac{x}{l(d+\delta)}\right)\ll x^{\varepsilon}\max \limits_{1\ll D\leq \frac{x}{N}\atop L^2D\ll x}\left|\sum_{d_1d_2\sim D}\sum_{l\sim L}\psi\left(\frac{x}{l(d_1d_2+\delta)}\right)\right|.
\end{eqnarray}%
By Lemma \ref{lemma:2}, we write
\begin{eqnarray*}
\sum_{d_1d_2\sim D}\sum_{l\sim L}\psi\left(\frac{x}{l(d_1d_2+\delta)}\right)=\mathfrak{S}_{\delta}^{\sharp}+\mathfrak{S}_{\delta}^{\dag}.
\end{eqnarray*}%
where
\begin{eqnarray*}
\mathfrak{S}_{\delta}^{\sharp}:=-\frac{1}{2\pi i}\sum_{1\leq |h|\leq H}\Phi\left(\frac{h}{H+1}\right)\frac{1}{h}\sum_{d_1d_2\sim D}\sum_{l\sim L}e\left(\frac{hx}{l(d_1d_2+\delta)}\right),
\end{eqnarray*}
and
\begin{eqnarray*}
\mathfrak{S}_{\delta}^{\dag}:=\sum_{d_1d_2\sim D}\sum_{l\sim L}R_H\left(\frac{x}{l(d_1d_2+\delta)}\right).
\end{eqnarray*}%
Noting that $0<\Phi(t)<1$ for $0<|t|<1$, by Proposition \ref{prop: bound for G_delta(x,H,D,L)}, we obtain
\begin{eqnarray}\label{equation:P1}
\mathfrak{S}_{\delta}^{\sharp}\ll x^{\frac{2\kappa+1}{2}}L^{-\kappa-\frac{1}{2}}D^{\frac{\lambda-3\kappa}{2}}H^{\frac{2\kappa+1}{2}}+x^{-\frac{1}{2}}H^{-\frac{1}{2}}(D^{\frac{3}{2}}L^{\frac{1}{2}}+D^{\frac{1}{2}}L^{\frac{3}{2}})+(xHL)^{\varepsilon}D.
\end{eqnarray}
By a similar argument, we derive that
\begin{align}\label{equation:P2}
\mathfrak{S}_{\delta}^{\dag}\ll x^{\frac{2\kappa+1}{2}}L^{-\kappa-\frac{1}{2}}D^{\frac{\lambda-3\kappa}{2}}H^{\frac{2\kappa+1}{2}}+x^{-\frac{1}{2}}H^{-\frac{1}{2}}(D^{\frac{3}{2}}L^{\frac{1}{2}}+D^{\frac{1}{2}}L^{\frac{3}{2}})+(xHL)^{\varepsilon}D+DLH^{-1}\log{D},
\end{align}
where the term $DLH^{-1}\log{D}$ comes from the contribution of $h=0$.
Combining (\ref{equation:P1}) and (\ref{equation:P2}) with \eqref{equation:Smax} and applying the exponential pair $$(\kappa,\lambda)=A^2B(0,1)=(1/14,11/14),$$ we obtain
\begin{eqnarray*}
&&\sum_{d\leq \frac{x}{N}}\tau(d)\sum_{l\leq \sqrt{\frac{x}{d}}}\psi\left(\frac{x}{l(d+\delta)}\right)\nonumber\\
&\ll& x^{\varepsilon}\max \limits_{1\ll D\leq \frac{x}{N}\atop L^2D\ll x}\left\{x^{\frac{4}{7}}L^{-\frac{4}{7}}H^{\frac{4}{7}}D^{\frac{2}{7}}+x^{-\frac{1}{2}}L^{\frac{1}{2}}D^{\frac{3}{2}}H^{\frac{1}{2}}+DLH^{-1}\log D\right\}.
\end{eqnarray*}%
Chosen $H=x^{-\frac{4}{11}}D^{\frac{5}{11}}L$, we have
\begin{align}\label{equation:PP}
\sum_{d\leq \frac{x}{N}}\tau(d)\sum_{l\leq \sqrt{\frac{x}{d}}}\ll x^{\frac{10}{11}}N^{-\frac{6}{11}}+x^{1+\varepsilon}N^{-\frac{5}{4}}
\end{align}
Plug \eqref{equation:PP} into \eqref{eq: def of S_delta(s,N)}, we have
\begin{eqnarray}\label{equation:SSS}
T_{\Delta}(x)\ll x^{\frac{10}{11}+\varepsilon}N^{-\frac{6}{11}}+x^{1+\varepsilon}N^{-\frac{5}{4}}+xN^{-1}.
\end{eqnarray}
Insert this into \eqref{T(x)=T+T(delta)}, we obtain
\begin{align}
T(x)=C_1x\log{x}+C_2x+O\left(x^{\frac{10}{11}+\varepsilon}N^{-\frac{6}{11}}+x^{1+\varepsilon}N^{-\frac{5}{4}}+xN^{-1}+N^{1+\varepsilon}x^{2\varepsilon}\right).
\end{align}
Then Theorem \ref{theorem:1} follows from taking $N=x^{\frac{10}{17}}$.\\
\section{Acknowledgements}
The authors would like to thank Professor Kui Liu for his suggestions to improve this paper.
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{"url":"https:\/\/zbmath.org\/?q=an:0741.33011","text":"## Generalized Euler integrals and $$A$$-hypergeometric functions.(English)Zbl\u00a00741.33011\n\nIn their previous works the authors have introduced a holonomic system of linear differential equations associated to a certain finite set $$A\\subset \\mathbb{Z}^ n$$. They called this system the $$A$$-hypergeometric system, and its solutions the $$A$$-hypergeometric functions. The basis in the space of these functions consisting of series of hypergeometric type has been constructed. In the present paper the authors study integral expressions for these series. The main result is that for any $$A$$- hypergeometric system a complete set of its solutions can be represented as integrals of the form $\\int\\prod_ i P_ i(x_ 1,\\dots,x_ k)^{\\alpha_ i} x_ 1^{\\beta_ 1}\\dots x_ k^{\\beta_ k} dx_ 1\\dots dx_ k,$ where $$P_ i$$ are polynomials; the integrals are considered as functions of the coefficients of $$P_ i$$. The authors call these integrals generalized Euler integrals. They generalize the classical Euler integral for the Gauss hypergeometric function to the case of several variables.\n\n### MSC:\n\n 33C70 Other hypergeometric functions and integrals in several variables\nFull Text:\n\n### References:\n\n [1] Aomoto, K, On the structure of integrals of power products of linear functions, Sci. papers college gen. ed. univ. Tokyo, 27, 49-61, (1977) \u00b7 Zbl\u00a00384.35045 [2] Bateman, H; Erdelyi, A, () [3] Bjork, J.E, Rings of differential operators, () \u00b7 Zbl\u00a00198.35903 [4] () [5] Brylinsky, J.-L, Transformations canoniques et transformation de Fourier, Ast\u00e9risque, no. 140, (1987) [6] Gelfand, I.M, A general theory of hypergeometric functions, Doklady akad. nauk SSSR, 288, 14-18, (1986), [In Russian] [7] Gelfand, I.M; Graev, M.I, A duality theorem for general hypergeometric functions, Doklady akad. nauk SSSR, 289, 19-23, (1986), [In Russian] [8] Gelfand, I.M; Graev, M.I; Zelevinsky, A.V, Holonomic systems of equations and series of hypergeometric type, Doklady akad. nauk SSSR, 295, 14-19, (1987), [In Russian] [9] Gelfand, I.M; Zelevinsky, A.V; Kapranov, M.M, Equations of hypergeometric type and Newton polytopes, Doklady akad. nauk SSSR, 300, 529-534, (1988), [In Russian] [10] Gelfand, I.M; Zelevinsky, A.V; Kapranov, M.M, Hypergeometric functions and toric varieties, Funktsional. anal. i prilozhen., 23, 12-26, (1989), [In Russian] [11] Gelfand, I.M; Zelevinsky, A.V; Kapranov, M.M, A-discriminants and Cayley-Koszul complexes, Doklady akad. nauk SSSR, 307, 1307-1310, (1989), [In Russian] [12] Gelfand, I.M; Zelevinsky, A.V; Kapranov, M.M, Discriminants of polynomials in several variables and triangulations of Newton polytopes, Algebra i anal., 2, 1-62, (1990), [In Russian] [13] Ginsburg, V.A, Characteristic varieties and vanishing cycles, Invent. math., 84, 327-402, (1986) \u00b7 Zbl\u00a00598.32013 [14] Kashiwara, M, Systems of micro-differential equations, (1983), Birkh\u00e4user Boston \u00b7 Zbl\u00a00566.32022 [15] {\\scM. Kashiwara and P. Shapira}, Microlocal study of sheaves. Ast\u00e9risque no. 128. [16] Kouchnirenko, A.G, Poly\u00e8dres de Newton et nombres de Milnor, Invent. math., 32, 1-31, (1976) \u00b7 Zbl\u00a00328.32007 [17] MacPherson, R; Vilonen, K, Elementary constructions of perverse sheaves, Invent. math., 84, 403-435, (1986) \u00b7 Zbl\u00a00597.18005 [18] Steenrod, N, Homology with local coefficients, Ann. of math., 44, 610-627, (1945) \u00b7 Zbl\u00a00061.40901 [19] Vasiliev, V.A; Gelfand, I.M; Zelevinsky, A.V, General hypergeometric functions on complex Grassmannians, Funktsional. anal. i prilozhen., 21, 23-38, (1987), [In Russian] \u00b7 Zbl\u00a00614.33008\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2022-08-18 09:56:12","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\": 0.9054399132728577, \"perplexity\": 4112.561688066603}, \"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-2022-33\/segments\/1659882573193.35\/warc\/CC-MAIN-20220818094131-20220818124131-00065.warc.gz\"}"} | null | null |
Visual Acuity – The Sense and Non-sense of Ultra High Definition Displays
August 27, 2014 June 24, 2021 Simon Baker
Article by Kid Jansen
Fovea and foveola
Normal vision
When Can a Computer Display be Considered Retina?
Font Sizes and Scaling
Retina Display Televisions
HD and FHD viewing distance recommendations
UHD viewing distance recommendations
Almost every time you see a news article about ultra high definition pop up on a news site with the option to comment on the article, you'll find that there's quite a discussion going on about the sense and the non-sense of ultra high definition (mostly the latter). We thought it would be nice to clear that up a bit. Often the problem lies with the fact that most people do not use all the relevant parameters. For instance they discard the usefulness of ultra high definition solely on the resolution itself. For a correct assessment of the usefulness all of the following parameters should be included:
Viewing distance
From the screen size and the resolution the pixel pitch can be determined. Combining that result with viewing distance gives the visual angle, or angular size of a pixel. As long as the angular size of a pixel is larger than the smallest discernible angular size (visual acuity) it's useful to increase the resolution for the same screen size and viewing distance. A widely used term nowadays for displays that have a smaller than discernible angular pixel size at a normal viewing distance, is "retina display", a term made up by Apple and first used for the display in the iPhone 4.
Visual acuity (VA) is a measure of the acuteness of foveal vision. The fovea is the part of the retina with the highest visual acuity. It sees only the central 2° of our vision and comprises less than 1% of the total retina, but more than half of the information processed by the visual cortex in the brain comes from that small area of the retina. To put that 2° angle in perspective: if you look straight ahead, your total horizontal field of view (peripheral vision) is about 190° and your binocular vision (seen by both eyes) is about 130°. Not just the acuity is, but also accuracy of colour vision is the highest in the fovea. This is also the reason why the CIE 1931 xy and CIE 1976 u'v' chromaticity diagrams (also discussed in the Pointer's gamut article) are based on measurements with only a 2° field of view.
In fact that 2° angle is still optimistic, because only the central region of the fovea, the foveola, can achieve maximum acuity. The foveola measures only about 350 µm in diameter and comprises only the central 1.2° of vision.
Rod and cone cell distribution for angle from fovea in horizontal direction
The figure shows that cone cell density outside the fovea is very low. The relation between visual acuity and cone density is near linear for normal lighting conditions.
Normal vision is defined as being able to distinguish a gap size of one arc minute (1/60°) in an optotype. The standard distance for this test is 20 feet or 6 meters. Normal vision is therefore often described as 20/20 or 6/6 vision. This basically says that one sees as well at 20 feet as an average person would at 20 feet. If your vision is twice as good it's 20/10 vision, meaning that you see as well at 20 feet as an average person would at 10 feet. Especially in Europe the most common denotation is the decimal value of that ratio, so normal vision is denoted as VA = 1.0.
The problem with normal vision is that it's pretty much the average across all ages and for both sexes. Young people generally have much better eyesight; in fact, the majority of people under 40 have a VA higher than 1.0. Males also tend to have a higher visual acuity than females. There are also groups of people with exceptionally high visual acuity on average, for instance fighter pilots and professional baseball players. Human visual acuity is diffraction limited by the pupil at around 2.5 (20/8 vision), but anything higher than 1.8 (roughly 20/11) is very rare.
The viewing distance for a computer display is a rather fixed value. It often hardly depends on the display itself, but almost solely on the depth of the desk it stands on. Most desks have a depth between 75 and 80 cm and generally viewing distance is slightly shorter than that. For now we'll assume a viewing distance of 70 cm. By combining that distance with the normal visual acuity the required pixel pitch can be determined:
The resolution for a given screen diagonal D in inches with an aspect ratio of a:b then is:
This is however assuming that the screen diagonal in inches is an accurate value. Unfortunately this is often not the case, most standard screen sizes are rounded values. The most accurate way to determine the size of the active part of the panel is by multiplying the resolution with the pixel pitch. This was done for all currently common display sizes.
Smallest fitting retina resolution for common display sizes for VA = 1.0
Standard size Aspect ratio Active screen size (mm) Horizontal field of view Required horizontal resolution Smallest fitting resolution
21.5" 16:9 475.2 x 267.3 37.50° 2334 2336 x 1314
23" 16:9 509.2 x 286.4 39.97° 2502 2528 x 1422
24" (23.8") 16:9 527.0 x 296.5 41.26° 2590 2592 x 1458
24" 16:10 518.4 x 324.0 40.64° 2546 2560 x 1600
Note: for the 29" 21:9 and 34" 21:9 displays the actual aspect ratios of 64:27 and 43:18 respectively were used to determine the smallest fitting retina resolution, not the approximated 21:9 aspect ratio.
This is of course assuming normal visual acuity. If VA = 1.6 the resolution will have to be 60% higher in both horizontal and vertical direction for the display to be considered retina. Especially with young people (under 25 years old) a VA of 1.6 isn't all that rare. Another thing to keep in mind is that because of the relatively short absolute viewing distance, sitting marginally closer can already have a fairly significant effect. Sitting at 65 cm is already a 7.7% increase in linear resolution compared to 70 cm, at 60 cm that's 16.7%.Those 4K 24" displays suddenly make a lot more sense.
There's certainly a benefit in having ultra-high definitions in many uses. This includes highly detailed CAD/CAM work, high resolution image editing, and high resolution (mostly 4K) games and movies where available. The increased resolution increases the clarity and detail of the images being shown, especially when you are viewing the screen from a relatively close distance. Nowadays you can get 4K resolution desktop monitors quite easily, with models available as small as 24" (Dell UP2414Q for instance) and a wide variety in the size range of 28 – 32". One thing you do need to keep in mind though when selecting a 4K resolution screen is your uses. At native resolution and without any scaling applied, text size and fonts are an area which many people struggle with, and when you're trying to cram that much resolution into such a (relatively) small sized display; the pixel pitch becomes very small.
Let's focus on native resolution and default scaling from the operating system for a moment. As an example, text and fonts on a 28" 3840 x 2160 resolution screen are very small indeed, arguably too small for most users. Running the screen at native resolution for office type work can be difficult. Those with very good eye-sight might find it okay, but for many the fonts will be uncomfortably small, making it hard on the eyes. Larger screen sizes of around 31.5 – 32" are a bit more comfortable than 28" screens when running at the same resolution, but even then it is still very small. As a comparison, the pixel pitch on a 31.5" 3840 x 2160 screen is 0.182 mm, which makes it considerably smaller than common 27" models with a resolution of 2560 x 1440 pixels, giving a pixel pitch of 0.233 mm. On the other hand, there are also plenty of people working on laptops with 15.6" FHD displays and even 13.3" FHD displays at native resolution and no scaling. The pixel pitch on a 15.6" FHD laptop display is nearly identical to that of a 31.5" 4K display and the pixel pitch of a 13.3" FHD display is even smaller than that of a 28" 4K display.
If the text size is too small for you, there are ways around that to some extent. To be fair, Ultra HD resolutions are not so much about providing more desktop real-estate, but about providing crisper images. The way this is handled is through Operating system scaling. Some users like to increase the font scaling within the operating system, up to 125% or 150% even. With the small pixels and high detail, the scaled fonts can look very sharp and crisp still and it is a solution to the problem of very small text. Scaling has improved in recent years and the latest Windows 8, Mac OS and Linux systems can handle scaling pretty well on the most part. This is where the ultra HD resolutions can offer real benefits.
However, some things don't get scaled correctly or at all by the operating system, so it's not a viable option for many users. For example some text / maps in games don't get scaled, leaving you with incredibly small details to try and read. Some programs also don't scale well, so you're left with a bit of a mess of some scaled content and some not scaled. It's not as easy certainly as having a screen with a comfortable native resolution and pixel pitch at default scaling. If you're considering a high resolution or 4K monitor you need to think about your individual uses. If you want to do a lot of office/text work, and are hoping the higher resolution will be useful, you may want to be careful. You will almost certainly need to ensure you have an Operating system which can scale the content for you effectively and without too many issues. Many users of course may not have the most up to date operating systems and software so it really will vary from one user to another. Try a screen out in person if you have chance, as 4K isn't for everyone yet.
Most viewing distance recommendations for televisions are based on the horizontal field of view or the screen diagonal. They don't take visual acuity and pixel pitch into account directly. Instead a new set of recommendations is made for each new definition format, often loosely based on the average visual acuity.
Recommendation standard(for 16:9 aspect ratio) Horizontal field of view Required horizontal resolution (VA = 1.0) Smallest fitting resolution
THX 40° 2504 2528 x 1422
SMPTE 30° 1844 1856 x 1044
2.5 x screen diagonal 19.78° 1200 1216 x 684
3.5 x screen diagonal 14.19° 858 864 x 486
The required horizontal resolution for retina is calculated as follows:
HR = horizontal resolution
HFOV = horizontal field of view
ARVA = angular resolution for visual acuity (for instance 1/60° for VA = 1.0)
Looking at the results in the table it's easy to see why many people think that ultra high-definition televisions have no added value over full high-definition televisions. It's not surprising that UHD doesn't have much added value over FHD with these recommendations, because they are designed with HD (720p) and FHD (1080p) in mind. You don't see many people now watching their FHD TV from a distance based on the viewing recommendation for NTSC (7x picture height with a 4:3 aspect ratio, 10.88° HFOV). Especially the SMPTE recommendation comes very close to the optimal HFOV for a FHD TV for VA = 1.0:
If you were to measure the viewing distance to your television and calculate the horizontal field of view (HFOV) with the formula below you'd probably find that that's a narrower angle than 30°, let alone 40°.
D = screen diagonal in inches
a:b = aspect ratio
VD = viewing distance in cm
To calculate the viewing distance for a certain display size and HFOV you can use the formula below:
For a 55" (140 cm) TV, which is already quite a bit larger than average, the viewing distance would be 167 cm for the THX recommendation (HFOV = 40°) and 227 cm for the SMPTE recommendation (HFOV = 30°). An easy approximation for the THX and SMPTE recommendations is multiplying the screen diagonal with 1.2 and 1.63 respectively (in the same units, for diagonal in inches and distance in cm multiply result by 2.54). Most people will already find this far too close. The immersive viewing experience might be fun for watching movies and series, but it's probably a bit much for anything else, the news for instance. Aside from personal preference there are various other reasons for a smaller TV, like budget and interior decorating. For a somewhat comfortable viewing distance of 250 cm, you'd need 82.2" (208.8 cm) TV for THX. Which simply is far more expensive than what most people could afford and far larger than what properly fits in the average house.
The fun really starts with the Ultra High Definition viewing distance recommendations. The previous part already showed that the SMPTE and THX recommendations have a wide field of view, which results in very large TV's for what most people consider a comfortable viewing distance. But those are still quite narrow though when you compare them with the 60° HFOV of 4K and the insanely wide 100° HFOV of 8K. Using the 250 cm viewing distance example the previous section ended with results in a 130.4" (331.2 cm) screen diagonal for 4K and a 269.2" (683.7 cm) screen diagonal for 8K. The viewing distance can also be written as percentage of the dimensions of the screen for a certain HFOV:
For 4K this gives a viewing distance of:
87% of screen width
154% of screen height
75% of screen diagonal
And for 8K this gives a viewing distance of:
75% of screen height
Especially for 8K you often see the viewing distance specified as percentage or fraction of the screen height. You might notice that the values for 4K are about twice as high as those for 8K. That's because the viewing distance recommendations for UHD are pretty much solely based on visual acuity and 8K is twice as wide and twice as high as 4K. For VA = 1.0 the optimal viewing angle would be 58.37° for 4K and 96.33° for 8K. These values were just rounded up to 60° and 100° respectively. At those angles the horizontal retina resolution for VA = 1.0 would be 3970 pixels for 4K and 8194 for 8K. If you think about that you only have maximum visual acuity in the foveola, 8K seems quite a waste; you can only see a circle with a 69 pixel diameter with maximum accuracy at the time out of 7680×4320.
Even if you move your focus by moving your eyes or turning your head you still can't see the maximum detail of resolutions wider than 3438 pixels if your head is straight in the front of the centre of the screen (VA = 1.0). Even for VA=1.6 that's still only 5500 pixels. For resolutions wider than that you'd really have to move your head sideways to see all the detail. To see all detail of 8K by just rotating your head and not moving lateral or axial you would need to have near perfect vision of VA = 2.23 (roughly 20/9 vision).
For HR an even number and VD a positive real number that inequality does not have a real solution when ARVA = 1/60° and HR > 3438 (for VA = 1.0), or when ARVA = 1/96° and HR > 5500 (for VA = 1.6).
On computer displays there is definitely something to say for 4K. You can display a lot of information simultaneously and you usually only have to focus on a small area at the time, which means the higher detail really has added value. Furthermore the short viewing distance allows a wide field of view without the need for extremely large displays. 8K might have its uses with very specific applications, but in general it would be excessive.
With televisions it's a different story. Many people probably aren't even making full use of their FHD TV yet. To really profit from 4K you'd need an extremely large screen, or sit extremely close. And 8K is just plain ridiculous. For a 250 cm viewing distance you'd need a 595 x 335 cm screen. There aren't that many people with a wall that big in their house and even if you had, you'd need a pretty impressive beamer and a very large projection screen (they obviously don't make TV's that big).
One of the reasons that 4K televisions sell relatively well might be that in the store people tend to look at it from a very short distance, at which they could easily see that 4K is sharper than FHD. If they would look at it from the same distance as the actual distance they would view it from at home, many would not be able to tell the difference (if all other aspects of the image reproduction were identical for both displays). Manufacturers know this, so from a marketing perspective 4K is very clever.
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\section{Introduction}\label{sec:intro}
Easton's theorem (see \cite[Theorem 15.18]{jech})
was a milestone in set theory, which showed that ZFC by itself does not impose severe limitations on the behaviour of the continuum function at regular cardinals.
However, when we bring large cardinals into the picture, the situation is more complicated. Often the mere violation of GCH at a large cardinal requires strong assumptions.
The prototypical example is the case of a measurable cardinal.
By results of Gitik \cite{gitik-negation-sch, gitik-equiconsistency}
(see also Mitchell \cite{mitchell-core-i}), the violation of GCH at a measurable cardinal
is equiconsistent with the existence of a measurable cardinal
$\kappa$ such that $o(\kappa) = \kappa^{++}$.
In this paper, we look at the possible behaviour of the continuum function in the presence of strongly compact cardinals that are not supercompact.
Our goal will be to work with strongly compact cardinals which possess no
non-trivial degrees of supercompactness.
There are fundamental open questions in this regard,
such as whether it is possible to force GCH at an arbitrary non-supercompact
strongly compact cardinal.
As motivation, let us mention that if we allow
enough supercompactness assumptions,
the continuum function at a non-supercompact strongly compact cardinal
can be manipulated fairly easily.
For instance, to realise a $\Delta_2$-definable Easton function $F$, we can use a result due to Menas \cite[Theorem, pages 83--88]{menas-spct},
which shows that it is possible to realise $F$ while preserving the supercompactness of a cardinal $\kappa$. We can then use Magidor's Prikry iteration \cite{magidor-identity-crises}
that destroys all measurable cardinals below $\kappa$.
In the resulting model, $\kappa$ is strongly compact, $\kappa$ is
the least measurable cardinal (and so is not $2^\kappa$-supercompact),
and $F$ is still realised.
In a similar vein, suppose that $F$ is an Easton
function definable by a $\Delta_2$ formula
in a model $V$ of ZFC + GCH in which $\lambda$ is a supercompact
limit of supercompact cardinals.
The aforementioned theorem of Menas shows that it is possible
to force over $V$ to obtain a model $V_1$ in which
$\lambda$ remains a supercompact limit of supercompact cardinals
and the Easton function $F$ has been realised.
In $V_1$, let $\kappa < \lambda$ be the least measurable
limit of supercompact cardinals.
Another theorem of Menas shows that in $V_1$,
$\kappa$ is both strongly compact and not $2^\kappa$-supercompact.
In particular, by starting with hypotheses stronger than the
existence of a measurable limit of supercompact cardinals,
it is possible to force and construct a model containing a
non-supercompact strongly compact cardinal in which $F$ has been realised.
Also, if we assume that the strongly compact cardinal has a sufficient degree of supercompactness,
there are positive results. In \cite[Theorem 4.5]{hamkins-lottery},
Hamkins shows that if $\kappa$ is both strongly compact and
$\lambda$-supercompact, then
$\kappa$ can be forced to have
its strong compactness and $\lambda$-supercompactness
indestructible under any $\kappa$-directed closed forcing that has size at most $\lambda$.
In particular, it is possible to realise suitable Easton functions in the interval $[\kappa,\lambda)$.
The structure of this paper is as follows.
Section \ref{sec:intro} contains our introductory remarks.
Section \ref{sec:prelim} contains a discussion of our notation, terminology,
and some earlier results
used later on.
We then separate our results into two categories,
depending on whether we are interested in preserving a single strongly compact cardinal or
more than one strongly compact cardinal.
Our results for one strongly compact cardinal
are found in Section \ref{sec:local}.
We first answer a long-standing open question
on the problem of whether it is possible to violate GCH at a strongly compact
cardinal using no stronger assumptions.
We show that just assuming $2^\kappa =
\kappa^+$ and $\kappa$ is strongly compact, it is possible to preserve
the strong compactness of $\kappa$ while forcing any desired value for $2^\kappa$.
This result is due to the third author.
We then address the question of what sort of
Easton functions can be realised in the presence of a certain
non-supercompact strongly compact cardinal.
We show that if $\kappa$ is the least measurable limit of supercompact cardinals
and $F$ is an arbitrary Easton function defined on regular cardinals greater
than or equal to $\kappa$,
then it is possible to force to realise $F$ while preserving the
fact that $\kappa$ is the least measurable limit of supercompact cardinals.
The techniques used, however, will of necessity destroy many supercompact cardinals.
We therefore also present another result along the same lines,
where the Easton function realised has restrictions placed on it,
but all supercompact cardinals are preserved.
Our results for more than one supercompact cardinal
appear in Section \ref{sec:global}.
We begin by showing how
to iterate the partial orderings used in Section \ref{sec:local}
so as to preserve all measurable limits of supercompact cardinals simultaneously,
while realising certain Easton functions
at all of them.
We then prove a theorem which gives a partial answer to the problem
of the simultaneous preservation of all supercompact and measurable limits
of supercompact cardinals while violating GCH at each of them.
Finally, Section \ref{sec:ques} contains some open questions.
In order to present our results in full generality, we will make the minimal number
of assumptions on the structure of the class of strongly compact and supercompact
cardinals in our ground model. However, if we force over a model in which GCH
and the property of {\em compactness coincidence} both hold\footnote{The
property of {\em compactness coincidence} states that the strongly compact and supercompact
cardinals coincide, except at measurable limit points.
Models satisfying compactness coincidence non-trivially were first constructed
by Kimchi and Magidor in \cite{kimchi-magidor}.
As we observe in the paragraph immediately preceding the statement of
Theorem \ref{thm:least-measurable-limit}, by work of Menas, a further
coincidence between these two classes is impossible.}
(such as the one constructed by the first author and Shelah
in \cite{apter-shelah-strong-equality}), then all strongly compact cardinals will be
preserved to our generic extension.
This is since the only strongly compact cardinals which exist
in a model satisfying compactness coincidence
are the supercompact
cardinals and the measurable limits of supercompact cardinals.
\section{Preliminaries}\label{sec:prelim}
Our notation and terminology on forcing are
standard and follow \cite{cummings-handbook}.
In particular, $p\leq q$ means {\em $p$ is stronger than $q$},
and we call a partial ordering {\em $\kappa$-directed closed}
if every directed subset of size less than $\kappa$ has a lower bound.
We will say that {\em F is an Easton function for the
model V of ZFC} if
$F$ satisfies the following conditions:
\begin{itemize}
\item Either $F \in V$ (if $F$ is a set) or $F$ is
definable over $V$ (if $F$ is a proper class).
\item $\dom(F)$ is a class of $V$-regular cardinals.
\item ${\rm rge}(F)$ is a class of $V$-cardinals.
\item For every $\kappa \in \dom(F)$, $F(\kappa) > \kappa$.
\item If $\kappa < \lambda$, $\kappa, \lambda \in \dom(F)$, $F(\kappa) \le F(\lambda)$.
\item For every $\kappa \in \dom(F)$, ${\rm cf}(F(\kappa)) > \kappa$.
\end{itemize}
\noindent A model $V^*$ of ZFC {\em realises an Easton function $F$} if
in $V^*$, for every regular cardinal $\delta$ in the domain of $F$,
$2^\delta = F(\delta)$.
We assume that
the reader is familiar with the large cardinal notions of measurability, strong compactness,
and supercompactness. See \cite{jech} for further details. As it is
a lesser known notion, we recall that a cardinal $\kappa$ is called \emph{tall}
if for every $\lambda\geq\kappa$,
there is an elementary embedding $j:V\to M$ with $\crit(j)=\kappa$,
$j(\kappa)>\lambda$,
and ${}^\kappa M\subseteq M$.
In \cite{hamkins-tall}, Hamkins made a systematic study of tall cardinals.
We will use the following facts about tallness.
\begin{prop}{\rm (}\cite[Corollary 2.7]{hamkins-tall}{\rm )}\label{prop:tall-limit}
If $\kappa$ is measurable and a limit of tall cardinals, then $\kappa$ is tall.
\end{prop}
\begin{prop}{\rm (}\cite[Corollary 2.6]{hamkins-tall}{\rm )}\label{prop:nice-emb}
If $\kappa$ is tall, then for every $\lambda\geq \kappa$,
there is a $\lambda$-tallness embedding $j:V\to M$ with $\crit(j)=\kappa$ such that there is no $\lambda$-tall cardinal in $[\kappa,\lambda]$ in $M$.
\end{prop}
When it comes to strong compactness, we are interested in functions with the Menas property, which is defined as follows.
\begin{defn}\label{defn:menas-property}
Suppose $\kappa$ is a strongly compact cardinal. A function $f:\kappa\to \kappa$ has the \emph{Menas property} if for all $\lambda\geq\kappa$, there is a $\kappa$-complete,
fine ultrafilter $U$ on $\mathcal{P}_\kappa \lambda$ such that for the ultrapower embedding $j_U : V \to M_U$,
$|[id]_U]|^{M_U}<j_U(f)(\kappa)$ holds in $M_U$.
\end{defn}
First used by Menas in \cite{menas-strong-compactness}, this property is quite helpful when lifting strong compactness embeddings through forcing. In \cite{hamkins-lottery}, Hamkins showed that fast function forcing at an arbitrary strongly compact cardinal adds a function with the Menas property.
\begin{prop}{\rm (}\cite[Theorem 1.7]{hamkins-lottery}{\rm )}\label{prop:fast-function-menas}
Suppose $\kappa$ is a strongly compact cardinal. Then the fast function forcing $\mathbb{F}_\kappa$ preserves the strong compactness of $\kappa$ and adds a fast function $f:\kappa\to \kappa$ that has the Menas property.
\end{prop}
Moreover, there are cases when ZFC implies the existence of such a function.
\begin{prop}{\rm (}\cite[Theorem 2.21
and Proposition 2.31]{menas-strong-compactness}{\rm )}\label{prop:menas-property}
Suppose $\kappa$ is a measurable cardinal which is a limit of strongly compact cardinals.
Then $\kappa$ is strongly compact, and
the function $f:\kappa\to \kappa$, where $f(\alpha)$ is the least strongly compact cardinal greater than $\alpha$, has the Menas property.
\end{prop}
By an easy adaptation of the previous proposition,
we can also obtain the following corollary, which will be used in our results.
\begin{cor}\label{cor:menas-property}
Suppose $\kappa$ is a measurable cardinal which is a limit of supercompact cardinals.
Then $\kappa$ is strongly compact, and
the function $f:\kappa\to \kappa$, where $f(\alpha)$ is the least supercompact cardinal greater than $\alpha$, has the Menas property.
\end{cor}
We will want to show at certain junctures that no
new instances of large cardinals are created
in certain forcing extensions.
This will follow by a corollary of
Hamkins' work of \cite{hamkins-approximation}
on the approximation and cover properties
(which is a generalization of his
gap forcing results found in \cite{hamkins-gap-forcing}).
This corollary follows from
\cite[Theorem 3 and Corollary 14]{hamkins-approximation}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{hamkins-gap-forcing, hamkins-approximation}
when appropriate.
Suppose $\mathbb{P}$ is a partial ordering
which can be written as
${\mathbb Q} \ast \dot {\mathbb R}$, where
$|{\mathbb Q}| \le \delta$,
${\mathbb Q}$ is non-trivial, and
$\Vdash_{\mathbb Q} ``\dot {\mathbb R}$ is
$\delta^+$-directed closed''.
In Hamkins' terminology of
\cite{hamkins-approximation},
$\mathbb{P}$ {\em admits a closure point at $\delta$}.
Also, as in the terminology of
\cite{hamkins-gap-forcing, hamkins-approximation} and elsewhere,
an embedding
$j : V \to \bar M$ is
{\em amenable to $V$} when
$j \restriction A \in V$ for any
$A \in V$.
The specific corollary of
Hamkins' work from
\cite{hamkins-approximation}
we will be using
is then the following.
\begin{thm}\label{tgf}
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\mathbb{P}$
that admits a closure point
at some regular $\delta < \kappa$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an elementary embedding
with critical point $\kappa$ for which
$M[j(G)] \subseteq V[G]$ and
${}^\delta{M[j(G)]} \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \restriction V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \restriction V$ is definable from the names
of those parameters in $V$.
\end{thm}
It immediately follows from Theorem \ref{tgf}
that any cardinal $\kappa$ which is either
$\lambda$-supercompact or measurable
in a forcing extension obtained
by a partial ordering that admits a closure point
below $\kappa$ (such as at $\omega$)
must also be
$\lambda$-supercompact or measurable
in the ground model $V$.
In particular, if $\bar V$ is a forcing extension
of $V$ by a partial ordering admitting a closure point
at $\omega$
in which each supercompact cardinal and
each measurable limit of supercompact cardinals is preserved,
the classes of supercompact cardinals
and measurable limits of supercompact cardinals in $\bar V$ remain
the same as in $V$.
\section{Results for one strongly compact cardinal}\label{sec:local
We begin by showing that we can force violations of GCH
at a strongly compact cardinal $\kappa$ without any stronger assumptions.
Theorem \ref{thm:sc-continuum} and Corollary \ref{cor:consistency}
are due to the third author.
Here, $\Add(\kappa, \delta)$ is the standard partial ordering for
adding $\delta$ Cohen subsets of $\kappa$.
\begin{thm}{\bf (Usuba)}\label{thm:sc-continuum}
Let $\kappa$ be a strongly compact cardinal.
There is then a forcing extension in which the strong compactness of $\kappa$ is indestructible under $\Add(\kappa,\delta)$ for all $\delta$.
\end{thm}
\begin{proof}
By forcing with the fast function forcing
${\mathbb F}_\kappa$ if necessary,
we can assume that there is a function $f^*:\kappa\to\kappa$ with the Menas property.
Define $\mathbb{P}=\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid \beta<\alpha < \kappa\rangle$,
an Easton support iteration of length $\kappa$, as follows.
Let $\mathbb{P}_0$ be the trivial forcing notion.
$\dot{\mathbb{Q}}_\alpha$ is then also a name for the trivial forcing notion,
unless $\alpha$ is inaccessible and $f^*``\alpha\subseteq \alpha$.
In this case, $\dot{\mathbb{Q}}_\alpha$ is a name for the lottery sum
$$\bigoplus_{\beta<f^*(\alpha)} \Add(\alpha,\beta),$$
as defined in $V^{\mathbb{P}_\alpha}$.\footnote{If
${\mathfrak A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\bigoplus {\mathfrak A} =
\{\langle \mathbb{P}, p \rangle \mid \mathbb{P} \in {\mathfrak A}$
and $p \in \mathbb{P}\} \cup \{1\}$, ordered with
$1$ above everything and
$\langle \mathbb{P}, p \rangle \le \langle \mathbb{P}', p' \rangle$ iff
$\mathbb{P} = \mathbb{P}'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\bigoplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{hamkins-lottery},
``holds a lottery among the posets in
${\mathfrak A}$'') and
then forces with it. The
terminology ``lottery sum'' is due
to Hamkins, although the concept
of the lottery sum of partial
orderings has been around for quite
some time and has been referred to
at different junctures via the names
``disjoint sum of partial orderings'',
``side-by-side forcing'', and
``choosing which partial ordering to
force with generically''.}
Let $G\subseteq \mathbb{P}$ be $V$-generic.
The arguments of \cite[Theorem 4.1]{hamkins-lottery}
show that $\kappa$ remains strongly compact in $V[G]$. We wish to show that in $V[G]$,
the strong compactness of $\kappa$ is indestructible
under $\Add(\kappa,\delta)$ for all $\delta$.
Fix $\delta$,
and let $g\subseteq \Add(\kappa,\delta)$ be $V[G]$-generic. If we let $Q=\bigcup g$,
then $Q : \kappa\times \delta\to 2$ is a function.
Let $\lambda>\max(\kappa, \delta)$ be a regular cardinal, and
fix a cardinal $\theta \ge 2^{\lambda^{< \kappa}}$.
Using the Menas property of $f^*$, let $j:V\to M$ be an ultrapower embedding by a $\kappa$-complete,
fine ultrafilter $U\in V$ on $\mathcal{P}_\kappa\theta$ with $\crit(j)=\kappa$ such that $|[id]_U|^M<j(f^*)(\kappa)$. Since there is no source of confusion, we will drop the subscript from elements of $M$ and denote them as $[h]$. As usual, $j``\theta\subseteq [id]$, so $|\theta|^M \le |[id]|^M$.
\begin{claim}\label{claim:pi}
There is in $M$ a function $\pi:[id]\to \theta$ such that for all $\alpha<\theta$, $\pi(j(\alpha))=\alpha$.
\end{claim}
\begin{proof}
For each $\alpha<\theta$, let $g_\alpha:\mathcal{P}_\kappa\theta\to V$ be a function such that $[g_\alpha]=\alpha$.
Without loss of generality, we can assume that $g_\alpha(x)$ is
defined for every $x \in \mathcal{P}_\kappa \theta$.
Let $h:\mathcal{P}_\kappa\theta\to V$ be the function given by
$$h(x)=\{\langle \alpha,g_\alpha(x)\rangle \mid \alpha\in x\}.$$
By its definition, $h(x)$ is a function with domain $x$.
It follows that $[h]$
is a function with domain $[id]$,
and for each $\alpha<\theta$, $[h](j(\alpha))=[g_\alpha]=\alpha$.
This completes the proof,
since we can easily use $[h]$ to define a function $\pi$ with the required properties.
\end{proof}
We now proceed by lifting $j$ through $\mathbb{P}\ast
\dot \Add(\kappa,\delta)$.
As usual, $j(\mathbb{P})$ can be factorised as $\mathbb{P}\ast \dot{\mathbb{Q}} \ast \dot{\mathbb{P}}_{tail}$,
where $\dot{\mathbb{Q}}$ is a name for the lottery sum $\bigoplus_{\beta<j(f^*)(\kappa)}\Add(\kappa,\beta)$,
and $\dot{\mathbb{P}}_{tail}$ is a name for the remaining stages through $j(\kappa)$.
Using $G$ as an $M$-generic filter for $\mathbb{P}$, we can form $M[G]$.
Also, since $\delta< \theta \le |[id]|^M < j(f^*)(\kappa)$,
we can choose to force below a condition in $\mathbb{Q}=(\dot{\mathbb{Q}})_G$
that opts for $\Add(\kappa,\delta)$.
Thus, we can use $g$ as an $M[G]$-generic filter for $\mathbb{Q}$. Furthermore, note that since $j(f^*)(\kappa)>|[id]|^M$,
$\mathbb{P}_{tail} = (\dot{\mathbb{P}}_{tail})_{G\ast g}$ is at least $(|[id]|^+)^M$-closed in $M[G][g]$.
Force over $V[G][g]$ to add a generic filter $G_{tail}$ for $\mathbb{P}_{tail}$.
Using $G_{tail}$ as an $M[G][g]$-generic filter for $\mathbb{P}_{tail}$, since $j `` G \subseteq G$,
we can lift $j$ in $V[G][g][G_{tail}]$ to
$$j:V[G]\to M[j(G)],$$
where $j(G)=G\ast g \ast G_{tail}$.
In order to further lift $j$ through $\Add(\kappa,\delta)$,
we will use a master condition argument.
Consider the function $\pi$ given by Claim \ref{claim:pi},
and note that $|[id]\cap j(\delta)|^M\leq |[id]^M|<j(\kappa)$.
Define in $M[j(G)]$ a function $q:\kappa\times ([id]\cap j(\delta))\to 2$
given by $q(\langle \beta,\gamma \rangle)=
Q(\langle \beta,\pi(\gamma) \rangle)$ if
$\pi(\gamma) < \delta$,
and $0$ otherwise.
Clearly, $q$ is a condition in $j(\Add(\kappa,\delta))$.
\begin{claim}\label{claim:q}
$q\leq j(p)$ for all $p\in g$.
\end{claim}
\begin{proof}
By elementarity and the fact that $\crit(j)=\kappa$, for each $p\in g$, $j(p)$ is a function with domain $j``\dom(p)=\{\langle \beta,j(\gamma)\rangle \mid \langle \beta,\gamma\rangle\in \dom(p)\}$.
Hence, $\dom(j(p))\subseteq \dom(q)$.
For $\langle \beta,j(\gamma)\rangle\in \dom(j(p))$,
we have $j(p)(\langle \beta,j(\gamma) \rangle)=
p(\langle \beta,\gamma \rangle)=
Q(\langle \beta,\gamma \rangle)=
Q(\langle \beta,\pi(j(\gamma)) \rangle)=q(\langle \beta, j(\gamma) \rangle)$.
\end{proof}
Force over $V[G][g][G_{tail}]$ to add a generic filter
$h\subseteq j(\Add(\kappa,\delta))$ containing $q$.
By Claim \ref{claim:q}, we can lift $j$ in $V[G][g][G_{tail}][h]$ to
$$j:V[G][g]\to M[j(G)][h].$$
Let $\vec{X}=\langle X_\xi\mid \xi< 2^{\lambda^{<\kappa}}\rangle \in V[G][g]$
be an enumeration of $\mathcal{P}(\mathcal{P}_\kappa\lambda)^{V[G][g]}$.
In $M[j(G)][h]$, consider the set $B=\{\xi\in [id]\mid [id]\cap j(\lambda)\in j(\vec{X})_\xi\}$.
Since $\mathbb{P}_{tail}\ast j(\dot \Add(\kappa,\alpha))$
is at least $(|[id]|^+)^M$-closed in $M[G][g]$,
$B\in M[G][g]\subseteq V[G][g]$.
Hence, $W=\{X_\xi \in \mathcal{P} (\mathcal{P}_\kappa \lambda)^{V[G][g]}
\mid j(\xi)\in B\} \in V[G][g]$,
and since $\theta \ge 2^{\lambda^{< \kappa}}$,
$W$ is easily seen to be a $\kappa$-complete, fine ultrafilter
on $\mathcal{P}_\kappa\lambda$.
Thus, $\kappa$ is $\lambda$-strongly compact in $V[G][g]$.
Since $\lambda$ can be chosen arbitrarily large, we have shown that $\kappa$ remains strongly compact in $V[G][g]$.
This completes the proof of Theorem \ref{thm:sc-continuum}.
\end{proof}
\begin{cor}\label{cor:consistency}
The existence of a strongly compact cardinal is equiconsistent with the existence of a strongly compact cardinal where GCH fails.
In particular, assuming $2^\kappa = \kappa^+$ and $\kappa$ is strongly compact,
it is possible to force to preserve the strong compactness of $\kappa$ while
also forcing any desired value for $2^\kappa$.
\end{cor}
We now proceed by looking at a specific case of a non-supercompact strongly compact cardinal, the least measurable limit of supercompact cardinals.
By (the proof of) \cite[Theorem 2.22]{menas-strong-compactness}, if $\kappa$ is
the least measurable limit of supercompact cardinals, then $\kappa$
isn't $2^\kappa$-supercompact. Thus, if $2^\kappa = \kappa^+$,
then $\kappa$ isn't $\kappa^+$-supercompact, i.e.,
$\kappa$ exhibits no non-trivial degree of supercompactness.
\begin{thm}\label{thm:least-measurable-limit}
Suppose GCH holds, $\kappa$ is the least measurable limit of supercompact cardinals,
and $F$ is an arbitrary Easton function.
There is then
a forcing extension in which
$\kappa$ remains the least measurable limit of supercompact cardinals,
$\kappa$ exhibits no non-trivial degree of supercompactness,
and $F$ is realised at all regular cardinals greater than or equal to $\kappa$.
\end{thm}
\begin{proof}
Let $f:\kappa\to \kappa$ be the function where $f(\alpha)$ is the least supercompact cardinal greater than $\alpha$.
We define
$\mathbb{P}=\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid \beta<\alpha < \kappa\rangle$, an Easton support iteration
of length $\kappa$.
We start by letting $\mathbb{P}_0 = \Add(\omega, 1)$.
For $0 \le \alpha < \kappa$, $\dot{\mathbb{Q}}_\alpha$ is then defined as follows:
\begin{enumerate}
\item If $\alpha$ is supercompact, but not the least supercompact
cardinal greater than an inaccessible limit of supercompact cardinals, $\dot{\mathbb{Q}}_\alpha$ is a name for the Laver preparation
\cite{laver-preparation}
of $\alpha$, defined using only $\sigma$-directed closed partial orderings.
Here, $\sigma<\alpha$ is the least inaccessible cardinal greater than the supremum of
the supercompact cardinals below $\alpha$, or the least inaccessible cardinal
if there are no supercompact cardinals below $\alpha$.
We explicitly note that since there is no
supercompact limit of supercompact cardinals below $\kappa$,
the first non-trivial stage in the realisation of $\dot{\mathbb{Q}}_\alpha$
can be assumed not to occur until after stage $\sigma$.
\item If $\alpha$ is an inaccessible limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\bigoplus_{\beta<f(\alpha)}\Add(\alpha,\beta)$.
\item In all other cases, $\dot{\mathbb{Q}}_\alpha$ is a name for the trivial forcing notion.
\end{enumerate}
Let $G\subseteq \mathbb{P}$ be $V$-generic.
In $V[G]$, we force with the Easton product $$\prod_{\delta \ge \kappa} \Add(\delta, F(\delta)),$$ where
$\delta \ge \kappa$ is a ($V$ or $V[G]$)-regular cardinal.
Let $g \times H$ be $V[G]$-generic over
$$\Add(\kappa, F(\kappa)) \times \prod_{\delta > \kappa} \Add(\delta, F(\delta)) =
\Add(\kappa, F(\kappa)) \times {\mathbb R} = \prod_{\delta \ge \kappa} \Add(\delta, F(\delta)).$$
Standard arguments (see \cite[proof of Theorem 15.18]{jech}) show that $F$ is realised in $V[G][g][H]$
at all cardinals greater than or equal to $\kappa$.
We wish to show that $\kappa$ remains the least measurable limit of
supercompact cardinals in $V[G][g][H]$ (so $\kappa$ is strongly compact in $V[G][g][H]$)
and also exhibits no non-trivial degree of supercompactness in $V[G][g][H]$.
We begin by showing that $\kappa$ remains a limit of supercompact cardinals in $V[G][g][H]$.
For this, note that in $V$, the set
$\{\alpha<\kappa\mid \alpha$ is supercompact and
is not the least supercompact cardinal
greater than an inaccessible limit of supercompact cardinals$\}$
is unbounded below $\kappa$. For each such $\alpha$, the
partial ordering $\mathbb{Q}_\alpha$ forces $\alpha$ to have its supercompactness
indestructible under $\alpha$-directed closed forcing.
Since all the stages of $\mathbb{P}$ above $\alpha$ are at least $\alpha$-directed closed,
$\alpha$ remains indestructibly supercompact in $V[G]$.
Moreover, since the forcing $\Add(\kappa,F(\kappa))\times {\mathbb R}$
is itself $\alpha$-directed closed in $V[G]$, $\alpha$ is supercompact in $V[G][g][H]$ as well.
We now show that $\kappa$ remains measurable in $V[G][g][H]$.
We first note that by Corollary \ref{cor:menas-property}, $f$ has the Menas property.
Further, in the definition of ${\mathbb P}$, if $\alpha$ is an inaccessible limit of supercompact cardinals,
the first non-trivial stage of forcing after stage $\alpha$ does not occur until after $f(\alpha)$.
Therefore, the proof of Theorem \ref{thm:sc-continuum} immediately yields that $\kappa$
is strongly compact in $V[G][g]$. However, by Easton's lemma \cite[Lemma 15.19]{jech},
$\mathbb{R}$ is $(\kappa^+, \infty)$-distributive in $V[G][g]$. This consequently implies that
$\kappa$ is measurable in $V[G][g][H]$.
To complete the proof of Theorem \ref{thm:least-measurable-limit}, it only remains to show that in
$V[G][g][H]$, $\kappa$ remains the least measurable limit of supercompact cardinals
and exhibits no non-trivial degree of supercompactness.
Towards a contradiction, suppose $\gamma < \kappa$ is a measurable limit of supercompact
cardinals in $V[G][g][H]$.
By its definition, ${\mathbb P} \ast (\dot {\prod_{\delta \ge \kappa} \Add(\delta, F(\delta)))}$
admits a closure point at $\omega$.
Hence, by our remarks in the paragraph immediately following Theorem \ref{tgf},
$\kappa$ exhibits no non-trivial degree of supercompactness in $V[G][g][H]$.
In addition, these same remarks imply that
$\gamma$ must be a measurable limit of supercompact cardinals in $V$, which contradicts
that in $V$, $\kappa$ is the least measurable limit of supercompact cardinals.
This completes the proof of Theorem \ref{thm:least-measurable-limit}.
\end{proof}
\begin{rmrk}
Our choice of $\kappa$ as
the least measurable limit of supercompact cardinals together with GCH
was in order to highlight the fact that we do not assume
any non-trivial degree of supercompactness for $\kappa$.
However, the same proof would go through if $\kappa$ were an arbitrary measurable limit of supercompact cardinals.
\end{rmrk}
\begin{rmrk}
The partial ordering ${\mathbb P}$ of Theorem \ref{thm:least-measurable-limit} will destroy the
supercompactness of any cardinal $\alpha < \kappa$ which is in $V$ the least supercompact cardinal
above an inaccessible limit of supercompact cardinals.
To see this, note that we can write ${\mathbb P} = {\mathbb P}^1 \ast \dot{\mathbb P}^2$, where
$\mathbb{P}^1$ is forcing equivalent to a partial ordering having size less than $\alpha$,
and $\dot{\mathbb P}^2$ is forced to add a subset of some $\gamma > \alpha$,
$\gamma$ below the least supercompact cardinal in $V$ above $\alpha$.
By \cite[Theorem, page 550]{hamkins-shelah}, it is then the case that in
$V[G]$, $\alpha$ is no longer $\gamma$-supercompact.
Hence, by the closure properties of $\prod_{\delta \ge \kappa} \Add(\delta, F(\delta))$,
$\alpha$ is no longer $\gamma$-supercompact in $V[G][g][H]$ as well.
However, if we are willing to impose some restrictions on our Easton function,
it is possible to prove a version of Theorem \ref{thm:least-measurable-limit} in which
all supercompact cardinals below $\kappa$ are preserved. In particular, we have the following theorem.
\end{rmrk}
\begin{thm}\label{thm:least-measurable-limit-ii}
Suppose GCH holds, $\kappa$ is the least measurable limit of supercompact cardinals,
and $F$ is an Easton function such that $F(\kappa)$ is regular and
$F(\delta) = F(\kappa)$ for every $\delta \in [\kappa, F(\kappa))$.
There is then
a forcing extension in which
$\kappa$ remains the least measurable limit of supercompact cardinals,
$\kappa$ exhibits no non-trivial degree of supercompactness,
$F$ is realised at all regular cardinals greater than or equal to $\kappa$,
and the supercompact cardinals below $\kappa$ are the same as in the ground model.
\end{thm}
\begin{proof}
We first note that since $\kappa$ is strongly compact, by \cite[Theorem 2.11]{hamkins-tall},
$\kappa$ is also tall.
Next, let $f:\kappa\to \kappa$ be the function where $f(\alpha)$ is the least tall cardinal greater than $\alpha$.
Since every supercompact cardinal is clearly also tall and $\kappa$ is a limit of supercompact cardinals,
$f(\alpha)$ has a value less than $\kappa$ for every $\alpha < \kappa$
and so is well-defined.
It is also the case that $f$ has the {\em Menas property for tallness} \cite[page 75]{hamkins-tall}, i.e.,
for every ordinal $\theta$, there is an elementary embedding $j : V \to M$ with ${\rm crit}(j) = \kappa$,
${}^\kappa M \subseteq M$, and $j(f)(\kappa) \ge \theta$.
To see this, let $\theta \ge \kappa$. Using Proposition \ref{prop:nice-emb}, we
fix for $\kappa$ a $\theta$-tallness embedding $j:V\to M$ such that:
\begin{itemize}
\item $\crit(j)=\kappa$.
\item $j(\kappa)>\theta$.
\item ${}^\kappa M \subseteq M$.
\item $j$ is given by a $(\kappa,\theta)$-extender embedding.
\item There is no $\theta$-tall cardinal in $M$ in the interval $[\kappa,\theta]$.
\end{itemize}
Since in $M$, $j(f)(\kappa)$ is the least tall cardinal greater than $\kappa$,
and because there are no $\theta$-tall cardinals in $M$ in the interval $[\kappa,\theta]$,
it follows that $j(f)(\kappa) > \theta$.
We will now proceed
along the same lines as the proof of Theorem \ref{thm:least-measurable-limit}.
We begin by
fixing for $\kappa$ an $F(\kappa)$-tallness embedding $j:V\to M$ such that:
\begin{itemize}
\item $\crit(j)=\kappa$.
\item $j(\kappa)>F(\kappa)$.
\item ${}^\kappa M \subseteq M$.
\item $j$ is given by a $(\kappa,F(\kappa))$-extender embedding.
\item There is no $F(\kappa)$-tall cardinal in the interval $[\kappa,F(\kappa)]$.
\end{itemize}
We next define
$\mathbb{P}=\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid \beta<\alpha < \kappa\rangle$, an Easton support iteration
of length $\kappa$.
We start by letting $\mathbb{P}_0 = \Add(\omega, 1)$.
For $0 \le \alpha < \kappa$, $\dot{\mathbb{Q}}_\alpha$ is then defined as follows:
\begin{enumerate}
\item If $\alpha$ is supercompact,
$\dot{\mathbb{Q}}_\alpha$ is a name for the Laver preparation
\cite{laver-preparation}
of $\alpha$, defined using only $\sigma$-directed closed partial orderings.
Here, $\sigma<\alpha$ is redefined as
the least tall cardinal greater than $\gamma$, the supremum of
the supercompact cardinals below $\alpha$, or the least
tall cardinal if there are no supercompact cardinals below $\alpha$.
As before, since there is no supercompact limit of supercompact cardinals below $\kappa$
and $\alpha < \kappa$ is supercompact, $\gamma < \alpha$.
Also, by \cite[Lemma 2.1]{apter-cummings}, $\sigma \in (\gamma, \alpha)$
(where we take $\gamma = \omega$ if there are no supercompact cardinals below $\alpha$).
Therefore, in analogy to the proof of Theorem \ref{thm:least-measurable-limit},
the first non-trivial stage in the realisation of $\dot{\mathbb{Q}}_\alpha$
can be assumed not to occur until after stage $\sigma$.
\item If $\alpha$ is an inaccessible limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\bigoplus_{\beta<f(\alpha)}\Add(\alpha,\beta)$.
\item In all other cases, $\dot{\mathbb{Q}}_\alpha$ is a name for the trivial forcing notion.
\end{enumerate}
Let $G \subseteq \mathbb{P}$ be $V$-generic, and let $g_0 \subseteq \Add(\kappa, F(\kappa))$
be $V[G]$-generic.
By clause (2) in the definition of $\mathbb{P}$, the choice of $j$,
and the fact $f$ has the Menas property for tallness,
it is possible to opt for $\Add(\kappa, F(\kappa))$ at stage $\kappa$ in $M$
in the definition of $j(\mathbb{P})$.
In addition, by clause (1) in the definition of $\mathbb{P}$, the first non-trivial stage
in $M$ in the definition of $j(\mathbb{P})$ after $\kappa$ does not occur until after stage $j(f)(\kappa)$,
the least tall cardinal in $M$ greater than $\kappa$.
This means that the proof of \cite[Theorem 3.13]{hamkins-tall} unchanged remains valid and
allows us to infer the existence of a $\kappa$-directed closed, $(\kappa^+, \infty)$-distributive,
cardinal and cofinality preserving partial ordering ${\mathbb R} \in V[G][g_0]$ such that
if $g_1 \subseteq {\mathbb R}$ is $V[G][g_0]$-generic, in $V[G][g_0][g_1]$,
$\kappa$ is a tall cardinal, and $2^\delta = F(\kappa)$ for every $\delta \in [\kappa, F(\kappa))$.
If we now let
$g_2 \subseteq \prod_{\delta \ge F(\kappa)} \Add(\delta, F(\delta))$ (the Easton product for
$\delta \ge \kappa$ a regular cardinal in any of the models $V$, $V[G]$, $V[G][g_0]$, or $V[G][g_0][g_1]$) be
$V[G][g_0][g_1]$-generic, then $F$ is realised in $V[G][g_0][g_1][g_2]$ at all regular cardinals $\delta \ge \kappa$.
In addition, the same arguments as found in the proof of
Theorem \ref{thm:least-measurable-limit} show that in $V[G][g_0][g_1][g_2]$,
$\kappa$ remains the least measurable limit of supercompact cardinals, and
$\kappa$ exhibits no non-trivial degree of supercompactness.
By the definition of $\mathbb{P}$, all ground model supercompact cardinals less than $\kappa$
have been made indestructible and hence are preserved to $V[G][g_0][g_1][g_2]$.
Consequently, by our remarks in the paragraph immediately following Theorem \ref{tgf},
the supercompact cardinals below $\kappa$ in $V[G][g_0][g_1][g_2]$ are the
same as in $V$.
This completes the proof of Theorem \ref{thm:least-measurable-limit-ii}.
\end{proof}
\section{Results for more than one strongly compact cardinal
}\label{sec:global}
In the previous section, we successfully violated GCH and even realised certain
Easton functions above one non-supercompact strongly compact cardinal,
the least measurable limit of supercompact cardinals.
We now present results in which we handle more than one measurable
limit of supercompact cardinals.
In what follows, let $A = \{\delta \mid \delta$ is a measurable limit of supercompact cardinals$\}$.
Define $\Omega = \sup(A)$ if $A$ is a set, or $\Omega = \textnormal{Ord}$ if $A$ is a proper class.
Let $f : \Omega \to \Omega$ be the function where $f(\alpha)$ is the least
supercompact cardinal greater than $\alpha$.
\begin{thm}\label{thm:global}
Suppose $V$ is a model of GCH containing more than one measurable
limit of supercompact cardinals.
Let $F$ be an Easton function defined on measurable limits of supercompact cardinals such that $F(\kappa)<f(\kappa)$
for any $\kappa\in \dom(F)$.
Then there is a forcing extension in which the measurable limits of supercompact cardinals
are the same as in $V$ and $F$ is realised.
\end{thm}
\begin{proof}
Intuitively, we will proceed by iterating the forcing notion used
in the proof of Theorem \ref{thm:least-measurable-limit}. More formally, let
$\langle \kappa_\alpha \mid \alpha < \Omega \rangle$ enumerate in increasing
order the measurable limits of supercompact cardinals.
We define
$\mathbb{P}=\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid \beta<\alpha < \Omega\rangle$, an Easton support iteration
of length $\Omega$.
We start by letting $\mathbb{P}_0 = \Add(\omega, 1)$.
For $0 \le \alpha < \Omega$, $\dot{\mathbb{Q}}_\alpha$ is then defined as follows:
\begin{enumerate}
\item If $\alpha$ is supercompact, but neither the least supercompact
cardinal greater than an inaccessible limit of supercompact cardinals
nor a supercompact limit of supercompact cardinals, $\dot{\mathbb{Q}}_\alpha$ is a name for the Laver preparation
\cite{laver-preparation}
of $\alpha$, defined using only $\sigma$-directed closed partial orderings.
Here, $\sigma<\alpha$ is the least inaccessible cardinal greater than the supremum of
the supercompact cardinals below $\alpha$, or the least inaccessible cardinal
if there are no supercompact cardinals below $\alpha$.
We explicitly note that since $\alpha$ is not a supercompact
limit of supercompact cardinals,
the first non-trivial stage in the realisation of $\dot{\mathbb{Q}}_\alpha$
can be assumed not to occur until after stage $\sigma$.
\item If $\alpha$ is a non-measurable inaccessible limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\bigoplus_{\beta<f(\alpha)}\Add(\alpha,\beta)$.
\item If $\alpha$ is a measurable limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\Add(\alpha, F(\alpha))$.
\item In all other cases, $\dot{\mathbb{Q}}_\alpha$ is a name for the trivial forcing notion.
\end{enumerate}
Let $G\subseteq \mathbb{P}$ be $V$-generic.
Fix $\kappa$ such that $\kappa$ is a measurable limit
of supercompact cardinals in $V$. Write $G$ as
$G_\kappa \ast g \ast G_{tail}$, where $G_\kappa$ is $V$-generic for
the forcing defined through stage $\kappa$, $g$ is $V[G_\kappa]$-generic for
$\Add(\kappa, F(\kappa))$ (the stage $\kappa$ forcing), and
$G_{tail}$ is $V[G_\kappa][g]$-generic for the rest of $\mathbb{P}$.
The proof of Theorem \ref{thm:least-measurable-limit} shows that
in $V[G_\kappa][g]$, $\kappa$ remains a measurable limit of supercompact cardinals, and $2^\kappa = F(\kappa)$.
By the definition of $\mathbb{P}$, because $F(\kappa) < f(\kappa)$ and the first non-trivial stage of
forcing after stage $\kappa$ does not occur until after $f(\kappa)$,
in $V[G_\kappa][g][G_{tail}] = V[G]$, $\kappa$ remains a measurable limit of supercompact cardinals,
and $2^\kappa = F(\kappa)$.
By our remarks in the paragraph immediately following Theorem \ref{tgf},
any cardinal in $V[G]$ which is a measurable limit of supercompact
cardinals must have been a measurable limit of supercompact cardinals in $V$.
Since standard arguments show that if $\mathbb{P}$ is a proper class, $V[G]$ is a model of ZFC,
this completes the proof of Theorem \ref{thm:global}.
\end{proof}
As was the case with Theorem \ref{thm:least-measurable-limit}, the proof of Theorem \ref{thm:global}
yields that any $\delta < \Omega$ which in $V$ is the least supercompact cardinal greater than an inaccessible
limit of supercompact cardinals has its supercompactness destroyed after forcing with $\mathbb{P}$.
It is possible, however, by making some slight changes in the definition of $\mathbb{P}$,
to prove an analogue of Theorem \ref{thm:global} in which
both the measurable limits of supercompact cardinals and the
supercompact cardinals which are not limits of supercompact cardinals are the same as in $V$.
In particular, suppose we assume:
\begin{itemize}
\item $A$, $\Omega$, and $\langle \kappa_\alpha \mid \alpha < \Omega \rangle$
have been defined as in the proof of Theorem \ref{thm:global}.
\item $f : \Omega \to \Omega$ is redefined as $f(\alpha)$ is
the least tall cardinal greater than $\alpha$.
\item
$\sigma$ is redefined as
the least tall cardinal greater than the supremum of the supercompact cardinals below $\alpha$,
or the least tall cardinal if there are no supercompact cardinals below $\alpha$.
\item We define a partial ordering $\mathbb{P}$ as in the proof of Theorem \ref{thm:global}, except that
in Case (1) of the definition of $\mathbb{P}$,
$\alpha$ can be {\em any} supercompact cardinal which is not a limit of supercompact cardinals.
\end{itemize}
We now have the following.
\begin{thm}\label{thm:global-ii}
Suppose $V$ is a model of GCH containing more than one measurable
limit of supercompact cardinals.
Let $F$ be an Easton function defined on measurable limits of supercompact cardinals such that $F(\kappa)<f(\kappa)$
for any $\kappa\in \dom(F)$.
Then there is a forcing extension in which
the measurable limits of supercompact cardinals
and the supercompact cardinals which are not limits of supercompact cardinals
are the same as in $V$,
and $F$ is realised.
\end{thm}
The proof of Theorem \ref{thm:global-ii} is essentially the same as the proof of Theorem \ref{thm:global},
with all references to the proof of Theorem \ref{thm:least-measurable-limit}
replaced by references to the proof of Theorem \ref{thm:least-measurable-limit-ii}.
We note only that if $\delta < \Omega$ is in $V$ a supercompact cardinal which
is not a limit of supercompact cardinals, the definition of $\mathbb{P}$
(specifically, the change made in Case (1))
shows that $\delta$ is preserved to the generic extension $V[G]$ by $\mathbb{P}$.
By our remarks in the paragraph immediately following Theorem \ref{tgf},
it now immediately follows that the supercompact cardinals which
are not limits of supercompact cardinals are the same in both $V$ and $V[G]$.
If
$A$ is a set instead
of a proper class (so that in particular, $\Omega$ is an ordinal), then
by the L\'evy-Solovay results \cite{levysolovay},
the supercompact cardinals above $\Omega$ in both $V$ and $V[G]$
are precisely the same.
The techniques used in the proofs of Theorems \ref{thm:global} and \ref{thm:global-ii} do not
seem to allow for the preservation of supercompact limits of supercompact cardinals.
Although we do not yet know a way of
accomplishing this in general, it is possible
to achieve this goal in a certain restricted situation.
More specifically, we have the following.
\begin{thm}\label{thm:spct-limit}
Suppose $V$ is a model of GCH in which $\kappa$ is the only supercompact limit of supercompact cardinals
and there is no inaccessible cardinal greater than $\kappa$.
Then there is a forcing extension in which
the supercompact cardinals and measurable limits of supercompact cardinals are the same as in $V$
(so in particular, $\kappa$ remains the only supercompact limit of supercompact cardinals),
and $2^\delta=\delta^{++}$ for every
$\delta$ which is either supercompact or a measurable limit of supercompact cardinals.
\end{thm}
\begin{proof}
Let $\langle \kappa_\alpha \mid \alpha \le \kappa \rangle$ enumerate in increasing
order the measurable limits of supercompact cardinals.
Let $f : \kappa \to \kappa$ be the function where $f(\alpha)$ is the least
tall cardinal greater than $\alpha$.
We define
$\mathbb{P}=\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid \beta<\alpha \le \kappa\rangle$, an Easton support iteration
of length $\kappa + 1$.
We start by letting $\mathbb{P}_0 = \Add(\omega, 1)$.
For $0 \le \alpha \le \kappa$, $\dot{\mathbb{Q}}_\alpha$ is then defined as follows:
\begin{enumerate}
\item If $\alpha < \kappa$ is supercompact,
$\dot{\mathbb{Q}}_\alpha$ is a name for the Laver preparation
\cite{laver-preparation}
of $\alpha$, defined using only $\sigma$-directed closed partial orderings.
Here, $\sigma<\alpha$ is the least tall cardinal greater than the supremum of
the supercompact cardinals below $\alpha$,
or the least tall cardinal if there are no supercompact cardinals below $\alpha$.
We explicitly note that as in the proof of
Theorem \ref{thm:least-measurable-limit-ii}, since $\alpha$ is not a supercompact
limit of supercompact cardinals,
the first non-trivial stage in the realisation of $\dot{\mathbb{Q}}_\alpha$
can be assumed not to occur until after stage $\sigma$.
\item If $\alpha$ is a non-measurable inaccessible limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\bigoplus_{\beta<f(\alpha)}\Add(\alpha,\beta)$.
\item If $\alpha \le \kappa$ is either supercompact or a measurable limit of supercompact cardinals,
$\dot{\mathbb{Q}}_\alpha$ is a name for $\Add(\alpha, \alpha^{++})$.
\item In all other cases, $\dot{\mathbb{Q}}_\alpha$ is a name for the trivial forcing notion.
\end{enumerate}
Let $G \subseteq \mathbb{P}$ be $V$-generic. Write
$G$ as $G_\kappa \ast g$, where $G_\kappa$ is $V$-generic
for the forcing defined through stage $\kappa$, and $g$ is
$V[G_\kappa]$-generic for $\Add(\kappa, \kappa^{++})$ (the stage $\kappa$ forcing).
The arguments found in the proofs of
Theorems \ref{thm:global} and \ref{thm:global-ii},
in tandem with the definition of $\mathbb{P}$, show that in $V[G_\kappa][g]$,
the supercompact cardinals less than $\kappa$
and measurable limits of supercompact cardinals are the same as in $V$, and
$2^\delta = \delta^{++}$ for every $\delta$ which is either supercompact or a measurable
limit of supercompact cardinals. In addition, since there are no inaccessible cardinals
greater than $\kappa$ in $V$, there are no inaccessible cardinals greater than $\kappa$
in $V[G_\kappa][g]$ as well.
Thus, the proof of Theorem \ref{thm:spct-limit} will be complete once
we have shown that $\kappa$ remains supercompact in $V[G_\kappa][g]$.
To do this,
fix an arbitrary
$\lambda \ge \kappa^{++}$,
and let $j:V\to M$ be a
$\gamma = 2^{\lambda^{< \kappa}}$-supercompactness embedding
with $\crit(j)=\kappa$.
In $V$, there is no inaccessible cardinal greater than $\kappa$,
and since ${}^\gamma M\subseteq M$,
in $M$, there is no inaccessible cardinal in
$(\kappa,\gamma]$.
Thus, we can write $j(\mathbb{P})$ as $\mathbb{P}\ast \dot \Add(\kappa,\kappa^{++})\ast \dot\mathbb{P}_{tail}
\ast \dot \Add(j(\kappa), j(\kappa^{++}))$,
where $\dot\mathbb{P}_{tail}$ is a name for a $\gamma^+$-directed closed forcing
whose first non-trivial stage occurs after $\gamma^+$.
Standard arguments (see, e.g., \cite[proof of the Theorem, pages 387--388]{laver-preparation})
now show that $\kappa$ is $\lambda$-supercompact in $V[G_\kappa][g]$.
Since $\lambda$ was chosen arbitrarily, $\kappa$ is supercompact in $V[G_\kappa][g]$.
This completes the proof of Theorem \ref{thm:spct-limit}.
\end{proof}
\section{Questions}\label{sec:ques}
The following questions remain open concerning strongly compact
cardinals and the continuum function.
\begin{qst}
If $\kappa$ is a strongly compact cardinal, can we force GCH at $\kappa$ while preserving the strong compactness of $\kappa$ without assuming any stronger hypotheses?
\end{qst}
\begin{qst}[Woodin]
If GCH holds below a strongly compact cardinal, does it hold above it too?
\end{qst}
Also,
our methods leave unresolved the problem of realising
an arbitrary Easton function in the presence of a strongly compact cardinal.
\begin{qst}
Suppose $F$ is {\em any}
Easton function and $\kappa$ is a strongly compact cardinal.
Under what conditions can we realise $F$ while preserving the strong compactness of $\kappa$?
\end{qst}
One of the challenges in the proofs of the theorems of Section \ref{sec:global}
that remains unresolved is the
preservation of {\em arbitrary} supercompact limits of supercompact cardinals.
\begin{qst}
Can we prove analogues of Theorems \ref{thm:global}, \ref{thm:global-ii}, and \ref{thm:spct-limit}
where {\em all} supercompact limits of supercompact cardinals are preserved?
\end{qst}
\bibliographystyle{abbrvurl}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,288 |
Q: Self hosted Kestrel with SSL on an IIS server side by side issue I have a development server which has IIS installed and multiple assigned static private IP's but I also want to use it to run a Kestrel self hosted web service. When I set the Kestrel service endpoint config to run HTTP on port 80 with a specific IP it runs fine side by side with IIS as long as I don't overlap the endpoints/bindings.
That's great, however when I try to set up SSL on Kestrel's service it will not start at all. It will fail with the following exception
System.Net.Sockets.SocketException (10013): An attempt was made to access a socket in a way forbidden by its access permissions.
This only happens when I set the endpoint's port to 443, 5001 will work, 446 will work, but not 443.
Here's an example endpoint config that fails for me.
"Kestrel": {
"Endpoints": {
"Http": {
"Url": "http://10.10.13.11:80"
},
"Https": {
"Url": "https://10.10.13.11:443",
"Certificate": {
"Location": "LocalMachine",
"Store": "My",
"Subject": "portaldev.mydomain.com",
"AllowInvalid": false
}
}
}
Only the port 443 is the issue, IIS also has all it's bindings set to other IP's, none of them attempt to reuse that 10.10.13.11 IP that the Kestrel service is using.
What could be going on here? What's it only with the SSL port and not the HTTP port? Do I need to give permissions to something to let it bind to that 443 port/ip socket?
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: How to get the substring of a string that contains HTML tags using C# .net? I have a substring what contains HTML tag and I need to shorten it but display it with the same formatting as it appears on the string.
It doesn't have to be exactly X characters long, but it should be short enough to be displayed inside a panel with a certain width and height?
Is there any way I can achieve this using c#?
What about using CSS? I.e. displaying the panel with a fixed height regardless of its content?
Thanks..
Example: I have the following panel containing a label that contains text with html tags:
I need to remove the scroll bar without making the panel longer but keeping this height & this width..
A: You could use a regex to find the contents of the specific tag. Use a .substring to shorten the result afterwards.
A example could be:
<h1>head</h1>
<p>contents</p>
Regex could be:
<p\b[^>]*>(.*?)</p>
Result would be:
<p>contents</p>
Now just exclude the start and end tag. as its a fixed length.
I found more interesting reading about changing the content between HTML tags. Take a read here (regex ftw!):
http://www.thatsquality.com/articles/how-to-match-and-replace-content-between-two-html-tags-using-regular-expressions
Another solution that might not drive you as crazy if you want to solve it in c#:
HTML Agility Pack
Take a look at the examples part of the site. Great little tool!
A: If you have following html code:
<div class="div1"> Some Really Bold String </div>
You can provide css to hide the scroll bars,
.div { overflow:hidden; height:200px; width:200px;}
height and width values are just for example purpose.
overflow:hidden does not let the content of the div to expand out side the div.
you will find more information on overflow here.
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A walking testament to the power of mortitheugy, Despoiler is as much a beast as it is an arcane implement through which the Void howls. The product of unorthodox experimentation, the beast is driven mad by the shrieking of damned souls. Consequently it fights in a near-frenzy, smashing victims to a pulp, though their deaths serve only to further fuel the din in its mind.
A player may field one Despoiler in a Skorne army. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,512 |
Home Forums > All Things Horror > General > Site Polls >
Will you see High Tension in theaters?
Discussion in 'Site Polls' started by rhett, Jun 6, 2005.
Poll closed Jul 6, 2005.
Yes, but I wish it wasn't dubbed and cut
32 vote(s)
Yes, I prefer the dubbing and cutting
No, I'll wait for the uncut R1 DVD
No, I have no interest in the film
No, I have already seen it/ own it on DVD
rxfiend Joe Six-Pack
Southern IN
Tien21 said:
Wow this movie only made 1.75 mil this weekend. I would have thought it would make more.
Lions Gate should have released it subbed and uncut. If their editing only brought 1.75; untouched, it could have at least brought that in, if not more (i know i would have made an effort to watch it on the big screen if it was in it's original form).
rxfiend, Jun 13, 2005
Tien21 Guest
It was just a really bad time to release this movie. American audiences generally won't like this movie in the first place. The average moviegoer doesn't like dubbing or subtitles, nor do they like confusing endings. The movie just isn't marketable in America, I must say. Its kinda sad really. If there is a twist in a movie they want it spoonfed to them like SAW. I don't really see The Devil's Rejects making all that much either, but compared to this it will make a truckload of money. LG promises to give it the widest release of all of their movies to date.
About the plot holes in High Tension(highlight)-I thought when the police saw her kill the clerk it was a plot hole only because it was through their perspective and not Marie's.
Tien21, Jun 13, 2005
IGotsNewShoes Guest
Well according to some
the whole movie is just Marie's version of what happened, the car chase couldn't have happened cause it was only her blah blah blah....yet when she's in the institution at the end, she has the injuries from the car wreck.
that's not a plot hole though.
IGotsNewShoes, Jun 13, 2005
Ash28M Active Member
IGotsNewShoes said:
Again I'm still not sure what's so confusing if you just think outside the box a little bit ..
The only one knows what exactly happened that night was Marie and in the state she was in. I'm not sure she even knows. So she could have sustained those injuries any number of ways including self inflicted.
This is why I have grown to love the ending and why I find it quite original. the movie can make as little sense or as much sense as your willing to invest in it.
Ash28M, Jun 13, 2005
Mok Family is Forever
By the way, the whole scene at the beginning..
She wakes up the car and tell her friend that that was a dream she just had. It actually wasn't reality in advance at that point. I guess she was psychic too
Mok, Jun 13, 2005
Mok said:
This scene was a mix of her recalling her dream to the investigators/Doctors along with director presenting some foreshadowing for the viewer watching the film.
wizzer get outta here ewe
Ash28M said:
dude, you should field the questions for the director at his Q and A sessions
wizzer, Jun 13, 2005
wizzer said:
Yeah sometimes i feel I've analyzed the twist more than I should give the director credit for
I thought she told her friend about the dream, no? In the car ride at the beginning.
Yes but this is all still in the context of her telling the story of what happened that night to the investigators..
But they caught her on surveillance.. :hum:
Agent Z "Get to the river...
There are moments in the film (moreso as it reaches the conclusion) where the narrative switches to reveals (like the police reviewing the surveillance tape, Marie's hand dropping the bullets out the window, and Marie killing Alex's family)
Agent Z, Jun 13, 2005
I saw this the other day and i was wondering if
She didn't even know alex i.e. the dream she tells her about at the beginning never happened. they didnt go to school together she just picked this house. that is why she had the van and why we see the van at the beginning with the killer getting head from the head. so the forshadowing and psychic stuff was all part of her story, maybe. I can't remember if alex ever calls her marie after the twist is revealed, so she may have never met alex until she is woken up and bound.
someone may have said this already. and do i make any sense?
soxfan666, Jun 13, 2005
soxfan666 said:
Good observation! I've had the same theory going through my head also. I think it's ultimately up to the viewer. I tend to think about them actually being friends beforehand, but you could make great cases for either scenario.
Cujo108 Guest
I actually never thought that was the case. At least to me it doesn't seem to be Aja's intentions. While she's obviously psychotic, I think Marie's love is too strong for her not to have been friends with Alex. I also don't think Alex would be coming to the mental hospital if she didn't know her. Just my 2 cents anyway.
Also, I don't think its at all a stretch that she got that truck from somewhere on the farm. It was said that they had just bought it recently. Who knows what was still lying around.
Cujo108, Jun 13, 2005
Cujo108 said:
That is why I tend to lean more towards them being friends beforehand, as it tends to hit a more emotional nerve and makes it all that much more effective.
tobaccoman White, Proud, and Stupid
But reading that
they might not have been friends ever
really does help cover up the so-called plot holes. Goddammit, every couple hours I think more and more about this.
tobaccoman, Jun 14, 2005
I'm going to go see it at its first matinee in hopes that there won't be anyone else in the theater--Any talkative teens or infants and I'm out to the box office for a full refund.
Now I'll finally get to see what's behind all of these blue boxes!!!!
Myron Breck, Jun 14, 2005
Umm, the first matinee was probably last Friday unless Atlanta became a inbreed shit-town. But by all means, if you haven't seen it, do so.
tobaccoman said:
:lol: Yeah, that sums it up for me.
I just meant the first matinee of the day--I figured a weekday would be better.
Sorry. I assumed everyone would assume. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,301 |
Q: Need to adapt MFC, c++ native application to OS Font Scaling, We need to adapt an existing MFC, C++ native application (GUI) so the text will enlarge with user preference in system preference:
Windows Settings|Ease of Access|Display|Make text bigger.
Nowadays application will scale their text depending on this user preference. I have found links explaining how to do this using C# or C++/WinRT:
https://learn.microsoft.com/en-gb/uwp/api/windows.ui.viewmanagement.uisettings.textscalefactor
https://learn.microsoft.com/en-us/windows/uwp/design/input/text-scaling
We would like to use a native C++ API in order to ease the development effort. Building bridges to C# or c++/WinRT would also complexify our solution.
We have already adapted our application to DPI scaling.
Thanks for your help
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,031 |
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Sheddocksley Bed and Breakfasts. Compare latest rates and Live availability for all your favourite places and places you have always wanted to go using our new My Shortlist feature. Click + to add to your Shortlist.
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Bed and Breakfasts in Sheddocksley, Aberdeenshire for 1 night from Sat Jan 22, 2022 to Sun Jan 23, 2022 within 10 miles, in an Average Nightly price range of £1 to £200.
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Saturday, 22 January 2022 Jan 2022
Week Month
Sure Hotel by Best Western Aberdeen
Aberdeen 0.9 1.45 3 22232425262728 £39.60 £39.60 £39.60 £39.60 £39.60 £39.60 £39.60 £ 39.60
Britannia Hotel Aberdeen
Aberdeen 1.34 2.16 3 22232425262728 £29.00 £29.00 £29.00 £29.00 £29.00 £29.00 £29.00 £ 29.00
Atholl Hotel
St Elmo Guest House
The Chester Hotel
Malmaison Aberdeen
Hilton Aberdeen TECA
The Craighaar Hotel
Aloft Aberdeen TECA
Aberdeen 1.99 3.2 3 22232425262728 £75.00 £75.00 £75.00 £75.00 £75.00 £75.00 £75.00 £ 75.00
Palm Court Hotel
The Dutch Mill Hotel
Aberdeen 1.99 3.21 N/A 22232425262728 £55.00 £55.00 £55.00 £55.00 £55.00 £55.00 £55.00 £ 55.00
Aberdeen Northern Hotel
Village Hotel Aberdeen
Rosemount Palace
Prime 2 Bedroom City Centre Apt
Great Western Hotel Aberdeen
Copthorne Aberdeen Hotel
Holiday Inn Express Aberdeen City Centre, an IHG Hotel
Luxury Scottish Apartment
Bright & Airy Two Bed Set In Granite
Charming one bedroom
Park Inn by Radisson Aberdeen
The Globe Inn
Cults Hotel
The Craibstone Suites
Mercure Aberdeen Caledonian Hotel
Siberia Bar & Hotel
The Coffee House Hotel
Aberdeen Marriott Hotel
Dyce 2.88 4.63 4 22232425262728 £74.00 £74.00 £74.00 £74.00 £74.00 £74.00 £74.00 £ 74.00
20 Summerhill Court, Aberdeen, AB15 6TW
Distance:0.9 miles | Star Rating:
Located 4.8 km from Aberdeen city centre, Sure Hotel by Best Western Aberdeen welcomes guests with a bar, as well as free WiFi. Each room is equipped ...more
Located 4.8 km from Aberdeen city centre, Sure Hotel by Best Western Aberdeen welcomes guests with a bar, as well as free WiFi. Each room is equipped with a flat-screen TV. Certain rooms feature a seating area for your convenience. Tea and coffee making facilities are provided. Each bedroom comes with an en suite bathroom equipped with a bath or shower, a hairdryer and free toiletries. There is a 24-hour front desk at the property. Sure Hotel by Best Western Aberdeen has a versatile conference and banqueting room. Aberdeen Art Gallery is 3.2 km from Sure Hotel by Best Western Aberdeen, while Aberdeen Harbour is 3.6 km from the property. The hotel is 6 km from AECC in Aberdeen. The nearest airport is Aberdeen Airport, 6 km from the property.
Saturday, 22 January 2022Jan 2022
Dates:Jan 22, 2022 - Jan 23, 2022 Sat Jan 22, 2022 - Sun Jan 23, 2022 Nights:1
Total cost: £ 39.60
In Shortlist
Malcolm Rd, Airport Hotel, Aberdeen, AB21 9LN
Distance:1.34 miles | Star Rating:
Just 3 miles from Aberdeen Airport, Britannia Hotel Aberdeen offers free parking. There is a restaurant and bar, and Aberdeen city centre is a 10-minu...more
Just 3 miles from Aberdeen Airport, Britannia Hotel Aberdeen offers free parking. There is a restaurant and bar, and Aberdeen city centre is a 10-minute drive. The modern rooms each have a private bathroom and satellite TV. All rooms have tea/coffee facilities, a hairdryer and work desk. A wake-up service is also available. Our restaurant and lounge are located on the ground floor of the building. Britannia Hotel has a 24-hour front desk. A events centre, P&J Live, is just a short walk from Britannia Aberdeen. Britannia Aberdeen Hotel is a 10-minute drive from the lively shopping centres and nightlife of Aberdeen. Aberdeen Harbour and Ferry Terminal are 15 minutes' drive away, offering crossings to Shetland.
54 Kings Gate, Aberdeen, AB15 4YN
Atholl Hotel is quiet, friendly and has a popular restaurant, and rooms with free broadband internet access. It is a few minutes west of Aberdeen city...more
Atholl Hotel is quiet, friendly and has a popular restaurant, and rooms with free broadband internet access. It is a few minutes west of Aberdeen city centre. Atholl Hotel is located off Anderson Drive, which is one of Aberdeen's main roads. There is easy access to Aberdeen Airport, railway station and city centre. Traditional Scottish food is served in the restaurant and lounge bar.
64, Hilton Drive, Aberdeen, AB24 4NP
Smoke free, guest house on city centre bus route #12, within 2 miles of shops, theatres, museums, University and Royal Infirmary. All rooms ensuite, ...more
Smoke free, guest house on city centre bus route #12, within 2 miles of shops, theatres, museums, University and Royal Infirmary. All rooms ensuite, with free WiFi and self-catering facilities included (microwave, fridge, kettle, etc.). Off-street parking
59 - 63 Queens Road , Aberdeen, AB15 4YP
The Chester Hotel is a listed building dating from the 19th century, combining original design with striking, contemporary features. Set on Aberdeen's...more
The Chester Hotel is a listed building dating from the 19th century, combining original design with striking, contemporary features. Set on Aberdeen's Queens Road, this beautiful hotel has a restaurant, free gym and private dining rooms, just 1.5 miles from Aberdeen Railway Station. Located in the recently refurbished wing of the hotel. Stylish, contemporary and comfortable with patio doors leading to gardens at the rear of the hotel. The stylish rooms each include free WiFi, TV, a fridge, a safe, desk and a modern bathroom with Noble Isle toiletries. Rooms also feature a Nespresso coffee maker and Twinings tea. The Gallery Bar offers a smart setting serving Scottish cuisine. There is also a bar serving whisky, international spirits, champagne, cocktails, fine wines and beers. Afternoon tea prepared by the Chester's pastry chefs is also available. Just a 20-minute walk from Aberdeen centre, the hotel is located 2 miles from Aberdeen Ferry Terminal. Free on-site parking is available, and Royal Aberdeen Golf Club is a 10-minute drive away.
49-53 Queens Road, Aberdeen, AB15 4YP
Malmaison Aberdeen is a stunning luxury hotel with sumptuous rooms, a sophisticated brasserie, a diverse spa and endless exquisite designs, style and ...more
Malmaison Aberdeen is a stunning luxury hotel with sumptuous rooms, a sophisticated brasserie, a diverse spa and endless exquisite designs, style and flair. Formerly the Queen's Hotel, the boutique Malmaison Aberdeen is part new-build, part original façade. Other special features include the gym and the wine-tasting cellar. In addition to the brasserie with its excellent wines, the Whisky Snug is a cosy corner where you can sip fine whiskies from over-sized tartan armchairs.
East Burn Road, Stoneywood, Aberdeen, AB21 9FX
Located in Aberdeen, Hilton Aberdeen TECA has non-smoking rooms, and free WiFi in public areas. The property is around 8 km from Aberdeen City Centre ...more
Located in Aberdeen, Hilton Aberdeen TECA has non-smoking rooms, and free WiFi in public areas. The property is around 8 km from Aberdeen City Centre and 9 km Aberdeen Beach. Located on the site of the new exhibition centre, P&J Live at TECA, the Hilton Aberdeen TECA is just steps from some of the biggest events in Scotland. All guest rooms have smart HDTV, tea and coffee making facilities, wardrobe, hair dryer, complimentary toiletries, air-conditioning and a work desk. Continental and buffet breakfast options are available each morning at Hilton Aberdeen TECA. Guests can enjoy a meal in the Quarter House Bar & Grill daily. Guests can make use of the on-site business centre or book a meeting in one of the seven meeting rooms. Speaking English, staff at the 24-hour front desk can help you plan your stay. Aberdeen Harbour is 8 km from Hilton Aberdeen TECA, while Aberdeen Airport is 3 km away and is easily accessible via public transport.
Waterton Road, Bankhead, Aberdeen, AB21 9HS
Situated just 5 minutes' drive from Aberdeen Airport, The Craighaar Hotel offers free parking and free Wi-Fi throughout. Full Scottish breakfasts are ...more
Situated just 5 minutes' drive from Aberdeen Airport, The Craighaar Hotel offers free parking and free Wi-Fi throughout. Full Scottish breakfasts are served in the mornings. The rooms at The Craighaar include a flat-screen Freeview TV with a DVD player, room service, and a free early morning newspaper. The rooms also benefit from en suite facilities and a hairdryer. Guests can enjoy dining cooked from fresh, locally-sourced produce at The Craighaar Hotel's restaurant with its à la carte menu or relax with drinks in the Club Lounge. The city of Aberdeen and Aberdeen Rail Station are just 15 minutes away by car. Duthie Park and Gardens are just 20 minutes' drive away, and the historic University of Aberdeen is just a 10-minute drive. The TECA facility is also just a 5-minute walk away.
Gough Burn Crescent, Aberdeen, AB21 9FY
Situated in Aberdeen, 750 metres from TECA, Aloft Aberdeen TECA features air-conditioned accommodation and a fitness centre. The property is set 8 km ...more
Situated in Aberdeen, 750 metres from TECA, Aloft Aberdeen TECA features air-conditioned accommodation and a fitness centre. The property is set 8 km from Beach Ballroom and 8 km from Bon Accord & St Nicholas. Guests can have a drink at the W XYZ bar. At the hotel, the rooms have a desk, a flat-screen TV and a private bathroom. A continental breakfast is available each morning at Aloft Aberdeen TECA. The nearest airport is Aberdeen Airport, 3 km from the accommodation.
81 Seafield Road, Aberdeen, AB15 7YX
Offering a restaurant, the 4-star Palm Court Hotel is located in Aberdeen, in the quiet West End district and 10 minutes' drive from local rail and bu...more
Offering a restaurant, the 4-star Palm Court Hotel is located in Aberdeen, in the quiet West End district and 10 minutes' drive from local rail and bus stations. It features an outdoor dining area with decking. The hotel offers en suite rooms with a plasma-screen TV, free WiFi, a hairdryer and ironing facilities. Traditional Scottish food is served in the bar and restaurant, The Bothy. This also serves a selection of rare malts and fine wines. Aberdeen Airport and the AECC Exhibition Cenrre are within 20 minutes' drive of the hotel.
The Dutch Mill Hotel 7 Queens Road, Aberdeen, AB15 4NR
Distance:1.99 miles | Star Rating: N/A
Located in Aberdeen, 3.7 km from Beach Ballroom, The Dutch Mill Hotel provides accommodation with a restaurant, a bar and a terrace (notice that The D...more
Located in Aberdeen, 3.7 km from Beach Ballroom, The Dutch Mill Hotel provides accommodation with a restaurant, a bar and a terrace (notice that The Dutch Mill Hotel has no parking site by the moment). The property is set 8 km from AECC, 2.8 km from Aberdeen Harbour and 3.3 km from Hilton Community Centre. The accommodation offers room service and free WiFi throughout the property. A Full English/Irish breakfast is available daily at the hotel. Popular points of interest near The Dutch Mill Hotel include Aberdeen Art Gallery & Museum, Bon Accord & St Nicholas and Central Library of Aberdeen. The nearest airport is Aberdeen Airport, 12 km from the accommodation.
1 Great Northern Road, Aberdeen, AB24 3PS
Whatever you're looking to discover in Aberdeen, the Northern Hotel is the ideal place to stay. The 32 bedroom hotel is your home away from home and h...more
Whatever you're looking to discover in Aberdeen, the Northern Hotel is the ideal place to stay. The 32 bedroom hotel is your home away from home and has many original features including a unique violin shaped ballroom, glass-walled lift, and circular bar. The bedrooms are modern and comfortable with all expected amenities including TV, telephone, tea/coffee making facilities, free Wi-Fi and much more. An on-site laundry is also available to guests for an extra fee. Aberdeen Airport is a 15-minute drive from Aberdeen Aberdeen Northern Hotel and the University of Aberdeen is a 10-minute walk from the property. The 727 bus stops right outside the hotel. This service runs every 10 minutes from Aberdeen Airport to the centre of town, and to the bus and train station. Historic Old Aberdeen, where St Machar Cathedral and the University of Aberdeen are located, is a 10-minute walk from the hotel.
Prime Four Kingswells, Aberdeen, AB15 8PJ
Offering a contemporary indoor pool and a spa and wellness centre, Village Hotel Aberdeen is located in Aberdeen's Kingswell area. This chic property ...more
Offering a contemporary indoor pool and a spa and wellness centre, Village Hotel Aberdeen is located in Aberdeen's Kingswell area. This chic property features a bar and the Village Grill restaurant. Each stylish room has a modern en suite bathroom, air conditioning, free WiFi, a flat-screen TV with Sky TV and the Sky Sports channel. The hotel offers onsite parking and free WiFi. At an extra cost, guests can use the Village Gym with a swimming pool, sauna and steam room. The hotel is located just under 6 miles and 15 minutes' drive from Aberdeen International Airport and Heliports. It is 6 miles from Aberdeen city centre with its railway and bus stations, and the Northlink Ferry Terminal.
Caroline Place, Aberdeen, AB25 2TH
The stone-built Rosemount Palace is a stunning converted church property offering en suite accommodation a 20-minute walk from the centre of Aberdeen....more
The stone-built Rosemount Palace is a stunning converted church property offering en suite accommodation a 20-minute walk from the centre of Aberdeen. The hotel provides free on-site parking and free WiFi throughout. Bedrooms at Rosemount Palace are all individually decorated in warm and rich tones. Each room comes with a flat-screen TV and satellite channels, tea/coffee-making facilities and a hairdryer. The Rosemount has its own spacious café, while the city centre has a variety of restaurants, shops and bars. The hotel is located a five-minute walk from the Royal Cornhill Hospital and Berryden Road Retail Shopping Park. Aberdeen Airport is a 15-minute drive away and Rosemount is within 10 minutes' drive of 4 sea-front golf courses.
24 Esslemont Avenue G, Aberdeen, AB25 1SN
Featuring a garden, Prime 2 Bedroom City Centre Apt features accommodation in Aberdeen with free WiFi and city views. The property is 2.2 km from Aber...more
Featuring a garden, Prime 2 Bedroom City Centre Apt features accommodation in Aberdeen with free WiFi and city views. The property is 2.2 km from Aberdeen Beach and 2.7 km from Beach Ballroom. Boasting a Blu-ray player, the apartment has a kitchen with a dishwasher, a microwave and a fridge, a living room with a seating area and a dining area, 2 bedrooms, and 1 bathroom with a shower and a bath. Towels and bed linen are available in the apartment. Guests at the apartment can enjoy a continental breakfast. Popular points of interest near Apartment 2 Bedroom City Centre include Aberdeen Art Gallery & Museum, Bon Accord & St Nicholas and Aberdeen Harbour. The nearest airport is Aberdeen Airport, 11 km from the accommodation.
Caroline Place, Aberdeen , AB25 2TH
EXCLUSIVE BED AND BREAKFAST IN ABERDEEN
239 Great Western Road, Aberdeen, AB10 6PS
Located in the West End of Aberdeen, and just 6 km from AECC in Aberdeen, The Great Western Hotel features free WiFi access and free, limited private ...more
Located in the West End of Aberdeen, and just 6 km from AECC in Aberdeen, The Great Western Hotel features free WiFi access and free, limited private parking. Guests can enjoy the on-site bar, and restaurant. The rooms are equipped with a flat-screen TV and tea/coffee making facilities. Each room has an en-suite. Aberdeen Art Gallery is 1.8 km from The Great Western Hotel, while Aberdeen Harbour is 2.1 km from the property. The nearest airport is Aberdeen Airport, 9 km from the property.
122 Huntly Street, Aberdeen, AB10 1SU
The Copthorne Aberdeen Hotel is located in the city centre, within a 10-minute walk from the main shopping area. It has a Grill restaurant, a bar and ...more
The Copthorne Aberdeen Hotel is located in the city centre, within a 10-minute walk from the main shopping area. It has a Grill restaurant, a bar and modern rooms with flat-screen TVs. All spacious rooms feature free tea and coffee making facilities and a trouser press. The en-suite bathrooms have a hairdryer, with some rooms including bathrobes. Guests can dine in the West End Bistro, which also serves a tasty breakfast each morning. The friendly and relaxed Mac's Bar offers light meals and drinks. There are 3 golf courses within 10 minutes' drive, and there are several bar and restaurants within 5 minutes' walk. Aberdeen Rail Station is 0.6 miles away. Adjacent to the hotel there is a public car park with applicable parking charges.
Chapel Street, Aberdeen, AB10 1SQ
In Aberdeen's city centre, this Holiday Inn is just 200 metres from the shops and restaurants of Union Street. Guests can enjoy hot buffet breakfasts ...more
In Aberdeen's city centre, this Holiday Inn is just 200 metres from the shops and restaurants of Union Street. Guests can enjoy hot buffet breakfasts and modern rooms with satellite TV. The Holiday Inn Express Aberdeen City Centre is just a 10-minute walk from Aberdeen Rail Station. Aberdeen Art Gallery is less than 10 minutes' walk away, while Aberdeen's bustling centre can be reached in less than a 15-minute walk. Each of the modern rooms feature a private bathroom with a power shower, a hairdryer, and free toiletries. Please note that if booking a room with a sofa bed for 2 people or less, the sofa bed will only be made up on request. Rooms also feature satellite TVs, and tea/coffee facilities. The modern lounge bar offers teas, coffees and drinks until late, 7 days a week. There is limited free parking at the hotel on a first come first served basis and cannot be booked. Public parking is available 100 metres away.
250 North Deeside Road, Aberdeen, AB15 9PB
Located in Aberdeen in the Grampian region, Luxury Scottish Apartment features a balcony. This apartment provides free private parking, a 24-hour fron...more
Located in Aberdeen in the Grampian region, Luxury Scottish Apartment features a balcony. This apartment provides free private parking, a 24-hour front desk and free WiFi. The apartment comes with 3 bedrooms, 4 bathrooms, bed linen, towels, a flat-screen TV, a dining area, a fully equipped kitchen, and a patio with garden views. For added convenience, the property can provide towels and bed linen for an extra charge. The apartment offers a terrace. A children's playground can be found at Luxury Scottish Apartment, along with a garden. Beach Ballroom is 8 km from the accommodation, while AECC is 12 km away. The nearest airport is Aberdeen Airport, 15 km from Luxury Scottish Apartment.
42 Broomhill Road, Aberdeen, AB10 6HT
Situated in Aberdeen, 3 km from Aberdeen Beach, 4.1 km from Beach Ballroom and 8 km from AECC, Bright & Airy Two Bed Set In Granite features accommoda...more
Situated in Aberdeen, 3 km from Aberdeen Beach, 4.1 km from Beach Ballroom and 8 km from AECC, Bright & Airy Two Bed Set In Granite features accommodation with a patio and free WiFi. This apartment is 2.1 km from Aberdeen Art Gallery & Museum and 2.3 km from Bon Accord & St Nicholas. The apartment features 2 bedrooms, a flat-screen TV and a fully equipped kitchen that provides guests with a microwave, a fridge, a washing machine, a stovetop and a toaster. Towels and bed linen are available. A continental breakfast is available each morning at the apartment. Popular points of interest near Bright & Airy Two Bed Set In Granite include Duthie Park, Aberdeen Music Hall and St Mary's Cathedral. The nearest airport is Aberdeen Airport, 12 km from the accommodation.
26 Broomhill Road First floor right, Aberdeen, AB10 6HS
Charming one bedroom is situated in Aberdeen, 2.9 km from Aberdeen Beach, and features a patio, garden, and free WiFi. The property is 4 km from Beach...more
Charming one bedroom is situated in Aberdeen, 2.9 km from Aberdeen Beach, and features a patio, garden, and free WiFi. The property is 4 km from Beach Ballroom and 8 km from AECC. The apartment features 1 bedroom, a flat-screen TV and a fully equipped kitchen that provides guests with a dishwasher, a microwave, a washing machine, a fridge and an oven. Towels and bed linen are available in this accommodation. Popular points of interest near the apartment include Aberdeen Art Gallery & Museum, Duthie Park and Aberdeen Music Hall. The nearest airport is Aberdeen Airport, 12 km from Charming one bedroom.
1 Justice Mill Lane, Aberdeen, AB11 6EQ
Park Inn by Radisson Aberdeen is located in the centre of the city centre, just off Union Street and towards the cosmopolitan west end of the city. Th...more
Park Inn by Radisson Aberdeen is located in the centre of the city centre, just off Union Street and towards the cosmopolitan west end of the city. The hotel offers free WiFi and is a perfect home base for exploring the Grampian Whisky and Castle trail. Public parking is available off-site. Guests can enjoy the terrace. The stylish, colourful rooms include a flat-screen TV, work desk, safe, and tea and coffee making facilities. Many offer views of the city. A room service menu is available until 21:30. Each room has an en-suite bathroom with complimentary toiletries provided. With a vibrant and warm atmosphere, RBG Bar & Grill serves an international menu. Guests can enjoy a cocktail at the bar or choose beverages from the extensive drinks menu. Park Inn Aberdeen is just 15 minutes' walk from Aberdeen Harbour and the Ferry Terminals. The hotel is within walking distance from the shopping and business districts while the Aberdeen Train Station is only a 10-minute walk away. The nearest airport is Aberdeen Airport, 9 km from the property.
13 North Silver Street, Aberdeen, AB10 1RJ
Boasting a beer garden and a bar, The Globe Inn is situated in Aberdeen, 2.7 km from Beach Ballroom. Attractively situated in the Aberdeen City Centre...more
Boasting a beer garden and a bar, The Globe Inn is situated in Aberdeen, 2.7 km from Beach Ballroom. Attractively situated in the Aberdeen City Centre district, the property is set 6 km from AECC and 600 metres from Aberdeen Art Gallery. The property is 500 metres from the city Aberdeen City Centre and 800 metres from Bon Accord & St Nicholas. At the inn, the rooms have a desk. Each room comes with a private bathroom. All units will provide guests with a fridge. A varied breakfast menu is available daily for an additional cost. At the accommodation you will find a restaurant serving Scottish, British and Local cuisine. A gluten-free option can also be requested. Aberdeen Harbour is 1.2 km from The Globe Inn. Aberdeen train station is 650 metres from the property while His Majesty's theatreis 200 metres away. The nearest airport is Aberdeen Airport, 11 km from the property.
328 North Deeside Road, Aberdeen, AB15 9SE
The Cults Hotel is just a 10-minute drive from Aberdeen's centre and offers modern, individually styled rooms. The friendly Lounge Bar serves a varied...more
The Cults Hotel is just a 10-minute drive from Aberdeen's centre and offers modern, individually styled rooms. The friendly Lounge Bar serves a varied menu and real ales, and there is free Wi-Fi. With contemporary décor, all well-equipped rooms feature a TV, free tea and coffee, a telephone and trouser press. Each luxury tiled bathroom includes a hairdryer and fluffy bathrobes. A popular public bar has a pool table, a large flat-screen TV and open fireplace. The Lounge Bar offers freshly prepared food with a terrace seating area outside. Until May 31st 2021 the offered breakfast will consist of a Bagged Continental Breakfast delivered to the room. The Cults has 2 golf courses within 10 minutes' drive. The local area is famous for its fishing, with the River Dee a 5-minute walk from the hotel. Clay pigeon shooting can also be arranged on request.
15 Bon Accord Square, Aberdeen, AB11 6DJ
These luxury suites look out over a quiet garden square in Aberdeen city centre. Each stylish suite has free Wi-Fi, a fully equipped kitchen, a king-s...more
These luxury suites look out over a quiet garden square in Aberdeen city centre. Each stylish suite has free Wi-Fi, a fully equipped kitchen, a king-size bed, and a flat-screen TV. There are 2 suites on the 1st Floor with a Balcony area (The Studio Balcony Suite and the 1 Bedroom Balcony Suite). In addition to this, there is the Patio Suite which is situated on the lower ground floor. Craibstone Suites are 200 metres from the shops, bars and restaurants of Union Street and 800 metres from Aberdeen Rail Station. The suites are in a historic building and mix original features with modern luxury. Each one has a flat-screen TV with over 50 channels. Craibstone Suites offer the facilities of a 4-star hotel with the convenience and privacy of an apartment.
10-14 Union Terrace, Aberdeen, AB10 1WE
Situated in the heart of Aberdeen, Mercure Aberdeen Caledonian Hotel offers an award-winning restaurant and modern rooms with free toiletries and dail...more
Situated in the heart of Aberdeen, Mercure Aberdeen Caledonian Hotel offers an award-winning restaurant and modern rooms with free toiletries and daily newspapers. Guests can enjoy afternoon tea, a drink, snack or light meal in the elegant Victorian hotel. The Café Bar Caley and Diamond Suite are open daily and have a relaxing ambience. The hotel also offer private dining and a pre-theatre menu. The modern rooms feature en suite facilities, flat-screen TVs and tea/coffee facilities. Wireless internet is accessible in all rooms, and 24-hour room service is also available. Mercure Aberdeen Caledonian Hotel is within easy walking distance of the city centre attractions, including the Satrosphere Science Centre, St Mary's Cathedral and His Majesty's Theatre. Aberdeen Rail Station is within 10 minutes' walk.
Belmont Street, Aberdeen, AB10 1JR
This modern hotel in the city centre of Aberdeen has free WiFi and a stylish decor. It is a short walk away from the local entertainment, nightlife, s...more
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\section{Introduction}
\label{intro}
Euclidean lattices, metrized free ${\mathbb Z}$-modules in real vector spaces, are central to number theory and discrete geometry. In addition to their arithmetic and geometric appeal, lattices are also key to discrete optimization, often providing solutions to classical optimization questions like sphere packing, covering and kissing number problems. They are also extensively used in applied areas, such as coding theory, cryptography and other areas of digital communications; see the famous book by Conway and Sloane~\cite{conway} for a wealth of information on the theory of lattices and their numerous connections and applications. In this paper, we focus on certain geometric properties of lattices that are of utmost interest for both, theory and applications. One of the main sources of lattices possessing these special properties is the classical Minkowski construction from ideals in rings of integers of algebraic number fields: not only does this construction allow for a nice and compact description of the resulting lattices, but also algebraic properties of an ideal often inform the geometry of the corresponding lattice.
Let $L \subset {\mathbb R}^n$ be a free ${\mathbb Z}$-module of rank $k \leq n$, then
$$L = B{\mathbb Z}^n = \operatorname{span}_{{\mathbb Z}} \{ {\boldsymbol b}_1,\dots,{\boldsymbol b}_k \},$$
where ${\boldsymbol b}_1,\dots,{\boldsymbol b}_n \in {\mathbb R}^n$ are ${\mathbb R}$-linearly independent basis vectors for $L$ and $B = ({\boldsymbol b}_1\ \dots\ {\boldsymbol b}_n) \in \operatorname{GL}_n({\mathbb R})$ is the corresponding basis matrix. Then for any $U \in \operatorname{GL}_n({\mathbb Z})$, $BU$ is also a basis matrix for $L$, i.e. $L = BU{\mathbb Z}^n$ for any $U \in \operatorname{GL}_n({\mathbb Z})$.
Let $f({\boldsymbol x}) = f(x_1,\dots,x_n) \in {\mathbb R}[x_1,\dots,x_n]$ be a positive definite quadratic form, i.e. $f({\boldsymbol x}) = {\boldsymbol x}^t Q {\boldsymbol x}$, where $Q$ is an $n \times n$ symmetric positive definite coefficient matrix for $f$. Then $Q = A^t A$ for some $A \in \operatorname{GL}_n({\mathbb R})$, so
\begin{equation}
\label{qf}
f({\boldsymbol x}) = (A{\boldsymbol x})^t (A{\boldsymbol x}).
\end{equation}
Thus we will use notation $f_A$ for the form $f$ as in~\eqref{qf} with coefficient matrix $A^t A$ for some $A \in \operatorname{GL}_n({\mathbb R})$.
We will use the term {\it lattice} to refer to a pair $(L,f_A)$, where the form $f_A$ is used to define the norm of vectors in $L$. If $A=I_n$, the $n \times n$ identity matrix, then $f_A$ is $\|\ \|^2$, the square of the usual Euclidean norm, in which case we simply write $L$ instead of $(L, f_{I_n})$. In general, let ${\boldsymbol x} = B {\boldsymbol y} \in L$, i.e. ${\boldsymbol y} \in {\mathbb Z}^n$, so
\begin{equation}
\label{qA}
f_A({\boldsymbol x}) = (AB{\boldsymbol y})^t (AB{\boldsymbol y}) = {\boldsymbol y}^t (AB)^t (AB) {\boldsymbol y} = \| (AB) {\boldsymbol y} \|^2.
\end{equation}
In other words, a lattice $(L,f_A)$ can be identified with the lattice~$AL$, which we will refer to as the {\it twist} of $L$ by $A$. Now, for a lattice $L$ we define its rank, $\operatorname{rk}(L)$, to be the cardinality of its basis, and its determinant
$$\operatorname{det}(L) := \left| \operatorname{det} (B^{\top} B) \right|^{1/2},$$
where $B$ is a basis matrix for~$L$.
There are two important classes of lattices we will discuss. A lattice $L$ is called {\it well-rounded} (abbreviated WR) if there exist $n$ linearly independent vectors ${\boldsymbol c}_1,\dots,{\boldsymbol c}_n \in L$ such that
$$\|{\boldsymbol c}_1\| = \dots = \|{\boldsymbol c}_n\| = |L|,$$
where $|L| := \min_{{\boldsymbol x} \in L \setminus \{{\boldsymbol 0}\}} \|{\boldsymbol x}\|.$ On the other hand, $L$ is called {\it stable} if for each sublattice $L' \subseteq L$,
$$\operatorname{det}(L)^{1/rk(L)} \leq \operatorname{det}(L')^{1/rk(L')},$$
and $L$ is called unstable otherwise. Well-roundedness and stability are independent properties for lattices of rank greater than two: WR lattices can be unstable and stable lattices do not have to be WR. On the other hand, in the plane WR lattices form a proper subset of stable lattices~\cite{lenny:stable}. There is an important equivalence relation on the space of lattices: two lattices are called {\it similar} if they are related by a dilation and an orthogonal transformation; geometric properties, like well-roundedness and stability are preserved under the similarity, hence we can talk about WR or stable similarity classes of lattices.
Both of these types of lattices are very important in reduction theory of algebraic groups. In particular, they were studied in the context of the diagonal group action on the space of lattices. Let
\begin{equation}
\label{A}
{\mathcal A} = \left\{ A = (a_{ij}) \in \operatorname{GL}_n({\mathbb R}) : a_{ij} = 0\ \forall\ i \neq j,\ a_{ii} > 0\ \forall\ i,\ \prod_{i=1}^n a_{ii} = 1 \right\}
\end{equation}
be the group of real positive diagonal matrices with determinant 1. This group acts on the space of lattices in~${\mathbb R}^n$ by left multiplication: $L \mapsto AL$ for each $A \in {\mathcal A}$ and lattice $L \subset {\mathbb R}^n$. A celebrated result of McMullen~\cite{mcmullen} in connection with his work on Minkowski's conjecture asserts that any bounded ${\mathcal A}$-orbit of lattices contains a well-rounded lattice. Inspired by McMullen's work, Shapira and Weiss proved~\cite{weiss} that the orbit closure of a lattice under the action of ${\mathcal A}$ also contains a stable lattice.
Throughout this note, let $K$ be a totally real number field of degree $n$ with the ring of integers ${\mathcal O}_K$, $\sigma_1,\dots,\sigma_n : K \hookrightarrow {\mathbb R}$ be the embeddings of $K$, and define the Minkowski embedding
$$\sigma_K = (\sigma_1,\dots,\sigma_n) : K \hookrightarrow {\mathbb R}^n.$$
Let $I \subset K$ be a full module, i.e. a ${\mathbb Z}$-module of rank $n$, for instance ${\mathcal O}_K$ or any (fractional) ideal. We define $L_K(I) := \sigma_K(I)$, i.e. the image of $I$ under Minkowski embedding, then $AL_K(I)$ is a lattice in ${\mathbb R}^n$ for any $A \in \operatorname{GL}_n({\mathbb R})$.
\begin{prop} \label{WR2} With notation as above, there exist $A, B \in {\mathcal A}$ such that the lattice $AL_K(I)$ is WR and $BL_K(I)$ is stable.
\end{prop}
\proof
By the similarity relation, we can assume that $L_K(I)$ is unimodular without loss of generality. Consider the orbit of $L_K(I)$ under the action of ${\mathcal A}$, i.e. the set
$${\mathcal A} L_K(I) = \left\{ A L_K(I) : A \in {\mathcal A} \right\}.$$
Since $L_K(I)$ comes from a full module in a totally real number field, Theorem~3.1 of~\cite{mcmullen} implies that the orbit ${\mathcal A} L_K(I)$ is compact. Then Theorem~1.3 of~\cite{mcmullen} implies that this orbit contains a WR lattice and Theorem~1.1 of~\cite{weiss} implies that it also contains a stable lattice, i.e. there exist some $A,B \in {\mathcal A}$ such that the lattice $AL_K(I)$ is WR and the lattice $BL_K(I)$ is stable.
\endproof
\begin{rem} The proofs of Theorem~1.3 of~\cite{mcmullen} and of Theorem~1.1 of~\cite{weiss} are not constructive, meaning that they do not help to explicitly find $A, B \in {\mathcal A}$ such that the lattice $AL_K(I)$ is WR and $BL_K(I)$ is stable.
\end{rem}
\medskip
Now let $I \subseteq {\mathcal O}_K$ be an ideal. Let $\alpha \in K$ be totally positive, i.e. $\sigma_i(\alpha) > 0$ for all $1 \leq i \leq n$, then the pair $(I,\alpha)$ gives rise to the {\it ideal lattice} $(L_K(I),f_{A(\alpha)})$, where
$$A(\alpha) = \begin{pmatrix} \sqrt{\sigma_1(\alpha)} & \hdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \hdots & \sqrt{\sigma_n(\alpha)} \end{pmatrix}.$$
Ideal lattices have been extensively studied by a number of authors, especially E. Bayer-Fluckiger (see~\cite{bayer2} for a survey of this topic); a systematic study of WR ideal lattices has been initiated in~\cite{kate-me} and~\cite{fletcher-jones}, where it was shown that WR ideal lattices $(L_K(I),\|\ \|^2)$ from quadratic number fields are relatively sparse. The situation is different if we allow a more general quadratic norm form $f_A$. Notice that $A(\alpha) \in {\mathcal A}$ if and only if $\alpha \in {\mathcal O}_K$ is a totally positive unit. More generally, ${\mathcal A}_1(K) := \left\{ A(\alpha) : \alpha \in K \text{ is totally positive} \right\}$ is also a multiplicative group of (positive) diagonal matrices so that
$${\mathcal A} \cap {\mathcal A}_1(K) = \left\{ A(\alpha) : \alpha \in K \text{ is totally positive and } {\mathbb N}_K(\alpha) = 1 \right\},$$
where ${\mathbb N}_K$ stands for the number field norm on $K$. In fact, notice that the set
$${\mathcal A}'_1(K) = \left\{ {\mathbb N}_K(\alpha)^{-1/n} {\mathcal A}(\alpha) : A(\alpha) \in {\mathcal A}_1(K) \right\}$$
is a proper subset of ${\mathcal A}$, while the lattices $(L_K(I),f_{A(\alpha)})$ and $(L_K(I),f_{{\mathbb N}_K(\alpha)^{-1/n} A(\alpha)})$ are scalar multiples of each other, and hence have the same properties.
\begin{quest}\label{question1} By Proposition~\ref{WR2}, we know that, given $L_K(I)$, there exists $A, B \in {\mathcal A}$ such that $AL_K(I)$ is WR and $BL_K(I)$ is stable, but do there exist such $A, B \in {\mathcal A}_1(K)$?
\end{quest}
In this note, we consider the two-dimensional situation, for which this question was answered in the affirmative for WR lattices in~\cite{bayer-nebe} (specifically, see Corollary~3.1). Further, the authors construct explicit examples of principal ideal lattices (i.e., those coming from full rings of integers) similar to the square and the hexagonal lattices. Of course, this immediately implies the same affirmative answer for stable lattices, since in two dimensions WR lattices are stable; in fact, unlike the WR situation, there are infinitely many stable ideal lattices from real quadratic number fields even without twisting by a matrix, as proved in~\cite{lenny:stable}. Our goal here is to further explicitly investigate well-roundedness and stability properties of planar ideal lattices. First we need some more notation.
\begin{defn} Given a particular basis matrix $B$ for a lattice $L$, we will say that $B$ is {\it WR twistable} (respectively, {\it stable twistable}) if there exists $A \in {\mathcal A}$ such that $A B {\mathbb Z}^n$ is WR (respectively, stable) with $AB$ being the shortest basis matrix, as discussed in Section~\ref{setup}. We will refer to the resulting lattices as a {\it WR twist} (respectively, {\it stable twist}) of the original lattice by the matrix $A$, respectively.
\end{defn}
While WR twistable bases have been discussed in~\cite{taoufiq}, to the best of our knowledge stable twistable bases have not previously been investigated. A criterion to determine whether a given basis for an ideal lattice from a real quadratic number field $K$ is WR twistable or not was given in~\cite{taoufiq}: this criterion in particular implies there can be only finitely many such bases for a fixed ideal lattice. Further, Proposition~7 of~\cite{taoufiq} asserts that a basis is WR twistable by some $A \in {\mathcal A}$ if and only if it is WR twistable by some $A(\alpha) \in {\mathcal A}_1(K)$, and an explicit construction of such corresponding twists is given.
In this note, we focus on the twistable properties of the particularly important kind of basis for ideals in quadratic number fields, the so-called canonical basis. We always talk about twists by matrices $A(\alpha) \in {\mathcal A}_1(K)$ for some totally positive $\alpha \in K$, referring to such twists as a twist by~$\alpha$.
Let $D \in {\mathbb Z}_{>0}$ be squarefree and let $K = {\mathbb Q}(\sqrt{D})$. Let $I \subseteq {\mathcal O}_K$ be an ideal. Notice that ${\mathcal O}_K={\mathbb Z}[\delta]$, where
\begin{equation}
\label{delta}
\delta = \left\{ \begin{array}{ll}
- \sqrt{D} & \mbox{if } D \not\equiv 1\ (\operatorname{mod} 4) \\
\frac{1-\sqrt{D}}{2} & \mbox{if } D \equiv 1\ (\operatorname{mod} 4) \\
\end{array}
\right.
\end{equation}
The embeddings $\sigma_1, \sigma_2 : K \to {\mathbb R}$ are given by
$$\sigma_1(x+y\sqrt{D}) = x+y\sqrt{D},\ \sigma_2(x+y\sqrt{D}) = x-y\sqrt{D}$$
for each $x+y\sqrt{D} \in K$. The number field norm on $K$ is given by
$${\mathbb N}_K(x+y\sqrt{D}) = \sigma_1(x+y\sqrt{D}) \sigma_2(x+y\sqrt{D}) = \left( x+y\sqrt{D} \right) \left( x-y\sqrt{D} \right).$$
Now $I \subseteq {\mathcal O}_K$ is an ideal if and only if
\begin{equation}
\label{I_abg}
I = \{ ax + (b+g\delta)y : x,y \in {\mathbb Z} \},
\end{equation}
for some $a,b,g \in {\mathbb Z}_{\geq 0}$ such that
\begin{equation}
\label{abg}
b < a,\ g \mid a,b,\text{ and } ag \mid {\mathbb N}(b+g\delta).
\end{equation}
Such integral basis $a,b+g\delta$ is unique for each ideal $I$ and is called the {\it canonical basis} for $I$ (see Section~6.3 of~\cite{buell} for a detailed exposition): it is important in the arithmetic theory of binary quadratic forms and quadratic number fields. For instance, canonical basis is used to determine reduced ideals and compute the ideal class group in quadratic fields (see, e.g., the classical work of H. H. Mitchell~\cite{mitchell} as well as a more recent paper~\cite{yamamoto}), and it is employed for number field computations in several modern computer algebra systems; it has also been used in the previous study of geometric properties of ideal lattices in the plane (see \cite{kate-me}, \cite{fletcher-jones}, \cite{lenny:stable}, \cite{taoufiq}). It is then easy to check that $L_K(I) = B {\mathbb Z}^2$, where
\begin{equation}
\label{B1}
B = \begin{pmatrix} a & b-g\sqrt{D} \\ a & b+g\sqrt{D} \end{pmatrix},
\end{equation}
when $D \not\equiv 1 (\operatorname{mod} 4)$, and
\begin{equation}
\label{B2}
B = \begin{pmatrix} a & \frac{2b+g}{2} - \frac{g\sqrt{D}}{2} \\ a &\frac{2b+g}{2} + \frac{g\sqrt{D}}{2} \end{pmatrix},
\end{equation}
when $D \equiv 1 (\operatorname{mod} 4)$. Then any other basis matrix for $L_K(I)$ is of the form $BU$ for some $U \in \operatorname{GL}_2({\mathbb Z})$. We can now state our result about WR $K$-twists of canonical bases of planar ideal lattices from real quadratic number fields.
\medskip
\begin{thm} \label{main_wr} Let $K = {\mathbb Q}(\sqrt{D})$ be a real quadratic number field and ${\mathcal O}_K$ its ring of integers. There can be at most finitely many ideals in ${\mathcal O}_K$, up to similarity of the resulting ideal lattices, with the canonical basis being WR twistable: if this is the case for some ideal $I$ with canonical basis $a, b+g\delta$, then
$$b < \left\{ \begin{array}{ll}
g\sqrt{D} & \mbox{if $D \not\equiv 1\ (\operatorname{mod} 4)$,} \\
\frac{(\sqrt{D}-1)g}{2} & \mbox{if $D \equiv 1\ (\operatorname{mod} 4)$.}
\end{array}
\right.$$
The canonical basis for ${\mathcal O}_K$ is WR twistable if and only if $K={\mathbb Q}(\sqrt{5})$.
\end{thm}
\begin{rem} In fact, the assertion that the canonical basis of ${\mathcal O}_K$ is WR twistable if and only if $K={\mathbb Q}(\sqrt{5})$ also follows from Corollary~2 of~\cite{taoufiq}.
\end{rem}
The set of similarity classes of planar WR lattices is parameterized by a curve, while the set of similarity classes of planar stable lattices is two-dimensional (see~\cite{lenny:florian} for details). The results of~\cite{lenny:florian} also indicate that the set of stable ideal lattices in the plane is likely order of magnitude larger than the set of WR ideal lattices: at least this is true for planar arithmetic lattices. With this in mind, it is somewhat surprising that the number of ideal lattices with stable twistable canonical basis is not much larger than with WR twistable canonical basis: this is an interesting property of the canonical basis.
\begin{thm} \label{main_stable} Let $K = {\mathbb Q}(\sqrt{D})$ be a real quadratic number field and ${\mathcal O}_K$ its ring of integers. There can be at most finitely many ideals in ${\mathcal O}_K$, up to similarity of the resulting ideal lattices, with the canonical basis being stable twistable: if this is the case for some ideal $I$ with canonical basis $a, b+g\delta$, then
$$b < \left\{ \begin{array}{ll}
\frac{2}{\sqrt{3}} g \sqrt{D} & \mbox{if $D \not\equiv 1\ (\operatorname{mod} 4)$,} \\
\frac{g}{2} \left( \frac{2}{\sqrt{3}} \sqrt{D} - 1 \right) & \mbox{if $D \equiv 1\ (\operatorname{mod} 4)$.}
\end{array}
\right.$$
The canonical basis for ${\mathcal O}_K$ is stable twistable if and only if $K={\mathbb Q}(\sqrt{5})$.
\end{thm}
\noindent
While our results prove finiteness of the number of ideals in each fixed number field with WR or stable twistable canonical basis, there are still more ideals with stable twistable than with WR twistable canonical basis as we demonstrate in Section~\ref{stable_proof}. We set some additional notation in Section~\ref{setup}, including convenient explicit criteria to check if the canonical basis of a planar ideal lattice is WR or stable twistable, or not. We prove Theorem~\ref{main_wr} and give some explicit examples of ideal lattices with WR twistable canonical basis in Section~\ref{wr_proof}. In Section~\ref{stable_proof} we prove Theorem~\ref{main_stable} and show examples of ideal lattices with stable twistable (but not WR twistable) canonical basis in Section~\ref{wr_proof}. Finally, in Section~\ref{remarks} we make some remarks on the relation between stable and WR twists of arbitrary ideal bases.
Let us conclude this section with a few additional words of motivation for our results. Our theorems reveal certain new properties of the canonical basis for ideals in quadratic number fields. Canonical bases have an advantage of being convenient for computations, including explicit computations of WR and stable lattices, as demonstrated in~\cite{kate-me}, \cite{fletcher-jones} and~\cite{lenny:stable}. Knowing which of those that are not WR or stable to start with can be twisted into WR or stable shortest bases is helpful. Further, this information may find some future applications when choosing ideal lattice bases for lattice codes constructions. In Section~\ref{comm_theory} we include some additional details on applications of the WR and stable twists of lattice bases in communication theory and in arithmetic theory of quadratic forms and lattices. We are now ready to proceed.
\bigskip
\section{Notation and setup}
\label{setup}
We start here with a basic review of some general lattice theory background. If rank of $L$ is $n \geq 2$, we define the {\it successive minima} of $L$ to be the real numbers
$$0 < \lambda_1 \leq \dots \leq \lambda_n$$
such that
$$\lambda_i = \inf \left\{ \mu \in {\mathbb R} : \operatorname{dim}_{{\mathbb R}} \{ {\boldsymbol x} \in L : \|{\boldsymbol x}\| \leq \mu \} = i \right\}.$$
Then $\lambda_1 = |L|_{\|\ \|^2}$, and $L$ is WR if and only if $\lambda_1 = \dots = \lambda_n$. Linearly independent vectors ${\boldsymbol x}_1,\dots,{\boldsymbol x}_n \in L$ such that $\|{\boldsymbol x}_i\| = \lambda_i$ are called {\it vectors corresponding to successive minima}. They do not necessarily form a basis for $L$. On the other hand, $L$ has a {\it Minkowski reduced basis} ${\boldsymbol v}_1,\dots,{\boldsymbol v}_n$, defined by the conditions that
$$\|{\boldsymbol v}_1\| = \lambda_1,\ \|{\boldsymbol v}_i\| = \min \left\{ \|{\boldsymbol x}\| : {\boldsymbol v}_1,\dots{\boldsymbol v}_{i-1}, {\boldsymbol x} \text{ are extendable to a basis for } L \right\}.$$
In general, $\|{\boldsymbol v}_i\| \geq \lambda_i$, and when $n \geq 5$ these inequalities can be strict, although a theorem of van der Waerden asserts that for all $i \geq 4$,
\begin{equation}
\label{vdw}
\|{\boldsymbol v}_i\| \leq \left( \frac{5}{4} \right)^{i-4} \lambda_i,
\end{equation}
and there is a conjecture that the constant in the upper bound of~\eqref{vdw} can be improved. When $n \leq 4$, a Minkowski reduced basis for a lattice $L$ always consists of vectors corresponding to successive minima. For each $n \geq 2$, there are finitely many inequalities that have to be satisfied by the vectors ${\boldsymbol v}_1,\dots,{\boldsymbol v}_n \in L$ to be a Minkowski reduced basis for $L$. The number of such inequalities depends only on $n$, but it grows fast with $n$. The explicit list of non-redundant inequalities is known only for $n \leq 7$ (it is called Tammela's list); see \S\S 2.2-2.3 of \cite{achill_book} for Tammela's list, as well as more information on Minkowski reduction and relation to successive minima.
Let $L = ({\boldsymbol v}_1\ {\boldsymbol v}_2)\ {\mathbb Z}^2$ be a planar lattice. The necessary and sufficient conditions for the basis ${\boldsymbol v}_1,{\boldsymbol v}_2$ to be Minkowski reduced (and hence to correspond to successive minima $\lambda_1,\lambda_2$) are as follows:
\begin{equation}
\label{mink-2}
\|{\boldsymbol v}_1\| \leq \|{\boldsymbol v}_2\|,\ 2 | {\boldsymbol v}_1^t {\boldsymbol v}_2 | \leq \|{\boldsymbol v}_1\| \|{\boldsymbol v}_2\|.
\end{equation}
In order for $L$ to be WR we need $\lambda_1=\lambda_2$, i.e.
\begin{equation}
\label{mink-2-WR}
\|{\boldsymbol v}_1\| = \|{\boldsymbol v}_2\|,\ 2 | {\boldsymbol v}_1^t {\boldsymbol v}_2 | \leq \|{\boldsymbol v}_1\|^2,
\end{equation}
and in order for $L$ to be stable we need
\begin{equation}
\label{mink-2-stable}
\sqrt{\operatorname{det}(L)} \leq \|{\boldsymbol v}_1\|,\ 2 | {\boldsymbol v}_1^t {\boldsymbol v}_2 | \leq \|{\boldsymbol v}_1\| \|{\boldsymbol v}_2\|.
\end{equation}
The angle $\theta$ between ${\boldsymbol v}_1$ and ${\boldsymbol v}_2$ must therefore be in the interval $[\pi/3,2\pi/3]$, and $| \cos \theta |$ is an invariant of the lattice. Two planar WR lattices $L_1,L_2$ are similar if and only if these values are the same (see~\cite{hex}). There is an easy way to ``deform" a non-WR lattice into a WR one.
\begin{lem} \label{WR_stretch} Let $L = ({\boldsymbol v}_1\ {\boldsymbol v}_2)\ {\mathbb Z}^2 \subset {\mathbb R}^2$ be a lattice of full rank, where ${\boldsymbol v}_1, {\boldsymbol v}_2$ is a Minkowski reduced basis matrix, so $\|{\boldsymbol v}_1\| = \lambda_1, \|{\boldsymbol v}_2\| = \lambda_2$ for the successive minima $\lambda_1 \leq \lambda_2$ of $L$. Let ${\boldsymbol u}_1 = \lambda_2 {\boldsymbol v}_1$, ${\boldsymbol u}_2 = \lambda_1 {\boldsymbol v}_2$, then the lattice $M = ( {\boldsymbol u}_1\ {\boldsymbol u}_2)\ {\mathbb Z}^2$ is a well-rounded lattice with successive minima equal to $\lambda_1 \lambda_2$.
\end{lem}
\proof
This follows immediately from Lemma~3.6 of~\cite{hex}.
\endproof
\begin{rem} In principle, there is a deformation of a lattice into a WR lattice in higher dimensions too, but it is more complicated. Such a deformation is described in Remark~3.3 of~\cite{lizhen_ji}.
\end{rem}
\smallskip
We now come back to the ideal lattices. Let $K={\mathbb Q}(\sqrt{D})$ be a real quadratic number field and $\alpha = p + q \sqrt{D} \in K$ be totally positive, then
$$A(\alpha) = \begin{pmatrix} \sqrt{p+q \sqrt{D}} & 0 \\ 0 & \sqrt{p - q\sqrt{D}} \end{pmatrix},$$
where $p \pm q\sqrt{D} > 0$; in fact, we can assume without loss of generality that $p,q$ are positive rational numbers. Let $B$ as in~\eqref{B1} or~\eqref{B2} be the canonical basis matrix for an ideal lattice $L_K(I)$, where $I \subseteq {\mathcal O}_K$ is the corresponding ideal. Then $B$ is WR twistable (respectively, stable twistable) if
$$C := A(\alpha) B = \left\{ \begin{array}{ll}
\begin{pmatrix} a \sqrt{p+\sqrt{D}} & (b - g\sqrt{D}) \sqrt{p+\sqrt{D}} \\ a \sqrt{p-\sqrt{D}} & (b + g\sqrt{D}) \sqrt{p-\sqrt{D}} \end{pmatrix} & \mbox{if $D \not\equiv 1\ (\operatorname{mod} 4)$,} \\
\begin{pmatrix} a \sqrt{p+\sqrt{D}} & \left( \frac{(2b+g) - g\sqrt{D}}{2} \right) \sqrt{p+\sqrt{D}} \\ a \sqrt{p-\sqrt{D}} & \left( \frac{(2b+g) + g\sqrt{D}}{2} \right) \sqrt{p-\sqrt{D}} \end{pmatrix} & \mbox{if $D \equiv 1\ (\operatorname{mod} 4)$.}
\end{array}
\right.$$
is a Minkowski reduced basis for WR (respectively, stable) lattice $C{\mathbb Z}^2$. These conditions can be described by explicit inequalities, stemming from~\eqref{mink-2-WR} and~\eqref{mink-2-stable}. First assume that $D \not\equiv 1 (\operatorname{mod} 4)$, then $B$ is WR twistable by $\alpha = p + q\sqrt{D}$ if and only if
\begin{eqnarray}
\label{C1}
a^2p = (Dg^2 + b^2)p - 2qbgD,\ \left| b^2-g^2D+a^2 \right| \leq ab,
\end{eqnarray}
and $B$ is stable twistable by $\alpha$ if and only if
\begin{eqnarray}
\label{C2}
-4 D ^2 g^2 q^2 + 6 b D g p q + D g^2 p^2 - 3 b^2 p^2 \geq 0, \nonumber \\
\min \left\{ a^2p, (Dg^2+b^2)p-2Dbgq \right\} \geq ag \sqrt{D(p^2 - q^2D)}.
\end{eqnarray}
On the other hand, if $D \equiv 1 (\operatorname{mod} 4)$, then $B$ is WR twistable by $\alpha = p + q\sqrt{D}$ if and only if
\begin{eqnarray}
\label{Cd1}
2a^2p = \frac{1}{2} \left( (D+1)p - 2Dq) \right) g^2 - 2b(Dq-p) g+2b^2 p, \nonumber \\
\left| 4a^2 + 4b^2 + 4bg - (D-1) g^2 \right| \leq 2a(2b+g),
\end{eqnarray}
and $B$ is stable twistable by $\alpha$ if and only if
\begin{eqnarray}
\label{Cd2}
-D^2 g^2 q^2 + 3 D g b p q + \frac{3}{2} D g^2 p q - 3 b^2 p^2 - 3 b g p^2 - \frac{3}{4} g^2 p^2 + \frac{1}{4} D g^2 p^2 \geq 0 \nonumber \\
\min \left\{ 2 a^2p, ag (p - Dq) + 2abp \right\} \geq ag \sqrt{D(p^2 - q^2D)}.
\end{eqnarray}
We can now use these criteria to analyze the WR and stable twistable properties of the canonical bases for ideals in real quadratic number fields.
\bigskip
\section{Proof of Theorem~\ref{main_wr}}
\label{wr_proof}
In this section we prove Theorem~\ref{main_wr} step by step. Let $K={\mathbb Q}(\sqrt{D})$ for a squarefree integer $D > 1$. Let $I \subseteq {\mathcal O}_K$ be an ideal with the canonical basis $a, b + g\delta$ as described in~\eqref{abg}. Suppose this basis is WR twistable by some totally positive $\alpha = p + q \sqrt{D} \in K$. First assume that $K = {\mathbb Q}(\sqrt{D})$ with $D \not\equiv 1\ (\operatorname{mod} 4)$, then by~\eqref{C1} we must have
$$p = \left( \frac{2bgD}{b^2+g^2D-a^2} \right) q \text{ and } \left| b^2-g^2D+a^2 \right| \leq ab.$$
We should remark that the first identity holds, unless the denominator $b^2+g^2D-a^2 = 0$, which is not possible: if this is the case, then~\eqref{C1} implies that $b=0$, and so $a^2=g^2D$, which contradicts $D$ being squarefree. Since $p$ must be positive, we have $a^2 < b^2+g^2D$. If $b^2 > g^2D$, we have
$$a^2 < (b^2 - g^2D) + a^2 \leq ab,$$
meaning that $a < b$, which contradicts the choice of the canonical basis. Hence we must have $b^2 < g^2D$: equality is not possible since $D$ is squarefree. If $I = {\mathcal O}_K$, then $a=1, b=0, g=1$, and so we must have $|1-D| \leq 0$, which is a contradiction. Hence ${\mathcal O}_K$ cannot have WR twistable canonical basis.
Now suppose that $D \equiv 1\ (\operatorname{mod} 4)$, then by~\eqref{Cd1} we must have
$$p = \left( \frac{2gD (2b + g)}{4b^2+g^2(D+1)+4b-4a^2} \right) q,$$
unless $4b^2+g^2(D+1)+4b-4a^2 = 0$, in which case $2b + g=0$: this is not possible, since $g \mid b$. Additionally,~\eqref{Cd1} implies that
\begin{equation}
\label{Cd-3}
\left| 4a^2 + 4b^2 + 4bg - (D-1)g^2 \right| \leq 2a (2b+g).
\end{equation}
Since $p$ must be positive, we have $a^2 < b^2+b+\frac{g^2(D+1)}{4}$. If $b^2 + bg > \frac{(D-1)g^2}{4}$, then we get
$$4a^2 < 4a^2 + 4b^2 + 4bg - (D-1)g^2 \leq 2a (2b+g),$$
and so $a < b+g/2$, which contradicts the fact that $g \mid a-b$. Hence we must have $b^2 + bg \leq \frac{(D-1)g^2}{4}$, which means that $b < \frac{(\sqrt{D}-1)g}{2}$: again, equality is not possible since $D$ is squarefree. If $I = {\mathcal O}_K$, then $a=1, b=0, g=1$, so $p = \frac{2Dq}{D-3}$. Hence $\alpha = q \left( \frac{2D}{D-3} + \sqrt{D} \right)$, which again is not totally positive unless $D=5$ (otherwise $D \geq 13$, and so $2D/(D-3) < \sqrt{D}$). If $D=5$, then we can take $\alpha = 5 + \sqrt{5}$, obtaining the matrix
$$A(\alpha) B = \begin{pmatrix} \sqrt{5+\sqrt{5}} & \frac{(1-\sqrt{5}) \sqrt{5+\sqrt{5}}}{2} \\ \sqrt{5-\sqrt{5}} & \frac{(1+\sqrt{5}) \sqrt{5-\sqrt{5}}}{2} \end{pmatrix}$$
with orthogonal columns, both of norm $= \sqrt{10}$. Hence ${\mathcal O}_K$ cannot have WR twistable canonical basis for any real quadratic number field $K \neq {\mathbb Q}(\sqrt{5})$.
Finally notice that if $I$ is an ideal with canonical basis $a, b+g\delta$ and $I' = \frac{1}{g} I$ is the corresponding ideal with canonical basis $\frac{a}{g}, \frac{b}{g} + \delta$, then the lattices $L_K(I)$ and $L_K(I')$ are similar, and so are there twists by the same element~$\alpha \in K$. Therefore our upper bounds on $b$ mean that there can be at most finitely many ideals in ${\mathcal O}_K$, up to similarity of the resulting ideal lattices, with the canonical basis being WR twistable.
\medskip
This finishes the proof of Theorem~\ref{main_wr}. Notice that this theorem implies that the canonical basis is rarely WR twistable, however such examples still exist. We demonstrate a couple here.
\begin{ex} \label{ex1} Let $D = 139 \equiv 3\ (\operatorname{mod} 4)$ and $K={\mathbb Q}(\sqrt{139})$. Let $I \subset {\mathcal O}_K$ be an ideal generated by the canonical basis
$$9,\ 7-\sqrt{139},$$
i.e. $g=1$, $b=7 < \sqrt{139}$, $a = 9 \mid {\mathbb N}(b-\sqrt{D}) = 7^2-139 = -90$. This basis is WR twistable by the totally positive element
$$\alpha = \frac{1946}{107} + \sqrt{139} \in K$$
with the resulting WR lattice having the Minkowski reduced basis matrix
$$\frac{1}{107} \begin{pmatrix} 9 \sqrt{208222+11449 \sqrt{139}} & (7 - \sqrt{139}) \sqrt{208222+11449 \sqrt{139}} \\ 9 \sqrt{208222-11449 \sqrt{139}} & (7+\sqrt{139}) \sqrt{208222-11449 \sqrt{139}} \end{pmatrix}$$
and cosine of the angle between these basis vectors being $-1/14$. The common value of the successive minima of this lattice is $\sqrt{\frac{315252}{107}} \approx 54.27964973...$
\end{ex}
\begin{ex} \label{ex2} Let $D = 141 \equiv 1\ (\operatorname{mod} 4)$ and $K={\mathbb Q}(\sqrt{141})$. Let $I \subset {\mathcal O}_K$ be an ideal generated by the canonical basis
$$5,\ 4+\frac{1-\sqrt{141}}{2},$$
i.e. $g=1$, $b=4 < \sqrt{141}/2$, $a = 5 \mid {\mathbb N}\left( b+\frac{1-\sqrt{D}}{2} \right) = \frac{(8+1)^2-141}{4} = -15$. This basis is WR twistable by the totally positive element
$$\alpha = \frac{1269}{61} + \sqrt{141} \in K$$
with the resulting WR lattice having the Minkowski reduced basis matrix
$$\frac{1}{61} \begin{pmatrix} 5 \sqrt{77409+3721 \sqrt{141}} & \left( \frac{9 - \sqrt{141}}{2} \right) \sqrt{77409+3721 \sqrt{141}} \\ 5 \sqrt{77409-3721 \sqrt{141}} & \left( \frac{9 + \sqrt{141}}{2} \right) \sqrt{77409-3721 \sqrt{141}} \end{pmatrix}$$
and cosine of the angle between these basis vectors being $2/9$. The common value of the successive minima of this lattice is $\sqrt{\frac{63450}{61}} \approx 32.25157258...$
\end{ex}
\bigskip
\section{Proof of Theorem~\ref{main_stable}}
\label{stable_proof}
Here we prove Theorem~\ref{main_stable}. As above, let $K={\mathbb Q}(\sqrt{D})$ for a squarefree integer $D > 1$ and let $I \subseteq {\mathcal O}_K$ be an ideal with the canonical basis $a, b + g\delta$ as described in~\eqref{abg}. Suppose this basis is stable twistable by some totally positive $\alpha = p + q \sqrt{D} \in K$. First assume that $K = {\mathbb Q}(\sqrt{D})$ with $D \not\equiv 1\ (\operatorname{mod} 4)$. If we consider the first condition of~\eqref{C2} as a quadratic inequality in $p$, then its leading coefficient is negative unless $b \leq g \sqrt{D/3}$ and it has negative discriminant unless $b \leq 2g\sqrt{D/3}$. Hence the canonical basis can be stable twistable only if
$$b < \frac{2}{\sqrt{3}} g \sqrt{D}.$$
As always, equality is not possible since $b$ is an integer and $D$ is squarefree. Now assume that $I={\mathcal O}_K$, then $a=1,b=0,g=1$, so the inequalities of~\eqref{C2} become:
$$D(p^2-4Dq^2) \geq 0,\ p \geq \sqrt{D(p^2-Dq^2)}.$$
Combining these inequalities, we obtain $(1-3D)p^2 \geq 0$, which is not possible. Hence ${\mathcal O}_K$ cannot have a stable twistable canonical basis.
\medskip
Now suppose that~$D \equiv 1\ (\operatorname{mod} 4)$. Considering the first condition of~\eqref{Cd2} as a quadratic inequality in $p$, we see that its leading coefficient is negative unless $b \leq \frac{g}{2} \left( \sqrt{D/3} - 1 \right)$ and it has negative discriminant unless $b \leq \frac{g}{2} \left( 2 \sqrt{D/3} - 1 \right)$. Hence the canonical basis can be stable twistable only if
$$b < \frac{g}{2} \left( 2 \sqrt{D/3} - 1 \right).$$
Again, equality is not possible since $b$ is an integer and $D$ is squarefree. Now assume that $I={\mathcal O}_K$, then $a=1,b=0,g=1$, so the inequalities of~\eqref{Cd2} become:
$$\frac{p^2}{4} (D-3) + \frac{3}{2} D pq - D^2 q^2 \geq 0,\ 2p > \sqrt{D (p^2-q^2D)}.$$
These inequalities lead to
$$q \left( \frac{D \sqrt{4D-3}-3}{D-3} \right) < p < q \left( \frac{D}{\sqrt{D-4}} \right),$$
which means that we must have $\frac{D \sqrt{4D-3}-3}{D-3} < \frac{D}{\sqrt{D-4}}$. This only holds for $D=5$, which, as we know, yields even a WR twist. By the same reasoning as in the proof of Theorem~\ref{main_wr}, our upper bounds on $b$ mean that there can be at most finitely many ideals in ${\mathcal O}_K$, up to similarity of the resulting ideal lattices, with the canonical basis being stable twistable. This finishes the proof of Theorem~\ref{main_stable}.
\bigskip
Finally, let us demonstrate a couple examples of stable twists of ideal lattice canonical bases that are not WR twistable.
\begin{ex} \label{ex1-1} Let $D = 1327 \equiv 3\ (\operatorname{mod} 4)$ and $K={\mathbb Q}(\sqrt{1327})$. Let $I \subset {\mathcal O}_K$ be an ideal generated by the canonical basis
$$39,\ 38-\sqrt{1327},$$
i.e. $g=1$, $b=38$, $a = 39 \mid {\mathbb N}(b-\sqrt{D}) = 38^2-1327 = 117$. Notice, in particular, that
$$g \sqrt{D} = 36.42801120... < b < 42.06344416... = \frac{2}{\sqrt{3}} g \sqrt{D},$$
hence this basis cannot be WR twistable, by Theorem~\ref{main_wr}. On the other hand, this basis is stable twistable by the totally positive element
$$\alpha = 63 + \sqrt{1327} \in K$$
with the resulting stable lattice having the Minkowski reduced basis matrix
$$\begin{pmatrix} 39 \sqrt{63 + \sqrt{1327}} & (38 - \sqrt{1327}) \sqrt{63 + \sqrt{1327}} \\ 39 \sqrt{63 - \sqrt{1327}} & (38 + \sqrt{1327}) \sqrt{63 - \sqrt{1327}} \end{pmatrix}$$
and cosine of the angle between these basis vectors being $0.4951063950...$. The determinant of this lattice is $146048.2881...$ and values of the successive minima are $\sqrt{147442}$ and $\sqrt{191646}$, respectively.
\end{ex}
\begin{ex} \label{ex1-2} Let $D = 125173 \equiv 1\ (\operatorname{mod} 4)$ and $K={\mathbb Q}(\sqrt{125173})$. Let $I \subset {\mathcal O}_K$ be an ideal generated by the canonical basis
$$183,\ 182 + \frac{1-\sqrt{125173}}{2},$$
i.e. $g=1$, $b=182$, $a = 183 \mid {\mathbb N} \left( \frac{(2b+1) - \sqrt{D}}{2} \right) = 2013$. Notice, in particular, that
$$\frac{g (\sqrt{D}-1)}{2} = 176.3989824... < b < 203.7653503... = \frac{g}{2} \left( \frac{2}{\sqrt{3}} \sqrt{D} - 1 \right),$$
hence this basis cannot be WR twistable, by Theorem~\ref{main_wr}. On the other hand, this basis is stable twistable by the totally positive element
$$\alpha = 611 + \sqrt{125173} \in K$$
with the resulting stable lattice having the Minkowski reduced basis matrix
$$\begin{pmatrix} 183 \sqrt{611 + \sqrt{125173}} & \left( \frac{365}{2} - \frac{\sqrt{125173}}{2} \right) \sqrt{611 + \sqrt{125173}} \\ 183 \sqrt{611 - \sqrt{125173}} & \left( \frac{365}{2} + \frac{\sqrt{125173}}{2} \right) \sqrt{611 - \sqrt{125173}} \end{pmatrix}$$
and cosine of the angle between these basis vectors being $0.4853755919...$. The determinant of this lattice is $32252383.1...$ and values of the successive minima are $\sqrt{33252444}$ and $\sqrt{40923558}$, respectively.
\end{ex}
\medskip
\section{Further remarks on stable and WR twistable bases }
\label{remarks}
Here we include some further heuristic remarks on stable and WR twistable bases beyond the canonical ones discussed above. Let
$${\mathbb H} = \{ x+iy \in {\mathbb C} : y>0 \}$$
be the upper half-plane. Following~\cite{lenny:florian}, define
$${\mathcal D} := \{ \tau = a + bi \in {\mathbb H} : -1/2 < a \leq 1/2, |\tau| \geq 1 \}$$
and
$${\mathcal F} := \{ \tau = a + bi \in {\mathbb H} : 0 \leq a \leq 1/2, |\tau| \geq 1 \},$$
so ${\mathcal F}$ is ``half" of ${\mathcal D}$. Then ${\mathcal D}$ is the standard fundamental domain of ${\mathbb H}$ under the action of $\operatorname{SL}_2({\mathbb Z})$ by fractional linear transformations and ${\mathcal F}$ is the space of similarity classes of planar lattices. As in~\cite{lenny:florian}, this can be illustrated by Figure~\ref{fig:domain} with the subsets of WR and stable similarity classes marked accordingly.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{domain.png}
\caption{Similarity classes of lattices in ${\mathbb R}^2$ with WR and stable subregions marked by colors.}\label{fig:domain}
\end{figure}
Let $K$ be a real quadratic field and $I \subseteq {\mathcal O}_K$ an ideal. For each lattice $(L_K(I),f_{{\mathcal A}(\alpha)})$ for a totally positive $\alpha \in K$, let us write $\left< L_K(I),f_{{\mathcal A}(\alpha)} \right>$ for its similarity class. As shown in~\cite{bayer-nebe}, sets of similarity classes of the form
$${\mathcal G}(I):= \left\{ \left< L_K(I),f_{{\mathcal A}(\alpha)} \right> : \alpha \in K \textrm{ totally positive} \right\}$$
correspond to closed geodesics in ${\mathbb H}/\operatorname{SL}_2({\mathbb Z})$, and every closed geodesic will necessary intersect the WR locus ${\mathcal W}$ in the upper-half plane (the blue arc in Figure~\ref{fig:domain}); denote by ${\mathcal I} := {\mathcal W} \cap {\mathcal G}(I)$ the set of such intersection points.
The set ${\mathcal I}$ is completely defined by a relation described in~\cite{taoufiq}. Let $B=\{x,y\}$ be a basis for the ideal $I$, and define
$$F(B) = {\mathbb N}(x)^2+{\mathbb N}(y)^2 + {\mathbb N}(x) {\mathbb N}(y)- {\mathbb N}(I)^2 \Delta_K /4,$$
where $\Delta_K$ is the discriminant of the number field $K$. We say that two bases $B$ and $B'$ are {\it equivalent} if $F(B)= F(B')$: this is indeed an equivalence relation. Then the set ${\mathcal I}$ is in bijective correspondence with the equivalence classes of bases $B$ with $F(B) < 0$. This implies finiteness of $|{\mathcal I}|$. Furthermore, as the geodesics ${\mathcal G}(I)$ are closed, the number of arcs ``inside" the stable locus is at most twice $|\mathcal{I}|$, except when the only WR twist of our ideal lattice is orthogonal as in Figure~\ref{WR-orth}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{D5.jpg}
\caption{Geodesic ${\mathcal G}(I)$ (blue) intersects the WR locus (red) in one point (orthogonal lattice), which is also a stable twist.}\label{WR-orth}
\end{figure}
For example, if $I={\mathcal O}_K$ this will happen if and only if $D=s^2 +1$ (respectively $D=s^2 +4$) for $D \not\equiv 1\pmod 4$ (respectively $D \equiv 1 \pmod 4$). In general, this will be the case if for every basis $B=\{x,y\}$ of $I$ such that ${\mathbb N}(x) < {\mathbb N}(I) \sqrt{\Delta_K}/4$, we have ${\mathbb N}(x) = -{\mathbb N}(y)$.
Figure~\ref{WR-59} illustrates an example of the intersection of ${\mathcal G}(I)$ with the WR locus for $I={\mathcal O}_K$, where $K={\mathbb Q}(\sqrt{59})$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{D59.jpg}
\caption{Geodesic ${\mathcal G}({\mathcal O}_{{\mathbb Q}(\sqrt{59})})$ intersects the WR locus.}\label{WR-59}
\end{figure}
Before intersecting the WR locus, the geodesic ${\mathcal G}(I)$ crosses ``continuously" the stable locus, which yields infinitely many stable twists: in other words, this insures the existence of infinitely many totally positive $\alpha \in K$ such that ${\mathcal A}(\alpha)B$ is stable for a fixed WR twistable basis.
In summary, every WR twistable basis will give rise to infinitely many stable twists, except in the cases where the ideal $I$ admits only the orthogonal WR twist. The converse is not true: there exist bases that are stable twistable, but not WR twistable, as demonstrated by Examples~\ref{ex1-1} and~\ref{ex1-2}.
\bigskip
\section{Applications of WR and stable lattice twists}
\label{comm_theory}
In this section we discuss some applications of WR and stable twistable lattice bases. First we highlight a connection between such bases and the theory of error control in wireless communication. We assume a single-input single-output (SISO) Rayleigh flat fading channel model:
\begin{equation}
\label{model}
{\boldsymbol y} = H{\boldsymbol x} + {\boldsymbol v},
\end{equation}
where ${\boldsymbol x}$ is the transmitted codeword taken from some finite codebook in ${\mathbb C}^n$, $H = \operatorname{diag}(h_1,\dots, h_n)$ describes the random channel response, and ${\boldsymbol v} \in {\mathbb C}^n$ is a random additive white Gaussian noise with variance $\sigma_{{\boldsymbol v}}^2$.
To study the communication reliability of a given code $C$ we consider the codeword error probability $P_e(C)$. The goal is to choose $C$ to be a subset of a lattice that minimizes $P_e(C)$. Considering $\Lambda \subset {\mathbb R}^n$, it is proved in \cite{oggier} that the lattices (of fixed volume) that minimize $P_e(C)$ for Rayleigh fading channel are those with maximal $d_{\min}(\Lambda)$, where
$$d_{\min}(\Lambda) = \min_{{\boldsymbol x} \in \Lambda \setminus \{{\boldsymbol 0}\}} \prod_{i=1}^n |x_i|.$$
This criterion is restricted to the so called fully diverse lattices, i.e. the lattices with non-vanishing~$d_{\min}(\Lambda)$. Let us prove that the existence of twistable bases allows one to restrict this optimization problem to the set of WR or stable lattices without loss of generality.
\begin{prop}
Let $\Lambda$ be a lattice with maximal $d_{\min}(\Lambda)$ in its dimension. Then there exists a WR lattice $L$ and a stable lattice $M$, such that
$$d_{\min}(\Lambda)=d_{\min}(L)=d_{\min}(M).$$
\end{prop}
\proof
Let $\Lambda\subset {\mathbb R}^n$ be a fully diverse lattice. Notice that $d_{\min}(\Lambda)$ is invariant under the action of ${\mathcal A}$, i.e $d_{\min}(A \Lambda) = d_{\min}(\Lambda)$ for any $A \in {\mathcal A}$. In \cite{mcmullen} Mcmullen showed that if the orbit closure $\overline{{\mathcal A}\Lambda}$ is compact then ${\mathcal A}\Lambda$ meets the set of WR lattices. Hence we only need to show that the full diversity of $\Lambda$ will ensure the compactness of $\overline{{\mathcal A}.\Lambda}$, and this is a straightforward application of the Mahler compactness criterion. Namely, for a set $E$ of unimodular lattices we have
$$\overline{E}\textrm{ is compact} \iff \lambda_1(L)>0 \textrm{ for all } L\in E.$$
If $\Lambda$ is a fully diverse lattice, then $d_{\min}(\Lambda) > 0$. Hence, by the AM-GM inequality, we have
$$0 < \sqrt{n} \prod_{i=1}^{n} |x_i| \leq \|{\boldsymbol x}\|$$
for any ${\boldsymbol x} \in \Lambda$. In particular, taking ${\boldsymbol x} \in \Lambda$ such that $\|{\boldsymbol x}\| = \lambda_1(\Lambda)$, we see that $\lambda_1(L) > 0$ for any $L\in {\mathcal A} \Lambda$.
For stable lattices the argument is the same replacing the result of~\cite{mcmullen} with the analogous result of~\cite{weiss} which guarantees that if the orbit closure $\overline{{\mathcal A} \Lambda}$ is compact then ${\mathcal A}\Lambda$ meets the set stable lattices.
\endproof
\begin{rem} Notice that $d_{\min}(\Lambda)>0$ for $\Lambda$ arising from a real number field. Margulis conjectured that the converse is also true. More precisely, if a lattice $\Lambda \subset {\mathbb R}^n$ for $n \geq 3$ has $d_{\min}(\Lambda) >0$ then $\Lambda$ comes from an order in a number field.
In other words, Margulis's conjecture asserts that all fully diverse lattices come from orders in totally real number fields, and this motivates the choice of number theoretic constructions in this context.
\end{rem}
\medskip
Let us also briefly discuss the use of WR and stable twists in the arithmetic theory. Let $K$ be a real number field of degree~$n$. The Euclidean minimum of $K$ is defined as
$$M(K) := \inf \left\{ \alpha \in {\mathbb R}_{>0} : \forall x \in K~\exists y \in {\mathcal O}_K \textrm{ such that } |{\mathbb N}(x-y)| \leq \alpha \right\}.$$
The Euclidean minimum measures how far is $K$ from having a Euclidean algorithm. In fact, if $M(K) < 1$ then ${\mathcal O}_K$ is a Euclidean ring. Furthermore, for an ideal $I$ in ${\mathcal O}_K$ we can define
$$M(I) := \inf \left\{ \alpha \in {\mathbb R}_{>0} : \forall x \in K~\exists y \in I \textrm{ such that } |{\mathbb N}(x-y)| \leq \alpha \right\}.$$
Now, for a lattice $\Lambda \subset {\mathbb R}^n$, its covering radius is
$$\mu(\Lambda) := \sup_{{\boldsymbol y} \in {\mathbb R}^n} \min \left\{ \| {\boldsymbol x} - {\boldsymbol y} \| : {\boldsymbol x} \in \Lambda \right\}$$
and the Hermite thickness of $\Lambda$ is
$$\tau(\Lambda) := \frac{\mu(\Lambda)^2}{\operatorname{det}(\Lambda)^{1/n}}.$$
Then for an ideal $I \subseteq {\mathcal O}_K$, define
$$\tau_{\min}(I):= \min\{\tau(L_K (I),f_{A(\alpha)}) : \alpha \in K~\textrm{totally positive}\}.$$
\begin{thm} [\cite{bayer}]
For all number fields $K$ of degree $n$,
$$M(I) \leq \left( \frac{\tau_{\min}(I)}{n} \right)^{n/2} \sqrt{|\Delta_K|}\ {\mathbb N}(I),$$
where $\Delta_K$ is the discriminant of $K$.
\end{thm}
This theorem motivates the study of the covering radii in the orbit of the action of ${\mathcal A}_1(K)$ on $L_K (I)$.
Furthermore, it has been proved that $\mu(\Lambda) \leq \frac{\sqrt{n}}{2}$ for any unimodular WR lattice of dimension $n\leq 9$ (not true for $n\geq 30$), and it is conjectured to be true for stable lattices in any dimension.
Combining these observations, we have
$$M(K) \leq \frac{\sqrt{|\Delta_K|}}{4}$$
when $K$ is a real quadratic number field. Notice that this bound automatically implies that $\mathbb{Q}(\sqrt{d})$ is a Euclidean domain for $d=5,2,3,13$.
\medskip
{\bf Acknowledgment.} We would like to thank Laia Amoros, Camilla Hollanti and David Karpuk for many helpful conversations on the subject of this paper. A part of the work was done during a stay of the first author at Claremont McKenna College (CMC): M.T. Damir would like to thank CMC for its hospitality. We also thank the anonymous referee for some helpful suggestions.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,118 |
PSG's defense was unable to contain Lille forward Nicolas Pepe, who scored the second goal and then set up the fourth and fifth to underline Lille's unselfish teamwork.
PSG needs a win at Nantes on Wednesday to guarantee the league title, although a draw would likely be enough at the end of the season given PSG's huge advantage on goal difference.
"You can lose, but you must lose in a certain manner - we cannot lack character like we did here", said the France worldwide.
He told reporters: "It's too much, we miss so many players, it's not possible like that".
"It's not normal. We lack personality, it's one of our flaws, we have to correct this". "We made a good start to the match, with four goals, two for us and two offside".
"Today we played like amateurs".
"They didn't have any chance in the first half, and there was Kehrer's opportunity in the beginning of second half. To win here, we would have needed more quality".
"Nobody said anything before because we were winning". | {
"redpajama_set_name": "RedPajamaC4"
} | 7,610 |
import ornicar.scalalib
package object chess
extends scalalib.OrnicarValidation
with scalalib.OrnicarCommon
with scalalib.OrnicarNonEmptyLists
with scalaz.NonEmptyLists
with scalaz.Strings
with scalaz.Lists
with scalaz.Booleans {
val White = Color.White
val Black = Color.Black
type Direction = Pos ⇒ Option[Pos]
type Directions = List[Direction]
object implicitFailures {
implicit def stringToFailures(str: String): Failures = str wrapNel
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,326 |
Cybocephalus antonius is een keversoort uit de familie Cybocephalidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1962 door Endrödy-Younga.
Cybocephalidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 563 |
\section{Introduction}
Numerical relativity is an essential tool to study many processes involving
strong gravitational fields. In four space-time dimensions, processes of this
sort, such as black hole (BH) binary evolutions, are of utmost importance for
understanding the main sources of gravitational waves, which are expected to be
detected by the next generation of ground based [Laser Interferometer
Gravitational-Wave Observatory (LIGO), VIRGO] and space based [Laser
Interferometer Space Antenna (LISA)] interferometers. Long-term stable numerical
evolutions of BH binaries have finally been achieved after four decades of
efforts \cite{Pretorius:2005gq,Campanelli:2005dd,Baker:2005vv}. The numerical
modelling of generic spinning BH binaries in vacuum Einstein gravity is an
active field of research, with important consequences for gravitational wave
detection in the near future.
Numerical relativity in a higher dimensional space-time, instead, is an
essentially unexplored field, with tremendous potential to provide answers to
some of the most fundamental questions in physics. Recent developments in
experimental and theoretical physics make this a pressing issue. We refer, in
particular, to the prominent role of BHs in the gauge-gravity duality, in
TeV-scale gravity or even on their own as solutions of the field
equations. These are some of the most active areas of current research in
gravitational and high energy physics.
\subsection{Motivation}
\renewcommand{\theenumi}{\textit{\roman{enumi}}}
\begin{enumerate}
\item \label{item:1} \emph{AdS/CFT and holography.} In 1997--98, a powerful new
technique known as the AdS/CFT correspondence or, more generally, the
gauge-string duality, was introduced and rapidly developed
\cite{Maldacena:1997re}. This holographic correspondence provides an effective
description of a non-perturbative, strongly coupled regime of certain gauge
theories in terms of higher-dimensional classical gravity. In particular,
equilibrium and non-equilibrium properties of strongly coupled thermal gauge
theories are related to the physics of higher-dimensional BHs, black branes
and their fluctuations. These studies revealed intriguing connections between
the dynamics of BH horizons and hydrodynamics \cite{Son:2007vk}, and offer new
perspectives on notoriously difficult problems, such as the BH information
loss paradox, the nature of BH singularities or quantum gravity.
Numerical relativity in anti-de Sitter backgrounds is bound to contribute
enormously to our understanding of the gauge-gravity duality and is likely to
have important applications in the interpretation of observations
\cite{Mateos:2007ay,Hartnoll:2009sz,Amsel:2007cw,Gubser:2008pc}. For
instance, in the context of the gauge-gravity duality, high energy collisions
of BHs have a dual description in terms of \textit{a)}~high energy collisions
with balls of de-confined plasma surrounded by a confining phase and
\textit{b)}~the rapid localised heating of a de-confined plasma. These are the
type of events that may have direct observational consequences for the
experiments at Brookhaven's Relativistic Heavy Ion Collider (RHIC)
\cite{Amsel:2007cw,Gubser:2008pc}. Numerical relativity in anti-de Sitter is
notoriously difficult, and so far only very special situations have been
handled \cite{Pretorius:2000yu,Witek:2010qc}. The phenomenologically most
interesting case is a five dimensional space-time, $AdS_5$, and therefore the
higher dimensional extension of numerical relativity is necessary.
\item \label{item:2} \emph{TeV-scale gravity scenarios.} An outstanding problem
in high energy physics is the extremely large ratio between the four
dimensional Planck scale, $10^{19}$ GeV, and the electroweak scale, $10^2$
GeV. It has been proposed that this \textit{hierarchy problem} can be resolved
if one adopts the idea that the Standard Model is confined to a brane in a
higher dimensional space, such that the extra dimensions are much larger than
the four dimensional Planck scale (they may be large up to a sub-millimetre
scale) \cite{Antoniadis:1990ew,ArkaniHamed:1998rs,Antoniadis:1998ig}. In a
different version of the model, the extra dimensions are infinite, but the
metric has an exponential factor introducing a finite length scale
\cite{Randall:1999ee,Randall:1999vf}.
In such models, the fundamental Planck scale
could be as low as 1 TeV. Thus, high energy colliders, such as the Large
Hadron Collider (LHC), may directly probe strongly coupled gravitational
physics \cite{Argyres:1998qn,Banks:1999gd,Giddings:2001bu,Dimopoulos:2001hw,
Ahn:2002mj,Chamblin:2004zg}. In fact, such tests may even be routinely
available in the collisions of ultra-high energy cosmic rays with the Earth's
atmosphere \cite{Feng:2001ib,Ahn:2003qn,Cardoso:2004zi}, or in astrophysical
BH environments \cite{Banados:2009pr,Berti:2009bk,Jacobson:2009zg} (for
reviews see \cite{Cavaglia:2002si,Kanti:2004nr,Kanti:2008eq}). From Thorne's
hoop conjecture it follows that, in this scenario, particle collisions could
produce BHs \cite{Giddings:2001bu,Dimopoulos:2001hw}. Moreover, the production
of BHs at trans-Planckian collision energies (compared to the fundamental
Planck scale) should be well described by using classical general relativity
extended to $D$~dimensions
\cite{Banks:1999gd,Giddings:2001bu,Dimopoulos:2001hw,Feng:2001ib,Ahn:2003qn,
Ahn:2002mj,Chamblin:2004zg,Cardoso:2004zi,Cavaglia:2002si,Kanti:2004nr,
Kanti:2008eq,Solodukhin:2002ui,Hsu:2002bd}. The challenge is then to use
the classical framework to determine the cross section for production and, for
each initial setup, the fractions of the collision energy and angular momentum
that are lost in the higher dimensional space by emission of gravitational
waves. This information will be of paramount importance to improve the
modelling of microscopic BH production in event generators such as
\textsc{Truenoir}, \textsc{Charybdis2}, \textsc{Catfish} or \textsc{Blackmax}
\cite{Dimopoulos:2001hw,Frost:2009cf,Cavaglia:2006uk,Dai:2007ki,Dai:2009by}.
The event generators will then provide a description of the corresponding
evaporation phase, which might be observed during LHC collisions.
The first models for BH production in parton-parton collisions used a simple
black disk approach to estimate the cross section for production
\cite{Giddings:2001bu,Dimopoulos:2001hw}. Improved bounds have been obtained
using either trapped surface methods to estimate the cross section for BH
production
\cite{Eardley:2002re,Kohlprath:2002yh,Yoshino:2002br,Yoshino:2002tx} or
approximation schemes \cite{D'Eath:1992hb,D'Eath:1992hd,D'Eath:1992qu,
Cardoso:2002ay,Berti:2003si,Cardoso:2005jq} to evaluate the gravitational
energy loss. Only recently exact results for highly relativistic collisions
where obtained in four dimensions, using numerical relativity techniques
\cite{Sperhake:2008ga,Shibata:2008rq,Sperhake:2009jz}. No such exact results
are yet available in the higher dimensional case. To obtain them is one of our
main goals and the present paper introduces a formalism to achieve that.
\item \label{item:3} \emph{Higher dimensional black holes.} Asymptotically flat
higher dimensional black objects have a much richer structure than their four
dimensional counterparts. For instance, spherical topology is not the only
allowed topology for objects with a horizon. One can also have, \emph{e.g.},
black rings, with a donut-like topology. Remarkably, these two different
horizon topologies coexist for certain regions in phase-space
\cite{Emparan:2008eg}. The stability of general higher-dimensional BHs is now
starting to be explored. Generically it has been conjectured that for $D\ge
6$ ultra-spinning Myers-Perry BHs will be unstable \cite{Emparan:2003sy}. This
instability has been confirmed by an analysis of linearised axi-symmetric
perturbations in $D=7,8,9$ \cite{Dias:2009iu}. Clearly, the study of the
non-linear development of these instabilities requires numerical methods, such
as the ones presented herein. A study of this type was very recently presented
for a non axi-symmetric perturbation in $D=5$ \cite{Shibata:2009ad}, where it
was found that a single spinning five dimensional Myers-Perry BH is unstable,
for sufficiently large rotation parameter (thereby confirming previous
conjectures \cite{Cardoso:2006sj,Cardoso:2009bv,Cardoso:2009nz}).
Not much is known about general equilibrium states in anti-de Sitter
backgrounds. The gauge-gravity duality and the hydrodynamic limit have been
used to predict the existence of larger classes of BHs in anti-de Sitter
backgrounds, including non axi-symmetric solutions
\cite{Cardoso:2009bv,Cardoso:2009nz}. However, these have not yet been found.
\end{enumerate}
Finally, there are issues of principle, as for example testing Cosmic Censorship
in BH collisions \cite{Sperhake:2008ga,Sperhake:2009jz} which require
state-of-the-art numerical simulations.
\subsection{Space-times with symmetries}
From what has been said, the extension of four dimensional numerical Relativity
is mandatory. Some pioneering works have been concerned with the non-linear
development of the Gregory-Laflamme instability \cite{Gregory:1993vy} of cosmic
strings \cite{Choptuik:2003qd} and gravitational collapse, with spherical
symmetry \cite{Sorkin:2009bc}, axial symmetry \cite{Sorkin:2009wh} or even
static situations \cite{Headrick:2009pv}. Another numerical code, based on the
cartoon method \cite{Alcubierre:1999ab}, was developed and tested for five
space-time dimensions in Ref.~\cite{Yoshino:2009xp}. See also
Ref.~\cite{Nakao:2009dc} for a discussion of slicings of $D$ dimensional black
holes. The (phenomenologically) most interesting large extra dimensions models
are, however, in higher than five space-time dimensions (see for instance
\cite{Kanti:2004nr}). Moreover, the ultra-spinning instabilities of Myers-Perry
BHs should occur in $D\ge 6$. Thus, our approach here is to develop a framework
and a numerical code that can, in principle, be applied to different space-time
dimensions with little adaptations. This may be achieved by taking the
$D$~dimensional vacuum space-time to have an isometry group fit to include a
large class of interesting problems. If this isometry group is sufficiently
large, it allows a dimensional reduction of the problem to 3+1 dimensions,
wherein it appears as (four dimensional) general relativity coupled to some
\textit{quasi-matter} terms.\footnote{Hereafter, we dub the source terms of the
lower dimensional Einstein equations as \textit{quasi-matter}, since its
energy-momentum tensor is not that of canonical matter.} Thus, the different
space-time dimension manifests itself only in the different quasi-matter content
of the four dimensional theory. We emphasise, in this context, that full blown
$4+1$, $5+1$, \emph{etc.\ }numerical simulations without symmetry are currently
not possible due to the computational costs, so that our approach pushes
numerical relativity in higher dimensions to the outmost practical limits of the
present time. Moreover, an obvious advantage of this approach is that we can
use existing codes with small adaptations: the four dimensional equations need
to be coupled to the appropriate quasi-matter terms and some issues related to
the chosen coordinates must be addressed, as we shall see. Finally, the lessons
learnt in treating our effective gravity plus quasi-matter system might be of
use in dealing with other four dimensional numerical relativity problems with
sources.
\subsection{Axial symmetry $SO(D-2)$ and $SO(D-3)$}
\label{sec:axial-symmetry}
We consider two classes of models, which are generalisations of axial symmetry
to higher dimensional space-times: a $D\ge5$ dimensional vacuum space-time with
an $SO(D-2)$ isometry group, and a $D\ge6$ dimensional vacuum space-time with an
$SO(D-3)$ isometry group. The former class allows studies of head-on collisions
of non-spinning BHs. In order to end up with a $3+1$ dimensional model we use,
however, only part of this symmetry: we perform a dimensional reduction by
isometry on a $(D-4)$-sphere which has an $SO(D-3) \subset SO(D-2)$ isometry
group. The latter class allows to model BH collisions with impact parameter and
with spinning BHs, as long as all the dynamics take place on a single
plane.\footnote{This follows from the fact that the angular momenta of the black
holes are parallel to the orbital angular momentum.} In this case we perform a
dimensional reduction by isometry on the entire $SO(D-3)$ isometry group. This
class includes the most interesting physical configurations relevant to
accelerator---and cosmic ray---physics (in the context of TeV-scale gravity),
and to the theoretical properties of higher-dimensional black objects (such as
stability and phase diagrams).
We formulate the evolution equations in the Baumgarte, Shapiro, Shibata and
Nakamura (BSSN) formulation \cite{Shibata:1995we,Baumgarte:1998te}, together
with the moving puncture approach \cite{Campanelli:2005dd,Baker:2005vv}. This
is known to provide a stable evolution scheme for vacuum solutions in four
dimensions, and therefore it is the natural framework for our Einstein plus
quasi-matter system. The quasi-matter terms however, exhibit a problem for
numerical evolution, well known from other numerical studies using coordinates
adapted to axial symmetry, which is sourced by the existence of a coordinate
singularity at the axis. In our formulation, this problem appears when a certain
3+1 dimensional Cartesian coordinate vanishes, $y=0$. We present a detailed
treatment of this problem, introducing first regular variables, then analysing
one by one all potentially pathological terms in our evolution equations and
finally presenting a method to heal all of them. The resulting equations have no
further (obvious) problems for numerical evolution and could, in principle, be
implemented in any working 3+1 dimensional numerical relativity code.
Here we present numerical results using the \textsc{Lean} code
\cite{Sperhake:2006cy}, developed by one of us. We stress that the formalism
developed here is valid in general $D$. However, long term stable evolutions
typically require some experiments with free parameters in the gauge conditions
and also possibly with constraint damping. For $D=5$ we show that, if
appropriate gauge conditions are chosen, the numerical evolution for
Brill-Lindquist initial data describing a single BH is stable and the
constraints are preserved in the evolution, within numerical error. As another
test, we evolve the same initial data in a geodesic slicing gauge. This gauge is
inappropriate for a long term evolution; but it allows us to compare the
numerical evolution with the analytic solution for a single Tangherlini BH in
$D=5$. We find excellent agreement between the two. We also present some
preliminary results for $D=6$.
This paper is organised as follows. In Section~\ref{sec:3+1-dimens}, we discuss
the $D$ dimensional ansatz, perform the dimensional reduction by isometry,
perform the Arnowitt-Deser-Misner (ADM) split and present the BSSN formulation
of our equations. In Section~\ref{sec:initial-data}, the construction of
Brill-Lindquist initial data in $D$ dimensions is discussed and a procedure to
match it to our $3+1$ formulation is given. In
Section~\ref{sec:numerical-treatment} we present the numerical treatment and
results. We draw our conclusions and discuss implications of our results for
future work in Section~\ref{sec:final-remarks}. A considerable part of the
technical details for the numerical treatment is organised into three
appendices. In Appendix~\ref{axis} we motivate and discuss the introduction of
regular variables at $y=0$ and present all relevant equations in terms of these
variables. In Appendix~\ref{troubleterms} we explain how to tackle all the
problematic terms at $y=0$ in these equations. Finally, in
Appendix~\ref{geoslice}, we discuss the construction of the geodesic slicing
which is used to compare analytical with numerical results.
\section{The effective 3+1 dimensional system}
\label{sec:3+1-dimens}
The starting point of the formalism used here is a dimensional reduction from
$D$~dimensional general relativity in vacuum to a four dimensional model. The
isometry group of $D$~dimensional Minkowski space-time is $ISO(1,D-1)$;
solutions of general relativity (or of other metric theories of gravity)
generically break this symmetry into a subgroup. For instance, the isometry
group of a Schwarzschild (or, for $D>4$, Tangherlini \cite{Tangherlini:1963bw})
BH is $SO(D-1)\times \mathbb{R}$, whereas for a head-on collision of two
non-rotating BHs it is $SO(D-2)$: indeed, neither the time direction nor the
direction of the collision correspond to symmetries, but a rotation of the
remaining $D-2$ spatial directions leaves the space-time invariant. The total
space-time can then be considered as the semi-direct product of a three
dimensional space-time ${\cal N}$ with the sphere $S^{D-3}=SO(D-2)/SO(D-3)$. A
coordinate system for ${\cal N}$ can be given, for example in the case of a
head-on collision of two BHs, by the time $t$, the coordinate $z$ along the
collision axis, and the distance from that axis.
One can take advantage of this symmetry to reduce the space-time
dimensionality. This can be accomplished by writing Einstein's equations in
$D$~dimensions in a coordinate system which makes the symmetry manifest,
allowing for a lower dimensional interpretation of the $D$~dimensional
Einstein's equations (in the spirit of Kaluza-Klein reduction). We remark,
however, that we are not performing a compactification; rather, we perform a
dimensional reduction by isometry, as first proposed by Geroch
\cite{Geroch:1970nt}. The extra dimensions manifest themselves in the lower
dimensionality as a source of Einstein's equations, defined on the lower
dimensional manifold.
In principle, one could use the symmetry in a more na\"ive way, assuming that
the solution does not depend on the coordinates parameterizing the sphere and
simply evolving the relevant components of the $D$~dimensional Einstein's
equations. The perspective provided by dimensional reduction, however, has two
advantages: \textit{(i)} all quantities have a geometrical interpretation, and
this allows for a deeper understanding of the problem and a better control of
the equations; \textit{(ii)} it is possible to use, with minor modifications,
the numerical codes which have already been written to implement Einstein's
equations in a four dimensional space-time. Therefore, we do not use the entire
$SO(D-2)$ symmetry of the process, but only a $SO(D-3)$ subgroup. This reduces
the space-time on a $(D-4)$-sphere and yields a four dimensional manifold.
In the original proposal of Geroch \cite{Geroch:1970nt} the symmetry space was
$SO(2)$. This approach has been applied to numerical relativity, see for
instance \cite{Sjodin:2000zd,Sperhake:2000fe,Choptuik:2003as}; a five
dimensional extension, with the same symmetry space, has been derived in
\cite{Chiang:1985rk}. A generalisation to coset manifolds (like the sphere
$S^n$) was given by Cho in \cite{Cho:1986wk,Cho:1987jf}, but in these papers the
complete form of Einstein's equations was not presented. Here we provide the
explicit form of Einstein's equations for symmetry spaces $S^n$ together with
their numerical implementation.
\subsection{$4+(D-4)$ split}
We now describe in detail the reduction from $D$~to $4$ dimensions. In order to
highlight the particular classes of BH binaries we are able to study with this
framework, it is convenient to begin this discussion with the isometry group of
the $S^{D-3}$ sphere, i.~e.~with the $3+(D-3)$ split.
A general $D$~dimensional space-time metric may be written in the form
\begin{equation}
d\hat{s}^2=\hat{g}_{MN}dx^Mdx^N=g_{\bar{\mu}\bar{\nu}}(x^M)dx^{\bar{\mu}}
dx^{\bar{\nu}}+\Omega_{\bar{i}\bar{j}}(x^M)
\left(dx^{\bar{i}}-A_{\bar{\mu}}^{\bar{i}}(x^M)dx^{\bar{\mu}}
\right)\left(dx^{\bar{j}}-A_{\bar{\nu}}^{\bar{j}}(x^M)
dx^{\bar{\nu}}\right) \,,
\end{equation}
where we have split the space-time coordinates as
$x^M=(x^{\bar{\mu}},x^{\bar{i}})$; $M,N=0,\dots, D-1$ are space-time indices,
$\bar{\mu},\bar{\nu}=0,1, 2$ are three dimensional indices and
$\bar{i},\bar{j}=3,\dots D-1$ are indices in the remaining $D-3$ dimensions. We
may think of the space-time as a fibre bundle; $\{x^{\bar{i}}\}$ are coordinates
along the fibre and $\{x^{\bar{\mu}}\}$ are coordinates on the base space.
We are interested in studying $D$~dimensional space-times with an $SO(D-2)$
isometry group. This is the isometry group of the $S^{D-3}$ sphere, which
justifies why we are performing a $3+(D-3)$ splitting of the $D$~dimensional
space-time. Thus, we assume that $\xi_a$, $a=1,\dots, (D-3)(D-2)/2$, are Killing
vector fields,
\begin{equation}
\mathcal{L}_{\xi_a} \hat{g}_{MN}=0 \,,
\label{kvf}
\end{equation}
with Lie algebra
\begin{equation}
\left[\xi_a,\xi_b\right]=\epsilon_{ab}{}^c\xi_c \, ,
\label{algebra}
\end{equation}
where $\epsilon_{ab}{}^{c}$ are the structure constants of $SO(D-2)$. Because
the fibre has the minimal dimension necessary to accommodate $(D-3)(D-2)/2$
independent Killing vector fields, we may assume without loss of generality that
the Killing vector fields have components exclusively along the fibre:
$\xi_a=\xi_a^{\bar{i}}\partial_{\bar{i}}$. Furthermore, we may normalise the
Killing vectors so that they only depend on the coordinates of the fibre,
\emph{i.e.\ }$\partial_{\bar\mu}\xi_a^{\bar i}=0$. Then Eq.~\eqref{kvf} gives
the following conditions
\begin{align}
\label{Omega}
\mathcal{L}_{\xi_a} \Omega_{\bar{i}\bar{j}} & =0 \, , \\
\label{comm}
\mathcal{L}_{\xi_a} A_{\bar{\mu}}^{\bar{i}} & =0 \, , \\
\label{g}
\mathcal{L}_{\xi_a} g_{\bar{\mu}\bar{\nu}} & =0 \, .
\end{align}
These expressions can be interpreted either as Lie derivatives of
rank-$2$ tensors defined on the $D$~dimensional space-time, or as Lie
derivatives of a rank-$2$ tensor, a vector and a scalar, which are
defined on $S^{D-3}$.
Conditions \eqref{Omega}-\eqref{g} have the following implications:
\begin{enumerate}
\item[]
\begin{equation}
\Omega_{\bar{i}\bar{j}}=
f(x^{\bar{\mu}})h_{\bar{i}\bar{j}}^{S^{D-3}} \, ,
\end{equation}
because, from \eqref{Omega}, $\Omega_{\bar{i}\bar{j}}$ admits the maximal number of Killing vector fields and thus must be the metric on a maximally
symmetric space at each $x^{\bar{\mu}}$. Due to \eqref{algebra} this space must be the
$S^{D-3}$ sphere.
$h_{\bar{i}\bar{j}}^{S^{D-3}}$ denotes the metric on an $S^{D-3}$ with unit
radius;
\item[]
\begin{equation}
g_{\bar{\mu}\bar{\nu}}=
g_{\bar{\mu}\bar{\nu}}(x^{\bar{\mu}}) \, ,
\end{equation}
because the Killing vector fields $\xi_a$ act transitively on the fibre
and therefore the base space metric must be independent of the fibre
coordinates;
\item[]
\begin{equation}
A_{\bar{\mu}}^{\bar{i}}=0\, ,
\label{novectors}
\end{equation}
because Eq.~(\ref{comm}) is equivalent to
\begin{equation}
[\xi_a,A_{\bar\mu}]=0\label{comm1} \ ,
\end{equation}
and there exist no non-trivial vector fields on $S^{D-3}$ for $D\ge 5$
that commute with all Killing vector fields on the sphere.
\end{enumerate}
We remark that \eqref{comm1} corresponds to the statement, expressed in
\cite{Cho:1986wk} in group theoretical language, that the gauge group for a
theory reduced on a coset space $G/H$ is the normaliser of $H$ in $G$; in the
case of a sphere, where $G=SO(D-2)$ and $H=SO(D-3)$, the normaliser vanishes and
then there are no ``gauge vectors'', \emph{i.e.}, no non-vanishing metric
components $g_{\bar\mu\bar i}$. If the normaliser of $H$ in $G$ is
non-vanishing, such metric components appear, and with dimensional reduction
they yield vector fields which contribute to the stress-energy tensor in the
reduced theory. For example, in the case of head-on collision, if $D=4$, the
isometry space is $SO(2)$ and the quasi-matter of the reduced theory consists of
a scalar field and of a vector field (as in \cite{Geroch:1970nt} and in
\cite{Sjodin:2000zd,Sperhake:2000fe,Choptuik:2003as}); if $D>4$, the isometry
space is $SO(D-2)/SO(D-3)$, and the quasi-matter of the reduced theory consists
of a single scalar field. In the remainder of this work we focus on this
subclass of space-times, which already contains a vast class of physically
relevant problems, and postpone a discussion of the general case
with $A_{\bar{\mu}}^{\bar{i}}\neq0$ (\emph{i.e.}, with $g_{\bar\mu\bar i}\neq0$)
to future work.
In practice, we are actually interested in performing a $4+(D-4)$ split of the
$D$ dimensional space-time. This may be done as follows. The metric on a unit
$S^{D-3}$ may always be written in terms of the line element on a unit
$S^{D-4}$, denoted by $d\Omega_{D-4}$, as follows,
\begin{equation}
h_{\bar{i}\bar{j}}^{S^{D-3}} dx^{\bar{i}}dx^{\bar{j}}=
d\theta^2+\sin^2\theta d\Omega_{D-4} \, ,
\end{equation}
where $\theta$ is a polar-like coordinate, $\theta\in [0,\pi]$. Now we
introduce four dimensional coordinates, $x^\mu=(x^{\bar{\mu}},\theta)$,
$\mu=0,1,2,3$, and define a four dimensional metric
\begin{equation}
g_{\mu\nu}dx^\mu dx^\nu=g_{\bar{\mu}\bar{\nu}}dx^{\bar{\mu}}dx^{\bar{\nu}}
+f(x^{\bar{\mu}}) d\theta^2 \, ,
\label{4d}
\end{equation}
as well as a new conformal factor
\begin{equation}
\lambda(x^{\mu})=\sin^2\theta g_{\theta\theta} \, .
\label{conformal}
\end{equation}
Then, the most general $D$~dimensional metric compatible with $SO(D-2)$
isometry is, for $D\ge 5$
\begin{equation}
d\hat{s}^2= g_{\mu\nu}dx^\mu dx^\nu + \lambda(x^{\mu}) d \Omega_{D-4} \, .
\label{ansatz}
\end{equation}
Without specifying \eqref{4d} and \eqref{conformal}, the geometry \eqref{ansatz}
has only a manifest $SO(D-3)$ symmetry. We now perform a dimensional reduction
on a $(D-4)$-sphere. This yields, from the $D$~dimensional vacuum Einstein
equations, a set of $3+1$ dimensional Einstein equations coupled to
quasi-matter. If $SO(D-2)$ is the full isometry group, the quasi-matter terms do
not contain independent degrees of freedom; rather, they may be completely
determined by the $3+1$ dimensional geometry, via \eqref{conformal}. In this
case, we could perform a dimensional reduction on a $(D-3)$-sphere, which has
the full isometry group $SO(D-2)$. This would yield a 2+1 dimensional
system. The former method allows, however, the use of existing numerical codes,
with small changes, which justifies our choice.
The equations derived with dimensional reduction on a $(D-4)$-sphere can be
applied, of course, to describe also space-times in which the \emph{full}
isometry group is $SO(D-3)$. This is the isometry group of a class of BH
collisions with impact parameter and with spin: the collisions in which the two
BHs always move on the same 2-plane and the only non trivial components of the
spin 2-form are on that same 2-plane -- see Fig.~\ref{headon_impact}. With our
framework we are able, therefore, to describe not only head-on collisions of
spinless BHs but also a class of collisions for spinning BHs with impact
parameter. As follows from the discussion of \eqref{novectors}, the ansatz
\eqref{ansatz} describes general space-times with $SO(D-3)$ isometry in $D\ge
6$. We remark that the models with $D\ge6$ are actually the most interesting
for phenomenological studies of large extra dimensions models (see for instance
\cite{Kanti:2004nr}).
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\textwidth]{headon_impact}
\caption{$D$~dimensional representation, using coordinates $(t, x^1,x^2,\dots,
x^{D-3},x^{D-2},z)$, of two types of BH collisions: (left panel) head-on for
spinless BHs, for which the isometry group is $SO(D-2)$; (right panel) non
head-on, with motion on a \textit{single} 2-plane, for BHs spinning in that
\textit{same} plane only, for which the isometry group is $SO(D-3)$. The
figures make manifest the isometry group in both cases.}
\label{headon_impact}
\end{figure}
\subsection{Dimensional reduction on a $(D-4)$-sphere and $3+1$ split}
In the following we take \eqref{ansatz} as an ansatz, which has a manifest
$SO(D-3)$ isometry. The $D$~dimensional pure Einstein theory reduces then to a
four dimensional theory of gravity coupled to a scalar field
$\lambda(x^\mu)$. We remark that in this theory $\lambda$ and $g_{\mu\nu}$ are
viewed as independent degrees of freedom; the relations \eqref{4d},
\eqref{conformal} select a subset of the solution space. The solutions belonging
to this subset have enhanced isometry $SO(D-2)$ and correspond to some of the
physical processes we want to study (for instance, head-on collisions of
spinless BHs).
The $D$~dimensional Einstein-Hilbert action reduces to
\begin{equation}
\mathcal{S} = \frac{1}{16\pi G_4}\int d^4x\sqrt{-g} \lambda^{\frac{D-4}{2}}
\left[
R + (D-4) \left(
(D-5) \lambda^{-1} - \lambda^{-1} \Box \lambda
- \frac{D-7}{4} \lambda^{-2} \partial_\mu \lambda \partial^\mu \lambda
\right)
\right] \, ,
\end{equation}
where the $D$~dimensional Newton's constant $G_D$ is related to the four
dimensional one $G_4$ by the area of the unit ${D-4}$ dimensional sphere:
$G_4=G_D/A^{S^{D-4}}$. Explicitly, the $D$~dimensional Einstein's equations in
vacuum yield the following system of four dimensional equations coupled to a
scalar field:
\begin{align}
R_{\mu\nu} & =\frac{D-4}{2 \lambda}\left(\nabla_\mu\partial_\nu \lambda
- \frac{1}{2\lambda} \partial_\mu\lambda \partial_\nu \lambda \right) \, ,
\label{4deinstein} \\
\nabla^\mu \partial_\mu \lambda & = 2(D-5)
- \frac{D-6}{2\lambda} \partial_\mu \lambda \partial^\mu \lambda \, .
\label{scalar}
\end{align}
In these equations, all operators are covariant with respect to the four
dimensional metric $g_{\mu\nu}$. The energy momentum tensor is\footnote{We use
the standard form of the Einstein equations $G_{\mu\nu}=8\pi T_{\mu\nu}$ and
choose geometrised units throughout.}
\begin{equation}
T_{\mu\nu}=\frac{D-4}{16\pi \lambda}\left[\nabla_\mu\partial_\nu \lambda
- \frac{1}{2\lambda} \partial_\mu\lambda \partial_\nu\lambda
- (D-5)g_{\mu\nu} + \frac{D-5}{4\lambda} g_{\mu \nu} \partial_\alpha
\lambda \partial^\alpha \lambda \right] \, .
\label{emtensor}
\end{equation}
With this four dimensional perspective, the usual $3+1$ split of space-time
\cite{Arnowitt:1962hi, York1979} can be performed (see, \textit{e.g.\
}\cite{Gourgoulhon:2007ue,Alcubierre:2008}). For this purpose, we introduce the
projection operator $\gamma_{\mu\nu}$ and the normal to the three dimensional
hyper-surface $\Sigma$, $n^\mu$ ($n^\mu n_\mu = -1$),
\begin{align}
\label{eq:2}
\gamma_{\mu\nu} = g_{\mu\nu} + n_\mu n_\nu \, ,
\end{align}
as well as the lapse $\alpha$ and shift $\beta^\mu$,
\begin{align}
\label{eq:4}
\partial_t = \alpha n + \beta \ ,
\end{align}
where $t$ is the time coordinate. The four dimensional metric is then
written in the form
\begin{equation}
\label{4dinitial}
ds^2=g_{\mu\nu}dx^\mu dx^\nu=-\alpha^2dt^2+\gamma_{ij}
(dx^i+\beta^idt)(dx^j+\beta^jdt) \, , \qquad i,j=1,2,3 \, .
\end{equation}
As usual, we introduce the extrinsic curvature $K_{ij} = -\frac{1}{2}
\mathcal{L}_n \gamma_{ij}$, which gives the evolution equation for
the $3$-metric,
\begin{align}
\label{eq:gammaevol0}
\left(
\partial_t - \mathcal{L}_\beta
\right) \gamma_{ij} = - 2\alpha K_{ij} \, .
\end{align}
The time evolution for $K_{ij}$ is given by
\begin{align}
\label{eq:Kevol00}
\left(
\partial_t - \mathcal{L}_\beta
\right) K_{ij} = -D_i \partial_j \alpha
+ \alpha \left(
{}^{(3)}\! R_{ij} + K K_{ij} - 2K_{i k} K^{k}{}_j
\right)
- \alpha \gamma^\mu{}_i \gamma^\nu{}_j R_{\mu\nu} \, ,
\end{align}
where $D_i$ is the covariant derivative on the hyper-surface. The last
term, $ \gamma^\mu{}_i \gamma^\nu{}_j R_{\mu\nu}$, vanishes for vacuum
solutions. In the present case, it is given by the projection of equation
\eqref{4deinstein},
\begin{align}
\gamma^\mu{}_i \gamma^\nu{}_j R_{\mu\nu} = \frac{D-4}{2 \lambda} \left(
\gamma^\mu{}_i \gamma^\nu{}_j \nabla_\mu\partial_\nu \lambda
- \frac{1}{2\lambda} \partial_i\lambda \partial_j \lambda \right) \, .
\end{align}
Using the formula
\begin{align}
\label{eq:proj}
D_\alpha D_\beta \lambda
= -K_{\alpha\beta} n^\sigma \partial_\sigma \lambda
+ \gamma^\mu{}_\alpha \gamma^\nu{}_\beta \nabla_\nu \partial_\mu
\lambda \, ,
\end{align}
and defining the variable
\begin{align}
\label{eq:Kphi0}
K_\lambda \equiv - \frac{1}{2} \mathcal{L}_n \lambda =
- \frac{1}{2} n^\mu \partial_\mu \lambda \, ,
\end{align}
we obtain
\begin{align}
\gamma^\mu{}_i \gamma^\nu{}_j \nabla_\nu \partial_\mu \lambda =
D_i \partial_j \lambda - 2 K_{ij} K_\lambda \, .
\end{align}
Thus, \eqref{eq:Kevol00} becomes
\begin{equation}
\label{eq:Kevol0}
\begin{split}
\left(
\partial_t - \mathcal{L}_\beta
\right) K_{ij} & = -D_i \partial_j \alpha
+ \alpha \left(
{}^{(3)}\! R_{ij} + K K_{ij} - 2K_{i k} K^{k}{}_j
\right) \\
&{}\quad - \alpha \frac{D-4}{2\lambda} \left( D_i \partial_j \lambda
- 2 K_{ij} K_\lambda - \frac{1}{2\lambda} \partial_i \lambda
\partial_j \lambda \right)\, .
\end{split}
\end{equation}
To summarise, the evolution equations for the 3-metric and extrinsic
curvature are \eqref{eq:gammaevol0} and \eqref{eq:Kevol0}.
If the isometry group is $SO(D-3)$, the quasi-matter field $\lambda$ represents
an independent degree of freedom, and we need to solve the evolution equations
for $\lambda$ and $K_\lambda$. Even in the case of the larger isometry
$SO(D-2)$, the evolution equations for $\lambda$ and $K_{\lambda}$ are useful as
they enable us to test Eq.~(\ref{conformal}) and thus provide a check of the
numerical evolution. The evolution equation for $\lambda$ is \eqref{eq:Kphi0}
\begin{equation}
\label{phievo}
\left(
\partial_t - \mathcal{L}_\beta \right)
\lambda = - 2 \alpha K_\lambda \, .
\end{equation}
Eq. \eqref{scalar} provides an evolution equation for
$K_\lambda$. The contraction of Eq. \eqref{eq:proj} with $g^{\alpha\beta}$,
yields
\begin{align}
\Box \lambda = \gamma^{ij}D_i\partial_j \lambda - 2 K K_\lambda
- n^\mu n^\nu \nabla_\nu \partial_\mu \lambda \, .
\end{align}
Noting that
\begin{align}
\mathcal{L}_n K_\lambda = n^\mu \partial_\mu K_\lambda
= -\frac{1}{2} n^\mu \nabla_\mu n^\nu \partial_\nu \lambda
- \frac{1}{2} n^\mu n^\nu \nabla_\mu \partial_\nu \lambda \, ,
\end{align}
and
\begin{align}
n^\mu \nabla_\mu n^\nu = \frac{1}{\alpha} D^\nu \alpha \, ,
\end{align}
we obtain
\begin{align}
- n^\mu n^\nu \nabla_\mu \partial_\nu \lambda = 2 \mathcal{L}_n K_\lambda
+ \frac{1}{\alpha} D^\nu \alpha \partial_\nu \lambda \, .
\end{align}
Noticing also that $D^\nu \alpha \partial_\nu \lambda = \gamma^{ij}\partial_i \alpha \partial_j
\lambda $, we write
\begin{align}
\Box \lambda = \gamma^{ij}D_i\partial_j \lambda - 2 K K_\lambda
+ 2 \mathcal{L}_n K_\lambda + \frac{1}{\alpha} \gamma^{ij}\partial_i \alpha \partial_j
\lambda \, .
\end{align}
Moreover, from equation
\begin{align}
D_\mu \lambda = \gamma^\nu{}_\mu \partial_\nu \lambda
= \partial_\mu \lambda - 2 n_\mu K_\lambda \, ,
\end{align}
we get
\begin{align}
\partial_\alpha \lambda \partial^\alpha \lambda = \gamma^{ij}\partial_i \lambda \partial_j
\lambda - 4 K_\lambda ^2 \, ,
\end{align}
so that the evolution equation for $K_\lambda$ is
\begin{align}
\label{eq:evolphi0}
\frac{1}{\alpha} \left(
\partial_t - \mathcal{L}_\beta
\right) K_\lambda
= - \frac{1}{2\alpha} \gamma^{ij}\partial_i \lambda \partial_j \alpha
+ (D-5) + K K_\lambda + \frac{D-6}{\lambda} K_\lambda^2
- \frac{D-6}{4\lambda} \gamma^{ij} \partial_i \lambda \partial_j \lambda
- \frac{1}{2} D^k \partial_k \lambda \, .
\end{align}
Equations \eqref{phievo} and \eqref{eq:evolphi0} are the evolution
equations for the quasi-matter degrees of freedom.
\subsection{BSSN formulation}\label{BSSN}
For numerical implementation, let us now write the evolution equations
in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation
\cite{Shibata:1995we,Baumgarte:1998te}. Instead of evolving the variables
$\gamma_{ij}$ and $K_{ij}$, we introduce a conformal split of the physical
3-metric $\gamma_{ij}$ as
\begin{equation}
\gamma_{ij}\equiv \frac{1}{\chi}\tilde{\gamma}_{ij} \, .
\end{equation}
The conformal factor
\begin{equation}
\chi=\left({\rm det} \gamma_{ij}\right)^{-1/3} \, ,
\end{equation}
is chosen such that $\det\tilde{\gamma}_{ij}=1$ holds at all times. The
extrinsic curvature is split into a conformal traceless part,
$\tilde{A}_{ij}$, and its trace, $K$, as
\begin{equation}
\tilde{A}_{ij}\equiv \chi \left(K_{ij}-\frac{\gamma_{ij}}{3}K\right) \, .
\end{equation}
Moreover, we introduce the contracted conformal connection
\begin{equation}
\tilde{\Gamma}^i=\tilde{\gamma}^{jk}\tilde{\Gamma}^i_{jk} \, ,
\end{equation}
where
\begin{eqnarray}
\Gamma^k_{ij} = \tilde{\Gamma}^k_{ij} - \frac{1}{2\chi}
\left(\delta_i{}^k \partial_j \chi + \delta_j{}^k \partial_i \chi
-\tilde{\gamma}_{ij}\tilde{\gamma}^{kl}\partial_l
\chi \right) \ \Rightarrow \
\Gamma^k = \chi \tilde{\Gamma}^k + \frac{1}{2}
\tilde{\gamma}^{kl}\partial_l \chi \, ,
\end{eqnarray}
as an independent variable. In terms of the BSSN variables
$\chi,\tilde{\gamma}_{ij},\tilde{A}_{ij},\tilde{\Gamma}^k$, the evolution
equations are
\begin{subequations}
\begin{align}
\left( \partial_t - \mathcal{L}_\beta \right) \tilde \gamma_{ij} & =
- 2 \alpha \tilde A_{ij}\, , \\
\left( \partial_t - \mathcal{L}_\beta \right) \chi & =
\frac{2}{3} \alpha \chi K\, , \\
\left( \partial_t - \mathcal{L}_\beta \right) K & =
[\dots] + 4 \pi \alpha (E + S)\, , \\
\left( \partial_t - \mathcal{L}_\beta \right) \tilde A_{ij} & =
[\dots] - 8 \pi \alpha \left(
\chi S_{ij} - \frac{S}{3} \tilde \gamma_{ij}
\right)\, , \\
\left( \partial_t - \mathcal{L}_\beta \right) \tilde \Gamma^i & =
[\dots] - 16 \pi \alpha \chi^{-1} j^i\, ,
\end{align}
\end{subequations}
where $ [\dots] $ denotes the standard right-hand side of the BSSN equations in
the absence of source terms (see \emph{e.g.\ }\cite{Alcubierre:2008}); the
source terms are determined by
\begin{align}
E & \equiv n^\alpha n^\beta T_{\alpha\beta} \, , \\
j_i & \equiv - \gamma_i{}^\alpha n^\beta T_{\alpha \beta} \, , \\
S_{ij} & \equiv \gamma^\alpha{}_i \gamma^\beta{}_j T_{\alpha \beta}\, , \\
S & \equiv \gamma^{ij} S_{ij}\, ,
\end{align}
where the energy momentum tensor is given by Eq.~\eqref{emtensor}.
A straightforward computation shows that
\begin{subequations}
\label{matterterms}
\begin{align}
\begin{split}
\frac{4 \pi (E + S)}{D-4} & = -(D-5) \lambda^{-1} + \frac{1}{2}
\lambda^{-1} \chi^{3/2} \tilde \gamma^{ij} \tilde D_i
\left(
\chi^{-1/2} \partial_j \lambda
\right) \\
&{} \quad
+ \frac{D-6}{4} \lambda^{-2} \chi \tilde \gamma^{ij} \partial_i
\lambda \partial_j \lambda
- \lambda^{-1} K K_\lambda - (D-5) \lambda^{-2} K_\lambda^2 \, ,
\end{split} \\
%
\begin{split}
\frac{8\pi \chi \left( S_{ij} - \frac{S}{3} \gamma_{ij}
\right)}{D-4} & = \frac{1}{2} \chi \lambda^{-1} \tilde D_i
\partial_j \lambda + \frac{1}{4} \lambda^{-1} \left(
\partial_i \lambda \partial_j \chi + \partial_j
\lambda \partial_i \chi - \tilde \gamma^{kl} \tilde
\gamma_{ij} \partial_k \lambda
\partial_l \chi \right) - \frac{1}{4} \chi
\lambda^{-2} \partial_i
\lambda \partial_j \lambda \\
&{} \quad - \lambda^{-1} K_\lambda \tilde{A}_{ij} -
\frac{1}{6}\tilde \gamma_{ij} \lambda^{-1} \chi^{3/2} \tilde
\gamma^{kl} \tilde D_k \left( \chi^{-1/2} \partial_l \lambda
\right) + \frac{1}{12} \tilde \gamma_{ij} \lambda^{-2} \chi \tilde
\gamma^{kl} \partial_l \lambda
\partial_k \lambda \, ,
\end{split} \\
\frac{16 \pi \chi^{-1} j^i}{D-4} & = 2 \lambda^{-1}
\tilde \gamma^{ij} \partial_j K_\lambda
- \lambda^{-2} K_\lambda \tilde \gamma^{ij} \partial_j \lambda
- \tilde \gamma^{ik} \tilde \gamma^{lj} \tilde{A}_{kl} \lambda^{-1}
\partial_j \lambda-\frac{\tilde{\gamma}^{ij}}{3}K\lambda^{-1}
\partial_j\lambda \, ,
\end{align}
\end{subequations}
where $\tilde{D}_i$ is the covariant derivative with respect to
$\tilde{\gamma}_{ij}$.
Finally, the evolution equations for $\lambda$ and $K_\lambda$ are
\begin{subequations}
\label{kl}
\begin{align}
\left( \partial_t - \mathcal{L}_\beta \right) \lambda & = - 2
\alpha K_\lambda , \\
%
\begin{split}
\left( \partial_t - \mathcal{L}_\beta \right) K_{\lambda} &
= \alpha \bigg\{ (D-5) + \frac{6-D}{4} \left[ \lambda^{-1}
\chi \tilde{\gamma}^{ij}\partial_i \lambda
\partial_j \lambda - 4 \lambda^{-1} K_\lambda^2
\right] \\
&{} \quad + K K_\lambda - \frac{1}{2} \chi^{3/2} \tilde
\gamma^{kl} \tilde D_k \left( \chi^{-1/2} \partial_l \lambda
\right) \bigg\} - \frac{1}{2} \chi \tilde{\gamma}^{ij}\partial_j
\alpha \partial_i \lambda \, .
\end{split}
\end{align}
\end{subequations}
As stated before, in the case of head-on collisions of spinless BHs the full
symmetry of the $D$~dimensional system we want to consider makes
equations~\eqref{kl} redundant, by virtue of~\eqref{conformal}. This allows to
determine the quasi-matter degree of freedom in terms of the three dimensional
spatial geometry, at each time slice. Indeed, we have only used an $SO(D-3)$
subgroup in the dimensional reduction we have performed. The extra symmetry
manifests itself in the fact that $\gamma_{ij}$ possesses, at all times, (at
least) one Killing vector field. If one chooses coordinates adapted to this
Killing vector field, $\partial/\partial \theta$, the metric can then be written
in the form \eqref{4d}, and then the quasi-matter degree of freedom can be
determined from the spatial geometry by \eqref{conformal}. In the numerical
implementation, one can either determine, at each time-step, the scalar field
through (\ref{conformal}), or impose (\ref{conformal}) only in the initial data,
and then evolve the scalar field using Eq.~(\ref{kl}).
\section{Initial data}
\label{sec:initial-data}
Following the approach in \cite{Yoshino:2005ps,Yoshino:2006kc}, we now
derive the initial data of the evolution.
\subsection{$D$~dimensional Hamiltonian and momentum constraints}
\label{sec:init-data-equations}
Let $\bar \Sigma$ be a $(D-1)$-dimensional space-like hyper-surface with
induced metric $\bar \gamma_{ab}$ and extrinsic curvature $\bar K_{ab}$
in the $D$~dimensional space-time.
The space-time metric has the form
\begin{align}
d\hat{s}^2 = \hat g_{MN} dx^M dx^N = -\alpha^2 dt^2
+ \bar \gamma_{ab} \left(
dx^a + \beta^a dt
\right) \left(
dx^b + \beta^b dt
\right),
\label{metricinitial}
\end{align}
where lower case latin indices take values $a=1,\dots,D-1$.
The constraint equations are
\begin{align}
\label{eq:hamiltonian2}
& \bar R + {\bar K}^2 - \bar K_{ab} {\bar K}^{ab} = 0 \, , \\
\label{eq:momentum2}
& \bar D_a \left( \bar K^{ab} - \bar \gamma^{ab} \bar K \right) = 0 \, ,
\end{align}
where $ \bar R$ is the Ricci scalar of the hyper-surface $\bar \Sigma$, $\bar K$
is the trace of the extrinsic curvature and $\bar D_a$ is the covariant
derivative with respect to $ \bar \gamma_{ab} $.
We conformally decompose the spatial metric
\begin{align}
\label{eq:conformal}
\bar \gamma_{ab} & = \psi^{\frac{4}{D-3}} \hat \gamma_{ab}\, ,
\end{align}
which introduces the conformal factor $\psi$, and split the extrinsic curvature in trace and trace-free parts,
\begin{equation}
\bar K_{ab} \equiv \bar A_{ab} + \frac{\bar K}{D-1} \bar \gamma_{ab}\, ,
\end{equation}
where $ \bar \gamma^{ab} \bar A_{ab} = 0$. Define $\bar A^{ab} \equiv
\bar \gamma^{ac} \bar \gamma^{bd} \bar A_{cd}$; define also the quantity
\begin{equation}
\hat A^{ab} \equiv \psi^{2 \frac{D+1}{D-3}} \bar A^{ab} \, ,
\end{equation}
and lower its indices with $\hat \gamma_{ab}$,
\begin{equation}
\hat A_{ab} \equiv \hat \gamma_{ac} \hat \gamma_{bd} \hat A^{cd}
= \psi^2 \bar A_{ab} \, .
\end{equation}
Assuming that the ``conformal metric'' $\hat \gamma_{ab}$ is flat, which is a
good approximation for the class of problems we want to study, we impose the
``maximal slicing condition'' $\bar K = 0 $. Then, the Hamiltonian and momentum
constraints become
\begin{align}
& \hat\nabla_a \hat A^{ab} = 0\, , \label{eq:vacuum_hamilton} \\
& \hat \triangle \psi + \frac{D-3}{4(D-2)} \psi^{- \frac{3D -5}{D-3} }
\hat A^{ab} \hat A_{ab} = 0 \, ,
\label{eq:vacuum_momentum}
\end{align}
where $\hat\nabla$ is the covariant derivative with respect to $\hat
\gamma_{ab}$ and $\hat \triangle$ is the flat space Laplace operator.
\subsection{Brill-Lindquist initial data and matching to four dimensions}
\label{sec:brill-lindq-init}
The simplest way to solve the constraints
\eqref{eq:vacuum_hamilton}-\eqref{eq:vacuum_momentum} is to require the
extrinsic curvature to be zero
\begin{equation}
\bar K_{ab}=0 \, .
\label{initialD}
\end{equation}
This is sufficient to model the evolution of a single BH or even of $N$
non-spinning, non-boosted BHs. The constraints reduce to a simple harmonic
equation for the conformal factor, $\hat \triangle \psi =0$, which we solve in
cylindrical coordinates $\{x^a\}=(z,\rho,\theta, \dots)$, where `$\dots$'
represent the coordinates on the $(D-4)$-sphere,
\begin{equation}
\hat\gamma_{ab}dx^adx^b=dz^2+d\rho^2+\rho^2\left(d\theta^2+\sin^2\theta
d\Omega_{D-4}\right) \, .
\label{initialspatial}
\end{equation}
This choice of coordinates makes manifest the symmetries we want to
impose. Observe that $\theta$ is a polar rather than an azimuthal coordinate,
\emph{i.e.\ }$\theta\in [0,\pi]$. Next, we introduce ``incomplete'' Cartesian
coordinates as
\begin{equation}
x=\rho \cos\theta \, , \qquad y=\rho \sin\theta \, ,
\label{inccartesian}
\end{equation}
where $-\infty<x<+\infty$ and $0\le y<+\infty$; we can then write the
$D$~dimensional initial data as \eqref{initialD} together with
\begin{equation}
\bar \gamma_{ab}dx^adx^b=\psi^{\frac{4}{D-3}}\left[dx^2+dy^2+dz^2+y^2
d\Omega_{D-4}\right] \, ,
\label{smhigher}
\end{equation}
where $\psi$ is a harmonic function on \eqref{initialspatial}.
If we compare the space-time metric \eqref{metricinitial} at the initial time
slice, for which the spatial metric is given by \eqref{eq:conformal}
and \eqref{smhigher}, with the generic form that has an $SO(D-3)$ symmetry and is
given by \eqref{ansatz}, \eqref{4dinitial}, we see that the initial data
for the four dimensional variables are
\begin{equation}
\label{k0}
\gamma_{ij}dx^idx^j=\psi^{\frac{4}{D-3}}\left[dx^2+dy^2+dz^2\right] \, ,
\end{equation}
and
\begin{equation}
\lambda=y^2\psi^{\frac{4}{D-3}} \, .
\label{iniscalar}
\end{equation}
It remains to determine the initial conditions for $K_{ij}$ and
$K_\lambda$. Using a set of $D$~dimensional coordinates that make manifest the
$SO(D-3)$ isometry, such as the one used in \eqref{smhigher}, the vanishing of
the extrinsic curvature $\bar{K}_{ij}$ is equivalent to
\begin{equation}
\label{k1}
K_{ij}=0 \, ,
\end{equation}
whereas the vanishing of the components of $\bar K_{ab}$ along the
$(D-4)$-sphere implies that
\begin{equation}
\label{k2}
K_\lambda=0 \, .
\end{equation}
Equations~\eqref{k0}--\eqref{k2} represent the Brill-Lindquist initial data
in our framework.
\subsubsection{Evolution of a single black hole}\label{single}
As one test of our framework we study the case of a single, non-spinning
BH. Even though the space-time is static, the slicing evolves when using the
puncture gauge.
The solution for the conformal factor, which shall be used in the numerical
tests to be presented below, is given by
\begin{align}
\label{eq:psiBL}
\psi \equiv 1 + \frac{\mu^{D-3}} {4 \left[x^2+y^2+(z-z_{BH})^2
\right]^{(D-3)/2}} \, ,
\end{align}
where the ``puncture'' \cite{Brandt:1997tf} is placed at $x=y=0$ and
$z=z_{BH}$. In this formulation, there is an interesting signature that the BH
we wish to evolve is higher dimensional: the fall off of $\psi$, which is that
of a harmonic function in $D-1$ spatial dimensions. Because the Tangherlini
solution \cite{Tangherlini:1963bw} may be expressed, in the same coordinate
system as used in \eqref{smhigher}, as
\begin{equation}
d\hat s^2=-\left(\frac{4R^{D-3}-\mu^{D-3}}{4R^{D-3}+\mu^{D-3}}\right)^2dt^2
+\left(1+\frac{\mu^{D-3}}{4R^{D-3}}\right)^\frac{4}{D-3}
\left(dx^2+dy^2+dz^2+y^2d\Omega_{D-4}\right) \, ,
\end{equation}
where $R=\sqrt{x^2+y^2+z^2}$, we conclude that the parameter $\mu$ appearing in
the initial condition \eqref{eq:psiBL} is the same which appears in this form of
the Tangherlini solution. It is related to the ADM mass by
\begin{equation}
\mu^{D-3}=\frac{16\pi M_{ADM}}{(D-2)A^{S^{D-2}}} \, .
\end{equation}
Note, however, that this form of the Tangherlini solution is not appropriate for
a comparison with the numerical data. Indeed, the evolution does not, in
general, preserve the conformally flat slicing of the initial condition, which
is the slicing used in this form of the Tangherlini solution. We shall return to
this issue in Section~\ref{numres}.
\subsubsection{Head-on collision of black holes}
\label{sec:head-on-init}
As another test of our formulation, and in particular of the numerical code's
long term stability, we also evolve a head-on collision of non-spinning
non-boosted BHs. In this case, the initial data for the conformal factor are
given by
\begin{align}
\psi \equiv 1 + \frac{\mu_{\rm A}^{D-3}} {4 \left[x^2+y^2+(z-z_{\rm A})^2
\right]^{(D-3)/2}}
+ \frac{\mu_{\rm B}^{D-3}} {4 \left[x^2+y^2+(z-z_{\rm B})^2
\right]^{(D-3)/2}} \, .
\end{align}
This conformal factor is used in Section~\ref{numres}.
\section{The numerical treatment}
\label{sec:numerical-treatment}
Our numerical simulations have been performed by adapting the \textsc{Lean} code
\cite{Sperhake:2006cy}, initially designed for 3+1 vacuum space-times. The
\textsc{Lean} code is based on the \textsc{Cactus} computational toolkit
\cite{cactus}. It employs the BSSN formulation of the Einstein equations
\cite{Shibata:1995we,Baumgarte:1998te}, uses the moving puncture method
\cite{Campanelli:2005dd,Baker:2005vv}, the \textsc{Carpet} package for mesh
refinement \cite{Schnetter:2003rb,carpet}, the spectral solver described in
\cite{Ansorg:2004ds} for 3+1 initial data and Thornburg's {\sc AHFinderDirect}
\cite{Thornburg:1995cp,Thornburg:2003sf}. Details about \textsc{Lean} may be
found in \cite{Sperhake:2006cy}. Here we focus on the numerical issues generated
by the quasi-matter terms arising from the dimensional reduction by isometry.
We expect that the quasi-matter field $\lambda$ has a $y^2$ fall off as
$y\rightarrow 0$, that is, on the $xz$ plane. This leads to divisions by zero on
the right-hand side of the BSSN evolution equations,
cf. \eqref{matterterms}. Since we expect all variables to remain regular on the
$xz$ plane, all divisions by $y$ need to be cancelled by a corresponding fall
off behaviour of the numerators. At $y=0$, however, in order to implement this
behaviour numerically, we need to isolate the irregular terms and evaluate
expressions such as
\begin{equation}
\lim_{y\rightarrow 0} \frac{f}{y}\, ,
\end{equation}
where $f$ is some example function which behaves like $y^n$ with $n\ge 1$ near
the $xz$ plane. It is necessary, for this purpose, to formulate the equations in
terms of variables which are manifestly regular at $y=0$. We also prefer to
apply a conformal re-scaling of $\lambda$ and use the evolution variable
\begin{equation}
\label{kappavariable}
\zeta \equiv \frac{\chi}{y^2} \lambda \, .
\end{equation}
As in \eqref{kl}, in order to obtain a first order evolution system in time, we
introduce an auxiliary variable (see Appendix~\ref{axis}):
\begin{eqnarray}
K_{\zeta} \equiv -\frac{1}{2\alpha y^2}(\partial_t-\mathcal{L}_\beta)
(\zeta y^2)=-\frac{1}{2\alpha} \left( \partial_t \zeta
- \beta^m \partial_m \zeta + \frac{2}{3}\zeta
\partial_m \beta^m - 2\zeta \frac{\beta^y}{y} \right) \, .
\label{eq:Kkappa_v2}
\end{eqnarray}
The third term on the right-hand side arises from the fact that $\zeta$ is not a
scalar, but a scalar density of weight $-2/3$. The inclusion of this term might
not be necessary for a stable numerical implementation. For consistency with
the rest of the BSSN variables, however, we decide to keep this form of
$K_{\zeta}$.
The quasi-matter terms \eqref{matterterms}, the quasi-matter evolution equations
\eqref{kl} and the constraints are recast in terms of $\zeta$ and $K_{\zeta}$ in
Appendix~\ref{axis}. In particular, we notice that
\begin{eqnarray}
K_{\lambda} = \frac{y^2}{\chi} K_{\zeta} + \frac{1}{3}
\frac{y^2\zeta}{\chi} K\, .
\label{useful}
\end{eqnarray}
A detailed analysis of the equations in terms of the variables $\zeta$ and
$K_{\zeta}$ shows how all terms with an explicit dependence on $1/y^n$, $n\ge 1$
may be treated for numerical implementation. This is discussed in
Appendix~\ref{troubleterms}.
\subsection{Numerical results in $D=5$}\label{numres}
We first address the question of longevity of our simulations in $D=5$. It is
also of interest in this context to test the code's capability to successfully
merge a BH binary. For this purpose we have evolved a head-on collision starting
from rest. The initial conditions are those from Section~\ref{sec:head-on-init}
with
\begin{eqnarray}
&& \mu^2_{\rm A} = \mu^2_{\rm B} \equiv \frac{\mu^2}{2} \, , \\
&& z_A = -z_B = 3.185~\mu \, ,
\end{eqnarray}
and we use the grid setup (cf.~Sec.~II E of Ref.~\cite{Sperhake:2006cy})
\begin{equation}
\left\{(512,~256,~128,~64,~32,~16,~8) \times (2,~1),~h=1/32\right\} \, , \nonumber
\end{equation}
in units of $\mu$. The gauge variables $\alpha$ and $\beta^i$ are evolved
according to the modified moving puncture conditions \eqref{eq:dtalpha} and
\eqref{eq:dtbeta} with parameters $\eta_K = \eta_{K_\zeta}=1.5$ and $\eta=0.75$.
We employ fourth order discretization in space and time and impose a floor value
\cite{Campanelli:2005dd} for the variable $\chi=10^{-4}$.
\begin{figure}[h!tb]
\centering
\subfloat[Head-on $\chi$]{
\includegraphics[clip=true,width=0.46\textwidth]{headon_stable}
}
\quad
\subfloat[Head-on $K_{\zeta}$]{
\includegraphics[clip=true,width=0.46\textwidth]{headon_stable_Kz}
}
\caption{The BSSN variable $\chi$ (left panel) and the quasi-matter momentum
$K_{\zeta}$ (right panel) are shown along the axis of collision
for a head-on collision at times $t=0$, $5$, $20$, $40$
and $256~\mu$. Note that $K_{\zeta}=0$
at $t=0$. }
\label{headon_stable}
\end{figure}
In Fig.~\ref{headon_stable} we show the conformal factor $\chi$ and the momentum
$K_{\zeta}$ along the axis of collision at various times. At early times, the
evolution is dominated by the adjustment of the gauge (cf.~the solid and
short-dashed curves). The two holes next start approaching each other
(long-dashed and dotted curves) and eventually merge and settle down into a
single stationary hole (dash-dotted curves). We have not observed any signs of
instability and decided to stop the simulation at $t=256~\mu$. It is reassuring
to notice that the framework can handle the merger in as robust a fashion as has
been demonstrated by various numerical groups for BH binaries in 3+1 dimensions.
We have also used the head-on collision to test the relation between the scalar
field $\lambda$ and the $3+1$ metric discussed in Sec.~\ref{sec:axial-symmetry}
for the case that $SO(D-2)$ is the full isometry group. We have verified for
this purpose that Eq.~\ref{conformal} remains satisfied to within a relative
error of $10^{-3}$ in the immediate vicinity of the puncture and at most
$10^{-5}$ everywhere else.
In order to further test our numerical framework, we have performed simulations
of a single BH, using the initial data described in Section~\ref{single} and the
grid setup
\begin{equation}
\left\{(512,~256,~128,~64,~32,~16,~8,~4,~2) \times (),~h\right\} \, ,\nonumber
\end{equation}
in units of $\mu$ with resolutions $h_{\rm c}=1/32$ and $h_{\rm f}=1/48$. In
Fig.~\ref{constraintsplot} we show the Hamiltonian constraint and the $y$
component of the momentum constraint at evolution time $t=28\mu$. By this time
there is hardly any more gauge dynamics going on. One can see that there is some
noise, but the overall convergence is acceptable. For the Hamiltonian constraint
the convergence is essentially 4th order and for the momentum constraint it
decreases slightly towards 2nd or 3rd order in patches. From experience in 3+1
dimensional numerical relativity this is perfectly acceptable, especially given
the fact that prolongation in time is second-order accurate.
\begin{figure}[h!]
\centering
\subfloat[Hamiltonian constraint]{
\includegraphics[clip=true,width=0.46\textwidth]{hamiltonian}
}
\quad
\subfloat[$y$-component of the
momentum constraint]{
\includegraphics[clip=true,width=0.46\textwidth]{momentum}
}
\caption{Constraints at time $t=28\mu$, for the evolution
of a single Tangherlini BH in five dimensions.}
\label{constraintsplot}
\end{figure}
A different test of our numerical code was performed in order to compare the
analytical Tangherlini solution with our numerical results. The challenge to do
this comparison, at the level of the line element, is to write the well known
analytical solution in the same coordinate system in which the numerical
evolution is occurring. One way around this problem is to fix the numerical
gauge as to match a known coordinate system for the analytic solution. Following
\cite{Yoshino:2009xp} we fixed the gauge parameters to be
\begin{equation}
\alpha=1 \, , \qquad \beta^i=0 \, , \, i=1,2,3 \, ;
\end{equation}
this corresponds to \textit{geodesic slicing}. The $D$~dimensional Tangherlini
solution may be expressed in a coordinate system of type \eqref{metricinitial}
with $\alpha=1,\beta^a=0$, $a=1,\dots, D-1$. This coordinate system may be
achieved by setting a congruence of in-falling radial time-like geodesics, each
geodesic starting from rest at radial coordinate $r_0$, with $r_0$ spanning the
interval $[\mu,+\infty[$, and using their proper time $\tau$ and $r_0$ as
coordinates (instead of the standard $t$, $r$ Schwarzschild-like coordinates). A
detailed construction of the Tangherlini solution in these coordinates is given
in Appendix~\ref{geoslice}. The line element becomes
\begin{equation}
ds^2=-d\tau^2+\frac{\left(r_0(R)^2+\left(\frac{\mu}{r_0(R)}\right)^2
\tau^2\right)^2}{r_0(R)^2-\left(\frac{\mu}{r_0(R)}\right)^2\tau^2}
\frac{dR^2}{R^2}+\left(r_0(R)^2-\left(\frac{\mu}{r_0(R)}\right)^2
\tau^2\right)d\Omega_3 \, ,
\label{geodesicmetric}
\end{equation}
where $r_0(R)$ is given by Eq.~\eqref{rzerotor}.
The numerical evolution in this gauge is naturally doomed. Geodesics hit the
physical singularity at finite proper time. Thus, this slicing is inappropriate
for a long term numerical evolution. As long as the evolution does not break
down, however, there is perfect control over the slicing, and hence the
numerical and analytical evolution can be compared with ease. This is shown in
Fig.~\ref{gammaxx}, where we have plotted one metric component
$\tilde{\gamma}_{xx}$ along the $x$ axis (left) and $\zeta/\chi$ (right), for
various values of $\tau$ using both the analytical solution and numerical
data. The agreement is excellent for $\tilde{\gamma}_{xx}$ and good for
$\zeta/\chi$. The latter shows some deviations very close to the puncture, but
we believe that it is not a problem for two reasons: \textit{(i)}~the agreement
improves for higher resolution; \textit{(ii)}~the mismatch does not propagate
outside of the horizon.
\begin{figure}[h!]
\centering
\subfloat[$\tilde{\gamma}_{xx}$ along the $x$-axis]{
\includegraphics[clip=true,width=0.46\textwidth]{hxx}
}
\quad
\subfloat[$\zeta/\chi=\lambda/y^2$ along the $y$-axis]{
\includegraphics[clip=true,width=0.46\textwidth]{kappa}
}
\caption{Numerical values versus analytical plot (solid lines)
for various values of $\tau$, for
the single Tangherlini BH
in five dimensions. The horizontal axes are labelled in
units of $\mu$.
}
\label{gammaxx}
\end{figure}
It is easy to interpret the behaviour observed for $\tilde{\gamma}_{xx}$. The
geodesic that starts from $r=r_0$ (in Schwarzschild-like coordinates) hits the
physical singularity of the Tangherlini solution within proper time
$\tau=r_0^2/\mu$. Moreover, this happens at
\begin{equation}
R=\frac{\mu}{2}\frac{1}{\sqrt{\tau/\mu}\pm\sqrt{\tau/\mu-1}} \, .
\label{hitssing}
\end{equation}
The earliest time at which the slicing hits the singularity is $\tau=\mu$, which
happens at $R=\mu/2$. On the $x$-axis $R=x$ and indeed one sees in
Fig.~\ref{gammaxx} that $\tilde{\gamma}_{xx}$ diverges at $x=\mu/2$. The
divergence then extends to both larger and smaller values of $x$, as expected
from \eqref{hitssing}.
\subsection{Preliminary numerical results in $D=6$}\label{numres6}
A quick glance at the evolution equations (\ref{evotildek}) and
(\ref{evomomentumk}) of the scalar field $\zeta$ as well as the source terms
(\ref{eq:ESnew})-(\ref{eq:jinew}) indicates that $D=5$ may be a special case. In
all these expressions there exist terms which manifestly vanish for $D=5$. In
contrast, there exist no terms which manifestly vanish for any dimension $D \ge
6$. The purpose of this Section is to extend the test of our framework to a
case which involves all source terms.
We have indeed noticed one fundamental difference between simulations in $D=5$
and those using $D\ge6$. Whereas we have been able to obtain stable simulations
of single BHs lasting hundreds of $\mu$ for the former case by modifying the
moving puncture gauge conditions, we have not yet succeeded in doing so for
$D\ge 6$. While the lifetime of the simulations in $D\ge 6$ shows a dependence
on the exact nature of lapse and shift, all simulations developed instabilities
on a timescale of about $10~\mu$. Resolving this issue requires an extensive
study involving a large number of experiments with gauge conditions, constraint
damping and possibly other aspects of the formulation. Such a study is beyond
the scope of this work and deferred to a future publication. The results
presented in this Section still provide valuable information. Most importantly,
they demonstrate the internal consistency of the code for $D\ge 6$ and thus
minimise the possibility of a simple error in the implementation. Furthermore
they exhibit clearly that our framework and in particular our regularisation of
the variables as discussed in Appendix~\ref{troubleterms} is in principle
suitable for simulations in arbitrary dimensions.
We first consider the convergence of the constraints analogous to the results
displayed in Fig.~\ref{constraintsplot} for $D=5$. Compared to those
simulations, the only change we have applied in $D=6$ is to set the gauge
parameters to $\eta_K=\eta_{K_\zeta}= \eta=2$. This choice enables us to evolve
single BHs to about $10~\mu$ when instabilities cause the runs to abort.
\begin{figure}[htb]
\centering
\subfloat[Hamiltonian constraint]{
\includegraphics[clip=true,width=0.46\textwidth]{hamd6_t08}
}
\quad
\subfloat[$y$-component of the momentum constraint]{
\includegraphics[clip=true,width=0.46\textwidth]{momd6_t08}
}
\caption{Constraints at time $t=8\mu$, for the evolution
of a single Tangherlini BH in six dimensions.}
\label{constraintsd6}
\end{figure}
In Fig.~\ref{constraintsd6} we show the Hamiltonian and the $y$-component of the
momentum constraint at $t=8~\mu$ along the $y$-axis. As for $D=5$, the high
resolution result is amplified by a factor $1.5^4$ expected for fourth order
convergence \cite{Alcubierre:2008}. While the convergence appears to be closer
to second order in some patches of the momentum constraint, the results are
clearly compatible with the numerical discretization.
For the second test, we compare the numerical evolution of a single $D=6$
Tangherlini BH with the analytic solution, using geodesic slicing. This
comparison is more difficult in the present case than in $D=5$,
\begin{figure}[htb]
\centering\includegraphics[clip=true,width=0.46\textwidth]{geodesic_d6_hxx}
\caption{Numerical values versus the semi-analytic
expression of $\tilde{\gamma}_{xx}$ (cf.~Appendix~\ref{geoslice})
along the $x$-axis for the single Tangherlini BH in six dimensions.
}
\label{geodesic_d6_hxx}
\end{figure}
because the line element analogous to \eqref{geodesicmetric} cannot be obtained
in a simple analytic form. In Appendix~\ref{geoslice} we demonstrate how a
semi-analytic solution can be obtained for the metric. In
Fig.~\ref{geodesic_d6_hxx} we compare this expression with the three dimensional
numerical values at times $\tau=0.5~\mu$, $0.7~\mu$ and $0.72~\mu$. The
agreement is excellent and demonstrates that our code works well at least up to
the point where instabilities set in. As mentioned above, resolving these
stability problems will be of the highest priority in future extensions of our
work.
\section{Final Remarks}
\label{sec:final-remarks}
In this paper we present a framework that allows the generalisation of the
present generation of 3+1 numerical codes to evolve, with relatively minor
modifications, space-times with $SO(D-2)$ symmetry in $5$ dimensions and
$SO(D-3)$ symmetry in $D\ge 6$ dimensions. The key idea is a dimensional
reduction of the problem along the lines of Geroch's \cite{Geroch:1970nt}
procedure that recasts the $D$~dimensional Einstein vacuum equations in the form
of the standard four dimensional equations plus some quasi-matter source terms.
The resulting equations can be transformed straightforwardly into the BSSN
formulation that has proved remarkably successful in numerical evolutions of BH
configurations in 3+1 space-times. We have isolated several issues related to
the regularisation of the variables used in our formulation and demonstrated how
all difficulties related to the coordinate singularity arising out of the use of
a ``radius-like'' coordinate can be successfully addressed in a numerical
implementation. We have further illustrated how initial data for single,
non-spinning BHs as well as BH binaries with vanishing initial extrinsic
curvature can be adapted straightforwardly to the formulation presented in this
paper. More generally, the class of problems that may be studied with our
framework includes head-on collisions in $D\ge 5$ and a subset of BH collisions
with impact parameter and spin in $D\ge 6$.
As might be expected, stable evolutions of such space-times require some
modifications of the underlying methods of the so-called {\em moving puncture}
technique, especially with regard to the gauge conditions used therein. We have
successfully modified the slicing condition via incorporation of the canonical
momentum of the quasi-matter field in order to obtain long-term stable
simulations in $D=5$ dimensions. Unfortunately, these modifications do not
appear sufficient to provide long-term stability for arbitrary values of the
dimensionality $D$. We will address this important issue in the form of a
systematic study in future work.
We have tested our framework by adapting the {\sc Lean} code and performed a
variety of single BH space-times. Most importantly, we have demonstrated the
internal consistency of our numerical framework in $D=5$ and $6$ dimensions by
showing convergence of the Hamiltonian and momentum constraints as well as
comparing numerical results with (semi-)analytic expressions for a single
Tangherlini BH in geodesic slicing. We have further shown for $D=5$ that the
head-on collision of a BH binary successfully merges into a single hole which
settles down into a stationary state and can be evolved numerically for long
times, hundreds of $\mu$ in the present example.
A complete study of such BH binary evolutions requires the implementation of
gravitational wave extraction in arbitrary dimensions as well as the
generalisation of apparent horizon diagnostics beyond $D=4$. Both are currently
being implemented in the {\sc Lean} code and will be discussed in detail in
future work.
In spite of several open questions, we believe that our formalism will open up a
vast range of uncharted territory in BH physics for contemporary numerical
relativity. The list of possible applications and extensions of our framework
is too large to be included here, and we merely mention strong hyperbolicity
studies of the BSSN formulation with sources and systematic investigation of BH
binary dynamics in $D$~dimensions. These studies are under way and will be
reported elsewhere.
\begin{acknowledgments}
We would like to thank L. Lindblom and M. Sampaio for discussions. We also
thank the participants of the V Iberian Cosmology Meeting, the XII Marcel
Grossmann Meetings, the Spanish Relativity Meeting and the I and II Black
Holes Workshop for useful feedback. M.Z. and H.W. are funded by FCT through
grants SFRH/BD/43558/2008 and SFRH/BD/46061/2008. V.C. acknowledges financial
support from Funda\c c\~ao Calouste Gulbenkian through a short-term
scholarship. V.C. and C.H. are supported by a ``Ci\^encia 2007'' research
contract. A.N. is funded by FCT through grant SFRH/BPD/47955/2008. This work
was partially supported by FCT - Portugal through projects
PTDC/FIS/64175/2006, PTDC/FIS/098025/2008, PTDC/FIS/098032/2008
PTDC/CTE-AST/098034/2008, CERN/FP/109306/2009, CERN/FP/109290/2009 as well as
NSF grants PHY-090003, PHY-0900735, PHY-0601459, PHY-0652995
and the Fairchild foundation to Caltech.
Computations were performed on the TeraGrid clusters ranger and kraken and at
Magerit in Madrid.
The authors thankfully acknowledge the computer resources, technical
expertise and assistance provided by the Barcelona Supercomputing Centre ---
Centro Nacional de Supercomputaci\'on.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,741 |
\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{A}{dvances} in
virtual reality (VR) and augmented reality (AR) headsets
have fueled interest in 3D graphics for information visualization
and `immersive analytics'
\cite{marriott2018,fonnet2019,ens2021,kraus2022}. %
For datasets with a natural 3D embedding,
such as 3D medical images or 3D models of buildings,
there is clear value in 3D visualization.
On the other hand, for abstract data such as
networks \cite{vonlandesberger2010} %
or multidimensional multivariate data \cite{wong1997},
the use of 3D is often advised
against \cite{munzner2014commentsOn3D}
due to previous studies
that have found 2D to be better (e.g., \cite{sedlmair2013scatterplots,jansen2013evaluating}).
One counter-example is the task
of path tracing in networks,
which was shown in a carefully designed experiment
\cite{ware2008} to be less error-prone when using a 3D layout
with stereo and motion parallax depth cues.
Practical implications remain unclear:
should networks be embedded in 3D?
The lack of clear implications is
partly because
the previous study did not allow the user
to control their view in 3D,
nor leverage interaction with an input device,
nor benefit from modern edge-routing \cite{dwyer2009fast}
in 2D (edges in \cite{ware2008} were simply drawn
as straight line segments, resulting in more occlusion).
We extend this previous work by experimentally comparing path tracing under new conditions that are more relevant to modern VR/AR headsets,
and find that 3D remains advantageous over 2D in terms of error rate. %
\begin{figure*}[!t]
\centering
\includegraphics[width=1.9\columnwidth]{fig/teaser3.png}
\caption{
In our two studies, users had to find the distance (in edges)
between the two nodes indicated in red.
Left and Middle: the \VRTwoHilite\ and \VRThreeHilite\ conditions,
where a mouse moved a green cursor, highlighting edges incident
on the node under the cursor.
Right: the \ARTouch\ condition,
where the user could touch a 3D printout,
and a Microsoft HoloLens augmented reality (AR) headset indicated the nodes.
}
\label{fig:teaser}
\end{figure*}
Our work is also informed by the recent trend
of physicalization of data \cite{jansen2015phys,dataphyswiki},
e.g., via 3D printing. The ability to touch a
tangible rendering of data
can yield advantages
over an equivalent virtual 3D visualization
\cite{jansen2013evaluating}.
This is likely in part because the user's fingers
can mark elements in a physicalization,
to ``remember'' a location %
and facilitate comparison with other elements. %
To date, however, there have been no empirical evaluations
of physicalizations of networks with 3D layouts.
Physicalizations also open the intriguing possibility of being augmented
with virtual information displayed using an AR headset.
Our 2nd study is the first to experimentally evaluate
a physical network with a 3D layout,
and also the first to use AR to augment physicalized networks, an example of what we call
{\bf augmented physicalization}.
\section{Background}
The choice between visualizing data in 2D and 3D
is most controversial when the data is abstract,
having no intrinsic embedding.
Previous works offer much advice
\cite[Section 3]{munzner2008pitfalls},
\cite{munzner2014commentsOn3D,
brath2014dealwithit,
ware2000attitude,lee2022,
shneiderman2003better}.
Problems with 3D \cite{brath2014dealwithit,sedlmair2013scatterplots}
include occlusion hiding information,
ambiguous depth,
distortion due to perspective,
complex navigation,
and difficulty reading text.
Advantages of 3D include having an additional visual channel for encoding a variable,
ability to have multiple views in 3D space \cite{collins2007vislink,marriott2018},
and depth cues sometimes making it easier to find information \cite{zou2022stereo}.
VR and AR \cite{kim2018} headsets
provide immersion, 3D input,
and enhanced depth cues, in particular head-coupled motion
and stereo disparity \cite{mcintire2014goodbadugly,mcintire2014stereoreview,zou2022stereo}.
Recent uses of these
platforms for information visualization
include
ImAxes \cite{cordeil2017imaxes} and DataHop \cite{hayatpur2020datahop}.
Input to such systems is often via handheld
controllers or whole hands.
Some systems use tangible input devices \cite{cordeil2017tangibleimmersive}
designed to better match
visualization tasks.
Recent examples include
a tangible cutting plane \cite{bach2018hologram}, %
tangible axes \cite{cordeil2020embodied, smiley2021madeAxis},
and a globe of the earth \cite{satriadi2022},
which is both a tangible input device
and a physicalization
\cite{jansen2015phys}
of geographic data.
Other systems
present virtual information on top of
a tangible physicalization {\em without}
using a headset \cite{gillet2005,taher2015emerge,taher2017emerge}.
The next two sections focus on previous
empirical evaluations of
2D vs 3D embeddings of networks,
and visualization vs physicalization of data.
\subsection{Comparing Networks in 2D and 3D}
Network visualization constitutes a large literature
\cite{vonlandesberger2010,dibattista1999,kaufmann2001,yoghourdjian2018,burch2020}. %
Some previous works have evaluated networks
embedded in 3D
\cite{kwon2016, cordeil2017cavehmd, drogemuller2020}
but without focusing on the question of comparing
a flat 2D layout (on a plane) vs fully 3D layout
(with nodes distributed throughout a volume).
In Kwon et al.\ \cite{kwon2016},
the nodes of the network were laid out on a curved surface,
whereas the other works \cite{cordeil2017cavehmd, drogemuller2020}
did not employ a flat 2D layout.
Irani and Ware \cite{irani2003geon} compared
network-like structures rendered with 2D and 3D depth cues,
but always with a flat layout of the nodes.
Other works have compared flat 2D and fully 3D layouts
for tasks related to highlighted subsets of nodes \cite{alper2011stereohiliting}
and counting clusters of nodes \cite{greffard2011community, greffard2014beyond}.
Our work extends previous studies of {\em path tracing}, also called
path finding or path following, where users identify a sequence of nodes.
This is a standard task \cite{lee2006task} with networks,
used in multiple previous studies
\cite{kwon2016,cordeil2017cavehmd,drogemuller2020,james2020,drogemuller2021}
that were not focused on comparing flat 2D vs fully 3D layouts.
Path tracing has also been used to compare monoscopic and stereoscopic
viewing of structures resembling angiograms
\cite{sollenberger1993,vanbeurden2010}.
Studies comparing 2D and 3D layouts for path tracing within networks
are reported in \cite{ware1996evaluating,belcher2003,ware2008}.
Two of these \cite{ware1996evaluating,belcher2003} used networks with random layouts, making them less relevant to real visualizations.
The most recent \cite{ware2008},
summarized below,
is also the most carefully designed.
Unlike our current work, none of the previous studies
involving path tracing
employed edge-routing in their 2D network visualizations.
\subsubsection{Ware and Mitchell (2008)}
Ware and Mitchell \cite{ware2008} report two studies,
and we focus on the first of these, which we abbreviate as W+M.
For each trial, two nodes were highlighted.
Users had to indicate if the shortest path between the two
nodes was 2 or 3 edges, a forced choice response.
There were
5 viewing conditions:
2D layout,
or a 3D layout with \{monoscopic, stereoscopic\} projection
$\times$ \{no motion, motion in the form of automatic rotation at 10$^\circ$ per second\}.
Users could not actively change their view,
either by moving their head
nor through any input device.
Viewing time was limited to 5 seconds per trial
(i.e., a rotation of 50$^\circ$ in the conditions with `motion').
Results showed that the highest error rate occurred in the
2D and 3D monoscopic conditions;
and the lowest error rate was with 3D stereoscopic + motion,
demonstrating an advantage of the fully 3D condition over 2D.
W+M focused on
``visual searches that could be conducted rapidly'' \cite{ware2008}.
Our studies are designed to be more realistic
and relevant to VR/AR headsets.
Our participants
can freely change their view of the network
by moving their head and hand.
In W+M,
the user's field-of-view (FOV) was $\approx$
26$\times$16$^\circ$ per eye,
much smaller than the FOV of the VR headset used
in our Study 1,
and slightly smaller than the AR headset in our Study 2.
In addition, our 2D conditions use
a state-of-the-art routing
algorithm \cite{dwyer2009fast} (Figures~\ref{fig:teaser}(Left) and Figure~\ref{fig:study1-2D}),
to make better use of space
and reduce ambiguity.
Our experimental task involves paths
that are longer.
Our Study 1 also involves
conditions with interactive highlighting,
double the number of participants of W+M,
and was preregistered (Section~\ref{sec:study1predictions}).
\subsection{Evaluating Data Physicalizations}
Jansen et al.\ \cite{jansen2013evaluating}
evaluated physical barcharts.
Their first study
compared
4 conditions:
2D virtual barcharts,
3D virtual barcharts displayed monoscopically
and stereoscopically (rotation performed with a mouse in both 3D virtual conditions),
and 3D physical barcharts that users could touch.
In terms of time, 2D was the best,
but more interestingly,
3D physical was the 2nd best.
Their second study investigated
why 3D physical might be better than 3D virtual,
comparing 4 conditions:
(1) virtual 3D monoscopic with mouse for rotation,
(2) virtual 3D monoscopic with a prop for more direct rotation,
(3) physical 3D without being allowed to touch,
and (4) physical 3D with touch allowed.
In terms of time,
the 4th condition was best,
and the 3rd condition was 2nd best.
Drogemuller et al. \cite{drogemuller2021} evaluated
networks with a flat, 2D layout,
ranging from 16 to 24 nodes in size,
with %
3 tasks,
comparing 4 conditions:
virtual on-screen (``graphical-only''),
and physical printouts
that could be seen (``visual-only'')
or touched (``haptic-only'') or both (``visual-haptic'').
Users preferred the physical printouts that could be seen
and touched, but within the path tracing task,
no differences are reported in error rates between
graphical-only, visual-only, or visual-haptic.
Our work is the first to empirically
evaluate physicalized networks with 3D layouts.
Also, unlike previous work,
our physicalized networks were augmented with an
AR headset to indicate end-nodes.
\section{Overview of Both Studies}
The following questions motivate our work:
is path tracing easier in networks presented in 3D than in 2D
when edge-routing is used in 2D,
and when the user can interact with the network using a pointing device?
Also, is path tracing easier with a physical 3D representation?
In both our studies, the task was to find the length (between 1 and 5 edges) of a shortest path between two end-nodes indicated by the system.
The user's non-dominant hand (NDH) held and repositioned the network,
because this matches the use of the NDH in the kinematic chain model \cite{guiard1987},
and because there is some evidence that rotation
via a handheld prop
is superior to using a mouse for the same purpose \cite{jansen2013evaluating},
and because it provides an easy-to-understand way to
simultaneously
pan and zoom within a 2D layout,
by simply translating the layout sideways or holding it closer or father away.
In addition, in some conditions, the user's dominant hand (DH) could
move a mouse cursor over nodes (causing incident edges to highlight)
or touch a physical 3D printout of the network.
In all conditions of both studies, the NDH
activated a trigger button to open a radial button
to provide the user's answer from 1 to 5.
The use of the NDH in this way allowed the user to complete each trial without
the ``homing time'' of moving a hand back and forth between two places.
(Had we instead used the DH to open the radial menu, then the \ARTouch\ condition in Study 2
would have required having the user move their DH between the physical network and a button to open
the menu.)
A radial menu was used so that every answer would take the same amount of time to select.
A single set of networks was used for both studies,
from which
networks were randomly chosen
for each condition and each user.
\subsection{Network Size, Topology and Layout}\label{sec:networks}
We generated 10 networks.
For each network, we computed its layout in 3D,
and projected the 3D node positions down to a plane to obtain a layout in 2D.
Each network can be displayed in virtual 2D or 3D,
and was also
3D printed using stereolithography (SLA) with a white plastic.
Each network has 70 nodes
and 140 edges (hence an average degree of 4),
and was generated with a Watts-Strogatz \cite{watts1998} small-world synthesis algorithm.
The algorithm begins by constructing a regular ring lattice of 70 nodes
each with degree 4.
Each edge is then randomly rewired with 20\% probability.
The average degree distribution that resulted over the 10 networks
was: 1, 15.7, 38.2, 12.9, 1.8, and 0.4 nodes
of degree 2 through 7, respectively.
Each of the 70 nodes was assigned a unique 2-character string label such as ``AA'', ``FE'', or ``HL''.
Layout of nodes was performed in two passes.
The first pass uses stress majorization
(equation 12 in \cite{gansner2004}) to position the nodes in 3D.
Projecting node positions down to a 2D plane results in overlap between labels,
hence a second pass applies repulsive forces between nodes whose labels overlap in the 2D plane, pushing nodes away from each other in the horizontal plane.
The new positions are saved in both 2D and 3D.
Thus, the 2D layout uses the projected coordinates of the 3D layout.
Next we compute the layout of edges.
In the 3D case, each node is modeled as an
elongated box (7$\times$7$\times$15 mm)
with a text label on one side,
and each edge is modeled as a single segment (3~mm thick),
with each edge's endpoint connected to the center or extremity
of the node's box in such a way as to avoid
extending through the labeled face of the box.
In 2D, nodes are 7$\times$11~mm rectangles,
and each edge is a multi-segment polygonal line (0.5 mm thick),
whose layout is computed using
the MSAGL (Microsoft Automatic Graph Layout)
library \cite{msagl}
based on \cite{dwyer2009fast}.
The networks displayed in 3D virtual form have the same geometry
as the physically printed networks: the same node dimensions, same edge thickness, same color (white),
and same font used for labels.
The 2D virtual form also uses the same color and font.
The number of nodes for the networks
was chosen based on physical limits of common 3D printers.
First, we wanted networks that would fit within a
6$\times$6$\times$6 inch volume,
to accommodate lower-end printers.
In addition, a 6-inch width
fits within the HoloLen's FOV
at a distance of 30~cm, well within arm's reach.
Second, we wanted edges to be 3~mm thick to avoid fragility.
Third, we wanted nodes big
enough to accommodate embossed text labels
that would be clear even on lower-end
FDM
(Fused Deposition Modeling) printers.
We implemented a custom font (each character a 5$\times$4 bitmap)
allowing us to print text labels on nodes 7~mm high
with 1~mm stroke thickness.
We found that 70 nodes resulted in a network
of reasonable complexity that fit the size constraints.
Testing revealed that embossed text on mono-color 3D printouts
is both difficult to read and to paint,
and because multi-color 3D printing is
much more expensive, we finally used 2D printed stickers
with the same custom font to label each node.
\section{Study 1: 2D and 3D Virtual}
Study 1 was preregistered \cite{mcguffin2022prereg},
done in VR,
and crossed the dimensionality \{2D, 3D\} of the network layout
with the use of a mouse to highlight edges.
\subsection{Main Conditions} %
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/study1_2D_mouse.png}
\caption{
In study 1, in the \VRTwoHilite\ condition,
the user can move the mouse cursor over a node, causing
incident edges to highlight in green (Top Left).
Once the user has determined the distance in edges between the
two nodes indicated in red (AJ and GG), they hold down the trigger
button with their non-dominant hand (NDH), causing a radial menu to appear
(Top Right). By tilting their held down and right, they select
``3'' in the menu, and then release the trigger button
to complete their answer. In this example, their answer is wrong,
so the system displays error feedback:
a correct shortest path of length 4 is highlighted
(Bottom Left). Another example of error feedback
(Bottom Right) is for the network
shown in Figure~\ref{fig:teaser}(Left), where the shortest path has length 5.
}
\label{fig:study1-2D}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/study1_3D_mouse.png}
\caption{
Study 1, the \VRThreeHilite\ condition.
The first three images show a trial ending with error feedback
displaying a correct shortest path of length 3 (Bottom Left).
The last image (Bottom Right) shows a correct shortest path of length 5 for the network shown in Figure~\ref{fig:teaser}(Middle).
}
\label{fig:study1-3D}
\end{figure}
The previous W+M study \cite{ware2008}
found that 3D outperformed 2D,
but it is plausible that this could change
if the 2D condition is improved with edge-routing,
or if the user can use a pointing device for simple interaction with
the network.
The simplest interaction we could think of
that might help with path tracing is for the edges incident on a node
to highlight when the user hovers over that node with a
pointing device.
This led to our choice of conditions for Study 1.
The independent variable \variable{MainCondition}\ has
four possible values:
\begin{itemize}
\item \VRTwo: Virtual network displayed in VR with 2D layout. The non-dominant hand (NDH) holds a controller to position the network with 6 degrees of freedom (DoF).
\item \VRTwoHilite: Same as previous, but
with a mouse in the dominant hand (DH) used to highlight edges.
\item \VRThree: Virtual network displayed in VR with 3D layout.
The NDH holds a controller to position the network with 6 DoF.
\item \VRThreeHilite: Same as previous, but
with a mouse in the DH used to highlight edges.
%
%
%
\end{itemize}
We decided to not compare with 2D conditions on a desktop screen
(without VR headset) because this would have introduced confounds
in having different display hardware,
and also potentially different input devices.
The VR handheld controller in the user's NDH provides
an easy-to-understand and quick way to
reposition a 3D network layout, and also
simultaneously pan and zoom
when examining a 2D network layout,
and there is no equivalent NDH input on standard
desktop PCs.
\subsection{Task} %
The experiment consisted of a sequence of trials
where the task
required the user to find the length, in edges,
of a shortest path between two nodes (the path's {\em end-nodes}) in the network.
The independent variable \variable{PathLength}\ ranged from 1 to 5,
where 1 means the end-nodes are neighbors.
The shortest path was not necessarily unique.
At the start of each trial, the two end-nodes were indicated
with red callout line segments,
as well as with red circular rings.
The user then examined the network by repositioning and rotating it with their
NDH.
Bringing the network closer to their eyes allowed them
to effectively ``zoom in'' to see more detail.
In some conditions (\VRTwoHilite\ and \VRThreeHilite), the user could also move a mouse with their DH, causing a cursor to hover over different nodes.
Whichever node was under the mouse cursor was highlighted
in green,
as were all the edges incident on that node. %
Users were told that using the mouse in these conditions was not mandatory,
but that it might allow them to more easily find the shortest
path and confirm its length.
The mouse buttons served no purpose.
Once the user thought they knew the answer, they pressed a trigger button using
the index finger of their NDH to open up a radial menu containing the answers
1 through 5 (Figures~\ref{fig:study1-2D}(Top Right) and \ref{fig:study1-3D}(Top Right)).
To select within this menu, the user tilted their head slightly
in the direction of the desired answer and released the trigger button.
Releasing the trigger button without tilting their head dismissed the radial menu
and allowed the trial to continue.
The trial ended only when the user made a selection within the menu.
A text message appeared immediately after to inform the user if their answer
was correct or not, and in the latter case,
the text also indicated the correct answer,
and the system highlighted a shortest path
(Figures~\ref{fig:study1-2D}(Bottom) and \ref{fig:study1-3D}(Bottom)).
The system then moved on to the next trial.
Users were aware that they could take their time during the warmup trials
at the start of each condition,
but after these warmup trials,
they were instructed to complete trials as quickly as possible with no errors.
The following was also explained to each user.
The correct path length is always at most 5,
and, sometimes, the shortest path is quite difficult to find.
After 20 seconds into a trial, the text instructions displayed by the headset turn red. Once this happens, the user is free to continue searching for a shortest path if they so desire, but a reasonable strategy after 20 seconds would also be to simply estimate an answer such as 5, or perhaps 4, even if the user has not found a path of that length.
(This was explained to avoid having users spend too much time searching for difficult paths.)
On the other hand, if a user answers quickly and incorrectly,
the system imposes a ``punishment'' of a delay of up to 15 seconds
before proceeding to the next trial.
(If $t$ was the time in seconds taken by the user to give an incorrect answer,
the precise delay imposed after the trial
was $\max(\min(20-t,15),5)$, i.e., a decreasing ramp function
clamped between 15 and 5 seconds.)
This disincentivizes a user from answering sloppily to complete the experiment faster.
The software asked users to take breaks between conditions.
Users could indicate that they were ready to proceed by pressing a key
on a keypad with their DH.
A Lego brick attached to the key made it easier to feel when the user
was wearing the VR headset.
\subsection{Pilot and Predictions}\label{sec:study1predictions} %
A pilot was performed with 6 users,
after which minor tuning to the protocol was made,
and a preregistration\cite{mcguffin2022prereg}
was archived to declare the number of users to recruit,
criteria for including participants,
predictions to test,
and the R script for plotting data and testing predictions.
Two predictions were preregistered:
first, that the error rate (averaged over trials of \variable{PathLength}\ 2, 3 and 4, and averaged over conditions with and without the mouse)
would be smaller in 3D than in 2D;
and second, that the error rate (averaged over trials of \variable{PathLength}\ 2, 3 and 4, and averaged over 2D and 3D conditions)
would be smaller with the mouse than without the mouse.
The R script tests each of these predictions by computing a single error rate for each
user and each subset of conditions, not including warmup trials,
and then performing a paired sample $t$-test. %
The W+M study \cite{ware2008} found that 3D yielded
a smaller error rate, possibly because
stereo and motion disambiguate edges.
A second mechanism that could play a role is
that shortest paths in 3D tend to follow a more straight line
(hence, are easier to perceive)
than in 2D.
This 2nd mechanism may not have been at play in W+M
because they
``selected paths in such a way that the mean Euclidean distance between start nodes and end nodes was the same'' \cite{ware2008},
regardless of whether the path was 2 or 3 edges long.
However, both mechanisms could benefit 3D in our study,
since we do not hold this Euclidean distance constant.
The edge-routing in our 2D layouts can also make 2D layouts
easier to read,
but the paths are still ``less straight'' than in 3D.
The reason our predictions about error rates exclude
\variable{PathLength}\ 1 and 5 is that those cases tend to be easier for users: in the case of 1, the nodes are often clearly adjacent,
and in the case of 5, the user knows that the length cannot be greater than
5 and can therefore guess a length of 5 when the path is difficult to find.
\subsection{Mouse Cursor in 2D and 3D} %
To allow the user to hover over a node for interactive highlighting,
we wanted to use the same pointing device in the 2D and 3D conditions,
for simplicity and consistency across conditions.
Although a 6 DoF handheld controller could have been used for pointing,
the mouse is very often used for raycast pointing in 3D,
whereas controllers are rarely used for pointing in 2D.
Furthermore, a 2D mouse may be easier and less tiring to control
than a handheld controller, because the user's arm can partially rest on the
desk, and the depth dimension is automatically handled by the software.
We therefore implemented a variant of raycast pointing with a 2D mouse.
In our variant of raycast pointing,
we wanted the user
to be able to position the mouse cursor over a node with their DH,
and then move or rotate the network with their NDH while the mouse cursor
remains `stuck' on the same node.
Therefore, the mouse cursor's position is stored in the network's local 3D space,
and the DH applies relative displacements.
Whenever the DH moves the mouse, a proportional translation is applied to the mouse
cursor parallel to the camera plane (i.e., the plane perpendicular to the camera's forward direction).
After applying this translation, a ray is cast from the camera position through the cursor,
and if this ray encounters one or more nodes, our software finds the intersection between the ray
and the node closest to the camera, and moves the cursor to that intersection.
The user therefore sees the cursor automatically jump forward or backward
to stick to the nearest node, but these jumps only happen if the mouse is being moved.
If the user is only using their NDH to reposition or rotate the network,
no automatic jumps happen, and the cursor retains its position in the network's local space.
Like EZCursorVR \cite{ramcharitar2018},
the size of our mouse cursor is scaled to be bigger when the cursor
is farther from the camera, so that the projected size of the cursor on the camera plane
appears constant.
\subsection{Hardware} \label{sec:study1hardware} %
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/study1_hardware.png}
\caption{
The equipment for Study 1:
HTC Vive headset and controller (held in the NDH),
mouse (for the DH)
and keypad (to advance to the next trial after a break).
}
\label{fig:study1hardware}
\end{figure}
We used an HTC Vive headset,
which has a 2160$\times$1200 resolution (1080$\times$1200 per eye)
and $\approx$110$^\circ$
FOV. %
We measured a framerate of 90 fps.
The handheld controller was held in the user's NDH.
The headset was connected to a PC with an Intel i7-8700K 6-core CPU at 4.7 GHz,
water cooling,
and Nvidia GeForce GTX 1080 Ti GPU.
A Logitech M100 mouse
(with default acceleration settings in Microsoft Windows 10)
on a gaming mouse pad was held in the user's DH.
\subsection{Measurement of rotation} \label{sec:rotationCalculation}
In the 3D conditions, we %
measured how much the user looked at the network from different points of view. %
Let $H_t$ be the position of the user's head at time $t$,
and let $N_t$ be the pose (position and orientation) of the network at time $t$.
We define a direction vector $d = d(H,N)$ as a function of $H$ and $N$,
where $d$ points from the network to the head, and where the components of $d$
are computed in the local space of the network
(i.e., a change in the position of the head, or in the position of the network,
or in the orientation of the network, will each change the components of $d$).
We compute $d_t = d(H_t,N_t)$ for each frame during a trial,
and at the end of the trial, we compute the mean direction $\bar{d}$,
and compute and record the standard deviation of the angles between all the $d_t$ and $\bar{d}$.
This standard deviation is an overall measure of how much the user looked at the network from
different points of view.
We also computed how much the user rotated the network with their hand,
versus how much they moved their head to look at the network from different points of view,
expressed as two percentages that sum to 100\%.
To compute these percentages,
we compute the directions
$d_{t,H} = d(H_t,N_{t-1})$ and $d_{t,N} = d(H_{t-1},N_t)$
that would have resulted if only the head, or only the network, respectively,
had moved.
We find the angle $\alpha_H$ between $d_{t-1}$ and $d_{t,H}$,
and the angle $\alpha_N$ between $d_{t-1}$ and $d_{t,N}$,
and then define the contribution of the head motion to the rotation as $\alpha_H / ( \alpha_H + \alpha_N )$,
and the contribution of the network's motion as $\alpha_N / ( \alpha_H + \alpha_N )$.
These fractions are computed for each frame of the trial (except the first frame),
averaged over the entire trial,
and recorded as percentages.
\subsection{Protocol} \label{sec:study1protocol} %
Equipment was disinfected prior to each user session.
At the start of each session,
after signing a consent form,
users had their
interpupillary distance (IPD) measured, %
as well as their
stereo acuity,
which was assessed
using the `circle test' of the FLY stereo acuity test
by Vision Assessment Corporation. %
(This resulted in a score on a scale of 10, corresponding to
a disparity of
400, 200, 160, 100, 63, 50, 40, 32, 25, or 20 seconds of arc.)
Users then filled out a pre-questionnaire,
and were also shown several printouts of example trials,
with conditions in random order,
to explain the task and test their understanding of
path length.
The equipment for the experiment was then explained to the user,
the headset was adjusted for comfort, and the IPD of the headset
was set to the value measured earlier.
After the trials were completed, a post-questionnaire was filled out.
\subsection{Users} %
A sample size of 34 users was chosen in the preregistration,
not including the pilot participants. This number was chosen to achieve a power of 0.8 at $\alpha=0.05$ for a medium effect size of 0.5, as calculated using the G*Power software\cite{gpower} and also with an online calculator \cite{dhand2014}.
Of the 34 users,
24 were male, 10 female;
28 right handed, 4 left handed, 2 ambidextrous, but all with a habit of using the mouse with their right hand;
age 19 to 45 years (average 24.7);
IPD 54 to 70mm (average 63.7);
stereoacuity test scores 3/10 to 10/10 (average 8.5).
\subsection{Design} %
Each user experienced the
4 levels of \variable{MainCondition}\ \{\VRTwo, \VRTwoHilite, \VRThree, \VRThreeHilite\}
in random order. For each \variable{MainCondition},
the user performed 10 warmup trials in random order
(5 levels of \variable{PathLength}\ $\times$ 2 repetitions)
with the warmup network,
followed by 20 trials in random order
(5 levels of \variable{PathLength}\ $\times$ 4 repetitions)
with another network chosen at random,
followed by another 20 trials with another network.
There were a total of
34 users
$\times$ 4 levels of \variable{MainCondition}\
$\times$ 2 networks
$\times$ 5 levels of \variable{PathLength}\
$\times$ 4 repetitions per trial
= 5440 trials, not counting warmup trials and not counting pilot data.
Each session with a user lasted $\approx$ 1.5 hours.
\subsection{Results} %
For these results,
advice was adapted
from Dragicevic \cite{dragicevic2016fair}.
Although we report some $p$ values,
we do not emphasize null hypothesis significance testing (NHST) %
as we wish to avoid misleading, dichotomous thinking
(``Tip 25'' in \cite{dragicevic2016fair}).
We present effect sizes visually
and with CIs (Tips 15, 16),
where the CIs are computed using one (averaged) value for
each (user, condition) pair (Tip 9);
and we clearly distinguish between pre-experiment predictions and
post-hoc exploratory data analysis,
to avoid
HARKing (Hypothesizing After the Results are Known) and $p$-hacking.
All CIs are 95\%.
The CIs for error rates were calculated using
bootstrapping, to prevent them from falling outside the [0,100\%] range.
For other variables such as time, CIs were calculated using
the $t$-distribution.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/study1-prediction1-cropped.png}
\caption{
Error rates in study 1, %
excluding \variable{PathLength}\ 1 and 5.
``2D'' is the union of \VRTwo\ and \VRTwoHilite;
``3D'' is the union of \VRThree\ and \VRThreeHilite.
Each dot is the average for one user,
and the bars show 95\% confidence intervals (CIs),
computed from the 34 users.
A paired $t$-test yields $p<0.0000005$.
The 3D conditions resulted in a lower error rate than 2D,
confirming the first of our preregistered predictions.
}
\label{fig:results-study1-prediction1}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/study1-prediction2-cropped.png}
\caption{
Error rates in study 1, %
excluding \variable{PathLength}\ 1 and 5.
``WithHilite'' is the union of \VRTwoHilite\ and \VRThreeHilite;
``WithoutHilite'' is the union of \VRTwo\ and \VRThree.
A paired $t$-test yields $p<0.09$,
providing limited evidence of our second preregistered prediction,
that the mouse improves error rates.
}
\label{fig:results-study1-prediction2}
\end{figure}
Figures~\ref{fig:results-study1-prediction1} and
\ref{fig:results-study1-prediction2}
show the results of testing the preregistered predictions.
The fact that the zero line falls far outside
the CI of the difference in the first figure,
and barely intersects the CI of the difference in the second
figure,
is reflected by the very small $p$ value in the first case
and a $p$ value somewhat larger than 0.05 in the second case.
We thus have strong evidence that 3D results in a lower error rate,
and limited evidence that the error rate is reduced by
interactive highlighting of edges with the mouse.
\subsubsection{Subjective Feedback and Exploratory Data Analysis} %
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/study1-errorRate2-cropped.png}
\caption{
Error rates in study 1 by \variable{MainCondition},
including all \variable{PathLength}\ values 1-5.
}
\label{fig:results-study1-errorRate2}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/subjectiveResults.png}
\caption{
Subjective results of each study.
}
\label{fig:subjective}
\end{figure}
In subjective comments,
10 out of the 34 users described 3D positively as compared to 2D, saying that 3D makes the task easier, simpler, less confusing, fun, making pathways clearer, only requiring rotation. One of these users mentioned longer paths as being clearer in 3D than in 2D.
Figure~\ref{fig:results-study1-errorRate2} shows
error rates by \variable{MainCondition}, suggesting
that interactive highlighting of edges
with the mouse helped in 2D but not in 3D.
This is further supported by comments made by users
and by the subjective results in
Figure~\ref{fig:subjective}.
(For ease of comparison, the results
of Study 2, discussed later,
are presented alongside several figures.)
There, we see that \VRTwo\ was
the least favorite condition,
and that \VRTwoHilite\ required
less mental effort,
produced less frustration,
and better enabled the task.
However, \VRThree\ was the most favorite
condition, not \VRThreeHilite.
14 out of the 34 users described the highlighting of edges with the mouse in 3D (\VRThreeHilite) in negative terms, such as being not intuitive, requiring extra motion and time, or difficult to position the cursor in the depth dimension.
Note that users were given no explanation of
how the mouse worked in 3D,
and it is possible that several users were confused by
it because they were moving their NDH and DH at the same time.
Despite this, Figure~\ref{fig:subjective} also shows that
6/34 = 18\% of users chose \VRThreeHilite\ as their favorite condition,
and 8 users described it as helpful, useful, faster, requiring less rotation of the network, and making pathways more visible.
Figures~\ref{fig:results-errorRate}
and \ref{fig:results-time}
present the error rates and
times in more detail.
Notice that both the times and error rates appear
smaller in 3D than in 2D.
In Figure~\ref{fig:results-errorRate},
if we examine the 2D conditions in Study 1,
it seems that the mouse helped with \variable{PathLength}\ 2 and 3
but hindered performance with \variable{PathLength}\ 4.
This may be because the interactive highlighting sometimes
misled %
users into following suboptimal paths.
This is partially supported by
comments made by
3 users, who talked about the edge highlighting causing them to focus more on those edges, inducing a different ``mental exercise'' than without edge highlighting, and encouraging the user to explore the network by ``testing'' different edges.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/errorRate.png}
\caption{
Error rates in more detail.
%
Each dot is the average for one user,
and the bars show 95\% CIs,
computed from the 34 or 12 users, in studies 1 and 2, respectively.
In the 2D conditions (green and orange) of study 1, notice that edge highlighting with the mouse appears to have helped for \variable{PathLength}\ 2 and 3 but not 4.
}
\label{fig:results-errorRate}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/time.png}
\caption{
Duration of trials.
%
}
\label{fig:results-time}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/rotation.png}
\caption{
How much users rotated their view of the network (Section~\ref{sec:rotationCalculation}).
%
}
\label{fig:results-rotation}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/rotationFraction.png}
\caption{
How much of the rotation was due to hand motion,
as opposed to head motion
(Section~\ref{sec:rotationCalculation}).
%
}
\label{fig:results-rotationFraction}
\end{figure}
As mentioned earlier (Section~\ref{sec:study1predictions}), in the
W+M study, ``the mean Euclidean distance between start nodes and end nodes was the same'' \cite{ware2008}.
Because our paths could be up to 5 edges long,
this kind of control was not feasible in our experiment,
nor would it have yielded realistic tasks.
However, to check how much the Euclidean distance
may have influenced user responses in our Study 1,
we computed two additional variables for each trial:
first, $\Delta$ = response $-$ \variable{PathLength},
so that $\Delta$ is negative, zero, or positive when the user's response
is under, equal to, or over the correct \variable{PathLength}, respectively.
For example, if the user's response is 5 when the \variable{PathLength}\ is actually
3, then $\Delta = 2$.
Second, we found all shortest paths in the network and,
for each \variable{PathLength},
we found the average and standard deviation of the
Euclidean distances (from end-node to end-node) of those paths.
This allowed us to compute a $z$-score for any pair of nodes,
as a way to
compare their Euclidean distance to that of other pairs of nodes
with the same topological distance.
So in a given trial, if the shortest path between the end-nodes
is 3 edges, but the $z$-score for those end-nodes is greater than 1,
this means that shortest paths of 3 edges in that network tend to have
end-nodes that are closer (in the Euclidean sense),
and we might expect such a $z$-score to bias the user
toward over-estimating the \variable{PathLength}\ in that trial.
To test for such a bias, we checked for a correlation between $\Delta$ and the $z$-score.
The correlation test yielded $R < 0.005$ and $p > 0.05$,
hence no evidence of the Euclidean distance biasing the user toward erroneous responses.
During the conditions without mouse,
where the DH was free,
3 out of the 34 users (plus 1 other user from the pilot) were either observed lifting their DH toward the controller during trials, as if trying to touch
the virtual network, and/or described imagining touching or
wanting to touch the virtual network during the post-questionnaire discussion. One of these users said this may have been because of their previous experience with an Oculus Quest 2 which displays the user's hands, and another user suggested using AR to allow the user to see their own hands.
\subsection{Discussion} %
As discussed in Section~\ref{sec:study1predictions},
the lower error rate in 3D may be due in part to the shortest paths
being more straight in 3D, because the layout algorithm has more freedom to position nodes.
For each of the shortest paths in the trials of Study 1,
we computed the following ratio: the sum of the lengths of the edges in the path divided by the Euclidean distance
between the two end-nodes of the path.
This ratio is high if the path is circuitous (i.e., winding), but close to 1 if the path is straight.
The average ratio for our 2D trials was 1.509, but for 3D it was 1.400, thus more straight.
The difficulty of a more winding path may be related to studies finding that participants
take longer to trace curves connecting two targets when the curve is longer,
even when the Euclidean distance between the targets is the same \cite{crundall2008}.
Figure~\ref{fig:results-rotationFraction} suggests
that most of the benefit of rotation (i.e., of motion parallax) comes from
motion of the hand rather than of the head.
Figures 6 and 7 in \cite{ware2008} suggest that motion provides
at least as much benefit as stereo.
Taken together, these suggest that a user would benefit simply from the ability to rotate
a 3D visualization with their hand,
without any headset, stereo, or head-coupled perspective.
We observed that some users wished they could touch the networks.
Previous work \cite{drogemuller2021} found that users preferred physical networks that could be touched.
This motivates our next section.
\section{Study 2: 3D Virtual and Physical}
Study 2 compared virtual and physical representations of networks,
using a VR and an AR headset, respectively,
and the same task as Study 1.
Study 2 was more exploratory,
where the number of participants
was determined by convenience,
and not preregistered.
\subsection{Pilot and Choice of Main Conditions} %
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/prototype_phys_edgeHilite2.png}
\caption{
Prototype implementation of highlighting of physical edges. Nodes FO and AA are indicated with red circular rings. The mouse cursor (green) hovers over HJ (Left) and FL (Right), causing three incident edges to highlight in each case.
This was not used in our studies due to insufficient
tracking accuracy.
}
\label{fig:physicalEdgeHiliting}
\end{figure}
To display virtual information on top of a 3D printed physical network,
which we call {\bf augmented physicalization},
we need some way for the AR headset to know where the network is located.
We tested 3 different ways of tracking the physical network,
including using the headset's built-in camera and a combination of 3 external cameras.
Section~\ref{sec:study2hardware} describes our ultimate tracking method.
Figure~\ref{fig:physicalEdgeHiliting} shows a prototype \ARHilite\ condition,
where the user's DH moves a mouse, and the AR headset displays a virtual cursor and virtual
highlighting on parts of the network.
Unfortunately, we were unable to achieve the accuracy
necessary to clearly highlight individual nodes or edges on a physical network.
We suspect that part of the problem is due to small errors in the IPD
used to generate the stereo rendering on the headset, making virtual imagery slightly
misaligned with physical objects.
We thus dropped the \ARHilite\ condition and do not display the circular rings
around nodes in any conditions of Study 2.
Nevertheless, the end-nodes in Study 2 are indicated with callout line segments.
In VR, these callouts are precisely located, but in AR, there are errors of $\approx$2-3cm
in their apparent locations.
Despite having only approximately
correct positions in AR,
the callouts do help the user find the correct physical end-nodes faster than if the user had no visual aid,
and constitute an example of augmented physicalization.
We expect that the approximate locations in AR
will slow down users compared to VR,
but it is plausible that this will have
minimal impact on the user's error rate in AR.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/study2.png}
\caption{
The first three images show a trial in the \ARTouch\ condition of study 2.
The user touches the indicated nodes
FG and FE and answers ``2'' (Top Right)
only to be shown the error feedback
``Error, answer should be 1''
(Bottom Left).
The last image (Bottom Right) shows the variant of the
\VRThree\ condition that was used in study 2.
In both conditions, nodes are indicated with red callout line segments but not
circular rings, due to insufficient tracking accuracy in AR.
}
\label{fig:study2}
\end{figure}
We ran a pilot with 3 users
and 4 conditions: \VRThree, \VRThreeHilite, \AR, and \ARTouch.
In the \AR\ condition,
the user repositions the network with their NDH but may not touch the network with their DH.
In \ARTouch\ (Figure~\ref{fig:study2}), the user is encouraged to touch the network with their DH.
For each of the 4 conditions,
the user performed 10 warmup trials,
followed by 20 trials with each of two networks.
The ordering of headsets was random, as was the ordering of the pair of main conditions
within each headset.
In contrast with Study 1,
each user was given more explanation of how the mouse worked in 3D for the \VRThreeHilite\ condition.
Sessions lasted 2 hours per user.
All users chose \AR\ as their least favorite condition;
one user found it uncomfortable to wear the HoloLens for so long;
and two users reported that the tracking accuracy
got worse over time (this was probably due to the users
holding the network in different positions during
calibration and during trials).
Therefore, some changes were made for the final Study 2 experiment:
we eliminated the \AR\ condition, which was the least favorite condition of the users
and less realistic than \ARTouch;
we also eliminated the \VRThreeHilite\ condition,
because most users in Study 1 did not find the mouse in \VRThreeHilite\ useful,
and we wanted Study 2 to involve the same number of trials with each headset.
We also increased the number of opportunities for the user to take breaks in both conditions,
and opportunities to redo the calibration
during the \ARTouch\ condition.
With only 2 main conditions,
we could also slightly increase the number of trials per condition while also
decreasing the total duration of each user session (Section~\ref{sec:study2design}).
Thus, Study 2 had two values for \variable{MainCondition}:
\VRThree, and \ARTouch.
As detailed in the next section, the AR condition suffered from a smaller FOV,
latency, and tracking error,
compared with the VR condition.
This creates confounds,
however whichever condition outperforms the other,
the results could be informative:
if \ARTouch\ yields a lower error rate,
despite the shortcomings of the AR system,
this demonstrates the importance of physical realism and/or
the ability to touch the network;
and if \VRThree\ yields a lower error rate,
despite affording no way
to interact with the network,
this shows how important it is for AR systems to be improved
to reach their full potential.
\subsection{Hardware} \label{sec:study2hardware} %
For the \VRThree\ condition, the same headset and controller were used
as in the previous study.
For \ARTouch,
we used a Microsoft HoloLens headset,
which has 1268$\times$720 pixels per eye. %
The FOV for displaying virtual information is limited to $\approx$31$\times$17$^\circ$,
however the physical world is visible through a much wider FOV.
The virtual images are rendered at a fixed focal distance
of $\approx$2 meters\cite{hololensFocalDistance}
however stereo and vergence depth cues create the illusion of virtual imagery at any distance.
We measured a framerate of 30 fps.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/study2_hardware.png}
\caption{
The equipment for the \ARTouch\ condition of Study 2.
Top: the network holder, held in the NDH,
with a trigger button.
Bottom left:
registration rig with 2 fake hands,
Microsoft HoloLens headset,
network holder,
and keypad.
Bottom right: during a trial.
}
\label{fig:study2hardware}
\end{figure}
3D printed networks were held in a ``network holder'' by the user's NDH.
A Polhemus Patriot reported the positions and orientations (in the Patriot's local coordinate system) of two sensors
attached to the network holder.
Because the HoloLens has built-in functionality to detect hands and report their 3D position in the headset's coordinate system,
we constructed a rig with two fake hands (Figure~\ref{fig:study2hardware}(Bottom Left)) whose positions were fixed with respect to the Patriot,
allowing us to determine the position of the network with respect to the headset with an accuracy of $\approx$5cm.
The user also performed a simple calibration procedure
to further improve the accuracy of the tracking to $\approx$2-3cm.
The Patriot was connected to the same PC mentioned in Section~\ref{sec:study1hardware}.
This PC processed the position and orientation information
from the Patriot and transmitted it via UDP packets over wifi to the HoloLens. Although the Patriot can read information at 60 Hz,
we only transmitted 10 packets per second to the HoloLens,
because a higher rate led to dropped packets.
By studying footage recorded through the HoloLens, we estimate that the latency between moving the network holder
and the HoloLens updating its rendered virtual imagery was $\approx$150-250ms,
despite using a dedicated high bandwidth wifi router (ASUS RT-AC5300). %
Having two different headsets in Study 2, with differences in
FOV and other characteristics,
necessarily introduces confounds.
An alternative approach would have been to use the HoloLens in both conditions of Study 2,
however this would have meant imposing a limited FOV that is not representative of the state-of-the-art in VR.
Although the differences in headsets will certainly create differences
in the time taken for trials,
we are more interested in differences in error rate
between the main conditions,
and it is plausible that error rate will be less affected
by the differences in the headsets.
\subsection{Use of blur in virtual feedback}
Like most headsets,
our VR and AR headsets
each reproduce correct stereo disparity
and vergence depth cues,
but not correct accommodation depth cues,
due to the virtual imagery being rendered at a fixed focal distance.
In VR, this results in the well known ``vergence-accommodation conflict'', which is often barely noticeable.
However, with see-through AR headsets like the HoloLens,
there is a further challenge:
if virtual imagery (such as a virtual highlight) is rendered at the same location as
a physical object (such as part of a physical network),
it is impossible for the user to accommodate (i.e., ``focus on'')
both simultaneously. Although both may appear to be 30cm from
the user's eyes in terms of stereo disparity and vergence depth cues, the focal distance of all virtual imagery rendered by the HoloLens is $\approx$2 meters\cite{hololensFocalDistance}.
To avoid having users focus on such virtual feedback,
making the physical network appear blurry,
we render callouts with a blur
effect (Figure~\ref{fig:study2}), i.e., without sharp edges.
For consistency, this was done in both \VRThree\ and \ARTouch.
\subsection{Users} %
12 new users were recruited:
9 male, 3 female;
all right handed;
age 20 to 42 years (average 27.3);
IPD 57 to 70mm (average 62.7);
stereoacuity test scores 3/10 to 10/10 (average 8.9).
As in the previous study (Section~\ref{sec:study1protocol}),
users were shown several printouts of example trials,
with conditions in random order,
to explain the task,
and each headset was adjusted to the user's IPD
before beginning warmup trials.
\subsection{Design}\label{sec:study2design} %
Each user experienced the
2 levels of \variable{MainCondition}\ \{\VRThree, \ARTouch\} in random order.
For each \variable{MainCondition},
the user performed 10 warmup trials in random order
(5 levels of \variable{PathLength}\ $\times$ 2 repetitions)
with the warmup network,
followed by 15 trials in random order
(5 levels of \variable{PathLength}\ $\times$ 3 repetitions)
with another network chosen at random,
followed by another 5 warmup trials with the warmup network,
15 trials with another network,
5 warmup trials with the warmup network,
and another 15 trials with another network.
Each subsequence of warmup trials gave the user an opportunity to
take a break, adjust the headset, and redo the calibration if they wished.
There were a total of
12 users
$\times$ 2 levels of \variable{MainCondition}\
$\times$ 3 networks
$\times$ 5 levels of \variable{PathLength}\
$\times$ 3 repetitions per trial
= 1080 trials, not counting warmup trials and not counting pilot data.
Each session with a user lasted $\approx$ 1.5 hours.
\subsection{Results} \label{sec:study2results} %
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{fig/results/distance.png}
\caption{
How far users held the network from their head.
%
}
\label{fig:results-distance}
\end{figure}
We find no evidence of a difference in error rate
between the \ARTouch\ and \VRThree\ conditions
(Figure~\ref{fig:results-errorRate}).
In \ARTouch, users took more time
(Figure~\ref{fig:results-time}),
which is explained by the difficulty that users had
in finding the end-nodes at the start of each trial.
Users also
rotated their view less
(Figure~\ref{fig:results-rotation})
and rotated less with their hand and more with head motion
(Figure~\ref{fig:results-rotationFraction}),
which is explained by user feedback
indicating that the network holder in \ARTouch\ was
not as easy to rotate as the handheld controller
in the \VRThree\ condition.
Users also held the network further away from their
head
(Figure~\ref{fig:results-distance}),
which is explained by the more limited FOV in AR.
All 12 users preferred the \VRThree\ condition
to \ARTouch\ (Figure~\ref{fig:subjective}).
The reasons given by users for this preference were:
the inaccurate positioning of callouts in AR (mentioned by 9 out of the 12 users);
the limited FOV of the AR headset (mentioned by 4 users);
the AR headset being less comfortable (4);
the network holder in the \ARTouch\ condition being more difficult to rotate than the VR controller (4);
the \VRThree\ condition providing better visual contrast
between the network and the black background (4);
and the latency with the AR system (1).
Based on user feedback, users often performed
each trial of \ARTouch\ in two stages: first, identifying the end-nodes,
and second, finding the shortest path between them.
The first was difficult because of the limited FOV and the
positional error in the callouts.
Once the end-nodes were identified, users would often ``mark'' them
by touching their fingertips to one or both end-nodes
and maintaining contact while
looking for the shortest path.
Several other behaviors were observed with users' fingers:
touching intermediate nodes along a path (observed in 9 out of the 12 users),
touching multiple nodes simultaneously (8 users),
pivoting the network around the hand while maintaining contact with one or more DH fingers (7),
pointing at nodes without touching them (6),
and grabbing a node with 2 fingers (3).
We also noted which fingers of the DH were employed during the \ARTouch\ condition.
Letters t, i, m, r, p denote thumb, index, middle, ring,
and pinky fingers, respectively,
and multiple letters indicate combinations,
such as ti for thumb + index.
Fingers i, m, p were often employed individually.
We also observed simultaneous uses
of fingers:
im (used by 7 out of the 12 users),
ti (6 users),
tm (5),
ip (2),
ir (1),
mp (1),
tim (1),
imr (1).
Other notable behaviors observed were:
touching edges in addition to nodes;
tapping a sequence of nodes along a path;
sliding a fingertip along the edges of a path;
walking two fingers (index and middle) along the nodes of a path like legs;
touching one node and pivoting the hand around the network, while maintaining contact with the finger;
touching three nodes at once (either with tim or with imr).
One user %
explained that they used 2 fingers simultaneously to trace 2 different paths
between end-nodes.
Another %
said they would have preferred the \ARTouch\ condition over \VRThree\ if the problems with FOV and tracking accuracy were fixed.
Another user %
said that touch had value for tracking the path with the finger.
Another %
stated that they really wanted to touch the network in the \VRThree\ condition, and wanted to see their fingers in VR, and that they were running their fingers through space where the path would have been during the \VRThree\ condition.
Another user (during the pilot) %
actually preferred the \ARTouch\ condition to \VRThree\ despite the limitations of the AR system, stating that the physical network allowed the task to be done faster and more simply.
\subsection{Discussion} \label{sec:study2discussion} %
In the \ARTouch\ condition, users employed their fingers
in a variety of manners.
The different uses of hands
has been studied before in visualizations \cite{walny2017}
and physicalizations \cite{taher2017emerge,drogemuller2021}.
Other work \cite{stival2019} %
has proposed a taxonomy
of uses of hands for grasping.
We are unaware of a taxonomy of more general uses of hands
relevant to data physicalizations.
Users preferred \VRThree\ over \ARTouch, but the reasons
given for this are almost entirely due to technological limitations of the AR platform.
Physicalizations that can be touched have been shown to
be preferred \cite{drogemuller2021} or yielded better performance \cite{jansen2013evaluating}
than 3D visualizations.
Hence, next steps could be to either improve the AR platform,
or improve the VR platform to better support the way users leverage their fingers.
We now consider each of these.
Our AR platform suffers from inaccurate alignment of virtual and physical objects.
Before using the Polhemus Patriot for tracking, we tried
using the RGB camera on the HoloLens
as well as multiple external cameras for object tracking,
but none of these methods achieved acceptable accuracy.
An alternative approach would be to use
video pass-through AR (either with a headset or not,
as in \cite{gillet2005}) to avoid the need for highly accurate
tracking, and enable virtual highlighting of physical
nodes and edges (Figure~\ref{fig:physicalEdgeHiliting})
with pixel-precise alignment,
correct occlusion cues,
and no conflicting accommodation (focal) distances.
To improve the VR platform to better support finger interaction,
we notice that users in our \ARTouch\ condition primarily
used fingers to mark parts of the network.
For example, users would often touch one or more fingertips against parts of the network, and then pivot a hand
(to examine the network from a different view)
while {\em maintaining contact} with fingertips, which was {\em made easier by friction}.
This suggests an interaction technique where the user can use their DH to define one or more ``sticky marks'' on the network,
that remain even while the hands continue to move.
We also observed users sliding their fingers along edges,
which suggests support for ``slippery marks''
whose positions are updated to slide along edges,
maybe as if being pulled by strings attached to the DH.
Although physical fingers are limited to maintaining one mark per finger,
this needn't be the case with virtual marks: different fingers might be pinched against the thumb to
invoke different commands to create, modify, or delete each of many marks. %
For example, when the user's DH approaches a virtual network,
a rope cursor \cite{guillon2015rope} from the DH's thumb could extend to the nearest node,
and a DH pinch against the thumb by the index, middle, or ring finger could create a sticky mark, create a slippery mark, or delete the mark, respectively, at that node.
Such finger-based interaction might be generally useful across many tasks
beyond finding shortest paths.
\section{Conclusions}
Our Study 1 provides strong evidence that path tracing
is less error prone in 3D layouts than in 2D layouts
(Figure~\ref{fig:results-study1-prediction1}),
despite the use of edge-routing in 2D.
The use of mouse-driven interactive highlighting in 2D
reduces the error rate in 2D, but not as much as
using a 3D layout (Figure~\ref{fig:results-study1-errorRate2}).
3D was also the most preferred layout (Figure~\ref{fig:subjective})
Our Study 2 found no evidence of different error rates
between the virtual (VR) and physical (AR) conditions
(Figure~\ref{fig:results-errorRate}).
VR was preferred by users,
but this was largely due to technological
limitations of the AR platform,
and users touched the physicalized networks in a variety of ways.
\section{Future Directions}
Section~\ref{sec:study2discussion} described ideas for future work.
In addition, to benefit from the advantages of 2D and 3D,
techniques could be designed allowing a user to rapidly
switch between 2D and 3D layouts,
possibly mixing styles in a hybrid that is
2D for most of the network
but 3D in certain parts where the user is more interested in perceiving shortest
paths.
New fabrication methods might also enable physicalized networks
that contain embedded buttons, touch sensors, and/or lights,
for richer interaction.
\ifthenelse{\boolean{IncludeAppendix}}{
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,081 |
Victor Lafosse foi um ciclista francês. Ele terminou em último lugar no Tour de France 1924.
Ciclistas da França
Naturais da França | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 36 |
Nash Healey Guide | Austin-Healey Club of America, Inc.
The history of the Nash Healey starts in the mid 40s, WWII had just ended, and service men and women were coming home and bring with them something that was new to America, sports cars from Europe.
Donald Healey was building a 4 cylinder Riley powered sports car called the Healey Silverstone. This car was doing well in races in Europe and caught the attention of Briggs Cunningham. Donald was commissioned to build a Healey Silverstone with a new engine from America, a Cadillac V8 engine. This car was very successful and Donald wanted to purchase additional V8 engines from General Motors, so he took a trip to America on the Queen Elizabeth. Looking back into history, this trip would turn out to be an important event in the continuation and success of Donald's car company.
In a chance meeting aboard the Queen Elizabeth, Donald would meet George Mason, the President of the Nash Motor Company. Donald shared that he was on his way to see if he could secure V8 engines from General Motors for the Healey Silverstone. Mr. Mason offered engines if Donald was unsuccessful with GM. Mason knew that having a "sports car" with the name Nash on it would be a nice addition to the Nash Family of cars. It was through this meeting that the Nash Healey was born, and a long-term friendship was established. This chance meeting would turn out to be profitable for both men.
A prototype with a 102" wheelbase was designed by Donald and constructed using a body provided by Panel Craft of England. A 234.8-cubic-inch (3.8. litre) high-compression straight 6-cylinder Nash Ambassador engine was fitted with an aluminum head and two 1¾" S.U. Carburetors. A three speed manual transmission with Overdrive was standard. The prototype was shown publicly for the first time at the Paris Motor Show in early fall of 1950. Production of the Panel Craft bodied Nash Healey began in late 1950 and continued until March 1951 in Warwick, England. The Nash Healey was built in England, but was for export only to the United States, and a total of 104 Roadsters were built. Some people think that the 1953 Corvette made by Chevrolet was the 1st American Sports Car, the Nash Healey beat it by 2 years.
From April 1951 until January 1952, the Nash Healey was not made. 1950 was the first entry of a Nash-Healey at Le Mans. It finished the 24 hours in 4th. In 1951, a Nash Healey finished 6th. Two special-bodied Nash Healeys entered and qualified to race at Le Mans in 1952. One dropped out due to engine troubles, the other finished 1st in class, and 3rd overall, beating out cars by Aston Martin, Cunningham, Ferrari and Lance. Only two factory prepared Mercedes-Benz 300SL coupes completed more total miles in the 24 hours than the Nash Healey.
Also in 1952, the Nash Motor Co. had acquired the services of Pinin Farina of Turin, Italy to redesign the body for the Nash and Nash Healey. The Nash Healey was truly an international car. The engines were manufactured by Nash at the Kenosha, Wisconsin plant, and then shipped to England where they were installed in the chassis with the "trailing link" front end suspension. The chassis and engine were then shipped to Turin, Italy were they were married to the custom hand built body by Pinin Farina. The new Nash-Healey was shown for the first time at the Chicago Automobile Show in February, 1952. Only 150 Pinin Farina bodied Nash Healey Roadsters were built that year.
1953 saw the addition of a longer 108 inch wheelbase hardtop coupe named the Le Mans. This was done to honor the fact that the Nash Healey placed 1st in class in the 1952 Le Mans Race. Only 162 Nash Healeys were built in 1953, 62 of them being the new Le Mans Hardtop. It was also during this year, that the engine was increased to a 258.6-cubic-inch (4.1-litre) and the 2 S.U. Carburetors were replaced with 2 HY Carter Carburetors. The 1953 Nash-Healey Hardtop, also known as the Le Mans Coupe won first place at the 1953 Italian International Concours D'elegance held at Tresa, Italy.
In 1954 the roadster was no longer being produced and the Le Mans Coupe went through some minor restyling. Less than 100 cars were built. Production costs were higher than the price for which Nash was able to sell the car in the states, and production stopped. Records indicate that 506 Nash Healers were made between December 1950 and August 1954.
This chance meeting aboard the Queen Elizabeth turned out to be profitable for both Donald Healey and George Mason. For Donald, it allowed him to get out of debt and to make the Healey 100. For George Mason, the meeting led to a friendship that one day allowed him to be introduced to Len Lord, President of Austin Motorcar Company. George Mason reached an agreement with the Austin Motorcar Company to produce the Nash Metropolitan. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,598 |
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